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Full text of "Airborne radar"

PRINCIPLES OF.©UrDED MiSSlLE DESI© 



WOODS HOLE 
OCEANOGRAPHIC INSTITUTION 



LABORATORY 
BOOK COLLECTION 









AIRBORNE 
RADAR 




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PRINCIPLES OF 

GUIDED MISSILE DESIGN 



Editor of the Series 

Grayson Merrill, captain, u.s.n. (Ret. 

General Manager, Astrionics Division, Fairchild Engine 
and Airplane Corporation 



GUIDANCE — by Arthur S. Locke and collaborators 

AERODYNAMICS, PROPULSION, STRUCTURES AND 
DESIGN PRACTICE— 6y E. A. Bonney, M. J. Zucrow, 
and C. W. Besserer 

OPERATIONS RESEARCH, ARMAMENT, LAUNCH- 
ING— 6y G. Merrill, H. Goldberg, and R. H. Helmholz 

MISSILE ENGINEERING HANDBOOK— 6y C. W. Bes- 
serer 

DICTIONARY OF GUIDED MISSILES AND SPACE 
FLIGHT — Edited by Grayson Merrill 

SPACE FLIGHT— 6y K. A. Ehricke 

Vol. I: ENVIRONMENT AND CELESTIAL 
MECHANICS 

Vol. II: DYNAMICS Vol. Ill: OPERATIONS 
SYSTEMS PRELIMINARY DESIGN— 6y J.J. Jerger 
AIRBORNE RADAR— by D. J. Povejsil, R. S. Raven, and 

P. Waterman 
RANGE TESTING— 6y J.J. ScavuUo and Eric Burgess 
FOUNDATIONS OF SEARCH THEORY— by N. S. Potter 
PRINCIPLES OF INERTIAL NAVIGATION— 6y Richard 

Parvin 



PRINCIPLES OF 

GUIDED MISSILE DESIGN 

Edited by 

Captain Grayson Merrill, u.s.n. (Ret.) 



DONALD J. POVEJSIL 

Director, New Products Services; 

Formerly Manager, Weapons Systems Engineering 

Air Arm Division, Westinghouse Electric Corporation 

Pittsburgh, Pa. 

ROBERT S. RAVEN 

Advisory Engineer, Weapons Systems Engineering 
... . .- ^ , , Westinghouse Electric Corporation 
Air Arm Division, Baltimore, Md. 



Ml'-. 



.PETER WATERMAN 

Head, Naval Research Laboratories, 
Radar Division, Washington, D. C. 




o 



AIRBORNE 
RADAR 



MARINE 
BIOIO<3ICAJ- 
UBORATORY 



LIBRARY 



WOOBS HOLE, MASS. 
W. H. 0. I. 



D. VAN NOSTRAND COMPANY, INC. 

PRINCETON, NEW JERSEY • TORONTO • NEW YORK • LONDON 



D. VAN NOSTRAND COMPANY, INC. 

120 Alexander St., Princeton, New Jersey {Principal office) 
24 West 40 Street, New York 18, New York 

D. Van Nostrand Company, Ltd, 
358, Kensington High Street, London, W.14, England 

D. Van Nostrand Company (Canada), Ltd. 
25 Hollinger Road, Toronto 16, Canada 



Copyright © 1961, by 
D. VAN NOSTRAND COMPANY, Inc. 



Published simultaneously in Canada by 
D. Van Nostrand Company (Canada), Ltd. 



Library of Congress Catalogue Card No. 61-8542 



No reproduction in any fortn of this book, in whole or in 
part {except for brief quotation in critical articles or reviews), 
may' be made without written authorization from the publishers. 






PRINTED IN THE UNITED STATES OF AMERICA 



CONTRIBUTORS 



P. J. Allen • Naval Research Laboratory^ Washington^ D. C. 

(Coauthor, Chapter 10) 
G. S. AxELBY • PFestinghouse Electric Corp., Air Arm Division^ 

Baltimore, Md. 

(Coauthor, Chapters 8 and 9) 
B. L. CoRDRY • Bendix Aviation Corp., Radio Division, Baltimore, 

Md. 

(Coauthor, Chapter 14) 
W. R. Fried • General Precision Laboratory , Pleasantville, N. Y. 

(Coauthor, Chapter 14) 
S. F, George • Naval Research Laboratory , Washington, D. C. 

(Coauthor, Chapter 6) 
M, Goetz • Westinghouse Electric Corp., Central Laboratories, 

Pittsburgh, Pa. 

(Coauthor, Chapter 13) 
D. J. Healey III • Westinghouse Ekctric Corp., Air Arm Division, 

Baltimore, Md. 

(Author, Chapter 7; Coauthor, Chapter 8) 
L. Hopkins • Raytheon Manufacturing Corp., Maynard Labora- 

tory, Maynard, Massachusetts 

(Coauthor, Chapter 6) 
D. D. Howard • Naval Research Laboratory , Washington, D. C. 

(Coauthor, Chapter 8) 
A. Kahn • Westinghouse Electric Corp., Air Arm Division, 

Baltimore, Md. 

(Coauthor, Chapter 12) 
M. Katzin • Electromagnetic Research Corp., Washington, D. C. 

(Author, Chapter 4) 
R. H. Laprade • Westinghouse Electric Corp., Air Arm Division, 

Baltimore, Md. 

(Coauthor, Chapter 14) 
T. Moreno • Varian Associates, Palo Alto, California 

(Author, Chapter 11) 
R. M. Page • Naval Research Laboratory, Washington, D. C. 

(Coauthor, Chapter 6) 



p. 


M. 


. Pan 


D. 


J. 


POVEJSIL 


R. 


S. 


Raven 



R. M. Sando 

F. Stauffer 

M. Taubenslag 

J. W. Titus 
P. Waterman 
M. S. Wheeler 

C. F. White 
H. Yates 



CONTRIBUTORS 

Westinghouse Electric Corp., Air Arm 'Division, 
Baltimore, Md. 
(Coauthor, Chapter 10) 

Westinghouse Electric Corp., New Products Serv- 
ices, Pittsburgh, Pa. 

(Editor and Coauthor, Chapters 1, 2, 6, and 12) 
Westinghouse Electric Corp., Air Arm Division, 
Baltimore, Md. 

(Editor and Author, Chapters 3 and 5; Coauthor, 
Chapters 8 and 12) 

Westinghouse Electric Corp., Air Arm Division, 
Baltimore, Md. 
(Coauthor, Chapter 13) 

Westinghouse Electric Corp., Air Arm Division^ 
Baltimore, Md. 
(Coauthor, Chapter 14) 

Aeronica Manufacturing Corp., Aerospace Divi- 
sion, Baltimore, Md. 
(Coauthor, Chapter 12) 

Naval Research Laboratory , Washington, D. C. 
(Coauthor, Chapter 13) 

Naval Research Laboratory, Washington, D. C. 
(Editor and Coauthor, Chapters 1 and 2) 
Westinghouse Electric Corp., Air Arm Division, 
Baltimore, Md. 
(Coauthor, Chapter 10) 

Naval Research Laboratory , Washington, D. C. 
(Coauthor, Chapters 8 and 9) 
Barnes Engineering Co., hifrared Division, Stam- 
ford, Conn. 
(Coauthor, Chapter 6) 



FOREWORD 

"Airborne Radar" is the eighth volume in the series Principles of Guided 
Missile Design. Other titles in the series are Guidance; Aerodynamics, 
Propulsion, Structures and Design Practice; Operations Research, Armament, 
Launching; Missile Engineering Handbook; Dictionary of Guided Missiles 
and Space Flight; Space Flight I — Environment and Celestial Mechanics; 
Space Flight II — Dynamics; Space Flight III — Operations; Preliminary 
Systejn Design; and Range Testing. 

The purpose of the series as a whole is to give a basis for instruction 
to graduate students, professional engineers, and technical officers of the 
armed services so that they can become well grounded in the technology 
of guided missiles and space flight. This book concerns itself with one 
of the most important systems used now in missilery and to be used soon 
in space flight — airborne radar. As this is written the world is commenc- 
ing a great search for a defense against intercontinental ballistic missiles; 
airborne radar will play a key role in this defense. 

In the interests of brevity, this book presumes considerable knowledge 
on the reader's part — namely, knowledge of the basic principles of elec- 
tronics and electromagnetic propagation and familiarity with the associ- 
ated language. It also presumes a knowledge of the weapons systems 
employing airborne radars. Certain prior issues of the series will be found 
especially valuable as references; these are the Dictionary oj Guided Missiles 
and Space Flight; Guidance; and Operations Research, Armament, Launch- 
ing. 

Criticisms and constructive suggestions are invited. With this aid and 
by keeping abreast of the state of the art we hope to make timely revisions 
to this volume. 

Grateful acknowledgment is made to the many authors and publishers 
who kindly granted permission for the use of their material and to the 
Department of Defense, whose cooperation made possible a meaningful 
text without violation of security. 

The opinions or assertions contained herein are the private ones of the 
authors and the editor and are not to be construed as official or reflecting 
the views of any government agency or department. 

Grayson Merrill 
Editor 
Wyandanch, Long Island, New York 
November J 960 



PREFACE 

The basic purpose of this book is to present a balanced treatment of the 
airborne radar systems design problem. Primary emphasis is placed upon 
the interplay between radar techniques and components on one hand, and 
the types of weapons systems which employ airborne radars on the other. 
Radar design details have been eliminated for the most part except for 
illustrative examples which show how a design detail can exert an impor- 
tant influence on the operation of a complete weapons system. 

Although the treatment is directed at airborne radars, this volume will 
be found extensively applicable to surface radars as well. Since the latter 
enjoy a less severe environment, especially with regard to relative target 
motion, stability of platform, and space and weight restrictions, the prin- 
ciples governing their design will implicitly be covered here. 

Because this book attempts to bridge the gap between the abstractions 
of overall system design and the hard realities of hardware design, it con- 
tains material which will interest almost anyone involved in the study, 
design, or application of airborne radars. For example, although the re- 
ceiver designer will not learn much that is new to him about circuitry 
design, he can learn a great deal about how the design of a receiver should 
be planned for optimum benefit to the overall system. Similarly, engineers 
and scientists charged with responsibility for monitoring the efi^orts of 
airborne radar subcontractors can find this book most useful in determin- 
ing the type of direction they should give to subcontractors to ensure 
eventual compatibility of the airborne radar with the complete weapons 
system. 

A particular eflfort has been made to present facts and combinations of 
facts which have not enjoyed prior publication in book form. This has 
been done at the expense of excluding a great deal of historical and back- 
ground information already available in the printed literature. 

This book possesses close ties with two previous volumes in the series: 
GUIDANCE by A. S. Locke et al. and OPERATIONS RESEARCH, 
ARMAMENT, LAUNCHING by Grayson Merrill, Harold Goldberg, 
and Robert H. Helmholz. Radar techniques and problems are presented 
in greater detail than was possible in GUIDANCE; similarly the problem 
of translating operational studies into detailed airborne radar requirements 
is covered in greater detail than was possible in OPERATIONS RE- 
SEARCH. The basic theme of these earlier volumes — the importance of 
the systems approach — is continued in this volume. 

xi 



xii PREFACE 

Many of the authors have had previous association on team efforts aimed 
at the development and production of complex airborne radar equipments. 
The technical approaches presented thus represent tools forged on the 
anvil of experience — tools which have facilitated the solution of many- 
difficult problems. It is the authors' hope that succeeding generations of 
system designers may use these tools to their advantage in designing the 
even more complex systems to come. 

Acknowledgments 

In addition to the authors, there are many individuals and organizations 
to whom acknowledgment must be made for an active part in the writing 
of this book. Members of the Naval Research Laboratories, the Naval 
Air Development Center, the Fairchild Astrionics Division and the West- 
inghouse Electric Corporation assisted with suggestions, criticisms, a.nd 
technical readings. Deepest thanks must be extended to the government 
laboratories (NRL and WADC) and the corporations (Barnes Engineering, 
Bendix Aviation, General Precision Laboratory, Raytheon Manufacturing, 
Varian Associates, and Westinghouse Electric) who provided encourage- 
ment and assistance to the contributors. 

The assistance of Mr. R. G. Clanton of Westinghouse was vital to the 
preparation of the examples employed in Chapter 2. In addition, Mr. 
Clanton's many helpful suggestions and detailed reviews of the remainder 
of this chapter are most gratefully acknowledged. Mr. R. H. Laprade 
had the responsibility of reviewing all the material relating to propagation 
in addition to his contribution to Chapter 14. 

Mr. A. Stanley Higgins, Mr. Melvyn Goetz and Dr. J. F. Miner of West- 
inghouse rendered invaluable services in overseeing the myriad details in- 
volved in the editing and production of the final text. The Westinghouse 
Electric Corporation deserves special thanks for the assistance provided on 
drawings, typing, and the reproduction of the many drafts of the manu- 
scripts. 

D, J. POVEJSIL 

R. S. Raven 
P. Waterman 



CONTENTS 

List of Contributors vii 

Foreword ix 

Preface xi 

1 ELEMENTS OF THE AIRBORNE RADAR 
SYSTEMS DESIGN PROBLEM 

1-1 Introduction . 1 

1-2 Classifications of Radar Systems 2 

1-3 Installation Environment 3 

1-4 Functional Characteristics of Radar Systems 4 

1-5 The Modulation of Radar Signals 16 

1-6 Operating Carrier Frequency 26 

1-7 The Airborne Radar Design Problem 27 

1-8 The Systems Approach to Airborne Radar Design ... 30 

1-9 Systems Environments 35 

1-10 Weapons System Models 36 

1-11 The Basic Statistical Character of Weapons System Models 39 

1-12 Construction and Manipulation of Weapons System Models 41 

1-13 Summary 44 

2 THE DEVELOPMENT OF WEAPONS SYSTEM 
REQUIREMENTS 

2-1 Introduction to the Problem 46 

2-2 Formulating the System Study Plan 48 

2-3 Aircraft Carrier Task Force Weapons System 50 

2-4 The Target Complex 55 

2-5 The Operational Requirement 57 

2-6 The System Concept 57 

2-7 The System Study Plan 5^ 

2-8 Model Parameters 60 

2-9 System Effectiveness Models 61 

2-10 Preliminary Design of the Airborne Early Warning System 67 

2-11 AEW System Logic and Fixed Elements 70 

2-12 AEW Detection Range Requirements 73 

2-13 AEW Target Resolution Requirements 75 

xiii 



xiv CONTENTS 

2-14 Interrelations of the AEW System, the CIC System, the In- 
terceptor System, and the Tactical Problem 79 

2-15 Accuracy of the Provisional AEW System 80 

2-16 Information-Handling Capacity of the Provisional AEW 

System 84 

2-17 Velocity and Heading Estimates 85 

2-18 AEW Radar Beamwidth as Dictated by the Tactical Problem 89 

2-19 Factors Affecting Height-Finding Radar Requirements . . 92 

2-20 Summary of AEW System Requirements 96 

2-21 Evaluation of Tentative Design Parameters with Respect to 

the Tactical Problem 98 

2-22 Interceptor System Study Model 100 

2-23 Probability of Reliable Operation 103 

2-24 Probability of Viewing Target— Vectoring Probability . . 104 
2-25 Analysis of the Vectoring Phase of Interceptor System Op- 
eration 106 

2-26 AT Radar Requirements Dictated by Vectoring Considera- 
tions Ill 

2-27 Analysis of the Conversion Problem 116 

2-28 Lock-on Range and Look-Angle Requirements Dictated by 

the Conversion Problem 130 

2-29 AI Radar Requirements Imposed by Missile Guidance Con- 
siderations 135 

2-30 Summary of AI Requirements 136 

2-31 Summary 137 

3 THE CALCULATION OF RADAR DETECTION 
PROBABILITY AND ANGULAR RESOLUTION 

3-1 General Remarks 138 

3-2 The Radar Range Equation 138 

3-3 The Calculation of Detection Probability for a Pulse Radar 141 
3-4 The Effect of Scanning and the Cumulative Probability of 

Detection 156 

3-5 The Calculation of Detection Probability for a Pulsed- 

Doppler Radar 162 

3-6 Factors Affecting Angular Resolution 168 

4 REFLECTION AND TRANSMISSION 
OF RADIO WAVES 

4-1 Introduction • 174 

4-2 Reflection of Radar Waves 175 

4-3 Effect of Polarization on Reflection 179 

4-4 Modulation of Reflected Signal by Target Motion . . . 180 



CONTENTS XV 

4-5 Reflection of Plane Waves from the Ground 181 

4-6 Effect of Earth's Curvature 190 

4-7 Radar Cross Sections of Aircraft 192 

4-8 Amplitude, Angle, and Range Noise 198 

4-9 Prediction of Target Radar Characteristics 208 

4-10 Sea Return 211 

4-11 Sea Return in a Doppler System 217 

4-12 Ground Return 219 

4-13 Altitude Return 222 

4-14 Solutions to the Clutter Problem 224 

4-15 Attenuation in the Atmosphere 227 

4-16 Attenuation and Back-scattering by Precipitation . . . 230 

4-17 Attenuation by Propellant Gases 231 

4-18 Refraction Effects in the Atmosphere 233 

5 TECHNIQUES FOR SIGNAL 
AND NOISE ANALYSIS 

5-1 Introduction 238 

5-2 Fourier Analysis . 238 

5-3 Impulse Functions 243 

5-4 Random Noise Processes 245 

5-5 The Power Density Spectrum 248 

5-6 Nonlinear and Time-Dependent Operations 253 

5-7 Narrow Band Noise 258 

5-8 An Application to the Evaluation of Angle Tracking Noise 264 

5-9 An Application to the Analysis of an MTI System . . . 269 

5-10 An Application to the Analysis of a Matched Filter Radar . 272 
5-11 Application to the Determination of a Signal's Time of 

Arrival 281 

6 GENERIC TYPES OF RADAR SYSTEMS 
AND TECHNIQUES 

6-1 Introduction 292 

6-2 Basic Principles 293 

6-3 Monopulse Angle Tracking Techniques 300 

6-4 Correlation and Storage Radar Techniques 305 

6-5 FM/CW Radar Systems 311 

6-6 Pulsed-Doppler Radar Systems 320 

6-7 High-Resolution Radar Systems 333 

6-8 Infrared Systems 338 

7 THE RADAR RECEIVER 

7-1 General Design Principles 347 

7-2 The Interdependence of Receiver Components .... 352 



XVI CONTENTS 

7-3 Receiver Noise Figure 353 

7-4 Low-Noise Figure Devices for RF Amplification .... 356 

7-5 Mixers 357 

7-6 Coupling to the Mixer 361 

7-7 IF Amplifier Design 362 

7-8 Considerations of IF Preamplifier Design 368 

7-9 Overall Amplifier Gain 373 

7-10 Gain Variation and Gain Setting 375 

7-11 Bandwidth and Dynamic Response 375 

7-12 Sneak Circuits 377 

7-13 Considerations Relating to AGC Design 379 

7-14 Problems at High-Input Power Levels 380 

7-15 The Second Detector (Envelope Detector) 382 

7-16 Gating Circuits 386 

7-17 Pulse Stretching 387 

7-18 Connecting the Receiver to the Related Regulating and 

Tracking Circuits 388 

7-19 Angle Demodulation 389 

7-20 Some Problems in the Measurement of Receiver Character- 
istics 390 

8 REGULATORY CIRCUITS 

8-1 The Need for Regulatory Circuits 394 

8-2 External and Internal Noise Inputs to the Radar System . 395 

8-3 Automatic Frequency Control 401 

8-4 Variation of Transmitter Frequency with Environmental 

Conditions 402 

8-5 Magnetron Pulling . . . . ' 403 

8-6 Static and Dynamic Accuracy Requirements 405 

8-7 Continuous-Correction AFC 407 

8-8 Limit-Activated AFC 412 

8-9 The Influences of Local Oscillator Characteristics . . . 413 

8-10 Relation to Receiver IF Characteristics 414 

8-11 Discriminator Design 414 

8-12 Instantaneous AFC 415 

8-13 Problems of Frequency Search and Acquisition .... 416 

8-14 Automatic Gain Control 416 

8-15 Linear Analysis of AGC Loops 419 

8-16 Static Regulation Requirements of AGC Loops .... 420 

8-17 Dynamic Regulation Requirements of AGC Loops . . . 422 

8-18 AGC Transfer Characteristic Design Considerations . . . 423 

8-19 The Modulation Transmission Requirement 424 

8-20 Design of an AGC Transfer Function 425 



CONTENTS xvii 

8-21 The IF Amplifier Control Characteristic 427 

8-22 The Angle Measurement Stabilization Problem .... 429 

8-23 AI Radar Angle Stabilization 433 

8-24 Aircraft Motions 433 

8-25 Stabilization Requirements 439 

8-26 Search Pattern Stabilization 440 

8-27 Search Stabilization Equations 440 

8-28 Static and Dynamic Control Loop Errors 442 

8-29 Search Loop Mechanization 448 

8-30 Stabilization During Track 452 

8-31 Possible System Configurations 453 

8-32 Accuracy Requirements on the Angle Track Stabilization 

Loop 457 

8-33 Dynamic Stability Requirements on Angle Track Stabiliza- 
tion 464 

8-34 Stabilization Loop Mechanization 468 

9 AUTOMATIC TRACKING CIRCUITS 

9-1 General Problems of Automatic Tracking 471 

9-2 Automatic Angle Tracking 474 

9-3 External Inputs: Undesired and Desired 475 

9-4 Requirements in Angle Tracking Accuracy 479 

9-5 Angle Tracking System Organization 480 

9-6 Tracking Loop Design 485 

9-7 Angle Tracking Loop Rate Errors 486 

9-8 Angle Tracking Loop Position Errors 489 

9-9 Angle Tracking Loop Mechanization 492 

9-10 Introduction to Range and Velocity Tracking 498 

9-11 x'\utomatic Range Tracking 498 

9-12 Servo System Transfer Function Relationship to Input Time 

Function for a Range Tracking System 502 

9-13 Range Tracking Design Example 505 

9-14 Practical Design Considerations 508 

10 ANTENNAS AND RF COMPONENTS 

10-1 Antennas: Introduction to Radar Antennas 512 

10-2 Some Fundamental Concepts Useful in the Development of 

Radar Antenna Requirements 513 

10-3 The Paraboloidal Reflector as a Radar Tracking Antenna . 515 

10-4 System Requirements for Radar Antennas 518 

10-5 Pattern Simulation as a Link Between System Requirements 

and Antenna Characteristics 520 



xviii CONTENTS 

10-6 Several Anomalous Effects in Antennas for Tracking Systems 523 

10-7 The Linear Array as a Fan Beam Antenna for Surveillance 524 

10-8 Two-Arm Spiral Antennas 528 

10-9 Radomes 531 

10-10 Introduction to Transmission Lines and Modes of Propaga- 
tion 535 

10-11 Types of Transmission Lines and Modes of Propagation . . 536 

10-12 Standing Waves and Impedance Matching 540 

10-13 Broadband System Design 543 

10-14 Pressurization 545 

10-15 Miscellaneous Microwave Components 546 

10-16 Microwave Ferrite Devices and Their Application . . . 557 

10-17 Microwave Dielectric, Magnetic, and Absorbent Materials . 565 

10-18 The Duplexing Problem 566 

10-19 Duplexing Schemes 567 

10-20 Special Problems of Coherent Systems 573 

10-21 Solid-State Amplifiers 574 

11 THE GENERATION OF MICROWAVE POWER 

11-1 The Magnetron 580 

11-2 The Klystron 590 

11-3 Traveling Wave Tubes for High Power 597 

11-4 Modulation Techniques for Beam-Type Amplifiers . . . 599 

11-5 A Typical Radar System Employing a High-Gain Amplifier 601 

11-6 Backward Wave Oscillators — Carcinotrons 602 

11-7 The Platinotron 603 

12 DISPLAY SYSTEM DESIGN PROBLEMS 

12-1 Introduction 607 

12-2 Uses of Display Information 608 

12-3 Types of Displays 613 

12-4 Types of Input Information 619 

12-5 The Cathode Ray Tube 621 

12-6 Important Characteristics of Electrical-to-Light Transducers 627 

12-7 Important Characteristics of the Human Operator . 634 

12-8 Development of Requirements for a Display System . . . 651 

12-9 Special Display Devices 655 

12-10 Special Displays 673 

13 MECHANICAL DESIGN AND PACKAGING 

13-1 The Influence of Environment on Design 680 

13-2 Military Specifications 682 

13-3 Temperature 683 



CONTENTS xix 

13-4 Solar Radiation 692 

13-5 Nuclear Radiation 692 

13-6 Vibration and Shock 694 

13-7 Acoustic Noise 704 

13-8 Acceleration 707 

13-9 Moisture 708 

13-10 Static Electricity and Explosion 710 

13-11 Pressure 711 

13-12 Maintenance and Installation 712 

13-13 Transportation and Supply 714 

13-14 Potential Growth 715 

13-15 ReUabihty 715 

14 AIRBORNE NAVIGATION AND GROUND 
SURVEILLANCE RADAR SYSTEMS 

14-1 Introduction to Doppler Navigation Systems 726 

14-2 Basic Principles of Doppler Radar Navigation .... 728 

14-3 System Considerations 733 

14-4 Major Characteristics and Components of a Doppler Radar 

Navigation System 736 

14-5 Doppler Navigation System Errors Caused by Interactions 

with the Ground and Water 746 

14-6 Modifying the Radar Range Equation for the Doppler Navi- 
gation Problem 749 

14-7 Low Altitude Performance and the "Altitude Hole" . . . 752 

14-8 Doppler Navigation System Performance Data .... 755 

14-9 Introduction to Weather Radar 759 

14-10 Meteorological Effects at Microwave Frequencies . . . 760 
14-11 Designing Airborne Radar Systems Explicitly for Weather 

Mapping 764 

14-12 Modifying the Radar Range Equation for the Weather 

Problem 764 

14-13 Relative Importance of Design Variables in Airborne Weather 

Radar 766 

14-14 Design Features 769 

14-15 Introduction to Active Airborne Ground Mapping Systems 772 

14-16 Basic Principles 772 

14-17 System Considerations 774 

14-18 Major Characteristics and Components 778 

14-19 Modifying the Radar Range Equation for the Active Ground 

Mapping Problem 782 

14-20 Resolution Limits in Ground Mapping Systems .... 784 



XX CONTENTS 

14-21 Future Possibilities in Airborne Active Ground Mapping 

Systems 787 

14-22 Introduction to Infrared Reconnaissance 787 

14-23 Basic Principles Concerning IR Ground Mapping . . 788 

14-24 System Considerations 791 

14-25 Major Systems Features 798 

14-26 New Developments 801 

Index 805 



J. POVEJSIL • P. WATERMAN 



CHAPTER 1 

ELEMENTS OF THE AIRBORNE RADAR 
SYSTEMS DESIGN PROBLEM 

1-1 INTRODUCTION 

This book presents, and illustrates by examples, the basic information 
and procedural techniques required to plan and execute the design of an 
integrated airborne radar system. Basically, this design problem has three 
parts: (1) the development of radar system performance requirements 
based on the operational requirements of the overall, weapons system; 
(2) the development and application of specific radar techniques that will 
meet the performance requirements within the limitations imposed by 
laws of nature and the state of the art; (3) the evaluation of the proposed 
radar system to determine whether or not it meets the requirements of 
the overall weapons system. 

In each part of the design problem, the systems concept is employed; 
i.e., the airborne radar system is viewed as an integral part of a complete 
weapons system rather than as a separate entity. 

The systems concept will be developed by the case study method. A 
hypothetical weapons system model will be constructed. This model will 
then be analyzed in relation to the operational requirements in order to 
derive the specific characteristics of the various system environments which 
have an important bearing on the airborne radar system design. Those 
areas which sensitively affect the overall system capability will then be 
developed. 

By using the derived characteristics, it will be shown how airborne radar 
systems may be selected and designed to fulfill the overall system require- 
ments and be compatible with the system environments. 

As an example, the air defense system of a naval carrier task force will be 
considered. Two types of airborne radar systems are included in this 
weapons system. 

1. An airborne early warning (AEW) system for alerting the air 
defense of a fast carrier task force. 

2. An interceptor defense system, utilizing the primary information 
generated by the AEW system. 

1 



2 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

The analysis of these examples presents a basic method of approach which 
involves the concept of balancing various system elements — a procedure 
that can and should be used in the design of any airborne radar system. 

Perhaps the most important concept that must be grasped by the radar 
designer is this: A radar is usually a small but vitally important part of a 
dynamic system, i.e., a system whose basic characteristics and parameters 
are constantly changing functions of time. Because of its role as the 
"eyes" of the system, the dynamic performance of the radar must be 
related to — and to a large extent, governed by — the dynamic perform- 
ance required of the entire system. For this reason, the radar designer must 
possess the capability for understanding and analyzing the overall weapons 
system in addition to his specialized knowledge of the details of radar 
systems analysis and design. 



1-2 CLASSIFICATIONS OF RADAR SYSTEMS 

In order to provide background for the discussion of the systems aspects 
of airborne radar design, the basic characteristics and uses of radar systems 
are described. Many of the descriptive terms commonly used in radar 
system technology are defined. Some of the simpler mathematical expres- 
sions that arise in radar work are presented. 

Radar is a word derived from the function performed by early radar 
systems — RAdixo Detection And, /hanging. The word was meant to denote 
systems that transmitted and received radio signals. Today the meaning 
of the word has been extended to include a wide variety of systems that 
employ microwave techniques. It encompasses systems using received 
energy originating in the system (active systems), systems using received 
energy originating at the target (passive systems), systems using received 
energy originating at a transmitter separate from the receiver or target 
(semiactive systems), and systems emitting electromagnetic radiations for 
various purposes (transmitting or illuminating systems). Many complex 
weapon systems include combinations of these basic types. For example, 
an electronic countermeasures system may be composed of a passive radar 
system that detects the presence of hostile electromagnetic radiation and 
utilizes this intelligence to control the action of a jamming system to combat 
the enemy radiation. 

The profusion of radar systems in use today requires that some logical 
means of classification be employed. One such means that has achieved 
general acceptance classifies a radar system according to the four character- 
istics: 

1. Installation environment 

2. Function(s) 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 3 

3. Types of modulation intelligence carried on the transmitted and 
received radiations and the types of demodulation processes used 
to extract information from the received signals 

4. Operating carrier frequency 

Reference to these four characteristics is usually made in any general 
qualitative description of a radar — e.g. an (1) airborne (2) intercept search 
and track (3) conical-scan pulse radar (4) operating at X Band. 

1-3 INSTALLATION ENVIRONMENT 

The most common types of radar system installations are: 

1. Ground-based 3. Airborne (piloted aircraft) 

2. Ship-based 4. Airborne (missile) 

Procurement agencies, in general, have been divided into groups according 
to installation environment in order to simplify their diversity of interest. 
Such a division facilitates the proper treatment of the complex problems 
associated with the development and design of a radar system for a partic- 
ular installation environment, but does not always provide the cross 
fertilization of experience needed to take advantage of progress in any one 
particular line. 

1-4 FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 

Some basic functions which may be performed by radar systems are: 



1. 


Search and detection 


6. 


Communication 


2. 


Identification 


7. 


Radiation detection (Ferret) 


3. 


Tracking 


8. 


Illumination 


4. 


Mapping 


9. 


Information relay 


5. 


Navigation 


10. 


Jamming 



11. Scientific research (e.g. radio astronomy) 

A given radar system may perform only one of these functions. More 
frequently it will perform two or more. Multimode operation is particularly 
characteristic of airborne radar systems where space, size, and weight 
limitations dictate that maximum capability and flexibility be obtained 
from each pound of radar equipment. 

The specification of the functions that must be performed by the radar 
systems equipments is a major product of the system study that must 
precede equipment design. This system study must also produce quanti- 
tative performance goals for each of the required functions. 

In cases where multimode operation is required, the system study must 
set up a definitive specification of primary and secondary modes. This 



4 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

definition can serve as the basis for arbitrating conflicting design require- 
ments resulting from the multimode requirement. 

The specific functions performed by a radar system are outlined below 
in somewhat more detail. 

Search and Detection. An important function of a radar system is 
to interrogate a given volume of space for the presence (or absence) of a 
target of tactical interest. One very common method by which a radar 
system may be used to perform this function is shown in Fig. 1-1. In this 



,-»- Search Radar 




Scan Pattern 



ar Search and Detection. 



example, RF (radio-frequency) energy is generated in the radar system 
(active system). This energy is focused into a highly directional beam by 
an antenna and propagated through space. Should there be an object of 
appropriate characteristics within the radar beam, a portion of the electro- 
magnetic energy impinging on the object will be scattered away from it. 
A portion of this scattered energy finds its way back to the point of trans- 
mission where it may be detected by a receiver. 

In order to extend the space coverage of the radar system, it is customary 
to scan a predetermined volume of space in a cyclic manner by changing 
the direction of propagation as indicated in Fig. 1-1. 

Identification. The system may be required to operate in an area 
where both friendly and unfriendly aircraft or targets possibly exist. 
A requirement will then arise to search the area and identify any targets 
as friend or foe (IFF). When it is performing the search and detection 
function, the radar system generates answers to a specific question: Is 
there — or is there not — a target of tactical interest within a given volume 
of space? The basic characteristics of a radar — or any detection device 
■ — are such that both correct and incorrect answers to this question may 
be generated. There are, in fact, four possible sets of circumstances: 

1. There is a target within the searched volume and its presence is 
detected by the radar. 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 5 

2. There is a target within the searched volume, but for one reason 
or another its presence is not detected by the radar. 

3. There is, in fact, no target within the searched volume and none 
is indicated by the radar. 

4. There is no target within the searched volume; however, the 
presence of a target is indicated by the radar. 

In cases (1) and (3) the radar provides the proper answer to the question. 
In case (2) the radar /^z7j to provide the proper answer by failing to provide 
any information whatsoever. In case (4) the radar provides the wrong 
answer by providing spurious information. 

The manner in which the identification function is performed varies 
widely according to the type of radar and the tactical use to which it is 
put. In some cases, the detection and identification functions may be 
combined by a logical nonmechanical process which uses a suitable choice 
of a detection criterion and a prior knowledge of the probable target 
characteristics. For example, in the search and detection system, Fig. 
1-1, one might specify that the appearance of a target indication on 
each of three successive scan cycles constitutes a detection — the assump- 
tion being that it is not likely that a spurious indication would be repeated 
on three successive scan cycles. One might further stipulate that any 
target thus detected shall be considered an enemy target if it is approaching 
at predetermined altitudes, speeds, or courses. 

The identification function is sometimes performed by a completely 
separate radar system designed specifically to accomplish some part of 
the identification problem. Many forms of identification-friend-or-foe 
(IFF) systems fall into this category: e.g., in Fig. 1-2 a presumably friendly 




Fig. 1-2 IFF System. 

aircraft is equipped with a passive receiver that detects the search radar 
signals. These signals are used to initiate transmission of a coded signal 
back to the search radar location. This coded signal is correlated with 
the search radar target return signal to establish and define the presence 
of a "friendly" aircraft. 



6 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

The foregoing discussion of the search, detection, and identification func- 
tions points out an important characteristic that affects the performance 
of these functions. This characteristic is the implied uncertainty that 
the desired result will be obtained in a given case. The element of uncer- 
tainty requires a statistical approach to the problem of understanding 
and analyzing the detection and identification characteristics of a radar 
system. 

Tracking. A radar system may be designed with the capability of 
measuring the relative range, range rate, and bearing of any object which 
scatters microwave energy impinging on it. When a radar makes any or 
all of these measurements on a more or less continuous basis (depending 
upon whether it is also searching), it is said to be tracking the target. The 
tracking function can provide information for: 

1. A continuous display or record of relative target position as a 
function of time 

2. Calculation of relative target motion 

3. Prediction oi future relative target position 

The range measurement is achieved by measuring the elapsed time 
between a transmitted signal and the reception of the portion of the 
transmitted energy that is scattered by the target back along the direction 
of transmission and multiplying it by a constant representative of the 
average propagation velocity. The radar energy is propagated at the 
speed of light {c = 328 yd/jusec). Thus the time required for the radar 
energy to travel from the transmitter to the target and back to the trans- 
mitter location is 

. = ?^. (M) 

c 

The range to the target may be expressed 

R = '^= I64t yards (1-2) 

where R = range to target in yards 

/ = time in microseconds between transmission and reception 

c — propagation velocity in yards per microsecond. 

The closing velocity along a line from the radar to the target (range rate) 
can be measured by means of the frequency difference between the trans- 
mitted and received signals caused by the relative target motion. This 
doppler effect will be discussed in Paragraph 1-5. 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 7 

Angular bearing of the target is measured by utilizing a directive beam 
like that shown in Fig. 1-1. With this arrangement a target return is 
obtained only when the beam is pointed in the direction of the target. 
Thus by measuring the angular position of the beam with respect to some 
reference axis when a target return is present, a measure of relative target 
bearing from the radar system is obtained. The accuracy of this measure- 
ment depends to a large extent on the parameters associated with the 
detailed design. The nature of this dependence and the means that may 
be used to improve the accura.':y of angular measurement will be developed 
in later portions of this book. 

Target motion relative to the tracking radar platform may be computed 
with measured range information and the time derivatives of the measured 
range and angle information. Analysis of the two-dimensional case dis- 
played in Fig. 1-3 illustrates the basic principle. 

The relative velocity of the target, 
Vtr can be represented by two com- 
ponents — one parallel to the line-of- 
sight, Vtrp, and the other normal to 
the line-of-sight, Vtru- These quan- 
tities may in turn be expressed 



Line • of • Sight 
to Target 



Vt 



V, 



(R) = R (1-3) 



Target Velocity 
Relative to Target 



R(f) 



(1-4) 



where R = range rate along the line- 
of-sight and 4> = space angular rota- 
tion of the line-of-sight. 

Range-rate information can be ob- 
tained by differentiation of the radar 
range measured. It can also be 
measured directly by doppler fre- 
quency shift as previously indicated. 

Commonly, the space angular rate 
of the line-of-sight is measured by 
an angular-rate gyroscope mounted 
on the antenna of a tracking radar. 

The relative target velocity information may be utilized in several ways. 
For example, this information coupled with a knowledge of the tactical 
situation can provide a means for identifying targets of tactical interest. 
In addition, the computation of the components of relative target velocity 
makes it possible to predict the future target position relative to the radar 




Fig. 1-3 Relative Target Motion: Two- 
Dimensional Case. 



8 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

platform. This capability is essential to the solution of the fire-control 
problem. 

An analysis of the two-dimensional fire-control problem (Fig. 1-4) 
illustrates the basic principles. If an aircraft is armed with a weapon 
which is fired along the aircraft flight line, the weapon position relative 
to the interceptor // seconds after firing can be expressed 

Rf^ = V,tf (1-5) 



L=Lead Angle 
= Line of sight Angular Rate 




Fig. 1-4 Air-to-Air Fighter-Bomber Duel Fire-Control Problem in Two-Dimen- 
sional Coordinates Relative to the Weapon Firing Aircraft. 



where Rfw = future relative weapon position 

Vq = average velocity of weapon relative to fighter velocity 
// = weapon time of flight (i.e. time elapsed after weapon firing). 

The fire-control problem is solved when the future relative range of the 
weapon coincides with the future relative range of the target Rft, i.e. Rfw 
= Rft (at some value of //). 

The predicted future relative target range may be expressed in terms 
of its components relative to the line of sight 

Rft^ = R- VTRvtf = R- Rtf (1-6) 

Rftn = VTRjf = R4>tf (1-7) 

Similarly, relative weapon range may be expressed 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 

Rfwp = ^(/f cos L 
Rfwn = y^tf sin L 
Equating components, we obtain the basic fire-control equations 
VqIj cos L = R — Rtf (time of flight equation) 



_R^ 



(lead angle equation). 



9 
(1-8) 

(1-9) 

(1-10) 
(1-11) 



Mapping. The microwave energy scattering characteristics of 
physical objects provide a wide range of characteristic returns. The 
differences between these returns make it possible to use a radar system to 
obtain a map of a given area and permit the interpretation of the results 
through an understanding of the characteristic returns. The mapping 
function is accomplished by "painting" (scanning) a designated area with 
a radar beam of appropriate characteristics. Two common means for 
performing this function are shown in Fig. 1-5. 

In the first method, Fig. 1-5A, the picture is "painted" by rotating the 
antenna beam around an axis perpendicular to the area to be mapped. 
The resulting picture is a circular map whose center, disregarding trans- 




FiG. 1-5 Radar Mapping: (a) Forward-Look System, Variant of the Plan Position 
System, (b) Side-Look System. 



lational motion, is the radar's position. The coordinates of the display are 
conveniently in terms of angle and range. The title "Plan Position" is 
applied to this type of map. A variant of this scheme would be a system 
that mapped only a sector of the circle — for example, a sector just for- 
ward of the radar aircraft (Forward-Look System). 



10 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



In the second method, Fig. 1-5B, fixed antennas are mounted on each side 
of the aircraft. The motion of the aircraft with respect to the ground 
provides the scanning means. Thus the picture obtained by this radar is 
a continuous map of two strips on either side of the aircraft flight path. 
In each case the detail is very diflFerent than that obtainable from photo- 
graphs of the same terrain under conditions of good visibility. Never- 
theless, a considerable amount of potentially useful tactical information 
can be obtained from such pictures. The distinction between land and 
water areas is particularly striking, and prominent targets — large ships , 
airfields, and cities — can also be clearly distinguished. 

The basic capabilities of radar provide several attractive features in 
the performance of the mapping function. The range to the target is 
directly measurable. Smoke, haze, darkness, clouds, and rain do not pro- 
hibit taking useful radar pictures (depending on the radar parameters 
chosen). A camouflaged target that might be exceedingly difficult to 
distinguish by visual means is often readily unmasked by a radar picture. 
Finally, a radar picture does not necessarily have the same problems of 
perspective that tend to distort a visual picture. 

The change of target characteristics with frequency can be employed to 
provide increased contrast. The basic principle is illustrated in Fig. 1-6, 
which shows hypothetical backscattering curves for the sea and a target. 
If the mapping is performed at two frequencies,/i and/2, and if the returns 
at these frequencies are transformed into green and blue, respectively, 
on a visual display, then the target will appear green and the sea blue. 
This color transposition utilizes the human eye's ability to discern color 
differences (see Paragraph 12-7), thereby improving the contrast in cases 
where a relationship similar to Fig. 1-6 exists. 

By the use of the doppler (velocity 
discrimination) eff"ect, a mapping 
system may also be provided with 
the capability for distinguishing 
moving targets that have a compo- 
nent of velocity along the sight-line 
of the radar. This is known as woy- 
ing target indication (MTI). 

Another type of radar mapping 
does not involve the generation and 
transmission of microwave energy 
by the radar. Rather, it utilizes the 
fact that all bodies — as a conse- 
quence of their temperature and em- 
issivity characteristics — emit energy in the microwave spectrum. By using 
highly directional antenna and a receiver that is sensitive to these radia- 




TRANSMITTED FREQUENCY-f 

Fig. 1-6 Utilizing the Change of Target 
Characteristics with Frequency to En- 
hance Mapping. 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 11 

tions, a given area may be mapped by scanning the area and correlating 
the signals received with the antenna position. This method — often 
referred to as microwave thermal mapping (MTR) — is similar in concept 
to the various forms of infrared mapping. The only difference is the 
frequency spectrum covered. The use of microwave frequencies sometimes 
alleviates the severe weather limitations of the much higher-frequency 
infrared spectrum. Counterbalancing this advantage is the inherently 
poorer resolution obtained at microwave frequencies and the vastly smaller 
amounts of thermal radiation energy at these lower frequencies. 

Navigation. The mapping capability can be used to perform a portion 
of the navigation function, particularly under conditions of poor visibility. 
Prominent land masses, land-water boundaries, and objects located in a 
relatively featureless background such as an aircraft carrier at sea are 
usually readily distinguishable — even on a radar picture obtained from 
a radar system not specially designed to perform the mapping function. 

By a proper choice of radar parameters, cloud formations that represent 
a potential flight hazard can readily be detected by a radar of appropriate 
design. Radar systems designed specially to perform this function have 
become standard equipment on many transport and military aircraft. A 
typical radar picture obtained from such a system is shown in Fig. 14-15. 
Information such as this represents a valuable navigational aid. It can 
permit the successful completion of many missions that might otherwise 
be aborted because of weather uncertainty. Radars designed for other 
purposes can provide this information as an auxiliary function. 

Another radar navigational aid is the radar beacon system (Fig. 1-7). 
In this system an airborne radar transmits microwave energy at a specified 
beacon frequency. When some of the energy is received by a beacon station 
tuned to this frequency, this energy is, in effect, amplified greatly and 
transmitted back to the interrogating aircraft. There is preset, fixed time 
delay 4 between the reception and the transmission in the beacon. Thus 
if the total time between interrogation of the beacon and the reception of 
the beacon reply is ti /xsec, the range to the beacon is 

R = ^{ti- 4) 



R = 164(/i - 4) yards (1-12) 

where c jl = \ propagation speed of light in yd/jusec 
ti = propagation transit time 
4 = beacon delay time. 



12 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 




(b) 



Beacon Station at a 

Known Geographical 

Location 



Fig. 1-7 Radar Beacon System. 

The angular position of the beacon relative to the aircraft is measured 
by the airborne radar. Since the pilot knows his own heading in space and 
the geographical position of the beacon, the knowledge of relative range 
and bearing of the beacon permits him to determine his own geographical 
location. 

It is quite common for an airborne radar to have a beacon mode as an 
auxiliary function. Despite the apparent simplicity of the mode, the 
proper integration of this function into an airborne radar system is often 
difficult, particularly if early systems planning neglects to include the 
cooperative beacon itself. Variations of the beacon mode of operation are 
also quite common in guided missile applications. 

An airborne radar possesses an inherent capability for providing still 
another type of navigational information — true ground speed — achieved 
through the use of the doppler effect mentioned above in the discussion 
of the tracking function. This application will be discussed in detail in 
Chapter 6. 



Communications. The transmitted radar signal may also be used as 
a carrier for the transmission of communications intelligence. While such 
transmission is limited essentially to line-of-sight because of the inherent 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 13 

nature of microwave propagation (see Chapter 4), it has a number of 
potential advantages: (1) high directivity, increasing the security of the 
communications link; (2) dual utilization of the same antenna and carrier 
power source; and (3) relative predictability of the transmission character- 
istics. 

Radiation Detection. The radiation detection or passive listening 
function that may be performed by a radar system has already been men- 
tioned in the preceding discussions of IFF, ECM, beacon, and com- 
munications systems. 

A passive radar system consists of only a receiving channel or channels 
designed to detect and — in some applications — to track microwave 
energy that is emitted or scattered by a separate source. Passive radars 
cannot measure range without auxiliary devices. 

There is a variety of means for obtaining range measurements from a 
passive system — e.g., triangulation using several passive tracking systems 
at different locations; but all these methods are complicated and inaccurate 
when compared with the convenience of range measurement in an active 
radar system. 

Several important functions may be performed by passive radar systems 
in addition to those already discussed. In the Ferret application, radar 
receivers tuned to cover a wide band of frequencies are used to detect 
enemy radiations, thereby providing intelligence data on the characteristics 
and capabilities of enemy radar systems. Such information is of great 
value in determining the tactics and countermeasures to be employed in 
subsequent operations. 

A variation of the above application is one in which the enemy radiation 
is used as a source upon which a guided missile homes — a system known 
colloquially as a "radar buster." Despite their simplicity of concept, such 
systems may present formidable systems engineering and design problems. 
The multiplicity of enemy signal sources, the intermittency of trans- 
mission from a scanning source, and the importance of having a stand-by 
mode of operation in the event that the enemy ceases to radiate for exten- 
sive periods of time, all contribute to the difficulties. 

A special case of the "radar buster" passive radar homing system is the 
"home-on-jam" system. This system might be used as an alternative 
mode of operation for an active radar system. When the active radar is 
jammed, the jamming source could be detected and tracked by the passive 
system. 

A passive radar system also forms a vital part of a semiactive guidance 
system. This application is discussed later in this chapter. 

Illumination. A common form of radar system is the semiactive 
system. The functional operation of such a system is shown in Fig. 1-8 



14 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 




Fig. 1-8 Semiactive Guidance System. 

and is described elsewhere in greater detail.^ In this system, the target is 
illuminated by a source of microwave energy. A portion of this energy 
is scattered by the target and may be detected and tracked by a passive 
receiver located at some distance from the transmitting source. 

Semiactive systems find their greatest use in guided missile systems, 
where it is often desirable to retain the basic advantages of an active 
system without incurring the weight penalty and transmitting antenna 
size restrictions that would result from placement of the transmitter in the 
guided missile. 

It is possible to obtain a crude measurement of range in a semiactive 
system if the missile is illuminated by the same energy transmission as the 
target. The accuracy of this range measurement is greatest when the 
illuminator, missile, and target are in line as shown in Fig. 1-9. In this 



&^ 



r) 



Rf^ct, 



L_ U ' 



Fig. 1-9 Rane;e Measurement in a Semiactive system. 



case, the target receives energy from the interceptor-borne radar /o Msec 
following transmission. The illuminating energy is also received directly 

lA. S. Locke, Guidance (Principles of Guided Missile Design Series), D. Van Nostrand Co., 
Princeton, N. J., 1955. 



1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 15 

by a rearward-looking antenna on the missile t\ ^tsec after transmission. 
The missile, by measuring the time difference between these two signals, 
can obtain the range to the target; thus 

Rft = Ct2 

Rfm = ctx. 

Since Rmt = R/t — Rfm 

then Rmt = c{t2 - /i) (1-13) 

where c = speed of propagation in yd/jusec = 328 yd/Msec. 

The relative velocity between the missile and the target can be obtained 
by analogous means, using the frequency difference between the direct 
and reflected signals. This frequency difference is caused by the doppler 
effect. 

Information Relay. From a systems standpoint, it is often desirable 
to display and utilize radar information at a different location from the 
point of collection of the information. Typical of such an application is 
the air surveillance system shown in Fig. 2-15. Data are collected by a 
number of airborne early warning (search radar) systems located in such 
a manner as to provide the required coverage. It is desirable to assemble, 
correlate, and assess the data at a central location (Fleet Center) in order 
to provide a complete picture of the tactical situation. From this analysis, 
instructions and data can be relayed to the operating elements. This type 
of operation is typical of airborne, ground, or ship-based combat information 
centers (CIC). 

Jamming. Radars may also be used to transmit microwave energy 
with the object of confusing or obscuring the information that other radars 
are attempting to gather. Jamming is of two fundamental types: (1) 
"brute force" and (2) deceptive. 

Brute force jamming attempts to obscure as completely as possible the 
information contained in other radar signals by overpowering these signals. 

Deceptive jamyning, on the other hand, endeavors to create mutations 
in the information contained in other radar signals to render them less 
useful tactically. 

Both types of jamming are aided by their one-way transmission char- 
acteristic as contrasted with the two-way transmission characteristic of 
active radar. This feature allows a jammer to operate successfully with 
a few watts of transmitted power against a radar transmitting hundreds 
of thousands of watts of peak power. 



16 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

Despite this formidable advantage, the design of a jamming radar 
system can be one of the most perplexing of all radar systems problems — 
from the points of view of both systems engineering and hardware design. 
This arises from the vast multitude of possibilities with which a jamming 
system must cope. 

Scientific Research. Airborne radars are frequently utilized to 
gather basic scientific data such as atmospheric transmission characteristics, 
target reflectivity, and ground reflectivity and emission characteristics. 

The coming space age opens up several interesting possibilities. It is 
very probable that the first glimpse of the surface characteristics of the 
planet Venus will be provided by a radar picture taken from an inter- 
planetary vehicle. The use of radar techniques would permit the penetra- 
tion of the optically opaque atmosphere which completely obscures this 
enigmatic planet, as well as provide a quantitative evaluation of its atmos- 
pheric components. This could be accomplished by measurement of the 
attenuation of the radar energy as a function of frequency. As will be 
discussed in Chapter 4, water vapor, oxygen, and carbon dioxide exhibit a 
marked efl^ect upon radar energy transmission characteristics at certain 
frequencies. Passive radar techniques (microwave thermal mapping) 
could be employed to ascertain the surface temperature distributions and 
the heat balance. This type of scientific data would be invaluable for the 
determination and prediction of weather conditions. 

1-5 THE MODULATION OF RADAR SIGNALS 

A radar system may perform a number of functions (Paragraph 1-4) 
that involve the collection or transmission of intelligence for some defined 
tactical objective. The intelligence is carried by modulations of the radar 
microwave signal. The means used to create these modulations and the 
means employed to extract information from them (demodulation) form 
a convenient and mathematically useful way to describe and classify radar 
systems. 

As will be seen in later portions of this book, the key to the understanding 
and proper design of a radar system is a knowledge of the modulation proc- 
esses that can take place. The various processes of modulation and de- 
modulation are conveniently explained by the use of simple generic repre- 
sentations of the three basic elements of a radar system: (1) the transmitter, 
(2) the target, and (3) the receiving system. 

A simple transmitting system is shown in Fig. 1-10. It consists of a 
means for generating alternating current power, a means for carrying this 
power to an antenna, and an antenna that radiates some portion of this 
power into the surrounding space. 



Amplitude Control 








Frequency Control 


Modulator 




£=Acos(wf+</));: 




Phase Control 






















AC 
Power 
Source 















K---' 



Lobe ^ 



Fig. 1-10 Simple Transmitting System. 



17 



The generating device may be visualized as producing a sine wave out- 
put of constant amplitude and frequency. 

E{t) = ^cos (a;o/+ <i>). (1-14) 

If this power is in turn applied to an antenna which radiates a portion 
equally in all directions (omnidirectional), we have the simplest sort of 
radar transmitter. We may proceed to refine the system by modulating 
the radiation in different ways. 

Space Modulation. The radiated energy may be space-modulated by 
an antenna possessing directivity. Such a characteristic is shown in Fig. 
1-10; the radiated energy is concentrated into a lobe by means of a parabolic 
reflector. 

Three other types of modulation — amplitude ^ frequency , and phase — 
may be introduced by suitable operations upon the power generator. 

Amplitude Modulation. If the output of the transmitter is ampli- 
tude-modulated at an angular frequency coi with fractional modulation m^ 
it then has the form 

E{t) = Aq {\ -\- m cos coi/) cos {ui4 + (^) 

= A^ cos (wo/ + <^) H 2~^ cos [(wo 

, mA u 
^ 2 



OJl) / + 0] 

cos [(coo + wi) / + 0]. (1-15) 



Note that this type of modulation produces sidebands in the generated 
voltage; i.e., the generated voltage has frequency components which 
differ from the carrier angular frequency wo by plus-or-minus the modulat- 
ing angular frequency wi. The transmitted spectrum for the case of 100 
per cent modulation {in = 1) has the form shown in Fig. 1-11. The voltage 
amplitude of each sideband in this case is one-half that of the carrier, and 
the power in each sideband is one-quarter of the carrier power. Obviously 



18 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



FREQUENCY (ANGULAR) 

Fig. 1-11 Generated Frequency Spec- 
trum for 100 Per Cent Sinusoidal Am- 
plitude Modulation of Carrier. 



as m decreases the power in the side- 
bands decreases and becomes a lesser 
fraction of the carrier power. 

A common type of amplitude mod- 
ulation arises from a modulating 
signal of the form shown in Fig. 1-12. 
Essentially, this signal turns the 
transmitter on and off on a periodic 
basis. Accordingly the output is a 
train of pulses of the carrier fre- 
quency. Since this modulating signal 
is periodic, it may be expressed as 
a Fourier series with a fundamen- 
tal frequency equal to the pulse 



TIME 

Fig. 1-12 Pulse Modulation. 

repetition frequency (PRF — 1 /T^), where Tr is the time between 
successive pulses.^ Thus, this type of modulation gives rise to a large 
number of sidebands separated from the carrier frequency by multiples 
of the pulse repetition frequency. The amplitude spectrum of such a 
modulated wave is shown in Fig. 1-13. As can be seen, the pulse width r 
determines the amplitude of each of the sidebands. 

Radar systems employing the type of amplitude modulation just de- 
scribed are known as pu/se-type radars. Pulse radars, however, are not 
limited to this type of modulation, as will be described in later paragraphs. 



Frequency Modulation. Another major type of modulation is 
frequency modulation. In this case, the argument of the cosine function 
in Equation 1-14 is varied in such a manner as to cause the instantaneous 
frequency to be altered in accordance with the modulating signal. When 

^Actually, the pulse amplitude modulated AF wave can be represented by a Fourier series 
with a fundamental frequency equal to the pulse repetition frequency only when the carrier 
frequency oin is an integral multiple of the PRF. 



1-5] 



THE MODULATION OF RADAR SIGNALS 



li 



^T 



o o o 

3 33 
FREQUENCY *■ 



■ III 



Fig. 1-13 Amplitude Spectrum of a Pulse Train. 



19 



the latter is a cosine wave of angular frequency coi and the peak excursion 
of the modulated transmitting angular frequency is Ao), the transmitter 
output is 



£(/) 



.-^ cos {(joot -\ sin coi/ -j- 0) 



(1-16) 



whose envelope has a constant value. A typical frequency-modulated 
wave is shown in Fig. 1-14. 



■Ao)- 



FREQUENCY ^ 

Fig. 1-14 Typical FM Spectrum for High-Modulation Index (Aoj/coi > 10). 



A key parameter in an FM system is the ratio 
— = modulation index. 



(1-17) 



If this index is relatively high — say 10 or greater — the output spectrum 
has the form shown in Fig. 1-14. As can be seen, a single modulating 
frequency gives rise to a large number of sidebands separated from the 
carrier frequency by harmonics of the modulating frequency coi. The 
sidebands of primary importance lie within a bandwidth Aco centered about 
the carrier frequency coo- 



20 FXEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

Various types of transmitter frequency modulations are commonly 
employed. Pulse-width modulation and pulse-time modulation are used to 
transmit information on a train of pulses. In Chapter 6 it will be seen 
how range can be obtained from a continuous-wave (CW) radar by fre- 
quency modulation of the transmitted frequency. 

Frequency modulation of the transmitted signal often occurs inadvert- 
ently owing to the characteristics of the transmitter. The magnitude of 
this effect must be carefully controlled by the designer. 

Phase Modulation. Phase modulation is similar to frequency 
modulation in that the instantaneous phase angle is varied from some mean 
value. With phase modulation by a single cosinusoid of frequency coi and 
phase deviation <^, the transmitter output is 

E{t) = A cos (coo/ + A0 cos CO i/). (1-18) 

The difference in the arguments of the cosine functions of Equations 1-16 
and 1-18, while not important for audio systems, is important elsewhere 
where the waveshape must be controlled. 

Subcarriers. The foregoing discussion has shown that it is possible 
to modulate the transmitted radar signal in four basic ways • — space, 
amplitude, frequency, and phase. At this juncture, it is appropriate to 
consider just why one would want to modulate the transmitted signal. 
The purpose of these modulations is to create information subcarriers, i.e., 
an angle information subcarrier, a range information subcarrier, etc. The 
target information is contained in modulations of these subcarriers (and 
also the carrier frequency) that are created by the target itself and is 
derived upon return of the signal to the receiver by correlation with the 
transmitted subcarriers. 

Target Modulations. In order to understand the basic processes 
involved, it is now appropriate to investigate the modulations of the main 
carrier and its associated subcarriers that are created by the target. First 
of all, the amplitudes of the transmitted radar signals that are reflected 
back to the transmitting location are vastly reduced — perhaps by a factor 
of 10^" on a power basis. Moreover, the reflecting characteristics of the 
target are, in general, a function of frequency. Thus, the amplitudes of 
each carrier frequency in the reflected wave may not be modulated by 
equal amounts. 

Additional amplitude modulations are created by characteristic time 
variations of the target reflective characteristics. Chapter 4 will cover 
this phenomenon in detail. It will suffice for the moment to state that this 
effect introduces additional modulation which broadens each of the returned 



1-5] 



THE MODULATION OF RADAR SIGNALS 



21 



sidebands to an extent depending upon the rate of target reflection char- 
acteristic fluctuations. 

The target reflection entails phase changes with reference to the trans- 
mitted signal incident to the finite time required for propagation of micro- 
wave energy to and from the target. These phase changes occur in all 
the frequencies of the transmitted wave. The phase changes are linear 
with frequency and have a proportionality constant which depends upon 
the distance to the target. The phase modulation that occurs in the 
portion of the transmitted signal that is reflected back from the target 
provides the basic means for measuring range to the target. Pulse radars, 
for example, measure the phase (or time) difference between transmitted 
and received pulse trains. 

Phase modulations of a somewhat different sort may result from the 
motion of the target in conjunction with the space-modulation character- 
istic of the radar. As an example of this process, consider a radar which 
scans a directional beam through an angle of 360° once each second. If 
there is a stationary target at an angle of Qt with respect to the reference 
axis, a return from the target will be obtained as the radar beam sweeps 
past this point. The amplitude of the return signal will have the general 
shape of the radar beam resulting in a return signal having the envelope 
shown in Fig. 1-15. Thus, the scanning process gives rise to an angle in- 







A. 



A 



[- — 1 sec — H 



27r 47r 

ANGLE (rad) 



^Lll sec-^ 
A , A 



2 

TIME (sec) 



, J\ Stationary 
67r Target 



I A Moving 
3 Target 



Fig. 1-15 Effect of Target Motion. 



formation subcarrier which has a fundamental frequency of 1 cps. The 
angle information is carried on the phase angle of this subcarrier funda- 
mental. 

Now let us assume that the target flies in a circle around the radar station 
in the same direction the beam is revolving, at a speed of 1 revolution 



22 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

every 10 seconds (i.e., one-tenth of the scanning velocity). The return 
signals now have the form shown in the lower diagram. The effect of the 
target motion has been to shift the fundamental frequency of the angle 
information subcarrier by 10 per cent to 0.9 cps for a target moving in 
the same direction as the scan. This process might also be viewed as in- 
troducing a time-varying phase shift in the 1-cps subcarrier (phase modula- 
tion). How the target modulates the information subcarriers is one of the 
most important problems of radar design. The choice of frequency and 
bandwidth for the subcarrier frequency information channels is largely 
governed by these characteristics. 

One other target modulation — the doppler frequency shift mentioned 
in preceding paragraphs — is of fundamental importance. 

Motion of the target along the direction of propagation (see Fig. 1-16) 



^=> 



OAAAAAAAAA. 

•« 

Fig. 1-16 The Doppler F.ffect. 

causes each frequency component of the transmitted wave that strikes 
the target to be shifted by an amount 

/d = {VtIc)/ (1-19) 

where Vt = the velocity of the target 

c = the velocity of light 

/ = the radio frequency. 

When this signal is reflected or reradiated back to the radar, the total 
frequency shift of each component is 

fn = {2FtIc)/. (1-20) 

The frequency modulation caused by target motion is important; an 
entire family of radars known as doppler radars has been developed to 
exploit this characteristic. However, whether use is made of this char- 
acteristic or not, the doppler shift occurs in all signals reflected from 
objects that possess relative radial motion. 

Thus, it can be seen that the target generates a large number of am- 
plitude, phase, and frequency modulations of the transmitted signal. 



1-51 



THE MODULATION OF RADAR SIGNALS 



23 



These modulations create information sidebands about the carrier and 
subcarrier frequencies. The designer's problem is to determine how this 
information may be extracted from the target return signal. 

Extraction of Target Intelligence from Radar Signals. One thing 
is common to all the many techniques for extracting target information 
from a radar return signal. This is the concept of taking a product between 
the target return signal and another quantity which serves as the reference 
for the particular piece of information being extracted from the target 
return. Thus, the generic building block for a radar receiving system is a 
product-taking device, as shown in Fig. 1-17. 







Reference 


Incoming Signal 




Product Signal 











Fig. 1-17 Generic (Product) Building Block for a Radar Receiver. 

Conceptually, the simplest product-taking device is a network — or 
filter — composed of linear impedances which can be characterized by a 
transfer function F{jui). Each frequency component of the incoming signal 
is multiplied by the vector transfer function of the network corresponding 
to the frequency (see Fig. 1-18). The output product is a signal containing 



Input 



F(;co) 



Output 



[Output] =[f(; CO)] X [input] 
Fi/co) - Fl(j} 

Fig. 1-18 Impedance Products. 

the same frequencies as the input; however, the amplitude and phase of 
each frequency component may be changed with respect to the input. 
In this type of product device, the references are the characteristics built 
into the filter. 

The second type of product-taking device is the nonlinear impedance. 
A simple example of such a device is shown in Fig. 1-19. The operation 
of the device is such that positive inputs are faithfully reproduced at the 
output while negative inputs are completely suppressed. Thus, for an 



24 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



Input 


■5 
B- 



/ 


Output 


V-/ \^ 








Input 





Fig. 1-19 Product Obtained from a Nonlinear Impedance. 

input sine-wave, the output consists of only the positive half-cycles as 
shown. 

It is interesting to observe that this process may be put into the form 
of the generic device of Fig. 1-18 merely by considering the output to be 
the product of the input and a reference square wave of the same frequency 
and phase as the input. Such a representation is shown in Fig. 1-20. 

Thus, the input may be defined as 



:rLrLn 

TT 2ir 3ir Att Stt 




Fig. 1-20 Nonlinear Impedance as a Product-Taking Device. 

Ei = A sin CO/. 
The reference signal may be expressed by a Fourier series 

sin Sco/- , sin 5a)/ 



Er 



H (sin co/ 



+ 



+ -) 



and the product has the Fourier series form 

zr w r^ A , A . lA V 

Ei X /t,. = - + ;r- sin a)/ ) 



Inixit 



An' 



(1-21) 
(1-22) 

(1-23) 



The consequences of this product process are quite evident from Equation 
1-23. Although the input contains only one frequency, the output has a 
d-c component, an input-frequency component, and components at all 
the even harmonic frequencies of the input. 

Now, if the amplitude of the incoming wave A, instead of being constant 
as implied, were amplitude-modulated at a frequency aj;„, such that 



1-5] THE MODULATION OF RADAR SIGNALS 25 

A = A^{\ ■\- 771 COS co„/) (1-24) 

where aj„ = modulating frequency (aj^ ^ w) and m — modulation ratio 
{m < 1), then, each of the terms of the product (Ei X Er) would contain 
modulation sidebands. For example, the d-c term would now become 

Ao . mAo n nc\ 
1 cos co,„/ (1-25) 

T IT 

and the fundamental frequency term would become 

'-y sin w/ H ^ [sin(co -(- w™)/ — sin(w — w™)/] (1-26) 

and so on for the higher harmonics. 

If this (Ei X Er) product were then passed through a filter, F(ju), 
which eliminated the d-c term and the fundamental frequency u and all 
its harmonics, the final output would be 

(Ei X Er) X F(jc^) = '-^ cos co„,/. (1-27) 

TT 

Thus, we observe, the frequency and the magnitude of the modulation 
intelligence are recovered from the incoming wave by the product-taking 
procedures. The procedures just described are often referred to as de- 
modulation or detection. 

A third type of product-taking device closely resembles the basic model 
of Fig. 1-17. The incoming signal is multipled by a reference signal gener- 
ated within the radar receiver. One form of this process is known as 
7nixing or heterodyning. In this process, a cross-product is taken between 
the incoming signal and a locally generated signal. This process converts 
the microwave signal to a much lower frequency, which may be filtered 
and amplified by relatively simple electronic techniques. 

Two general forms of microwave mixing are commonly used, noncoherent 
mixing and coherent mixing. In coherent mixing, the phase of the locally 
generated signal is made to have a known relationship to the phase of the 
transmitted signal. This type of mixing makes it possible to detect the 
phase and frequency modulations introduced by target motion. 

The extraction of angle and range information from the received signals 
is almost always accomplished by a cross-product of the received intel- 
ligence and an internally generated reference signal. 

The detailed analysis of the various means for extracting target intel- 
ligence from radar signals — and the problems that arise in these processes 
— forms a major portion of this book. Chapter 3 and Chapters 5 through 9 
are all concerned with various phases of these problems. 



26 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

1-6 OPERATING CARRIER FREQUENCY 

The operating frequencies for radar systems cover an extremely wide 
band, ranging from below 100 to above 10,000 Mc. This range is divided up 
into bands designated P, L, S, X, K, Q, V, and W as shown in Fig. 1-21. 



100 80 60 50 40 30 20 

I I I I I I I I \ L 



WAVELENGTH - Cm 
10 8 6 5 4 3 

I I I I I I I L 



1 0.8 0.6 

Mill I 



0.4 0.3 

J I 

100 



0.5 0.6 



3 5 6 10 

FREQUENCY - KMc 



20 30 



50 60 



BAND DESIGNATION 



TV 



m 



i 



I 



in 



USE ALLOCATION 



□ 



Allocated to Armed Forces and Other 
Departments of the U.S. Government 

Allocated to Radio Navigation, Radio Location, and Civilian 
Radar. Sometimes Used By Military Equipments. 

Allocated to Television, Common Carriers, Domestic Public, 
Industrial Safety, and International Control. Military Equipments 
Precluded from These Bands Except at Times of National Emergency. 

Fig. 1-21 Operating Radar Frequency. 



The specific frequencies available for airborne radar systems are, in general, 
regulated by the Federal Communications Commission during times of 
comparative peace. 

The operating carrier frequency has a profound effect on the following 
characteristics of a radar system: 

1. Size, weight, and power-handling capabilities of the RF com- 
ponents (see Chapter 11) 

2. Propagation of RF energy (see Chapter 4) 

3. Scattering of RF energy (see Chapter 4) 

4. Doppler frequency shift from a target moving relative to the radar 
direction of propagation. 

These characteristics vary quite radically over the range of radar 
operating frequencies — enough, in fact, that it becomes convenient to 
classify a radar according to its operating carrier frequency. This method 
of classification is commonly used by microwave component designers 



]-7] THE AIRBORNE RADAR DESIGN PROBLEM 27 

because the design problems and the techniques used to solve them are 
strongly dependent upon the operating frequency. 

This method of classification is also important to the system designer 
because the operating frequency determines certain of the radar's reactions 
to its physical and tactical environment. For example, an atmosphere 
heavily laden with moisture is more or less opaque in some bands to the 
highest radar operating frequencies, whereas the transmission of the lower 
frequencies is little affected. 

In airborne applications, the smaller size of the higher-frequency radar 
components has favored the use of S, X, and K bands despite their limita- 
tions with respect to weather and moving target indication, as discussed 
in Chapters 4 and 6. 

1-7 THE AIRBORNE RADAR DESIGN PROBLEM 

Preceding sections discussed general radar characteristics. The following 
problem is of paramount importance: How does the radar designer select 
and employ the right combination of these characteristics to achieve an 
acceptable performance level in a given weapons system application? 

The design problem may be divided into two basic parts, problem 
definition and problem solution. 

Problem Definition. The airborne radar design problem is defined by 
the weapons system application. In such applications, an airborne radar 
combines with other system elements — human operators and the airborne 
vehicle and its associated propulsion, navigation, armament, flight control, 
support, and data processing systems ■ — to form a closely integrated 
weapons system designed to perform a specific mission. To achieve a 
given performance level, the weapons system requires certain performance 
characteristics from the airborne radar. 

The radar designer's first task is to examine the requirements and 
characteristics of the complete weapons system. From this analysis, the 
nature of the airborne radar's contributions to overall weapons system 
performance (mission accomplishment) may be obtained. Typical examples 
of the parametric relationships developed in such a study are shown in 
Fig. l-22a. From such curves, the radar requirements for a desired level 
of mission accomplishment may be obtained. In addition, the sensitivity 
of mission accomplishment to changes in radar performance is displayed, 
thereby providing the designer knowledge of the relative importance of 
each performance characteristic. 

The derivation of such relationships must be relatively uninhibited by 
known limitations in the radar state of the art. That is to say, the range of 
values considered for each of the radar's performance capabilities need 



28 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



Detection Range 
Fig. l-22a 



Tracking Error 



Other 



Resolution Element 
Size 
Typical Relations Between Mission Accomplishment and Radar 
Performance in the Operating Environment. 



not bear any relation to a presently realizable radar system. The purpose of 
this analysis is to define the radar problem solely as it is dictated by the 
weapons system problem. Whether the radar problem thus defined is 
technically reasonable for a given era is determined in the next major 
step of the design process. 

Problem Solution. The systems analysis defined the required radar 
performance. Now, the designer must attempt to solve the defined problem 
by (1) hypothesizing a radar system of given general characteristics and 
(2) examining the interrelationships between radar parameters and radar 
performance. Typical examples of the interrelationships developed by 
such a study are shown in Fig. l-22b for the case of a single radar param- 







l_. 



Fig. l-22b Typical Interrelations of Radar Frequency and Radar Performance 

Parameters. 



eter — operating frequency. Similar parametric relations are derived 
for each radar parameter that exercises important influences on radar 
performance. 



1-7] 



THE AIRBORNE RADAR DESIGN PROBLEM 



29 



The information thus derived is examined and correlated to find — if 
possible — the combinations of radar parameters which fulfill the pre- 
viously derived radar performance requirements. Then and only then can 
the designer proceed in an intelligent manner to design the radar hardware 
for fabrication, evaluation, and service use. Often the proper combinations 
cannot be found. State-of-the-art limitations, laws of nature, and other 
factors may conspire to prevent a successful problem solution using the 
assumed radar concept. In these cases, the parametric information generated 
for the problem definition and the problem solution provide readily avail- 
able means for ascertaining the most promising course of action — whether 
it be a change in radar concept, the initiation of a new component develop- 
ment, or a change in the overall weapons system concept. In extreme cases, 
a failure to find a radar solution may justify abandonment of a weapons 
system concept; in other cases an early display of seemingly irreconcilable 
deficiency may provide the spur for the generation of a bold new radar 
concept that performs as required. 

Summary and Discussion. Airborne radar performance usually 
exercises a decisive influence on overall weapons system performance. 
The approach to the design problem must therefore be an overall systems 
approach, even though the radar is only a weapons system component. 

The two basic steps in the design process are problem definition and 
problem solution as illustrated in Fig. 1-23. The first step derives the radar 



Weapons 
System 
Problem 
















Radar 

System 

Requirements 


Radar 
Problem 
Definition 




















Radar 
State of the Art 

and 

Weapons System 

Schedule 

Limitations 






Radar 
System 
Design 








Radar 

^ Problem 

Solution 













Fig. 1-23 The Airborne Radar System Design Problem Approach. 

requirements imposed by the complete weapons system and neglects 
possible limitations of radar techniques. The second step is concerned with 



30 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



fulfillment of the requirements considering limitations of radar state of 
the art, schedules, and other factors germane to the problem of achieving 
a useful operational capability for the overall weapons system. 

In actual practice, the design process is long enough and complicated 
enough to justify subdivision of the two basic phases described. The next 
paragraph will discuss the complete design cycle of a typical weapons 
system development with particular emphasis on the role of the radar 
designer as a vital part of the designers' team. 

1-8 THE SYSTEMS APPROACH TO AIRBORNE RADAR 
DESIGN 

A representation of a typical weapons system design cycle is shown in 
Fig. 1-24. Each step represents a subdivision of the problem definition 

GENERATED SYSTEM CAPABILITIES 



Operational 
Requirements 



£3: 



^-^.^- 



iviission 
Accomplisiiment 



System 
Concept 



Tectinical 

Design 

Objectives 



Technical 
Analysis 



^T 



Detail 

Technical 

Requirements 



Mechanization 
Requirements 



Evaluation 



System 
Mechanization 
Requirements 



Equipment 
Development 



Evaluation 



Model 
Equipment 



Evaluation 
Equipment 



Prototype 
Equipment 



Service 
Equipment 



Service 
Equipment 



Fig. 1-24 Weapons System Design Cycle. 



or a problem-solving step just discussed. On the left side of the figure are 
displayed the sequential steps of an orderly development process from the 



THE SYSTEMS APPROACH TO AIRBORNE RADAR DESIGN 



31 



initiation of an operational requirement to the fabrication of service 
equipment designed to fulfill the requirement. On the right of the figure 
are the definitive outputs or accomplishments resulting from this develop- 
ment. In the middle of the figure are the evaluation processes which meas- 
ure the level of accomplishment attained in the problem-solving phases 
of the development. The outputs of these evaluations may also be used 
to modify succeeding phases of the development process. 

The diagram also indicates feedbacks from the various development 
phases into preceding phases. These reflect the fact that as more is learned 
about the system, prior concepts must be modified and expanded to ensure 
that the system development objectives are current and realistic. 

Viewed in its entirety, the indicated procedure provides a basis for 
playing current accomplishment against the requirement to obtain a 
continuous rating factor representing the generated system capability. 

The step-by-step processes for executing the system development plan 
may be summarized as follows. 

Operational Requirement. The overall system objective is set forth 
in an operational requirement . This requirement usually outlines the 
military task(s) which the weapons system must perform. It will also 
specify — or at least indicate — the level of fnission accomplishment which 
the system must achieve to accomplish the desired military objective. 
The mission accomplishment requirement often has the general form dis- 
played in Fig. 1-25. The weapons system must be operative in a given time 





Desired Level 


1 




\ 


1 


V-Obsolescen 

1 Minimum \ \ 
J Acceptable \ \ 
Level \ \ 
\ \ 



a TIME (YEARS) b 

Fig. 1-25 Operational Requirement. 



period a-l?. The desired level of mission accomplishment represents the 
best estimate of what the military planners believe is necessary to achieve 
unquestioned military superiority in a given area of interest. This goal 
may be variable over the expected operational use cycle (as shown) by 
reason of anticipated introduction of new techniques by the enemy. 

The minimum acceptable level of mission accomplishment represents a 
capability which the military planners believe is still useful enough to 



32 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

justify the weapons system cost. Thus, although the system design will 
endeavor to meet the desired goal, some degradation may be acceptable 
if such degradation can be shown to be unavoidable. 

Unexpected developments in technology or in the long-range strategic 
situation can cause radical changes in the operational requirement during 
the design cycle. For this reason, the radar designer must constantly 
monitor the operational requirement to ensure quick reaction to such 
changes. 

System Concept. The operational requirement defines a military 
problem. The next step is to define a system concept which provides bases 
to presume a weapons system potential capability compatible with the 
operational requirement. This step usually is implemented in the following 
way. Various possible systems are postulated. Technical military agencies 
examine these in the light of available or projected technical capabilities 
to determine which provides the best foundation for a subsequent develop- 
ment. Weapons system contractors may assist this study phase by pro- 
viding new ideas, state-of-the-art evaluations, etc.; however, the basic 
responsibility for decision and action invariably rests with the military. 

Once a decision is made on the type of system desired, the basic features 
of the selected system are set forth in the form of technical design objectives 
These comprise the performance specification of the overall weapons 
system and 

1. The system effectiveness goal related to the operational require- 
ment 

2. The basic system philosophy, i.e., mode of operation 

3. The system environment as defined by tactics, logistics, climate, 
etc. 

4. The characteristics of major system elements 

5. The system design, development, and evaluation program 

6. Fundamental state-of-the-art limitation in various portions of 
the system 

Unless he has already participated in the definition of the system concept, 
the radar designer's T^^rj/ task is to become familiar with these conceptual 
characteristics of the overall weapons system. They define the elements of 
his problem which are relatively fixed and with which his design must be 
compatible. 

Technical Analysis. The systems problem and its boundary con- 
ditions having been defined and understood, the radar designer now is 
ready for the next step — the construction of a weapons system model that 
will define the radar problem. This model is used to determine the quan- 



1-8] THE SYSTEMS APPROACH TO AIRBORNE RADAR DESIGN 33 

titative interrelationships of the radar and other system elements. From 
manipulation of these interrelationships the designer must obtain the true 
technical requirements of the radar necessary to attain a proper balance 
between mission accomplishment and the operational requirement (see 
Fig. 1-24). 

The detailed technical requirements include specification of 

1. Functional capabilities 

2. Radar range and angle coverage requirements 

3. Information handling, transfer, and display requirements 

4. Radar information accuracy requirements 

5. Radar environmental requirements 

6. Radar system reliability requirements 

7. Radar maintenance, stowage, and handling requirements 

In this stage of the design, the emphasis is placed on the job the radar 
must do and the environment in which it must operate. As previously 
mentioned, this analysis should not be inhibited by the introduction of 
state-of-the-art radar limitations. The problem in this phase is to ascertain 
what requirements the radar must satisfy to allow the defined system con- 
cept to demonstrate an adequate system capability. 

This type of analysis is not popular with radar designers. Often it 
results in establishing technical requirements beyond the scope of known 
radar technology. The radar designer is forced to admit a set of require- 
ments he does not believe he can meet. 

However, the wise weapons systems contractor will demand that such an 
analysis be performed and demonstrated by the radar designer for two 
reasons: 

1. It will ensure that the radar designer really understands the 
problem before he tries to solve it. 

2. The earlier a potential source of system degradation is known, 
the easier it is to correct by invention or by modification of the 
development program. 

Because this step is so important to the radar designer, Chapter 2 is 
devoted to a detailed discussion and example of the processes involved in 
the derivation of technical requirements for a radar system. 

Mechanization Requirements. The next step in the design process 
is to synthesize a realizable radar system to meet the derived technical 
requirements within the limitations imposed by development time, state 
of the art, and delivery considerations. Various radar systems are designed 
on paper and analyzed in detail to demonstrate their performance relative 



34 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

to the previously established requirements. Often, it is found that the 
requirements are not compatible with realizable radar systems. This 
discrepancy may be corrected in some cases by using the interrelation- 
ships of the radar and the overall system derived in the previous phase to 
find a different balance of radar requirements that will still permit mission 
accomplishment. In other cases, a known degradation in performance — 
relative to the initial weapons system goal — may have to be accepted. 
A somewhat happier situation arises when it is found that a realizable radar 
system can provide greater capabilities than those required by the initial 
weapons system concept and operational requirement. In this case, the 
initial system concept could be enlarged and improved or, conversely, the 
development objectives could be revised with a resulting economy of design. 
The indicated processes of evaluation and feedback are shown in Fig. 
1-24. The flexibility achieved by the feedback process is the real strength 
of the systems approach. There are few weapons systems that cannot 
benefit from modification of the originally established concepts and tech- 
nical requirements. Circumstances change — often in a highly unpre- 
dictable manner — over the five-to-ten-year development period of a 
weapons system. That is why the radar designer must continue to treat 
this problem on an overall weapons system basis throughout the life of 
the project. 

Equipment Development, Evaluation, and Use. Similar com- 
ments concerning the value of the systems approach apply to the vitally 
important task of building, evaluating, and using the equipment in accord- 
ance with the requirements derived in the first three phases. The problems 
in the latter phases can be formidable. For some perverse reason the actual 
equipment in certain critical areas may not perform in the manner pre- 
dicted, particularly with respect to reliability. A vital part of the system 
approach is the process of rapid isolation and correction of system deficien- 
cies in these phases and the anticipation of potentially critical areas. 
Since it is often difficult to distinguish between a genuine system deficiency 
and a temporary bottleneck, the judgment and experience of radar systems 
engineers who have also participated in the requirements derivation phase 
is most important. There are so many problems in these phases that it 
is easy to concentrate effort on the wrong ones. 

One problem — reliability — dominates these last phases. This is the 
most vexing, most difficult, and most important problem in the design of a 
radar system. Radar systems have never been simple; in the future, their 
complexity may be expected to increase. The most common failure of the 
systems approach to the radar design problem has been the tendency to 
maximize system capability by specifying unnecessarily exotic radar 
requirements which lead to reliability problems in the mechanization phases. 



1-9] SYSTEMS ENVIRONMENTS 35 

In the technical requirements analysis phase, all possible ingenuity should 
be employed to minimize radar complexity for a given weapons system capa- 
bility. This is one of the most important reasons why the radar system 
designer must analyze the radar problem as a weapons system problem. 
Important performance benefits can be achieved by making the proper 
reliability-complexity trade-off early in the game. 

Summary. The systems approach to radar design implies that the 
radar is considered in its relation to the construction and objectives of the 
entire weapons system during all phases of its conception, design, construc- 
tion, and use. The radar systems designer must participate in all phases 
of the development; he must demonstrate a good understanding of the 
overall system and the characteristics of its other components prior to 
laying out a radar design. 

To increase the reader's understanding of the basic features of the systems 
approach to airborne radar design, several of the points presented will be 
amplified in succeeding paragraphs. These include a more precise definition 
of the system environment and its eflFects upon the problem, and a brief 
discussion of the concepts and processes involved in the construction of a 
weapons system model. 

1-9 SYSTEMS ENVIRONMENTS 

In this book, the "expected tactical conditions of operation" include 
all of the following environments. 

1. Tactical Environment — The salient elements of this environment 
are the speed, altitude, operating characteristics, and mission profile of the 
airborne portion of the weapons system; the composition, operating char- 
acteristics, and relative position of the ground-based portion of the weapons 
system; and the characteristics (speed, course, altitude, number, physical 
size) of the target complex. 

2. Physical Enivronment — The salient elements involved are tem- 
perature, pressure, humidity, precipitation, fog, salt spray, wind, clouds, 
sand, and dust. In systems requiring a human operator, the physical 
environment will include factors affecting his ability to operate the system. 
Among these are habitability, ease of operation, length of attention span 
required, and the physical readiness and mental acumen required. 

3. Airframe Enviromnent — The salient elements involved are volume 
and configuration of allotted space within the airframe, weight limitations, 
vibration, and shock. 

4. Electronic Environment — This includes all the external sources of 
electromagnetic radiations and electromagnetic radiation distortions and 
anomalies. Examples are ground, sea, and cloud clutter; radiation from 



36 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

Other systems; electronic countermeasures; propagation anomalies, at- 
mospheric attenuation; and target radar reflective characteristics. 

5. Logistics Environment — This includes all salient considerations of 
the parts of the weapons system that affect reliability, maintenance, han- 
dling, stowage, supply, replacement, and transport. 

6. Weapons System Integration Environment — This is the environment 
formed by other systems with which the system under consideration must 
be compatible. An example would be the environment formed by a ground- 
to-air missile weapons system operating in the vicinity of a proposed 
interceptor weapons system. 

These environments should be looked upon as boundary conditions 
imposed upon the systems design problem. The concept is shown diagram- 
matically in Fig. 1-26. Each element of the system design must be com- 




System 
Environment 



Logistics 
Environment 



Weapons System 
Integration Environment 

Fig. 1-26 The System Environment. 



patible with the requirements and limitations imposed by all of these 
environments. 

1-10 WEAPONS SYSTEM MODELS 

A weapons system model is a simplified representation of the actual 
system which can be used to predict the changes in system performance 
when one or more of the components which make up the system are 
changed. For example, a common problem in radar system design is to 
determine the effect of radar detection range on the performance of the 
overall system. Such a problem would be solved — as will be shown in 
Chapter 2 — by constructing a model containing radar detection range as 
a variable parameter. The radar detection range would be related by 
appropriate means to the pertinent characteristics of the overlapping 
environments which make up the system complex. Within the model. 



1-10] 



WEAPONS SYSTEM MODELS 

Radar Range 



37 



Target Inputs 



TTT 

Fixed Elements 



Mission Accomplishment 




RADAR DETECTION RANGE 



Fig. 1-27 Diagrammatic Representation of a Model Used to Determine Sensitivity 
of Mission Accomplishment to Variations in Radar Detection Range. 



the elements of the system model interact to provide a generated mission 
accomplishment as shown in Fig. 1-27. From such a model, the effect of 
radar range upon system effectiveness could be obtained for various inputs 
(targets), yielding the characteristic type curves shown. 

It is far more convenient to conduct an experiment with a model of a 
system than with the system itself. This is particularly true with a military 
weapons system where the testing of actual hardware is enormously 
expensive. Moreover, system hardware usually is not available until 
long after the original concepts. Theoretical models are required to predict 
expected system performance. 

There are three classes of models which can be used in systems analysis 
work: iconic, analogue, and symbolic. Briefly, we may define the character- 
istics of these model types in the following ways: 

1. An iconic model represents certain characteristics of a system by 
visual or pictorial means. 

2. An analogue model replaces certain characteristics of the system 
it represents by analogous characteristics. 

3. A symbolic model represents certain characteristics of a system 
by mathematical or logical expressions. 



38 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

Iconic Models. An iconic model is the most literal. It "looks like" 
the system it represents. Iconic models can quickly portray the role that 
each subsystem plays in the operation of the overall system. It is therefore 
particularly well adapted to illustrating the qualitative aspects of system 
performance, such as information flow and functional characteristics of 
various portions of the system. 

The iconic model is not well adapted to the representation of dynamic 
characteristics of the system because it does not reveal the quantitative 
relationships between various elements of the system. For the same reason, 
it is not very useful for studying the efi^ects of changes in the system. 
Because of its pictorial value, most system analyses usually begin with the 
construction of an iconic model (block diagram) in order to establish the 
characteristics of the system and to provide the investigator with a realistic 
frame of reference for subsequent studies. This process will be illustrated 
by examples later in this book. 

Analogue Models. Analogue models are made by transforming 
certain properties of a system into analogous properties, the object being 
to transform a complicated phenomenon into a similar form that is more 
easily analyzed and manipulated to reveal at any early time the initial 
elements of system performance. For example, fluid flow through pipes 
can be replaced by the flow of electrical current through wires. A slightly 
more abstract example would derive from the problem of calculating the 
probability of a mid-air collision in a situation where only the laws of 
chance were operative — i.e., where no special equipment or techniques 
were used to prevent collisions. The imaginative investigator might per- 
ceive that this problem bears a striking similarity to the problem of calculat- 
ing the mean free path of a gas molecule. Having established the validity 
of this insight, we would then be free to make appropriate transformations 
between the two problems and apply the kinetic theory of gases to his 
problem. 

Unlike the iconic model, the analogue model is very effective in repre- 
senting dynamic situations. In addition, it is usually a relatively simple 
matter to investigate the efi^ects of changes in the system with an analogue 
model. For these reasons, analogue models form very powerful tools for 
the solution of complex system problems — particularly problems involving 
many nonlinearities. 

The great utility of analogue models is evidenced by the large-scale 
analogue computer installations that form a part of almost every major 
weapons system engineering organization. 

Symbolic Models. The symbolic model represents the components of 
a system and their interrelationships by mathematical or logical symbols. 



1-11] STATISTICAL CHARACTER OF MODELS 39 

This type of model is the most abstract. When such a model is formulated 
and used without incurring prohibitive mathematical complexity, it is 
the most useful model for obtaining quantitative answers to systems 
problems. 

Many problems may be solved by either analogue or symbolic models. 
Where a choice exists, it is preferable to employ the symbolic model, for it 
allows one to examine the effect of changes by a few steps of mathematical 
deduction. This process was implied in the example of the mid-air collision 
analogy, just cited. Here the problem was transformed into an equivalent 
gas dynamics analogue. However, for such a problem we should not con- 
struct a complex instrument and make measurements — it is far simpler 
to use the symbolic models already established for the kinetic behavior 
of gases. Further, we would gain greater insight into the basic nature of 
the problem in this way than would be obtained by empirical methods. 

The utility of symbolic models is particularly evident for problems in- 
volving probability concepts. Often, answers may be obtained in closed 
form for problems that would otherwise require many repeated tests of 
an analogue model. 

The primary disadvantage of symbolic models arises from limitations in 
available mathematical and computational techniques for obtaining 
answers from the model. State-of-the-art improvements in applied 
mathematics and large digital computers are relieving this problem. 
Despite these advances, however, there will always be a great premium 
on the ability to construct symbolic models that strike at the heart of a 
problem and eliminate nonessentials that merely increase complexity. 

1-11 THE BASIC STATISTICAL CHARACTER OF WEAPONS 
SYSTEM MODELS 

The model approach consists of abstracting from a complex system 
certain persistent and discernible relations and using these relations to 
construct a system model. Frequently, owing either to the inherent 
nature of the process being examined or to the complex nature of the 
process, the relations must be expressed in a statistical form. That is to 
say, certain portions of the system — and as a result of this, the system 
itself — will not possess a unique output for a given input. Rather, the 
output must be expressed as a spectrum of possible events where each 
event has a certain probability of occurrence. 

Two simple examples may serve to illustrate the nature of the phenomena 
involved: 

Example 1 — Measurement Uncertainties. Measurements of time, 
distance, temperature, etc., always possess a certain error tolerance. For 
example, a large number of distance measurements made with the same 



40 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 




L tcAL 



MEASURED LENGTH 



Fig. 1-28 Measurement Errors Obtained in Determining Length. 



instrument measuring the same distance might give rise to a distribution 
of values about some mean value as shown in Fig. 1-28. If these measure- 
ments are compared with a standard, we see that two sources of error 
exist: (1) a calibration or bias error, and (2) a random error. The calibra- 
tion error — so long as it remains fixed or if its variations can be predicted 
— is obviously a correctable source of inaccuracy. However, the random 
error is just that — random. Any given measurement may be in error by 
an amount determined by the character — usually Gaussian — of the 
randomness. 

Measurement uncertainties are a vitally important problem in any 
weapons system analysis. Unlike many engineering problems, where 
measurements may be made with whatever degree of preciseness is neces- 
sary to render inconsequential the measurement error, weapons systems 
habitually are required to work with measurement uncertainties that 
exercise a profound and usually decisive influence upon their performance. 

Example 2 — Dice Throwing. The cast of a die is an example of a process 
that is, theoretically, completely predictable; however, because of the 
extreme complexity of the mechanisms that govern its behavior, the 
whole process is more easily handled by probability concepts. For example, 
if we knew the exact orientation of the die, its velocity, direction of motion, 
physical size, shape and weight distribution, condition, characteristics 
and orientation of the die table, etc., we could predict with certainty which 
of the die faces would appear on top. However, the amount of information 
that must be obtained and processed to arrive at this result is usually 
prohibitive. It is much easier — and, also, as in the case of representations 
of this type, less profitable — to characterize such a process by saying 
that for any input (legal throw) the system (balanced die cube) may 
produce any output from 1 to 6 with equal probability. 

Weapons systems contain many processes of similar brand. Ballistic 
trajectories are a prime example. The behavior of a human being in a 



1-12] CONSTRUCTION AND MANIPULATION OF MODELS 41 

control loop is another. Thus the basic parameters of the model usually 
take the form of distribution functions. 

Attention is invited to the Operations Research volume^ of this series 
for a detailed treatment of the theory of probability as applied to weapons 
system evaluation. 



1-12 CONSTRUCTION AND MANIPULATION OF WEAPONS 
SYSTEM MODELS 

Before analyzing the structure of a model, let us review some of the 
peculiar characteristics of a weapons system. 

1. A weapons system is an organization of men and equipment designed 
for operation and use against specific classes of enemy targets. To carry 
out its overall function — usually the destruction of the enemy target — 
it must carry out many complex subfunctions. Each functional activity 
converts certain quantitative inputs into outputs. The entire weapons 
system is merely a series — or series-parallel — arrangement of these 
subfunctions connected in such a manner as to permit achievement of the 
overall system objective. As an example of a typical organization, an air- 
to-air intercept system might be characterized by the sequence of opera- 
tional functions shown in Fig. 1-29. Also shown are the major equip- 
ments that are involved in the performance of each function. 

The input to the system is an enemy target — the output is the destruc- 
tion of the enemy target. Similarly, each operational function can be 
viewed as an input-output device. The subject of input-output relations 
brings us quite naturally to a consideration of another distinguishing 
characteristic of a weapons system. 

2. A weapons system is a dynamic or time response system. Both the 
system inputs and outputs have time variables. This fact makes it neces- 
sary to treat a weapons system in terms of the time delays that it introduces 
between the input (enemy target) and the output (action against the 
enemy target). The likelihood of mission accomplishment usually is 
strongly dependent upon the ability of the system to respond to an input 
within a specified period of time. 

Each operational function can contribute to the overall time response 
characteristic. For example, a finite time is required to process target 
intelligence and tactical situation information for the purpose of assigning 
a weapon to the target. Upon being assigned, the interceptor aircraft 
requires a certain amount of time to take off and fly to the target location, 
etc. Thus, the concept of partitioning the system into subfunctions — • 

^Grayson Merrill, Harold Greenberg, and Robert H. Helmholz, Operations Research, Arma- 
ment, Launching (Principles of Guided Missile Design Series), D. Van Nostrand Co., Inc., 
Princeton, N. J., 1956. 



42 



ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 



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1-12] CONSTRUCTION AND MANIPULATION OF MODELS 43 

each characterized by a time delay response characteristic — is basic to 
our modeling approach. 

Bearing these observations in mind, we may outline the structural 
composition of a weapons system model as follows: 

1. Input Variables — The input variables include all the characteristics 
of the enemy target complex — size, number, location, speed, defense 
capability, etc. — that are pertinent to the operation of the weapons 
system. Also included are the elements of the fixed environment — the 
physical environment, the environment provided by other weapons sys- 
tems, etc. — that affect the operation of the weapons system. 

2. Mission Accomplishment Goals — This is a quantitative expression 
of the desired system output. Usually, it derives from the operational 
requirement. This quantity and the input variables define the problem 
that the weapons system must solve. 

3. System Logic — The system logic describes the system organization 
and the flow of information through the system; i.e. how the system oper- 
ates on input data, what sequence of operations takes place, what the 
pre-established tactical doctrine is for a given set of input variables, etc. 
This structural element of the mathematical model provides the means 
for breaking up the overall system function into logically consistent sub- 
functions. 

4. System Configuration Parameters — These include the basic elements 
and characteristics of the system needed to implement the system logic — 
the geometry of the system, the number of weapons, the weapon character- 
istics, and the capabilities and characteristics of each of the system elements 
such as aircraft, radars, etc. 

5. Model Parameters — The basic model parameters are the time delays 
that are defined for each of the system subfunctions on the basis of the 
input parameters, the system logic, and the system configuration param- 
eters. For reasons that were discussed in Paragraph 1-11, the time 
interval associated with the performance of each function must usually 
be expressed as a probability distribution of time delays rather than as 
fixed time delay. 

Often, range to the target is used instead of time as the means for 
expressing the basic parameters of the model. This is merely another 
way of expressing the time delay, since range and time are related through 
the relative velocity between the target and the weapon. 



Suboptimization. Most weapons systems are so complex that it is 
not possible to construct a single model that includes all possible parameters 
and variables. Instead, many different models must be constructed, each 
designed to explore a certain facet of the system operating and its relation 



44 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 

to the whole. In the example of the interceptor system previously referred 
to, we might have the following models. 

1. System Flow Model illustrating the qualitative aspects of system 
operation. 

2. Overall System Effectiveness Model indicating in gross terms the 
defense level that could be provided against a multiplane attack. 

3. Early Warning Detection and Tracking Model indicating the 
quantitative aspects of the problem of early-warning detection, 
identification, tracking, and target assignment. 

4. Interceptor-Bomber Duel Model, including (a) Vectoring Model, 
(b) Airborne Radar Detection Model, (c) Attack Phase Model, 
and (d) Target Destruction Model. 

Each model must be logically consistent with the others, to maintain 
a unified systems approach. Constructing these models shows the same 
stages as constructing the overall system, namely, 

1. Define input variables. 

2. Define performance criteria ("mission accomplishment goals"). 

3. Outline system logic. 

4. Define the configuration parameters of the part of the system 
being analyzed. 

5. Derive the quantitative relationships on the basis of the input 
parameters, system logic, and configuration parameters. 

This process of analyzing only a small portion of the system problem 
at any one time is known as suboptimization. Successive suboptimizations 
of various portions of the system can form a step-by-step approximation 
which converges to the results that would be obtained if one could analyze 
the entire system with one model. 

Counterbalancing the obvious analytical advantages of the suborpti- 
mization technique is the fact that it gives rise to a serious bookkeeping 
problem. The results of each suboptimization process must be logically 
consistent with the rest. As we successively suboptimize various portions 
of the system, the assumptions of previous suboptimizing routines may be 
changed. We must recognize such changes and modify previous sub- 
optimization routines to be consistent with these changes. Summary 
tables, information flow diagrams, and functional block diagrams form 
indispensable tools for the bookkeeping process. 

1-13 SUMMARY 

In this chapter, we have covered the general characteristics of radar 
systems; the environments in which they operate; the functional capabilities 



1-13] SUMMARY 45 

they provide; the basic means by which intelligence is carried on and 
extracted from microwave radiations. 

In addition, we have emphasized the role of the radar as a device in- 
timately connected with other devices to form an overall system. The 
requirements for the radar must be derived from a logical study of the per- 
formance of the complete weapons system in its expected tactical condi- 
tions of operation. A knowledge of broad weapons system modeling tech- 
niques, technical competence in the diverse aircraft and guided missiles 
arts, and a capacity for logical reasoning are required for this process in 
addition to a detailed knowledge of radar systems. 

The next chapter will demonstrate how the systems approach is applied 
to typical examples of airborne radar system design. Succeeding chapters 
will cover the detailed problems of radar design and their relationship to 
the system problem. 



J. POVEJSIL • P. WATERMAN 



CHAPTER 2 

THE DEVELOPMENT OF WEAPONS SYSTEM 
REQUIREMENTS 

2-1 INTRODUCTION TO THE PROBLEM 

Assume that a radar design group is presented with the following 
problem. 

Specify, design, and build the following radar systems for use in a fast attack 
carrier task force environment in the time period Jgxy to I9^x. 

1. Airborne Intercept (AI) radar and fire-control system for all-weather 
transonic interceptors, and 

2. Airborne Early Warning (AEW) radar system for installation in a 35,000 
lb gross weight AEW aircraft. 

This chapter will demonstrate how the first part of this problem — 
specification of radar requirements — can be solved using the general 
approach outlined in Chapter 1. (Later chapters will discuss the additional 
problems involved in mechanizing a radar system to meet a set of derived 
requirements.) 

Although a specific problem is treated, the method of approach has 
applicability to all radar specification problems. Particular emphasis is 
placed on the processes of obtaining a good understanding of the overall 
weapons system problem which dictates the radar requirements. 

Methods for formulating a system study plan and for constructing 
system models are illustrated by examples. The type of information which 
must be collected and analyzed by the radar design group prior to attempt- 
ing the radar design also receives attention and comment. 

The hypothetical problem used as a vehicle for this discussion has been 
greatly simplified; however, it is still a complicated problem and the reader 
will be required to perform the same sort of rechecking, cross-referencing, 
and backtracking that is required to understand and follow an actual 
systems analysis problem. To facilitate this process, much of the reference 
data is displayed on charts and diagrams which typify the pictorial 
(iconic) representations that should be constructed for an actual systems 
problem. 

46 



2-1 



INTRODUCTION TO THE PROBLEM 



47 



Wherever possible, mathematical complexities have been eliminated, 
minimized, or referenced. However, the reader will find that knowledge of 
simple probability theory, operational calculus, and feedback control 
theory provides a key to better understanding of a weapons system study. 

Readers having some familiarity with the radar design problem may be 
somewhat surprised — even appalled — at the amount of study which 
must be done prior to considering the radar design problem itself. They 
may be tempted to say: "Surely, you do not expect a radar designer to 
concern himself with such matters? These are the responsibility of the 
agency that prepares our specification!" 

The authors' reply to these remarks is based on their own not always 
pleasant personal experiences and close observation of the experiences of 
many other radar system designers. 

Seldom, if ever, does the basic specification given to a radar designer 
provide the necessary input information for a successful system design. 
It is a starting point — nothing more. Until the radar designer understands 
the nature of his contribution to the solution of the overall system problem 
he has little chance of designing a radar system that will operate satis- 
factorily as part of a complete weapons system. 

In recognition of this fact, the weapons system contractor not only 
should encourage, he should require the radar designer to demonstrate his 
understanding of the problem by preparing a formal document embodying 
the type of analysis and development to be demonstrated in this chapter. 
This will tend to ensure that the purposes and objectives of the weapons 
system are not made subservient to the preconceived biases of the radar 
designer. 



What is the 
problem? 



What is 

known 

about the 

problem? 



What remains to be 

understood about 

the problem? 



What is the 
plan for 
action? 



Fig. 2-1 Steps Leading to the Formulation of the System Study Plan. 



48 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

2-2 FORMULATING THE SYSTEM STUDY PLAN 

Reduced to its barest essentials, any system may be visualized as the 
logical process of asking and answering the sequence of four questions out- 
lined in Fig. 2-1. This basic pattern is repeated — again and again — 
throughout the course of the study. 



Operational 

Requirements 

(O.R.) 



Carrier 

Task Force 

Weapon System 

Model 



Fixed 
Elements 



Variable 
Elements 



Interceptor 



tJJ 



Subsystem Analysis 



Generated Capabilities 

and 
System Deficiencies 



Mission 
Accomplishments 



Step 1 
System 
Analysis 



Target 
Complex 



Complete Accomplishment of O.R. 




VARIABLE ELEMENTS 



Target 
Characteristics 



Radar Req'ts 
Step 2 



Al Radar and 
Fine Control 
Req'ts - Step 3 



■ Early Warning Detection 
Range from Fleet Center 

Interceptor Kill 
Probability 

No. of Interceptors 
Engaging Raid 



Fig. 2-2 Master Plan for Weapons System Analysis 



First, the problem is defined. For the hypothetical carrier task force air 
defense system, this problem definition takes the form of the following 
question : 



2-2] FORMULATING THE SYSTEM STUDY PLAN 49 

"What does the weapons system require of the airborne radars (AI and 
AEW) to achieve a satisfactory level of mission accomplishment?"^ 

The basic elements of this problem are shown in Fig. 2-2. The operational 
requirement defines a weapons system problem. By the processes described 
in Paragraph 1-8, the operational requirement leads to the establishment 
of the concept of a system which depends for its operation upon the charac- 
teristics of a number of subsystems — aircraft, missiles, detection devices, 
and shipboard installations, for example. 

As is apparent from this figure, the solution to the radar requirements 
problems will involve consideration of many complex characteristics and 
relationships external to the airborne radars. Hence, the middle two 
questions (Fig. 2-1) and their answers are fundamental to further progress. 
For the hypothetical problem, the known and unknown elements of the 
problem are displayed in Figs. 2-2 through 2-8. These will be discussed in 
greater detail in subsequent paragraphs. 

The plan for action will develop quite naturally from the indicated 
sequence of questions and answers. Elements of the problem that are not 
known or understood must be investigated in greater detail and related to 
the known elements. In some cases, adequate information may not be 
available on the unknown elements of the problem — or the path to under- 
standing may be blocked by the inherent difficulty of the problem. These 
cases will require that arbitrary assumptions be made in order that the 
analysis can proceed. 

When a sufficient understanding of the overall problem is obtained by 
analysis (or assumption), weapons system models are constructed. These 
models have as variable parameters the performance characteristics of the 
airborne radar known to be important to weapons system operation. 

Using the techniques of Operations Research and Systems Analysis, these 
models are "played" against the target inputs (Fig. 2-2). The level of 
system performance (Mission Accomplishment) obtained is compared with 
the operational requirement, thereby generating a measure of the system 
capabilities (or deficiencies). By such processes, mission accomplishment 
may be related to the radars' performance characteristics (see inset, Fig. 
2-2) thereby providing a means for obtaining the true requirements of the 
airborne radars as dictated by the weapons system requirements. 

It is important to emphasize once again that the derivation of require- 
ments should not be aflFected by state-of-the-art considerations in radar 
technology. The purpose of the analysis is to define the radar problem, not 
to solve it. Only after this task is completed is the radar designer free to 
turn his attention to the job of designing and building a specific radar 
system to meet the requirements imposed by the weapons system problem. 

iThe problem of defending a carrier task, force is quite analogous to the defense of a city 
or important military base. 



50 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

2-3 AIRCRAFT CARRIER TASK FORCE AIR DEFENSE 
SYSTEM 

The fixed elements of the carrier task force environment, the target 
complex, and the operational requirement provide the basis for the deriva- 
tion of radar requirements. We shall assume that all elements of the task 
force air defense system are fixed, except the following: 

1. The Airborne Early Warning Aircraft Radar and Data Processing 
System, 

2. The Airborne Interceptor Radar and Fire-Control System. 

Referring to the system study plan of Fig. 2-2, the first task is to de- 
termine: (1) what is known about this weapons system, and (2) what 
remains to be known or understood. 

The overall weapons system is broken down into four major operating 
elements: (1) Ship Weapons System, (2) AEW Aircraft System, (3) Inter- 
ceptor Aircraft System, and (4) Air-to-Air Missile System. The character- 
istics of each element — and the interrelationship between elements — 
which exert a sensitive influence on the overall system performance are 
shown in Figs. 2-3 through 2-6. The known characteristics are checked; 
graphical or tabular description of these appear in the same figures. 

The unknown (unchecked) characteristics constitute the items a knowl- 
edge of which must be gained from the system study. These include the 
characteristics of the AEW and AI radar systems; they also include 
important interrelationships among the various system elements, known 
and unknown. For example, the individual characteristics of the interceptor 
aircraft and the air-to-air missile are known; the manner in which these two 
elements combine and interrelate with the interceptor fire-control system 
to produce a weapons system capability is not known and must be derived 
by study. 

The list of sensitive parameters is not complete. At the outset of a 
system study it is not possible to name all the parameters that may be 
important. As more is learned about the system through study, these must 
be added and considered in their proper relationship with other parameters. 

Carrier Task Force Weapons System (Fig. 2-3). This system 
includes two aircraft carriers and three missile-firing cruisers in the dis- 
position shown. (This configuration is, of course, fictitious and is used to 
illustrate the possible elements of this problem. Elements of the task force 
not germane to this example are excluded.) 

Two large carriers constitute the main ofi^ensive and defensive elements. 
The carriers are separated (5-20 n. mi.) for tactical reasons. Mass attacks 



2-3] 



AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM 



51 



Sensitive Parameters and Elements 

Tactical Doctrine 

Task Force Structure Y 

Air Defense Tactics >/ 

Aircraft Handling and Availabilityv^ 

Reaction Time y^ 

Data Collection 
Early Warning Range 
Information Rate 
Communications v^ 
Accuracy 
Reliability 

Vectorin g 
Accuracy 

Communication v^ 
Interceptor 
Effectiveness 
Speed v^ 



Task Group Disposition 



Likely Direction 
I of Raid 




100 200 300 
(n.mi) 

I = Carrier 

i = AEW 

6 = 6 CAP 

o = Missile Cruisers 




Geometry of the Defense Zones 



/ Missile 
: Zone 

_ CIC 



■CAP- 



Deck Launch Interceptors 
«^AEW 



f-^lnitial 
I Attack 
I Boundary 



—AEW 
Detection 
Boundary 



50 100 150 200 250 

RANGE FROM TASK FORCE CENTER (n.mi.) 



Down for Maintenance 

and Repairs 

(18) 



Aircraft Availability and Tactics 

76 

60 
50 
40 
30 
20 
10 



CAP Guardin g Flank ( 6) 
CAP Engaging Raid (6") 



(36) 
Deck -Ready 
Launch Rate, 
2 per Minute 



7 



Note: Tactics Require One CAP 
to Maintain Station 
During a Raid to 
Guard Against Attacks 
from Other Direction 

Total Available to 

Engage Raid 

(42) 



Communications 

AEW 1000 Bits Per Second Per Channel 



Reaction Time 

3 Minutes from Instant of Early Warning Signal 



Fig. 2-3 Carrier Task Force Weapons System Characteristics. 



are assumed to occur only on the side of the task force which is exposed to 
enemy territory. 

Local defense of the task force is augmented by missile-firing cruisers 
flanking, and somewhat forward of each carrier. These provide a point- 
defense system with an altitude capability of 50,000 ft extending out to 
about 40-50 n. mi. in front of the carriers. A third missile-firing crusier 



52 



THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



guards against sneak attacks from the rear as well as turn-around and 
reattack tactics. 

Early warning detection and interceptor vectoring information is pro- 
vided by airborne early warning aircraft (AEW). The task force is assumed 
to have a capability for maintaining three AEW aircraft aloft on a 24-hour 
basis. The primary functions of the AEW system are: 

(1) To provide detection of enemy aircraft at sufficient range forward of 
fleet center to permit interception by piloted aircraft. 



Sensitive Parameters and Elements 
Speed w^ Detection Range 
Altitude v^ Information Rate 
Endurance v Resolution 
Maneuver Display 

Reliability Measurement Accuracy 

Target-Handling Capacity 
Communications y 
Integration with CIC y' 
Aircraft Availability v^ 
Human Operator Characteristics 
Environment v^ 
Navigation Accuracy v 



Navi g ation and Communications 

Communication Channel Capacity 

1000 Bits per sec 

Navigation Accuracy (Relative to CIC) 

1 n.mi. rms 



Mission Profile 



^20K - 



350 Knots 



■7- \ 200 Knots 

■i<A~wl^y Patrol 



5 lOK 




50 100 

RANGE FROM FLEET CENTER (n.mi.) 



Inte g ration with Combat 
Information Center (CIC) 



Target 




^^^^^^"AEW 



Aircraft Availabilit y 
6 



Environment 




Deck-Ready 



On-Station Stand-I 



On Station 



A. Shock and Vibration 

B. Temperature 

C. Pressure 

D. Humidity 

E. Space and Weight 

F. Environmental Noise 

Interference 
Clutter 
Weather 
Internal 



Fig. 2-4 Airborne Early Warning System. See Fig. 2-18 for Tactical Deployment. 



I 



2-3] AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM 53 

(2) To provide target data for the ship-based combat information center 
(CIC), which supplies vectoring information to piloted interceptors 
and guidance information to ship-based missile radars. 

Two combat air patrols (CAP) are maintained. Each CAP contains 6 
all-weather interceptor aircraft. In addition, interceptors may be launched 
at a maximum rate of 1 per minute from each of the two carriers during 
attack conditions. Aircraft availability limits the total number of deck- 
launched interceptors to 36. 

During an attack, only one CAP (6 aircraft) engages the raid; the other 
is held in reserve to guard against attacks from other directions. Thus, a 
maximum of 42 interceptors can be used to engage a raid. 

The interceptors are armed with air-to-air guided missiles and are 
required to perform the interception function at altitudes from sea level to 
60,000 ft. 

An optimum battle-control and communications system is assumed. 
This is to say, the deployment of interceptors by CIC is such that any 
interceptor which enters the interceptor zone is able to make an attack so 
long as there are targets within the zone. As will be shown later, system 
performance is sensitively affected by this assumption, which represents a 
condition most difficult to realize in practice. 

Airborne Early Warning System (Fig. 2-4). The basic functions of 
the AEW system have been described. 

The carrier-based aircraft available for this purpose is assumed to be 
capable of housing an antenna with a maximum dimension of 12 ft in the 
mushroomlike appendage shown. 

The exact disposition of the AEW aircraft and the AEW radar and data 
processing requirements will be determined by study. 

A major unknown is the contribution of AEW target information accu- 
racy to interceptor effectiveness. 

Interceptor Aircraft System (Fig. 2-5). The known and unknown 
characteristics are defined as shown. The determination of detailed radar 
requirements will require an analysis of the dynamics of the closed-loop 
system formed by the target, the interceptor, the pilot, and the AI radar 
and fire-control system. 

The interrelationships among the aircraft system, the air-to-air missile 
system, and attack tactics are also unknown and must be analyzed as a 
prelude to the ascertainment of radar requirements. 

A major unknown, to be determined by the system study, is the con- 
tribution of vectoring accuracy to interceptor effectiveness. 



54 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



Sensitive Parameters and Elements 
Speed v^ 



Altitude v^ 
Range y 
Climb Capability -Z' 
Maneuver X 
Environment -/ 



Angle Coverage 
Vectoring Accuracy 
Attack Tactics 
Fire Control 
Weapon s/ 



Pilot Characteristics 
Control Dynamics 



Maneuverin g Characteristics 
Maneuvering Capabilities 2.0 g 

Max. Angle of Attack 6° 

Transient Angle of Attack 2° 

Transient Roll Rate 60Vsec 



Mission Profiles 
Combat Air Patrol (CAP) Mission 

ffefurn 

y^ 1200 ips 
2.8 Hours at 850 fps 




-15 Minutes Combat 
at Vc = 1200 fps 



100 200 300 

RANGE FROM FLEET CENTER (n.mi.) 



Deck Launch Mission 
Return 




100 200 

RANGE FROM FLEET CENTER (n.mi.) 



Weapons 

2 Semiactive Air-to-Air 
Guided Missiles Launched 
in a Salvo following 
Acquisition of Target by 
Missile Seekers 



Environmental Factors 

A. Shock and Vibration 

Catapult Launch 

Landing 

Flight and Maneuver 

Missile Launch 

B. Temperature 

C. Pressure 



D. Humidity 

E. Noise: 

Clutter 
Vl/eather 
Interference 
Internal 

F. Size and Weight 



Interceptor Control System Dynamics 
TL Tracking (\ 




^^ 



Target Input 



. Ji 



Tracking Error 



Tracking 
Radar 



r 



LS = Line of Sight 

TL = Tracking Line 

■^ = Interceptor Heading 

i/^(- = Computed Correct Heading 

Vp = Interceptor Velocity 

V^ = Target Velocity 

R = Range 

Steering Error 



Computer — »Q--^-»|~ Pilot | — 1\ Aircraft 



Aircraft Radar 
Interaction 



Fig. 2-5 Interceptor System Characteristics. 



Air-to-Air Missile System (Fig. 2-6). The sensitive parameters of 
the air-to-air missile system, are shown. The major unknown parameters 
involve interrelationships of the missile with the aircraft and the radar and 
fire-control system. 



THE TARGET COMPLEX 



55 



Sensitive Parameters and Elements 
Performance y^ 
Kill Probability v^ 
Guidance y^^ 

Radar and Computer Integration »^ 
Illumination Accuracy y 
Illumination Power v 



Guidance 




2 4 6 8 10 

SEEKER ACQUISITION RANGE (n.mi.) 




ILLUMINATOR TRACKING ACCURACY (deg rms) 




12 3 4 5 
DOWN RANGE (n.mi.) 



Kill Probability (Including Reliability) 
Single Shot - 0.50 
2-Missile Salvo - 0.75 



Radar and Computer Integration 
Seeker Angle Slaving Signal 



Seeker Range Slaving Signal 



J^^ 



Launching Signal 



T^ 



Seeker Acquisition Signal 



Fig. 2-6 Air-to-Air Missile System Characteristics. 



2-4 THE TARGET COMPLEX (FIG. 2-7) 

Target Complex. The characteristics of the target complex include 
its parameters and its mission as shown in Fig. 2-7. 

The 50-nautical mile (n. mi.) air-to-surface missile (ASM) carried by the 
hostile aircraft requires that interception take place outside of a 50-n. mi. 
circle around the aircraft carriers. 

The target's 2-g maneuver capability will exercise an important influence 
on the radar and fire-control system design. It will be assumed for this 
example that the enemy aircraft is provided with the capability for op- 
timum timing of this maneuver. Also, it is assumed that the target does 
not employ electronic countermeasures (ECM). 



56 



THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



Sensitive Parameters and Elements 
Speed ^ Maneuver -/^ 
Altitude ^ Defense "^ 
Number v^ Radar Size 
Tactics v^ Weapon y^ 



Raid Geometr y 

|-* 95 n.mi. H 



20 Aircraft 



VM\ 



_j L 5 n.mi. 
~^ '^Spacing 



Direction of Raid 



IVlission Profile 



g 50 



K 300 fps (2g Evasive 

Action) 



1000 



01 (\ L 

700-1000 V 100 50 

RANGE FROM TARGET (n.mi.) 




Target 



Radar Cross Section 



90° 




180° 1 



Fig. 2-7 Target Complex Characteristics. 



The radar cross-section characteristics of the hypothetical target are 
only generally known and are shown in Fig. 2-2. Paragraph 4-7 explains 
the factors contributing to characteristics of this type. Paragraph 4-9 
discusses how the target radar area may be estimated for purposes of 
preliminary design. 

The turboprop propulsion system of the enemy aircraft was chosen to 
introduce into the model the effects of the modulation characteristics of the 
reflected radar energy (Paragraphs 4-7 and 4-8). This can be an important 
radar design consideration. 

The target is assumed to carry a high-yield nuclear weapon. Destruction 
resulting from impact on the target aircraft is assumed to cause a detona- 
tion capable of producing a destructive overpressure within a 1000-ft radius 
sphere around the target. Ignoring time effects, 1000 ft thus defines the 



2-6] THE SYSTEM CONCEPT 57 

point of allowable minimum approach of the interceptor to the target 
aircraft. 

The number of target aircraft (20) and their spacing (5 n. mi.) is charac- 
teristic of a raid designed to present a difficult problem to the model air 
defense system. 

In an actual problem, a number of different target complexes would have 
to be defined in this way. The behavior of the system would be analyzed 
for the several inputs and the design parameters chosen on the basis of the 
response to all expected target complexes with emphasis on the most 
effective configuration. For simplicity in this example, we will confine our 
attention to the single problem defined; however, the sensitivity of system 
performance to changes in this input (i.e. target speed and number) will be 
examined. 



2-5 THE OPERATIONAL REQUIREMENT: MISSION 
ACCOMPLISHMENT GOALS (FIG. 2-8) 

The operational requirement defines a military problem which must be 
solved by the combination of known and unknown weapons system 
elements previously described. Bases for judging the military usefulness 
of any system proposed as an answer to the operational requirement are 
also shown. 

2-6 THE SYSTEM CONCEPT 

The operational requirement defined a weapons system problem. The 
procedures for solution of this problem are determined by the system con- 
cept or logic. 

Within the framework of the system elements already defined, the 
system logic for the interceptor system may be developed by listing the 
sequence of events which lead to the interception of the target by the 
missile-armed interceptor. The following events would normally be 
expected to occur in sequence: 

a. Early warning detection 

b. Identification 

c. Threat evaluation 

d. Weapon assignment 

e. Interceptor direction or vectoring 
/. AI radar search and detection 

g. AI radar acquisition 

h. Airborne weapons system tracking control and missile launching 

i. Air-to-air missile guidance 

j. Missile detonation and target destruction (without self-destruction) 



58 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



Function High-Attrition Air Defense 
Against Medium Bombers 

Features (1) Compatibility with Surface-to-Air 
Missile System 

(2) Compatibilty with Fleet 
Elements, Logistics, and Tactics 

(3) Compatibility with Transonic 
Interceptor Aircraft -- 30,000 Lb 
Gross Weight 



Mission Accomplishment Goals 




20 40 60 80 100 
FREQUENCY OF OCCURRENCE 



Mission Accomplishment Goals (See Inset Figure) 



(1) 



(2) 



Fig. 2-8 Operatioric 



A 20-Plane Raid at 50,000 ft and 800 fps shall be Employed as 
the Input for Judging System Performance. 
System Performance shall be Judged as Satisfactory if the 
System can be Demonstrated to Attain or Exceed the Following 
Performance Levels (As Indicated by the Shaded Area in 
the Figure) 

(a) 50 per cent Probability of Killing all 20 Targets 

(b) 90 per cent Probability of Killing 16 Targets 

(c) 99 per cent Probability of Killing 12 Targets 

All Kills are to be Accomplished at a Minimum Distance of 
50 n.mi. from Fleet Center 

.1 Requirements, Attack Carrier Task Force Interceptor Ai 
Defense. 



k. Return to base 

/. Transfer of residual target elements to the surface-to-air missile 
support defense system. 

A diagrammatic summary representation of the overall tactical situation 
is shown in Fig. 2-9. 

The statistical nature of the system operation is shown by this diagram. 
For various reasons — interceptor availability, time limitations, system 
failures, system inaccuracies — a certain percentage of the interceptors fail 
to complete each of the successive steps required for interception. 

Thus, any interceptor chosen at random from the total complement has a 
certain probability that it will kill a target. This probability is the product 
of the individual probabilities that it will pass successfully through each 
successive stage of the interception. 

This line of reasoning points out the necessity for obtaining a proper 
balance between the performance of various elements of the system. A 



2-7] 



THE SYSTEM STUDY PLAN 59 

Zone of CIC 
-Vectoring Inaccuracy 





Zone of 
AFCS Inaccuracy 



rrv 



'=^X 



Search 




Zone of Missile 
Guidance Inaccuracy 



Fig. 2-9 A Tactical Situation. 



very low probability of success for any phase of the intercept mission can 
render pointless any efforts to achieve very high probabilities in other 
phases and thus would serve as a guide to more effective development 
emphasis. 



2-7 THE SYSTEM STUDY PLAN 

The known (fixed) and unknown (variable) elements of the problem have 
now been defined. Referring to Fig. 2-2, it is seen that the next step is to 
analyze the interrelationships between the fixed and variable elements to 
determine the contribution of each variable element to mission accomplish- 
ment. From such analyses, a quantitative understanding of system 
operation will be obtained and — eventually — radar requirements will 
evolve. 

The unknown or variable elements may be broken into two basic cate- 
gories: (1) weapons system variables, and (2) subsystem variables. 

The primary weapons system variables are: 

1. The early warning detection range measured from fleet center. 

2. The number of interceptors which may engage the specified target 
complex 

3. The effectiveness (kill probability) of each interceptor which 
engages the target complex 



60 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

All of the other unknown parameters (subsystem variables) affect one 
or more of these three basic weapons system variables. This observation 
makes it possible to organize the study plan on a step-by-step basis as 
follows : 

Step I: Construct a model of the overall weapons system using the three 
primary weapons system variables as adjustable parameters. Assume 
values for each of these adjustable parameters and calculate the resulting 
system performance. Compare this performance with the desired level of 
mission accomplishment; use any discrepancy between the two to adjust 
parameter values for another tentative design. Testing of the model 
continues until the following information is derived. 

1. All combinations of the adjustable weapons system parameter 
values that will allow achievement of the mission accomplishment 
goal. 

2. The sensitivity of system performance to changes in the values of 
the adjustable parameters. 

Additional information — useful for obtaining a good understanding of the 
overall problem — is obtained by ascertaining the sensitivity of system 
performance to changes in the fixed elements. 

Step 2: Assume fixed values for the three weapons system variables of 
Step 1 that permit the system to achieve the desired level of mission 
accomplishm.ent. Construct a model (or models) which expresses the 
relationships between the adjustable (unknown) AEW parameters (beam 
width, information rate, radar detection range, etc.) and the assumed 
weapon system parameters. Test this model for various assumed combina- 
tions of AEW parameters. Establish acceptable combinations of AEW 
parameter values and the sensitivity of system performance to parameter 
changes. Derive a specific set of AEW requirements. 

Step 3: Using the values for the unknown system variables derived in 
Step 1 and 2, repeat Step 2 for the adjustable parameters of the interceptor 
weapons system. Derive a specific set of requirements for the AI radar and 
fire-control system. The suggested order of Steps 2 and 3 is somewhat 
arbitrary; a reasonable case might be made for reversing this order. As a 
general rule, where a choice exists, it is wise to select an order which places 
the most difficult subsystems first, since this will maximize the number of 
adjustable parameters available for its preliminary design. 

2-8 MODEL PARAMETERS 

The interrelations between major system parameters and the contribu- 
tions of each parameter to overall effectiveness may be developed through 



2-9] 



SYSTEM EFFECTIVENESS MODELS 



61 



the use of mathematical models. These interrelationships form the quanti- 
tative bases for the choice of basic system parameters. 

The following paragraphs will develop a number of models designed to 
expose some of the more important aspects of the task force air defense 
problems. The techniques used to develop these models are illustrative of 
the means by which any complex system problem may be broken down to 
forms that can be handled by analytical means. 



2-9 SYSTEM EFFECTIVENESS MODELS 

The operational requirement (Fig. 2-8) specified the effectiveness of the 
interceptor air defense system in terms of the degree of success which must 
be achieved with a required reliability. For example, the probability of 
destroying at least 16 of the 20 targets should exceed 90 per cent. 

The first step of the systems analysis must determine the nature of the 
relationships between system eflFectiveness and the fixed and variable 
elements of the defense system (Fig. 2-2). The following examples demon- 
strate how such an analysis may be carried out. 

Assume that 40 interceptors may be brought to bear against the 20- 
target raid previously assumed as the threat (A^ = 40). Each interceptor 
can make only one attack with its two-missile salvo. Thus, when one 
attack against a target fails, another interceptor will be assigned to that 
target until either 40 attacks have been made or all the targets have been 
destroyed. 

20 

18 

16 

Q 
Ul 

d 14 
\2 12 

C3 

^ 10 







■---. 


#2^ 




^^ 


' ^v. 


\ 




















"v 


N^ 










Specified Effectiveness from 
Operational Requirement 


^ 






( 


Fig. 2 - 8) Shown by Dashed Lines 

1 1 1 1 


\ 














1 

Case^l 


Case*'2 _ 
25 




Number of Interceptors = 


40 



















































































20 40 60 

RELIABILITY (PER CENT) 



100 



Fig. 2-10 System Effectiveness Operating Characteristics. 



62 



THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



If the effectiveness of each interceptor is assumed to be P = 0.5, the 
system operating characteristic shown in Fig. 2-10 may be calculated by 
the application of simple probability theory. ^ As can be seen, the assumed 
parameters allow the operational requirement to be met. 

To obtain a complete picture, other possibilities may be assumed and 
analyzed in the same fashion. For example, the effectiveness level provided 
by 25 interceptors, each with a kill probability of 0.7, is also shown in the 
figure. This combination of parameter values fulfills only part of the 
operational requirement. The probability that more than 16 targets will be 
destroyed is less than that for the previous assumptions; thus, this system 
would impose increased requirements on the back-up surface-to-air missile 
system. 

Continuing in this fashion, trade-off curves between the number of 
interceptors and the interceptor kill probability can be determined for each 
point of the operational requirement. Such a curve is shown in Fig. 2-11. 
Here all the combinations of the interceptor kill probability and number of 
interceptions are shown which will kill at least 16 out of 20 targets with a 
90 per cent reliability. 





\ 








1 1 
No. of Targets 


-20 




> 


\ 




QJ LlI 

— ° 


Interceptors -36 


Missiebaiv 


\ 




3 
3 








Kill 


\ 


Prob 


ability 


' \ 


V 
















\ 


















li 


V 














f 


= 1^1 


\ 


1 












< 




II- 

oK 
^1 


5 










a. 

< 




1 












' lol 

Icvil 

1 1 




3 





10 20 30 40 50 60 70 80 
N-NUMBER OF INTERCEPTIONS 



Fig. 2-11 Interceptor Kill Probability vs. Number of Interceptions Required to 
Kill 16 or More Targets with 90 Per Cent Probability. 

^Grayson Merrill, Harold Goldberg, and Robert H. Helmholz, Operations Research, Jrma- 
ment. Launching {Yv\nc\p\&s oi Guided Missile Design Series), D. Van Nostrand Co., Inc., 1956. 
Chapters 6 and 7 provide an excellent discussion of the mathematical techniques involved. 



2-9] SYSTEM EFFECTIVENESS MODELS 63 

Also shown are the limiting effects of the fixed problem elements pre- 
viously outlined in Figs. 2-3 through 2-6. For example, the missile salvo 
kill probability is 0.75; obviously the interceptor kill probability cannot 
exceed this value. If the raid is engaged by six CAP interceptors and all 36 
of the deck-ready interceptors, an interceptor kill probability of 0.48 is 
required. If the tactics are changed to allow both CAP patrols to engage 
the raid in addition to the 36 deck-ready interceptors, the individual inter- 
ceptor kill probability required drops to 0.42. If the total complement of 
66 interceptors could be used, a kill probability of only 0.3 would be 
required. 

The basic parametric relationships between the number of interceptors 
and system effectiveness now are established. The next phase of the sys- 
tems analysis must determine the relationships between the number of 
interceptors and the other fixed and variable elements. Completion of this 
phase will provide the basic parametric data which will, in turn, allow 
intelligent selection of the following system parameters (see Fig. 2-2). 

1. Number of interceptors (A^) 

2. Interceptor effectiveness (Po) 

3. Early-warning range {Raew)- 

The number of interceptors which can be used to defend a given raid, 
and thus the required interceptor kill probability, is a function of initial 
interceptor deployment, detection ranges, reaction times, and target and 
interceptor speeds. These factors can be conveniently summarized in a 
diagram similar to Fig. 2-12, which shows the sequence of events in a typical 
raid. The interceptor and target performance characteristics were given 
in Figs. 2-5 and 2-7. We assume, as an illustrative case, that the AEW 
detection range is 250 n. mi. from the fleet center. Since the target has a 
speed of 800 fps (474 knots), it will arrive at the fleet center 32 minutes 
after detection. The target track is shown in Fig. 2-12 as the straight line 
connecting 250 n. mi. at zero time to 32 minutes at zero range. 

The CAP interceptors stationed 100 n.mi. from the fleet center are 
vectored to intercept the raid following a 3-minute time delay consumed by 
the process of identification, acquisition, and assignment. The track of the 
CAP aircraft is constructed as a line with a slope equal to the reciprocal of 
their speeds (1200 fps or 710 knots). We observe that the intersection of 
the two tracks occurs at 175 n.mi., the maximum range at which the raid 
can be engaged. 

In accordance with defined tactical doctrine (Fig. 2-3), only one combat 
air patrol (6 aircraft) is committed to the raid. The remaining CAP 
maintains its station to guard against the possibility of attacks from other 
directions. 



64 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

40 




50 100 150 200 250 

DISTANCE FROM FLEET CENTER (n.mi.) 

Fig. 2-12 An Attack Diagram. 



300 



From Fig. 2-5 it is seen that an additional 3-minute delay is created by 
the interceptor climb and acceleration characteristics. This makes the 
total effective reaction time of the deck-launched aircraft equal 6 minutes. 
After the initial reaction, interceptors will be launched at a rate of 1 per 
minute from each of the two carriers until either no more interceptors are 
left or it is obvious that the interceptors will not be able to intercept the 
targets outside of the surface-to-air missile zone corresponding to a 50-n.mi. 
radius from the fleet center. In our example, this latter consideration is the 
limiting factor, and it is possible to launch only 32 interceptors from the 
carriers. Thus, a total of 38 attacks can be made against the raid with the 
assumed deployment, tactical doctrine, and equipment performance. This 
is close to our previously assumed case with 40 interceptors, and the 
required interceptor kill probability will be slightly greater than 0.5. The 
air battle takes place during a 16-minute time period to enemy penetration 
of the missile defense zone barrier. The maximum time that any interceptor 
must fly at Mach 1.2 is 11 minutes (for the first two deck-launched inter- 
ceptors), which is well within the interceptor performance capabilities as 
displayed in Fig. 2-5. 

This model may be used to examine the effect of variations in the 
system parameters. The results of such an analysis are shown in Fig. 2-13, 
where trade-off curves relating pertinent factors are given. If the early 
warning range is increased to 300 n.mi., 50 interceptors can engage the raid. 
With 50 interceptor attacks, the required interceptor kill probability will 
be reduced to 0;42. However, with the tactical doctrine assumed, the 
maximum interceptor complement available to counter an attack is limited 
to 42. Thus as early warning range increases, aircraft availability in this 



2-9] 



SYSTEM EFFECTIVENESS MODELS 



65 



^t3 



Po=0.7 
p„-0 6 




^ 


Y- 


.Ofiej^tionaL 


'°'1 




\ , 


1 












0=0.3 
















/ 


y / 











20 40 60 80 100 120 

NUMBER OF INTERCEPTORS ENGAGING 

TARGETS 





AEW^ 


250 n.ml. 
08 M 





















1.2 1.0 0.8 

TARGET SPEED, M 



0.6 




80 

£g 60 

li 40 

p20 





200 250 

AEW RANGE (n.mi.) 



300 









AEW 


1 
= 250 n.mi. 






















A 

























2 4 6 8 10 12 

INTERCEPTOR SPEED, M 



Fig. 2-13 Sensitivity of System Effectiveness to Number of Interceptors, AEW 
Range, and Interceptor Kill Probability Po- 



model becomes the limiting factor. This limitation might indicate that a 
trade-off of parameter values elsewhere in the problem should be examined 
to exploit the potential advantage which might accrue from a range 
increase.^ Conversely, a 50-n.mi. decrease in early warning range would 
require that interceptor kill probability be increased to 0.70 — a value that 
is almost equal to the kill probability of the missile salvo alone — in order 
to maintain the system effectiveness required. 

Increases in target velocity have much the same effect as decreases in 
early warning range. If the target velocity were to increase by 10 per cent, 
only 34 interceptions could be made. Thus, to maintain the same defense 
level under these conditions, interceptor kill probability would have to be 
raised to 0.58 or early warning range would have to be increased by about 
30 n.mi. Increases in interceptor velocity have the same general effect as 
increases in early warning range; i.e., aircraft availability limits the useful- 
ness of such increases. System sensitivity to this change is relatively small, 
however, for extremely high interceptor speeds. 

The time delays defined for the model made no allowance for any time 
delay introduced by the vectoring process. The assumption is that the 
vectoring system guides each interceptor on a straight-line path to the 

^For example, we might explore the possibility of using the other CAP aircraft which were 
assumed to maintain their stations. 



66 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

earliest possible interception point. Other types of vectoring guidance — 
for example, a tactic whereby it is attempted to guide the interceptor on a 
tail-chase attack — introduce additional time delays, and these reduce the 
total number of interceptions which can be made with a given early warning 
range. 

System tactics also exercise other important influences. For example, 
a 3-minute dead time delay was assumed between early warning detection 
and assignment of the first interceptors to specific targets. This type of 
operation places a high value on the time delay. Each minute of time delay 
requires 8 additional miles of early warning detection range to maintain a 
fixed number of interceptions. 

The effect of this time delay would be different if the system tactics 
called for launching of interceptors to begin before evaluation was com- 
pleted. This operation, however, incurs a risk that interceptors may be 
launched unnecessarily. In this latter case, it would be necessary to 
evaluate the consequences of a false alarm as a function of threat evaluation 
time; i.e., the penalties of launching interceptors when the threat does not 
materialize following an early warning detection in terms of fuel loss, 
vulnerability to attacks from other directions, etc. Some of these consider- 
ations may seem to go a little far afield, but the answers to such questions 
are of great importance to the radar designer because they affect what his 
equipment must do. To simplify our example, we assume that no inter- 
ceptors are launched until evaluation of the threat is completed. 

As a second example of the effect of tactics, we might consider the target 
assignment procedure. In our example, we assume that an optimum 
assignment procedure could be used. That is to say, each of the 40 inter- 
ceptors was able to make an attack during the course of the air battle — 
except in the cases where all 20 targets were destroyed by less than 40 
attacks. This assumption assumes a very sophisticated battle control and 
communications system. Another method of assignment could be as 
follows: the first 20 interceptors are assigned — one-on-one — to the first 
20 targets. The following interceptors are assigned as back-up interceptors 
on the same basis — i.e. interceptor 21 to target 1 , interceptor 22 to target 2, 
etc. For 40 interceptions, this would mean each target could be attacked 
twice. In some cases, however, the target would be killed by the first 
interceptor thereby leaving the back-up interceptor without a target to 
attack, resulting in a potential inefficiency. On the other hand, two attacks 
may not suffice to kill the target since each attack has less than unity 
success probability. 

With these alternate tactics, a substantially greater interceptor kill 
probability would be required to meet the operational requirement for the 
case of 40 interceptors reaching the attack zone. This value has been 
determined to be 0.7 as compared with 0.5 when optimum target assign- 



2-10] DESIGN OF AIRBORNE EARLY WARNING SYSTEM 67 

ments are made. The advantages of the optimum assignment tactic are 
apparent. 

In this paragraph we have shown how the effectiveness of an interceptor 
system may be analyzed for an assumed mode of operation and assumed 
values for system parameters. We have also illustrated the concept of 
obtaining the "trade-off" between various system parameters and the 
effects of changes in system logic. The examples chosen are merely illustra- 
tive of the information that must be generated for an actual problem to 
enable the radar system designer to understand his part of the overall 
problem. 

Using the assumed or derived values for the overall system parameters, 
and the defined system logic, we shall now derive the requirements for the 
AEW and AI radar systems. The first phase of these analyses (Steps 2 and 
3 of Master Plan of Fig. 2-2) is to establish the allocation of responsibility 
between these two systems. 



2-10 PRELIMINARY DESIGN OF THE AIRBORNE EARLY 
WARNING SYSTEM 

The AEW system must contribute to the solution of the air defense 
problem in several ways as may be seen from the operational sequence 
given in Paragraph 2-3. 

1 . The targets must be detected at sufficient range from task force center 
to permit fulfillment of the required system kill probability. 

2. The targets must be identified and evaluated. This means that their 
identity, number, position, heading, speed, and altitude must be obtained; 
this information must be evaluated in terms of the implied threat to the 
task force; and weapons must be assigned, if necessary. This process must 
be completed within a delay time that is compatible with early warning 
detection range and the characteristics of the interceptor defense system. 

3. The AEW system must provide information which can be used to 
vector the interceptors toward their assigned targets so that the inter- 
ceptors may detect and acquire the targets with their own AI radars. The 
type of vectoring guidance employed must be compatible with system re- 
sponse times permitted by the early warning detection range. The accuracy 
of vectoring guidance must be compatible with the input accuracy require- 
ments of the interceptor aircraft, AI radar, and fire-control system. 

4. The AEW system must provide information that may be used for 
overall battle control and surveillance. 

The basic plan to be used for the AEW system analysis is shown in 
Fig. 2-14. Also shown are the interrelations between: (1) the AEW system 
and the overall problem, and (2) the AEW system and interceptor system. 



68 



THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 





















Pp (Required Kill 






1 Probability/ Interceptor) 








Manned 




Interceptor 










AEW System 
Model 




Interceptor 
Effectiveness 


^ 


Air Defense 




N (Achieved] ^ ^N (Required) 




System 








>. 


Model 




Requirement 




, 


A 


j- 






1 


B 


.i 




















CO 


"oj 










, ,Q 




Fixed Elements 




Variable Elements 




Fleet Disposition 


System Logic 




Operational Doctrine 




Vectoring Technique 




AEW Aircraft Charac- 
teristics 




AEW Aircraft Deployment 
AEW Radar Detection 




Target Characteristics 




Range 
Resolution 




Speed 




Information Rate 




Altitude 




Tracking Accuracy 




Number 




Stabilization 




Spacing 




Display 


N = Number of Interceptors 


Ship-Based CIC 




Data Interval 


Vectored into Attack 


Evaluation Time 




Smoothing Time 


Position 


Delay 




Beamwidth 




Interceptor Availability 
Navigation Accuracy 




Frequency 




















Inputs to Interceptor System 
Study (Step 3) 








Vectoring Technique 


AEW Svstem 


Vectoring Accuracy 












Specif 


ication 











Fig. 2-14 Plan for Analysis of AEW System Requirements (Step 2 of Master 
Plan — Fig. 2-2). 



The number of interceptors A^ which can be directed against the separate 
elements (single surviving targets) of the 20-plane raid during the air 
battle is selected as the criterion for judging AEW system performance. 

As already shown (Paragraph 2-9) the required number of interceptions 
depends upon the kill probability per interceptor. If we assume a value 
of 0.50, the required number of interceptions is 40 (Fig. 2-11). T\\& task 
force early warning range needed to meet this requirement is 255 n.mi."* 
This is one possible combination of parameters satisfying the operational 
requirement and the interceptor availability limitation. We may select 
this combination as a design point — keeping in mind that it is 
possible to trade off interceptor effectiveness and number of interceptions 
should subsequent analysis indicate this to be desirable. For purposes of 

''More correctly, the specification of detection range should include the required minimum 
probability of obtaining that range, e.g. 90 per cent probability of detection when the target 
has closed to 255 n.mi. This point is covered in more detail in Paragraph 2-12. 



2-10] 



DESIGN OF AIRBORNE EARLY WARNING SYSTEM 



69 



ready reference, the selected system parameters and the predicted system 
performance compared with the operational requirement are shown in 
Table 2-1. 

Now, the problem is to find the combination of variable elements which 
in combination with the fixed system elements will allow the desired value 
of A^ to be achieved. 

The first phase of this process is to hypothesize a specific AEW system 
that provides the required functional capabilities by techniques that 
experienced judgment deems reasonable. The specific parameters of the 
assumed system are then derived from the overall problem requirements. 

State-of-the-art and schedule limitations are not considered in this 
analysis (see Paragraph 1-8). The only restrictions arise from the fixed 
problem elements, laws of nature, and the basic nature of the assumed AEW 
system concept. The latter element is variable. In an actual design study, 
a number of possible AEW system concepts would be examined in this 
manner with the object of determining which provided the best solution 
to the system problem. We shall investigate only one possibility to 
illustrate the nature of the analysis problem. The AEW system selected 
as an example is not intended to be an optimum solution to the AEW 
problem presented by the hypothetical air defense system being examined 
— or to any other AEW problem. It is presented only to illustrate the 
types of problems that must be considered in any AEW system design; 
t\\e. form of the specification for an AEW system; and the nature of the 
interrelationships of AEW parameters and other system elements. 



Table 2-1 SUMMARY OF SYSTEMS ANALYSIS 



Selected 
System Parameters 


Predicted System 
Performance 


Operational 
Requirement 


Number of interceptions 

A^=40 
Kill probability per 
interceptor 

P, = 0.5 
Early warning detection 
range 
Raew = 255 n.mi. 


Minimum 

Number of Minimum 
Targets Probability, 
Killed % 


Minimum 

Number Minimum 
of Targets Probability, 
Killed % 


20 


55 


20 


ss 


16 


90 


16 


90 


14 


100 


12 


100 


See Figs. 2-10 and 2-11 


See Fig. 2-8 


Fixed system parameters 
as defined in Figs. 2-3 
to 2-6 


Target parameters 
as defined in Fig. 2-7 











Note. Selected parameters allow the operational requirements to be met or exceeded. 
Sensitivity of the system performance to parameter changes are shown in Figs. 2-11 and 2-13. 



70 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

2-11 AEW SYSTEM LOGIC AND FIXED ELEMENTS 

A hypothetical AEW system that represents a possible answer to the 
air defense problem being considered is shown in Figs. 2-15 and 2-16. 




AEW Position 
Line 



Fig. 2-15 AEW Operation Illustrating Azimuth Location and Height-Finding 
Means and AEW Aircraft Relations to Fleet Center. 



Two interrelated airborne radars are employed in each AEW aircraft: 
(1) a fan beam which is rotated through 360°, and (2) a pencil beam which 
is nodded up and down past the target to measure height. 







Range, R 








Targets 


AEW 
Radar 


Azimuth, 
Height, h 


AEW 
Commun. 
System 


R,d,h 

AEW Position 


GIG 
Commun. 
System 




Systems 




'1 














t 










AEW 
Display 




AEW 

Navigation 

System 





GIG 
Computer, 
Data Store 
and Fighter 
Direction 
Center 



Fighter 
Direction 
Commun. 
System 



Target 
Position 
Target 
Velocity 
Target 
Heading 



Tactical 
Display 

and 
Decision 



Tactical Doctrine 



Fig. 2-16 Early Warning and Vectoring System Information Flow. 



2-11] AEW SYSTEM LOGIC AND FIXED ELEMENTS 71 

Initial detection of the target is provided by the fan-beam radar. This 
equipment also measures slant range to the target, R, and target azimuth 
position 6 with respect to a reference direction. 

The height-finding radar is positioned in azimuth with information 
obtained from the fan-beam radar. It measures the elevation angle of the 
target y with respect to the horizontal. The sine of the target elevation 
angle, multiplied by the range R and modified by AEW aircraft altitude 
and an earth curvature correction, provides a measure of target altitude h. 
The measured target data are displayed in the AEW aircraft for monitoring 
purposes. 

The AEW system encodes the measured range, azimuth, and height 
information and transmits this intelligence to the CIC in the form of a 
digital message. AEW aircraft position — as obtained from the navigation 
system — is also transmitted via the digital communications link. 

The task force is provided with means for ascertaining AEW aircraft 
position relative to the combat information center (CIC) but with an error 
dependent upon the specific defense problem. A standard deviation of 1 
n.mi. in both the rectangular coordinates is assumed for our analysis, as 
defined in Fig. 2-3. 

Several AEW aircraft are employed — the number and disposition will 
be derived in the succeeding paragraph. The information from all AEW 
aircraft is presented on a master tactical display in CIC to permit overall 
battle control and surveillance. 

Each AEW aircraft measures range and azimuth of all aircraft within its 
zone of surveillance. Height measurements are made only on the designated 
targets; the interceptors are commanded to climb to target altitude, so there 
is no reason (in this example) for measuring the interceptor altitude. 

CIC System Information Processing. The polar coordinate (R, 6) 
information gathered by the AEW radars is transformed into a common 
rectangular (cartesian) coordinate system by the CIC computer to facilitate 
the generation of target heading and velocity information. Rectangular 
coordinates have an advantage over polar coordinates because constant- 
velocity, straight-line flight paths can be represented by x and y velocity 
components which also remain constant. Thus, if the position, P(/), of a 
constant velocity straight-line target at any time t is designated in rec- 
tangular coordinates, then 

m-lox-{-joy (2-1) 

where ^, jy = target position in rectangular coordinates at time ( 

to, jo = unity vectors along the Xo and jo axes of the stationary 
rectangular coordinate system. 



72 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

The instantaneous velocity of this target may be expressed as the time 
derivative of P(/), or 

P{t) = V{t) = hx +7or = "i^Vx +]fsVy = constant (2-2) 

and the position of the target at any time r seconds later can be written 

Pit + r) = m + rFit) = Ux + tF,) = joiy + tF,). (2-3) 

Thus, the computation of the velocity components and the prediction of 
future target position can be done by relatively simple means once the 
present position information has been transformed to rectangular co- 
ordinates.^ 

CIC Command Functions. The position and velocity information 
computed by the CIC is first used for purposes of assessing the threat on 
the basis of numbers, position, and velocity. Then, it is employed to com- 
pute a vectoring guidance course for each interceptor assigned to engage 
specific target aircraft. The guidance information is transmitted to the 
interceptor and displayed there by appropriate means. 

Overall battle control is maintained by CIC using a master tactical 
display in combination with a pre-established operating doctrine. The 
tactical doctrine — target assignment, force deployment, etc. — applicable 
to the threat situation is formulated by the CIC officer and is used to 
monitor and adjust the processing of information in CIC. 

The CIC computer also generates commands which are transmitted to 
the AEW for the purpose of designating targets for the height-finding 
radars. 

Vectoring Guidance (Fighter Direction). The type of vectoring 
guidance employed is dictated by the requirements of the tactical problem 
and should be uniquely controlled by the weapons system requirements. 
In the hypothetical example, a high premium was placed on the ability 
to bring the interceptors into a position to fire their missiles as quickly as 
possible. In fact, the calculation of the number of interceptors that could 
engage the threat (40 for 255 n.mi. AEW range) was based on the implied 
assumption that each interceptor flew in a straight line from fleet center 
to a point where it could engage its assigned target (see Fig. 2-12). 

The type of fighter direction best fulfilling this requirement is collision 
vectoring. Its basic principle is shown in Fig. 2-17. For a target at Pi 
traveling with velocity Ft and an interceptor at P2 traveling with velocity 

•''This advantage does not always lead one to choose rectangular coordinates tor the proc- 
essing of radar information. For example, in Paragraph 1-4 and Fig. 1-4 the use of polar 
coordinates is indicated. 



2-12] 



AEW DETECTION RANGE REQUIREMENTS 



73 



Fixed Reference Direction 
(e.g. North) 




Fig. 2-17 Collision Vectoring Geometry. 

Vp, the interceptor will close with the target in the shortest possible 
time if the following relationship is satisfied: 



sin L = {Vt IVp) sin Oj 



(2-4) 



where each of the variables may be defined from the figure. 

The quantities, dr, V-r, and Bls are obtained from the AEW and CIC 
systems and transmitted to the interceptor. A computer in the interceptor 
uses this information along with its measurement of its own velocity Vp 
to calculate the proper lead angle L. Then, it adds this angle to the space 
line-of-sight direction, 6 is, to obtain the desired space heading of the inter- 
ceptor. The pilot flies the aircraft to maintain this heading and commences 
to search for the target with his AI radar oriented along the line of sight. 

Target altitude also is transmitted to the interceptor. The interceptor 
climbs to this altitude using his altimeter as a reference. In the hypothetical 
system, vectoring guidance information is transmitted at a rate equal to 
the scanning rate of the AEW fan-beam radars. Vectoring guidance is 
continued until the interceptor acquires the target with its own AI radar. 

The choice of this vectoring technique has a profound effect on the inter- 
related requirements of the interceptor AI radar and the AEW and CIC 
systems. A further description of the vectoring problem and the manner 
by which vectoring errors affect AI radar requirements is contained in 
Paragraph 2-25. 



2-12 AEW DETECTION RANGE REQUIREMENTS 

The foregoing discussion has established that an early warning range of 
255 n.mi. is compatible with the system effectiveness goal and the assumed 
fixed elements of the problem. 

It has been assumed that the early warning coverage need be provided 
only in the area of most likely attack. The operational doctrine established 
that a calculated risk would be taken that the defined mass attack would 



74 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

not, in this case, approach the task force from the side farthest from enemy 
bases. 

Carrier deck space and AEW aircraft cycle time limitations dictate that 
the required coverage be provided by a maximum of 3 AEW aircraft. 
Another systems consideration governs the choice of detection character- 
istics of the early warning radars: back-up or overlapping coverage where 
the loss of an AEW aircraft due to enemy action or equipment failure leaves 
the task force undefended. 

The required detection range and AEW aircraft spacing for the assumed 
system may be analyzed by the simple geometrical model of Fig. 2-18. 




Range Requirement 



, Reserve \ 
AEW #1 / AEW \ AEW #2 

-^1 ^ ]^ 

\ (Nonradiating) ; 
During Normal Operation 



Guided Missile 





Fig. 2-18 



Possible AEW Aircraft Detection Range, Coverage, and Disposition 
to Provide 255-n.mi. Early Warning Range. 



The arrangement shown represents one possible answer to the hypothetical 
system requirements. This deployment shows 2 AEW aircraft, each capable 
of detecting enemy targets at ranges of 150 n.mi. with a 360° search sector. 
The two operating AEW aircraft are positioned with respect to task force 
center so that detection occurs at a distance of 255 n.mi. or more from 
task force center in the directions from which enemy raids are expected. 
A third AEW aircraft is positioned as shown for use as a back-up or 
ready replacement for either of the other two aircraft. This aircraft does not 
radiate during normal operation, in order to make its detection and de- 
struction by the enemy more difficult. 



2-13] AEW TARGET RESOLUTION REQUIREMENTS 75 

The configuration developed in this manner is one of many that could 
be developed as a possible problem solution. An actual study would exam- 
ine a number of such configurations. This example is chosen to illustrate 
some of the quantitative and qualitative aspects of the system problems 
that must be considered in an AEW design. 

The specification of detection ranges must take account of the un- 
certainty attending the detection process. For identical tactical situations, 
detection by radar will take place within a band of possible ranges, such as 
are shown by the distribution density function in Fig. 2-19. The proba- 



Probability that Target is Detected 
before Range Closes to R 
(Cumulative Probability)- 



f 1.0 



§0.5 

Q. 



Probability that Target 

is Detected between 

Rand R+c/R 




RANGE, R -W-dR 



Fig. 2-19 Characteristic Forms of Radar Detection Probability Distributions and 
Cumulative Detection Probability Curves. 



bility that the target will be at some time detected bejore it closes to a given 
range R — customarily called the cumulative probability of detection — 
is the integral of the distribution density function taken from i? to °o ; its 
usual form is also shown in Fig. 2-19. 

A preliminary requirements study such as we are performing generally 
expresses the radar detection requirement in terms of the range for 90 
per cent cumulative probability of detection. Accordingly, the detection 
requirements for the AEW radars may be defined as: 

Search Sector — 360° azimuth — Sea level to 50,000 ft. 

Detection Range — 90 per cent cumulative probability of detection of 

the specified enemy targets at 150 n.mi. 

2-13 AEW TARGET RESOLUTION REQUIREMENTS 

A primary function of the AEW system is to provide an early description 
of potential targets. The description might include range, bearing, eleva- 
tion, and number of targets. This information is employed in the threat 
evaluation phase and, later, to vector interceptors against specific targets. 
In both phases, the ability of the AEW system to resolve separate target 
elements is of fundamental importance. 



76 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Threat assessment and tactical decision must occur within 3 minutes 
following initial detection at 255 n.mi. from fleet center. During this 
time, enough information must be obtained to allow the CIC system to 
compute an estimate of the position, speed, altitude, heading, and number 
composition of a potential target complex. The specified targets can travel 
about 25 n.mi. in 3 minutes; therefore this information must be gathered 
and processed when the target is at ranges of 125 to 150 n.mi. from the 
AEW aircraft in order to provide sufficient problem lead time. 

As a first step, the sensitivity of system performance to the resolution 
quality of airborne early warning information must be examined. Then 
the problem of providing the necessary resolution by appropriate AEW 
radar design parameters may be treated. 

The defined target complex consists of 20 targets spaced 5 n.mi. apart 
(see Fig. 2-7). Now, consider the following problem: Suppose that 
during the threat assessment phase only 10 separate targets are indicated 
by the AEW radar information (such a condition could be caused by in- 
sufficient resolution in the AEW system — i.e. a circumstance which could 
cause two or more targets to appear as only one target on the radar dis- 
play). What effect does this condition have upon overall system operation ? 

This question may be answered by considering the effect of this condition 
upon each phase of the air-defense operation. 

First of all, the 6 CAP aircraft would be directed to engage the threat 
elements. Simultaneously, deck-ready interceptors would be launched at 
the rate of 2 per minute. To ensure high target attrition, tactical doctrine 
might dictate that at least 2 interceptors be employed for every potential 
target. This would require launching at least 14 deck-ready aircraft in 
response to a 10-target threat. Thus, for the first 10 minutes following 
initial detection (3 minutes delay time plus 7 minutes for launching 14 
deck-ready interceptors), the conduct of the air battle would be in no 
wise different from what would have taken place if all 20 targets had been 
indicated initially. 

During this 10-minute interval, the threat will have closed to about 
175 n.mi. from fleet center. At this range the 6 CAP aircraft will engage 
separate elements of the raid (see Fig. 2-12). For these interceptions to 
be vectored successfully, at least 6 of the separate target elements must be 
resolved and tracked by this time. 

In addition, if we assume that the number of deck-ready aircraft kanched 
is a direct function of the number of known targets, it is necessary to begin 
to distinguish more than 10 objects by the time the threat has reached 
175 n.mi. from fleet center (or 75 n.mi. from the AEW aircraft). In fact, 
to prevent delay in deck-ready aircraft launchings, the number of targets 
counted must increase at a minimum rate of 1 per minute until all 20 are 
separately resolved. 



2-13] AEW TARGET RESOLUTION REQUIREMENTS 77 

Thus, it is seen that the fact that only 10 of the 20 targets were resolved 
initially does not in itself degrade system performance. So long as resolution 
is sufficient to resolve additional targets faster than the interceptor launch- 
ing rate, system performance will not be affected for the assumed tactical 
doctrine.^ A further examination would disclose that as few as 5 targets 
could be indicated by the initial early warning information provided that 
the subsequent "break-up" of targets was sufficient to keep pace with deck- 
ready interceptor launch rate. 

The vectoring phase imposes additional resolution requirements. The 
assumed tactical doctrine requires that individual interceptors be directed 
against individual targets. Thus, both targets and interceptors must be 
separately resolved and tracked in this phase. 

An inspection of the tactical geometry of Figs. 2-12 and 2-18 discloses 
that contact between targets and interceptors will take place at ranges 
that are seldom greater than 75 n.mi. from an AEW aircraft. The majority 
of contacts will be less than 50 n.mi. from an AEW aircraft. Thus, if 
the AEW radar resolution and the interceptor tactics are chosen to ensure 
that substantially all the targets and all the interceptors can be separately 
resolved at ranges of 75 n.mi. or less from the AEW aircraft, little or no 
degradation in system performance will result if at least 5 separate targets 
are indicated at the early warning range (150 n.mi.). 

Now the foregoing tactical requirements may be translated into radar 
performance requirements. With a radar, it is possible to measure three 
quantities directly (see Paragraph 1-4) — range, angle, and velocity along 
sight-line to target. Resolution between targets may be done on the basis 
of any or all of these. 

Fig. 2-20 shows a particularly difficult case that could exist for the hypo- 
thetical threat. The target threat complex is approaching along a radial 
line which passes through the AEW aircraft and fleet center. The location 
of each threat element relative to the AEW aircraft is shown in the ex- 
panded view. As can be seen, the angular differences between adjacent 
threat elements are of the order of 4°. The range difference between ad- 
jacent elements varies from about 2.5 n.mi. for the extreme outer threat 
elements to less than 1 n.mi. for the central elements. In the case of the 
two center elements, the range difference is zero. The differences in radial 
velocity components of adjacent elements vary from about 20 fps for the 
outer elements to fps for the central elements. 

From the diagram, it is seen that an angular resolution capability of 4° 
or less will provide the stipulated tactical capability. However, this is 
not the only means for meeting the requirement. A range resolution ca- 

^This analysis does not consider the possible benefits of finer resolution to the assignment 
procedure. These might be significant in a practical case and should be taken into account. 
The analysis of this problem is too complex for consideration in this example. 



78 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Target Complex Broken up into Resolution Cells 5° x 1 n.mi. 
\~ Unresolved ,Z~Zr7-r~c^ 



20 Targets Spaced 5 n.mi., 
Apart Resolved into 17 Elements 



75 n.mi. 



' / 

^^ Resolution Cell 

I / 5° X 1 n.mi. 

I / 

I / 

I / 
I / 
I / 
I / 
I / 



^5° Azimuth 
Beamwidth 

12-Msec Pulsewidth = l n.mi. 



AEW Aircraft 
To Fleet Center 
Fig. 2-20 AEW Radar Resolution Capacity at 75 n. 



pability of 0.5 n.mi. would permit resolution of 19 out of the 20 targets 
at 75 n.mi. range. (The two center targets would appear as one if range 
resolution were used.) At 150 n.mi., 15 targets would be seen. 

Similarly, various combinations of range and angular resolution capa- 
bilities may be employed. For example, the diagram shows that an angular 
resolution capability of 5° coupled with a range resolution of 1 n.mi. will 
allow resolution of 18 of the 20 targets at 75 n.mi. range. At 150 n.mi., 
15 separate targets will be indicated. 

Several other factors must be considered in a practical treatment of the 
resolution problem. The individual threat elements will be unable to main- 
tain perfect station-keeping with respect to each other. Errors in relative 
heading, velocity, and position will exist at any given time. This will cause 
the actual target positions and velocities to be distributed around the values 
shown in Fig. 2-20. We shall assume that these errors are small relative to 
the size of a resolution element for the example. However, if these errors 
were of the order of magnitude of a resolution element, they could cause 
substantial modification of the tactical resolution capability. 



2-14] INTERRELATIONS OF AEW, CIC, INTERCEPTOR SYSTEMS 79 

Resolution of the interceptors may be accomplished by a combination of 
tactical doctrine and AEW radar resolution capability. For example, the 
deck-launched interceptors are launched at the rate of 2 per minute. Thus, 
between successive pairs of interceptors there is a range difference of about 
12 miles. Each pair of interceptors may be instructed to maintain a given 
relative spacing (e.g., 5 mi. or more). Assignment doctrine, in turn, must 
be adjusted to be compatible with these tactics. If these steps are taken, 
the AEW radar resolution capability dictated by the threat will also be 
adequate for resolution of the separate interceptors. 

For an actual AEW design problem, many combinations of range and 
angular resolution would be examined for a number of different threat con- 
figurations and approach geometries. Such analysis would serve to place 
upper bounds on the required resolution capability and would establish the 
allowable trade-offs between range and angular resolution for the particular 
system problem. The principles and types of reasoning used for the single 
case examined in this paragraph could be employed for the more compre- 
hensive analysis required for an actual design. In the example problem, 
it was seen that angular resolution capabilities of less than 5° and range 
resolutions of 0.5 to 1.0 n.mi. represented potentially useful ranges of 
values. The final choice will depend upon the influence of other functions 
and problems of the AEW radar system design. 

2-14 INTERRELATIONS OF THE AEW SYSTEM, THE CIC 
SYSTEM, THE INTERCEPTOR SYSTEM, AND THE 
TACTICAL PROBLEM 

Target tracking will follow detection. The tracking information is 
first utilized for threat assessment; later, tracking of both targets and 
interceptors provides the information needed for fighter direction. 

Three interrelated characteristics of the AEW /CIC system are of funda- 
mental importance in determining the contributions of this system to 
overall mission accomplishment. 

(1) Detection range 

(2) Accuracy 

(3) Information handling capacity (number of separate tracks, 
etc.) 

In Paragraph 2-9, the detection range was found to be one of the critical 
factors in determining the level of mission accomplishment along with the 
individual interceptor effectiveness and the number of interceptors available 
for defense. Implicit in the analysis, however, were the assumptions that 
AEW /CIC system accuracy or data-handling capacity did not limit over- 
all system performance. We must now determine the specific characteristics 
that the AEW /CIC system must possess to make these assumptions valid. 



80 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

2-15 ACCURACY OF THE PROVISIONAL AEW SYSTEM 

For the hypothetical problem under consideration, there are 20 targets 
and 40 interceptors — all of which could conceivably be in the zone of 
coverage of a single AEW aircraft. Thus each AEW aircraft must be 
capable of keeping track of 60 objects. Height measurements must be made 
on a maximum of 20 objects (targets only). 

One facet of the accuracy problem — resolution — and its relation to 
the overall system problem has already been discussed in Paragraph 2-13. 
In addition to separating the 60 objects, the AEW/CIC system must also 
track each object, i.e. determine its position relative to some reference 
coordinate system and — for each of the attacking aircraft — its heading, 
velocity, and altitude. As has been described, this information is utilized 
to direct specific interceptors on collision courses with specific targets. The 
required accuracy of this guidance depends upon the characteristics of the 
interceptor system, particulary upon the AI radar and fire-control system. 
The accuracy of the AEW/CIC system determines the accuracy with which 
the interceptors can be vectored, and the vectoring error in turn determines 
the required lock-on range of the AI radar. This last factor is a critical 
item and may be severely limited by fixed elements of the problem and use 
environment. Thus the AEW accuracy can only be firmly specified after 
a study of the vectoring problem has determined the trade-off relation 
between vectoring error and the required AI lock-on range.'' 

Unfortunately, because of the complex interrelations between AEW/CIC 
system errors and vectoring errors, the analysis in Paragraphs 2-22 to 2-28 
cannot be made abstractly but will require, as inputs, provisional assump- 
tions of the AEW /CIC system design and accuracy. Thus, in this and some 
of the following paragraphs, we shall assume tentative values for the required 
AEW/CIC system accuracy and carry on our study of the preliminary 
design of the AEW radar on the basis of these assumptions. We should 
bear in mind, however, that these provisional values may lead to an un- 
acceptable requirement for the AI lock-on range, in which case the analysis 
would have to be repeated for a modified AEW/CIC system design. 

The AEW radar measures the relative position — azimuth and range — 
of a target with respect to itself once per revolution of the fan beam. The 
accuracy of each measurement, as it is seen in CIC, is limited by a number 
of factors. The most significant of these are: 

1. Beamwidth 3. Data quantization 

2. Range aperture (pulse length) 4. Data stabilization 

5. Time delay errors 

''An illustrative analysis of this kind has been carried out in Paragraphs 4-6 and 4-7 of 
Merrill, Greenberg, and Helmholz, of), cit. 



2-15] 



ACCURACY OF THE PROVISIONAL AEW SYSTEM 



81 



Measurement Errors due to Beamwidth and Range Aperture. 

Referring to Fig. 2-21, if a large number of individual measurements were 
made on a target at point T, the measured values could be plotted as prob- 
ability density distributions of azimuth and range values about the point 
T. For any single measurement, the indicated target position might lie 
anywhere in the region encompassed by these distributions, e.g., the point 
A indicated in the figure. 



T = Actual Position of Target 
A= Measured Position of 

Target 
t= Azimuth Measurement 

Error of AEW Radar 
e„ = Range Measurement 

Error of AEW 
a^= Standard Deviation of 

Azimuth Measurement 

Errors of AEW Radar 

0), = Standard Deviation of 
Azimuth Measurement 
Errors of AEW Radar 

R = Range from AEW to 
Target T 



AEW Radar Location 
Fig. 2-21 Representation of AEW Measurement Errors. 




It is convenient to describe the measurement errors by their standard 
deviations or root mean square (rms) errors. The magnitudes of these rms 
errors are closely related to the resolution capabilities previously dis- 
cussed; however, the reader should be careful not to confuse the resolution 
of two targets and the accuracy in tracking one. 

Accuracy describes the radar's ability to measure the position of a single 
target; as a rough approximation the standard deviation of the errors of a 
single measurement of target position may be considered to be about one- 
quarter of the resolution capability. Actually, besides being related to the 
beamwidth, the measurement error is a function of the signal-to-noise ratio 
and the number of hits per beamwidth. An analysis of these relations is 
given in Paragraph 5-1 1 . As an example, with a 5° azimuth beamwidth and 
a pulse width of 12 Msec corresponding to 1 n.mi. (i.e. resolution capabilities 
shown to be adequate in the preceding paragraph), the approximate ac- 
curacy of a position measurement for a single scan will have rms values of 



(5)(0.25)(150) 



57.3 



3.3 n.mi. (at 150 n.mi. range) 



(7 A = 1-65 n.mi. (at 75 n.mi. range) 



(2-5) 



82 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

(Tff = (0.25) (164/) 

= (0.25) (164) (12) = 494 yd = 0.25 n.mi. (independent of range). 

Several factors act to make the effective errors somewhat larger than the 
basic errors given in Equation 2-5. 

Ouantization Errors. Errors may be introduced by the process of 
quantizing or "rounding off" measured values of range and angle. This is 
done both at the AEW radar and in the CIC computer in order to minimize 
the amount of data which must be processed and transmitted over the as- 
sociated data links. 

The process may be visualized as follows. The space around a reference 
point (in this case either the AEW aircraft or the CIC) is broken into "cells" 
of arbitrary size and shape. An object anywhere in one of the cells is as- 
signed a position description corresponding to the position of the center of 
the cell. This process introduces rms errors which are approximately one- 
quarter of the cell dimensions. These errors are independent of the meas- 
urement errors. Thus, if the quantization level (cell size) of the AEW 
system is chosen to be equal to the rms measurement errors (1.25° and 0.25 
n.mi. in our case), the equivalent rms error in the AEW data will be in- 
creased by only about 4 per cent. 

The insensitivity of the equivalent errors to the quantization level of the 
foregoing example shows that coarser range quantization could be employed 
if desired. For example, if the space around CIC were broken into cells 1 mi. 
on a side, the equivalent rms range error would be increased by about 40 
per cent to a value of 0.35 n.mi. when the polar data from the AEW system 
were transformed to rectangular coordinates in CIC. 

Because the angle measurement is considerably coarser than the range 
measurement, the 1-n.mi. CIC data quantization cells would make almost 
no contribution to the rms angular error data received from the AEW air- 
craft. Accordingly, the following quantization levels may be chosen as 
reasonable: 

AEW: Azimuth 1.25° 

Range 0.25 n.mi. 

CIC: X coordinate 1 n.mi. 

Y coordinate 1 n.mi. 

These levels are compatible with a range resolution requirement of 
1 n.mi. (Paragraph 2-13). If finer range resolution were to be employed, 
the CIC quantization levels would have to be reduced accordingly. 

Stabilization Errors. Another important possible source of error is 
the rolling and pitching motion of the AEW aircraft due to maneuvers and 
wind gusts. It was required (Paragraph 2-11 and Fig. 2-15) that AEW 



2-15] ACCURACY OF THE PROVISIONAL AEW SYSTEM 83 

measurements be referenced to a fixed angular coordinate reference system. 
Errors may be induced in the azimuth measurements by aircraft motions 
if the AEW radar is not space stabilized. Therefore, the preliminary design 
must consider how AEW aircraft motions affect the AEW radar system 
measurements. If the effects are substantial, means must be provided for 
correcting the errors thus introduced. For the purpose of this analysis it 
will be initially assumed that stabilization errors do not degrade the angular 
accuracy by more than 10 per cent. The effect of this assumption may be 
examined in more detail when the operation of the system has been more 
completely analyzed and understood. 

Time Delay Errors. A possible source of additional position error on 
a moving target is the fact that time may elapse between the measurement 
of target position by the AEW radar and the registration and use of this 
information in the CIC. If the time delay is td, the amount of position 
error is simply 

e = Vtd n.mi. 

where V = velocity of object being tracked. 

Since the data-handling system must process information at least as fast 
as it is coming into the system, the maximum possible value of the time 
delay would be approximately equal to the time, tsc, for the AEW fan beam 
to make a 360° scan. For example, if the scan time were 6 seconds, the 
maximum error against an 800 fps target caused by time delay would be 
approximately 0.8 n.mi. For 1200 fps interceptors, this error would be 
50 per cent larger or about 1.2 n.mi. 

Three courses of action are open to the designer with respect to this error. 
(1) The error may be tolerated if it does not appreciably affect system 
performance. (2) Scan speed and data processing speed may be increased 
to reduce the error magnitude to an acceptable level. (3) The position 
information may be up-dated by using estimates of velocity and heading 
of the object being tracked along with a knowledge of the time delay, to 
produce a term which cancels the time-delay error. 

For preliminary design of the overall AEW system, it will be assumed 
that time-delay error does not increase the total position error by more than 
10 per cent. The effects of this assumption upon system operation and the 
detailed requirements of AEW radar can be examined when more is under- 
stood about the interrelationships among various parameters of the air 
defense system. At that time a decision can be made about the course of 
action to be taken to correct the time-delay error. 

Summary of Assumed Accuracy Characteristics. For purposes of 
analysis, the AEW^ radar is assumed to have the following characteristics: 



84 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Beamwidth 5° 

Pulse length 12 /xsec 

Scan time 10 sec 

These characteristics coupled with assumed values for quantization levels 
in the AEW and the CIC systems and the assumed limits for stabilization 
errors and time-delay errors lead to the following accuracy characteristics 
for the provisional AEW system. 

a A = (3.3) X (1.04) X (1.10) X (1.10) = 4.15 n. mi.rms 

(at 150 n.mi. range) 

CA = (1.65) X (1.04) X (1.10) X (1.10) = 2.07 n.mi. rms 

(at 75 n.mi. range) 

aR = (0.25) X (1.04) X (1.41) X (1.10) = 0.41 n. mi.rms 

(unaffected by maneu- 
vers and range). 

(2-7) 

The total rms position errors may be expressed as the vector sum of the 
range and azimuth errors or 

<TT = (c7a' + (TR^y = 4.17 n. mi. (150 n.mi. range) 

= 2.09 n. mi. (75 n.mi. range). (2-8) 

One source of error — the 1 n.mi. rms navigation error of the AEW 
aircraft (see Paragraph 2-11) — has not been included in this analysis. 
This error is not significant when all of the target and interceptor tracking 
data come from a single AEW aircraft and when the navigation error 
changes vary slowly with time. When these conditions prevail, each piece 
of data in CIC will be biased by the same error. The relative errors between 
pieces of data are therefore unaffected. As we shall see, it is these relative 
errors that determine tracking and vectoring accuracy. 

The navigation error does become important for targets and interceptors 
which are tracked by both AEW radar aircraft. Such an overlapping zone 
is shown in Fig. 2-18. A large navigational error would complicate the 
problem of correlating data from the same target. However, since the 
navigation error is less than the measurement errors and less than aircraft 
separation distances, no great amount of difficulty can be expected for 
this hypothetical case. 

2-16 INFORMATION-HANDLING CAPACITY OF THE 
PROVISIONAL AEW SYSTEM 

An important aspect of system design relates to its data-handling 
capabilities. Both the data link for transmitting information between the 



2-17] VELOCITY AND HEADING ESTIMATES 85 

AEW aircraft and the CIC and the data-processing computer at the CIC 
will have limited capacities for handling data. The maximum per channel 
capacity of the data link has been specified (Fig. 2-3) as 1000 bits^ per second. 

There will be 60 objects (40 interceptors and 20 targets) in the field of 
the AEW radar. To identify each object requires 6 bits as shown in the 
footnote. The azimuth location of each target is determined to the nearest 
multiple of 1.25°. This will require 9 bits per object. Range information to 
the nearest 0.25 n.mi. from zero to 150 n.mi. requires 10 bits per object. 
If we add these items and multiply by 60, we find that the amount of 
information needed to specify the range and azimuth of the 60 objects on 
a single scan of the radar is 1500 bits. In addition, the elevation of the 20 
targets must be determined. The accuracy and quantization level of the 
target altitude has not yet been specified. Here, we shall assume that target 
altitude is determined to the nearest 0.25 n.mi. = 1500 ft, the same as in 
range. To specify a target altitude from zero to 50,000 ft, then, requires 
6 bits, and all the altitude data for 20 targets comprise 120 bits. The total 
information load on one scan, then, is 1500 + 120 = 1620 bits. In order 
to incorporate self-checking codes and message redundancy in the data link 
to increase reliability, this figure should be about doubled. Thus, in a round 
figure, the data link must transmit about 3000 bits per scan to the CIC. 

The actual information rate will, of course, depend upon the scan time. 
It is generally desirable to make the scan time relatively short in order to 
increase the accuracy of the heading and velocity estimates. A study in 
Chapter 3 indicates that the cumulative detection range tends to be 
relatively independent of scan time, although a broad optimum may exist. 
Yet the scan time cannot be made indefinitely small, because of the 
limitations of mechanical design and the increase in the data rate. We 
have chosen a provisional scan time of 6 seconds for the basic AEW radar. 
This radar then scans at a rate of 60° /sec. The information rate which the 
data link must handle is 500 bits /sec. This figure is well within the capacity 
of the defined data link system. 



2-17 VELOCITY AND HEADING ESTIMATES 

The position data are used in the CIC to compute estimates of target 
heading and velocity. This may be done in a variety of ways. One of the 
simplest can be illustrated with the aid of Fig. 2-22. 

^A "bit" represents a binary digit, i.e. either zero or one. Transformations from decimal to 
binary are made in the following manner: the number 60, for example, may be expressed 

60 = (1 X 25) + (1 X 24) + (1 X 23) + (1 X 22) + (0 X 2^) + (0 X 20) 

In binary form, the number 60 is the six-digit number formed by the multipliers of the powers 
of2: 



60 decimal = 111100 binary. 



86 



THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



; Target 
Positions for Eacii 
Look" 




Fig. 2-22 Positk 



CIC Location 

Data Used in Computing Esti 
Velocity. 



lates of Target Heading and 



Assume that the specified 800-fps target is initially detected at point 1. 
Since the assumed AEW radar scanning time is 6 seconds, the next look at 
the target will occur when it reaches position 2. It will be at position 3 on 
the third look, and so on. At the end of seven looks, the target will be at 
position 7. The position measurement on each look is characterized by an 
assumed radial error with a standard deviation of or. This error may be 
broken into components parallel and normal to the target path, where the 
standard deviation of each component is^ 



ap = ot/VS n.mi. 
o"7v = ot/V2 n.mi. 



(2-9) 
(2-10) 



If the errors of each position measurement are assumed to be independ- 
ent, the relative errors between any two measurements have the standard 
deviations 



(TPIP2 — 



v(ty-(iy= 



ar n.mi. (2-11) 

(yN\N2 = or n.mi. (2-12) 

A very simple procedure for determining the target velocity and heading 
can be based on the extreme position measurements. The estimated 



^Breaking the errors into equal components ignores the influence of the fact that the range 
and azimuth errors for a given target measurement are markedly different (Equations 2-7 and 
2-8). If this factor is considered, the mathematical complexity of the problem is greatly 
increased; the final answer expressing the probable position errors for any randomly chosen 
target position relative to the CIC is not changed substantially. 



(TP1P2 

ntsc 


_ ar 
~ ntsc 


0'N1N2 


_ 0T_ 


l^isc 


ntsc 



2-17] VELOCITY AND HEADING ESTIMATES 87 

velocity will be the difference of the extreme measurements divided by the 
total time, ni^c, where n is the number of scans and 4c is the scan time. The 
estimated direction is simply that determined by the two extreme measure- 
ments. The velocity and heading errors are expressed in terms of the 
parallel and normal components of the relative position errors. Thus 

(2-13) 

'^^ = ^7^ = — (2-14) 

where F = true velocity of object being tracked. 

For example, assume that position measurements are made on an 800-fps 
target 75 n.mi. from the AEW aircraft. This range represents a likely value 
of the maximum range at which accurate tracking will be required for the 
generation of vectoring information as explained in Paragraph 2-13. 

With the assumed parameters of the provisional AEW system, the total 
position error of a single measurement was derived to be Equation 2-8: 

ar = 2.09 n.mi. = 2 n.mi. (2-15) 

Thus, the standard deviations of the two components of the relative 
error between two measurements are from Equations 2-11 and 2-12: 

ap.Po - 2 n.mi. (rms) = 12,000 ft (rms) (2-16) 

(^NiNi = 2 n.mi. (rms) = 12,000 ft (rms). (2-17) 

Thus the rms errors in the estimated velocity and heading are calculated 
by Equations 2-13 and 2-14 to be 

av = 12,000/(6) (10) = 200 fps (rms) (2-18) 

cT^ = 12,000/(800) (6) (10) = 0.25 rad = 14.5° (rms). (2-19) 

Accordingly, we see that the accuracy of the velocity and heading 
estimates depends upon the following factors: 

1. Accuracy of each position measurement 

2. Number of position measurements used for estimates of velocity 
and heading 

3. Elapsed time between position measurements 

4. Velocity of object being tracked 

In addition, accelerations of the object being tracked during the time 
interval, ntsc, also give rise to additional errors in the position and velocity 
estimates. 



88 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

There are several means for improving the tracking accuracy. Each of 
these involves a trade-off between improved accuracy and greater informa- 
tion-handling complexity. For instance, in the simple case we have just 
been discussing, when only the extreme position measurements are used 
all the information associated with interior measurements is lost. If all 
the position measurements were used to determine the velocity and heading 
estimates, the errors would be substantially smaller. Thus, the error 
estimates in Equations 2-18 and 2-19 are somewhat pessimistic. In some 
cases, a trade-off between maneuvering and nonmaneuvering target track- 
ing accuracy also is required. 

The velocity and heading estimates may be used in several ways. First 
of all, this information is used to compute vectoring guidance for the 
interceptors. The collision vectoring equation (Equation 2-4) illustrates a 
typical application. Part of the vectoring problem involves prediction of 
the future positions of the targets and interceptors. Prediction for a single 
scan is also used to update the position information. An example of such a 
prediction process is shown in Fig. 2-23. The AEW/CIC system indicates 
target position as point A. The velocity and heading estimates are used to 
generate a track AA^ On the next scan, target position is indicated as 
point B. The position data are corrected to this point and a new continuous 
track BB^ is estimated, etc. Thus, at any time between measurements a 

A,6,C,D = Measured Target Positions 

a\b\c\d^ = Estimated Target Positions 
f^c Seconds After 
Measurements of 
A,6,C,D Respectively 




Fig. 2-23 Prediction Process Employing Scan-to-Scan Correction of Position Data. 

position measurement is available which accounts for the change in target 
position since the last measurement was made. This type of information 
processing (updating) greatly reduces the time-delay error discussed in 
Paragraph 2-15. 



2-11 



AEW RADAR BEAMWIDTH AS DICTATED BY PROBLEM 



89 



The updating process has another advantage. It produces an estimate of 
target position on the next scan (see Fig. 2-23). This estimate greatly 
facilitates the problem of maintaining the identity of a target from scan to 
scan because it provides a better idea of where each target is going to be the 
next time the AEW radar looks at it. The heading and velocity information 
is used to obtain these predictions. 

Prediction must be paid for, though, and the longer the prediction time, 
the larger the errors in the predicted position. Fig. 2-24 illustrates how the 



Error in Present 
Position 



Errors in Future 
Positions 




-• 8n.mi. — ^- Sn.mi. — A 



Vj = 800 f ps 
a, - 200 fps 



2n.mi. 



Fig. 2-24 Growth of Position Error with Prediction Time. 

indeterminacy volume of the predicted position expands with the prediction 
time. This figure was determined on the basis of the following expressions 
for the future parallel and normal rms errors app and aNF in terms of 
prediction time T and the present position, velocity, and heading errors. 

(2-20) 

(2-21) 



(JPF 

ctnf 



+ av'T- 



A&M^V^^^^ 



With the same target velocity and system characteristics used previously, 
the position error expands from 2 n.mi. rms to 4.46 n.mi. rms with a pre- 
diction time of 2 minutes. 



2-18 



AEW RADAR BEAMWIDTH AS DICTATED BY THE 
TACTICAL PROBLEM 



On the basis of target resolution requirements (Paragraph 2-13) a value of 
5° was selected for the fan beamwidth of the provisional AEW radar design. 
Subsequent estimates of accuracy and information handling characteristics 
were based on this value (Paragraphs 2-15 to 2-17). 



90 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

The selected value of 5° was representative of a likely allowable upper 
limit for the AEW radar design on the basis of resolution. The use of larger 
beamwidths would complicate the problem of target resolution since the 
angular beamwidth would then be appreciably larger than the angular 
separation of the targets (Fig. 2-20) at the maximum vectoring range 
(75 n.mi.). 

Since the system accuracy is almost a direct linear function of beamwidth, 
the estimated accuracy of the provisional AEW design represents the 
poorest that might be obtained from a potentially suitable AEW design. 
Thus the accuracy performance characteristics of the provisional AEW 
design will tend to place the most severe requirements on the interceptor 
system. If the interceptor system can be built to meet these requirements, 
the same interceptor system will be more than adequate for smaller values 
of AEW radar beamwidth. On the other hand, if the selected value of 5° 
makes AI radar requirements unreasonable, the maximum permissible 
AEW radar beamwidth may have to be reduced. The objection to a 
reduced beamwidth is the larger antenna which it entails and the penalty 
thus imposed upon the AEW aircraft. 

In this chapter, only the interrelationships of AEW radar beamwidth 
with the tactical problem are discussed. As will be seen in Chapter 3, radar 
beamwidth also enjoys close interrelationships with other parameters and 
performance characteristics of the radar system. Among these are (1) 
detection range, (2) information rate, (3) operating frequency, (4) antenna 
size, and (5) stabilization requirements. 

In addition, AEW radar beamwidth affects the response of the radar 
system to electromagnetic disturbances arising in the tactical operating 
environment. Enemy countermeasures, radar returns from clouds and 
ground, and radiations from other AEW aircraft are representative of such 
phenomena. Strictly speaking, the consideration of these factors should 
be made at the same time as the resolution and accuracy requirements 
studies since they are an important part of the AEW radar's relationship 
with the overall tactical problem. For simplicity, the discussion of these 
factors is deferred until Chapter 14 because a knowledge of radar techniques 
and propagation phenomena is necessary to make such a discussion mean- 
ingful. 

To summarize, then, AEW radar fan beamwidth is dictated by three 
primary tactical considerations: resolution, vectoring accuracy, and inter- 
action with electromagnetic disturbances. Resolution considerations have 
been shown to dictate a value of about 5° or less. Vectoring accuracy 
requirements are unknown at the present time. In order to proceed with 
the problem, the vectoring accuracy obtainable with a 5° beam will be used. 
Subsequent analysis of the AEW and AI systems will disclose whether 
vectoring accuracy dictates a narrower beam. Electromagnetic disturbance 



2-18] 



AEW RADAR BEAMWIDTH AS DICTATED BY PROBLEM 



91 



effects are unknown. The details of this problem will be largely neglected 
in the development of system requirements in this chapter. 

Vertical Beamwidth. Vertical beamwidth also is an important 
factor. The AEW radar must detect and track the specified 50,000 ft 
altitude targets. It should also have a capability for detecting and tracking 
targets at all other reasonable values of altitude, since the specified threat 
could not be considered realistic if there were significant holes in the early 
warning coverage at other altitudes which could be exploited by the enemy. 

The characteristics of the threat determine the required vertical coverage. 
If it is assumed that the primary threat (Mach 0.8, 50,000 ft) could also 
attack from lower altitudes — for example, 10,000 to 50,000 ft — then, 
AEW coverage must be provided over this range of altitudes. The coverage 
must be sufficient that targets are not lost for appreciable periods of time. 
For example. Fig. 2-25 shows that vertical coverage of 45° upward and 18.3° 

Altitude of 
Primary Specified 
^Threat 



50,000 - 




Possible 
Altitude Range 

of Targets 
10,000 - 50,000 ft 



-10 -5n.mi. 5 10 

RANGE FROM AEW AIRCRAFT - n.mi. 

Fig. 2-25 AEW Vertical Coverage Diagram — Example. 



downward can create a zone 10 n.mi. in diameter where the primary target 
(50,000 ft, 800 fps) can be lost from view. In the worst case, this would 
involve loss of the 800-fps target for a period of slightly greater than 1 
minute. With the assumed target spacing — 5 n.mi. — a maximum of two 
targets would be within this zone at any one time. 

By the time targets enter this zone, the estimates of their velocity and 
heading have been obtained quite accurately since they have been under 
surveillance for almost 150 n.mi. These estimates may be used to update 
the target position during the blind time, thereby reducing the effect of 
the blind zone on system performance. Moreover, the tracking of objects 
entering the zone is being done at very short ranges, and this greatly 
improves the position accuracy of the data obtained just before the target 
enters the zone. On these bases, it is reasonable to assume that dead zones 
of the order of 10 n.mi. do not sensitively affect system performance, since 
surveillance is lost for a relatively short time. Thus, vertical coverages 



92 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

of the order of that indicated in Fig. 2-25 should be adequate for the tactical 
problem. 

One other problem related to vertical beamwidth is of great importance 
to the AEW problem, namely ground and sea return. Fig. 2-25 shows that 
the fan beam intersects the surface of the water — or land, as the case may- 
be — at all ranges greater than 10 n.mi. Thus reflections from the ground 
will compete with target signal reflections at all ranges greater than 10 n.mi. 
This fact requires that means be provided in the AEW system to distinguish 
between returns from the surface of the earth and target returns. 

Interactions between the radar, the target, and the ground constitute 
a very complicated problem. The polarization of the radar transmission, 
surface characteristics, and AEW and target altitudes all interrelate to 
produce nulls and reinforcements which influence the system capability. 
These factors are discussed in some detail in Chapter 4. 

In an actual design study, the quantitative aspects of this problem should 
be carefully studied and set forth at this point of the systems requirements 
development. The interrelations of the tactical geometry and propagation 
and scattering characteristics must be ascertained to define the magnitude 
of the problem implied by the requirements for distinguishing between 
ground and target returns. 

2-19 FACTORS AFFECTING HEIGHT-FINDING RADAR 
REQUIREMENTS 

The height-finding radar for the example problem is positioned in the 
aircraft nose. It is directed to point in a given azimuth direction at a target 
located by the fan-beam radar. It is then nodded up and down to determine 
the elevation of the target with respect to a horizontal reference in the AEW 
aircraft. The nodding action causes the target return to vary as a function 
of the space (or angle) modulation characteristic (elevation) of the height- 
finding beam. The particular type of space modulation characteristic that 
is used depends upon the accuracy requirements of the height finder. 

The requirements of the height-finding radar are dictated primarily by 
the following tactical considerations: 

1. The characteristics of the expected threat including possible varia- 
tions from the specified values. These characteristics include speed, 
altitude, and number of aircraft. 

2. Height-finding requirements during threat evaluation. 

3. Height-finding requirements during vectoring. 

4. Height-finding requirements dictated by the need to supply early 
information to the ground-to-air missile system. 

To meet these requirements within the limitations of the hypothesized 



2-19] FACTORS AFFECTING HEIGHT-FINDING RADAR 93 

system logic, the height finder must operate with azimuth input commands 
obtained from the AEW fan-beam radar. 

The possible dimensions of the height-finding radar are limited by the 
dimensions of the AEW aircraft nose. For purposes of specification, it will 
be assumed that the aircraft in the example may accommodate an antenna 
with a maximum dimension of 3 ft. It is also assumed that such an antenna 
may be gimbaled so as to be capable of performing the height-finding 
function on objects within a ±80° horizontal zone around the AEW 
aircraft's nose. The vertical coverage of the height finder must be matched 
to the primary vertical pattern coverage of the fan-beam azimuth search 
system. 

The assumed placement of the height-finder places a limitation on tactical 
usage. To evaluate a target the AEW aircraft must be pointed within 
±80° of the line of sight to the object whose height is being measured. 
Thus, by virtue of the assumed system logic, the AEW aircraft must 
maneuver to perform its mission. The required maneuver must be within 
the performance characteristics of the AEW aircraft. In addition, the 
effect of the maneuver upon the stabilization problem must be evaluated. 



Requirements Dictated by Threat Evaluation. Height-finding 
information need not be obtained at the same rate as position information 
for the specified threat, since its altitude does not change during the attack. 
In fact it need be measured only once during the specified attack. Once 
again, the possibility of other attack situations must be considered. If the 
enemy aircraft were capable of making an abrupt altitude change during 
the attack, the height-finding system must be able to detect such a change 
in time for appropriate defensive measures to be taken. 

Immediately following detection, the height-finding radar is required to 
begin measuring target altitudes for purposes of raid evaluation. Ideally, 
the evaluation of target altitudes should take place within the time allowed 
for threat evaluation (assumed to be 3 minutes in the example). If we 
allow an average time of 1 minute^" for the AEW aircraft turning to face 
the raid, a total time of 2 minutes is available to measure target altitudes 

lOThe AEW aircraft speed is 200 knots. At this speed, and at a bank angle of 10°, the AEW 
aircraft can turn at the rate of about l°/sec. 

where F = velocity in knots, </> = roll angle (degrees), andi/' = horizontal turning rate (°/sec). 
Since the height-finding radar coverage is ±80° or 160°, the maximum turn required to bring 
a target under height-finder surveillance is 100°. Thus, 100 seconds would be required in the 
worst case for a 10° bank angle. On the average less than half this time would be required, 
since the orientation of the AEW aircraft relative to the raid is random. Thus, the assumption 
of 1 minute does not imply extreme maneuvers by the AEW aircraft. 



94 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

and transmit this information to CIC for decision and target assignment. 
Since there are 20 targets, a maximum time of 6 seconds per target is 
permissible. During this phase, it is sufficient to know whether the targets 
are high, medium, or low altitude. 

Requirements Dictated by Vectoring. During vectoring, inter- 
ceptors are vectored to the measured altitudes of the assigned targets (for 
the assumed system logic of the hypothetical example). A height-finding 
error can limit system performance in the following ways. 

1. It can cause the interceptor to fly at an unnecessarily high altitude, 
thereby degrading speed and maneuvering capability. 

2. It can cause the interceptor to approach the target with an altitude 
differential which its weapon (assumed to be a guided missile) 
cannot overcome. 

3. It increases the zone of probable target positions which must be 
searched by the AI radar. 

The first limitation can be attenuated somewhat by the use of tactical 
doctrine based on a prior knowledge of threat characteristics. For example, 
if the probable threats are known to have a performance ceiling of 50,000 ft, 
there would be little point in directing the interceptor to fly at 60,000 ft 
even though the height finder indicated such an altitude. 

The second limitation must be related to weapon characteristics and 
aircraft and fire-control system characteristics. An inspection of the missile 
performance (Fig. 2-6) shows that the weapon can itself correct substantial 
altitude errors by its maneuvering capability. For a weapon traVel of 3.2 
n.mi. or more, altitude errors to 2 n.mi. can be corrected if the weapon is 
fired horizontally. A further attenuation of the effects of altitude can be 
obtained from the fire-control system. Following AI radar lock-on, the 
pilot obtains a reasonably precise measurement of relative target elevation. 
This may be used by the fire-control system to point the aircraft up or 
down as required to eliminate an elevation error. Of course, the required 
climb or dive angle must be compatible with aircraft performance character- 
istics. 

The third limitation — the required AI radar search zone needed to 
encompass the height-finding inaccuracies — is also most important. As will 
be demonstrated later, the range performance of a radar system is strongly 
influenced by the volume it must search. 

On the basis of these considerations, a height-finding error of approxi- 
mately 0,5 n.mi. (3000 ft) standard deviation at a range of 75 n.mi. 
represents a reasonable first approximation to the height-finding accuracy 
requirement. This corresponds to a maximum error of about 1.5 n.mi. — 
a value which is still within the guided missile performance capabilities. 



2-19] FACTORS AFFECTING HEIGHT-FINDING RADAR 95 

It will be assumed that the same height-finding information rate (one 
measurement on each target aircraft every 2 minutes) will be maintained 
during vectoring. 

Requirements Dictated by the Surface-to-Air Missile System. 

Height-finding data can be used to direct the search and tracking system 
associated with the surface-to-air missiles to those regions of the airspace 
where targets are most likely. For the system of the example, such informa- 
tion can be most useful, since the primary target for the ground-to-air 
missile system is a missile launched from the hostile aircraft at a range of 
about 50 n.mi. The relatively smaller size of the missile makes knowledge 
of where to look for it most desirable. For such an operation, provision 
must be made for the proper transfer of data within the CIC system. It is 
not likely, however, that the requirements of this function are more severe 
than the interceptor vectoring height-finding requirements. For the 
purposes of the example, this will be assumed to be the case. Once again, 
this is an area which deserves more detailed scrutiny in an actual design 
study. 

Requirements Dictated by the Stabilization Problem. Height- 
finding — even when the requirements are as coarse as indicated for the 
hypothetical problem — involves measuring rather small angles. The 
assumed system logic requires that elevation angle of the target be meas- 
ured with respect to the horizontal plane. In addition, the height-finder 
must be commanded to the measured space azimuth position of a particular 
target. 

Some idea of the problem may be obtained by translating the derived 
0.5-n.mi. rms interceptor vectoring height-finding requirement into an 
equivalent angle for 75 n.mi. range. This angle may be expressed 

^, = ^ = 0.067 rad = 0.38° (rms). (2-22) 

This is a total error — including the accuracy of the radar, the stabiliza- 
tion errors, mechanization errors, and quantization errors. If the latter 
two errors are assumed negligible and if the stabilization and radar meas- 
urement rms errors {au and dhm respectively) are assumed equal, normally 
distributed, and independent then 

c^a™ = <r,, = 0.38/V2 = 0.27°rms. (2-23) 

Thus, to meet the height-finding requirement of 75 n.mi., the height- 
finding system must be stabilized to within 0.27° of true vertical. This 
accuracy must be maintained despite aircraft pitching or rolling motions. 

In addition, the azimuth beamwidth of the height-finding radar must be 
large enough to include the uncertainty of the azimuth fan beam. It should 



96 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

not be appreciably larger than that of the fan beam or it will have difficulty 
resolving between adjacent targets that are resolved by the fan beam. 
Thus, as a first approximation, the azimuth beamwidths of the height 
finder and the fan-beam radar may be made approximately equal. 

The elevation beamwidth depends upon the required accuracy. An 
approximation may be obtained by using the same expression employed for 
fan-beam azimuth accuracy (Paragraph 2-15). 

Cn = 6n/4 degrees (rms) for a single measurement (2-24) 

an = Qnl'^yln degrees (rms) for n measurements (2-25) 

where n = number of measurements averaged to obtain a single estimate 

9n = elevation beamwidth of height-finding radar. 

Since 6 seconds can be taken for the height-finding measurement, it is 
reasonable to assume that five to ten separate measurements could be made. 
If, e.g., nine measurements are made, the required beamwidth is found by a 
manipulation of the above equation as 

e„ = 4V^ c7„ = (4) (3) (0.27) = 3.2°. (2-26) 

Actually, techniques known as beam splitting can be employed to obtain 
greater angular accuracy than is implied by Equation 2-26. Accordingly, 
the derived result is only one of the possible solutions to the height-finding 
problem. 

2-20 SUMMARY OF AEW SYSTEM REQUIREMENTS 

The preceding discussions have shown some major considerations 
involved in the design of a typical AEW system. Numerical examples 
illustrated the various interrelations and were chosen in such a manner as 
to be applicable to the solution of the hypothetical air defense problem we 
have been considering. 

We may now use all of this information to compile an estimate of the 
basic characteristics of an AEW system which represents a reasonable 
answer to the overall system problem. These estimated characteristics may 
then be employed to provide the basic input data needed to specify the AI 
radar and fire-control system. All during this process, we estimate — as 
best we can — the overall system performance to ensure that we do not 
depart from the mission accomplishment objectives. As already mentioned, 
in an actual overall systems study, we would repeat this process several 
times to obtain a better feeling for the trade-offs between the AEW system 
and the AI system. However, for all cases, the basic considerations and 
the method of attack on the problem would remain very much the same; 
only the assumed system logic and specific parameter values would undergo 
appreciable change. 



2-20] SUMMARY OF AEW SYSTEM REQUIREMENTS 97 

One important characteristic of a systems problem has been implied by 
the foregoing discussion; it is worth mentioning explicitly at this point to 
impress the reader with its importance. In a systems study designed to 
derive basic system requirements, it is often necessary to make arbitrary 
decisions on the basis of incomplete quantitative results. The system logic 
described in Paragraph 2-1 1 for the hypothetical AEW system is an example 
of such a decision. Many choices could have been made; however, in order 
to get on with the problem one choice had to be made and then followed to 
its logical conclusion. Conceivably, it would develop that this was the 
wrong choice, in which case we should have to repeat the entire process 
for a more satisfactory initial hypothesis. 

Keeping in mind the provisional nature of a system specification at this 
stage of the analysis, we may specify the basic parameters of the AEW 
system as in Table 2-2. The number of the paragraph which discusses 
each parameter is included for convenience. 



Table 2-2 TENTATIVE AEW RADAR PARAMETERS 

Detection Range 90 per cent probability of detection at 150 n.mi. (Paragraph 

2-12) 

Number of Targets 20 hostile targets, 40 interceptors (Paragraph 2-13) 

Threat Evaluation Range 125-150 n.mi. (Paragraph 2-11) 

Nominal Vectoring Range 75 n.mi. (Paragraph 2-13) 

Azimuth Coverage 360° (Paragraph 2-11) 

Elevation Coverage ..... AS° up, 18.3° down. Operation at 20,000 feet (Para- 
graph 2-18). 

Range Resolution 1 n.mi. (Paragraph 2-13) 

Angle Resolution 5° maximum (Paragraph 2-13) 

Range Accuracy a a = 0.25 n.mi., rms (Paragraph 2-15) 

Angular Accuracy (Ta = 1-25° rms (Paragraph 2-15) 

Quantization Levels (Paragraph 2-15) 

AEW Azimuth 1.25° 

AEW Range 1 n.mi. 

CIC System 1.0 n.mi. 

Stabilization Errors Less than 10 per cent of measurement error (Para- 
graph 2-15) 

Time Belay Errors Less than 10 per cent of measurement error (Para- 
graph 2-15) 



98 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Total System Position Error (Paragraph 2-15, Equation 2-8) 

At 75 n.mi. 

or = 2 n.mi. rms (5° beam). 
At 150 n.mi. 

(It = 4.2 n.mi. 

Height-Finding Radar (Paragraph 2-19) 

Threat Evaluation Range 125-150 n.mi. 

Nominal Vectoring Range IS n.mi. 

Azimuth Coverage ± 80° from aircraft nose 

Elevation Coverage 45° up, 18.3° down 

Beamwidth — Elevation 3.2° 

Height Finding Error 0.5 n.mi. rms or 3000 ft rms 

Beamwidth — Azimuth Approx. 5°, to match fan beamwidth 

Stabilization Data stabilized to within 0.27° rms of true 

vertical 

2-21 EVALUATION OF TENTATIVE DESIGN PARAMETERS 
WITH RESPECT TO THE TACTICAL PROBLEM 

We have discussed the general problems of AEW radar design; we also 
have hypothesized an AEW System which provides answers to certain of 
these problems (detection range, resolution, target counting, etc.). Now 
we shall hypothesize reasonable means for processing the radar information 
to provide headirig and velocity information in the tactical environment 
of the example. The accuracy of the heading and velocity estimates 
obtained — coupled with the position accuracy — form inputs for the study 
of the interceptor system effectiveness. 

Three minutes (180 seconds) are available to evaluate the threat fol- 
lowing detection. From Equations 2-13 and 2-14 we see that the standard 
deviations of the heading and velocity measurements obtained by using 
the position measurements made at the beginning and end of the 3-minute 
interval are 

ayr = ^;^"-""- = 0.0233 n.mi./sec = 142 fps rms (2-27) 

180 sec 

(4.2) (6080) (57.3) „ ^^^ ,. oox 

""'' = (800)(180) = ^^-^ ''''' ^^-^^^ 

where 6080 = conversion factor between knots and fps 

57.3 = conversion factor between radians and degrees. 

This accuracy is sufficient to provide a basis for evaluating the threat 
within the 3-minute period. Actually, the accuracy is somewhat better than 
is indicated by these figures. As already mentioned, the range resolution 
capability of the radar allows fifteen of the twenty targets to be resolved at 



2-21] EVALUATION OF TENTATIVE DESIGN PARAMETERS 99 

the range of required detection. The remaining five would appear as one 
or two large targets until they reached a range close enough to the AEW 
radar to be resolved. Tracking can begin on each of the targets indicated 
by the AEW radar. The average standard deviations of the raid considered 
as a whole would tend to approach the standard deviations of one track 
divided by the square root of the number of separate target tracks. 

By the time the targets have closed to 75 n.mi., it is possible to make 
further refinements in the measurements of target velocities. Two addi- 
tional 3-minute intervals are available for this purpose. Neglecting the 
decrease in position error for each interval and considering that the meas- 
urement made in each interval is independent of the previous measurement, 
the error can be reduced by the square root of 3 by averaging the three 
readings taken over a 9-minute period. This process yields an error of 

142 fps error 142 ^ in i . ^o on\ 

<rvT = I . = = —j= = 82 fps = 49 knots. (2-29) 

VNo. of velocity measurements V3 

Smoothing times consistent with this magnitude are allowable for 
velocity measurements because it is not reasonable to expect large changes 
in target velocity. 

A somewhat different situation attends the measurement of heading. 
The target can make heading changes at a maximum rate of 3° per second. 
Thus, it is not desirable to use long smoothing times for heading informa- 
tion. In fact, a major problem in the design of the data-processing system 
is to choose an observation time and smoothing technique for heading 
information that provide a satisfactory compromise between maneuvering 
and nonmaneuvering targets. This is a complicated problem which cannot 
be considered here in detail. However, the basic nature of the problem 
will be indicated. 

The development so far has considered the very simplest type of heading 
measurement; the target position is measured at two different times, / and 
f + ntsc, and the heading is determined by the direction of the straight-line 
passing through these points (Fig. 2-22). 

At a range of 75 n.mi., with an observation time nisc equal to 60 seconds, 
this technique gives rise to an error (Equation 2-19) in measured heading 
equal to 

(2) (6080) (57.3) ., ^o .. .r,. 

""'^ = (800)(60) = ^^-^ ^^-^^^ 

where the constants 6080 and 57.3 have been previously defined. 

Now, let us assume a scan time isc of 6 seconds. The heading of the 
previous expression was calculated on the basis of information obtained 
from two scans, which we may relate to each other by calling the first scan 
number 1, and the second scan, occurring 60 seconds later, number 11. A 



100 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

similar computation may be made using scan number 2 and scan number 12. 
If the errors for the two computations are independent, we may improve the 
heading approximation by ^2 by sim.ply averaging the two computations to 
yield for a straight-line target: 

a^T = 14.5 /V2= 10.2°. (2-31) 

Such improvement is obtained at the expense of increased dynamic lags 
when the target maneuvers. 

Many other smoothing schemes could be used. However, for present 
purposes, it is reasonable to assume that the hypothetical AEW system can 
provide heading information with an accuracy of the order of 10° (standard 
deviation). The suitability of this figure will depend upon the sensitivity 
of interceptor performance to this figure. 

From the foregoing analysis, the accuracies of the AEW radar system 
with which the interceptor system must be compatible are approximately 

Position Error: or = 2 n. mi. radial error, rms (2-32) 

Velocity Measureynent Error: avr = 50 knots rms (2-33) 

Heading Measurefnent Error: g^t = 10° rms (2-34) 

For the vectoring problem we are interested in the relative position 
inaccuracy between the interceptor and the target. The total relative 
radial position error, <trt, between two objects is 

(TRT = ^^ (TT = 2.8 n. mi. (2-35) 

As will be shown later (Paragraph 2-25) it is convenient to express the 
total relative position error in terms of two components: (1) a component 
(TRR along the line of sight between target and interceptor and (2) a com- 
ponent, (TRa^ normal to the target sight-line, 

where (trr = aRT/yjl = 2 n.mi. (2-36) 

aRa = cTRT/^I2 = 2 n.ml (2-37) 

The position, velocity, and heading information is employed to vector 
interceptors on a collision course with assigned targets (Paragraph 2-11 and 
Equation 2-4). 

2-22 INTERCEPTOR SYSTEM STUDY MODEL 

The design goal for the kill probability of a single interceptor has been 
derived as 0.5. We shall now study the problem of specifying the require- 
ments of an airborne intercept (AI) radar and fire-control system that will 
allow the interceptor to achieve this goal within the limitations imposed by 
other system elements and the operational environment (Step 3, Fig. 2-2). 



2-22] INTERCEPTOR SYSTEM STUDY MODEL 101 

As before, the first step is to formulate a master plan for the analysis. 
This master plan shows the fixed and variable elements of the interceptor 
system problem; it must also show the method by which the problem can be 
handled on a step by step (suboptimization) basis without losing the 
relation of each step to the overall problem. Such a master plan is shown 
in Fig. 2-26. It is merely a variant of the plans for steps 1 and 2 showing 
the details of the interceptor weapons system analysis. 

The system effectiveness goal Pq, and the fixed elements of the system, 
including AEW and vectoring system characteristics, have been derived or 
defined in preceding analyses. These are shown in Fig. 2-26 as providing 
the effectiveness criteria and inputs for the interceptor system analysis. 

The output of the system model is Pa (achieved). The variable elements 
are manipulated in such a manner as to make Pa (achieved) equal Pa 
(required). The combinations of variable element values for which this 
condition is realized form the basis for the interceptor system specification. 

The separate steps of the interceptor system analysis can be derived from 
the basic system logic and a careful consideration of the factors affecting 
each phase of interceptor system performance. The interceptor reaching 
the defense zone goes through three discrete phases in attacking a target 
(see Fig. 2-9): (1) a vectoring phase which terminates in AI radar lock-on, 
(2) a tracking phase which terminates in weapon launch, and (3) a missile 
guidance phase which terminates in the destruction of the target. 

The performance in each phase of operation may be characterized by the 
probability that — for a given set of fixed and variable elements — the 
phase will be completely successful. These probabilities and the factors^^ 
which determine their values are shown in Fig. 2-20 as: 

Pm = probability that the two-missile salvo will kill the specified target 
(already specified as 0.75) 

Pc = probability that the interceptor will proceed from the point of 
AI radar lock-on to a point where the missile salvo may be 
launched with a kill probability of 0.75 

P„ = probability that the vectoring system will operate to bring the 
interceptor to a position and orientation where it may detect, 
identify, and lock on the target with its own AI radar. 

^^Only the most significant factors are shown for this hypothetical example. An actual 
analysis might include many more. However, the same basic model would be applicable and 
the approach to the problem — though more complicated mathematically — could be much 
the same as will be used for this hypothetical example. Another important fact: Often It is 
difficult to establish all of the vital factors affecting a given phase of system operation — some 
of these are products of the analysis itself. This model has considerable flexibility in that such 
additions can be made by simply reanalyzing the phase(s) affected. 



102 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 




2-23] PROBABILITY OF RELIABLE OPERATION 103 

Throughout all of the phases of operation, an equipment failure can cause 
the interception to fail. To account for this, we define a fourth probability: 

Pr = probability that the interceptor system equipment (fire-control 
system, aircraft, communications, etc.) will operate satisfac- 
factorily until weapon impact on the target. 

The interceptor kill probability Po may be defined as the likelihood that 
the complete sequence of events will be completed successfully for any inter- 
ceptor operating under the expected tactical conditions. Mathematically, 
this statement has the form: 

Po (achieved) = PmXPcXP.X Pr. (2-38) 

Thus, the basic model is established. We will now demonstrate how 
quantitative models may be derived and manipulated for each phase of 
operation to produce specifications for the variable elements of the inter- 
ceptor weapons system (AI Radar, Computer, Display, and Missile 
Guidance Tie-in). First, we make an estimate of the expected contribution 
of each phase of system operation to the overall kill probability. For 
instance, in the hypothetical example, we may substitute specified input 
values in Equation 2-38 and write 

Po = 0.50 = O.lSPcPvPr (2-39) 



0.667 = PcPvPr. 

Any combination of P^, P„, and Pr which yields this result will satisfy 
the requirement. For preliminary design purposes, we shall select one of the 
possible combinations to provide a criterion for the performance of each 
phase: 

Pr = 0.85 

P. = 0.95 (2-40) 

Pc = 0.825 

Note that Po = (0.85) (0.95) (0.825) = 0.50. 

This choice is somewhat arbitrary. In an actual analysis a number of 
different combinations might be assumed to establish trade-offs between 
the contributions of each phase of system operation. 

2-23 PROBABILITY OF RELIABLE OPERATION 

The analysis of the tactical situation showed that the total air battle 
lasted less than one-half hour. The total number of interceptors in the air 



104 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

during a battle is 48 — 12 combat air patrol and 36 deck launch — out of a 
total complement of 66. The combat air patrol interceptors maintain 
station for 2.8 hours. 

These considerations coupled with the interceptor kill-probability goal 
are the principal factors that determine required interceptor system 
reliability during the attack operational situation. 

The AI radar and fire-control system may be expected to be the primary 
contributors to interceptor system unreliability — recognizing that the 
guided missiles' reliability has already been included in the specified missile 
kill probability. On this basis, we shall assume for purposes of specification 
that failures in the AI radar and fire control will cause two-thirds of the 
aborts due to equipment failure. Since we specified the overall reliability 
of the interceptor system as 0.85, the reliability requirement for the AI 
radar and fire-control system is 0.90. 

A reliability requirement has little meaning unless the element of time is 
included. Based on the large number of CAP interceptors that must be 
kept continuously aloft, it is specified that the reliability requirements shall 
be met for any 3-hour operating period. Chapter 13 will discuss the 
implications of this requirement, the type of design techniques that must 
be employed to meet it for the defined environmental conditions, and the 
means for determining whether a given radar can meet such a requirement. 

2-24 PROBABILITY OF VIEWING TARGET — VECTORING 
PROBABILITY 

The study plan — as extracted from the master plan of Fig. 2-26 — is 
shown in Fig. 2-27. The object of the study is to derive the combinations 
of the variable elements that will permit achievement of the assumed 
performance goal and to ascertain the sensitivity of vectoring probability 
performance to changes in the system parameters. 

From Fig. 2-26 it can be seen that several variable factors — notably 
lock-on range and look-angle (maximum gimbal angle) — are common to 
conversion and vectoring probability. Accordingly, we cannot develop 
firm requirements for these in this phase of the study. Rather, the results 
will be expressed as a spectrum of possibilities, all of which satisfy the 
viewing probability requirement. Later we shall determine the portion of 
these possibilities which also satisfy the conversion probability require- 
ments. 

Search for the target and its detection obviously must precede AI radar 
lock-on. Thus, these factors are functions of the lock-on range and cannot 
be specified until lock-on range is specified. 

AI radar search data display and stabilization and search doctrine are 
dictated almost entirely by vectoring phase considerations. Thus, these 
may be specified by the analysis of the viewing probability problem. 



2-24] 



PROBABILITY OF VIEWING TARGET — VECTORING 



105 



Probability of 

Viewing 
System Study 

iVIodel 
(Vectoring Prob.) 



Defined by 
Prior Study' 



P, (Achieved) 



P, = 0.95 



Assumed 

System 

Goal 

(Para. 

2.11) 



Fixed Elements 



Vectoring Method 

Vectoring Accuracy 

Assignment Doctrine 

Target Characteristics 

Target Aspect 

Interceptor Char. 

Pilot Characteristics 



Variable Elements 



Al Radar 
Lock-On Range 
Detection Range 
Search Range 
Look Angle 
Display 

Search Doctrine 



Output of Study 



Fig. 2-27 Plan for the Study of Viewing (Vectoring) Probability. 

One of the fixed elements of the problem, target aspect, deserves some 
discussion preparatory to the systems analysis. The target aspect or angle 
off the target's nose at the beginning of vectoring is a function of the 
geometry of the attack situation. Primary emphasis is placed on forward 
hemisphere attacks; the first twenty interceptors are vectored into such 
attacks on the twenty targets. The remaining interceptors are sent to 
back up the first twenty. Some of these will be initially vectored to targets 
that are destroyed by earlier interceptors. In such cases, the interceptor 
will be assigned to a new target in order to utilize fully the total interceptor 
fire power. These attacks may require the interceptor to approach the 
target on the beam or from the rear hemisphere. In addition, some of the 
forward hemisphere attacks will be aborted before missile launching because 
of a failure to see the target or to make the proper conversion. In such cases, 
the interceptor can turn around and employ its speed advantage to attack 
one of the targets from the rear. These considerations indicate that all 
initial angles off the target's nose must be considered. The interceptor kill 
probability should be realized or exceeded for all possible approach angles; 
i.e., the interceptor should have "around the clock" capability. 

Of paramount importance to both the vectoring and conversion phases is 
the manner in which the fixed and variable problem elements combine to 
produce distributions of possible aircraft headings at any point in space. 
We may visualize this problem from Fig. 2-28. At any selected point (R,d) 
relative to the target, the uncertainties of the vectoring system may cause 
the heading of an interceptor passing through that point to assume any 
value within the bounds shown. The spectrum of possible headings usually 



106 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

enjoys approximately a normal dis- 
tribution about some mean heading 
as indicated. If the transition from 
the vectoring phase to the tracking 
phase is made at this point (AI radar 
lock-on), this distribution defines the 
range of initial conditions for the 
conversion phase. In addition, for 
any point in space the distribution 
defines the likely angular positions of 
the target with respect to the inter- 
ceptor flight path. The maximum 
look-angle required for the AI radar 
is largely determined by this consid- 
eration coupled with the viewing 
probability requirement. For exam- 
ple, if the interceptor heading in Fig. 

2-28 were along line OA, a look-angle of approximately 90° would be required 

for the AI radar to "see" the target. 




Fig. 2-28 Distribution of Interceptor 
Headings Due to Vectoring Errors. 



2-25 



ANALYSIS OF THE VECTORING PHASE OF 
INTERCEPTOR SYSTEM OPERATION 



Analysis of the vectoring phase must yield the following information: 

(1) The distributions of aircraft headings as functions of lock-on range 
and angle off the target's nose 

(2) The AI radar characteristics required for compatability with the 
operation of the vectoring phase; i.e. display requirements, 
stabilization requirements, look-angle requirements. 

Vectoring System Logic. The flow of information and allocation of 
function for the vectoring system are shown in Fig. 2-29. This diagram 
expresses the system logic outlined in Paragraph 2-11 for the assumed 
AEW/CIC system. 

System Configuration Parameters. The basic factors governing the 
operation of the vectoring system may be ascertained from preceding 
definitions of target inputs and fixed parameters and the design objectives 
established for the vectoring system. These factors are summarized in 
Table 2-2 and Paragraph 2-21. 

Collision vectoring was specified to minimize the average target penetra- 
tion. The equation defining this vectoring method was derived as 

sin Ld = {VtIVf) smd (2-4) 



2-25] 



ANALYSIS OF THE VECTORING PHASE 



107 





























x> 








<u 








n: 


<1> 


-s. 






u 


M 
















<i> 


< 


-^ 


q: 


B 


-?. 




> 














TO 




O" 


E 












(^ 


1- 




















V 








cS 









OJ o 



apniiiiv 
jojdaojsiui 



§U!PB9H 

JO}daoj9}u| 



P98ds 

J0}d93J9}U| 



w E M 
^ o '^ E 

<t5 ^ 






Sc^ 



108 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

where Ld = the collision course lead angle for perfect collision vectoring 

6 = the angle off the target's nose. 

A plot of the required collision course lead angle versus angle off target's 
nose is shown in Fig. 2-30. The vectoring system computes this angle from 

50 



Qo 40 



;2 30 



q 10 







^^ 




Vp = 1200 fps 
V^ = 800 fps 




/ 


/ 


"\ 


\ 






/ 






\ 




/ 










\ 


/ 










\ 



30 60 90 120 

ANGLE OFF TARGET NOSE, 6 (deg) 



150 



180 



Fig. 2-30 Collision Course Lead Angles Versus Angle off Target's Nose. 

the AEW radar measurements. It transforms this lead angle into a space 
heading command which is transmitted to the interceptor. The pilot flies 
the aircraft so that the heading as measured by the aircraft compass 
corresponds to the vectoring system heading command. 

Distribution of Aircraft Headings due to Vectoring Errors. 

Because of errors in the vectoring system measurements, the commanded 
heading does not always correspond to the correct collision-course lead 
angle. In addition, the ability of the pilot to follow the commanded heading 
is limited by the resolution of his display, compass accuracy, the aircraft 
stability and control characteristics, and the distracting effects of the search 
and acquisition functions he must perform just prior to lock-on. 

The diagram of Fig. 2-31 may be used for an analysis of the heading error 
distributions. The uncertainties of the vectoring system cause errors to 
develop in a sequence that may be examined as follows: 

The interceptor-target sight line established by the vectoring system may 
differ from the true sight-target line by an amount which can be expressed 
approximately as 



2-25] ANALYSIS OF THE VECTORING PHASE 

'Space Reference 



109 




for 



Error 
Signal 



Fig. 2-31 Vectoring Error Geometry. 
A^i = ARa/R 

ARa « R 



(2-41) 



where ARa = component of the relative position error between target and 
interceptor which is normal to the sight line. 

The vectoring system computes a desired interceptor lead angle Lc with 
respect to the erroneous sight-line angle. From Fig. 2-31 the computed lead 
angle is 



dd 



dLn 
dd 



dLp 

dVr 



sin Ld = Vt/^f sin 6 
Then, 

[ l^F cos Ld\ L ^^'' <^°s Ld 



AxPt + 



Fp cos Ld 



(2-42) 

(2-4) 

AFt. 
(2-43) 



110 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 
The commanded heading differs from the correct heading by 

.c = £c + 0, - io = [l + P-'^] A«, + \^^^] ^H 
I ^F COS Ld] L^fcosLdJ 

+ r^^i^lA^.. (2-44) 

l^F cos Ln] 

The error signal presented to the pilot is the difference between the 
commanded heading and the actual heading, 

A^pF - ipFc - ^F. (2-45) 

It is assumed that the pilot follows the commanded heading with an error 
whose standard deviation is 5°. 

The total heading error with respect to the correct heading is then 

\_RFf cos Ld] ll^F cos LdJ [^fcosLdJ 

(2-46) 

since the closing rate, R, may be expressed 



Cf.r 



-R = VtCosO -\- Vf cos Ld. (2-47) 

If the vectoring errors are assumed to be independent, we may write the 
standard deviation of the collision course heading error as 

[V^^rrJ -^[FFCosLn""''') ^\Ff cos Ln''''') + "'H 

(2-48) 

where aa = iARa)/R. 

The evaluation of this expression for various values of lock-on range from 
8 to 30 n.mi. is given in Fig. 2-32 for the estimates of measurement uncer- 
tainty derived for the AEW system (Paragraph 2-21). The curves may be 
interpreted in the following manner. For range to the target R and an 
angle off the target's nose 6: if the proper collision-course lead angle for 
this condition is Ld (Fig. 2-30) then the vectoring errors will cause the 
interceptor lead angles to be normally distributed about the value Ld with 
a standard deviation of cr^.r degrees. The magnitude of the heading error 
increases very rapidly as the range decreases. This will be shown to have 
detrimental effects on the AI radar gimbal angle requirements for short- 
range lock-ons and on the ability to convert a short-range lock-on into 
a successful attack. 

The large magnitude of the heading errors for forward-hemisphere 
attacks is characteristic of any guidance system employing "prediction". 
Collision vectoring is such a system; it attempts to guide the interceptor 
towards a point in space where the target will be at some future time. 



2-26] AI RADAR REQUIREMENTS DICTATED BY VECTORING 111 

28 




30 60 90 120 150 180 

ANGLE OFF TARGET'S NOSE AT LOCK-ON (deg) 

Fig. 2-32 Standard Deviation of Heading Error vs. Angle off Target's Nose at 

Lock-on. 



Prediction guidance systems require the use of velocity as well as position 
information. For this reason they are most sensitive to closing speed. This 
phenomena was indicated in Fig. 2-24. The reader might satisfy himself 
on this point by analyzing the errors for a pursuit vectoring system, i.e., a 
system where the interceptor is commanded to point at the target. This 
analysis would disclose that the heading error distributions for all angles 
are about equal to the tail-chase distributions for collision vectoring. Thus 
the tactical advantage of collision vectoring is bought at the price of 
increased AI radar and vectoring system requirements. 



2-26 AI RADAR REQUIREMENTS DICTATED BY 
VECTORING CONSIDERATIONS 

The vectoring situation gives rise to several requirements that must be 
fulfilled by the AT radar. 



112 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Look- Angle Requirements. If lock-on is to occur at any selected 
range in the 8-30 n.mi. interval, the radar must be able to "look" at the 
target. That is to say, the maximum look-angle of the AI radar antenna 
must be sufficient to encompass the distributions of probable target angular 
positions relative to the interceptor heading. {Look-angle is often referred to 
as gimbal angle or train angle). 

A typical situation is shown by Fig. 2-33. The possible positions of the 
target relative to the interceptor are shown as a distribution of angular 
positions around the lead angle Ld that would exist for perfect vectoring. 



Note: Lq- Collision Course 
Lead Angle 
tg=Lead Angle Limit 

Total Area Under 





LOOK ANGLE, I 

Fig. 2-33 Probability Density Distribution of Target Angular Positions Relative 
to Interceptor Heading. 



For any range and angle off the nose, such a figure could be formulated from 
the data in Figs. 2-28 and 2-32 in the preceding paragraph. The probability 
that the AI radar can look at the target at this range and angle is simply 
the area under the curve that lies between the AI radar look angle limits Lg. 
The look-angles required to ensure that 95 per cent of the targets are 
within the AI radar field of view are displayed in Fig. 2-34. The prices of 
short-range lock-ons and "around the clock" attack capability are evi- 
denced by the large radar gimbal angles required to maintain 95 per cent 
probability. When the lock-on range satisfying the conversion probability 
requirement is found, Fig. 2-34 may be used to determine the AI radar 
gimbal angle dictated by vectoring considerations. 



Display Requirements. From Fig. 2-29, we see that the vectoring 
system transmits required attack altitude, time to collision, and range 
relative to the interceptor — in addition to the heading commands already 
discussed — to provide tactical situation information to the pilot. All the 
vectoring information plus pitch and roll information must be presented on 



2-26] AI RADAR REQUIREMENTS DICTATED BY VECTORING 

100 



o< 80 

LU UJ 

S"^ 60 
cc < 

LlJ O 

!^ 40 
o o 



113 



ii 20 



R= 


8 n.mi.^ 










R = l 


n.mi.O- 
5 n.mi.^X 

u3 




^ 






/^ X 


^ 


\^R=20 


n.mi. 


\ 






y 


\^R=25 
^R=30 


n.mi. 
n.mi. 


\ 


\ 












^ 



30 60 90 120 150 

ANGLE OFF TARGET'S NOSE, d (deg) 



180 



Fig. 2-34 



Maximum Look Angles Required for 95 Per Cent Probability of Seeing 
Assigned Target with Collision Vectoring. 



an integrated display which allows the pilot to fly the aircraft in response 
to the vectoring commands. 

During the last part of the vectoring phase, the pilot must detect and 
acquire the target. The displays required for these functions must also be 
integrated with the other vectoring displays to permit proper utilization of 
the information. The considerations governing the design of a display 
system to meet such requirements are treated in Chapter 12. This is one 
of the most difficult design problems for any radar system; it is particularly 
so for an AI radar because of the limited space and multiplicity of functions 
the pilot must perform. Display integration, like reliability, is easier to 
specify than to achieve. 



Search Volume Requirements. Radar search is accomplished by 
scanning a prescribed volume of space as was shown in Chapter 1 (Fig. 1-1). 
Target position uncertainty relative to the interceptor determines the 
required dimensions of this volume. 

For a given lock-on range, the azimuth look-angle needed to accom- 
modate 95 per cent of the expected tactical situations is shown in Fig. 2-34. 
Since the search and acquisition procedures precede AI radar lock-on, it is 
necessary to ascertain whether a target that is within the field of view at a 
given range would also have been continuously within the field of view at 
greater ranges. An inspection of Fig. 2-32 shows this to be the case. The 
largest gimbal angles are required by the shortest ranges. Thus, only the 



114 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

required lock-on range need be known to specify a maximum gimbal angle 
satisfying search, acquisition, and lock-on requirements. 

The sensitivity of viewing probability to gimbal angle may be obtained 
by examining similar curves for viewing probabilities of 90, 80, and 70 per 
cent (Figs. 2-35 to 2-37). For example, a 67° maximum look-angle is 
required to achieve 95 per cent viewing probability at 10 n.mi. range and 
75° off the target's nose (Fig. 2-34). For a 90 per cent probability under 
the same conditions, a 60° maximum look-angle is required (Fig. 2-35). 
This heavy price suggests that a different allocation of viewing and conver- 
sion probabilities might yield a result nearer the optimum. 

The required elevation angular coverage is determined by the elevation 
uncertainty of the vectoring system. As already derived (Paragraph 2-19), 
the elevation measurement error has a standard deviation of 0.5 n.mi. 
Thus the probability is virtually unity that the target height is within three 
standard deviations (1.5 n.mi.) of the vectoring radar system measurement. 
At a range of 10 n.mi. an AI radar elevation coverage of 17° (0.3 radian) 
is required to encompass this uncertainty. This requirement varies 
inversely with the required lock-on range and may be expressed 

6(7//(57.3) , 
R'l — " ^^^^ 



Search pattern elevation coverage 



(2-49) 



The maximum range dimension Ri of the search volume is the range at 
which search begins. Its value depends on the required lock-on range and 



100 



§1 80 
Q Q 

LU UJ 

<=y^ 60 



UJ O 

en rn 



40 



20 















R = l 
R = 15n. 


R=8 n.rr 
) n.mi.-i 
mi.-v Jy 


'A 


■->- 






^^\^ 


^ 


^ 










^ 


-R = 20n.tT 
-R = 25 n.rr 
-R = 30 n.rr 


i. 
ii. 




V 


y^ 










> 



30 60 90 120 150 

ANGLE OFF TARGET'S NOSE, B (deg) 



180 



Fig. 2-35 M 



aximum 



Look Angles Required for 90 Per Cent Probability of Seeing 
Assigned Target with Collision Vectoring. 



2-26] 



AI RADAR REQUIREMENTS DICTATED BY VECTORING 115 

100 



CC DC 
UJ UJ 

5S5 

LlI 00 

q: < 

LiJ C3 
_l Z 

< CO 

^ u. 
OO 

3e 






60 



40 















R = 
R=l 
R=l 


8 n.mi.-A^ 
n.mi.-O 










5 n.mi.-N^ 


\ 




V 




^ 




■--R=20n. 
^R=25n. 


mi. 
mi. 


\ 


\ 






^R=30n. 


mi. 




\ 



30 60 90 120 150 180 

ANGLE OFF TARGET'S NOSE, 6 (deg) 

Fig. 2-36 Maximum Look Angles Required for 80 Per Cent Probability of Seeing 
Assigned Target with Collision Vectoring. 

100 



Q< 



60 


















R- 












R=l 
R=l 


n.mi.-\s 
5 n.mi.n,\ 


^ 


^ 






/^ 


S 


20 n.mi. 




\ 


V 


r^ 


\^R=25 n.mi. 
^— R=30 n.mi. 






\ 



30 60 90 120 150 

ANGLE OFF TARGET'S NOSE, d (deg) 



180 



Fig. 2-37 Maximum Look Angles Required for 70 Per Cent Probability of Seeing 
Assigned Target with Collision Vectoring. 



the radar characteristics. For most radars, a value of two times the required 
lock-on range is adequate. 



116 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Stabilization Requirements. The angular requirements for the 
search pattern were derived with the tacit assumption that the search 
pattern was space-stabilized in roll and pitch about the aircraft flight line. 
That is to say, the volume of space illuminated by the radar is independent 
of aircraft angles of attack and roll. This assumption results in a con- 
siderably smaller search pattern than would be the case if these motions 
were allowed to displace the search pattern. This effect is illustrated by 
Fig. 2-38. Search pattern stabilization also makes the radar search display 

140 



Q 

oi 100 



60 



40 



20 



























N, 


^Unstabilized Searc 
/ Value Caused by 7C 
/ Roll Angle plus 


1 
)° 










\ 


< 


Angle of Attack 
















^ 














/ 


1 
^Unstabiiized Search 
Value Caused by 
Angle nf Attack 












/ 
























































Stat 


ilized Sec 


irch^ 







20 



40 



60 80 100 120 

ANGLE OFF TARGET'S NOSE (deg) 



140 



160 



180 



Fig. 2-38 Elevation Search Angle Requirements for 10 n.mi. Lock-on (Stabilized 
and Unstabilized Search). 

problem easier to solve, as will be shown in Chapter 8. For these reasons 
it is required that AI radar search pattern be stabilized in roll and pitch 
about the aircraft flight line. 

Summary. A summary of the AI radar requirements dictated by 
vectoring considerations is shown in the overall requirements summary, 
Paragraph 2-30. 



2-27 ANALYSIS OF THE CONVERSION PROBLEM 

The plan for analyzing the conversion problem is shown in Fig. 2-39. 
Analysis of the conversion phase must yield the following information 
relevant to the AI radar design: 



2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



117 



Defined by 
Prior Study" 



Probability of 

Conversion 

System Model 



P (Achieved) 



P, = 0.825 



Fixed Elements 



Heading Errors 
at Lock-on 
Target Characteristics 
Interceptor Charac- 
teristics 

Pilot Characteristics 



Assumed 

System 

Goal 

(Para. 2.22) 



— System 
Deficiencies 



Variable Elements 



Al Radar 

Lock-on Range 
Look Angle 
Stabilization 
Display 
Tracking Ace. 
Dynamic Range 

Attack Doctrine 

Missile Launch 
Requirements 



Study Output 
Fig. 2-39 Plan for the Analysis of Conversion Probability. 

1. The minimum required AI radar lock-on range 

2. The fire-control computer requirements 

3. The attack display requirements 

4. The missile launching and illumination requirements 

5. The radar tracking and stabilization requirements 

Attack Phase System Logic. The flow of information and the 
allocation of function during the attack phase are shown in Fig. 2-40. 
Following AI radar lock-on, the AI radar measures target range, lead angle. 



Target 
Input 



AI 
Radar 



Measured 

Target 

Info 



Aircraft 
Flight 1^— 
Data 



(Speed, Altitude, etc.) 



Fire 

Control 

Computer 



Auxiliary 
Signals 

1 

1. 



Error 
Signal 



Display 



Aircraft 

Control I — I Aircraft 

System 



Aircraft Heading 



Aircraft 
Heading 






Missile 
System 



Fig. 2-40 Interceptor System Logic Diagram During Attack Phase. 



118 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

angular velocity, and range rate along the line-of-sight. This information is 
utilized in conjunction with aircraft flight data (speed, altitude, etc.) to 
compute an attack course that permits the weapons to be launched with a 
high kill probability (see Paragraph 1-4 and Figs. 1-3 and 1-4). Deviations 
between the computed attack course and the actual interceptor flight path 
are presented to the pilot as a steering error signal. The pilot — -or auto- 
pilot — flies the aircraft to reduce the steering error to within the limits 
required by the weapon characteristics. 

Guided Missile Launching Zone Parameters. The allowable 
launching ranges and angular error launching tolerances for the inter- 
ceptor's guided missile may be obtained from a graphical representation of 
the launching problem. This analogue model — shown schematically in 
Fig. 2-41 — utilizes the fixed parameters of the target, interceptor, and 



Launch Point 

fn 



,Maximum-G Missile Trajectory 



-Maximum Missile 
Range Envelope 



Impact Point for 

Straight-Line / 
Target Trajectory^ 


^ 


yfl Target Position 
/ at Launch 

^Maximum-G Target 
Trajectories 



Fig. 2-41 Graphical Determination of Launching Zones. 



guided missile as defined in Figs. 2-5 to 2-7. The launching tolerances 
calculated from this model represent the permissible deviations from perfect 
solutions to the fire-control problem. 

We may construct and employ this model to analyze the problem in the 
following manner: 

STEP 1 . A value of range to impact point that lies between the maximum 
and minimum missile downranges is chosen. A semicircle with 
a radius equal to this range is drawn around the impact point. 

STEP 2. The missile time of flight corresponding to the chosen down- 
range is read from the missile performance diagram. The target 
position at weapon launch can then be plotted P^rff units back 
of the impact point. 



2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



19 



STEP 3. 



STEP 4. 



The target is assumed to have two possible types of trajectories 
during the weapon time of flight: (a) a straight line, and (b) a 
maneuver at the maximum permissible target aircraft load 
factor. These trajectories can be plotted as functions of time 
after weapon launch. 

Now we may plot the missile performance diagram on a 
transparent sheet, using the same scale as the target and firing 
circle diagram. The origin of the missile performance diagram 
is made to coincide with a point on the firing circle. The missile 
performance diagram overlay can then be rotated with respect 
to the target and firing circle diagram to determine the maxi- 
mum aiming errors that would still permit interception of the 
target by the guided missile. An interception is defined as any 
point within the missile performance contour where a missile 
time-of-flight line and the time marker on the target trajectory 
are equal. This procedure may be repeated for a number of 
points on the firing circle. 



STEP 5. The foregoing steps may be repeated for a number of assumed 
ranges-to-impact and for all the assumed altitude and speed 
conditions. Using maximum allowable aiming error as a 
parameter, we may plot range against angle off the target's nose 



50 



40 



30 



20 



10 



V 




10 sec- 


1 
50,000 ft Alti 
Vf^M 1.2 
V^^M 0.8 


tude 


p7 


v\>^ 


Isec 
6 sec 








\ \ 


^^ a12 


tt 


rO 


U / 


\ \ 


^ 


^\ 


^ 


^ 


i%f 


b^v^ 


^^^ 


^ 


erj\ 


K>^J 




^ 


/ / 


§ 


^yi -—-—. 


, 


^ 


■^7 


s 



12 LU 

10 o 

s 

'I 
4 5 

S 




Head-On 



30 



60 



90 120 150 

Beam 
ANGLE OFF TARGET'S NOSE, d (deg) 



180 

♦ 

Tail-On 



Fig. 2-42 Missile Launch Zones and Launching Tolerances; 50,000-Ft Altitude. 



120 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

at launch. The results of such a process as applied to our 
fictitious problem are shown in Fig. 2-42. 

This analysis shows that the allowable launching tolerance varies quite 
widely, depending upon the launching range. The tolerances on heading at 
launch are quite tight for very large or very small ranges. They are 
comparatively liberal for intermediate ranges. For example, if missile firing 
occurs from 20,000 ft range at 90° off the target's nose, an error of 12° is 
permitted. The missile time of flight for this instance is 12 seconds. 

The usable minimum missile launching range is determined by the 
requirement that the interceptor not pass closer than 1000 ft to either the 
impact point or the target in order to preclude self-destruction. Using the 
defined maneuvering capabilities of the interceptor, the minimum launching 
range or breakaway barrier dictated by this requirement may be calculated 
by graphical techniques similar to those used for the latmChing tolerance 
determination. The result of such an analysis (for a nonmaneuvering 
target) is shown superimposed on the missile launch zone diagram (Fig. 
2-42). 

Thus the allowable missile launching ranges and angular aiming errors — 
as limited by the characteristics of the target, interceptor, and guided 
missile and the target avoidance problem — are determined for each angle 
off the target. Note that the allowable angular launching tolerances are 
appreciably smaller than an inspection of only the missile performance 
diagram would indicate — 5° to 10° compared with 10° to 30° for the missile 
itself (Fig. 2-6). This is a typical result of a study which examines the 
guided missile performance in its expected tactical environment. It can 
be seen that the allowable launching tolerances determine the required 
accuracy of the AI radar and fire-control system. This is why the AI radar 
designer must be certain the missile performance is defined for operation 
in the expected tactical environment. 

Fire-Control System Parameters — Attack Doctrine. All the 

basic information needed for fire-control system specification is now 
available. 

The fire-control system must be compatible with five requirements or 
limitations: (1) minimum average penetration distance; (2) "around-the- 
clock" launching capability; (3) collision vectoring; (4) missile launching 
tolerances; and (5) interceptor maneuver limits. 

A modified form of collision guidance — known as lead collision — 
provides a reasonable answer. For any tactical situation this guidance 
system attempts to direct the interceptor on a straight-line course to a point 
where the missiles may be fired with high kill probability. The straight-line 
characteristic reduces penetration, reduces intc-ceptor maneuver require- 
ments, and allows missile launching to take place at any angle off the 



2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



121 



target's nose. In a lead-collision system, missile launching occurs auto- 
matically at such a range that the missile time of flight to the impact point 
equals a preset constant. The value of this constant may be chosen to 
utilize the best characteristics within the allowable launching zones. For 
a given angle off the target, the lead angles required in a lead-collision 
system correspond closely to the collision vectoring lead angles — a fact 
which is helpful in solving the conversion problem. 

Lead-collision geometry is shown in Fig. 2-43. Solution of the fire-control 
triangle yields 



Relative Range at Impact 




V^iT-t,) 



Missile Average Velocity 

Relative to interceptor 

During Time of Flight, ff AV Interceptor 

T = Time to Go Until Impact 

Fig. 2-43 Lead-Collision Geometry: Two-Dimensional. 



R^ y^T cos 6+ VtT cos L^- V^tf cos L (2-50) 

VtT sin d = {V,^T+ V^tf) sin L. (2-51) 

The component of relative velocity along the line-of-sight is 

R ^ -Frcosd - Ff cos L. (2-52) 

The component of relative velocity perpendicular to the line-of-sight is 

Rd = Ft sin d - Fp sin L. (2-53) 

By definition of a lead collision course 

// = a preset constant. (2-54) 

From the definition of missile characteristics for straight-line flight 
(Fig. 2-6) 

F^ =/(Ff, altitude,//). (2-55) 

Thus, for a fixed time of flight and known speed and altitude conditions 

FrJf = Ro = constant. (2-56) 



122 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

Using Equations 2-52, 2-53, 2-56 to eliminate velocity terms in Equations 
2-50 and 2-51 and rearranging terms, we obtain 

R -\- RT- RocosL = (2-57) 

sin L = {RT!Ro) d. (2-58) 

The fire-control system must solve two problems. It must provide (1) a 
signal for automatically firing the missiles at the correct point, and (2) an 
aiming error signal for the pilot or autopilot. 

The i\I radar measures range, range rate, lead angle, and space angular 
velocity of the line of sight (R^f, Rm, L:,t, ^.i/)-'' Aircraft speed and altitude 

may be combined with known missile performance at the preset time of 
flight to obtain Rq (see Equations 2-54 to 2-56). 

The measured target inputs and the computed missile characteristics 
may be substituted into Equations 2-57 and 2-58 to obtain 

„ — Rm + Ro cos L^f ^ , . ... > ,-, -n\ 

Jc — -■ (computed time-to-go until nnpact) {^-^yj 

Rm 

sin Lc = I R.\[-^] 6m (computed correct lead angle). (2-60) 



Firing occurs when 



// (preset). (2-61) 



A steering error signal is obtained by taking the differences between the 
sines of computed and measured lead angles and multiplying this difference 
by a sensitivity factor (Ro cos L) /(Ro -\- VfT). This factor causes the 
computed angular error signal to be a close approximation of the actual 
angular aiming error. Thus, the computed steering error is 

tHc = [R^^ cos Lm\Ri^ + /VTc)][sin L, - sin L.m]. (2-62) 

Both the azimuth and elevation error signals are computed from an 
expression of this form. 

Equations 2-59 to 2-62 define the fire-control and tracking problems that 
are to be solved by the AI radar and fire-control system. The precision 
required of this solution is determined by the angular aiming tolerances 
corresponding to the selected value of preset time-of-flight. 

For the purpose of developing a representative set of accuracy specifi- 
cations we shall select 10 seconds for the preset time of flight //, which 
corresponds to a relative displacement at impact of Ro — 6800 ft. An 
inspection of Fig. 2-42 shows this is a reasonable choice since firing will occur 
near the center of the allowable launch zone for all angles off the nose at 

'-The subscript M denotes a measured quantity. 



2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



123 



firing. The maximum allowable launching error for a 10-second time of 
flight may be plotted from the data of Fig. 2-42 as shown in Fig. 2-44. As 



10 



<o 

Op: 

































n 
































































































































/" 






N 




























/ 










\ 
























/ 














s 




















/ 






















^ 




= 




^ 








































































































- 


Missile Flight Time = 10 sec 
Maneuvering Target 


























































































L_ 









30 60 90 120 150 

ANGLE OFF TARGET'S NOSE (deg) 



180 



Fig. 2-44 Maximum Allowable Launching Errors. 



can be seen, head-on and tail-on attacks impose the most severe require- 
ments upon overall aiming accuracy. 

The functions and overall accuracy required from the fire control system 
have now been defined. The next problem is to specify how this error is 
to be divided among the possible sources of error in the system. 

Fire-Control System Error Specification. The sources of system 
error can be listed as follows: 

(1) AI radar measurements [Rm^ Rm, Om, Lyi) 

(2) Flight-data measurements (altitude, speed) 

(3) Fire-control computation 

(4) Pilot-airframe-display interaction 

There are two general types of errors — predictable bias errors and 
random errors. 

Predictable bias errors arise from the dynamic response characteristics 
of the measuring device. For example, in the lead-collision system specified, 
the variables R, R, L, and 6 can change rapidly as the launching point is 
approached (Fig. 2-45). ^^ Dynamic lags in the measuring devices will cause 
measurement errors whose values may be predicted from a knowledge of 
the input parameters and the dynamics of the measuring device. The 

i-In this application, the system must continue to track after missile launch to provide 
illumination for the missile seeker. Thus, dynamic lags are also important after missile launch 
(r< 10 seconds); it will be noticed in this connection, that the dynamic inputs are quite 
severe for this case. This point will be discussed further in Paragraph 2-29. 



124 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



50 
^^ 40 
- 30 
■ o, 20 

10 



-1200 
-1100 
-1000 
-900 
-800 
-700 



^ 60 

t 50 
o 

-S 40 

^ 30 
o 

£ 20 

^ 10 



:110° 

















































^===== 


__ 









-1.0 
-0.8 o 
-0.6 ^ 

X) 

-0.4 I 
-0.2 



10 20 30 40 

TIME TO GO (seconds) 



50 



60 



:110 



-< — 5,^^ 








A 












/ 




■"■ 




— 6 




/' 












/ 












/ 

























20 30 40 

TIME TO GO (seconds^ 



50 



-1.0 
-2.0 
-3.0 
-4.0' 
-5.0 



60 









y^ 


k* 








y 


/"Range 








^L^ 





, 




V^ 


^ 




Le 


ad Angle 






y^ 

























-S 


40 


III 

















< 


20 


n 




2 


10 





20 30 40 

TIME TO GO (seconds) 



50 



60 



Fig. 2-45 Dynamic Variation of Lead Collision Fire Control Parameters. 

measurement errors affect the firing time and the steering error as calculated 
in the fire-control computer by Equations 2-59 through 2-62. 

Because dynamic lag errors are predictable, it is theoretically possible 
to eliminate them entirely by suitably clever design. However, the more 
usual approach is to limit the magnitude of these errors to some finite value. 
As a general rule of thumb, it is desirable that the total value of the predict- 
able bias error contribution obey the following inequality: 

B <yj2(r (2-63) 

where B = total predictable bias error 



<j == standard deviation of the total random error. 



2-27] ANALYSIS OF THE CONVERSION PROBLEM 125 

and computed quantities in this expression were correct and if the pilot flew 
the aircraft in such a manner as to reduce the computed error to zero, then 

Random errors arise from several main sources. First of all are the 
measurement uncertainties caused by the basic limitations of the measuring 
device. The angular measuring accuracy of a radar, for example, is limited 
by beamwidth as was indicated in the discussion of AEW radar require- 
ments.'^ Mechanical and electrical component tolerances also contribute 
to errors of this type. 

The system noise sources also contribute to random errors. For example, 
the finite dimensions of a radar target introduce time-dependent uncer- 
tainties into the measurements of range and angle (see Paragraph 4-8). 
Similarly the vagaries of airflow past the aircraft may introduce random 
noise errors into flight data measurements. These latter would aff^ect the 
computation of Rq- 

Random aiming errors also are caused by the pilot's inability to guide 
the aircraft on exactly the course indicated by the displayed error infor- 
mation. Paragraph 12-7 will discuss this problem in some detail. Generally 
speaking, however, if the pilot is presented with an error signal which is 
band-limited to about 0.25 rad sec'^ and if the error signal, is contaminated 
by random noise which is bandlimited to about 1 rad /sec, then the pilot can 
steer the aircraft with a random error which has a standard deviation 
approximately equal to the standard deviation of the noise. Thus the pilot's 
contribution to the total aiming error may be written: 

(Tpf = (tn (2-64) 

where cpf = standard deviation of the pilot's flyability error 

o-iv = rms value of the noise on the error signal display. 

To illustrate how the error specification might be developed we shall 
consider two cases: (1) a head-on attack and (2) an attack which begins at 
an angle oflF the target's nose at launch of 80°. 

The method for attacking the problem can be outlined as follows. As 
already mentioned. Equation 2-62 is designed to provide a reasonable 
approximation of the actual heading error. In fact, if all of the measured 

i^Actually, as will be indicated in Chapter 5, the problem is a good deal more complicated 
than is indicated by this statement. Signal-to-noise ratio and observation time also strongly 
affect the angular accuracy. However, for fixed values of these latter parameters, the state- 
ment is substantially correct. 

'^The bandwidth of the error signal depends upon the type of attack trajectory flown. A 
lead-collision course is a straight line; hence the effective bandwidth of the input guidance 
signals is very low. Curved-course trajectories such as lead-pursuit have higher effective 
guidance signal bandwidths. Chap. 12 of the "Guidance" volume of this series presents an 
excellent discussion of the concept of treating a guidance trajectory in terms of its frequency 
spectrum. 



126 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

perfect aiming would result. Practically, however, the measured and 
computed quantities (Ro, Lm, Tc, Lc) are not correct for reasons previously 
discussed. Thus the computed error eH,c differs from the actual aiming 
error. The contribution of each source of error to this difference may be 
expressed. 

Aen = (den,c/dx)Ax (2-65) 

where Aen = the steering error due to error in the quantity x 

den.c/dx = partial derivative of the steering error with respect to the 
quantity x 

Ax = error in the measurement of the quantity x. 

As an example, the sensitivity of the steering error to an error in measured 
lead angle may be derived from Equations 2-62 and 2-59 as: 



den.c/dLM = (deH.c/dLm) + {den ,c / dr ,c){dT .c / dLxj) 



[ Rn cos Lm 1 [" 
F,T+Ro\[ 



r RmG ■ r 
cos L\i : — sm Lm 

Rm 



(2-66) 



It should be noted that the sensitivity is a variable quantity during an 
attack course; it also varies from one course to another. Consequently the 
sensitivities must be examined for the range of attack courses. In this 
discussion we will confine our attention to the two courses assumed (head-on 
and 80° off the nose). 

The values of the input variables and their derivatives are shown in 
Fig. 2-45. The error sensitivity factors for each of the assumed attack 
courses are shown in Fig. 2-46. It will be noted that dynamic variations of 
the input quantities are greatest for the attack which terminates near the 
target's beam; thus, predictable bias errors arising from dynamic lags will 
be greatest for this course. On the other hand, the effect of errors in angular 
rate and lead angle is greatest for head-on attacks. This fact is particularly 
significant because angular rate errors tend to be the most important source 
of system errors. 

Using the foregoing error data, an error specification may be derived in 
the following manner. For a head-on attack, the total system aiming error 
must be held below 7° to ensure that the missile will hit a maneuvering 
target (see Fig. 2-44). For purposes of deriving a tentative specification, 
we may split this error among the various error sources by appropriate 
manipulation of the following expression: 

Total system error = pilot requirement + 2(6e//,c/^>^i)Axi 



+ 2V2[(d6/,.c/a^,)<r.v,P (2-67) 



2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



127 





(08S/S9P) ie/^"'? 

















"- o 3^ 














00 


II II II 


<!> 


> > 1^ 




o -> 


III 

°^^l^ 


SO 





IX> lO ^ 00 CM 

o o o o o 

(§9p/sap) ^e/^"9e 



o o o o 

I I I I 

(y/§3P) °ye/^"9e 




(09s/§9p/§9p)V/^"9e 




(§9p/S9P)*^7er"5e 






in 












■- O CU 






g 

II 


g ^ O 

II II 7i 


11 


>":> kT 



09S/S9P/39P) '^ee/^^^e 




(y/§9P)'^ae/^"3e 




09s/ii/S9p)^ae/^"3e 



128 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 
where pilot requirement = maximum allowable indicated error at firing 
^{deH,cldXi)Axi = summation of predictable bias errors 



2^j'E[{^eH,c /dXi)aXiY = twice the standard deviation of the total 
random error. 

We will assign a value of 2° to the pilot requirement; i.e., the pilot is 
required only to bring the indicated error within a value of 2° to ensure 
successful missile launching. This error is, in effect, treated as an allowable 
predictable bias error and it is desirable that its allowable value be made 
as large as possible because this will reduce the total time needed to reduce 
an initial steering error at lock-on (see Fig. 2-49 below). 

For the head-on case, predictable bias errors due to dynamic lags present 
no problem because the input quantities (R, d, Lm) are relatively constant 
over the entire attack course and the system is relatively insensitive to 
mechanization approximations used in the computation of Rq (relative 
range of the guided missile at impact). Thus, predictable bias errors (other 
than pilot bias) can be assigned a value of zero for the head-on case. The 
remaining error tolerance (5°) can be split up among the sources of random 
error as shown in Table 2-3. It will be noted that no tolerances are given 
for range and time-to-go quantities; their effect on the head-on attack 
problem is too insignificant to provide a satisfactory basis for specification. 

The allowable random angular errors (^m, Lm, and pilot steering) are 
equally divided between the azimuth and elevation channels by dividing 
the total allowable error by -^2. This analysis shows that the radar must 



Table 2-3 



MEASUREMENT ACCURACY REQUIREMENTS 
FOR HEAD-ON ATTACKS 



Source 








Allowable 


of 




Allowable Steering 


Error 


rms Error 


Error 


Sensitivity 


Error Contribution 


Specification 


per Channel 


{=<i) 


{dtH.c/d.i) 


{deu,c/d.i) X 2(7, i 


(Txi 


(Azimuth and 
Elevation) 


Rm 












Rm 












Ro 












T 












Bm 


14.2 


4.25 


0.157sec 


0.11°/sec 


Lm 


0.36 


0.11 


0.15° 


0.11° 


Pilot 


= 1 


2.8 


1.4° 


1° 


Total random error 


= 5° 






Pilot bias 


2° 






Total error 


7° 







2-27] 



ANALYSIS OF THE CONVERSION PROBLEM 



129 



provide angular rate and angle information which has rms errors in each 
channel of about 2 mils /sec (0.11° /sec) and 2 mils (0.11°) respectively. 
Referring to Equation 2-64 and the accompanying discussion, it is also seen 
that the computer filtering system must be designed to limit the rms noise 
on the indicator to a value of about 1.0° rms in order to meet the pilot 
steering accuracy requirement. 

The other attack course (80° off the nose at lock-on) may be analyzed in 
a similar fashion. For this case the maximum allowable error is about 10.7° 
(80° off the nose at lock-on will result in about 90° off the target's nose 
at time of firing). Using the allowable errors already established for the 
head-on case, the values of the allowable predictable bias errors and the 
values of the random range and time errors may be established as shown 
in Table 2-4. It should be emphasized that this allocation can be adjusted 
to suit the designer's convenience, provided the total error allowance is not 
exceeded. 

Chapter 9 will present a discussion of how error specifications and 
dynamic input requirements derived in this manner can be used to dictate 
the detailed requirements of the range and angle tracking loops of the radar. 



Table 2-4 MEASUREMENT ACCURACY REQUIREMENTS 
FOR BEAM ATTACKS 









Allowable 












Random 










Allowable 


Bias Error 




Random 






Predictable 


Contri- 


Predictable 


Bias 


Source 


Sensi- 


Bias Error 


bution 


Bias 


Error 


of 


tivity 


Contri- 


{deH,c/d.i) 


Error 


Specification 


Error 


d^H,c/d:c.i 


bution 


X ld,i 


Specification 


CTxi 


Rm 


0.0014 


0.25 


1.85 


179 


662 


Rm 


0.0086 


0.25 


1.85 


29 


108 


Ro 
T 

(computation) 


0.0023 


0.25 


1.85 


109 


400 


I 


0.30 


1.85 


0.30 


0.925 




Per 












Total 


Channel 


Om 


10 


2 


3.0 


0.2 


0.15 


0.11 


Lm 


0.285 


0.15 


0.086 


0.525 


0.15 


0.11 


Pilot 


= 1 




2.8 


2 


1.4 


1.0 



Units: degrees, seconds, feet. S = 5.2. Vs( )^ = S.S. 



130 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



2-28 



LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS 
DICTATED BY THE CONVERSION PROBLEM 



The establishment of the weapon firing range as a function of target 
aspect angle completes the information needed to calculate lock-on range 
requirements for the conversion probability of 0.825. Fig. 2-47 displays 



Distribution of 
Vectoring Headings 

Perfect 

-Lead Collision 

Course 



f, Contours 



10 sec 
800 fps 
1200 fps 




RANGE (n. mi.) 
Fig. 2-47 Interceptor System Model for Conversion Problem. 



the essential elements of the problem. If lock-on occurs at {R, d) the 
heading error that must be corrected has two components: (1) the vectoring 
uncertainty and (2) the difference between the correct collision-course lead 
angle at {R, d) and the correct lead collision-course lead angle at {R, d). 
We shall assume that the distribution of vectoring errors is centered about 
the correct collision lead angle for point (i?, d). The error which is present 
at lock-on must be reduced below the allowable 2° pilot bias error prior to 
reaching the missile firing range at 10 seconds time-to-go. We shall assume 
that the time available for reduction of the steering error at lock-on is equal 
to the time available to an interceptor passing through the point {R, 6) on 
a lead collision course prior to reaching the missile launch range. Fig. 2-47 
illustrates the situation. Contours indicating the time from {R, d) to missile 
release are shown, as well as a typical heading distribution at {R, d) which 
will arise at lock-on. (The distribution of aircraft headings relative to a 
perfect collision vectoring course is defined in Fig. 2-32.) 

The lead collision-course lead angle is a function of both time-to-go and 
aspect angle. Fig. 2-48 illustrates the variation in lead-collision lead angle 
as a function of time-to-go and aspect angle. For a given {R, d) value, the 
lead collision-course lead angle always is less than the correct collision- 
course lead angle. 

The time required to reduce an initial steering error is shown in Fig. 2-49 
for various initial values of steering error. The primary factor contributing 



2-28] 



LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS 

45 



131 





35 


^ 

3 


30 


LlT 

a 

1 


25 
20 


< 


15 



1 
Collision 


-.niirsfi y 












Lead Angles ~ 

1 


Y 




\ 




me to bo 
f=40 
hf=30 
-f=20 






/ 


^ 




^ 








'/A 






N 


'n, Vi 








l/f 


V 






^ 


L\i 










/// 


/ 








\ 






// 






/F = 1200fF 
^r=800fps 
f =10 sec 


s 




/ 










\ 


\ 


/ 
















\ 



20 40 60 80 100 120 140 160 180 
TARGET ASPECT ANGLE (deg) 

Fig. 2-48 Lead Collision Lead Angles. 



40 



30 



20 



-\ 


^/^ 


"/ 




35° 
30° 
25° 

<~ir\° 


Initial 
Steering 






-C^ 






/ — 15° 
/ — 10° 

. 5° 


E 


ror 




\ 


\ 


^ 


S^ 








^ 


< 


^ 


^ 


^ 


Allowable 
BiasE 


Pilot 
Irror 



15 
TIME (sec) 



20 



25 



30 



Fig. 2-49 Time for Example Interceptor to Reduce an Initial Steering Error. 



to this time is the limitation on aircraft maneuverability (2 g's). Data of 
this type are usually obtained from simulation studies of aircraft-pilot- 
display performance. 

The analysis procedure to determine the probability of conversion is 
indicated by the flow diagram shown in Fig. 2-50. As an example of how 
such a calculation might be made, we may consider the following case. 



132 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 



Choose 
Aspect Angle 



Pick a 
Lock ■ on Range 



Determine 

Collision Course 

Lead Angle 

(Fig. 2-48) 



Determine Vectoring 

Distribution About 

Collision Lead Angle 

(Fig. 2-32) 



1 £ 



Establisii 
Distribution of 
Heading About 
Collision Course 



Establish Distribution 

of Headings About 

Lead Collision Course 

(Fig. 2-51) 



Determine 

Lead Collision 

Lead Angle 

(Fig. 248) 



r 



Establish 

Percentage of 

Distribution Which 

Can Convert 

Probablity of 
Successful Conversion 



Determine 
T-ff 
(Fig. 247) 



^ 



Determine Magnitude 

of Steering Error 
Which Can Be Reduced 
(Fig. 249 ) I 



Fig. 2-50 Conversion Probability Analysis Plan. 



Aspect angle at lock-on = 60° 

Lock-on range = 8 n.mi. 

Collision-course lead angle = 35° 

Vectoring distribution, ae.r = 21.5° 

Lead collision-course lead angle, L = 26.5° 

T - tf = 20.5 sec 

Maximum correctable steering error = 36° 

Fig. 2-51 shows the distribution of heading errors relative to the correct 
collision and lead-collision courses. Since an error of 36° may be corrected, 
any initial heading which results in a lead angle between 62.5° and —9.5° 
may be converted into a successful missile launch. Thus, the probability 
of conversion is equal to the shaded area sh jwn, which may be determined 
as 88.5 per cent. 



2-28] 



LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS 



133 



Limits of Correctable 
Steering Error ±36° 

H- 



R=8 n.mi, 
e.=60° 



Heading Error 
Distribution o-=21,5' 



Area of 
Siiaded Region =Pc 




LEAD ANGLES 



62.5 



Fig. 2-51 Method for Calculating Conversion Probability. 



The calculation of conversion probability by this technique is approxi- 
mate. Certain kinematic effects such as the change of collision-course 
lead-angle with time-to-go and the effects of initial steering error on the 
ultimate attack course flown by the pilot are neglected. Evaluation of these 
effects requires elaborate simulation programs. In a practical case, it is 
usually desirable to investigate these areas by more elaborate techniques. 

This simplified analysis, repeated for many values of lock-on range and 
aspect angle, culminates in curves like those in Fig. 2-52. Notice that as 
one would expect, the head-on attack provides the most stringent require- 
ments for lock-on range. The assumed system requirement stated that the 
conversion probability must be at least 0.825 for any aspect angle. The 
corresponding viewing probability requirement was 0.95. Thus for this 
hypothetical system approximately 10 n.mi. lock-on range is required to 
achieve the requisite conversion probability. 

The look-angle requirements are dictated by vectoring considerations, 
since the collision-course lead-angle is greater than the lead-collision-course 
lead-angle for the same range and aspect angle (see Fig. 2-48). From 
Fig. 2-34 we may determine that a 10 n.mi. lock-on range and a vectoring 
probability requirement of 0.95 combine to dictate a look-angle capability 
of 67°. 



134 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 
100 



90 



> 
o 70 



o 
^ 60 



40 









6=75°-* 


y 




r 


^ 


:^ 


























1 
0=6Os 


/ 




<d 


-6= 
































J 


/ 


i 


J" 


1 1 1 

Probability Goal ^ 

— t-^-H 














■ — 


- - 


— 






i 




— • 


— 




















6=3 


1 


'1 




































/ 






































L 


1 




































// 






































/ 






































/ 






































1 








































































































\a 







































4 6 8 10 12 14 16 18 20 22 

RANGE (n.mi.) 

Fig. 2-52 The Probability of Conversion After Lock-on. 



The discerning reader will note that a trade-off analysis could be made 
between lock-on range and look-angle. For example, a longer lock-on range 
would allow a smaller look-angle. When space in the aircraft nose is at a 
premium, it may be easier to increase lock-on range than to provide large 
look-angles. In addition, the derived look-angle specification (67°) is 
pessimistic. At the aspect angle at which look-angle is critical {d = 75°) a 
lock-on range of 10 n.mi. yields a conversion probability of 100 per cent. 
This reduces the vectoring probability requirement for this attack from 
95 per cent to 78.5 per cent. The look-angle requirement corresponding to 
this vectoring probability may be read from Fig. 2-36 as 53°. This is quite 
a significant relaxation of requirements and illustrates the advantages to be 
gained by examining the interrelationships among the system factors. 

In summary, the lock-on requirements are established by the head-on 
attack situation, and the look-angle requirements are establised by the 
beam aspect approach situation. These requirements are: 



Required lock-on range 



Required look-angle 



10 n.mi. with 90 per cent cumulative 
probability 

±53° in azimuth and elevation 



The look-angle capability must be provided in both azimuth and eleva- 
tion because the aircraft will roll to angles approaching 90° during the 
conversion and vectoring phases. 



2-29] REQUIREMENTS BY MISSILE GUIDANCE CONSIDERATIONS 135 

The required detection range is found by specifying a mean lock-on time 
and adding the range closed between the target and the interceptor during 
this time. For example, the closure rate in a head-on attack is 2000 fps 
(0.33 n. mi. /sec). For mean lock-on times of 6 and 12 seconds the required 
detection ranges are therefore 12 and 14 n.mi. respectively. In each of 
these cases, the required cumulative probability of detection is defined as 
90 per cent.'^ 

2-29 AI RADAR REQUIREMENTS IMPOSED BY MISSILE 
GUIDANCE CONSIDERATIONS 

The requirements dictated by missile guidance considerations can be 
derived from Fig. 2-6 and the previous analysis of the tactical situation. 

The AI radar illuminated the target continuously during the missile flight 
time; the missile seeker tracks the reflected signal and homes on the target 
on a proportional navigation course. From Fig. 2-6 it is seen that if the 
AI radar tracking accuracy is better than 0.35° rms, the AI radar will not 
cause degradation of missile performance. The specified tracking accuracy 
of 0.15° rms is well within these limits. However, dynamic lag errors pose 
an additional complication. The data of Fig. 2-45 show that very rapid 
changes in angular rate and range occur near the end of missile flight 
(T^O). The dynamic responses of the range and angle tracking loops 
must be sufficient to maintain AI radar range lock-on and limit the angle lag 
error. The exact determination of the allowable lag error would require 
a more detailed study of interrelations between the missile seeker and the 
AI radar. However, a value of about 0.25° would represent a reasonable 
estimate. 

The maximum range to the target for which illumination must be 
provided is obtained on the head-on attack (4.4 n.mi.). Fig. 4-6 shows that 
120 kw of peak pulse power is required to ensure seeker lock-on at this range 
with a 24-inch antenna. A larger antenna would reduce the power require- 
ment and vice versa. 

The frequency of the seeker (X band) and the type of seeker (pulse radar 
semiactive) are major factors governing the choice of AI radar frequency 
and type, since a separate illuminating system would have to be provided 
if the two were different. As will be indicated in Chapter 6, the choice of a 

'^The derivation of the radar detection and lock-on requirements made no mention of 
probability. As discussed in Paragraph 2-12, it is customary to express ranges so derived as 
the range at which the radar should have 90 per cent cumulative probability of detection 
(on lock-on). This assumption puts a safety factor into the analysis, since a radar which meets 
this requirement will yield a slightly better probability of conversion than a radar which 
always locked on at exactly 10 n.mi. This comes about because 90 per cent of the lock-ons 
occur at ranges greater than 10 n.mi.; the resulting improvement in conversion probability 
for these cases more than offsets the decreased conversion probability of the 10 per cent which 
occur at ranges less than 10 n.mi. 



136 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 

pulse radar is entirely reasonable for the high-altitude (i.e. clutter-free) 
operation required in this tactical application. 

To assist lock-on of the missile seeker, the AI radar also is required to 
provide range and angle slaving signals to the missile seeker. The angular 
accuracy is not particularly critical, since the missile seeker beamwidth is 
relatively wide, perhaps of the order of 10° to 12°. Range accuracy, on the 
other hand, can be fairly critical. If the missile seeker is assumed to operate 
with a pulsewidth of 0.5 /xsec (250 ft) and a l-jusec (500-ft) range gate, then 
range errors in excess of about 150-200 ft can begin to affect seeker lock-on 
capability. The range error specification previously derived (Table 2-4) 
dictated allowable errors of about this magnitude (bias error plus la value 
of random error). In a practical case, this condition would dictate a more 
comprehensive analysis of seeker AI radar interrelations. 

2-30 SUMMARY OF AI REQUIREMENTS 

Reliability: 90 per cent for 3-hour operation 

Search Pattern: 60° azimuth; 
17° elevation; 
Stabilized in roll and pitch 

Search Range: 20 n.mi. 

Search Display: Vectoring heading command 
Time to collision 
Attack altitude 
Range to target 
Interceptor roll and pitch 
AI radar target detection information 

Detection Range: Yl n.mi. (90 per cent probability) with 6-second 
lock-on time 
14 n.mi. (90 per cent probability) with 12-second 
lock-on time 

Lock-on Range: 10 n.mi. at a closing speed of 2000 fps with 90 per cent 
probability 

Look-Angle: ±60° in azimuth and elevation 

Required Computation: Lead collision (see Equations 2-59 to 2-62) 

Required Accuracies: See Table 2-4, Paragraph 2-28 

Dynamic Inputs: See Fig. 2-45 



2-31] SUMMARY 137 

Maximum Allowable Angle Tracking Lag: 0.25° during missile guid- 
ance phase only 

Stabilization: Compatible with accuracy requirements and maneu- 
vering characteristics listed in Table 2-5 

Display: Steering error signal display 

Aircraft roll and pitch (see Chap. 12) 

Additional tactical information as shown to be necessary 
(see Chap. 12) 

Noise filtered to 1° rms 

Maximum signal information delay 0.5-1.0 second 
Frequency: X band 

Power: Greater than 120 kw peak with a 24-inch diameter antenna 

Radar Type: Pulse 

2-31 SUMMARY 

The foregoing analyses have demonstrated the vast amount of systems 
analysis that must precede the design of a successful airborne radar system 
for a particular application. The length of this chapter is in itself testimony 
to the possible complexities of such analyses. The drawing together and 
rationalization of the important factors of an airborne radar application 
problem is as difficult as it is necessary to proper system design. Moreover, 
work of this nature should continue in parallel with the radar system design 
to ensure that the radar design problem is always viewed in the light of 
the most advanced understanding of the overall weapons system problem. 

Succeeding chapters of this book will break the radar design problem 
into its component parts, with the general objective of showing how each 
element of the radar — transmitter, propagation path, target, receiver, 
data processing, and display — may be related to the overall functions and 
requirements of the system. Where appropriate, the examples developed 
in this chapter will be employed to develop further examples. 



R. S. RAVEN 



CHAPTER 3 

THE CALCULATION OF RADAR DETECTION 
PROBABILITY AND ANGULAR RESOLUTION 

3-1 GENERAL REMARKS 

In establishing the preliminary design of a radar subsystem to meet 
overall weapons system requirements, the designer must first choose the 
basic radar organization or configuration. He then endeavors to select the 
radar parameters so as to provide the required performance with practical 
equipments. In order to do this rationally, he must have reliable methods 
for estimating the performance of hypothetical systems. In this chapter, 
calculations in the critical areas of detection performance and angular 
resolution will be discussed. The former is a particularly complicated area 
of analysis because of the statistical problems introduced by receiver noise 
and target fluctuations. The effects of multiple looks at a target and 
operator performance further complicate the situation. Techniques for 
taking these factors into account for a conventional pulse radar and a pulsed 
doppler radar will be developed. 

The definition of angular resolution and the factors which might act to 
degrade it will be discussed briefly. These factors include the effects of 
unequal target sizes, signal-to-noise ratio, receiver saturation, pulsing, and 
system bandwidth. 

3-2 THE RADAR RANGE EQUATION 

A primary basis for the choice of radar system parameters is the radar 
range equation. In one form or another, this relation gives the power 
received from a radar target or the ratio of this signal power to the power 
of competing noise or other interference from which the signal must be 
distinguished. We shall briefly consider the origin of the range equation. 

We suppose that a radar transmitter radiates power denoted by Pt 
isotropically (uniformly in all directions). At a range R, then, the power 
density or power per unit area will be 

p 

Power density of an isotropic radiator = T~U2' (^-1) 

138 



3-2] THE RADAR RANGE EQUATION 139 

Normally, the transmitter is not an isotropic radiator but possesses a 
directivity or power gain due to the influence of an antenna. The power 
gain on transmission is denoted by Gt and the resulting power density at the 
range R is 

P G 

Power density with an antenna = D = - — ^- (3-2) 

This power is incident upon some sort of target which reflects a portion to 
the receiver. The target will be characterized by an idealized or eff"ective 
cross-sectional area a. This area is defined to reradiate isotropically all the 
incident energy collected. The target cross section will be dependent on the 
radar frequency being used and the aspect from which the target is viewed. 
It is normally determined experimentally and often represents a large 
unknown factor in radar system calculations. By definition, the power 
collected by the target is Da. When this power is reradiated isotropically, 
the power density at the receiver, which is assumed to be located near or at 
the transmitter, is simply 

Power density at the receiver = - — ^r;; = , . ,„„. • (3-3) 

■iirK- (4x)-K* 

The effective area of the receiving aperture is denoted by Jr- The power 
intercepted by the effective area of the receiving antenna is simply the 
product of this area and the power density. The receiving area is related to 
the receiving gain Gr and the wavelength X by the following relation.^ 

^r = ^- (3-4) 

We shall assume, as is normally the case, that the same antenna is used for 
reception and for transmission. In this event, the receiving gain will equal 
the transmission gain or Gr =" Gt — G. 

The power received by the receiver will be simply the product of the 
power density at the receiver and the receiving area. Combining Equations 
3-3 and 3-4, the received signal power will be 

Received signal power = ^S' = , yn^ (3-5) 

This expression represents one version of the radar range equation. It 
shows how the received power varies with target range and size and with 
the wavelength and power gain of the antenna. The received power can 
represent either average power or peak power, depending upon what the 
transmitted power Pt represents. 

'See Paragraph 10-1 for a further discussion and references. 



140 THE CALCULATION OF RADAR DETECTION PROBABILITY 

An extensive discussion of target cross section is given in Chap. 4. The 
radar cross section of aircraft targets is discussed in Paragraph 4-7 and some 
typical examples are shown in Figs. 4-20, 4-21, and 4-22. The effective 
cross sections of sea and ground surface reflections are discussed in Para- 
graphs 4-10 through 4-13. In this connection, a normahzed cross section 
is defined as the radar cross-sectional area per unit surface area. This 
quantity is denoted by o-" and is usually referred to as sigtna zero. With the 
illuminated surface area denoted by A, the radar cross section and sigma 
zero are related by 

ex = a'A. (3-6) 

The area of the resolution element on the ground is a function of the pulse 
length, depression angle, and antenna beamwidths and is given by Eq. 
4-60a and b. Examples showing the variation of sigma zero with environ- 
mental conditions and radar frequency are given in Figs. 4-34 through 4-43. 
The radar range equation is often expressed as the ratio of the received 
power reflected from the target to the power of some interfering signal. 
Most commonly, the interfering signal is random noise generated within 
the receiver; it might also be ground or sea clutter, atmospheric reflections 
or anomalies, or some sort of jamming. Internal receiver noise is often 
referred to as thermal noise, not necessarily because it arises physically from 
electronic agitation but because in characterizing it a comparison is made 
with noise which does arise from this source. Normally, internal receiver 
noise determines the maximum range of the radar system; and even when 
other sources of interference predominate, it provides a useful reference 
point. The equivalent input noise power of a receiver is normally expressed 
in the following form.^ 

Equivalent input noise power = A^ = FkTB watts 

= 4 X \0--'FB watts 

where F = noise figure — the factor by which the equivalent input noise 
of the actual receiver exceeds that of an ideal reference 

k = 1.37 X 10-23 joule /°K = Boltzmann's constant 

T = absolute temperature of noise source — arbitrarily, 290° K 

B = equivalent rectangular bandwidth of the receiver in cycles per 
second. 

The ratio of the signal and noise powers as given by Equations 3-5 and 
3-7 yields the signal-to-noise ratio, S /N. 

Signal-to-noise ratio = S/N' = . )3pkRTR'^ ^'^'^■^ 

^See Paragraph 7-3 for a further discussion of receiver noise and the origin of this expression. 



3-3] DETECTION PROBABILITY FOR A PULSE RADAR 141 

This expression is also referred to as the radar range equation. The receiver 
bandwidth B is normally determined by the IF amplifier in pulse radar 
systems, although in some cases subsequent filtering or integration is 
interpreted as equivalent to a narrowing of the noise bandwidth. 

Another convention is to solve Equation 3-8 for the range when the 
signal-to-noise ratio is unity. This range is called the idealized radar range 
and will be denoted by i?o: 

Idealized radar range = Ro = yj ^^^yj^jpp (3-9) 

With this definition, the expression for the signal-to-noise ratio given in 
Equation 3-8 takes the following simple and useful form: 

Signal-to-noise ratio = S/N = (Ro/R)'. (3-10) 

To provide an illustration of the use of Equation 3-9, let us suppose that 
an airborne radar system possesses the following parameter values: 

Pt = peak power = 200 kw a = target cross section = 1.0 m^ 

G = antenna gain = 1000 = 30 db F = noise figure = 10 db 

X = wavelength = 3 cm — 0.03 m 5 = IF bandwidth = 1 Mc/sec 

It is convenient to express each parameter in decibels relative to a con- 
venient set of units and then simply to add these figures with appropriate 
signs to obtain the logarithm of the idealized range, thus: 

Pt = 83.0 db (milliwatts) F = -10.0 db (unity) 

C = 60.0 db (unity) kTB = 114.0 db (milliwatts) 

X2 = -30.5 db (meters^) Ro* = 183.6 db (meters^) 

(T = db (meters^) i?o = 45.0 db (meters) = 3.89 X 

(4ir)3 = -32.9 db (unity) 10* meters 

= 20.4 n. mi. 

3-3 THE CALCULATION OF DETECTION PROBABILITY 
FOR A PULSE RADAR 

Target detection is a radar function of primary importance and a 
necessary preliminary to other important functions such as tracking, 
resolution, and discrimination. In this paragraph, we shall discuss the 
detection process and describe methods for estimating its reliability as a 
function of the radar system and target parameters. Although the develop- 
ments in this paragraph will relate primarily to pulse radar systems, the 
principles apply generally to any type of radar system used for detection. 



142 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



Factors to be considered include the model assumed to represent system 
operation, the effect of the operator, the effect of the target's closing 
velocity, and the effect of fluctuations in target size. 

Model of System Operation. The notation which we shall adopt in 
this paragraph is listed below. 

S /N = signal to noise power ratio 

tsc = scan time 

u = video voltage at pulse inte- 
grator output 

V = video voltage at square law 
detector output 

rj = false alarm number 

6 = antenna beamwidth 

r = pulse length 



a = received signal voltage 
(peak) 

fr = pulse repetition frequency 

A^ = noise power 



n = number of pulses illumi- 
nating a target during scan 

Ro = idealized range 

R = actual range 

R = range rate 

AR = range decrement between 
scans 

S = received signal power fpeak) 

The radar system model providing the basis for our analysis of the 
detection process is shown in Fig. 3-1. The target is assumed to be an 
aircraft at a range R closing on the radar system at a constant range rate R. 



coc = angular carrier frequency 
(RF or IF) 

cos = scan speed 



Target 
(possibly fluctuating) 

Range =R 
Range Rate =R 



Scanning 
Antenna^/\ 



Noise 



\JI 



Predetection Square - Law Pulse Deci 

Amplifier " ■ i-h,_x ^..i-H-r, — 



Integrator 



Threshold 



RECEIVER 



TRANSMITTER 



I Pulse Rate = f , 



Pulse Length = r 



Fig. 3-1 Radar System Model Assumed for Analysis of Detection Process. 

Two cases are distinguished: (1) a target of constant size and (2) a target 
whose size fluctuates in accord with a Rayleigh distribution as is discussed 
in Paragraphs 4-7 and 4-8. 

A pulse radar with a small duty cycle (on the order of a thousandth or 
less) is assumed. The target is illuminated periodically by a scanning 



5-3] DETECTION PROBABILITY FOR A PULSE RADAR 143 

antenna. The antenna pattern is approximated by a constant gain over the 
antenna beamwidth 9, and zero gain outside of this region. The received 
signal on a single scan will consist of n pulses. In the absence of target size 
fluctuations, these pulses will all be of the same size. The number of pulses 
is given by the product of the repetition frequency /r and the beamwidth, 
divided by the scan velocity co^: 

Number of pulses in a scan = n = frQ/cjis- (3-11) 

The received signal is assumed to be a pulsed sinusoid. The signal power 
during a pulse is denoted by S, and the internal noise power referred to the 
same point in the system is denoted by A^. In the case of a fluctuating 
target an average signal power S will be defined. 

The essential parts of the receiver for this analysis consist of a predetec- 
tion amplifier, a square law detector, a pulse integrator, and a decision 
threshold. 

The predetection amplifier is normally the intermediate frequency (IF) 
amplifier, and it is assumed to be matched to the envelope of the pulse 
shape. That is, the bandwidth of this amplifier is approximately equal to 
the reciprocal of the pulse length. Noise with a uniform power density is 
assumed to be introduced into the system at the input to this amplifier. 
The power spectrum of the noise at the amplifier output will thus be equal 
to the power transfer function of the amplifier. The peak signal-to-noise 
ratio at the output of the predetection amplifier is S jN, as was indicated 
above. 

A square-law detector is assumed to generate a video voltage equal to the 
square of the envelope of the predetection signal plus noise. In this case, 
the development in Paragraph 5-7 is applicable and can be used to establish 
the amplitude distribution and power-density spectrum of the video signal 
plus noise. The assumption of a square-law detector rather than a linear 
detector is primarily for mathematical convenience. It does not represent 
a serious restriction because the basic results are only slightly dependent 
on the detector law. 

The pulse integrator combines the n pulses received during a scan over the 
target. In Paragraph 5-10 it will be shown that the linear operation which 
gives the greatest signal-to-noise ratio for a signal consisting of n pulses 
corresponds to the addition of these pulses to form a sum signal. Accord- 
ingly, in order to provide the greatest possible signal-to-noise ratio at the 
decision threshold — and thus the greatest reliability of detection — these 
n pulses are assumed to be added together by a pulse integrator.^ 

^In many practical sj^stems, integration is provided by the memory of the human operators 
or by retention of the signal on the face of the cathode ray display tube. In such cases, the 
integration is not a perfect summing process, and degradation in the S/N ratio is experienced. 
This degradation is discussed later in this chapter and also in Chapter 12, on radar displays. 



144 THE CALCULATION OF RADAR DETECTION PROBABILITY 

The decision element in the radar system is assumed to be simply a thresh- 
old or bias. When the integrated video voltage exceeds this threshold, 
detection is said to have occurred. When this voltage fails to exceed the 
threshold, no detection occurs. The bias may be exceeded for one of two 
reasons. (1) The integrated video signal-plus-noise may exceed the bias; 
in this case, "target detection" takes place. (2) The integrated noise alone 
may exceed the bias; in this case, a"false alarm" takes place. Fig. 5-16 shows 
how a decision threshold or bias is used to distinguish between the dis- 
tribution of signal plus noise and noise alone. The selection of the threshold 
level thus will represent a compromise between the desire for maximum 
sensitivity to integrated signal plus noise and the system penalties incurred 
by false alarms. 



Method of Analysis. Using the radar system model already described 
and the mathematical theory presented in Chapter 5, we will trace the 
progress of noise and signal plus noise through the elements of the receiver. 
The objectives of this analysis are to derive the target detection and false-alarm 
probabilities as functions of S jN ratio, threshold level, and the amount of 
integration. 

The analysis will be performed for both constant and fluctuating radar 
targets to determine the. probability of detection on a single scan.^ 

Finally, the concept of single-scan probability of detection will be 
employed in Paragraph 3-4 to develop the multiple-scan probability of 
detection for a moving target. This quantity — also called the cmnulative 
probability of detection — is the one most directly related to system perform- 
ance in the tactical-use environment.^ For example, the detection ranges 
specified for the examples in Chapter 2 were expressed in terms of the 
cumulative probability of detection. 

From the standpoint of clear exposition, it is rather unfortunate that 
a true understanding of the radar detection problem is wrapped in com- 
plexities of statistical theory which do not convey to the practicing designer 
a real feel for the problem. The author has attempted to alleviate this 
problem by confining some of the more detailed mathematical derivation 
to Chapter 5; the analysis that follows herein applies some of the results 
of these derivations as they pertain to the assumed model. 

Signal Analysis. As previously mentioned, the input to the square- 
law detector consists of noise with a power spectrum equal to the power 
transfer function of the amplifier and — when a signal is present — a signal 

''This quantity is often called the "blip-scan" ratio or the "single glimpse" detection prob- 
ability. 

5See Paragraph 2-12 and Fig. 2-19. 



3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



145 



with a peak power of S. We are now interested in finding analytical 
expressions for the square-law detector output under the conditions of 
noise-only inputs and signal-plus-noise inputs. 

It is conventional and convenient to approximate the video voltage in the 
absence of the signal by a series of independent samples which are spaced 
at intervals equal to the reciprocal of the predetection bandwidth. Such 
an approximation is shown in Fig. 3-2. This approximation is based upon 




TIME 



Fig. 3-2 Representation of Continuous Video Voltage by a Sequence of Sample 



the famous sampling theorem which states: If a function /{i) contains no 
frequencies higher than W j2 cps, it is completely determined by its ordinates at 
a series of points spaced 1 IW seconds apart. ^ In this connection, the envelope 
of the noise in a predetection band of width W cps can be shown to be 
equivalent to a low- frequency function limited to frequencies less than W 11 
cps, and it can be represented by a series of samples spaced by T = 1 IW 
seconds. 

Since the spectrum of the predetection filter is matched to the spectrum 
of the pulse envelope, it will have a width approximately equal to the 
reciprocal of the pulse length, r. In this case, the samples will be spaced by 
intervals equal to t. It can also be shown by an appropriate application of 
the material in Chapter 5 that the statistical fluctuations in these samples 
are independent. In this case, each sample can be considered a separate 
detection trial, and the input to the decision element during an observation 
period can be regarded as a series of independent trials for which methods 
of analysis are well known. For instance, if the probability of exceeding the 
threshold is ^, the probability of exceeding the threshold at least once in 
m trials is 



Probability of at least one success in m trials 



1 - (1 - pY. (3-12) 



Further, the average number of trials between successes is the same as the 
average number of trials per success, which is equal to the reciprocal of the 
probability on a single trial, 1 \p. When there is no signal present so that 
any exceeding of the threshold represents a false alarm, the number 1 jp 
is called t\i& false -alarm number. 

^C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10-21 (1949). 



146 THE CALCULATION OF RADAR DETECTION PROBABILITY 

The video signal and noise samples are statistical variables. Their 
probability density functions are determined in Paragraph 5-7. We denote 
the video voltage out of the square law detector by v. This voltage is equal 
to the square of the video envelope r in Equations 5-78 and 5-79 in Para- 
graph 5-7. Making the transformations y = r^ ^ = a} 11^ and A^ = cr^ in 
these equations provides the probability density functions of the video 
voltage for signal plus noise and noise alone. 

Probability density video voltage signal plus noise = 

P.M^) = :^ exp 1^2^ - ^j hiyj'h^Sjm (3-13) 

Probability density video voltage noise alone = 

P.(.)=2^exp[^j. (3-14) 

The video voltage when no signal is present is thus represented by a series 
of independent samples at intervals of r = 1 jW which are chosen from a 
statistical population with the probability density of Equation 3-14. When 
the signal is present, the sample is chosen from a population with the 
probability density of Equation 3-13. 

An interpretation of these expressions may be given as follows. For a 
given value of noise power A^ and a given value of signal power S the 
probability that the video voltage will have a value between v and v -]- dv 
may be expressed as Ps+N{v)dv. 

Next we examine the effects of integration. We denote the sum signal 
at the pulse integrator output by u. 

« - ^1 + ^2 + ^3+ ••• + Vn (3-15) 

The components of the sum Vk are independent because they are separated 
in time by the repetition period while the correlation time of the video 
voltage is approximately the pulse length r, which is on the order of micro- 
seconds. 

Probability density functions giving the distribution of the signal plus 
noise and noise alone of the sum signal out of the integrator are required 
in order to determine whether a decision threshold will be exceeded. These 
probability density functions are denoted by 

Probability density integrator output, signal plus noise = Ps-\-n{h) (3-16) 

Probability density integrator output, noise alone = PNi'i)- (3-17) 

The probability density function of the integrator output when a signal 
is present is quite complicated, and we will not attempt a detailed study of 
this function here.'' Some calculations are greatly simplified, however, by 

''For such a study see J. I. Marcum, // Statistical Theory of Target Detection by Pulsed Radar, 
RM-754; and /I Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix, 
RM-753, The RAND Corp., Santa Monica, Calif. 



3-3] DETECTION PROBABILITY FOR A PULSE RADAR 147 

the adoption of a suitable approximation to P s+n{u). Such an approxima- 
tion can be based on the assumption that n(S /N) ^ I. In this case, the 
distribution of u is very nearly normal. This is so because when S /N ^ 1, 
V itself tends to be normally distributed (see Equation 5-80), while with 
n y> I, the distribution of the sum u tends to normality by the central limit 
theorem.^ 

The mean and standard deviation of the video voltage v can be found 
from Equation 5-81 or Fig. 5-12. 



Video d-c voltage = v = 1{N + S) 


(3-18) 


ideo rms a-c voltage = o-^ = 2^N(N -\- 2S). 


(3-19) 



The mean and standard deviation of the sum signal u will be larger by «, the 
number of components in the sum, and by the square root of w, respectively. 

Integrator output, d-c voltage = « = 2n{N + S) (3-20) 



Integrator output, rms a-c voltage = o-„ = 2^1nN(N -i- 2S). (3-21) 

The probability density function of the integrator output for noise alone 
can be established by standard statistical procedures to have the following 
form. 

''"^"^ = Wuhlji^)"' ^~""'- P-22) 

For the statistics-minded, we may note that each variable Vk is given by the 
sum of squares of two independent normal variables Xk andjy/c- Thus, u will 
be the sum of the squares of 2n normal variates, and Pn{u) will be the 
probability density function of a chi-squared distribution with 2n degrees 
of freedom.^ 

The Decision Element. The threshold type of decision element 
assumed for this system corresponds closely to the detection operations 
which would be performed by an automatic system such as might be 
employed in the terminal seeker of a guided missile. In many important 
cases, however, the human operator is the decision element. It is postulated 
that the human operator does something very similar to the threshold type 
of decision element. The functions of the human operator, though, would 
probably deviate somewhat from those performed by an ideal decision 
threshold. For instance, the threshold of human operators appears to vary 

8J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, pp. 46-52, McGraw-Hill Book Co., 
Inc., New York, 1950. 

^See P. G. Hoel, Introduction to Mathematical Statistics, pp. 134-136, John Wiley & Sons, Inc., 
New York. 



148 THE CALCULATION OF RADAR DETECTION PROBABILITY 

randomly from look to look because of their inability to judge accurately 
the signal strength or to remember exactly the threshold level. Their 
average threshold would also tend to increase with fatigue and inattention. 
The net result of these deviations generally seems to be a loss in detection 
efficiency of the human operator in comparison with that of a mathematical 
threshold. This degradation is often introduced through an "operator 
factor" or efficiency factor, ^q. The probability of detection obtained on the 
basis of some threshold assumption is simply multiplied by />(, to give the 
"realistic" probability of detection. Values ranging all the way from 0.05 
to 0.8 have been specified for this factor at one time or another. 

It is quite possible that a degradation of this kind represents certain 
detection operations quite well where the operators become fatigued or 
bored. On the other hand there are many detection situations where the use 
of an "operator factor" is very dubious. One such situation is that of the 
operator of an AI radar on a vectored, 10-minute interception mission. It is 
somewhat ridiculous to suppose that an operator on such a mission would 
completely miss, say, 50 per cent of all targets no matter how brightly they 
are painted on his scope. Another situation where the "operator factor" 
concept is obviously not applicable is in connection with automatic equip- 
ments. Here, the detection is directly accomplished through the use of a 
threshold. 

In this chapter the "operator factor" concept will be abandoned in favor 
of simply introducing an operator degradation of the signal-to-noise ratio. 
This procedure is a standard one,^° and it leads to a somewhat simpler 
formulation of the cumulative probability of detection. A typical value for 
the degradation is given in the footnote reference as 2 db. 

The decision threshold is chosen to give a false-alarm probability, or 
probability of detecting a target when none is present, which is compatible 
with the cost of committing the radar or weapon system to such an alarm. 
When such commitment costs can be established numerically, a selection of 
false-alarm time can be made on the basis of minimizing total costs. Most 
commonly, though, such costs cannot be established and the false-alarm 
time is arbitrarily fixed after a thorough but subjective study of its effect 
on the operational performance of the system. 

The number of independent samples of signal-plus-noise in the false- 
alarm time is called the false-alarm number and is denoted by t?. With 
false-alarm times varying from seconds to hours and pulse lengths varying 
from fractions of a microsecond up to milliseconds, the false-alarm number 
might have approximate upper and lower bounds of 10'- to 10^. The 
probability of having a false alarm on a single trial is the reciprocal of the 
false-alarm number. This probability, the probability that a noise sample 

lew. M. Hall, "Prediction of Pulse Radar Performance," Proc. IRE (Feb. 1956) 234-231. 



3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



149 



will exceed a threshold ^, is given by the integral of the probability density 
function of the sum voltage u of noise alone. 

False alarm probability = - = / P]s[{u)du. (3-23) 

V Jb 

This integral has been evaluated, and the result is shown in Fig. 3-3 for a 
useful range of parameters. 



1000 



500 



100 



50 



20 



10 


















— =;^' 
















aW 
















AW 
















>| 
















# 












A 


f_ 














-^^ 

















/ ^f// 














/ */ff/ 










rj^lO A 


/ , 


//// 










77 = 10%\ 


//. 













7j = 10^V\ 
s \ \ 


V/Z/i 


'// 










r7=iu\0 


w 












------^=- 


5P 














:/—> 


7^ 


r/ ' 












/ / 


/ 


yj 1 












/ / 


1 


n 


False Alarm Number 






I 


// 


't 






' // 


L 


' 


'' False A 


larm Prob. 

1 III 




7 

/ 


/// 


11 











10 20 50 100 200 500 1000 

RELATIVE BIAS LEVEL, b/2 N 



Fig. 3-3 



Relation Between False Alarm Number and Bias Level with a Square- 
Law Detector. 



Single-Scan Probability of Detection. Having chosen the threshold 
level on the basis of a required false-alarm time, the probability of detecting 
a target Pd is found by integrating the probability density function of the 
signal-plus-noise sum voltage 



Probability of detection = Pa 



i: 



S+N 



{u)du. 



(3-24) 



150 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



This integral gives the probability of detection of a nonfluctuating target 
on a single scan. It has been evaluated numerically for a useful range of 
the false-alarm number and the number of pulses integrated. The results 
of this calculation are shown in Fig. 3-4." In this figure Pd is plotted as a 
function of the relative range RjRa whose reciprocal is equal to the fourth 
root of the signal-to-noise ratio. 



0.999 
0.998 
0.995 
0.990 
0.980 

0.950 
0.900 

0.800 
0.700 
0.600 
0.500 
0.400 
0.300 
0.200 

0.100 
0.050 

0.020 
0.010 
0,005 
0.002 
0.001 



n = 10 



n=100 



n=1000 



— 


Ill 


1 


ll 




W \ 






\ 
















11 




1! 




it 








\\\ 


















1 




11 




u 








v\ 


















11 




ill 




u 








\\\ 


















1 




11 




1 


























1 


1 


111 




1 




\ 




w 






















lu 








\ 




\ 






















ilu 








l\ 




\ 






















IW 








\ ^ 




\ 






\ 
















u 








\ 


\ 


\ 






\ 














,\ 


i\ 1 








\ 


\ 


\ 






V 














\ 


\\ 








\ \ 


A 










\- 




r 


— 






l\ 


w 






\\ 


\y 








u 


La 






\ 


\\ 


\ 




\\ 


\ 


\ 


\ 


s \\ 
W A 


.\^ 






\\ 


1 




\ 




\ 


\'\ 




fyT" 


\ 




\ 










\^ 




\i' 


Ki 


:>A 




\ 


\ 


\ 






\ 






4 


\ 


k\ 


\ 


\ 


,\ 


\ 




\ 










-m 


\.\\ 


1 




\ 


V\\\ 


\ 




V\ 






s, 








!MA!!A^l 


\ 


\ 


s Vl- 


\ ^ 


\ 




\ 






\ 








'°sk 


i-\ 


-^ 


°\ 




>\l 


\ 


A 




A 




^ 




^ 



0.6 0.8 1.0 1.2 1.4 1.6 
RELATIVE RANGE, R/Rq 
n=NO. OF PULSES INTEGRATED 
^=FALSE ALARM NUMBER 



1.8 2.0 



Fig. 3-4 Single-Scan Probability of Detection of a Nonfluctuating Target. 



We refer to Pd as the single-scan or single-glimpse probability to dis- 
tinguish it from the probability of detection when multiple looks are 
considered. Another term which is commonly used in this connection is 
blip-scan ratio which refers to the fraction of the time that an operator will 
see a blip in a single scan over the target. 

'^For a wider range of such curves and also a discussion of a method for calculating them, 
see J. I. Marcum, op. cit. 



3-31 DETECTION PROBABILITY FOR A PULSE RADAR 151 

To fix ideas, we consider an example of an AI (airborne intercept) radar. 
The idealized range Ro has been determined in Paragraph 3-3 to be 20.4 
n.mi. The radar parameters of interest are assumed as follows: 

Pulse length = 1 jusec Beamwidth = 4.15° 

Azimuth scan = 120° Scan time = 3 sec 



Elevation scan = 17° 



PRF = 500 pps 



The scan area is approximately 17° X 120° = 2040° squared. The beam 
area is approximately (■7r/4) 4.15^ = 13.6° squared. The number of beam 
areas within a scan area is 2040/13.6 = 150. With a scan time of 3 seconds 
and a PRF of 500 pps, the number of pulses received in a scan over the 
target is 3 X 500/150 = 10. We suppose that a false-alarm time of 100 
seconds is chosen. With the IF bandwidth matched to the pulse width 
(approximately equal to its reciprocal) there will be 10^ independent noise 
samples per second, and the false-alarm nuniber will be 100 X 10^ = 10^ 
Referring to Fig. 3-4, the relative range at which Pd = 0.9 for « = 10 
and 7] = 10^ is determined to be 0.72. The actual range giving 90 per cent 
probability of detection is thus 0.72 X 20.4 = 14.7 n.mi. A similar 
calculation gives the range corresponding to a detection probability of 
10 per cent as 17.5 n.mi. The complete probability of detection curve for 
this example is shown in Fig. 3-5 along with the single scan and cumulative 



Ro 
AR 


= 


20.4 n 
LOn. 


. mi. 
mi. 


















































^ 


■^ 


^ 










N 


^^ 






""1 1 1 1 

■—Single -Scan - 




























^ 








^ 
































N, 








i 






Target 1 




















Fluctuating . 
Target-^ 


/ 




\ 






Y 








1 


























\ 




\ 


\ 






. I , 






1— 






























\ 






s. 






































\ 


\ 




/ 






Target 


































N 


S 


s 
















































\ 


•\ 


\ 


s 














































\ 






aaJ 









10 15 

TARGET RANGE (n.mi.) 



20 25 

Fig. 3-5 Single-Scan and Cumulative Probability of Detection for Text Example. 



probability of detection curves for a fluctuating target when the same basic 
radar parameters are used. The derivation of methods for the calculation 
of these other curves is discussed in the rest of this paragraph and in 
Paragraph 3-4. 

The Effect of Target Fluctuations. The discussion thus far and the 
curves in Fig. 3-4 refer only to the case where the magnitude of the received 



152 THE CALCULATION OF RADAR DETECTION PROBABILITY 

signal is constant. The energy reflected from an aircraft in flight is not 
generally constant. Such an object is a complex reflector of electromagnetic 
waves. As it moves in flight, it vibrates and turns relative to the radar 
system, and various parts of the aircraft reflect signals with more or less 
random amplitudes and with slightly diflPerent doppler shifts. As a result, 
the signal reflected from the aircraft fluctuates and can be represented as a 
noise process. ^^ The power in the signal will be distributed similarly to the 
square of the envelope of narrow-band noise with a probability density 
function of the same form as that in Equation 3-14. This distribution is 
often called a Rayleigh distribution. The probability density function of 
the signal power S will thus have the following form : 

Probability density of signal power from fluctuating target = 

1 

P(S) = -^e-^'s. (3-25) 

The factor S in this expression represents the average signal power. 

Because the rate of turn of aircraft is relatively slow, the spectrum of the 
fluctuations in S is normally less than about 3 cps in width,^^ and S is 
reasonably well correlated over intervals of less than about 50 msec. We 
shall suppose that the observation time is less than this and assume that 
S is constant during a look, but independent from look to look. 

With the signal-to-noise ratio fluctuating from scan to scan, the proba- 
bility of detection will also fluctuate, and an average value of Pd must be 
calculated by weighting the various values of Pd by their probability of 
occurrence. Thus the average probability of det ction is of the form 

Average probability of detection for a fluctuating target = 

Pd= Pd(u,S)PiS)^S. (3-26) 

This integral can be evaluated approximately by making use of the 
previously noted Gaussian approximation to the distribution of integrated 
video signal plus noise. Using the values given in Equations 3-20 and 3-21 
for the mean and variance of the integrated video, the approximate proba- 
bility of detection is 

or, with an appropriate change of variable, (3-27) 

l^A detailed discussion of the fluctuations in apparent size of aircraft is given in Paragraph 
4-8. 

i^See Fig. 4-24 for an example. 



3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



153 



P^{ 



'' - wS 



exp ( - 

■2n{N+S) 



xy2)dx. 



2ylnN{N+2S) 

In Fig. 3-6, Pd{S) is plotted as a function of ^. For small values of S, PdiS) 
is very small, while for large S, Pd(S) is approximately unity. A transition 




S=b/2n -N 

SIGNAL POWER 



Fig. 3-6 Probability of Detection as a Function of the Signal Amplitude. 



occurs when S = b jln — N. The width of the transition region is inversely 
proportional to n. When n is large, then, it is reasonable to approximate 
Pd{S) as zero for S < b jln — A^and unity for S larger than this transition 
point. We shall subsequently indicate that this is a fair approximation even 
when n = \. With this approximation, the integral in Equation 3-26 is 
easily evaluated. 

- - P - - ibllnN - 1) 

P. = (1/6')/ exp (- SIS)ds = exp ^,„ '■ (3-28) 

Jb/2n-N o/l\ 

It is convenient for. some developments to work directly with the range 
to the target instead of the average signal to noise ratio. The expression 
in Equation 3-10 giving the signal to noise ratio as the fourth power of the 
ratio of an ideal range to the actual range provides this relation: 

S/N = (Ro/Ry (3-29) 

In addition, a factor K(n,r]) is defined 

K = (b/2nN - 1). (3-30) 

With this notation, the average detection probability can be represented in 
the following very simple form. 

Pd = ^-if(«/-Ro)\ (3_31) 

The K factor can be evaluated from the data in Fig. 3-3 giving the relation 
between the relative bias, the number of pulses integrated, and the false- 
alarm number. The results of such an evaluation are shown in Fig. 3-7, 
where the K factor is plotted versus the number of pulses integrated for 
representative values of the false-alarm number. 



154 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



N. 


















S\ 


















\\s 


















^X 


















\^ 




v 










2 


% 




\,^ 




Ns 








"'"-77=10 




S 




^N^ 








/- 77-10 






s 


^ s^ 


>v 




. 


y , „ 




\ 




N 'nP 


^S^v 




/. 


;'^ -77=10 




\ 




X ^ 


."0<V 




/ / 


'' -77=10 






S 


^; 


s^NS 


Ss^ 


' / ' 


_ ^ in 








\ 


s \^ 


nN 












X 


\^ 


s^ 


^s , 


/ -77 = 10 










s ^ 




vN^2 














^ 


"v 


V "^>^ 


^3>>1 












\ 




■^z 


'•J>0^^ 














^ 


T?r 


sjfl^^CN 


^ 












r^i iru-^-^^^i 1 Mill 












'^^ 


. "^^ 


-^ 


sw 














■^«. -^K-N^CiC^MI 














-^ 


>sj>::sy:W 
















^ 


^">^^^ 


















^^ " = v 


















^•>: 


















t 


















I. 



2 5 10 20 50 100 200 500 1000 

NO. OF PULSES INTEGRATED,n 

Fig. 3-7 The Factor K{n,r]) as a Function of n. 



It is of interest to note that we can infer from the slopes of the curves in 
this figure the trade-off of signal-to-noise ratio with n, the number of pulses 
integrated. From Fig. 3-7, a typical slope is about — 6 db for a factor of 10 
in the number of pulses integrated. This is equivalent to a variation of K 
with n of the following form: 

A^ ^ «-«-6. (3-32) 

Because the average probability of detection is a function of the ratio 
K/(S/N), a variation in K is equivalent to an inverse variation in the 
average signal to noise ratio. The trade-off between signal to noise ratio 
and n, then, is simply 



S/N^ 



(3-33) 



Because of the rather gross approximation which had to be made to 
obtain Equation 3-31, there is a question about its range of validity. A 
reasonable validation of this equation is obtained by comparing it with some 
examples of exact calculations and observing the error. This is done in 
Fig. 3-8 where the average detection probability as given by Equation 3-31 
has been plotted as a function of the normalized range K^'*{R/Ro). It is 
approximately a straight line on the normal probability coordinates used 
in that figure. Also plotted in Fig. 3-8 are the exact values of Pd for n = 1. 
This is the case when the approximations made introduce the greatest error. 



3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



155 





0.99 




0.98 




0.95 


Q? 


90 


■^ 




1— 






0.80 






O 


0.70 


rr 




U- 


O.bO 


■p- 




O 


O.bU 






h^ 


0.40 


1— 






0,30 



2= 0.10 



0.02 





\ 






















V 










^= In2.^-" 










s 


V 


























•^ 


^ 


























,^, 


^ 


























^ 


^ 




Exact curves 


















\ 








/ 




















\ 




/ 
























V 


























^d 


























\ 




9^ 










































App 


roxi 


Tiatior 


A 


K' 
























\ 


^ 



0.2 



1.6 



0.6 0.8 1.0 ^ 1.2 
NORMALIZED RANGE, K4 (R/Rg) 

Fig. 3-8 Average Detection Probability as a Function of Normalized Range. 



When n = I, the probability density function departs from normality to 
the greatest degree, and the width of the transition region in Fig. 3-8 is 
largest. The approximation will also be poorest when the false alarm 
number is small. Curves are plotted in Fig. 3-8 for the two values r{ = 10^ 
and 17 = 10*. These are considered small values for this parameter. With 
the false alarm number rj equal to 10^, the approximation is already quite 
good for values of Pd greater than 20 per cent. For larger values of r? or n, 
the approximation becomes very good.^*. 

It is of interest to compare the average detection probability on a 
fluctuating target with that obtained on a constant target of the same size. 
The detection probability on a nonfluctuating target was previously 
determined in the case of Ro = 20.4 n.mi., n = 10, and rj = 10^. For the 
fluctuating case, we first determine the value of K from Fig. 3-7 for these 
parameters. This value is found to be 4.5 db or a factor of 2.82. The fourth 
root of K is then 1.3. In Fig. 3-8, the normalized range for Pd = 0.9 is 
found to be 0.58. The actual range giving an average 90 per cent probability 
of detection is thus 

i^More detailed development of these ideas can be found in P. Swerling, "Probability of 
Detection for Fluctuating Targets," Research Memorandum 1217, The RAND Corporation, 
Santa Monica, Calif. (17 March 1954). 



156 THE CALCULATION OF RADAR DETECTION PROBABILITY 

/^9o% = 0.58 X 20.4/1.3 = 9.1 n.mi. (3-34) 

This range is substantially less than the range (14.7 n.mi.) which gives 
Pd = 0.9 in the case of a nonfluctuating target. From Fig. 3-8, the nor- 
malized range giving Pd = 0.1 is 1.23, which yields an actual range corre- 
sponding to an average detection probability of 10 per cent of 19.3 n.mi. 
This value corresponds to 17.5 n.mi. in the nonfluctuating case. Thus, 
while the fluctuations degrade the performance at high probabilities, they 
enhance the performance on small and distant targets. The complete curve 
of detection probability versus range in this case is plotted in Fig. 3-5. 

3-4 THE EFFECT OF SCANNING AND THE CUMULATIVE 
PROBABILITY OF DETECTION 

Because the beamwidth of a high-gain antenna is normally much smaller 
than the search area within which a target might appear, the beam must 
be made to scan over the area. For AEW or ground-mapping systems 
where the beam is narrow in only one dimension, this motion is generally 
very simple, either a wigwag or a complete rotation. For systems where 
the beam is narrow in both azimuth and elevation, the motion of the beam 
can become quite complex. 

The efi^ect of scanning is to provide multiple looks at the target, giving 
multiple chances for detection. In this case, it is the cumulative probability 
of detection which is most significantly related to the tactical use of the 
system. Complex scans can produce a nonuniform coverage of the scan 
area, with holes in the pattern and undesired modulation of the received 
pulse packet. 

Multiple-Scan Probability of Detection. In a typical detection 
situation, the radar will periodically scan the target and there will be a 
number of looks at the target when a detection can be made. Moreover, 
since the target will normally move during the scan time, the average 
signal-to-noise ratio and thus the average single glimpse probability will 
vary from scan to scan. This situation is conveniently described by the 
cumulative probability of detection. When the target is closing on the 
radar, the cumulative probability of detection at a given range is defined as 
the probability that the target is detected on or before reaching that range. 

We shall assume that the radar closes on the target at a constant rate 
— Ry and the scan time t^c is also constant. Thus, the range interval which 
is closed during a scan is given by 

Range decrement = A/^ = —Rtsc (3-35) 

If the first look occurs at the range i?i, then the ^'-th look will occur at the 
range 



3-4] EFFECT OF SCANNING ON DETECTION PROBABILITY 157 

R, = R^- (k - l)AR. (3-36) 

We shall limit our discussion to a consideration of fluctuating targets 
which can be handled rather generally thanks to the simplicity of the 
expression for the average probability of detection (Equation 3-31). At 
each look, the average probability of detection is given by Equation 3-37 
below. 

PdiRi.) = ^-^'(«^/«o)\ (3.37) 

The cumulative probability that a target is detected at the range Rk or 
before is denoted by Pc(Rk) and is given by the well-known expression for 
the probability of at least one success in a sequence of k trials: 

Pc{Rk) = 1 - n[l - P.iRd]. (3-38) 

An additional refinement needs to be introduced. Equation 3-39 implicitly 
assumes that the last look occurred at Rk-. Actually, the last look may occur 
anywhere between Rk and Rk-i = Rk -\- AR with equal probability. That 
is, there will be a random phase between the antenna scan and the relative 
motion of the target. To take this effect into account, an average value of 
the cumulative probability of detection must be computed: 



\rJo 



Fortunately, the calculations shown in Equations 3-38 and 3-39 do not 
have to be carried out every time the cumulative probability of detection is 
desired. With properly normalized variables, a universally applicable series 
of detection curves can be derived. 

In order to do this, a normalized range denoted by p is defined: 

p = K''*(R/Ro). (3-40) 

The normalized range decrement is defined similarly: 

Ap = K''\AR/Ro). (3-41) 

With these definitions, the average single-glimpse probability of detection 
takes the following form: 

Pa = e-"' (3-42) 

With this form for the single-glimpse probability of detection, universal 
curves of the average cumulative probability of detection have been 
calculated on the basis of Equations 3-38 and 3-39. These curves are plotted 
in Fig. 3-9. 

From the appearance of these curves, it would seem desirable to make 
the normalized decrement Ap as small as possible in order to obtain the 



158 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



>- 


1.0 


-J 


0.9 


CD 




^ 


0.8 






ccio." 


0.7 






11 


0.6 
0.5 


S Q 




ID , 


04 


"^ 




LlI 


O.S 






< 




on 


0.2 


> 






0.1 












































K, 


■-"^ 


N 


■^ 


\ 


N 


N 




^^^Ap = 0.4 
^/^Ap = 0.2 
^— An = 0.1 
















K 




\ 




k 


V 




















\ 




V 


A 


A 




\ / 


-Ap = 0.05 
-Ap = 0.025 


















\ 


< 


^ s 


r 


\ 




V 




















\ 


^ 


\ 


\ 




\ 




























\ 


\ 


\, 


\ 


\ 


I 




























\ 


\ 


\ 


\ 


\ 




























\ 


\ 


K 


V 


^\ 






























s 


s 


\^ 


^ 


\ 


































^ 


^ 


^ 









0.4 



0.6 



1.0 



1.2 



1.4 



1.6 



NORMALIZED RANGE, P=K4(R/Ro) 
Fig. 3-9 Universal Curves of Average Cumulative Probability of Detection. 

maximum range. This is only part of the story, though. Normally, 
AR and thus Ap would be made small by decreasing the scan time, which 
in turn is obtained by speeding up the scan. With a higher scan speed, the 
number of pulses returned on a scan over the target is reduced. This 
reduces the factor K approximately through the relation in Equation 3-32. 
The net result is to give an optimum value of scan speed or scan time which 
maximizes the range at which a given value of cumulative detection 
probability is obtained. With a slower scan than this optimum, the target 
closes too much between scans and there will be too few chances to detect it. 
With a faster scan, there are not enough hits per scan. The determination 
of this optimum scan time will be illustrated as part of the following 
example. 

To illustrate the use of the curves in Fig. 3-9, we shall continue with the 
AI radar example which we have previously used in Paragraph 3-3 to 
illustrate the calculation of the single-scan probability of detection for both 
constant and fluctuating targets. We assume that the target closes on the 
radar at 2000 ft /sec or about Mach 2. The scan time was assumed to be 
3 seconds. The range decrement is 3 X 2000 = 6000 ft or 1.0 n.mi. The 
value of K^''^ was previously determined to be 1.3 while the idealized range 
is 20.4 n.mi. The normalized range decrement is thus 



Ap = 1.3 X 



1 
20.4 



0.0635. 



(3-43) 



Referring to Fig. 3-9, the normalized range giving a cumulative proba- 
bility of detection of 90 per cent for Ap = 0.0635 is p = 0.87. The equivalent 
actual range is 



3-4] 



EFFECT OF SCANNING ON DETECTION PROBABILITY 
20.4 



i?9o% = 0.87 X 



1.3 



13.7 n.mi. 



159 



(3-44) 



The complete curve of cumulative probability of detection versus range is 
given in Fig. 3-5 along comparable single-scan curves for both a constant 
and fluctuating target. 

Finding an Optimum Scan Time. In order to demonstrate how 
an optimum scan time would be determined, the calculations made above 
for a scan time of 3 seconds will be repeated for scan times of 1, 2, 5, and 10 
seconds as well. The values of A^, K, K^'^, Ap, p^^% and R^(^% are given in 
Table 3-1. 









Table 3-1 








Scan Time 


n 


K 


A-l/4 


Ap 


P90% 


R^Q%,n.n 


1 sec 


3.33 


6.02 


1.57 


0.0256 


1.025 


13.3 


2 sec 


6.67 


3.63 


1.38 


0.0451 


0.93 


13.7 


3 sec 


10.0 


2.82 


1.295 


0.0635 


0.87 


13.7 


5 sec 


16.67 


2.04 


1.197 


0.0976 


0.785 


13.4 


10 sec 


33.33 


1.305 


1.058 


0.1725 


0.655 


12.65 



The detection ranges given in Table 3-1 are plotted in Fig. 3-10. From 
this figure and the table, it is apparent that the optimum scan time in this 



Id 

14 

il2 

^ - 




































- 


- 


- 






























































o 

2 
n 































































1 2 3 456789 10 

SCAN TIME (sec) 

Fig. 3-10 Example of Detection Range vs. Scan Time. 

case is the original choice of 3 seconds. Another observation which can be 
made in Fig. 3-10 is that the optimum is very broad, and it actually does not 
make a great deal of difference whether a scan time of 2 seconds or 5 seconds 
is selected if the rest of the system is made compatible. This kind of 



160 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



situation is often true in matters of this nature and is often not generally- 
recognized until after elaborate studies have been made, if at all. 

A question which often comes up in connection with discussions of 
beamwidth and scanning is, why scan at all? Why not simply use a wider 
beam and a fixed antenna? This thought has a good deal of merit to it. The 
loss in gain due to the use of a wide beam can be oflFset by the integration of 
a much larger number of pulses, and the actual detection ranges might very 
well be comparable. A narrow beam, though, has other advantages which 
make its use desirable. One of these is that upon detection, the location of 
the target is known at once so that tracking can commence immediately. 
Further, the resolution which can only be provided by a narrow beam is 
often a basic tactical requirement of the system (see Paragraph 2-13). In 
addition, a narrow beam is often required to give sufficient accuracy during 
track or to provide a means for narrowing the scan area and "search- 
lighting" a suspected target. 

Types of Scans. Fig. 3-11 shows some scan patterns which have been 
used with pencil beam systems. The most common type of scan is a simple 



X. 


" 


-^ 


r.x -. - 




X. 


1 


1 , 


" .< ' 






. .^ 


w 


r 




^ 



Multi ■ Bar Raster Scan 
(A) 




Two - Bar Scan with 

Conical Lobing 

(Palmer Scan) 

(B) 



Fig. 3-11 Some Possible Scan Patterns with a Pencil Beam System 




constant-velocity raster scan with a fly-back at the bottom of the pattern. 
With a large area to be covered, up to seven or eight bars might be required. 
Very often the basic scan is modified by a lobing motion. Conical or circular 
lobing may be used during track to generate angular error signals. During 
search the lobing may be left on, either because there is no convenient way 
in which to stop and start the lobing motor or because the larger equivalent 
beamwidth can be utilized to cut down on the number of scan bars. When 
this is done, the circular lobing motion combined with the constant-velocity 
azimuth motion produces a cycloidal scan of the beam centers (Fig. 3-llb). 
This type of scanning motion is often referred to as a Palmer scan because 
of its resemblance to a pennmanship exercise. The cardioid and spiral scans 
shown in Fig. 3-11 represent attempts to minimize the fly-back or dead 
time. They are not generally regarded as normal designs, but may be 
required for some applications. 



3-4] 



EFFECT OF SCANNING ON DETECTION PROBABILITY 



161 



The Number of Pulses per Scan. In the system model adopted in 
Paragraph 3-3, to develop an analytical method for calculating the proba- 
bility of detection, it was assumed that n equal-amplitude pulses were 
received on a scan over a target. With a complex scan and realistic beam 
shapes, the pulses received are not all of the same amplitude; neither is it 
clear just what n should be in many situations. For instance, with the 
Palmer scan illustrated in Fig. 3-11 b, the pulses received on a single scan 
over a target may be grouped into several separate packets by the cyclical 
motion imposed on the basic scan. The grouping and number of pulses in 
the individual packets can then change with the location of the target in the 
scan pattern. 

In order to analyze situations of this nature correctly and in detail, 
extensive analytical investigations are often required. More commonly, 
it is quite adequate to make reasonable approximations which will allow 
the methods developed in Paragraph 3-3 to be applied. This is what we 
shall do here. 

We consider first the problem of estimating the effect of the antenna 
beamshape in a linear scan over a target. We suppose that the antenna 
pattern has a Gaussian shape similar to that defined in Equation 3-45: 

Two-way power pattern of antenna '-^ ^-eVo.ise^ (3-45) 

where Q = angular position of the antenna 

= antenna beamwidth (half-power, one-way). 

We wish to approximate this antenna pattern by a uniform pattern so 
that the results of the preceding paragraph are applicable. This type of 
approximation is indicated in Fig. 3-12. In making this approximation, the 




Fig. 3-12 Rectangular Approximation to Gaussian Beam Shape (Equal Area 
Approximation) . 



total integrated power will be maintained constant. That is, the integral of 
the uniform approximation will be made equal to the integral of the antenna 
power pattern between the effective limits of integration. This will result 
in an equivalent loss in signal power for pulses in the uniform pattern in 
comparison with pulses in the center of the more realistic pattern. This loss 
is referred to as the scan loss. Following current practice, we suppose that 



162 THE CALCULATION OF RADAR DETECTION PROBABILITY 

the effective number of pulses integrated n are those contained within the 

antenna beamwidth. The optimum number of pulses to integrate will differ 

slightly from this.^^ 

The effective power within the antenna beamwidth will be proportional 

to the following integral. 

re/2 

Total received power = / d'-^'/o.ise' ^^ == 0.686. (3-46) 

7 -e/2 

Thus, where the maximum power of the pulses in a Gaussian beam is unity, 
the equivalent power of uniform pulses is only 0.68, giving a scan loss of 
1.7db.i« 

A second problem concerns the number of pulses integrated when the 
scan is complex. Where it is probable that there is a substantial non- 
uniformity in the pulse distribution, a pulse count should be carried out. 
That is, the actual number of pulses returned from typical target locations 
for a sample scan would be determined by counting them. More usually, 
it is adequate simply to use the average number of pulses per scan as was 
done in Paragraph 3-3 for the example illustrating the calculation of the 
single-scan probability of detection. The beam area was divided into the 
scan area to give the number of beams per scan. This number was in turn 
divided into the total number of pulses per scan to yield the received pulses 
per scan. 

3-5 THE CALCULATION OF DETECTION PROBABILITY 
FOR A PULSED DOPPLER RADAR 

With proper interpretation, the methods developed in Paragraphs 3-3 
and 3-4 are applicable to a variety of types of radar systems. To illustrate 
how this can be carried out, we shall develop some of the details of such an 
application to the gated pulsed doppler radar described in Paragraph 6-6, 
whose functional block diagram is given in Fig. 6-25. This type of radar 
transmits pulses at a very high repetition rate in order to avoid doppler 
frequency ambiguities. The duty ratio is also considerably greater than in 
a conventional pulse radar. All the possible target ranges (ambiguous) are 
gated into separate filter banks which cover the spectrum of possible 
doppler frequencies. The filters respond to the fundamental component of 
the gated doppler signal which is received. 

Single-Scan Probability of Detection. The idealized range for this 
type of system is essentially given by Equation 3-9. This is restated in 
Chapter 6 as Equation 6-39 with the effects of the signal and gating duty 

'^L. V. Blake, "The Number of Pulses per Beamwidth in a Scanning Radar," Proc. IRE, 
June, 1953. 

'^A scan loss of L6 db was obtained by L. V. Blake in the paper cited in footnote 15. 



3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 163 

cycles specifically incorporated. For a gated pulse doppler system the noise 
is gated with the same duty factor as the signal so that dg = ds in Equation 
6-39. 

Detecting only the fundamental doppler signal in a filter output corre- 
sponds to the case of detecting a single pulse in Paragraph 3-3. Thus the 
basic single-scan probability of detecting a fluctuating target should be 
given by the expression in Equation 3-31 with the factor K correspond- 
ing to the integration of a single pulse. 

In order to account for certain features of the pulsed doppler system, 
this basic probability must be modified somewhat. This modification is due 
to the straddling of a pulse by contiguous range gates and the eclipsing of 
part of the received pulse by the transmitted pulse. These effects act to 
decrease the single-scan probability of detection from its basic value. This 
reduction is denoted by the factor F(R). Thus the probability of detection 
of a fluctuating target by a pulsed doppler radar can be represented by 

Prf = F(/?)e-^'(«/«o)4. (3_47) 

The Straddling and Eclipsing Factor. Range gate straddling refers 
to the situation when the received signal simultaneously falls within two 
range channels. This situation is illustrated in Fig. 3-13, where the received 



n 



I I I I I I ^ I I I I I I I I I I Ml I 

123455789 123456789 12345 



I 

n 



Transmitted Pulses 

Received Pulses 

Fig. 3-13 Range Gate Straddling. 



pulse lies partly in channel 2 and partly in channel 3. Since the gating of 
only a fraction of the received pulse into a given channel is equivalent to 
decreasing the duty ratio by this factor and since the noise in that channel 
is undiminished, the signal power in the channel will be proportional to the 
square of the fraction of the pulse within the channel gate. Thus, if a 
fraction a of the received pulse falls within gate k and the fraction 1 — a 
falls within gate ^ + 1, the signal power in the first gate will be proportional 
to a^ while that in the second channel will be proportional to (1 — a)^. 
When half the pulse lies in each gate, there will be a loss of 6 db in each 



164 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



channel. Of course, there will be two chances to detect the target. If the 
probability of detecting the target in the first channel is denoted by Pi and 
that for the second channel by P2, the probability of detecting the target 
in at least one of the channels will be 

Pd2 = 1 - (1 - Pi)(l - P,) = P,^ p,- p,p,. (3-48) 

The straddling factor will be periodic in range, with a period equal to the 
pulse length in range units. In the example illustrated in Fig. 3-14, the 



Ro=25 n. mi. 
AR = 0.67 n. mi. 




10 15 

RANGE (n. mi.) 

Fig. 3-14 Sifigle Scan and Cumulative Probabilities of Detection for a Pulsed 
Doppler Radar. 

oscillations of the detection probability with a period of about 1 /12 n.mi., 
which are shown in the expanded view, represent the eflfects of straddling. 
When the received pulse straddles the transmitted pulse, eclipsing is 
produced because the receiver is gated off when the transmitter is on to 
prevent feed-through. The effect of eclipsing is much more severe than that 
•of gate straddling. When the received pulse is centered on the transmitted 
pulse, the signal received and the resulting probability of detection become 
zero and produce a blind range at which the system is completely insen- 
sitive. These blind ranges are periodically spaced at intervals equal to the 
repetition period measured in range units. The nulls in the curve in Fig. 
3-14 at intervals of slightly less than a nautical mile represent the effects of 
eclipsing. 



3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 165 

It is not completely correct to substitute the average values of the 
detection probability in each channel into Equation 2-48 when considering 
a fluctuating target because the signal will fluctuate similarly in the two 
channels. Instead, we should use the procedure previously used in Equation 
3-28 for finding the approximate average value of a single channel to deter- 
mine the average of the two-channel expression given in Equation 3-48. 

Equation 3-28 was derived on the basis of an approximation to the curve 
in Fig. 3-6 that Pa was zero out to the value 6" = KN and unity for higher 
values of S. With this approximation, Pi and P2 become 

^'^""^ X.SyKN/a"- (3-49) 

prn .^2vi~o, ^<AW/(i ~aY 

Adopting these approximate expressions, it is apparent that the product 
P1P2 is equal to Pi when a < 1 /2, and to P2 when a > 1 /2. This observation 
materially simplifies Equation 3-48, since only one term is retained: 

The average value of Pd2 in each case will be of the exponential form first 
given in Equation 3-31: 

- _ exp - K{R/R,)V/i.\ - ar-\ = {P,iy"'-'\ cc<h 

^''- ~ exp - K{R/R,Y{\/a^-) = (Prf)l/«^ a > f ^^'^'^ 

The minimum value is attained when a = \, and the received pulse is 
divided equally between the two channels: 

min Prf2 - P/. (3-52) 

Of more interest is the average value, which can be used as a smooth 
replacement for oscillatory curves similar to that in Fig. 3-14 in many cases. 
The function in Equation 3-51 could be integrated by numerical means. 
It is more expedient, though, and a good approximation to simply use the 
average of 1 /a^, which = 2 in Equation 3-51. Thus, 

ave P,2 - P/. (3-53) 

On the average, then, the effect of straddling can be interpreted as a 3-db 
loss in signal-to-noise ratio. _ 

It should also be noted that Pd2 is of the same general form as Pa itself. 
Thus, if the effects of eclipsing can be neglected, the methods developed 
in Paragraph 3-4 for determining the cumulative probability of detection 
and the normalized curves in Fig. 3-9 are quite applicable. 



166 THE CALCULATION OF RADAR DETECTION PROBABILITY 

An approach similar to that taken in this paragraph should be applicable 
to many similar problems. For instance, a multiple-PRF method of 
determining range in a high-PRF pulse doppler system is described in 
Paragraph 6-G. In order to determine range on a given scan over the target, 
it must be detected in all PRF's, and the return must not be eclipsed nor 
can there be interference with a return from another target with the same 
doppler shift but at a different range. Calculating the probability of 
measuring range in such a situation is quite complicated, but should be 
possible with the methods indicated. 

An Example. The following system parameters of a gated pulsed 
doppler radar are assumed to provide an illustration of the methods under 
discussion. 

Rq = idealized range = 25 n. mi. 

77 = false-alarm number = 10^ 

T = pulse width = l^sec 

fr = pulse repetition rate = 100 kc/sec 

d = duty ratio = 0.1 

n = pulses integrated = 1 

R = closing rate = 0.33 n. mi. /sec 

/sc = scan time = 2 sec 

AR = range decrement = 0.67 n.mi. 

For 7] = 10^ and n = I, the value of K is found from Fig. 3-7 to be 6 db, 
or i^ = 4. The basic single-scan probability of detection of a fluctuating 
target is thus 

p, = ,,-4(ff/25)^^ /^ in n.mi. (3-54) 

This probability has been plotted in Fig. 3-14 as the maximum value o( Pd2- 
Also plotted in that figure are Pd'^ and Pd^ corresponding to the minimum 
and average values of Pd2- The shaded area between the minimum and 
maximum values of Pdi is composed of many oscillations with a period of 
about 1 /12 n.mi. This is illustrated in the expanded view. At intervals 
of slightly less than a mile, one of these oscillations deepens into a complete 
null due to the eclipsing to give a narrow blind region. 

When the effect of the eclipsing is neglected, the cumulative probability 
is easily determined from Fig. 3-8. Remembering that straddling has the 
effect of doubling the effective value of A', the normalized range corre- 
sponding to Pd~ is defined by 



3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 167 

^ - '"{Is) - IT-9- (•«5) 

The normalized range decrement is thus 

The resulting cumulative probability is also plotted in Fig 3-14. 

Postdetection Filtering. It is not uncommon in pulsed doppler 
systems to use a predetection doppler filter which is considerably wider than 
the reciprocal of the observation time of the signal. Subsequent post- 
detection filtering is matched to the signal observation time to provide the 
maximum output signal-to-noise ratio. In this manner the number of 
doppler filters required can be materially reduced at the expense, of course, 
of velocity resolution. The filtering or integration is also somewhat less 
efficient because it is noncoherent representing an operation on the detected 
signal plus noise. 

An exact analysis of postdetection filtering is not possible in general, and 
we shall look for reasonable approximations. Postdetection filtering is 
essentially similar to video pulse integration, whose eff'ect on detection was 
discussed in some detail in Paragraph 3-3, and it is natural to use this 
approach in establishing the approximate effect of this operation. What 
we shall do is to derive an equivalent predetection bandwidth which 
provides approximately the same detection performance as the combination 
of pre- and postdetection filters which it represents. It is assumed that the 
target fluctuates from scan to scan but has a constant size during the 
observation time. 

The following notation is adopted: 

B = predetection bandwidth (band pass) 
^ = postdetection bandwidth (low pass) 
B' = equivalent predetection bandwidth (band pass) 
n = equivalent number of signal samples integrated 

The output of the bandpass predetection filter can be represented by a 
series of samples separated by 1 /B (seconds) as was indicated in Paragraph 
3-3 where the sampling theorem is quoted. Similarly, the output of the 
low-pass, postdetection filter can be represented by a series of samples 
spaced by 1/2^ (seconds). In order to provide signal integration the 
postdetection sampling time will be longer than that of the predetection 
signal. The ratio of these sampling times gives the number of predetection 
samples which are integrated in the postdetection filter: 

Equivalent number of samples integrated = n = Bjlb. (3-57) 



168 THE CALCULATION OF RADAR DETECTION PROBABILITY 

Now in general the predetection signal-to-noise ratio is proportional to 
the reciprocal of the predetection bandwidth: 

S/N'^'^- (3-58) 

Also, the equivalent signal-to-noise ratio of a fluctuating target is 
proportional to a power of the number of video pulses integrated as in 
Equation 3-33: 

S/N ~ ny. (3-59) 

The appropriate power y corresponds to the slopes of the curves in 
Fig. 3-7. 

The equivalent gain in signal-to-noise ratio obtained through postdetec- 
tion integration can now be expressed either as the ratio of the actual and 
equivalent predetection bandwidths or simply as w"^: 



D / D \ 7 

Equivalent gain in S/N ~ "m ~ ^^ = ( ^r ) • (3- 



60) 



The equivalent predetection bandwidth thus is given approximately by 

B' = (2^y B'-y. (3-61) 

In Equation 3-33, the value of 7 was found to be 0.6. If this is stretched 
a point and assumed to be 0.5, the following simple expression is obtained: 

B' = ^|2^f. (3-62) 

This approximation is often used for estimating performance where post- 
detection filtering is involved. 

3-6 FACTORS AFFECTING ANGULAR RESOLUTION 

In many applications, it is required that a radar system be capable of 
separating or distinguishing closely spaced targets. This capability is 
referred to as the resolution of the system. Targets may be resolved on the 
basis of any of their characteristics. Thus they may be distinguished in 
range, velocity, or angular position. This paragraph discusses angular 
resolution. ^^ In ground mapping, the radar's angular resolution provides a 
primary means of target discrimination. In AEW radar systems, the 
angular resolution of the system breaks up multiple target complexes into 
individual components to provide an estimate of the threat. In fire-control 
radar, the angular resolution must be sufficient to separate desired targets 
from interfering targets and clutter. 

I'^A similar discussion can be found in J. Freedman, "Resolution in Radar Systems," Proc. 
IRE 39, 813-1818 (1951), upon which parts of this section are based. 



3-6] FACTORS AFFECTING ANGULAR RESOLUTION 169 

Antenna Pattern Characteristics. Angular resolution is provided 
by the directive properties of the radar antenna. The greater the direc- 
tivity, the better the resolution. 

There is an enormous variety of types of microwave antennas in use 
today. The most widely used in airborne radar systems are those employing 
parabolic reflectors. The discussion will center about this type of antenna 
although many of the observations are applicable to a much wider class. 

Parabolic reflectors can be constructed whose characteristics closely 
approximate those of a uniformly illuminated aperture. The relative 
voltage pattern radiated (or received) by a uniformly illuminated circular 
aperture will have the following form^^ at long ranges. 

r^ , • , 2Ji[(7rD/X) sin^] ,^ ,^, 

Une-way voltage pattern, circular aperture = , ^ .^ , . — - — (3-63) 

(tt/J/a) sm p 

where D = aperture diameter 

X = wavelength 

d = angle relative to aperture normal 

Ji( ) = first-order Bessel function. 

For convenience, we represent the argument of this expression by x so that 
the one-way relative voltage pattern is 2Ji(;c) /x. 

The received voltage reflected from a point target to a uniformly illumi- 
nated circular aperture used both for transmission and reception will be 
given by the square of the function in Equation 3-63 or (2Ji{x) jxY. This is 
also equal to the one-way relative power pattern of such an antenna. This 
pattern is illustrated in Fig. 3-15 where it is referred to as the two-way 
voltage envelope generated by a scan over a single target. 

The antenna beamwidth is normally defined as the width between the 
half-power points of the one-way antenna pattern. This is indicated in 
Fig. 3-15. For a uniformly illuminated circular aperture the beamwidth is 
related to the diameter and wa.elength by 

Beamwidth, circular aperture = 58X/Z) degrees. (3-64) 

The envelope of the received power on the two-way power pattern is 
probably most significant for defining resolution. This is given by the 
square of the envelope plotted in Fig. 3-15 or {2]i{x) IxY. 

The antenna pattern and the beamwidth can be modified by illuminating 
the aperture in a nonuniform manner. A uniform illumination yields one 
of the narrowest beams, but the sidelobe level is relatively high. The 
sidelobes of the one-way power pattern in Fig. 3-15 are down 17.6 db from 
the peak. When the illumination is tapered or stronger in the center of the 

i«J. D. Kraus, Antennas, p. 344, McGraw-Hill Book Co., Inc., New York, 1950. 



170 THE CALCULATION OF RADAR DETECTION PROBABILITY 

Pattern of a Uniformly 

Illuminated Circular Aperture 

2Ji(x) 




-29 -30/2 -9 -0/2 0/2 30/2 20 

Fig. 3-15 Two-Way Voltage Envelope Generated by a Scan over a Single Target. 

aperture than at the edge, the sidelobe level can be minimized, but at the 
expense of a wider beamwidth. In actual practice it is customary to taper 
the illumination so that the effective beamwidth is about 20 per cent greater 
than indicated by Equation 3-64; i.e., the multiplying factor becomes 70 
rather than 58. 

Resolution Criteria. When two targets are separated sufficiently, 
they can be identified as two distinct targets. When they are brought 
together, their returns merge into a single unresolved return. There are a 
number of criteria for deciding just when there are two returns and when 
there is only one. Fundamentally, resolution should be defined relative to 
the discrimination abilities of the human operator in the particular system 
involved. In general, though, this is much too complex an approach because 
of the many factors aflecting human performance, and it is more convenient 
to adopt an arbitrary definition of resolution. In some cases, this will lead 
to a situation where targets which are defined to be unresolved can actually 
be observed as separate entities. Most of the definitions which have been 
suggested for angular resolution lead approximately to the same result: 
targets separated by about 1 beamwidth can be resolved. A beamwidth is 
normally defined as the width between half-power points of the main lobe. 
We shall adopt a very similar definition of resolution which has the con- 
venient virtue of yielding a resolution of 1 beamwidth for a uniformly 
illuminated circular aperture. We shall say that two point targets are 
resolved when the average minimum of the received power envelope in a scan 
over thejn is less than half the power from the maxiynum of the smaller of the two. 

This definition is illustrated in Fig. 3-16, the two-way voltage envelope 
received from two point targets which are just resolved. As indicated in the 
figure, the voltage pattern fluctuates markedly depending upon whether 
the returns are in phase or out of phase. When the received reflections are 



3-6] 



FACTORS AFFECTING ANGULAR RESOLUTION 

, Average 
^Out of Phase 
,ln Phase 



171 




-50/2 -29 -39/2 -9 -Q/2 9/2 9 39/2 29 5Q/2 

Fig. 3-16 Two-Way Voltage Envelope Generated by a Scan over Two Targets 
Separated by One Beamwidth. 



in phase, only a small notch separates the two targets — they have merged 
in a single return. When the received signals are out of phase, there is a 
sharp null midway between the two targets. An average envelope can be 
determined for a random phase between the two reflections. This average 
two-way voltage envelope is also shown in Fig. 3-16. The minimum of this 
average curve is 0.707 of the maxima corresponding to half of the maximum 
received power. Consequently, the case illustrated in Fig. 3-16 shows the 
envelope of two targets which are just barely resolved. These targets are 
separated by a single beamwidth. Thus the definition of resolution adopted 
conveniently yields one beamwidth for two targets of equal size. 

Degrading Influences. In most practical situations the resolution 
will be degraded somewhat by a variety of factors. One such factor is 
unequal strength of the targets. In Fig. 3-17, the two-way voltage envelope 



Target 1 



Target 2 




-39/2 -9 

Fig. 3-17 Average Two-Way Voltage Envelope Generated by a Scan over Two 
Separated Targets of Unequal Size (4 : 1 Power Ratio). 

is shown of two targets whose maximum received voltages have a 2-to-l 
ratio. The radar size of these two targets is normally expressed in terms of 



172 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



the ratio of their reflected powers, which is 4-to-l or 6 db. The minimum 
separating the two targets in Fig. 3-17 is 0.707 of the smallest maximum, so 
that these targets are just barely resolved. The target separation required 
to achieve this resolution is 1.21 beamwidths. Thus, with a 4-to-l size ratio 
for targets, the resolution is 21 per cent greater than for targets of equal 
size. This can become important when the target's size fluctuates randomly. 
Fig. 3-18 shows how the effective resolution angle varies with target power 
ratio. 



1 2.0 


p 


p 






— 


— 1 


— 


— 




n 






n 


~ 


— 




n 


— 


~ 


— 


1 1.8 


















































































1 1.6 


















































































REQUIR 
TION 






































































. 






— ■ 




— 


RATION 
RESOLU 

So 
















_^ 




— - 






























— 


-^ 






























KS 1.0 


^ 




■^ 





































2 4 6 8 10 12 14 16 18 20 
POWER RATIO OF TARGETS —db 

Fig. 3-18 Resolution as a Function of Target Power Ratio. 

Another factor which can affect resolution is the signal-to-noise ratio. 
The simplest way to account for this factor is to apply the already adopted 
definition for resolution to the received signal-plus-noise power envelope. 
The deterioration of angular resolution with signal-to-noise ratio which 
can be determined in this manner is shown in Fig. 3-19. 























\ 




















V 


























■ — 






























Res 


jiution 


with n 


noise 





























2 3 4 5 6 7 
SIGNAL ■ TO ■ NOISE RATIO ~db 



Fig. 3-19 Resolution of Two Equal Targets as a Function of Signal-to-Noise Ratio. 



3-6] FACTORS AFFECTING ANGULAR RESOLUTION 173 

Very large degradations of resolution can often be attributed to non- 
linearities in the receiving system. The dynamic range of many search 
radar systems is less than 10 db above the average noise level, and 20 db 
is rare. The apparent beamwidth when scanning a very strong target with 
a system which has limited dynamic range can be as great as twice the 
normal beamwidth. In such cases, it is quite possible for large targets to 
completely blank out smaller adjacent targets which might have been 
resolved with a linear system. 

Two other minor factors might be noted, the effects of pulsing and 
the system bandwidth. When only a limited number of pulses compose 
the envelope generated by a scan over the target, the exact form of the 
continuous envelope is somewhat indeterminate. As an extreme example, 
if only two pulses are received during a scan over a target, the question 
arises as to whether these are two pulses from a single strong target or from 
two weaker targets. The effect of pulsing can be regarded as a widening of 
the effective beamwidth. Equation 3-65 gives a simple and useful approxi- 
mation for the equivalent effective beamwidth in terms of the actual 
beamwidth and the angular interval between pulses: 

Effective beamwidth = V©' + ^^" (3-65) 

where 9 = antenna beamwidth 

A^ = angular interval between pulses. 

The antenna pattern described by Equation 3-63 and illustrated in Fig. 
3-15 assumed monochromatic radiation. In some applications where very 
wide bandwidths are required, the antenna beamwidth will be modified. 
Such an application might be the use of microwave radiometers for map- 
ping. When there is no chromatic aberration (approximately true when 
a parabolic reflector is used) and the average frequency is maintained 
constant, the increase in beamwidth with bandwidth is small. A maximum 
beamwidth increase of about 5 per cent is given for a bandwidth of 15 
per cent of the average frequency. 



M. K A TZI N 



CHAPTER 4 

REFLECTION AND TRANSMISSION OF 
RADIO WAVES 



4-1 INTRODUCTION 

In the propagation of radio waves between a transmitter and receiver, 
we are interested in the problems associated with power transfer between 
two terminals. This involves an antenna problemi at each terminal (that is, 
the transformation of electrical power into electromagnetic waves or vice 
versa) and the problem of determining how the waves propagate to the 
receiver. In the case of airborne radar, the receiving antenna is replaced by 
the target, and interest is centered in reradiation by the target in the reverse 
direction, back toward the transmitter. This reradiation phenomenon is 
usually called scattering. The radar case with which we shall be primarily 
concerned is a special case of scattering in which the angle between the 
propagation directions of incident and scattered fields is 180°. Scattering 
may be viewed as an antenna problem, too, for the incident field sets up in 
the target currents whose distribution depends on the target material and 
configuration and on the distribution of the incident field. If this current 
distribution is known, then the field reradiated by it can be determined just 
as though that current distribution were set up in an antenna. In propaga- 
tion back from the target to the radar, the scattered wave is involved with 
the same factors as in propagation from the radar to the target: the radar 
problem involves (1) two-way propagation, and (2) back-scattering by the 
target. Thus, in order to predict the strength of the echo received from a 
target it is necessary to determine the characteristics of the propagation 
mechanism and also the back-scattering properties of the target. 

The frequencies normally used for radar operation range from about 
100 Mc/sec on up, or wavelengths of 3 meters down to less than 1 cm (see 
Fig. 1-21). Consequently most targets are many wavelengths in dimension. 
An antenna of corresponding size would have an extremely sharp radiation 
pattern, so that the target, considered as an antenna, has a correspondingly 
sharp scattering pattern. It follows that in general the field scattered 
backward is very sensitive to target orientation. Targets which move, 
therefore, usually give a radar echo which varies with time. Since a 

174 



4-2] REFLECTION OF RADAR WAVES 175 

differential radial movement of a half-wavelength of the target or a portion 
of it is sufficient in many cases to produce a profound variation of echo 
amplitude, even such targets as trees, towers, and buildings, normally 
considered stationary, frequently give fluctuating echoes. For a given 
target, the rate of fluctuation usually will be proportional to radar fre- 
quency. 

The current distribution set up in the target depends on the distribution 
of the incident field. In many common situations, the incident field is 
rather uniformly distributed over the target aperture, so that the target 
may be considered to be illuminated by a uniform plane wave. Then the 
scattering characteristics of the target may be analyzed independently of 
the propagation factors. This is permissible in the case of most airborne 
or elevated targets. More generally, however, the incident field may be 
distributed nonuniformly over the target, because of the nature of the 
propagation phenomena obtaining between the transmitter and various 
portions of the target. A ship is an example of a target in which the incident 
field varies over the target aperture because of the interference between 
direct and surface-reflected rays, which gives a resultant amplitude that 
varies with height. In such cases the scattering properties of the target 
cannot be separated from the propagation factors, so that a specification 
of the target properties becomes more complicated and involves the 
propagation factors. This same type of complication is also involved in 
sea and ground return. 

The principal propagation factors which affect airborne radar are the 
following. 

1. Reflection from the ground 

2. Attenuation by liquid water drops in the air 

3. Absorption by atmospheric gases 

4. Refraction in the atmosphere 

This chapter will be devoted to a discussion of these factors and to a 
description of the characteristics of the principal radar targets of interest 
in airborne applications; viz., aircraft, sea return, ground return, and rain. 

4-2 REFLECTION OF RADAR WAVES 

The radar equation for free space is derived in Chapter 3 (Equation 3-9). 
It may be modified to account for the effect of obstacles such as the earth's 
surface or an inhomogeneous atmosphere by introduction of a quantity 
called the propagation factor., which is the factor by which the free-space 
field is to be multiplied to obtain the actual field. This factor, which we 
denote by F, is a complex quantity, or phasor, representing the modification 
in amplitude and phase of the free-space field by the actual propagation 



176 REFLECTION AND TRANSMISSION OF RADIO WAVES 

process. F may be a function of the range and other parameters of the 
particular situation at hand. Thus, the radar equation becomes 

The quantity a is variously called the radar area, radar cross section, echoing 
area, and back-scattering cross section. 

It is sometimes useful to relate a to another quantity known as the radar 
length, designated by /. This is a phasor which represents the ratio, in 
amplitude and phase, of the back-scattered field-at-unit-distance to the 
incident field strength. Its relation to cr is 

(T = 47r|/r". (4-2) 

The radar length bears a relation to the received field strength similar to 
that of radar area to received power in Equation 4-1. Thus, the received 
field strength Er is given by 

Er = IE,F-'- '-^- (4-3) 

where E^ = the transmitted field at unit distance (the far field extrapolated 
to unit distance from the transmitting antenna) 

K = Itv l\ = phase constant which expresses the relationship 
between distance and the phase angle of a transmission 
of wavelength X. 

The radar area a may be very much larger than the actual projected area 
of the target. This may be shown in the following way. If the target is 
large relative to the wavelength, then it is essentially correct to consider 
that it intercepts a power P' equal to the product of its projected area A' 
and the incident power density Wi, 

P' = A'JV,. (4-4) 

The currents set up in the target by the intercepted field will produce a far 
field which has a certain directive characteristic, just as if the target were 
an antenna with such a current distribution. Hence the target will have a 
directive gain which is a function of angle. If we call the directive gain in 
the radar direction G', then the effective power reradiated backwards will be 

P'G' = A'G'lVi = ctJV,. (4-5) 

Hence 

a - A'G'. (4-6) 

It is obvious that if G' is large, then a will be large relative to the actual 
projected area A' . As an example, consider a target in the form of a flat 



4-2] REFLECTION OF RADAR WAVES 177 

metallic sheet perpendicular to the direction of the incoming wave. If we 
neglect edge effects, the current density is of constant amplitude and phase 
throughout the sheet. Accordingly, the sheet reradiates like an antenna of 
aperture A' with a uniform amplitude and phase distribution. Since the 
gain of such an antenna is 

G' - 47r A'l\^ (4-7) 

(which is large \i A' l\'^ is large) we obtain from Equation 4-6 

(7 = 4x(/f 7X^)2, for A'/\ » 1 (4-8) 

One of the conditions assumed in 
deriving Equation 4-1 is that the x\i- ^^-^ 

variation in range R over the target \P-^^- "" 

results in a negligible variation of ^^-^^ 

the phase of the incident field. In ^^"^ xT 



order to obtain a numerical estimate \" 

of the significance of this limitation, 

we may consider, as an example, an 

airborne search radar viewing, in 

free space, a rectilinear target of 

length 2L at a range R, as illustrated pic. 4.1 Geometry for Limitations of 

in Fig. 4-1. The difference in range Plane-Wave Conditions. 

between a point at x on the target 

and the nearest point of the target is 

Ai? = (i?2 + ;,2)l/2 _ ^ ^ y,2i2R. (4-9) 

Assuming that the antenna may be treated as a point source, the round-trip 
phase difference between the fields reflected back to the source from these 
two points is 

A0 = IkLR = l-KX-'IXR. (4-10) 

From Equation 4-3 the contribution of a differential length of the target 
to the received signal in free space is 

^£. = 1^ ^-^2^(«+^«> --i/ ' (4-11) 

R~ 

where dl = differential radar length of the differential target element dx 
located at a distance x from the center of the target. 

If we denote the plane-wave radar length of the target by / and assume for 
simplicity that the radar length per unit length of the target is constant. 



178 REFLECTION AND TRANSMISSION OF RADIO WAVES 

1 



then, neglecting the slight effect of variations in the — term, the total field 
received will be 



R" 



e-^^'^ dl 






ILR" ' 



(4-13) 



where u = ILKXRy^, and C(u) and S(u) are the Fresnel integrals 
C(u) = / cos (x2V2) dz, 

S(u) = / sin (7r2V2) dz. 
Hence the effective radar length is 

/' = /[C^_,^]. (4.U) 

From Equations 4-2 and 4-14 the effective radar area a' may be derived as 

(4-15) 



C~{u) + S%u) 



Thus the radar length (and radar area a) becomes a function of range, 
especially for targets of great width at short range, the measurements being 




10^ 2 4 7 10'' 2 4 

Fig. 4-2 Radar Cross Section as a Function of Range. 



4-3] EFFECT OF POLARIZATION ON REFLECTION 179 

in terms of the wavelength. Fig. 4-2 shows a plot of a' I a versus i?/X for 
various values of 2Z,/A. 

4-3 EFFECT OF POLARIZATION ON REFLECTION 

Although the majority of radars utilize linear polarization, for certain 
purposes other polarizations are found to be advantageous. The use of 
circular polarization, for example, reduces rain clutter considerably. Since 
any state of polarization may be described in terms of two orthogonal 
polarizations (for example, horizontal and vertical, or right-hand and left- 
hand circular), we may denote an arbitrarily polarized incident wave by 
the matrix 



(f:) 



Er = (V] (4-16) 



in which the orthogonal components £i, E2 are complex quantities, or 
phasors. 

The radar area of a target depends on the polarization of the incident 
wave. A long thin (in comparison to X) wire is a good example, since its 
reflection is very small when the incident field is linearly polarized at right 
angles to the wire axis, and maximum when parallel to the axis. It is 
evident, therefore, that the radar area and radar length are dependent on 
the polarization of the incident field. 

For targets of complex shape, the total field strength incident at a given 
point of the target is the resultant of the primary field from the radar and 
the reradiated fields from other parts of the target. Especially in the case 
of targets of large size which are in part inclined to the wave front, some 
of the latter fields have a component of polarization orthogonal to that of 
the primary field. For targets of symmetrical shape (as viewed from the 
radar) this cross-polarized component balances out in back-scattering, but 
otherwise it usually does not. Hence, in general, the back-scattered field 
has a different polarization from the incident field. The coupling between 
the incident and scattered polarizations depends on the incident polariza- 
tion itself. As a result, the radar length is a tensor quantity, which may be 
written in matrix form as 



in which each of the components /n, etc. is a phasor. For example, if the 
1-polarization is horizontal and 2-polarization is vertical, /n represents the 
radar length of the horizontally polarized echo from a horizontally polarized 
radar, /12 is the radar length of the horizontally polarized echo from a verti- 
cally polarized radar, etc. The reflected field-at-unit-distance is then given 
by 



180 REFLECTION AND TRANSMISSION OF RADIO WAVES 

+ /l2£2\ 



~r '22 £,2/ 



By the reciprocity theorem, /21 = /i2- 

An interesting theorem follows from Equation 4-18: For any given target 
and aspect, there is a polarization of incident field which gives maximuyn echo, 
and another which gives zero echo. This can be seen readily as follows. By 
adjustments of the radar antenna system, the ratio EilE\ may be adjusted 
(in magnitude and phase) until the received polarization is orthogonal to 
that of the receiving system, so no signal will then be received from the 
target.^ Similarly a polarization may be chosen such that the polarization of 
the echo coincides with that of the receiver, so that a maximum echo will be 
received. 

The radar area a also may be written in the form of a matrix by replacing 
the quantities Imn in Equation 4-18 by 

^mn =47r|/™„|2. (4-19) 

Then 



(o"ii cri2\ 
C2I 0'22/ 



(4-20) 



However, one could not deduce the polarization theorem above from this, 
since the radar area is a scalar. 

4-4 MODULATION OF REFLECTED SIGNAL BY TARGET 
MOTION 

The radar area of a complex target such as an aircraft depends on its 
orientation, or aspect relative to the radar. An aircraft is subjected to roll, 
pitch, and yaw motions by atmospheric turbulence. In addition, it may 
have internal motions due to rotating propellers and surface vibrations. 
Its gross aspect will vary with time if the target aircraft is on a noncollision 
or maneuvering course. All of these factors will affect the instantaneous 
radar area, so that the radar echo will have corresponding time variations. 
Some of these effects will be considered in greater detail in Paragraphs 4-7 
and 4-8. 

Another important effect produced by target motion is the change in 
frequency due to the doppler effect which was discussed in Paragraph 1-5. 
If the radar and the target have a relative approach velocity V, and the 
transmitter frequency is/o, the echo frequency is (see Equation 1-19) 

/ = /o(l + IV I C) = /o + 2/7X0 = /o +/o. (4-21) 

Ut is possible to build a radar which transmits one polarization and receives, on two separate 
receivers, the transmitted polarization and its orthogonal. For such a system, the theorem 
applies to only one received polarization at a time. 



4-5] 



REFLECTION OF PLANE WAVES FROM THE GROUND 



181 



In ordinary (non-doppler) radar, this shift in frequency due to the average 
approach velocity of the target is not noticed in the case of a point target. 
For extended targets, such as rain clouds and the ground or sea, for which 
various portions of the target area fill all or an appreciable part of the radar 
beam, the approach velocity varies over the beam, so that the composite 
echo has a spectrum of frequencies. In a doppler radar this spectrum will 
be properly discernible as frequency shifts relative to the radar frequency. 
In a non-doppler radar, beats between the various frequencies will be pro- 
duced in the final detector, so that an echo spectrum will also be obtained. 
In the case of an aircraft, a turn, pitch, or yaw will also introduce doppler 
beats which are discernible in a non-doppler radar. For example, consider 
the effect of a turn, which imparts an angular velocity c3 of the target about 
its center of gravity. Two fixed points on the target a distance D apart will 
then have a relative radial velocity toward the radar of 

A/^ = coD cos (4-22) 

as can be seen from Fig. 4-3. Hence 
by Equation 4-21 the difference in 
doppler frequency between these 
two reflection points is 

Af=?^=?5°^. (4-23) 

A A 

Thus the doppler frequency will be 
proportional to radar frequency, to 




= SD cos d 



Fig. 4-3 Differential Doppler Effect 
Due to Turning of Target. 



the angular velocity of the target, 
and to the gross aspect of the 
target. 

These and other effects which result in fluctuations of the target echo will 
be discussed further in later sections. 



4-5 REFLECTION OF PLANE WAVES FROM THE GROUND 

The reflection of radar waves from the ground or sea surface is an 
important factor in a number of phenomena associated with airborne radar. 
Among these may be cited the lobe structure which is encountered in 
tracking low-altitude targets, height-finding errors for such targets, and the 
dependence of sea and ground clutter upon polarization and depression 
angle. In all these cases, an understanding of the basic phenomena can be 
obtained from a consideration of the reflection of plane waves from a plane 
homogeneous surface. 

The reflection of a plane wave from flat ground depends on the frequency, 
polarization, and angle of incidence of the wave, and on the electrical 
properties of the ground (dielectric constant and conductivity). A wave of 




182 REFLECTION AND TRANSMISSION OF RADIO WAVES 

complex polarization customarily is 
resolved into its orthogonal linearly 
polarized components parallel and 
perpendicular to the surface, which, 
in the case of reflection from the 
Fig. 4-4 Reflection at the Ground. ground, are horizontally and verti- 

cally polarized components, respec- 
tively. These components can be treated separately and recombined 
after determining the change in amplitude and phase of each on reflection. 
The reflection coefficients for horizontal and vertical polarizations are 
given by the well-known Fresnel equations^ 

sin d - (e - cos^ 0)1/2 



sin e + {e - COS" 0)1/2 
es'md - (e - cos^^)!/^ 



|p//k-^^^ (4-24) 



Iprk-'*^ (4-25) 



€ sm -f (e - cos20) 1/ 

where 6 = depression angle of the radar (see Fig. 4-4) 

e = complex dielectric constant of the surface. 

The complex dielectric constant e is given in terms of the permittivity and 
conductivity of the ground k and a by 

6 = --j— = e'-7V' (4-26) 

eo coeo 

where eo = permittivity of free space. 

Values of typical ground constants and reflection coefficients are readily 
available in the literature. ■'^"^ 

A dependence of the reflection coefficient on frequency enters Equations 
4-24 and 4-25 through the dependence of e" on frequency. In addition, 
however, the ground "constants" k and a themselves are functions of 
frequency, by virtue of the dispersion of water. This dispersion takes place 
just in the frequency region most used for airborne radar. The resulting 
dispersion of ground thus depends on its water content. For airborne radar 
this is particularly important for water surfaces. Figs. 4-5 and 4-6 show the 
variation of the dielectric properties of pure, fresh, and sea water with 

2See J. A. Stratton, Electromagnetic Theory, Sees. 9.4 and 9.9, McGraw-Hill Book Co., Inc., 
New York, 1941. 

3F. E. Terman, Radio Engineers' Handbook, pp. 700-709, McGraw-Hill Book Co., Inc., 
New York, 1943. 

■•C. R. Burrows, "Radio Propagation over Plane Earth-Field Strenirth Curves," Bell System 
T^f/^. J. 16, 45-75 (1937). 

5R. S. Kirby, J. C. Harman, F. M. Capps, and R. N. Jones, Effective Radio Ground-Conduc- 
tivity Measurements in the United States, National Bureau of Standards Circular 546. 



4-5] 



REFLECTION OF PLANE WAVES FROM THE GROUND 
120 

100 

80 

e' 60 

40 

20 





183 





— 


1 1 

— Pure and Fresh Water 

— Sea WatPt- i 


















\ 










\\> 


_^20°C 

/ 














o°c^^ 


\ 










\ 










V 



10 



10^ 



103 



104 



105 



FREQUENCY (Mc) 
Fig. 4-5 Dielectric Properties of Pure, Fresh, and Sea Water. 



10' 



Sea Water 



Tresh Water 6?S\t\ <^" 




10 102 103 

FREQUENCY (Mc) 

Fig. 4-6 Dielectric Properties of Pure, Fresh, and Sea Water. 

frequency, taken from Saxton.^ The curves for temperatures of 0° and 20°C 
bring out a dependence on temperature as well. 

Figs. 4-7 through 4-10 show the magnitude and phase angle of the 
reflection coefficient of sea water for a temperature of 10°C at several 
wavelengths. Similarly, Figs. 4-11 and 4-12 show the reflection coeffi- 
cients for two different types of ground. For most airborne radar work, 
solid ground may be treated as a pure dielectric. These figures bring out 
clearly the diflFerence between horizontal and vertical polarization. For 
horizontal polarization, there is only a slight variation in magnitude and 
phase of the reflection coefficient with depression angle. For vertical 
polarization, however, there is a marked variation, caused by a partial 
impedance match of the two mediums which occurs at the Brewster angle. 
The reflection coefficient reaches a minimum magnitude and has a phase 
angle of 90° at this angle (the Brewster angle itself depends on frequency). 

6J. A. Saxton, "Electrical Properties of Sea Water," Wireless Engineer 2^, 269-275 (1952). 



184 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



IPI 









H— 


^ 












'^ 


^ 






. 










10 m 
1 m- 


/ 


/^ 




>< 


-< 










/ 


/ 


y^ 




> 




' 


^ 


lOcml 










J. i-in 


/ 


/ 


/ 


^ 


^ 


^ 










/ 




// 


^W^ 














ly 


/ 


/ 
















\\ 


// 




















ly 


















V 









































10 20 30 40 50 60 70 



90 



Fig. 4-7 Magnitude of Reflection Coefficient for Sea Water (Temperature = 10°C) 
as a Function of Depression Angle. 



190 

180 

180 

160 

140 

120 

100 

80 

60 

40 

20 























1 cm 
1 m 
10 cm 
10 m 












^ 


_ 




^ 




















\ 




















\ 












































Icm 
1 






































1 


" 


Ik 


) cm 
















\ 


V 


















u 


1 m. 

V 


\\^ 
















\ 


1^ 


\ 
















10 


m 




■ 







10 20 30 40 50 60 70 80 90 



Fig. 4- J 



Phase of Reflection Coefficient for Sea Water (Temperature = 10°C) as 
a Function of Depression Angle. 



4-5] 



REFLECTION OF PLANE WAVES FROM THE GROUND 



185 




1234bb7 8u9 10 
6° 

Fig. 4-9 Expanded Plot of Fig. 4-7 for Depression Angles Between and lO'^ 



182 

'l81 

180 

180 

160 

140 

120 

100 

80 

60 

40 

20 

























Im- 
10 cm 
10 m. 


















-— 


-'^ 







- 










































\^ 


=^ 


^^ 


=s^ 
















\ 


\ 




"^ 


'^ 


^ 












\ 




\ 






\ 




^ 












\ 


\ 




\ 






\^? 


^=lc 


m 




\ 




\ 






\^ 


cm 




\ 






\ 






\ 


Im 


\ 


\ 










\ 


\ 1« 


ni 




\ 


^ 


\ 


N 










\ 


-^ 










^ 


=:^ 














' " 


^^ 

















12 3 4 5 6 7 

6° 



Fig. 4-10 Expanded Plot of Fig. 4-8 for Depression Angles Between 0° and 10° 



186 



REFLECTION AND TRANSMISSION OF RADIO WAVES 























\ 






















\ 
























h" 




















\ 


k 






















""'■^ 




-^ 






\ 








/ 


^ 










\ 


V 




/ 


^ 














\ 


/ 


/ 
















\ 


/ 


















v 


/ 

















10 20 30 40 50 60 70 



Fig. 4-11 Magnitude of Reflection Coefficient for Average Land (e' = 10, a 
1.6 X 10~^ mho/m as a Function of Depression Angle. 



200 
180 
160 
140 
120 
100 
80 
60 
40 
20 
n 






























<^H 


/- 


—10 


m 


1 m 


S3 


•-^10 


cm 
















^x= 


10 m 






























0V 












































































^■0 


m 
















k 















10 20 30 40 50 60 70 80 90 

0° 



Fig. 4-12 Phase of Reflection Coefficient for Average Land (i' = 10, a 
10~^ mho/m) as a Function of Depression Angle. 



1.6 X 



4-5] 



REFLECTION OF PLANE WAVES FROM THE GROUND 



187 



The behavior is more complicated when the ground is stratified. The 
cases which are important to airborne radar are that of a layer of ice on top 
of a water surface, and that of a layer of snow on land. Then multiple 
reflections can occur between the surface and subsurface boundaries, with 
a resulting modification of the effective reflection coefficients^; the effective 
reflection coefficient then becomes an oscillating function of the electrical 
thickness of the ice or snow covering. 

When the radar target is at a low altitude, a variety of phenomena are 
generated by the interference of direct and reflected waves. Referring to 
Fig. 4-13, if both the direct and indirect paths are illuminated equally by 
the radar antenna, the resultant field at the target is 




Image 
Fig. 4-13 Path Difference Between Direct and Indirect Paths. 



E = Ea{\ + p^-^-^^^) 
where Ed is the field due to the direct wave, and Ai? 



(4-27) 
Ih sin d is the path 



difference. 

The ratio EjEd, obtained from Equation 4-27, thus is the propagation 
factor F due to the presence of the ground. In Paragraph 4-2, where this 
factor was defined, it was pointed out that the received power from a radar 
target is modified by the factor \FY. Introducing the values for K and A/? 
in Equation 4-27, we obtain 

F = E/Ed = 1 + pe-''-"" ^*" '"-. (4-28) 

The most significant and striking phenomena resulting from the inter- 
ference of direct and reflected rays are the lobe structure and the polarization 
dependence below the first lobe. The formation of a set of lobes is easily seen 
from Equation 4-28. With fixed 6 and continuous increase oi h, the resultant 
field will pass through alternate maximums and minimums when the phase 

■^J. A. Saxton, "Reflection Coefficient of Snow and Ice at V. H. F.," Wireless Engineer 27, 
17-25 (1950). 



188 REFLECTION AND TRANSMISSION OF RADIO WAVES 

lag of the reflected wave is an even or odd multiple, respectively, of 180°. 
The height interval between an adjacent maximum and minimum is 

Ml = X/4 sin d. (4-29) 

This succession of maximums and minimums of the resultant field gives rise 
to the lobe structure in the vertical coverage of the radar, which is especially- 
important for search radars. The location of a given maximum or minimum 
is different for vertical and horizontal polarization because of the phase of 
the reflection coefficient p. For airborne radar work with pencil beam 
antennas, the lobe structure usually is of importance only for targets at 
small depression angles, since otherwise the narrow beamwidth of the 
antenna would not illuminate the indirect path strongly. The lobe structure 
is pronounced only if the value of Ah is large compared with the vertical 
extent of the target. If the target covers more than one lobe, it effectively 
averages out the field variation over the lobe. This actually produces a net 
increase of gain over the free-space field acting alone, which is due to the 
field reflected from the surface. 

A similar oscillation in the propagation factor is observed with fixed radar 
and target altitudes and a continuously varying range as the target passes 
through the lobe pattern. In this case the angle 6 can be expressed as 

H -\- h H -\- h 

^^^ ^ = ^WTUH ^ -R- ^""^^^ 

where H = radar altitude 
h = target altitude 
R = target range. 

Neglecting the change in the phase angle of the reflection coefficient p, the 
range interval between an adjacent maximum and minimum is 

where R is the mean range. Thus for a target flying at a constant height, 
the lobes become packed more densely as the range is decreased. The 
oscillations of received power caused by the lobes are superimposed on a 
free-space variation which is proportional to the inverse fourth power of 
range as indicated in Equation 4-1. 

The situation is somewhat different when the target lies below the first 
lobe. In this case, the angle Q will be small and an expansion of F in powers 
of sin Q can be used. To obtain this expansion, we note first that ph and pr, 
which are given in Equations 4-24 and 4-25, can be approximated as 

p// = -1 +2(6- l)-i/2sine (4-32) 

PK = -1 + 2e(e - ])-■/- sin Q. (4-33) 



4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 189 

Similarly, the exponential term in Equation 4-27 may be approximated as 
^-iiKh sin e ^ I _ j2Kh sin d. (4-34) 

Substituting these approximations into Equation 4-28, and retaining only 
the first-order terms in sin d, we obtain 

Fh = 2[(e - l)-^/2 -\-jKh] sin d (4-35) 

Fv = 2[e{e - l)-i/2 -\-jKh] sin 6. (4-36) 

Thus, for sufficiently small d, both Fh and Fv are proportional to sin 6. 
But sin 9, as shown by Equation 4-30, is inversely proportional to range. 
Hence below the first lobe it follows that F is also inversely proportional 
to range: 

F oc R-\ (4-37) 

Therefore the received echo power, which is given by Equation 4-1, will be 
inversely proportional to the eighth power of the range in the region below 
the first lobe: 

P,i oc R-^ (4-38) 

This is in contrast to the inverse fourth power of the range which holds for 
free space. The range at which the transition occurs from a fourth-power 
law to an eighth-power law for a target which spans more than one lobe 
will be discussed in Paragraph 4-10. 

Equations 4-35 and 4-36 show how the resultant (one-way) field varies 
with height below the first lobe. Very close to the surface, where the term 
is small in magnitude compared with the other term in the brackets, Fh 
and F^ become 

Fy = j^^JyT. si" ^- (4-40) 

Hence the ratio of the fields at the target with vertical and horizontal 
polarization will be 

Fv/Fh = 6. (4-41) 

If the radar area of the target is the same for these two polarizations, then 
the ratio of the received echo powers will approach |e|^. This difference is 
important in the case of sea clutter. 

As the height is increased, the term j2Kh eventually will become large 
relative to the other term in the brackets in Equations 4-35 and 4-36. Then 
the field at the target will be approximately proportional to height and will 
be almost the same for either polarization. 



190 REFLECTION AND TRANSMISSION OF RADIO WAVES 

The first-order expansions of F in powers of sin 6 are limited in their 
ranges of validity. This can be seen, for example, in Fig. 4-7. For vertical 
polarization, the range of validity is limited to angles smaller than the 
Brewster angle, while for horizontal polarization, the angular range is much 
greater. For airborne radar frequencies, the range is about d < 30° for 
horizontal polarization, and < 4° for vertical polarization. 

The results deduced above are based on the properties of plane waves. 
In the case of the spherical waves radiated by an antenna, there is a surface 
wave which should be added to the direct and reflected waves. For airborne 
radar frequencies, however, this generally is unimportant. 



4-6 EFFECT OF EARTH'S CURVATURE 

The effect of the earth's curvature is twofold. First, it alters the geometry 
so that the path difference between the direct and reflected waves is 
decreased, and second, it decreases the amplitude of the reflected wave. 

The change from the plane to the spherical geometry is equivalent to a 
reduction in the heights of radar and target, as illustrated in Fig. 4-14. 



Direct \Na\je /? 




Fig. 4-14 Curved Earth Geometry. 

The second ef^-ect of the earth's curvature is to decrease the amplitude of 
the reflected wave, because the incident waves within a small range of 
vertical angles are spread out, or diverged^ into a larger range of vertical 
angles on reflection from the convex surface of the earth. 

For all distances encountered in airborne radar work, the reduced heights 
Aj', Ao "^ay be calculated from 

7;; = /;i - A/;, = /;i - d,yia (4-42a) 

Ji'., = /;o - A/;2 = //2 - doVla (4-42b) 

where a is the earth's radius, and diidi) is the distance from the reflection 
point to the transmitting (receiving) point. As will be shown in Paragraph 
4-18, the effect of average or "standard" atmospheric refraction may be 



4-6] EFFECT OF EARTH'S CURVATURE 191 

allowed for by increasing the earth's radius by a factor 4/3 to an effective 
earth's radius 

a.. ^ ^/2a. (4-43) 

With this factor, and expressing heights in feet and distances in statute 
miles, the height reductions due to curvature take the simple form 

A/7 1,2 = ^1,2-/2. (4-44) 

For a given value of range R, which is practically the same as the total 
distance d = d\-\- di measured along the earth's surface, the determination 
of ^1 (or d^ leads to the cubic equation 

Idr^ - Ud;^ - \lalh^ + ^2) - ^-] dr + la,h^d = 0. (4-45) 

Once this is solved for di, h[ and h'^ may be calculated from Equation 4-42 
and the remainder of the geometry handled like a plane-earth problem. 
Since the solution of the cubic is laborious, it is usually simpler to employ a 
graphical solution by plotting h[ jdi and h'^ jdi versus di. The proper value 
of di occurs where these two quantities are equal, since this gives equal 
A^alues of before and after reflection. 

As mentioned above, reflection at a spherical surface reduces the reflec- 
tion coefficient from the plane earth value p to 

p' = pD (4-46) 

where D is the divergence factor. This is given by 

y^ajsmd 

(4-47) 



(•-^r 



For very small values of 6 the divergence factor causes reduction of the 
effective reflection coefficient p' given by Equation 4-46 to a small value. 
In fact, at the horizon (9 = 0) D = 0, so that there p' = 0. However, the 
representation of the propagation process in terms of only a direct and a 
reflected ray breaks down as the horizon is approached. Norton^ gives as 

the limit to which Equation 4-46 is restricted: 

h 






, . . . . (4-48) 

d\ 

Practically all airborne radar ranges will be within this limit as long as 
atmospheric refraction does not depart greatly from the standard condition. 

^K. A. Norton, "The Calculation of Ground-Wave Field Intensity over a Finitely Con- 
ducting Spherical Earth," Proc. IRE 29, 623-639 (1941). 



192 REFLECTION AND TRANSMISSION OF RADIO WAVES 

4-7 RADAR CROSS SECTIONS OF AIRCRAFT 

Because all aircraft have dimensions large in terms of the wavelengths 
used in airborne radar, the radar area of an aircraft target is very sensitive 
to its instantaneous aspect. Because of air turbulence, the aspect is subject 
to statistical variations of roll, pitch, and yaw. Consequently the radar area 
is a statistically fluctuating quantity and it is not possible to give a single 
number for the radar area of such a target. The quantities of chief interest 
are the probability distribution of amplitudes, the aspect and frequency 
dependencies, and the time characteristics, or spectra, of the fluctuations. 

The amplitude distributions and aspect and frequency dependencies of 
certain aircraft will be presented in this paragraph, while the fluctuations 
and their effect on tracking systems will be discussed in Paragraph 4-8. 
A summation of the echo characteristics and their association with the 
physical structure and dynamic behavior of the aircraft will then be 
presented in Paragraph 4-9. This should make it possible to predict, with 
useful accuracy, the main features of the radar properties of a new or 
unmeasured target aircraft. 

An appreciation of the complicated nature of the radar area of an aircraft 
and its association with the physical structure of the aircraft can be 
obtained from some of the results of basic investigations into the properties 
of radar echoes from aircraft carried out by the Naval Research Laboratory, 
and recently made available. ^~^^ Pulse-to-pulse measurements were made of 
both fighter and bomber categories, with propeller-driven and jet-propelled 
models in each category. The measurements were made on three fre- 
quencies, 1250, 2810, and 9380 Mc/sec, with the radars searchlighted on 
the target by an optical tracker, and pulsed simultaneously. No antenna 
scanning was used, so that the observed fluctuations were all attributable 
to the target. The data were analyzed to determine amplitude distributions, 
median radar area versus aspect, and frequency spectrum of the amplitude 
fluctuations. The particular series of measurements to be discussed was 
made at elevation angles less than 15°. These measurements will be 
discussed in some detail, since comparable data have not been published 
before. Many of the characteristics observed can be explained in terms of 
physical processes, so that from these it should be possible to predict the 
principal characteristics to be expected in other situations. 

9F. C. MacDonald, Quantitative Measurements of Radar Echoes from Aircraft III: B-16 
Amplitude Distributions and Aspect Dependence, NRL Report C-3460-94A/51, 19 June 195J. 

low. S. Ament, M. Katzin, F. C. MacDonald, H. J. Passerini, P. L. Watkins, Quantitative 
Measurements of Radar Echoes from Aircraft V: Correction of X-Band Values, NRL Report 
C-3460-132A/52, 24 Oct. 1952. 

"W. S. Ament, F. C. MacDonald, H. J. Passerini, Quantitative Measurements of Radar 
Echoes from Aircraft VIII: B-45, NRL Memorandum Report No. 116, 28 Jan. 1953. 

12W. S. Ament, F. C. MacDonald, H. J. Passerini, Quantitative Measureynents of Radar 
Echoes from Aircraft IX: F-5I, NRL Memorandum Report No. 127, 4 March 1953. 



4-7] 



RADAR CROSS SECTIONS OF AIRCRAFT 



193 



Fig. 4-15 shows the cumulative amplitude distribution of a 2-second 
sample of echoes from the B-36, plotted on so-called Rayleigh coordinates. 



30 
25 
20 
15 
o 10 



-10 





































B - 36 Run 10 




X 


'^ 






























Range 18,400 - 

19,400 yd 

Az. 5.9°- 6.1° 

El. 5.7°- 6° 








X 






>», 


1 


































^ 


"> 


^ 






































»>. 


"^ 


.X ., 






































^^ 


ki^x , 




































Mc ^ 


"^ 


X X .^ 








































^^ X 


; 










A. 


























^^^ 


\ 














> 


>. 










































>> 


*> 










































^ 


•x 






































^53.>» 








































^">s. 


^*^. 








































'^^ 








1 
































^ 



30 



25 ~h 

20 # 

15 2 

10 I 

■a 
5 



-5 

-10 



0.010.11 5 10 30 50 70 80 90 95 98 99 99.5 99.9 

PERCENTAGE OF THE TIME THE VALUES EXCEED THE ORDINATE 
Fig. 4-15 Cumulative Amplitude Distribution of B-36 Echo, Approach Aspect. 



The straight lines through the points represent the Rayleigh distribution. ^^ 
Even with such a short sample (in this case, of only 240 pulses), the fit to a 
Rayleigh distribution is quite good. From the data obtained, it was 
concluded that for a 2-second sample the echo amplitude (and thus the 
radar area) is Rayleigh distributed for most aspects, except at broadside 
aspect. At broadside the amplitudes were compressed into a rather narrow 
range. 

The Rayleigh distribution signifies that the target consists of a large 
number of elements whose relative phases are independent and vary 
randomly during the time of the observation. The number of independent 
elements which constitute a "large" number, however, need be only about 
four or five if their amplitudes are comparable. Thus the conclusion to be 
drawn from the B-36 amplitudes distribution is that, except at broadside, 
the target consists of just such a "large" number of independent scatterers, 
and that in 2 seconds their relative phases pass through substantially all 
possible combinations.^^ At broadside, however, the echo from the flat 

13J. L. Lawson and G. E. Uhlenbeck, "Threshold Signals" Mass. Inst. TechnoL, Laboratory 
Series 24, 53, McGraw-Hill Book Co., Inc., New York, 1950. 

i^Practically, "all possible combinations" probably is satisfied if the phases vary over one 
or two times 360°. 



194 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



fuselage is so large relative to the echoes from other parts of the aircraft 
that it predominates over them and a relatively small amplitude variation 
results. 

Fig. 4-16 shows a 5-second sample for the B-45 twin jet bomber, taken 
for an approach run in which the aspect varied by 2|°. Here the approach 
to the Rayleigh distribution is poor, the range of amplitude variation being 
much more compressed. However, a 5-second sample at another aspect, 
in which the aspect angle varied 4|° (Fig. 4-17) shows a much closer 




0.01 0.5 5 20 40 60 80 90 95 p 98 99 99.5 

PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE 

Fig. 4-16 Cumulative Amplitude Distribution of B-45 Echo, Approach Aspect; 
Small Range of Aspect Angle. 



approach to the Rayleigh distribution. From an examination of data taken 
over a wide range of aspects, it was concluded that samples in which the 
azimuth of the B-45 varied by more than 4° gave a satisfactory fit to the 
Rayleigh distribution. 

Fig, 4-18 shows a set of distributions for the F-51 single-engine propeller- 
driven fighter. Although the lower amplitudes follow the Rayleigh distribu- 
tion quite well (on 9380 Mc/sec the lower levels were lost in the noise at 
the range of the measurements plotted in this figure), there is a pronounced 
upswing at high levels above the values expected from the extension of the 




PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE 

Fig. 4-17 Cumulative Amplitude Distribution of B-45 Echo; Larger Range of 
Aspect Angle. 

Rayleigh line fitted to the lower levels. This is attributed to reflections 
from the propeller, which, for a rather large range of angles, are stronger 
than from the remainder of the aircraft. This is shown by the original 
pulse-to-pulse photographs shown in Fig. 4-19. Here every fifth pulse 
(repetition rate 120 cps) is much larger (on all three frequencies) than the 
intervening ones. The dominance of the propeller echo was found to be 
especially marked at oblique aspects of the aircraft between head-on and 
broadside, corresponding to the region where a portion of the blade is nearly 
normal to the line from the radar. 

To depict the gross aspect variation of cr, the median values over roughly 
5° of azimuth were plotted against azimuth angle. Figs. 4-20 to 4-22 show 
the results for the B-36, B-45, and F-51, respectively. In averaging over an 
angular range of this amount, sharp peaks of the aspect dependence are 
largely smoothed out. In all cases, however, a prominent and rather broad 
maximum occurs in the neighborhood of the broadside aspect. This is 
especially true in the case of the B-36 (which has a rather flat fuselage) as 
shown by the 9380-Mc plot in Fig. 4-20 (broadside data for 1250 and 2810 
Mc were saturated and so are absent from this figure). The F-51 has its 
broadside maximum at an azimuth of about 98°, probably owing to the 
tapered tail section of the fuselage. 



REFLECTION AND TRANSMISSION OF RADIO WAVES 




PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE 

Fig. 4-18 Amplitude Distribution of F-51 Echo, Showing Effect of Propeller 

Reflection. 

The data in Figs. 4-20 to 4-22 also give the frequency dependencies of the 
radar area. A single number for the average radar area was obtained for 
each frequency by averaging all the values of a (in square meters) plotted 
in each figure. These averages are controlled by the large peak in the 
neighborhood of the broadside aspect. Similar numbers were obtained for 
all aspects measured outside of the broadside region. These latter numbers 
give a measure of the frequency dependencies of the aircraft for most 
tactical applications. The results are shown in Table 4-1. The B-36 and 
B-45 averages are roughly independent of frequency, but the F-51 average 
a increases approximately proportional to frequency. 



-E 4-1 



COMPARISON OF B-36, B-45, AND F-Sl 
AVERAGE RADAR AREAS 



Frequency, 
Mc 


Average a (in db > Im"^) 
Excluding Broadside Region 


Average cr (in db > \m-) 
Including Broadside Region 


B-36 B-45 F-51 


B-36 B-45 F-51 


1250 
2810 
9380 


13.2 
16.4 
10. 


10.7 
10.6 
10.2 


-0.5 
3.6 
6.6 


« 


14.9 
14.5 
17.3 


-0.2 

5.6 
9.2 



'Broiulside ret^ion saturated. 



4-7] 



EFFECT OF EARTH'S CURVATURE 

1250 MC/S 



197 




l-SEC 



2810 MC/S 



9 380 MC/S 



Fig. 4-19 PuIse-to-Pulse Records of F-51 Echo, Showing Strong Propeller Echo 
Every Fifth Pulse. 



198 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



30 

^S 25 

I— 

S 20 
b 

g 
o 10 



15 


B-36 
9380 Mc 






















u 










Median a Vs. 
Aspect 




















d| 










10 


— i = 

o = 

D = 


3° Elevation 




















1 

1 • 


U 








5° 












• 










u 


^ 


• 


V 






5 


f 


o 






• 


ib°^„ 


n 


n 


n 


^• 


• D 


n 


• • 


a 

• 


•-x- 

? 


9° 






p ° 


»u •£ 


X 




o '^^ ° 


i ' 

fl rf 


•? 


f 




a • 


% 


o 





11° 




n 


i\ 


I 










• i 


,^ 






S3 






Q 








COL 































320°330°340°350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 



B-36 
2810 Mc 
































Median a Vs. 
Aspect 


• 




'^ • 






T 


k 
U 




i 


• 


• 




•□ 






x=3"Ele 

A = 4° 


ation 




•T 




° 


t D 






» » 


• 






u * 




• 




o-5°H 




□ 








3 O 


3 • 


• 


















»=8° 
-12° 



































320°330°340°350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°110°120°130° 





25 


a: 


20 


tZ 

2 


15 


_b 

q 


10 









= 3°E 

= 4° 

3 = 5°" 


levatic 


n 


o 


u= 9"Elev 
*=10° 


tion 






























■=1 


r 


u 


I 
4 * 


■ □ 


■ i 


• 


• 


• 1 


• 




1 




a 


r = 8° 




t . 


u 


^ ^ 


i J 


fi -J 


' / 


^'^ 






D • 

t 








▼ • 




"? 


° '□ 






B-36 
1250 Mc 












• 




3 
















Mec 


ianff 
Aspec 


Vs. 



















320°330°340° 350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 
Fig. 4-20 Plot of Median Echo of B-36 Averaged over 5° of Azimuth. 

4-8 AMPLITUDE, ANGLE, AND RANGE NOISE^s 

In a tracking radar, rapid variations in target aspect can affect the 
smoothness of tracking and hence its accuracy. The variations in the target 

J^Most of the material in this Paragraph has been derived from the following NRL reports, 
and from references in footnotes 11-14, which can be consulted for further details: 

J. W. Meade, A. E. Hastings, and H. L. Gerwin, N'oise in Tracking Radars, NRL Report 
3759, Nov. 15, 1950. 

A. E. Hastings, J. E. Meade, and H. L. Gerwin, Noise in Tracking Radars, Part II: Dis- 
tribution Functions and Further Power Spectra, Jan. 16, 1952. 

D. D. Howard and B. L. Lewis, Tracking Radar External Range Noise Measuretnents and 
Analysis, NRL Report 4602, Aug. 31, 1955."^ 

A. J. Stecca, N. V. O'Neal, and J. J. Freeman, J Target Sitnulator, NRL Report 4694, 
Feb. 9, 1956. 

A. J. Stecca and N. V. O'Neal, Target Noise Simulator -— Closed-Loop Tracking, NRL 
Report 4770, July 27, 1956. 

B. L. Lewis, A. J. Stecca, and D. D. Howard, The Effect of an Automatic Gain Control on 
the Tracking Performance of a Monopulse Radar, NRL Report 4796, July 31, 1956. 



4-8] 



AMPLITUDE, ANGLE, AND RANGE NOISE 



199 







1 






B 


45 


1 








■^ 


7 














7 = 2° El 

x = 3° _J 


evation 


12bUM 
Median 


a vs 


1 
Aspect 








v>^ 














A = 4° 

• = 6° 
° = 7° - 






















^hese 

Four L] 














» 




















Valu 
Lov 
Lirr 


s are 


D 






















T 


O 
O 


"^^' 


td" 


its 


Q 
















, 


• ^ 


» 


7 








"" 








• 


° 








® = 12° 










'^\ 




^ 


















^ 




































i 





1000 

100 

10 







B-/ 
281f 


























'n 














Mr, 






















o9P 
















IVled 


an (7 


vs. 


\spec 


t 
















0" 







































o 
o 




o 
u 


^1 














X 




° 


^ 


• 


^. 


o't 


J 


/ •* 


^°^o 








□ 

T 


• 




















'^ 








il^Di 




5 










a D 





















































































100 



10 ^ 































u 














938C 


c 






















c^ 


u 














Mc 


vs./ 


\spec 


















a 


o 














Med 


an a 


t 
















",o 


^ 


























„. 


□ 








o 




a 














o 






o 


^ 


• A 


^ i 


ni 


" 




,^ 


s^" 






'(^ 




























^-^8 


•°"^ 


$ ' 


» 








^.^ 


^o 








































° 





















































































1 

0.1 

1000 

100 



300 320 



340 



10 20 30 40 50 60 70 
AZIIVIUTH ANGLE (deg) 



90 100 120 



Fig. 4-21 Plot of Median Echo of B-45 Averaged over 5° of Azimuth. 

characteristics are referred to as target noise (or scintillation) and have an 
effect similar to noise originating in the tracking system. Target noise is 
subdivided into amplitude noise, angle noise (or glint), and range noise. 

Amplitude Noise. In a sequential lobing radar, the antenna beam 
is scanned over a small range of angles, and comparison is made of the 
signals received with the beam swung to opposite sides of the boresight axis. 
The difference between these signals is used to drive the antenna toward a 



200 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



F-51 

Average of the Median cr's for 


Each 




















□ 




Five Degrees of Azimulli 
a = 1250 Mc 






















D° 





^These Values arc 
Lower Limits 




= 2810 
= 9380 
























□ D 


8 a 


J 


a 












n 


nfi 


Ln O 30 






□ 




o 




o o 




a 




o ° 


n 












D 


ft a 








A 








a 


a 






a 




o ° 









A 
















i ^ 


A 












A ' 










































a 


a'^ 











































15 

'T 10 

1 5 

^o o | I I t^Yt I I I °l f- l ° l °1 ^1 I L°h- I I h ^ 

D0_ - OAA _Oo 

2 -10 

-15 

300 320 340 10 20 30 40 50 60 70 80 90 100 120 140 

AZIMUTH ANGLE (deg) 

Fig. 4-22 Plot of Median Echo of F-51 Averaged over 5° of Azimuth. 

position where the signals are equal. Since the two samples are not received 
at the same instant, any change in signal amplitude during the scanning 
cycle, caused by target fluctuation, will lead to an angle error indication 
even if the antenna is pointed correctly at the target. In order to make an 
optimum choice of the parameters of a sequential lobing system, therefore, 
it is necessary to have information on the amplitude fluctuation charac- 
teristics of target aircraft. 

Some of the causes of amplitude fluctuations already have been men- 
tioned in Paragraph 4-7, e.g. propeller modulation. A clear example of this 
is shown in Figure 4-23, which is the spectrum of amplitude fluctuations in 







1 


^" G 




" 1 4 


< ' 


T 


1 1 C 


s , 




" 1 t 


5 1 


A 


1 U^ \ l- 


^ ^ i 


ihVl^ " 


~Mr- -M^ 4l4^ Jt 


s ^B 


^W^M 


kJ v...y k , J iL ,/i ., 


^ 1 




Y] ^vyyw.^ ^^w/ IW^y Uk^^^v... 


* 1 




1 1 1 


100 200 250 300 350 
FREQUENCY (cps) 



Fig. 4-23 Spectrum of Amplitude Noise of SNB (Two-Engine Transport) Aircraft 
in an Approach Run, Showing Spectral Lines Due to Propeller Modulation (= 9400 

Mc). 

an approach run of an SNB (two-engine transport) taken at a radar 
frequency around 9400 Mc. The repetition frequency of the radar was 
1000 cycles, so spectral information up to 500 cycles is derivable. Here 
peaks occur at multiples of about 58 cps. The remainder of the spectrum 
consists of a continuous band'^ whose amplitude decreases with increasing 

l*The small ripples or scintillations are due to incomplete smoothing in the spectrum 
analyzer. 



AMPLITUDE, ANGLE, AND RANGE NOISE 



201 



frequency. The propeller modulation occurs at the blade frequency (engine 
rps X number of propeller blades), so the fundamental modulation fre- 
quency should be the same for all radar frequencies. This was seen to be 
the case for the F-51, as illustrated in Fig. 4-19.^'' 



It/) 



I 


n nc f \. 




\ 






XAh, jalI 


^ j '^ ^'^"^W'V-'-*--s---~ — ^ .^^ 







I I I I I r 

2 4 6 8 10 15 20 25 35 45 60 
f(cps) 



B - 45 RUN 10 1250 Mc/sec 
RANGE 13,600 YD 
AZIMUTH 96°20' - 98°42' 
ELEVATION 6°56' - 6°55' 




2 4 6 8 10 15 20 25 

f(cps) 



35 45 



B - 45 RUN 10 2810 Mc/sec 
RANGE 13,300 YD 
AZIMUTH 91° -94° 
ELEVATION 6° 55' 




2 4 6 8 10 



35 45 60 



B- 45 RUN 10 9380 Mc/sec 
RANGE 13,600 YD 
AZIMUTH 96° 20' - 98° 42' 
ELEVATION 6° 56' - 6° 55' 



Fig. 4-24 Spectrum of Amplitude Noise of B-45. 



For a jet aircraft, the mechanical vibrations of the salient reflection 
surfaces would be expected to be of high frequency and noiselike in nature 

'■'Because the radar samples the instantaneous radar area of the target at discrete intervals 
r times per second (r = the pulse repetition frequency), a frequency of s cps in the spectrum 
could result from beats between the actual frequency and the repetition frequency. Hence 
the actual frequency of the target radar area spectrum may be any value given by «r ± s, 
where n is an integer (including 0). For the F-51, the observed propeller modulation frequency 
was found to correspond to w = 7 or 8. 



202 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



so that a line frequency spectrum similar to the propeller modulation 
spectrum would not be expected. The amplitude modulation spectrum 
observed in such cases would be due chiefly to beats between the doppler 
frequencies of the salient reflection centers, as described in Paragraph 4-4. 
The frequency of this type of modulation, therefore, would be proportional 
to the radar frequency. Whether a continuous spectrum is obtained, or one 
with spectral lines superposed on a continuous spectrum, depends on the 
nature of the perturbations of the aircraft for straight-line flight and on the 
duration of the observation (that is, on the length of sample). 

Figure 4-24 shows a spectrum^^ of the amplitude noise of the B-45 for 
which the observation time was 5 seconds. The voltage-time plots from 
which the spectrum was prepared are shown in Fig. 4-25. The 9380-Mc 



h 



B ■ 45 Run 10 1250 Mc 
Range 13,600 ya 
Azimuth 96°20'- 98°42' 
Elevation 6''56'- 6°55' 



^ 



i:¥ 



ills 



V 




Fig. 4-25 Voltage-Time Plots of B-45 Fxho, Showing Low-Frequency Modulation. 



i^The resolving power of the spectrum analyzer used was about L5 cps. Also, its response 
begins to drop off below about 6 cps. 



4-8] AMPLITUDE, ANGLE, AND RANGE NOISE 203 

spectrum in Fig. 4-24 shows two peaks, not quite resolved completely, at 5 
and 6.6 cps. In Fig. 4-25 the presence of a fundamental period in this region 
can be seen clearly; and similar fundamental periods, but of progressively 
lower frequency, can be seen in the 2810- and 1250-Mc plots. From an 
examination of the drawings of the B-45, it was concluded that the observed 
spectrum could be explained as the doppler beats between reflections from 
the engine nacelle, the wing tank, and a portion of the fuselage near the tail 
which was broadside at this aspect, all caused by a yaw rate of 0.14° /sec. 
For a longer sample (i.e., longer period of observation) during which the 
yaw rate varied, the discrete frequencies varied with time so that the 
spectrum over such a time of observation was smeared out into a more or 
less continuous band. Whether one should be concerned about a continuous 
band or discrete frequencies in the design of a tracking radar depends, 
therefore, on the time constant of the system — in other words, on the 
passband of the servo loop. This problem will be discussed in more detail 
in Chapter 9. 

Angle Noise. In a simultaneous lobing system (to be discussed in 
more detail in Paragraph 6-2) the signals which are compared to obtain 
angle information arrive simultaneously; thus amplitude fluctuations of the 
target echo do not generate angle error signals. If the angle tracking servo 
loop of a simultaneous lobing system is opened and the target is tracked 
optically, it is found that error signals still occur. These must be caused, 
therefore, by wandering of the eflFective center of reflection of the target. 
The principle involved in the generation of angle noise may be explained 
in terms of a target which consists of two point reflection centers, whose 

relative amplitude and phase vary as 

the target aspect changes. Fig. 4-26 
'BoresightAxis illustrates this model. Consider the 

following type of tracking system. 

A dual-feed antenna produces lobes 
Fig. 4-26 Physical Arrangement for on each side of the boresight axis. 
Illustrating the Origin of Angle Noise. the gains being equal along that axis. 

The voltage from each lobe is passed 
through an amplifier and square-law detector, and their diff"erence is used 
to derive angular inform^'.tion. The sum of the detector outputs is used for 
the AGC voltage of the receiver, so that the angular deviation of the 
arriving signal is determined by the difi^erence divided by the sum of the 
detector outputs. 

Let the boresight axis be directed at the center of the target (midway 
between A and B) and let the angle of A (and of B) to the boresight axis 
be d. For small d, the slope of the antenna lobes can be considered constant, 
and will be denoted by-^. Denoting the received RF voltages due to A alone 




204 REFLECTION AND TRANSMISSION OF RADIO WAVES 

and B alone when ^ = by £a and Eb, and their phase difference by </>, 
the total RF voltage received by the upper lobe is 

Eu = Ea{\ + gd) + Eb{\ - ge)e^'^ (4-49) 

and by the lower lobe 

El = Ea{\ - ge) + Eb{\ + ge)e^\ (4-50) 

The difference channel voltage is then 

En = k{\Eu\' - \El\') (4-51) 

= AkgdiE/ - En"-) 
where k = amplifier gain, and the sum channel voltage is 
Es = k{\EuV + \ElV) 

= 2k[{EA' + Eb' + IEaEb cos 4>) + {gey{EA'' + En' - lE^En cos «^)] 
= 2^(£x- + Ej? + 2£A£yj cos 0). (4-52) 

The error voltage is 

Eu __ ^ Ea' — Eb~ ,. rn■^ 

Es ~ ^ Ea' + Eb' + IEaEs cos 0' ^'^"'^^ 

In the presence of only a single target, say at A, the error voltage would be 

d. (4-54) 



(t)' 



(4-56) 



Comparing this with Equation 4-53 we see that the apparent reflection 
center of the dual-reflector target lies at an angle d' to the boresight axis 

1 - {Eb/EaY ,, ... 

1 + {Eb/EaY + 2{Eb/Ea) cos (^ ■ ^^'^'^ 

Thus, d' depends on the ratio 

^ 1 - {Eb/EaY 

' 1 + {Eb/EaY + 1{Eb/Ea) cos 4> 

and therefore on the relative amplitude and phase of the two reflections. 
This ratio can be less than, equal to, or greater than unity in absolute value, 
and may be positive or negative. In other words, the apparent reflection 
center can lie anywhere within the target, or even completely outside it. 

From Equation 4-56, r = when E^a = En-, so that equal reflectors have 
an apparent center midway between them, regardless of their relative 
phase. For values of £«/£.! other than unity, the value of r depends on the 
relative phase, 0. The apparent reflection center lies ovitside the target, 
when \r\ > 1, which requires that 

Er Ei 

- cos = COS (tt -</))> 7^ or — • (4-57) 

r.,\ ■ rLB 



4- 



AMPLITUDE, ANGLE, AND RANGE NOISE 



205 



Thus this phenomenon occurs in the 
region of destructive interference be- 
tween the two reflections, as may be 
seen from the circle diagram in 
Fig. 4-27, which is drawn for 
Eb > Ea- 

In general, a target may have a 
number of reflection centers whose 
relative amplitudes and phases vary 
with the instantaneous target as- 
pect. Delano^^ has developed the 
theory for a target composed of an 
infinite number of statistically in- 
dependent point sources and has determined the statistical properties of the 




Fig. 4-27 Vector Diagram for Two- 
Target Example. 




Angle Noise vs Time from R4D at 0° 



V — 1 sec — ^ 






Angle Noise vs Time from R4D at 90' 






1 



Angle Noise vs Time from R4D at 180° 

Fig. 4-28 Angle Noise Samples for R4D. 

'9R. H. Delano, "A Theory of Target Glint or Angular Scintillation in Radar Tracking, 
Proc. IRE^l, 1778-1784 (1953). 



206 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



apparent center of reflection. For a row of reflection centers uniformly 
spaced along a length L perpendicular to the line-of-sight from the radar, 
for example, the fraction of time that a conically scanning radar points ofl" 



I I I I I 



(a) 
SNB 




(0 
SNB 180 



TRUE NOISE FREQUENCY 
SPECTRAL ENERGY DISTRIBUTION 



Fig. 4-29 Spectra of Angle Noise for SNB Aircraft. 



4-8] AMPLITUDE, ANGLE, AND RANGE NOISE 207 

the target is 0.134. For equal reflection centers uniformly spaced over a 
circular area, this fraction becomes 0.2. 

Data on angle noise have been collected at the U. S. Naval Research 
Laboratory in connection with investigations of tracking noise. This work 
has been done at a frequency of about 9400 Mc, so that it is applicable to 
airborne radar problems. Examples of angle fluctuations of the type 
discussed above are shown for an R4D in Fig. 4-28. The variations in 
apparent reflection center are seen to be greater than the linear dimensions 
of the aircraft. Also, the deviations at the 90° aspect are larger than at 0° 
and 180°. 

Fig. 4-29 shows the spectrum^" of angle noise for several runs of an SNB. 
The angular noise in this spectrum has been multiplied by the target range 
so that the spectral density is independent of range and expressed in yards 
per Vcps- The amplitude decreases fairly regularly with increasing fre- 
quency, so that the total noise power is finite. In many cases the spectrum 
can be fitted satisfactorily by a curve of the form 

A = A,{\ ^r/U)-'i'~ (4-58) 

which corresponds to the transfer characteristic of a single-section RC 
low-pass filter. 

Since the vertical span of an aircraft is much less than its horizontal span, 
angle noise of a single aircraft is much less in elevation than in azimuth. 
In low-angle tracking, however, reflection from the ground or sea has the 
effect of creating an image aircraft at an equal distance below the surface 
(see Fig. 4-13). If the angle between the target and its image is less than 
the elevation beamwidth, the two will not be resolved. Variations in the 
phase difference between direct and reflected rays then will cause the 
effective reflection center to wander between target and image or beyond 
them. This has been observed to be the case. As the range decreases, the 
angular fluctuations increase until target and image can be resolved and the 
tracking system locks on. However, it is possible for lock-on to occur on the 
image instead of the target! 

Multiple targets have a similar effect on azimuthal variations. Thus, 
multiple targets which are not resolved give rise to a much higher level of 
tracking noise than a single target. 

Range Noise. In addition to causing angle noise, fluctuations of the 
effective center of reflection of the target can give rise to fluctuations in 
range, or range noise. Fig. 4-30 shows typical time plots of range noise for 
several classes of target. Fig. 4-31 shows the range noise of a single SNB at 

^OThe observation time included in this spectrum is about 80 seconds, so that spectral 
frequencies below about 3^ cps are cut off by analyzer limitations. 



208 



REFLECTION AND TRANSMISSION OF RADIO WAVES 







k\l^ht\M^^^%^ PB4Y 



_i 



lE^^^^'^^Yy^^E^. 



(c) 
2SNB90' 



^1 



H 



^5 



fS 



n 



-||!B^i|i^ 



m 



ift 



ELAPSED TIME 



SNB 



SNB 



SNB 



Fig. 4-30 Sample Time Function Plots of Range Noise from the (a) PB4y (Four- 
Engine Bomber) at 180° Target Angle, (b) ANB (Two-Engine Transport) at 180° 
Target Angle, and (c) SNB pair. 

aspect angles of 0°, 90°, and 1 80°, and of two SNB's at °90. Both the range 
noise spectrum and its probability distribution are shown. 

The distribution of the apparent reflection center in range generally lies 
wholly within the target, as can be seen from the curves at the right in 
Fig. 4-31. In this respect, range noise differs from angle noise. The reason 
lies in the different methods used for error detection in angle and in range. 

As in the case of angle noise, multiple targets which are not resolved will 
give rise to much higher noise levels than a single target. 

4-9 PREDICTION OF TARGET RADAR CHARACTERISTICS 

Quantitative measurements of radar characteristics require special 
instrumentation which is not widely available and is costly. Furthermore, 
targets of interest may not be available for measurement. For example in 
a problem of the type outlined in Chapter 2, it is highly unlikely that such 
definitive target information will exist. Hence great importance attaches 
to methods whereby the characteristics of interest may be calculated. As 



4-9] 



PREDICTION OF TARGET RADAR CHARACTERISTICS 



209 



(a) 
SNB 0' 



(b) 
SNB 90° 




3.3 yd-] ^ KSNBH KSNBH 


1 u 


It <^ s 


:: y ii : 




^■° -5; 


^^ \ 


^"^ " N 









01 23456789 10 11 

TRUE NOISE FREQUENCY 
SPECTRAL ENERGY DISTRIBUTION 



YARDS 
AMPLITUDE DISTRIBUTION 



Fig. 4-31 Sample Spectra Obtained for the SNB at Target Angles of (a) 0' 
(b) 90°, (c) 180°, and for (d) the SNB pair. 



a result of theoretical studies, ^^ and measurement programs such as those 
referred to in Paragraphs 4-7 and 4-8, techniques have been developed with 
which rather good success can be expected in predicting the radar charac- 
teristics of a target or target complex if their basic configurations are known. 
A brief discussion will be given of methods which have been used for 
aircraft targets. 

Since the dimensions of aircraft are many wavelengths for airborne radar 
frequencies, the methods of geometrical and physical optics are sufficiently 
accurate for most purposes. The principal reflections, therefore, come from 
surfaces which have portions parallel to the wavefront. The aircraft then 
can be approximated by a small number of bodies of simple shapes, for 
which the radar lengths can be calculated. Over a small range of angles 
about any given aspect, the contributions from the mdividual bodies will 
pass through substantially all values of relative phase, so that the average 

2iSee, for example, Studies in Radar Cross-Sections, XV, University of Michigan, Engi- 
neering Research Institute, Report 2260-1-T, Appendix A, and further references therein. 



210 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



radar area and its rms spread may be calculated quite easily. In this way, 
these quantities may be determined over the range of aspect angles of interest. 
In addition to the average areas, it is possible to predict the target noise. 
After having replaced the aircraft by a finite number of simple shapes at 
fixed locations, the doppler frequencies generated by a known variation of 
angular velocity can be calculated in the manner described in Paragraph 
4-4, and the spectrum can be determined. This calculation can be expedited 
by the use of a target simulator. ^^ As an example, Fig. 4-32 shows the 





ll||,,U 


. iWk,i.. 


TUD 


3 ' Tfll 1 


i 1 '^^w^l^^^ 


"""^'Xl^llMtj^,,. ^^ 


?*?•'*•-*'*•«-'- 


II 



1.2 



4.8 6.0 7.2 

FREQUENCY (cps) 



8.4 



9.6 



10.8 



Fig. 4-32 Spectral Distribution of Angle Noise from a B-17 (Four-Engine Bomb- 
er) Aircraft (Actual Measurement). 





Jk-i 


u- n/k i 


1 ^^WE 


i ^3iu 


< y^y*^ 


I - ^*^^^-^-^ , : 


1 1 1 1 1 1 1 1 1 



1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 

FREQUENCY (cps) 

Fig. 4-33 Spectra! Distribution of Simulated Angle Noise from a B-17 Target 

Simulator. 



measured angle noise spectrum of a B-17, while Fig. 4-33 shows the simu- 
lated spectrum. The latter was obtained with a "target motion" composed 
of a random oscillation having a Gaussian distribution of velocities about 
zero mean, with a standard deviation of 0.3° /sec. 

22See footnote 17, NRL Reports 4694, 4770, 4796. 



4-10] SEA RETURN 211 

4-10 SEA RETURN23 

In detection or tracking of targets on or near the surface, it is necessary 
to be able to distinguish the target from the background clutter due to 
reflections from the surface itself. For example, the AEW system discussed 
in Chapter 2 was required to distinguish between sea return and target 
echoes. In order to design a radar for such an application, the mechanism 
of sea return and its relation to radar and tactical parameters must be 
understood. 

In operations over water, this sea return or sea clutter is caused by all 
elements of the surface within a resolution element of the radar. Since this 
surface area is a function of pulse length and antenna beamwidth, the radar 
area of such a target complex is a function of range and some of the radar 
parameters. If the area of surface illuminated by a resolution element is 
not too small, then the return can be considered to be from scatterers 
uniformly distributed over this area, or area extensive. It is then convenient 
to use a quantity, the radar area per unit area of sea surface, usually denoted 
by 0-°, which is independent of the radar parameters. Then the radar area 
(J is 

a = a^A, (4-59) 

where A is the physical area of a resolution element. For pulse radar 
with pulse length r and azimuth and elevation beamwidths 4> and 6, 
respectively, where A is the smaller of the two values: 

A = m li^ii = R^L {6 small) (4-60a) 

A = i?2$e/sin d {6 large) (4-60b) 

withL = Tf/2. 

The characteristics of sea return depend on a number of parameters. 
These are the depression angle, polarization, frequency, and the condition 
(or "state") of the sea. The last quantity includes the many factors which 
affect the contour of the surface, such as the wind (its speed, direction, and 
duration), swell (wave systems generated by distant storms), currents, 
shoals, breakers, and others. Large waves in themselves do not necessarily 
produce strong clutter, since a heavy swell, with little or no wind blowing, 
does not produce a high level of clutter. On the other hand, clutter springs 
up suddenly with a sudden onset of wind, even before waves of appreciabel 
height are built up. Thus clutter seems to be more intimately connected 
with the secondary wave structure due to the local wind than with the 
primary wave structure. Although the many factors which influence sea 

2^For a thorough discussion of the World War II investigations of sea return, see the account 
by H. Goldstein in D. E. Kerr (Ed.), Propagation of Short Radio Waves, pp. 481-581, McGraw- 
Hill Book Co., Inc., New York, 1951. 



212 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



clutter make the phenomenon a complicated one, by now most of these 
appear to be understood. 

The radar parameters which control sea clutter are the depression angle 
(6), polarization, frequency, antenna beamwidths, and pulse length. The 
last two have been discussed already and may be eliminated, when the 
return is area extensive. The other three are interrelated. 

Since sea return is back-scattering from the surface itself, factors which 
affect the illumination of the surface elements responsible for the back 
scattering have an important effect on o-°. Polarization, frequency, and 
wave height are such factors, and their effects are intertwined. Katzin^S-^ 
showed that a number of their effects could be explained on the basis of an 
illumination of the scattering elements which is the combination of direct 
and reflected waves, similar to that above a plane reflecting surface. 

As was shown in Paragraph 4-5, interference between the direct and 
reflected rays creates a lobe structure above the surface. Below the lowest 
lobe, Pr oc R~^ for a single target. For an extended target distributed in 
height from the surface upward, this relation still holds if the top of the 
target is below the first lobe. At nearer ranges, where the target subtends 
one or more lobes, the target in effect integrates the varying illumination 
over it, so that the deep ripples of the lobe pattern are smoothed out and 
Pr cc R"^. For pulsed radar, the illuminated area of the sea surface at small 
depression angles is proportional to range, in accordance with Equation 



4-60a above, so that the reflection 




Log(R\/4/i) 



Fig. 4-34 Composite Plot of Sea-Clutter 
Power at Three Frequencies and Six 
Altitudes from 200 to 10,000 ft: Coor- 
dinates Normalized to Test Interference 
Mechanism. 



mechanism just described should 
give an R~^ range variation at short 
ranges, and an R~'' range variation 
at long ranges. 

Fig. 4-34 shows a composite plot 
of sea-clutter power measured with 
horizontal polarization at various 
frequencies and altitudes to test the 
interference mechanism. Here the 
measured points clearly define a re- 
gion where Pr oc R~^, which changes 
into one where Pr oc R~''. The 
agreement with the type of behavior 
just discussed lends strong support 
to the reflection mechanism. Pre- 
sumably reflection takes place froni 
the region ahead of the wave crests 
in the manner indicated in Fig. 4-35. 



^'•M. Katzin, "Back Scattering From the Sea Surface," IRE Cofitrn/ion Record, 3, (1), 
72-77 (1955). 

25M. Katzin, On the Mechanisms of Radar Sea Clutter, Proc. IRE 45, 44-54 (1957). 




4-10] SEA RETURN 213 

As was shown in Paragraph 4-5, 
vertical polarization produces a 
much stronger field on and just 
above a reflecting surface than does 
horizontal polarization. Hence the 
scattering elements of the sea sur- 
face are more strongly illuminated if p^^. 4-35 Possible Geometry of Reflec- 
vertical polarization is used so that ted Wave from Sea Surface, 
sea clutter at low angles is much 

stronger with vertical than with horizontal polarization, assuming that the 
same for both polarizations. 

Because of the presence of reflected waves, the appropriate radar equa- 
tion for sea clutter is obtained from Equations 4-1 and 4-59: 

where F is a suitable average value of F'^. For a uniform distribution of a 
with height above the surface, Katzin^^ gives 

F = 6, R< Rt (4-62a) 

F=6{Rt/R)\ R> Rt (4-62b) 

where Rt is the transition range between the R~^ and R~^ regions. The 
simple plane surface reflection theory for a surface with a reflection coeffi- 
cient of — 1 gives 

Rt = 5hH/\ (4-63) 

where h is the radar height and H the height of the top of the target, which 
here is to be interpreted as the height of the wave tops above the equivalent 
reflecting plane. Since wave heights themselves are distributed in a 
statistical manner, and the location of the equivalent reflecting plane is not 
known, an empirical relation must be deduced from experiment. A limited 
amount of experimental evidence suggests the relation 

Rt = 2A//i/io/X (4-64) 

in which //i/iu is the crest-to-trough wave height exceeded by 10 per cent 
of the waves (a unit frequently used by oceanographers). 

A further consequence of the reflection interference phenomenon at very 
small depression angles is that the return no longer remains "area exten- 
sive." The appearance of the sea clutter on an A scope then breaks up into 
a series of discrete echoes or "spikes" which appear much like individual 
targets. These can persist at fixed ranges for periods of a number of seconds. 
Fig. 4-36 shows an example of this. Spikiness is explainable by the com- 



214 REFLECTION AND TRANSMISSION OF RADIO WAVES 




Bm$i^> ■ 



Fig. 4-36 Expanded A-Scope Photographs of Sea Clutter. The Saturated Echo 

in the Center of the Sweep Is from a Stationary Ship. Blanking Gates Near Both 

Ends of the Sweep Define the Base Line. Wavelength, 3.2 cm. 



bined effects of destructive interference below the first lobe and a statistical 
variation of wave heights. Because of the statistical distribution of wave 
heights, there are relatively few waves which exceed the average height, 
and these thus appear as isolated "targets. "-*r^ 

^For a summary of a series of observations of spiky clutter, see F. C. MacDonakl, Charac- 
teristics of Radar Sea Clutter, Part I: Persistent Target-like Echoes in Sea Clutter, NRL Report 
4902, March 19, 1957. 



4-10] 



SEA RETURN 



215 



The variation of o-*^ with depression angle is a function of wind speed. 
Fig. 4-37 shows measurements on vertical and horizontal polarization at 
24 cm as reported by MacDonald,^^ while Figs. 4-38 and 4-39 show measure- 



-20 



-30 



-60 























w 












t- 


^^' 




-^ 


/'^f 

^M 










A 


1 -<it 


^R 




x'a 


t 












/f 


.a 


>' 


















;.# 




















.^ 











0.1° 



1.0° 10' 

DEPRESSION ANGLE 



100^ 



Fig. 4-37 Sea Clutter, 1250 Mc: Solid Line = Transmitted and Received Vertical 
Polarization. Dotted Line = Transmitted and Received Horizontal Polarization. 



20 

10 



S-lO 
b 
-20 

-30 
-40 



15-20 
Knots 




Fig. 4-38 
Velocity; 



20° 40° 60° 80' 
ANGLE OF DEPRESSION 



cj"" as a Function of Wind 
X = 8.6 mm. 



90 


1.25 cm 




Sea Clutter 


10 


10 - 15 )( 




Knots — ^ / 





15-20 \ / / 




- Knots-. \ n 
"20-25 \ \ ///' / 


-10 




Knots — V \ /V/// / 




\ Y\/ / 


-V(l 


J"\^' / 




/C-'^/^ "' / Knots 


-3U 


C^Z^-^^ - 10 / 




Knots/ 


-40 


1 1 1 1 1 1 ri 1 1 



20° 40° 60° 8 
ANGLE OF DEPRESSION 



Fig. 4-39 
Velocity, X 



as a Function of Wind 
1.25 cm. 



ments on vertical polarization at X = 8.6 mm and 1.25 cm by Grant and 
Yaplee."* At small depression angles, a^ increases with wind speed, but near 
vertical incidence this trend is reversed and a^ decreases with increasing wind 

2^F. C. MacDonald, "Correlation of Radar Sea Clutter on Vertical and Horizontal Polar- 
ization with Wave Height and Slope," IRE Convention Record \ (1), 29-32 (1956). 

2*C. R. Grant and B. S. Yaplee, "Back-Scattering from Water and Land at Centimeter and 
Millimeter Wavelengths, Proc. IRE 45, 976-982 (1957). 



216 REFLECTION AND TRANSMISSION OF RADIO WAVES 

speed. It should be noted that near vertical incidence, c7° rises to as high as 
+ 15 db. 

These and other characteristics of sea clutter have been explained by a 
theory developed by Katzin.^^ This theory is based on scattering by the 
small facets of the sea surface as the basic scattering elements. At small 
depression angles, where none of the facets is viewed broadside, the facets 
which back-scatter most effectively are those with perimeters of about a 
half-wavelength. The back-scattering of a facet increases with its slope, so 
that those near the wave crests contribute most strongly, even if the 
illumination is constant with height. 

Although at small depression angles the back-scattering is at angles far 
removed from the facet normals, at large depression angles some of the 
facets are viewed broadside, so that these contribute most strongly in this 
region. The larger the facet the greater is the contribution. The angular 
dependence at large depression angles then is governed by the slope 
distribution of the facets. At airborne radar frequencies, the angular 
dependence of o-° should follow the slope distribution rather closely. This 
distribution is approximately Gaussian, but is more peaked and is skewed 
in the upwind-downwind direction. ^^ 

At small depression angles, the theory shows that o-" is directly propor- 
tional to wind speed, but at high angles it is inversely proportional to wind 
speed. These features of the theory seem to be in accord with available 
measurements. 

The evidence regarding the frequency dependence of o-'^ is not uniform. 
Katzin^^ stated that o-° (at small d) was roughly proportional to frequency 
in the frequency range 1.25-9.4 kMc, and gave the formula for a" upwind 
at small depression angles, 

(7« = (2.6 X \0-'W^i'\-' (4-65) 

where W is the wind speed in knots and X the wavelength in cm. (In this 
formula, the illumination factor F is included in o-".) Wiltse, Schlesinger, 
and Johnson^** found a" to be substantially constant in the frequency range 
10—50 kMc. Grant and Yaplee-* found cr" to increase with frequency 
range 9.4-35 kMc, the increase being about as the square of frequency at 
vertical incidence and about as the first power or less at 10° depression 
angle. Grant and Yaplee's measurements on the different frequencies used 
were made on different occasions, however, so that their results on the 
frequency dependence are subject to wider variations due to different 
surface conditions. It is quite possible that the frequency dependence of a" 

2^C. Cox and W. Munk, "Measurement of the Roughness of the Sea Surface from Photo- 
graphs of the Sun's Glitter," J. Opt. Soc. Aju. 44, 838-850 (1954). 

3"J. C. Wiltse, S. P. Schlesinger, and C. M. Johnson, "Back-Scattering Characteristics of 
the Sea in the Region from 10 to 50 kmc, Proc. IRE 45, 220-228 (1957). 



4-11] SEA RETURN IN A DOPPLER SYSTEM 217 

may vary somewhat from time to time, depending on the condition of 
the sea surface. 

4-11 SEA RETURN IN A DOPPLER SYSTEM 

The doppler shift due to relative motion of radar and target was discussed 
in Paragraph 4-4, and the echo frequency due to a transmitted frequency/o 
was given as 

/=/o + 2F/X (4-66) 

where V is the line-of-sight component of the approach velocity of radar 
and target. If both radar and target are in motion, then with respect to 
fixed coordinates V may be divided into two parts, one due to the radar 
velocity F^, the other to the target velocity Vf Equation 4-66 corre- 
spondingly may be written as 

/=/o +/.+/.. (4-67) 

If the angle of the target from the ground track of the radar is Xj then the 
doppler frequency due to the radar motion is 

/. = (2F./X)cosx=/icosx (4-68) 

/i = lVrl\. (4-69) 

If the target is the surface of the sea, then the angle x will vary over the 
portion of the surface which is illuminated by the radar, owing to the finite 
width of the antenna beam. Hence there will be induced by the motion of 
the radar a corresponding band, or spectrum, of doppler frequencies Jr. 
This may be called an induced doppler spectrum. 

Similarly, if the various portions of the surface are in relative motion, 
then even if the radar is stationary or the radar beam is so narrow that no 
appreciable variation in cos x takes place over the illuminated area, a range 
of doppler frequencies/^ will result from the intrinsic motion of the surface. 
This may be called the intrinsic doppler spectrum. 

The relative importance of the induced and intrinsic doppler components 
depends on the relative velocities and the geometry, as well as on the 
antenna beamwidth. Referring to Fig. 4-40, 

cos X = cos ^0 cos 00 

where 0o is the depression angle and 0o the azimuth angle of the surface 
target relative to the aircraft motion. 

Hence for a small azimuth deviation ±A(/) from the mean value 0o, we 
have 

cos X = COS 0o(cos 00 cos A0 ± sin 0o sin A0) 

(4-70) 
= cos ^o{[l - (A0)V2] cos 00 ± A0 sin0o}. 



218 REFLECTION AND TRANSMISSION OF RADIO WAVES 




Fig. 4-40 Geometrical Relations for Doppler Spectrum of Sea Return. 



Thus it is evident that the spread in cos x, and hence the width of the 
induced doppler spectrum, will be minimum along the ground track 
(00 = 0)- As an example, we consider an airborne X-band pulse radar 
(9400 Mc/sec) with a horizontal beamwidth of 3° (A</> = 1.5°), and an 
aircraft speed of 200 knots. Then from Equation 4-69, /i = 6.44 kc. At 
grazing depression angles along the ground track {do = 0), the induced 
doppler spectrum has a half-power width of 0.000343/i = 2.2 cps, while at 
45° to the ground track the half-power width is 0.0037 /i = 238 cps. 

In principle, the induced spectrum is known from information available 
at the radar and thus may be compensated, in part, by appropriate (though 
complicated) circuitry. There still remains the intrinsic spectrum, and a 
knowledge of this is necessary in order to determine the capabilities and 
limitations of doppler radar in target detection and tracking through 
clutter. 

Measurements of the intrinsic spectrum of sea clutter have been made by 
the Control Systems Laboratory of the University of Illinois. ^^ These were 
made with a coherent airborne radar operating on a wavelength of 3.2 cm. 
By making measurements along the ground track, the width of the induced 
spectrum was made small relative to that of the measured spectrum, so that 
the measurements yielded the intrinsic spectrum directly. By multiplying 
the frequencies by X/2 (see Equation 4-69) the results were converted to a 
velocity spectrum. 

^iThe information on the intrinsic doppler spectrum of sea clutter was furnished through the 
courtesy of the Control Systems Laboratory, University of Illinois. 



4-12] GROUND RETURN 219 

Frequency spectrums were obtained for 15-second samples of recorded 
data (corresponding to 3750 ft along the sea surface), and also frequency, 
B-scope records of the spectrum as a function of position of the illuminated 
patch on the sea surface (250 ft long). These will be referred to as the 
A display and the B display, respectively. 

For low sea states, the average spectrum had a Gaussian shape, and width 
between half-power points of 2 to 3 knots (60-100 cps at X band). The 
corresponding B display was generally smooth on upwind and downwind 
edges for all ranges. Fig. 4-41 (a) shows a sample of the A and B displays 
for a low sea condition (wave height 2 ft, wind 9 knots). The 3-db band- 
width of 82 cps in this sample corresponds to a velocity spread of 2.55 knots. 

As the wind increased and white caps became evident, the A display 
broadened asymmetrically to 5 knots or more. The B display then was 
broadened on the downwind edge to an extent which varied irregularly with 
range, but the upwind edge remained smooth. Fig. 4-41 (b) shows a sample 
of the A and B displays for a medium sea (wave height 5 ft, wind 16 knots). 
Here the 3-db bandwidth is 172 cps, corresponding to a velocity spread of 
5.35 knots. These characteristics suggest that the irregular downwind 
broadening was due to spray filaments or patches associated with the white 
caps, blown off the wave crests and moving downwind more rapidly than 
the crests. 

4-12 GROUND RETURN 

The applications of airborne radar over land cover an even broader field 
than operations over sea. As in the case of sea clutter, reflections from a 
land surface form a clutter background which tends to obscure the desired 
echo, e.g. from a target aircraft flying at low altitude. At small depression 
angles, ground return generally is considerably larger than sea return. 
Hence the problem of detecting ground targets obscured by ground clutter 
is correspondingly more difficult. 

In another type of radar application — ground mapping — the most 
important characteristic is the contrast obtainable between objects and 
their immediate surroundings as determined by the nonuniformity of the 
return. This characteristic governs the type of ground map which may be 
obtained by radar techniques, as was discussed in Paragraph 1-4. 

The ground return which competes with or obscures the target echo is 
confined to the return from ground elements at the same apparent range as 
the target. Such returns can be received either on the main beam or the 
sidelobes of the antenna pattern. A special form of sidelobe clutter — the 
altitude line — will be discussed in the next paragraph. 

In a pulsed radar the returns which arrive at times precisely separated by 
the interpulse period appear at the same apparent range. This gives rise to 



220 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



SAMPLE NO. 262 
Sept. 24,1954 
Altitude -lOOOff 
Ronge- 13,050yds. 
Depression- 1.46° 
Wave height-2.0ft. 
3db bandwidth-82cps 



WIND 
9 knots 
80° 



SAMPLE NO. 140 
Sept. 22,1954 
Altitude-2500ft. 
Range- 15,080 yds. 
Depression-3.l7° 
tieight-S.Oft. 



3db bandwidth -I72cps A 



Wind 16 knots ^^J^ 



<iep. 




2400 2500 

FREQUENCY cps 

(b) MEDIUM SEA STATE 



2200 2400 2600 2800 
FREQUENCY cps 



Fig. 4-41 Doppler Spectrum of Sea Clutter, Showing both the Spectrum Averaging 

over the Sample (a) and the Spectrum vs. Time or Range. A/C velocity refers to 

Ground Track. 



a form of clutter known as nndtipk-time-aroioid echo (MTAE). This 
clutter is therefore important for ground targets whose range is given by 

R„ = R.^nR^rs (4-71) 



4-12] GROUND RETURN 221 

where Rg = range of ground reflector 
Rt = target range 
Rprf = range corresponding to an interpulse period. 

Echoes from objects in the range interval Rt + Rprf {n = 1) are known as 
second-time-around echoes (STAE). It is not unusual for STAE to be 
comparable to or stronger than the desired target echo. The range of angles 
for which STAE may be troublesome depends upon the geometry and radar 
parameters of the particular system under consideration. Obviously, a 
knowledge of the characteristics of ground return is of importance in this 
and in other applications. 

Some measurements of ground return at wavelengths of 0.86, 1.25, and 
3.2 cm are given in a paper by Grant and YapleCj^^ who used vertical 
polarization. Fig. 4-42 shows their results for a tree-covered terrain with 
the trees in full foliage. It will be noted that a^ is very roughly independent 
of the angle of incidence. o-° also increases with the frequency, but even at 
X = 8.6 mm does not exceed — 13 db at any angle. Thus this type of terrain 
absorbs most of the incident energy. 



20 




Trees With Foliage 


10 


- 






- 


1.25 cm 





- 


8.6 mm \ 


-10 


_ 


X-U 


-20 


y^^^^^cxr 


-30 


- 


3.2 cm 


-40 


- 


III 



20 


- Tall Weeds or Flags 




— Apr 1955 Green Grass / 


10 


- Nov 1955 Dry Grass / 





/ 
_ 1 

/ 


-10 
-20 




-30 
-40 


1 1 1 1 1 1 1 1 1 1 




20° 40° 60° 80° 




ANGLE OF DEPRESSION 



20° 40° 60° 80° 
ANGLE OF DEPRESSION 

Fig. 4-43 Comparison of a^ for Green 
Fig. 4-42 o^ for a Tree-Covered Terrain. Grass and Dry Grass. 

Fig. 4-43 shows the results for ground covered with tall weeds and grass 
in the spring when the grass was green and the ground wet and marshy, and 
in the fall when the grass and ground were dry. Two effects are clearly 
evident from this figure. (1) There is a very large and rapid rise in o-° near 
vertical incidence, amounting to 15-20 db, when the ground is wet. (2) 
Although a" increases steadily with frequency under dry conditions, when 



222 REFLECTION AND TRANSMISSION OF RADIO WAVES 

the ground is wet the curve for 1.25 cm falls much below the other two. 
Grant and Yaplee state that this behavior was always found on 1.25 cm 
when the ground was wet, and suggest that this anomaly may be associated 
with the water vapor absorption peak near this wavelength (see Paragraph 
4-16). Aside from the large rise near vertical incidence, the remainder of 
the curve lies approximately 5 db higher when the ground is wet than when 
it is dry. 

The large increase of o-" near vertical incidence when the ground is wet 
can be explained as caused by patches of water which are viewed broadside, 
as in the case of the facets which have been proposed as the scattering 
elements for sea clutter. This emphasizes the importance of plane surfaces 
whose dimensions are comparable to or large compared with the wavelength 
when they are viewed broadside. Hence, in attempting to generalize on the 
basis of the rather meager experimental results which have been reported 
in the literature, this characteristic should be kept in mind. 

An important example of this is the case of cultural areas, especially 
cities. Here, in addition to the presence of large flat surfaces, such as 
building walls, windows, roofs, and streets, there are many possibilities for 
corner reflectors. Since corner reflectors have a large radar area over a wide 
range of angles, they have a very large effect on the radar return. For 
example, observations of ground painting by airborne radar^^ show that the 
signals from man-made structures are often too strong to be fully explained 
in terms of their size, and that a certain amount of corner-reflector action 
("retrodirectivity") in the targets must be present. This action is present 
principally at long ranges (small depression angles) and is responsible for 
sharp contrast in the return from arrays of buildings at long ranges. At 
short ranges, where the depression angle is outside the range of corner- 
reflector action, this contrast tends to fade. These principles have to be 
kept in mind, for example, in estimating the effect of STAE from a city on 
the performance of the radar in an AI, an AEW, or a target-seeking missile 
application. 

4-13 ALTITUDE RETURN 

In Paragraphs 4-10 to 4-12, we have discussed the back-scattering 
properties of the sea and ground in terms of the scattering parameter o-". 
This has been done in order that the properties could be applied to radars 
with a wide range of parameters. In order to determine the response for a 
particular radar, one needs to consider the radar parameters in connection 
with the scattering characteristics of the surface. One case which is of some 
importance is that of the altitude return in pulse radar. This is the signal 
received from the ground directly beneath the aircraft. On a PPI display 

32L. E. Ridenour (Ed.), Radar System Engineering, Vol. 1, pp. 100-101, McGraw-Hill Book 
Co., Inc., New York, 1947. 



I 



4-13] 



ALTITUDE RETURN 



223 



it gives rise to the "altitude circle," while on an A display it is referred to as 
the "altitude line". In many cases this return is prominent because of the 
marked increase of cr° which occurs for depression angles near 90° (see 
Figs. 4-37, 4-38, 4-39, and 4-43). 

To a radar altimeter the altitude return is the desired signal, while 
to target detection and tracking radars it is a source of interference or 
"clutter." Since the antennas of these two classes of radars have widely 
different beam patterns, the illumination of the ground as a function of 
angle may vary widely between different applications. A full discussion of 
the problem, therefore, is beyond the scope of the present treatment, so that 
only some of the principal factors will be discussed. 

The expressions (Equation 4-60) were given for the area of a resolution 
element on the surface. For small depression angles this area is proportional 
to range, while for large depression angles it is proportional to range 
squared. The distinction between these two in the case of the altitude line 
is actually a function of altitude. For example, if both the antenna beam 
and the pulse shapes are rectangular, and if cr'' is a slowly varying function 
of angle near vertical incidence (as in the case of Fig. 4-42, for example), 
then the illuminated area is beamwidth limited if the leading edge of the 
transmitted pulse passes the outer 
edge of the antenna beam before the 
trailing edge of the pulse reaches the 
ground. The received power of the 
altitude line then will vary as the 
inverse square of altitude in accord- 
ance with Equation 4-60b. Because 
of the inverse square relationship (as 
contrasted with an inverse fourth 
power relationship for a point target) 
the altitude line return can be very 
strong. This is particularly true for 
altitude line return from a flat calm 
sea which tends to act as a perfect 
reflector (see Figs. 4-37 and 4-38.) 
However, if the altitude or beam- 
width is great enough that the trail- 
ing edge of the pulse reaches the 
ground before the leading edge 
passes out of the antenna beam, then 
the return is pulse-length limited, 
and the received power of the alti- 
tude line will vary as the inverse 
cube of altitude in accordance with Equation 4-60a (see Fig. 4-44). 



20 



16 











/ 


K 








/ 


/ 










/ 


/ 










I 






t 


V 
1--QV'. 






1 








K 




1 1 1 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 
L/2/i 

I \ \ I \ \ 1 I 

0.04 0.08 0.12 0.16 0.20 0.24 0.28 

r(Msec)A(1000ft) 

Fig. 4-44 Angular Extent of Altitude 
Line vs. Pulse Length. 



224 REFLECTION AND TRANSMISSION OF RADIO WAVES 

Actually, neither the antenna beam nor the pulse shape is rectangular, 
and the scattering properties of the ground, even if they are area-extensive, 
may vary with angle, so that a continuous transition from an inverse square 
to an inverse cube relation takes place. More complicated situations occur 
when one or more large individual scatterers are located within the illumi- 
nated area. A more detailed discussion of this problem can be found in a 
paper by Moore and Williams. ^^ 



4-14 SOLUTIONS TO THE CLUTTER PROBLEM 

Having considered the characteristics of radar targets and of sea and 
ground clutter, we can now examine these together in order to find the most 
favorable solution to the clutter problem. There is no unique solution, since 
the factors involved depend on the operational problem and the limitations 
placed on the radar parameters. A full discussion of all the considerations 
and possible solutions is beyond the scope of this chapter, since the problem 
involves the overall system design and operational philosophy. We shall 
restrict ourselves to a consideration of certain features of the antisubmarine 
warfare (ASW) problem, in order to bring out some interesting possibilities 
based on sea clutter characteristics discussed in Paragraphs 4-10 and 4-11. 

In the first place, an early decision can be made regarding the polarization 
of the antenna. Both theory and experiment show that sea clutter levels are 
much lower on horizontal polarization than on vertical polarization. From 
Fig. 4-37 it is seen that this can amount to 10 db or more. Hence, unless 
the target shows a preference for vertical polarization by more than this 
amount, horizontal polarization clearly is to be chosen. Furthermore the 
discrimination based on target height, which will be discussed below, will 
be achievable only with horizontal polarization. 

The following discussion will be based on a flat earth and will illustrate 
the principles involved. The modifications necessary to take into account 
the effect of the earth's curvature have been described in Paragraph 4-6. 
These will affect the answer only quantitatively and will not change the 
nature of the results. 

The primary mission of airborne radar in ASW is search; tracking is a 
secondary mission. The object of system design and operation is to choose 
the radar parameters so that the probability of detection is optimized. 
Inevitably practical limitations will arise which restrict the ranges of certain 
of the parameters. Ordinary (non-doppler) pulse radar will be considered 
first, and then the additional improvement due to doppler radar will be 
discussed briefly. 

8^R. K. Moore and C. S. Williams, Jr., Radar Terrain Return at Near-Vertical Incidence, 
Proc. IRE ^5, 228-238 (1957). 



4-14] SOLUTIONS TO THE CLUTTER PROBLEM 225 

Since in search it is desirable to sweep out a large area, the problem is 
concerned primarily with small depression angles. The variation of received 
clutter power with range will then be of the form shown in Fig. 4-34, and 
will be given by Equation 4-61 : 



UtYR' 



(4-72) 



In this we may insert the values of A and F given by Equations 4-60a and 
4-62, respectively. The horizontal beamwidth <!> and the gain of the radar 
antenna may be expressed by 

$ = y^ (4-73) 

G = ^^ (4-74) 

where Iw and 4 are the horizontal and vertical antenna apertures, respec- 
tively, and ka and h are constants of the antenna design. If we adopt the 
form of relation given in Equation 4-65 for a^, 

(T« = ^oA (4-75) 

then Equation 4-61 becomes for the received clutter power 

^c - (^,^3) (4-76) 

where kc = kah^h/i^T)' 

Fc = 6, R< Re 
Fc-^6{Rc/R)\ R> Re 
Re = 2hHiiio/\ = transition range for clutter 

as in Equations 4-62 and 4-64. kc is primarily a function of local wind speed, 
while //i/io is dependent rather on wind history, but may be forecast with 
reasonably good accuracy.^* 

Similarly, for the power Pt of the target echo, we have from Equation 4-1 

If the target is a surface target of uniform section and height Ht, then F^ is 
to be replaced by F of Equation 4-62, with its transition range, Rt, given by 
Equation 4-63 

Rt-^^-^ (4-78) 

A 

3^W. J. Pierson, Jr., G. Neumann, and R. W. James, Practical Methods for Observing and 
Forecasting Ocean Waves, H. O. Pub. No. 603, U.S. Navy Hydrographic Office, 1955. 



226 REFLECTION AND TRANSMISSION OF RADIO WAVES 

Then Pt may be written as 

kTPlLV<TTFT 



Pt 



\'R' 



(4-79) 



in which h = kb'^/i4T)K 

Plots of Equation 4-76 for a specific sea condition and of Equation 4-79 
will then be as shown in Fig. 4-45. From this, it follows that the range scale 




Fig. 4-45 Target and Clutter Power Relations vs. Range. 



in which the target-to-clutter ratio may 



REGION 1 



REGION 2 



REGION 3: 



kahLR 

kahLR 



(4-80) 



may be divided into three regions, 
be expressed as follows: 

Pt 
Pc 

Pt 
Pc 

Pt 
Pc 

Obviously, a large antenna width and a short pulse length will increase 
the target-to-clutter ratio in all three regions. Furthermore, if or is inde- 
pendent of frequency, then so is Pt/Pc in regions 1 and 3. The locations of 
the transition ranges Rt and Re can be controlled by the height h of the 
radar. It is evident from Fig. 4-45 that the largest target-to-clutter ratios 
generally will be obtained in region 3. 

Since region 3 is one in which destructive interference operates on both 
the target and clutter signals, this is a region of relatively low signal 



\2hHuio) 

arL ( SHt Y 
kahLR\2H,i,,)- 



4-15] ATTENUATION IN THE ATMOSPHERE 227 

strength. Depending on the transmitted power and other radar parameters, 
therefore, the useful limit of region 3 will be set by the minimum power 
required to produce a signal detectable above the noise. This minimum 
power level is indicated by the horizontal dashed line labeled P^in in Fig- 
4-45. Since Pm\n depends on receiver bandwidth, effective antenna scanning 
rate and beamwidth, and other factors, changes which are made in /^/L in 
order to increase Pt /Pc will also increase P.nin- Although Pt /Pc (in regions 
1 and 3) does not contain an explicit frequency factor, both Pt and Pc 
contain the factor X~-, and so increase with frequency. 

Fig. 4-45 relates to a specific target area and sea condition. Obviously, 
one must consider a whole family of such curves, relating to various possible 
combinations of interest, in order to arrive at the optimum choice of param- 
eters. Some of these parameters depend on the operational philosophy 
(e.g. barrier patrol, hunt-and-kill). In addition, the effect of the earth's 
curvature, which will steepen the rates of signal decrease in region 3, will 
have to be taken into account. 

The above discussion refers to non-doppler radar. Doppler radar offers 
the additional possibility of increasing the target-to-clutter ratio by 
exploiting differences in the target and clutter spectrums. In order to 
achieve a gain in target-to-clutter ratio, it is necessary that the target 
doppler frequency spectrum lie outside the range of the induced doppler 
spectrum of the clutter. For the example given in Paragraph 4-11 (Vr = 
200 knots, A0 = 1.5°) each doppler component of the intrinsic doppler 
spectrum would be broadened by about 2 cps along the ground track and 
about 350 cps at right angles to the ground track. The corresponding 
effective velocity broadening would be about 0.1 and 11 knots, respectively. 
Thus, no significant improvement will be obtained at large angles to the 
ground track unless the radial component of target velocity exceeds 10-15 
knots, for the 3° beamwidth assumed. Smaller beamwidths would reduce 
this figure proportionately. 

In principle it is possible to improve the target-to-clutter ratio by 
exploiting the difference between the widths of the received target and 
doppler spectrums. This requires a "velocity" filter (or a set of them). A 
system employing such techniques is described in Paragraph 6-6, below. 



4-15 ATTENUATION IN THE ATMOSPHERE 

The atmosphere is almost perfectly transparent to radio waves until 
frequencies in the microwave region are reached. Attenuation of radio 
waves in the atmosphere is due to absorption by gases (oxygen and water 
vapor) and absorption and scattering by suspended particles (precipitation, 
dust). The first effect will be discussed here, and the second in Paragraph 
4-16. 



228 



REFLECTION AND TRANSMISSION OF RADIO WAVES 



The theory of microwave absorption by oxygen and water vapor has been 
developed by Van Vleck.^^ The oxygen absorption is due to a large number 
of overlapping resonance lines, resulting in peaks centered at wavelengths 
of 5 and 2.5 mm, while water vapor has an absorption peak at 1.35 cm. 
Fig. 4-46 shows the theoretical attenuation due to oxygen for paths at sea 



3 
100 -1 





1 







v\ 


/AVELENC 
3 


5TH ( 
0.5 


\) cm 
0.3 





.1_ 




_ 


















































































































2 








































































































































































o 




























1 
























































c 
























































o 


















i 




























\ 












Sea Level 


" .1 


M 










n 1 


iL 




















\ \ 










c 


















\\ 


























// 


V 


^-j 


























\, 












4 Kilometer 


'A 1 




s 












n ^^^ 








_- 




-^ 


/ 


/ 




\ 


\ 








^ 












/ 
















/ 










y 
















4- 










^: 


--^ 
















3- 




— 




— 


— 




— 






— 






— 


































3,0 


3 t- 
00 


\ i 


.8 

1 


0,000 




54 6 
60,0C 


8 ^ 

: 


00 


I 
0( 


\ £ 
30- 


8 



6,000 30,000 100,000 

FREQUENCY (f) Mc 
Fig. 4-46 Atmospheric Attenuation Due to Oxygen. 



level and at 4 km. Fig. 4-47 shows the theoretical water vapor attenuation 
in an atmosphere containing 1 per cent water vapor. The attenuation is 
closely proportional to the water vapor concentration. Experimental points 

35See reference of footnote 25, pp. 646-664. 



4-15] 



ATTENUATION IN THE ATMOSPHERE 



229 



3 
100 





1( 


) 5 


WAVELENGTH (\) cm 

3 1 0.5 0.3 

1 1 1 


0. 


1 — 


* 
























































































































Sea Lev 






































c 
























































































1 








4 Kilometers — 4 


1 




























c 






















A 






















Q 














/ 






















/ 








n 1 










A 




/ 






















/ , 


















\ 




' 


/ 


















/ 




















\ 


/ 


















i 














nm 








/ 


A 




















/ 






/ 
















/ 






/ ■ 
















/ 






















/ 






/ 








o 












/ 






















./ 


















/ 






















/ 






















/ 






















/ 






















/ 






1% Water Vapc 


r 


0001- 








/ 


















3 
3,00 


4 6J 


5 2 
10,000 


3 


4 6{ 
50,00 


3 ^ 

D 3 


> 

00,( 


4 6{ 
)00 — 


3 



Fig. 4-47 



6,000 30,000 100,000 
FREQUENCY (f) Mc 

Atmospheric Attenuation Due to Water Vapor. 



due to Tolbert and Straiton^^ show general agreement with the oxygen 
attenuation, but for water vapor the measured values are 2.5 to 4 times the 
theoretical values. Theissing and Caplan" also found that the water vapor 
absorption between the peaks was higher than Van Vleck's theoretical curve 
by a factor 2.7. The reason for this disagreement with the theory is not 
known. 

^^C. W. Tolbert and A. W. Straiten, "Attenuation and Fluctuation of Millimeter Waves," 
IRE National Convention Record 5 (1), 12-18 (1957). 

^''H. H. Theissing and P. J. Caplan, "Atmospheric Attenuation of Solar Millimeter Radia- 
tion," J. App. Phys. 27, 538-543 (1956). 



230 REFLECTION AND TRANSMISSION OF RADIO WAVES 

In the presence of absorption, an additional factor is required in the radar 
equation. This factor is 

lO-o.2adbR (4_81) 

where adt is the one-way attenuation in db per unit distance. 

4-16 ATTENUATION AND BACK-SCATTERING BY 
PRECIPITATION 

Solid particles suspended in or falling through the air can affect radar 
operation both by the attenuation to waves passing them, and by the 
clutter due to back-scattering from them. The attenuation is a combination 
of absorption by the particles and scattering out of the forward beam. The 
particles which are most frequently encountered are those due to precipita- 
tion — viz., water, snow, and ice (hail). Of these, only water absorbs 
strongly, so that its attenuation is caused mainly by absorption. We shall 
give here only some salient features of the attenuation and back-scattering 
by precipitation, since rather complete summaries have been given in the 
literature. ^^'^^ 

For liquid water drops, the attenuation caused by absorption is much 
larger than that caused by scattering. For small drops (7rD/X<5C 1), the 
absorption is proportional to D^ while the back-scattering is proportional 
to D^. Hence the attenuation through small rain drops is proportional to 
the total liquid water content, but the back-scattering is proportional to 
SD^. Thus the larger drops are much more effective in back-scattering than 
the smaller ones. 

Because of the dispersion of water in the microwave region (see Para- 
graph 4-15) the attenuation varies in a complicated way with frequency, 
and also with drop size. The total attenuation is the integrated effect of all 
the drops in the beam between the radar and target, and thus depends on 
the drop size distribution, the drop density (number of drops per unit 
volume), and the length of the path through the precipitation. Drop size 
distribution is known only imperfectly, since most measurements have been 
made by catching rain drops af the ground. The distributions are then 
usually related to the precipitation rate. These may not be the same as the 
distribution and drop density encountered aloft. A further complication is 
that the precipitation density usually is not uniform for any great distance 
through the precipitation region. Hence the calculations made on the basis 
of such measurements necessarily must be considered as only approximate 
estimates of the actual effects which may be experienced. 

^The wartime research is summarized on pp. 671-692 of the reference of footnote 33 above. 
^^K. L. S. Gunn and T. W. R. East, "The Microwave Properties of Precipitation Particles," 
^uart. J. Roy. Meteorol. Soc. 80, 522-545 (1954). 



-4-17] 



ATTENUATION BY PROPELLANT GASES 



231 



Calculations of attenuation and back-scattering (radar area) for spherical 
drops have been made by Haddock"*" on the basis of the drop size distri- 
butions of Laws and Parsons. "^^ These are reproduced in Figs. 4-48 and 4-49. 
The total radar area is found by 
multiplying the value found in Fig. 
4-49 by the volume of precipitation 
illuminated by a pulse length. If the 
entire antenna beam is filled with 
precipitation, then this volume is 
R^^QL. The curves in these figures 
may be extended to longer wave- 
lengths by assuming a dependence 
as X-*. 

Snow is a mixture of air and ice. 
Since the refractive index of ice is 
much smaller than that of water, the 
scattering and attenuation due to 
snow are less than those of a corre- 
sponding mass of water. However, 
when a snow flake begins to melt, it 
becomes coated with a thin film of 
water. The scattering and absorp- 
tion then become almost the same 
as a water particle of the same size 
and shape and thus increase greatly. 
This effect has been advanced as the 
explanation for the radar "bright 
band" observed at or near the freezing level 



10.0 



1.0 



0.1 



0.01 
5 
3 



0.001 



- 










// 




^^ 


150 

mm/hr 
100 - 
50 
25 












/// 


/ 




12.5 






Cloud Burst' 


^' 


'// 


/ 




4.0 - 












H/ 


/ 




2.5 
1.25 




1 1 , 

Excessive Rain^ 


^ 


lil, 


// 


/ 




- 




1 
Heavy Rain^ 


/ 


/ 

/ 


/ 


0.25 _ 


- 


l\ 


/loderate Rain-' 
Light Rain-' 


/ 


'1 , 

7 / 


/ 




- 




















_ 








^ 








_ 






Drizzle-' 
1 













100 5 3 



X(cm) 



Fig. 4-48 The Variation of Attenuation 
with Wavelength for Various Rainfall 
Rates. 



4-17 ATTENUATION BY PROPELLANT GASES 

In the transmission of information between a missile and its ground 
control station, the flame of the propellant gases lies in or near the path 
between the missile antenna and the ground station antenna. Attenuation, 
reflection, and refraction of the radio waves by the flame then are an 
important factor in determining the performance of the radio channel. A 
discussion of the nature of this problem appears in the Guidance volume of 
this series. ^^ 

^"F. T. Haddock, Scattering and Attenuation of Microwave Radiation Through Rain, Report 
of NRL Progress, June 1956. 

*1J. O. Laws and D. A. Parsons, "The Relation of Raindrop Size to Intensity," Trans. Am. 
Geophys. Union 24, 452-460 (1943). 

42A. S. Locke fEd.), Guidance, pp. 118-124, D. Van Nostrand Co., Inc., Princeton, N. J., 
1955. 



232 



REFLECTION AND TRANSMISSION OF RADIO WAVES 

10^ 



3 


I 








- 


2 


= 


^ 


— ~ 


^ 




in 


- 


'_ 


?^ 


^ 


150 mm/h, 
100 Cloud Burst - 
50 Excessive RainI 
?5 


1 
-1 


: 


X 


^ 


^I'X 


- 




\ 




12.5 Heavy Rain - 




= 






\\ 


4 Moderate Rain 
2.5 

-1.25 Ught Rain 

0.25 Drizzle 


1-3 


- 






\ 



0.3 0.5 1 3 

X(cm) 



Fig. 4-49 The Variation of Radar Cross Section of Actual Rain-Filled Space with 
Wavelength, for Various Rainfall Rates. 



The ionization processes which render flames conducting are still not 
completely understood. A recent summary of the subject by Calcote^' 
presents the status of the understanding of these mechanisms. He cites 
experimental evidence from the older literature of ion concentrations of 
10^^ cm~^ From the standpoint of radio wave attenuation, only the electron 
density is of importance, since the conductivity due to a constituent ion of 
a highly ionized gas is approximately inversely proportional to the mass of 
the ion (see, for example, Guidance,'^'^ p. 121, Equation 4-19). 

The ion density varies greatly with the type of fuel. Furthermore, the 
ion density is influenced markedly by small quantities of low-ionization 
potential contaminants. For example, only trace quantities of the alkali 
metals such as potassium and sodium are sufficient to increase greatly the 
ion densities. 

Information on the quantitative attenuations to be expected from jet 
flames can be pieced together from the literature. Adler'*^ measured the 
attenuation in acid-aniline jet flames in a waveguide and found an atten- 
uation of 0.033 db/m at 200 Mc. Since attenuation is approximately 
proportional to/^'^, this is equivalent to 0.25 db/m at X band. Adler also 
observed that the addition of slight amounts of sodium caused large and 
erratic increases in the attenuation. It is probable that much higher 
attenuations would occur in modern high-energy fuels. 

■•^H. F. Calcote, "Mechanisms for the Formation of Ions in Flames," Cotnbiistion and Flame 
1, 385-403 (1957). 

^''F. P. Adler, "Measurement of the Conductivity of a Jet Flame," J. AppL Phvs. 25, 
903-906 (1954). 



4-18] REFRACTION EFFECTS IN THE ATMOSPHERE 233 

Andrew, Axford, and Sugden^^ measured the attenuation at X band in 
the flame of a rifle flash. They found values in the brightest part of the flash 
of 0.6 db /cm. 

The results quoted show that the attenuation in the flames of propellant 
gases can be serious whenever the geometry is such that the flame is a large 
obstacle in the path between transmitter and receiver. For example, a flame 
length of 1 meter in the path could introduce an attenuation at X band in 
the order of 50-60 db. The eff"ects of the flame are likely to be most serious 
as the missile ascends into rarefied air and the size of the flame grows. This 
indicates that special thought should be given to the location and design of 
the antenna on the missile in order to avoid placing the flame directly in the 
propagation path. 

4-18 REFRACTION EFFECTS IN THE ATMOSPHERE 

In computing the power received from a target by means of the radar 
Equation 4-1, allowance was made for a process other than free-space 
propagation by means of the propagation factor F. A process which can 
produce profound modifications is refraction in the atmosphere. 

The atmosphere is a nonhomogeneous dielectric because of the variation 
of its pressure, temperature, and humidity. The variations actually are 
three-dimensional, but the most pronounced refraction eflfects are caused 
by variations in a vertical direction. 

In a homogeneous atmosphere, it is convenient to plot rays as straight 
lines and to show the earth's surface (assumed to be smooth) as a curve. 
If the atmosphere is not homogeneous it is then more convenient to use the 
earth's surface as a frame of reference. Rays which are straight lines in 
space then appear as curves when referred to the earth's surface as the 
abscissa. This is equivalent to the situation where the earth \s,flat and the 
(homogeneous) atmosphere has a constant positive gradient of refractive 
index. This is known as the earth-flattening procedure, in which the actual 
refractivity of the atmosphere is replaced by a modified refractive index. 
The modified index is denoted by M and is determined by the equation 

M = {n-\+ hi a) X 10« = A^ + ^^^ (4-82) 

a 

where h = height above the earth 

a = radius of the earth. 

Its unit of measurement is called the M unit. N is called the refractivity, 
and is the excess of the refractive index over unity, measured in parts per 

*^E. R. Andrew, W. E. Axford, and T. M. Sugden, "The Measurement of Ionization in a 
Transient Flame, Trans. Faraday Soc. A4t, ^HA31 (1948). 



234 REFLECTION AND TRANSMISSION OF RADIO WAVES 

million. Numerically, 10^ hja amounts to 0.048 M unit per foot. From 
Equation 4-82 

f = (:| + ^)x.O' (4-83) 

It follows from this that a homogeneous atmosphere {dn jdh = 0) has an 
M curve with dM jdh equal to lOV'^, and that an atmosphere with a 
constant gradient of refractive index is equivalent to a homogeneous 
atmosphere of effective radius <3e, where 

- = - + % (4-84) 

a, a dh 

In temperate climates an average value oi dn jdh is about —\/{4a). Hence 
from Equation 4-84 

ae = \a (4-85) 

which is the so-called "four-thirds earth." Such an atmosphere is known as 
the standard at7nosphere, and the corresponding M curve, which is a straight 
line of slope 0.036 M unit per foot, as the standard M curve. Actually the 
M curve is rarely a straight line except in a restricted height range. 

The M curve is useful in ray tracing, since a one-to-one correspondence 
exists between the change in slope of a ray over a height interval and the 
change in M. In fact, if represents the elevation angle, measured in mils 
(1 mil = 10~^ radian = 3.44 minutes of arc), at height /z where the modified 
index has the value M, and 0o, Mo are the corresponding quantities at a 
reference height h^ (such as the ground), then 



Q = V^o^ + 2(M- Mo). (4-86) 

It can be seen from this that a height interval over which M — Mo is 
negative will give rise to a decrease in the absolute value of the elevation 
angle. Also, if the M curve has a sufficiently large negative excursion 
(Mo — Mmin > ^0^/2), then the ray will become horizontal at a certain 
height, and then curve back to earth. Assuming no loss in reflection at the 
earth's surface, the process will be repeated over and over, and the ray will 
go through a succession of hops along the surface. The ray is then trapped 
between the earth's surface and the height at which it becomes horizontal. 
A region of the atmosphere within which certain rays are trapped is called 
an atmospheric duct. The multi-hop trajectory resembles somewhat the 
crisscross path between the walls in waveguide propagation; and like a 
waveguide, an atmospheric duct can trap only waves of frequency higher 
than a lower limit. For effective utilization of the duct, both the radar and 
the target should be within the duct. 



4-11 



REFRACTION EFFECTS IN THE ATMOSPHERE 



235 



Since 2{Mmin — M) seldom exceeds 100 M units, trapping occurs only 
for rays with maximum elevation angle (which may or may not occur at 
the ground) of the order of 10 mils, or about |°. Hence trapping is a 
phenomenon which occurs only in almost horizontal propagation. 

The refractivity of the atmosphere for radio frequencies under about 
4 X 10^ Mc*'' is given by the formula 

77.6/. , _ .^ ^^_g^^ 



A^ 



^'(p + 4.81 X 10^^ 



where T is the absolute temperature (°K), p the total pressure, and e 
the partial pressure of water vapor, both in millibars. The refractivity 
decreases with an increase in temperature, but increases with pressure, 
and is especially sensitive to variations in vapor pressure. 

The refractivity at a given point usually fluctuates with time, so that 
average values are used for drawing an M curve. The principal types of 
M curves observed are illustrated in Fig. 4-50. Curve (a) is the standard 




Fig. 4-50 Various Classes of M Curves. 



M curve already referred to. The substandard M curve, shown in (b), is so 
called because the rays are refracted less than in the standard case, and it 
generally results in lower field strengths. Curves (c) and (d) are types 
associated with surface ducts. The duct extends from the surface to the 
height hi, the "nose" of the M curve. In (e) the value of M at the surface 
is less than that at the nose, so that the duct then extends from hi to hi- 
This is called an elevated duct. Various combinations of types can take place, 
such as a surface duct (0 to hi) with an elevated duct (A2 to A3) shown in (f). 
From Equation 4-87, situations where the temperature increases with 
height together with a simultaneous decrease of vapor pressure lead to a 
strong decrease of M with height. Such situations are favorable for duct 
formation. Just such conditions occur at subsidence inversions. These 

«See Essen and Froome, Proc. Phys. Soc. London, B64, 873 (1951). 



236 REFLECTION AND TRANSMISSION OF RADIO WAVES 

usually give rise to elevated ducts, since inversion levels commonly occur 
at 5000 to 10,000 ft. In some cases subsidence inversions descend low 
enough to form a strong surface duct of the type shown in Fig. 4-50d. 

Inversions can also be produced by cooling of the ground at night through 
the process of radiation. In the absence of wind, the radiation inversion 
grows upward as the night progresses, forming a surface duct of the type 
shown in Fig. 4-50c. 

Strong surface ducts are formed when warm air from a large land mass 
moves out over water. The air in contact with the water is cooled and 
moistened. This cooling and pickup of moisture works its way upward with 
time by eddy diffusion. As a result, during the formative process an inclined 
duct usually results, which can extend 200 miles or more out to sea. Duct 
heights can extend up to 1000 feet or so, and hence can influence airborne 
radar operation. 

Weak surface ducts are formed over the open oceans in the trade-wind 
regions. Here the air is colder than the water, so that an increase of 
temperature with height is accompanied by a decrease of vapor pressure 
with height. Their effects on the refractivity thus oppose, as can be seen 
from Equation 4-87, but the influence of the moisture predominates. These 
ducts are very persistent, lasting almost all year round, incident to the 
persistence of the trade winds. The duct height is about 50-75 ft, so that 
they are not very important for airborne radar, except possibly in unusual 
situations. 

An adverse effect on airborne radar can occur when an elevated layer lies 
below the radar and the target. Then, in addition to a direct ray, a ray 
refracted by the layer can be received. At certain ranges, well within the 
horizon, the two rays can interfere destructively, resulting in a decrease 
in field. This is referred to as a radio hole. Radio holes have been observed 
in which the field strength falls by as much as 15 db over a one-way path, 
which would mean a 30-db drop for a radar path. Radio holes extend in 
range for 20 to 50 miles, and so can seriously decrease the range of an 
airborne radar. 

Radio holes have been shown''^ to be caused by only small departures of 
the M curve from a straight line. A layer in which the slope changes by 
as little as 10 per cent of the slope in adjoining regions can produce a radio 
hole. It has been estimated that layers of this kind are present at altitudes 
between about 5000 and 10,000 ft between 50 and 95 per cent of the time. 
Thus this phenomenon can have a profound effect on airborne radar. 

Many of the effects of the varying refractivity of the atmosphere can be 
deduced, and to a certain extent predicted, from climatological considera- 
tions. However, most of the propagation measurements which have been 

'^''Investigation of Air-to-Air and Air-to-Ground Experimental Data, Final Report Part III, 
Contract AF33(038)-U)91, School of Electrical Engineering, Cornell University, 10 Dec. 1951. 



4-18] REFRACTION EFFECTS IN THE ATMOSPHERE 237 

made to evaluate the effects of atmospheric refraction have been either 
between two ground stations, or between an aircraft and a ground station. 
Thus the situations which are encountered in the use of airborne radar 
have not been explored sufficiently to yield a quantitative understanding 
of the meteorological effects which may be encountered. 



R. S. RAV 



CHAPTER 5 

TECHNIQUES FOR SIGNAL AND NOISE 
ANALYSIS 

5-1 INTRODUCTION 

The performance of radar systems can often be determined only by 
tracing the received signal and corrupting noise in detail through the 
individual system components in order to establish the cumulative effect 
of each operation. In this chapter, some of the mathematical methods of 
signal and noise analysis which are appropriate for studies of this kind will 
be developed and their application illustrated with several examples. These 
examples will include a discussion of the characteristics of signal plus noise 
after undergoing some common nonlinear operations, the erratic perform- 
ance of an angle tracking system in response to internally generated noise, 
the clutter cancellation which can be achieved with a moving target 
indicator (MTI system) and the characteristics of a matched filter radar. 

Noise analysis embodies a generalization of classical Fourier methods 
which recognizes the statistical properties of random noise. Much of this 
material will be presented briefly. More detailed discussions can be found 
in the referenced literature. ^~^ 



5-2 FOURIER ANALYSIS 

To develop the theory and methods of noise analysis several basic ideas 
relating to the representation of functions in terms of their frequency 
components as Fourier integrals are required. This paragraph explains 
and illustrates the concepts of: 

1. Fourier integrals or transforms and inverse transforms. 

2. Energy density spectra. 

3. Transfer functions and impulse responses. 

'S. O. Rice, "Mathematical Analysis of Random Noise," Bell System Tech. J. 23. 

2J. L. Lawson and G. E. Uhlenbeck, "Threshold Signals," Chap. 2 (Radiation Laboratory- 
Series) McGraw-Hill Book Co., Inc., New York, 1950. 

"P. M. Woodward, Prohabilily and Information Theory .vith Applications to Radar, McGraw- 
Hill Book Co., Inc., New York, 1953. 

238 



5-2] FOURIER ANALYSIS 239 

We are familiar with the representation of periodic functions by Fourier 
series. A Fourier integral is a limiting case of such a series where the period 
becomes indefinitely long. The separation between components becomes 
indefinitely small as do their magnitudes. For properly restricted functions, 
however, the magnitude density possesses well-defined values and a Fourier 
integral exists. The restrictions on a function /(/) in order that it have a 
Fourier integral are that the integrals of both its square and its absolute 
value have finite values and that it possess only a finite number of dis- 
continuities in any finite interval. When these conditions are met, a 
function F{co) can be defined by the relation 



/. 



F{c^) = / Me-'-'^'dL (5-1) 

If we suppose that/(/) is a function of time, then F{oci) is the spectrum of 
/(/) and gives the density of its diflPerential frequency components in much 
the same way that a Fourier series gives the resolution of a periodic function 
into finite frequency components. The variable co is the angular frequency 
equal to 27r times the cyclical frequency. In general, F(ci;) may be complex. 

The time function /(/) is given by the integral of all the differential 
Fourier components in a manner very similar to the way in which the sum 
of all the components of a Fourier series represents a periodic function. 
Thus,/(/) can be represented in terms of F(co) by the integral 

m = ^_j_^ F{c.)e'-^dc.. (5-2) 



TT. 

The functions /(/) and F{co) are often regarded as constituting a Fourier 
transform pair which are mutually related by Equations 5-1 and 5-2. With 
this terminology, Equation 5-1 is said to transform /(/) into the frequency 
domain, while the operation indicated in Equation 5-2 constitutes the 
inverse transformation. The symmetry of these transforming operations 
is striking. 

As a concrete illustration of such a pair of functions, suppose that/(/) is 
zero for negative values of time while for positive values it is a decaying 
exponential: 

/(/) = e-', </< oo. (5-3) 

The spectrum is easily calculated: 

F{.^) = f (.-0 e-^-^dt = — ^- (5-4) 

In this case, the spectrum is complex. Upon performing the inverse 
operation indicated by Equation 5-2, the exponeni.ial function given by 
Equation 5-3 will again be obtained. We shall not carry out the details of 



240 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

this calculation, which involves treating cu as a complex variable and 
integrating around a semicircular contour in the complex plane. 

The square of the absolute value of the spectrum is important in the 
development of techniques for analyzing noise processes. This is the energy 
density spectrum giving the distribution of signal energy with frequency. 
This terminology is adopted because the function /(/) will normally be a 
voltage or its equivalent, and its square will be proportional to power. The 
integral of the square of/(/), then, will be proportional to the total energy. 
In the development to be given below, it will be shown that (in this sense) 
the square of the absolute value of the spectrum of/(/) gives a resolution of 
the energy into frequency components. This development is obtained by 
manipulating a general definition of the energy density spectrum as it is 
derived from Equation 5-1 : 

!F(co)|2 = F(co)F*(co) 

= / f{ti)e-^'^'^dtA f{t2)ei'''^dt^ (5-5) 

= j_ j_ At,)/(t,)e-^"^'r'.^ dt^dt2. 

Making the substitution t = ti — ti and dr = dti and interchanging the 
order of integration 

I^MP = l_^ e-^'^Ur j _J{t, + T)f{t,)dt, 

= j e-''''<p(T)dT. (5-6) 

The right-hand side of Equation 5-6 is of exactly the same form as 
Equation 5-1; that is |F(aj)|^ is expressed as the spectrum of the function 
<p(t) or its Fourier transform. If ^(t) satisfies the conditions prescribed for 
the existence of a Fourier integral, then the inverse operation given by 
Equation 5-2 is applicable, and <p{t) can be expressed by 

<p{r) = l_J(t + r)/(/) dt = ^l_^ \F(o:)\'- .^^ do:. (5-7) 

When T is set equal to zero, the following important special case is obtained. 

^(0) = /_y'w^^ = ^/_^ \F(-^)\' d^- (5-8) 

This relation is often referred to as ParsevaFs equality. It expresses the idea 
that the total energy of /(/) is equal to the sum of the energies of each 
component of the frequency representation oif{t). 



5-2] FOURIER ANALYSIS 241 

Continuing with the example adopted in Equation 5-3, the form of ^(r) 
should be easy enough to find in this case. 

/• oo 

<p{t) = / (e-^'+^^)(e-')dt, T < 

^(r) = ^-m/ e-'-'dt= i^-M (5-9) 

^(0) = i 
Also, the absolute value of the spectrum is easily obtained from Equation 
5-4 in this case: 

By virtue of the relationship indicated in Equations 5-6 and 5-7, the 
functions given by the two equations above must constitute a Fourier 
transform pair, and the total energy in the signal is |. 

We consider next the effect of transmission through a linear network on 
the time history and spectrum of a signal. Linear networks are conveniently 
characterized in terms of either their impulse response or their transfer 
function. The impulse response, sometimes called the network weighting 
function, is simply the transient output of the network for a unit impulse^ 
at the time / = 0. The transfer function is most commonly defined as the 
complex ratio of the network output to an input of the form exp {iwt). 
These two functions are closely related. In fact, the transfer function is 
the Fourier transform of the impulse response. This relation is made more 
understandable by noting that an impulse function has a uniform spectrum 
(see Paragraph S-S) and so represents an input of the required form where 
all the frequency components occur simultaneously with differential ampli- 
tudes. As an example, consider the single-section, low-pass, RC filter shown 
in Fig. 5-1. Suppose that the driving point impedance is zero and the load 
impedance is very large. Then the transfer function of this network is 
readily recognized as 

Transfer function = ^/^^^c'^{l ^ = y^— (5-11) 



R 
AAAAA/ 1— ^ Transfer Function: 



T 

X 



1+jRCc 
Impulse Response= (l/RC)e 
f>0 



Fig. 5-1 RC Filter. 
*See Paragraph 5-3 for a definition of an impulse function and a discussion of its properties. 



242 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

Similarly, the impulse response of the network is recognized as a decaying 
exponential. If the RC time constant is assumed to be unity, the impulse 
response and transfer function are identical with the functions given as an 
example in Equations 5-3 and 5-4 which make up a Fourier transform pair. 
We denote the transfer function of a network and the input and output 
spectra by Y(o3), Fi((i:), and Fdco), respectively. Since the input spectrum 
gives the resolution of the input into components of the form exp (jW) and 
the transfer function indicates how each such component is modified by 
transmission through the network, it is clear that the output spectrum 
should be given by the product of these two functions. This can be rigor- 
ously demonstrated.^ 

Fo(a)) = y(a;) F, (a;). (5-12) 

The relation between the input and output energy density spectra is 
easily found by multiplying each side of this equation by its conjugate: 

|Fo(a,)|2 = \Y{coy\F,ico)\\ (5-13) 

Thus the input and output energy density spectra are related by the 
absolute square of the transfer function, which might appropriately be 
called the energy transfer function or, if power spectra are being considered, 
the power transfer function. 

It is often convenient to express the time history of the output of a 
network purely in terms of the time history of the input and the impulse 
response of the network. This relation is easily determined by substituting 
for y(co) and Fi(aj) in Equation 5-12 their expressions as Fourier transforms 
o{ y{t) and/i(/), the filter impulse response and the input to the filter: 



/:/- 



Fo(co) = j_ j_^J'(/0/-:(/2)^-^"^''+'^>^/i^/2. (5-14) 

Substituting r = /] + /2 and dr = dti, and interchanging the order of 
integration, 



^o( 



0;)=/ e-^'^'drl fi{t->)y{r - t2)dt.. (5-15) 

The right-hand side of this expression is again in the form of Equation 5-1 ; 
that is, Fo{oi) is given as a Fourier transform. Thus we can formally make 
an inverse transformation of both sides to obtain the desired relation 
between the input and output time histories: 

/o(r) = / //(/)v(r - t)dt. (5-16) 

5See M. 1'". Gardner and J. 1"". Barnes, Transirn/s in Lineur Systems, Vol. 1, pp. 233-236, 
John Wiley & Sons, Inc., New York, 1942. 



5-3] IMPULSE FUNCTIONS 243 

5-3 IMPULSE FUNCTIONS 

Impulse or deltajunctions (so called because they are often denoted by the 
symbol 5) provide a most useful mathematical device in signal and noise 
studies. These functions can be visualized as the limiting form of a function 
whose integral is unity but which is 
concentrated at a particular value of 
its argument. Specific representa- 
tions of impulse functions may take 
a number of forms. One such form is 

shown in Fig. 5-2. In this figure a 

rectangular function of height A and ^ 

width 1 I A is shown centered at the pic. 5-2 Representation of an Impulse 
point t\. As A becomes large, the Function, 
function becomes very highly con- 
centrated at the point A. For any finite value of A, though, the integral of 
this function will be unity and independent of A. Thus, the limit of this 
integral as ^^ ^ <» exists and is equal to the value of the integral. In 
physical problems, it is conventional to suppose that these operations are 
interchanged and that an impulse function denoted by hit — /i) whose 
integral is unity is given by the limit of the function pictured in Fig. 5-2 as 
A — ^ oo . This certainly seems reasonable in view of the fact that for any 
finite A^ no matter how large, the integral is unity. Unfortunately, though, 
integration over the singularity produced when A -^ ^ cannot be justified 
in a mathematical sense, and these operations cannot correctly be inter- 
changed. Thus, although we shall formally regard impulse functions con- 
ventionally as being infinite in height with unit integrals, there is an implicit 
understanding that the limiting operation must, in actuality, be carried out 
after the finite function has been integrated. In this connection we note 
that impulse functions acquire physical significance only after being 
integrated and do not in themselves represent the end product of any 
calculation. 

With these provisos, we proceed to a discussion of some of the properties 
of impulse functions. Probably their most important characteristic is their 
sampling property. The integral of the product of a continuous function 
and an impulse is simply the value of the continuous function at the location 
of the impulse. We can establish this relation with the aid of the represen- 
tation pictured in Fig. 5-2: 

/" r (i+i/2^ 

/(/)6(/ - t,)dt = lim A J{t)dt = /(/:). (5-17) 

A^co Jt,-i/2A 

Additional properties can be established by finding the Fourier transform 
of an impulse function : 



244 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

h{t)e-^'''dt = 1. (5-18) 



/. 



Thus the spectrum of an impulse is constant or uniform. Such a spectrum 
is often referred to as "white noise" in view of the fact that al! frequencies 
are equally represented. If the spectrum of an impulse (unity) is multiplied 
by the transfer function of a network, the spectrum of the network output 
is seen to be simply the transfer function itself. Thus, formally at least, the 
transfer function of a network is the Fourier transform of the transient 
response of the network to an impulse function input as was noted in 
Paragraph 5-2. 

The constant spectrum given by Equation 5-18 does not have a finite 
integral and so does not properly have an inverse Fourier transform. We 
can, however, approximate this spectrum by one which is unity for |co| < A 
and zero for |aj| > A^ where A is very large but finite, and this approxi- 
mation will have an inverse transform. This inverse transform should have 
characteristics very similar to the finite impulse pictured in Fig. 5-2 and 
should approach an impulse function as ^^ — ^ <^ . This turns out to be true 
and gives us a second representation for impulse functions: 

./ N 1- 1 / ,^, 7 r sin At .. .^. 

bit) = hm ;:- / e'"' do: = lim (5-19) 

A-^o.livJ-A /l^co irt 

This expression occurs often in signal and noise studies. Many important 
functions cannot be transformed from the time to the frequency domain 
because the Fourier integral Equation 5-1 does not converge with time. 
Approximations to this integral, however, can often be derived on the 
same basis as for the uniform spectrum. When this is done, the resulting 
expression often contains expressions which can be interpreted as impulse 
functions in the limit. 

For example, it was just established that the Fourier transform of a 
constant/(/) = 1 is an impulse, 27r6(co), which denotes concentration of the 
frequency spectrum at zero frequency. 

Similarly, a sinusoid will have a spectrum which may be derived as 



lim / 

.l-^ccj-.l 



COS (j^it e '"^ dt — lim d 



sin (co + cji).y sin (co — o:\)A 



-|- coi o) — aj]rf 



(5-20) 



= f/7r[5(co + coi) + 5(co — wi)]. 



This expression indicates a spectrum which is concentrated at the positive 
and negative values at the frequency of the sinusoid. 

In deriving the impulse function representation given in Equation 5-19, 
a constant over the entire range of w was approximated by a truncated 
function which approached che constant function in the limit. Other 



5-4] 



RANDOM NOISE PROCESSES 



245 



approximations to the constant function will give different impulse function 
representations. A third representation of an impulse function can be 
obtained in this way by using the triangular approximation shown in Fig. 
5-3. As yf — > 00, this triangular function obviously approaches a constant. 




-2A 2A " 

Fig. 5-3 Triangular Approximation to a Constant Spectrum. 

The limit of the Fourier transform of this function will give the desired 
representation: 

It is apparent that there are a variety of specific representations of 
impulse functions. A familiarity with the forms of the representations, 
so that they may be recognized when they arise during the course of an 
analysis, is useful. A case of this kind occurs in Paragraph 5-5, where in an 
example of a noise process the expression in Equations 5-21 turns up as part 
of the power density spectrum (Equation 5-40). 

5-4 RANDOM NOISE PROCESSES 

In describing noise mathematically, it is useful to visualize a very large 
group or ensemble of noise generators with outputs x{t), x'{t), x"(t), .... 
The output of a specific noise 
generator may be any one of the en- 
semble functions with equal proba- 
bility. The totality of all possible 
noise functions is referred to as a 
random process. Such processes are 
described in terms of their statistical 
characteristics over the ensemble. 
Fig. 5-4 shows a few of the elements 
of a noise process. At any time / the 
mean value, the variance, or other 

statistical parameters can be determined. These parameters can all be 
derived from the probability density Junction of the process at that time 
which describes the distribution of values of the elements of the process. 




Fig. 5-4 Elements of a Noise Process 



246 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

The integral of the probability density function between any two values 
will give the fraction of the elements of the process which lie between those 
values. As an example, we consider the most common type of noise process, 
a Gaussian process, so called because the probability density is Gaussian 
or normal in form : 



Probability density function of a Gaussian noise process 



V2 



— exp;r-^ 



(5-22) 

This process has an average value of zero and a variance or mean square 
value of 0-^. Most important, the probability density is independent of time. 
For most of the noise processes which are of importance in engineering 
applications the statistical parameters are independent of time, and such 
processes are therefore called stationary processes. 

Fig. S-S shows the probability density function for a Gaussian process. 
Most of the elements of the process have values in the neighborhood of the 



exp (x2/2(t2) 




Fig. 5-5 Gaussian Probability Density Function. 



origin. Only a very few of the noise functions will be very large or very 
small at any particular time. If the process is stationary, the values of the 
component functions will have the same distribution at any time. 

As previously noted, Gaussian noise processes are very common in 
physical applications. They can be generated by the superposition of a 
large number of time functions with random time origins. An example is 
the shot noise generated in an electron tube. The random times of arrival 
of electrons at the plate produce the shot noise fluctuations in the plate 
current, which has the properties of Gaussian noise. A mathematical 
example of Gaussian noise is produced by the superposition of a large 
number of sinusoids of different frequencies and random phases. 

Also very useful and important is the joint probability density function 
of values of the process at two different times. For a stationary Gaussian 
process with zero mean, this joint probability density will have the following 
form. 

Second-order probability density function of a Gaussian noise process 



27ra-Vl 



exp 



■vi- + 2p.Vi.V2 - 
2<tH1 - p') 



(5-23) 



5-4] RANDOM NOISE PROCESSES 247 

In this expression Xi and X2 are values of the noise process at times /i and /2, 
0-2 is the variance of Xi and X2, and p is a factor indicating the degree of 
correlation between Xi and X2. This factor is called the normalized autocorre- 
lation function. It is defined in this case, where the mean is zero, as the 
average value of the product XiX^, divided by the average value o{ x"^ which 
normalizes it so that its range is from +1 to — 1. When /i and t^. are close 
together so that x-i and X2 have about the same values, the value of p will be 
close to unity, indicating a high degree of correlation. That is, when Xx is 
high, Xi is also very likely high; and when Xi is low, x^ is probably low. On 
the other hand, when /i and ti are sufficiently far apart for several oscil- 
lations of the noise functions to occur between them, Xi and x^ will tend to 
be uncorrelated and p will be close to zero. When the process is stationary, 
the autocorrelation function will be independent of the particular times 
/i and /2 and depend only upon their difference, which we denote by 

T = ti — /o. 

The significant and meaningful attributes of noise processes must be 
expressed as average values. The notation we shall adopt to indicate the 
average value of some function of the process is to simply bar that function. 
Thus the average value of the process itself is denoted by x- If the process 
represents voltages or currents, then x can be interpreted as the d-c level. 
For the process whose probability density is given by Equation 5-22, the 
average value corresponding to the d-c is zero. The mean square value of 
the process about the mean or the variance can similarly be regarded as the 
average power in a unit resistance. As previously noted, this quantity is 
denoted by cr^. 



a^ = {^x -xY -= x^ - x\ (5-24) 

Actually, the term power will often be used very generally to refer to the 
square of arbitrarily measured variables so that sometimes it cannot be 
identified with physical power, although the electrical terminology has been 
retained. As an example, suppose that an angle 6 is found to be oscillating 
with an amplitude A and a frequency co or 6 = A cos cot. In this case, we 
might say that the angle d has a power of A^ /2 although the dimensions 
of this quantity are certainly not watts. 

The average value of the product X1X2 is very significant in signal and 
noise studies. This quantity is called the autocorrelation function, and we 
shall denote it by ^(r), where r is the time difference ti — t\. For a station- 
ary process, X\ and Xi are uncorrelated when t is very large except for the 
average value or d-c component: 

^(±00)= p. (5-25) 

When T = 0, the autocorrelation function simply equals the average value 
of x^. Thus, the variance of the process is given by 



248 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

^2 = ^(0) _ ^(<x>). (5_26) 

The normalized autocorrelation function can be defined in terms of the 
function ^(t) by subtracting the d-c term and dividing by the variance: 

<p{t) - v?(°o) 

P{t) = —77^ 7— T- (5-27) 

(p(0) - <p(oo) 

For a Gaussian process with the joint probability density given in 
Equation 5-28, the autocorrelation would be computed in the following 
manner: 



^W = o__2./l ~^j_^j_^ '''''' 



exp I 2a'^(l - 2) J^.vi^.V2 (5-28) 



[ 

(t^p{t). 



This integral can be evaluated by completing the square in the exponent for 
one of the variables and transforming to a standard form. In the next 
paragraph, it will be shown that the autocorrelation function is very closely 
related to the power spectrum of the process. 

5-5 THE POWER DENSITY SPECTRUM 

It is possible to decompose random processes into frequency components 
in a certain sense, and this will provide a powerful analytic technique. For 
instance, it was previously mentioned that a Gaussian random process could 
be constructed by the superposition of a large number of sinusoids of 
varying frequency and random phase. This sort of a process can certainly 
be decomposed into frequency components. Of course, the average values 
of the in-phase and quadrature components at a given frequency will be 
zero because of the introduction of a random phase angle. The power at a 
given frequency, though, will be independent of phase and in general have 
a non-zero value. Thus a frequency decomposition could be carried out on 
a power basis. This possibility turns out to be valid for more general 
random processes and leads to the useful concept of the power density 
s-pectrum. Physically, the power density spectrum of a noise process corre- 
sponds to the average power outputs of a bank of narrow filters covering 
the frequency range of the process. 

To develop this idea, consider a stationary random process x{t). Subject 
to the restrictions noted in Paragraph 5-2, the portions of the elements of 
the process between — T and T possess Fourier transform spectra. 

By limiting the range we can ensure that the integrals of the squares ofthe 
elements of the process are finite. Over an infinite range these integrals 



5-5] THE POWER DENSITY SPECTRUM 249 

would not be finite and Fourier transforms could not be defined. Thus we 
have the spectra Xt(co) : 



/: 



Xt(c^) = / x{i) e-^'^'dt. (5-29) 

Energy spectra will be given by expressions similar to Equation 5-6. If 
these energy spectra are divided by the observation time 2T, power spectra 
will be obtained which we denote by A^t(co): 

NtW) = 2^ \Xt{o:)\' = ^ i^e-^'^^drlxit + T)x{t)dt. (5-30) 

The range of the last integral has been denoted by R. Because the elements 
of the process x{t) are in effect zero for |/| > T, the limits of integration will 
be from - T + r to T for r > and from - T to T + r for r < 0. In either 
case, the total range is 2T — |r|. 

We are, of course, primarily interested in the statistical average of 
the power spectrum since only average values represent meaningful and 
measurable attributes of the process. To compute the average value of 
Nrico), we average the product x (t -\- t) x (/) in the expression for 7Vr(aj) 
given by Equation 5-30. The average of this product is the autocorrelation 
function of the process which will depend only upon the time difference r 
if the process is stationary: 



A^r(co) = ;^ e-'^^'dr / <pir)d( 



Letting T— ^ oo, the factor involving Tin the integrand approaches unity, 
and we obtain the following expression for the average power density 
spectrum of the process : 

A^ = / ^(r)^-^"Vr. (5-32) 

This expression gives the power density spectrum as the Fourier transform 
of the autocorrelation function. These two functions form a Fourier 
transform pair and the knowledge of one is, at least in theory, equivalent 
to a knowledge of the other. The inverse of the relation in Equation 5-32 
gives 

^(r) = 2^ / A^^^'^Voj. (5-33) 

When T is set equal to zero in this relation 

^(0) = C72 -f .^2 = i- / W(^) du^. (5-34) 



250 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

Thus, the noise power or mean square vakie is equal to the sum of the power 
components at all frequencies. Equation 5-34 can be regarded as a general- 
ization of Parseval's equality given in Equation 5-8. 

At the end of Paragraph 5-2 it was pointed out that the absolute square 
of the transfer function of a network acts as a transfer function relating the 
input and output energy spectra. We have just defined power density 
spectra as the average of the energy spectra of the elements of the process 
divided by the observation time to give power. Thus the same relation 
must hold between the input and output power density spectra, A^i(co) and 
A^o(co), of a noise process being transmitted through a network with a 
transfer function y(co): 



Noio:) = \Y(w)['N iic^). (5-35) 

We might note at this time that it is normal practice not to use a bar to 
indicate specifically that the power density spectrum of a noise process is an 
average value unless the averaging takes place explicitly in the derivation of 
the power spectrum. Thus, the power density spectrum of a noise process 
would normally be denoted by A^(co) rather than N{co)- 

In order to illustrate some of these ideas, we shall make up a noise process 
and compute its power spectrum and autocorrelation function. We suppose 
the process to be composed of the sum of identical functions A(/) which occur 
at random times. Initially, we consider only functions which originate in 
the finite range — T to + T. We denote the average density of these 
functions by y and suppose that there are ITy = n functions in the finite 
range of interest. Denoting the origin of the ^-th function by 4, our 
approximation to a random process is given by the following expression. 

fn{t) =i:,h{t- /,) (5-36) 

Denoting the Fourier transform o( h{t) by H{co), the Fourier transform of 
/„(/) is given by 

F„(co) = [ h{t - t,)e-i'^^dt = //(co) i; ^-'•"'^. (5-37) 

The power spectrum is simply the absolute square of F(oo) divided by the 
observation time: 

^\FM\-^ = ^[//(c.)E.--^-][//*(co)2:%^-']. 

= ^j^{^)\'zi:^^''''"'-'''- (5-38) 

ZI 1 1 



5-5] 



THE POWER DENSITY SPECTRUM 



251 



To find the average power spectrum, we must average over each variable 4 
supposing it to be uniformly distributed between —T and -\-T: 



2j^l^«(co)p = 22^ 



H{o:)\'^ 



. I'T CT n n 



e-j'^itk-tDdt^ ... dtn. (5-39) 



The integrals of the terms in this sum will have two forms, depending upon 
whether k = I or not. When k = I, the average value of each term is unity. 
There are n such terms. When k t^ I, the average value of each term is 
(sin coT/coTy. There are n(n — 1) of these terms. Thus, the average power 
spectrum has the following form : 

«(w — 1) 



2T 



FM\' = m<^w 



— + 

2T^ 



{iry 



©(-7 



(5-40) 



As T -^ 00 , we note that the term involving the factor sin^ coT is of the same 
form as the definition of an impulse function given by Equation 5-21. The 
power density spectrum over all time, then, will have the following form: 



lim 



^\F{o^)\' 



|i7(co)|M7 + 2x7^5(0.)]. 



(5-41) 



The singular part of this spectrum corresponds to a concentration of 
power at zero frequency or the d-c component. If h{t) has no such d-c 
component, then //(O) will be zero and the impulse has no significance. The 
continuous portion of the power spectrum is seen to be proportional to the 
energy spectrum of A(/). The remarkable thing about this is that the form 
of the spectrum is independent of the average number of functions per unit 
time 7. 

As a concrete illustration, suppose that h{t) is given by the decaying 
exponential defined in Equation 5-3. An element of such a noise process 
might then look like the example shown in Fig. 5-G. The energy spectrum 
of the exponential function has already been computed in Equation 5-10. 




Fig. 



5-6 Element of a Noise Process Composed of Identical Exponential Functions 
with Random Time Origins. 



252 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

The power density spectrum of this process will thus have the following 
form: 

A» = -r^ [t + 27rT^5(co)]. (5-42) 

The autocorrelation function, which is just the Fourier transform of the 
power spectrum, has already been partially computed in Equation 5-9 and 
will have the form 

<p(t) = ^ ^-H + ^2, (5_43) 

We may note that since we have used an A(/) corresponding to the impulse 
response of the RC filter pictured in Fig. 5-1 , the noise process that has been 
defined can be generated approximately by short pulses occurring at 
random times which are modified by this filter. 

A physical interpretation of our model of a noise process is provided by 
shot noise, fluctuations in the number of electrons arriving at the plate of a 
vacuum tube per unit time. We shall use our model to show that the mean 
square fluctuation in electron current, AP, incident to the shot eflFect is 
given by 

(a7)2 = lelAF (5-44) 

where e = electronic charge 

/ = average current (d-c) 

AF = observation bandwidth. 

We suppose that each function h(i — tk) in the sum in Equation 5-36 
represents the arrival of one electron at the plate. In this case, the integral 
oi h{t) should equal the electronic charge e, and we assume this, or what is 
equivalent, that //(O) = e. The magnitude of both the square of the direct 
current and average of the square of the fluctuation or noise currents can be 
determined from Equation 5-41. The square of the direct current corre- 
sponds to the magnitude of the impulse function at zero frequency in that 
expression and is given by 

P = \HmW' = e'y'. (5-45) 

The mean square value of the noise currents corresponds to the integral of 
the nonsingular term |//(co)|^7, in Equation 5-41. We are unable to deter- 
mine this exacdy without knowing the form of the spectrum of a current 
pulse, //(co). If, however, we are interested in the output of a filter which is 
narrow compared to //(co) we can approximate the mean square current 
in the output of the narrow filter by the product of twice the filter band- 
width 2AF and the low-frequency power density of the electronic pulse 



5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 253 

power spectrum. The factor 2 is introduced to account for contributions 
from negative frequencies. Forming this product and substituting //e for 7 
in Equation 5-41, yields the following expression for the mean square 
noise current: 

(KTp = \H{W'i2^F = leHC^FI^ = 2e/AF. (5-46) 

Comparison with Equation 5-44 indicates that the noise process model used 
does indeed give the correct expression for shot noise. The forms of the 
functions h{t) are not significant in this derivation as long as their spectra 
are wide compared with AF. Similar discussions can be made in connection 
with many physical phenomena which generate noise by means of some 
random mechanism. 

5-6 NONLINEAR AND TIME-DEPENDENT OPERATIONS 

In tracing signals and noise through radar systems, we find that the 
operations of many components are either nonlinear or time-dependent. 
Examples of such operations are rectification by second detectors, auto- 
matic gain control, time and frequency discrimination, phase demodulation, 
and sampling or gating. In this paragraph, procedures which can be used 
in the analysis of such operations will be discussed briefly and illustrated 
with a few examples. 

A basic case is provided by a nonlinear device which has no energy storing 
capacity; that is, it is assumed to operate instantaneously. We suppose 
that the input to this device is a Gaussian noise process denoted by x\ the 
output noise process is denoted by jy. The functional relation between these 
processes is denoted by 

v=/W (5-47) 

The process y will be random but not in general Gaussian. The average 
values of_y and jy^ can be found as the weighted averages of/(x) and/^(^): 

y =W) = vi^/-y^^^'"'^'"^^^^ ^^'^^^ 

7 = a/+y^=n^) = j^ f nx)e-^'''~^' dx. (5-49) 



The power spectrum of the y process can be found by first finding its 
autocorrelation function and then computing the Fourier transform of this 
function. The autocorrelation of y is the average value of the product 
yiy^ = f{xv)f{xi)- The average will have to be computed relative to the 
joint Gaussian probability density function expressed by Equation 5-23. 
If this probability density function is denoted by P2{xi,Xi), the autocorre- 
lation of jy is given by 



254 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

<p{r) = 3^2 = j Axr)f(x2)P2{x,,X2)^x,dx2. (5-50) 

The power spectrum of y is simply the Fourier transform of ^(t). 

A Square Law Device. As a specific example, suppose that the 
nonlinear operation is provided by a square law device: 

y = x^ (5-51) 

This type of nonlinearity is often assumed to approximate the rectifying 
action of second detectors in radar receivers. The mean and variance ofjy 
are found by carrying out the operations indicated in Equations 5-48 and 
5-49: 

y = X" = cr'^ 
y = .^ = 3(,4 (5.52) 

The autocorrelation function is found by evaluating the following integral: 

\ —Xi^ + 2pXiX2 — Xi"^! /c cn\ 

exp 1^ 2.2(1 - p2) r^'^''' ^^-^^^ 

= (7^1 + 2p2). 

This integral is evaluated by completing the square of one of the variables 
in the exponent and transforming to standard forms. The constant term in 
<p{t) corresponds to the square of the average value of y and will contribute 
an impulse function at zero frequency to the power spectrum of jy. 

In general, the squaring operation will provide a widening of the con- 
tinuous noise spectrum as the various frequency components beat with 
themselves to produce sum and diflPerence frequencies. To show this and to 
illustrate this type of analysis generally, suppose the x process is similar to 
the one defined in Paragraph 5-5 (Fig. 5-6) by a sum of exponential func- 
tions. For simplicity, we assume that on the average only half of the 
exponential functions are positive while the other half are negative, so that 
the average value of the x process is zero. We assume further that the 
variance is unity. The power spectrum and autocorrelation of the x process 
will be given by Equations 5-42 and 5-43. There will be no d-c term, and 
in order to have unit variance 7 = 2: 

<p(t) = a-'p(T) = .-IH (5-54) 

Nic) = y^, (5-55) 

1 + CO- 



5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 255 

From Equation 5-50, the autocorrelation function of the y process will be 

<pAt) = jT^ = 1 + 2^-21^1. {S-SG) 

The Fourier transform of this expression gives the power spectrum of the 
y process : 



NyiiS) = lirdico) + 



4 + 



(5-57) 



Thus, in this case, the form of the continuous spectrum remained the same, 
but the bandwidth was doubled. 

Another case which is very common in radar applications corresponds to 
the assumption of a uniform spectrum of finite bandwidth for the x process. 
Such an assumption normally represents a simplifying approximation to 
the more complicated forms which actual spectra might take. Such a 



2W 



-2ttW 2irW 

Fig. 5-7a Uniform Spectrum {x Process). 



spectrum is shown in Fig. 5-7a. The autocorrelation function corresponding 
to this spectrum will be 

■2wW 



r2wW 

■J -2^W 



in IttIVt 



ItvWt 
The autocorrelation function of the y process will now be 



Ysin iTrWrV- 
\ 1-kWt J ■ 



(5-58) 



(5-59) 



At the end of Paragraph 5-3 it was indicated that the Fourier transform of 
a triangular function is of the same form as the trigonometric term above. 
Thus the continuous part of the power spectrum of y will be triangular. 
This spectrum is pictured in Fig. 5-7b and it is represented symbolically by 

Ny(a>) = 2Tra^8(o^) + i<r'/fV)il - |a;|/47r/F), Ico] < 4w^. (5-60) 



cr" (Impulse Strength) 




Fig. 5-7b Triangular Spectrum (y = x"^ Process). 



256 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

An application of this result is made in Paragraph 5-7 in course of a discus- 
sion of the effect of the second detector in a pulse radar. 



A Synchronous Detector. Another example which is of interest is 
that of a product demodulator or synchronous detector. Such a device or 
an approximation to such a device is a common component in many types 
of radar systems. It will provide an example of a time-dependent operator. 
In operation, a -product demodulator simply multiplies the signal or noise by 
a sinusoid. Thus if the input is x{t) , the output would be x(t) cos Ww/. When 
x(t) possesses a component at the angular frequency Wm, the dc in the output 
gives a measure of the phase between the input component and the refer- 
ence. We again assume that ;c is a Gaussian noise process with zero mean, 
autocorrelation ^(r), and power density spectrum 7V(co). The autocorre- 
lation of the output is given by 

yiy2 = ■'''1-V2 cos a)mt cos COm (/ + t) 

(5-61) 

= (i) <P{t) [cos OOmT + cos C0m{2t + t)]. 

The autocorrelation of the output evidently varies with time periodically 
at the angular frequency 2ajm- The spectrum of the output will likewise vary 
periodically. In most cases, however, the angular frequency 2aJm is outside 
the range of practical interest, and we can use the time average of the 
autocorrelation or spectrum for our purposes. On taking the time average, 
the periodic component disappears: 



1 [T 

Xr) = yiy2 = hm ;p^ / yiy2dt = (Dv'W cos w^r. (5-62) 



Zl 2r 



The wavy bar is used to indicate a time average. Bearing in mind that the 
autocorrelation function and power density spectrum (p{t) and A^(co) of the 
input noise are Fourier transforms, the Fourier transform of the expression 
above is easily computed to give the output power density spectrum in 
terms of that of the input: 



/:.' 



Nyiw) = h ^(r) cos co„t^-J"Vt 

-I v'(t)[^-'<"-"'"' + ^-'("+"'«)]^r (5-63) 

A product demodulation, then, operates to shift the input power density 
spectrum N{o}) into sidebands about the modulating frequency aj„ and the 
image of the modulating frequency —ojm- 



S-6] 



NONLINEAR AND TIME-DEPENDENT OPERATIONS 



257 



A Clamping Circuit. Clamping circuits, sometimes called pulse 
stretchers or boxcar detectors, are another common component of radar 
systems. They also provide an example of an operation with a periodic 
time dependency. Such a circuit clamps the output to a sampled value of 
the input for a fixed period of time; at the end of this period, the output is 
clamped to a new value of the input. The operation of such a circuit is 
shown in Fig. 5-8. Symbolically, the output of this device can be repre- 



Clamped Output 



nput Signal 




-7 H TIME 

Fig. 5-8 Operation of a Clamping Circuit. 



sented by 

y{t) = x{tk), t, < t < tk + i = tk-\- T. (5-64) 

Clearly the autocorrelation of_y(/) is dependent upon time. As with the case 
of the product demodulator, however, the time average of the autocorre- 
lation function and power density spectrum yield results which can be used 
for almost all applications. To determine the average autocorrelation of 
y{t), consider that when the delay ti — t\ = r, used in computing the 
autocorrelation functions, is a multiple of the sampling interval T, the 
average value of the productjyiV2 of the sampled and stretched process must 
be the same as the average value of the product x-^Xi because at the sample 
points ^1 = xi and jy2 = Xi- Thus, for t = kT, 

<Py{kT) = ^{kT). (5-65) 

When the time delay is intermediate between these isolated points, say 
kT < t < (k -\- l)T, the autocorrelation function of jy will sometimes be 
(p(kT) and sometimes (p(kT -\- T) depending upon the value of /. The 
fraction of the time during which <py{T) takes one of the other of these values 
is proportional to the relative values of r — kT and {k -\- \)T — r. Thus, 
the average value of ^2,(t) should vary linearly between its values at the 
discrete points where r = kT, and it will be composed of these points 
connected by straight lines. 

A limiting case of special interest occurs when the sampling frequency is 
much smaller than the width of the input spectrum. In this case, the 
autocorrelation function of the input is narrow compared with the sampling 
period. That is, values of the process which are separated by more than 
the sampling period are very nearly independent. Since in this case 



258 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



^v{T) ~ 0, the autocorrelation of the output is very nearly a triangle as is 
indicated in Fig. 5-9a. The Fourier transform of a triangular function has 




Delay Time - r 

Fig. 5-9a Autocorrelation Function of Pulse Stretcher Output with Wide-Band 

Noise Input. 



already been determined in Equation 5-21 
spectrum of the stretched process will be 



On using that result, the power 



Nyic.) 



,^( smcoT/2 y 



V coT/2 



(5-66) 



The autocorrelation and the power spectrum of the pulse stretcher output 
in this case are both shown in Fig. 5-9. We might note that one of the basic 
features of this sort of operation is to concentrate the noise in a wide input 
spectrum in a low-frequency spectrum of width approximately 1 jT cps. 



NyiOO) 




-6f -47r -27r 2ir Air Sir 

Nondimensional Angular Frequency.cof 

Fig. 5-9b Power Density Spectrum of Pulse Stretcher Output with Wide-Band 

Noise Input. 



5-7 NARROW BAND NOISE 

Signals in radar systems normally have the form of a radio-frequency 
carrier modulated by a low-frequency envelope which contains the essential 
intelligence. Such signals are filtered and amplified by tuned circuits with 
bandwidths just sufficient to pass the modulation sidebands. Noise asso- 
ciated with signals of this form or originating in circuits designed to amplify 
such signals will have a narrow spectrum centered about the carrier. In 
this paragraph, we shall develop some of the properties of narrow band 
noise and signal plus noise. 



5-7] NARROW BAND NOISE 259 

We suppose that the noise power is concentrated in the neighborhood of a 
carrier frequency coc. Such a noise process can be constructed by modulating 
a relatively low-frequency noise process by the carrier frequency. The 
carrier frequency signal can be represented by either the in-phase or 
quadrature component, and, in general, the narrow band noise will be 
composed of both components. Denoting the low-frequency noise processes 
corresponding to the in-phase and quadrature components about the carrier 
by x{t) and y{t), the narrow band noise process denoted by z{t) can be 
represented by 

z{t) = x{t) cos coc/ + y{t) sin oij. (5-67) 

In general, x{t) and jy(/) could be correlated and also might have dissimilar 
features. But in most problems of practical interest they will be independ- 
ent and have identical spectra and other statistical characteristics. If 
the X and y processes did not have the same spectra and autocorrelation 
functions, the narrow band process would depend upon time, as is apparent 
in Equation 5-68 below. Requiring x and y to be independent makes the 
spectrum of the narrow band process symmetrical about the carrier fre- 
quency coc- We assume that x and y are independent and have identical 
spectra. The autocorrelation function of the z process is computed as 
follows : 



(Pz{t) = [xi cos ixiJi + y\ sin Wct]\[x2 cos 0)^/2 + y^ sin coo/2] 

= (i) XiXi [cos OOcT -j- cos C0c(2/ + t)] 

+ (I) Jl3'2 [cos WcT — cos C0c(2/ -(- t)] 

+ {h) ^ [sin coeT + sin co.(2/ + r)] (5-68) 

— (I) yiXi [sin cocT — sin coc(2/ -{■ r)] 

= (p{t) cos WcT 

where ^(t) denotes the autocorrelation function of the x and y processes. 
The autocorrelation function ^z(r) is of exactly the same form as that of the 
output of a product demodulator discussed in the preceding paragraph and 
given in Equation 5-62. Thus the Fourier transform of .^^(t) giving the 
power spectrum of the z process will be related to the spectrum of the x and 
y processes, A^(co), in a manner similar to that indicated in Equation 5-63: 

A^.('^) = ihWio: - CO.) + A^(co -F COe)]. (5-69) 

From this expression, we see that the spectrum of narrow band symmetric 
noise has the same form as the low-frequency modulating functions, but is 
shifted to the vicinity of the carrier frequency. 

In a large class of radar systems, the transmitted and received signals 
have the form of an RF carrier amplitude modulated by a low-frequency 
waveform. In the majority of these systems, the modulation consists of a 



260 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

periodic pulsing of the carrier. In such systems, the signal, when it is 
present, is a constant amplitude sinusoid. Noise will normally be present 
with a spectrum centered about the carrier frequency and a width deter- 
mined by the amplifiers which are designed to transmit the modulation 
sidebands. The ratio of the bandwidth to the carrier frequency is normally 
very small. Thus the noise can be considered narrow band noise with the 
representation and characteristics described above. 

When a signal is present, it is assumed to be of the form 

Signal = a cos wj. (5-70) 

The peak signal power is denoted by 6" = a} jl.. The noise power is denoted 
by A^ = 0"^ so that the signal-to-noise power ratio when the signal is present 
is 

SIN = ay2a\ (5-71) 

With the narrow band noise represented as in Equation 5-67, the signal plus 
noise has the form 

Signal plus noise = {a -{- x) cos wj + jy sin o^J. (5-72) 

A typical power spectrum of a c-w signal plus noise is shown in Fig. 5-10. 



I 5 6 -Function Continuous 

t Signal ^^ Noise 

i I Spectrum i^^Spectrum 



— COc ^c 

Angular Frequency 
Fig. 5-10 Power Density Spectrum of CW Signal Plus Narrow-Band Noise. 



General operations upon radar signals to extract desired information or 
to transform the signals into a more useable form are often referred to as 
demodulation or detection operations as discussed in Chapter 1. The simplest 
and most common such operation consists in the generation of the envelope 
of a narrow band signal by means of a rectifier. In superheterodyne receivers, 
this operation corresponds to the action of the second detector. The 
envelope output of the second detector is most often referred to as the video 
signal since it is commonly used as an input of some sort of visual display. 
In the following brief analysis, we shall develop some of the more important 
features of video signals and noise. 

The envelope of narrow band signal plus noise can be exhibited by 
rewriting the expression in Equation 5-72 in the following form: 



Signal plus noise = yjia + x)^ + y'^ cos ccj + 



5-7] NARROW BAND NOISE 261 

'-^ -(5-73) 
a -^ xj 

Here an envelope function modulates a carrier frequency with random 
phase modulation depending upon x and y. We note in passing that a 
frequency discriminator would be sensitive to this phase modulation and 
that studies similar to those which we shall make of the video envelope can 
also be made of a discriminator output. 

We first determine the probability density function of the envelope which 
is denoted by r: 



Envelope = r = -yjia + x)'- -j- y^- (5-74) 

The random variables x and y are assumed to represent independent 
Gaussian noise processes with zero means and equal variances. The 
differential probability that they will be found in the differential area (jxdy 
is given by their joint probability times this differential area: 



1 \ —x^ 

dp = Pi(x)Pi{y)dxdy = ^ — -, exp 



2^2 ^^^ 2(7^ 



dxdy. (5-75) 



In order to determine the probability density function of the video en- 
velope, this expression will be transformed to polar coordinates and the 
average value for all angles found. This transformation is represented 
as follows: 

a -{- X = r cos 6 

y ^ rsmd (5-76) 

dx dy ^ r dr dd. 

Substituting these relations into the expression in Equation 5-75 and 
integrating over the variable d gives 



dp = Pi{r)dr = — exp 



2a' 



1 P'^ 
dr ^ / exp [ar cos e/a^dd. (5-77) 

2xJDo 



The integral in this expression can be recognized as a representation of a 
zero-order Bessel function of the first kind with imaginary argument^ 
denoted by loi^r/a^). The probability density function of r is thus of the 
following form: 



Pi(r) = —exp 



2(7^ 



- \h{ar/a'). (5-78) 



A curve showing Pi{r) for some representative values of S /N is given in 
Fig. 5-11. In the two extremes of very small and very large values of the 
signal-to-noise ratio, Pi(r) approaches the following forms: 

^J. L. Lawson and G. E. Uhlenbeck, op. cit., p. 173. 



262 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



17 

S 16 

or 

CO 

z 

Q 14 

5 13 



CQ 



12 







r 


2 2 












P{r)- 


-^"^ 


2<^^ '0la2^" 










/ 


\ 


^aV2(r2=0 








/ 


/ 




\ 










/ 






\ 


^oV? 


7^ = 1 






/ 








^ 








// 


/^ 






N 


is. 




K 








f 


^O- 




^^^^^ 


— 








^^=^s 



1 2 3 

RELATIVE VALUE OF THE ENVELOPE, r/a 



Fig. 5-11 Probability Density Functions of the Envelope of Narrow-Band Signal 

Plus Noise. 






^^^^^^Vlx^^^P 



^72(72 ,»1, 



(5-79) 
(5-80) 



The first of these forms is often called a Rayleigh probability density and 
corresponds to the case of noise alone. When the signal-to-noise ratio is 
large, the envelope has approximately a normal distribution as is indicated 
by Equation 5-80. 

As might be expected from the form of the probability density function 
of r, its basic statistical properties such as its autocorrelation or spectrum 
cannot be expressed simply in terms of elementary functions. Approximate 
expressions valid for either large or small values of the signal-to-noise ratio 
have been developed. '^ Instead of becoming involved with such approxi- 
mations, however, it is often either more convenient analytically or more 
realistic in a physical sense to assume that the second detector is a square 
law rectifier producing the square of the envelope rather than the envelope 
itself. In most problems where such an assumption is made, the variations 
of many phenomena with parameters of interest are relatively independent 
of the detector law. The statistical properties of the square of the envelope 
can be expressed in much simpler forms than those of the envelope itself 
because r^ is a simple second-degree polynomial function of a; and j'. Thus, 
the autocorrelation function of r^ will involve the average values of products 
of the form Xi^x^"^ ^.nA yi^yi^ which have already been evaluated in Para- 
''Uid., Chap. 7. 



5-7] 



NARROW BAND NOISE 



263 



graph 5-6 in connection with the discussion of a square-law device. Using 
the results of that paragraph, the autocorrelation of r- is computed as 
follows : 



ri^ra^ 



a'^ + la^Xx + la^xi + arx^- + arx-^- -f- a?-y ^ + a'-y-^ + 1ax\X'^ 



+ .yi-j'2- + 4^2—2 (5-81) 

= (^2 + 2cr2)2 + 4<72((72p2 + ^2^) 

= (2^2)2(1 + ^/yV)2 + (2a2)2[p2 + 2(.S'/A^)p]. 

The spectrum of the video signal plus noise has three components: (a) an 
impulse at zero frequency representing the d-c, (b) a continuous portion of 
the same shape as the spectrum of the component x and y processes repre- 
senting beats between the signal and the noise, (c) a continuous portion 
somewhat wider than the spectrum of the x and y processes representing 
beats between various parts of the noise spectrum. Fig. 5-12 illustrates the 



Impulse 



Signal Power = a^/2 
Noise Power ■=2D\N=u 
S/N= a2/2(7" 




27r(ay4) 
Impulse 



27rW' 



47r W 

Signal Plus Noise 



Angular ^ 27rW"^ 1^ 

Frequency 





27r (2(7^)2 
Impulse 








/ 


\ t 








1 


/ 


t 
D 

1 


1 





Noise Alone 



Angular Frequency 



Fig. 5-12 Power Density Spectra of the Square of the Envelope of a Sinusoidal 
Signal Plus Narrow-Band Noise. 



forms of the various spectra in a typical case. The spectrum of the x and y 
processes is assumed rectangular with bandwidth W. The density of the 
positive and negative portions of the narrow band spectrum is denoted by 
D. The d-c level is equal to twice the sum of the signal and noise powers. 
The portion of the continuous spectrum corresponding to (b) is rectangular, 
of half the width of the narrow band spectrum (considering only positive 



264 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



frequencies), and has a power density equal to the product of 8 times the 
signal power and D. The portion of the continuous spectrum corresponding 
to (c) is triangular, with a width equal to that of the narrow band spectrum 
and with a power density at zero frequency equal to the product of 8 times 
the noise power and D. 

5-8 AN APPLICATION TO THE EVALUATION OF ANGLE 
TRACKING NOISE 

In this paragraph, the techniques developed for tracing signals and noise 
through radar systems will be illustrated by a discussion of the performance 
of an angle tracking loop in a pulse radar as a function of the signal-to-noise 
ratio. A block diagram showing the elements of the receiver composing this 
angle tracking loop is given in Fig. 5-13. This diagram represents a pulse 

Scanning 
Reference 








Antenna 
Gimbal 




Antenna 




T . .. 

Demodulator 


^- 


Pulse 
Stretcher 






( 


Controller 

























ANGLE TRACKING LOOP 


) 










— w 












Mixer 




IF 
Amplifier 




2d 
Detector 




Range 
Gate 










t 
AFC 














t 

Gating 










AGO 

Filter 


Pulses 1 

















Fig. 5-13 Block Diagram of Angle Tracking Loop Employing Conical Scanning. 

radar with a pencil beam which is conically scanned to generate an angular 
error signal. 

A signal received from a target which is being tracked will have the 
following form: 

Received signal = a[\ -\- ke cos (ws/ + f)] cos Wct (pulse modulation) 
where a = signal amplitude (5-82) 

k = modulation constant of the antenna 
e = angular error magnitude 
<p = angular error direction 
Ws = scan frequency (rad/sec) 
ojc = carrier frequency (RF or IF, rad/sec). 



5-8] APPLICATION TO EVALUATION OF ANGLE TRACKING NOISE 265 

The angular error information is contained in amplitude modulation at the 
scanning frequency. We shall refer to this modulation as the a-c error signal. 
Its amplitude is proportional to the error amplitude, while its phase gives 
the error direction. The RF carrier of the received signal is transformed 
to an intermediate frequency in the mixer or first detector. The IF amplifier 
then provides the necessary gain and maintains the average level of the 
signal at a convenient constant value in response to the feedback signal 
from the AGC (automatic gain control) filter. The envelope of the IF 
signal is developed by the second detector, which is basically a rectifier. 
For our purposes we shall assume the second detector to be a square-law 
device whose characteristics have already been discussed to some extent in 
the preceding paragraph. A range gate selects only pulses occurring at the 
proper radar time for use in deriving the angle error. The range gate is 
positioned by an auxiliary range tracking loop which is not shown in Fig. 
5-13. The AGC loop maintains the d-c value of the video signal during a 
pulse at a constant value so as to preserve a fixed relation between per cent 
modulation at the scanning frequency and angular error, independently of 
the received signal strength. 

A pulse stretcher generates a continuous signal suitable for use in 
the low-frequency control circuits from the pulsed signal delivered by 
the range gate. The output of the pulse stretcher is delivered to a product 
demodulator or synchronous detector which develops a servo control signal 
from the a-c error. 

Internally generated noise arises primarily within the mixer and the 
first stages of the IF amplifier. The noise may be represented exactly 
as in Paragraph 5-7 (Equation 5-67). That is, in-phase and quadrature 
components at the carrier frequency are modulated by independent low- 
frequency noise processes which we denote by x and y. The noise power 
is denoted by cr^ so that the average signal-to-noise ratio will be 

Signal-to-noise ratio - SIN = a^/ld"^. (5-83) 

This is an average signal-to-noise ratio because, on a short term basis, the 
signal power is modulated by the a-c error signal. 

The signal plus noise during a pulse will be of the following form: 

Signal plus noise = [a{\ + ke cos (cos/ + <p)) + x] cos Wct + y sin oij. 

(5-84) 

The video envelope from the square law detector during a pulse consists of 
the sum of the squares of the in-phase and quadrature components: 

Video signal plus noise = r^ = a'^[\ -\- 2ke cos (cos/ + tp) 

+ kh"" C0S2 (oj,/ -f if)] 

-\- 2ax[l + ke cos (co^/ + cp)] 

+ ^2_^y. (5-85) 



266 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

In analyzing the effects of the remaining circuits in the loop on this signal, 
it is convenient to make certain simplifying approximations. First, it is 
assumed that the fractional modulation kt is small enough that its square 
may be neglected. Second, when determining the spectrum of the video 
noise, the target is assumed centered in the beam so that ke is zero. This 
reduces the expression above to the case already considered in Paragraph 
5-7 (Equation 5-74). 

The average value of the video signal plus noise during a pulse will consist 
of a d-c term and the a-c error signal: 

Average video = ^2 ^ ^2 j^ 2(j^ -[- la'ke cos (co./ + ip). (5-86) 

The AGC loop will act to maintain the value of the d-c part of the video at 
a constant level which we may conveniently assume to be unity. Thus, 
ideally, the effect of the AGC is to divide the video by its d-c level. We 
assume that the AGC loop does indeed operate in this manner, although 
in an actual system only an approximate quotient would be formed. This 
assumption is sufficiently accurate for our purposes. In this case the 
effective a-c error signal during a pulse becomes 



/ S/N \ 

\\ + s/n) 



A-C error signal = f , , c/at ) ^^e cos (co^/ + <p). (5-87) 

One effect of the noise is to introduce a factor depending upon the signal- 
to-noise ratio which attenuates the a-c error at low values of this ratio. 
The net result of this suppression of the signal by the noise is to decrease 
the gain around the angle tracking loop. 

A pulse stretcher is used to generate a signal suitable for use in the low- 
frequency control circuits from the pulsed signal delivered by the range 
gate. The pulse stretching operation will introduce some distortion of the 
angular error modulation, but because the scanning frequency is normally 
much smaller than the pulse repetition frequency, this distortion can be 
neglected and the pulse stretcher assumed to generate the fundamental 
component of the pulsed signal. Thus the a-c error signal delivered to the 
phase-sensitive demodulator is essentially of the form given in Equation 
5-87. 

We suppose the demodulator to be a simple product type consisting of a 
multiplication of the modulated error signal by a sinusoidal reference, 
(1 Ik) cos oist. The factor 1 jk is incorporated in order that the output may 
be equal to the angular error. The properties of such a device with noise 
inputs were established in Paragraph S-6. The demodulator output is 
filtered so that only the very low frequencies are retained (components of cog 
and above are eliminated) as the angular error signal. The development of 
the error signal in the demodulator can be represented by the following 
operations: 



(5-88) 

€ COS ip. 



5-8] APPLICATION TO EVALUATION OF ANGLE TRACKING NOISE 267 
Error signal = [ SiN r "^ ^^^ ^°^ ("«/ + ip)\{\IK) cos w,t 

= ( S/^ \ 

VI + S/NJ 

In this expression the wavy bar indicates the time average, which eliminates 
the fluctuating terms. The factor cos <p indicates that the error derived is 
the projection of the total error on the axis represented by one of the angle 
tracking loops. Complete directional control of the antenna requires it to be 
controlled in two directions, normally azimuth and elevation. The error 
signal for the other loop is obtained from a demodulator with a reference 
sin cos/. 

This error represents the input to the antenna controller which moves the 
antenna in order to null the error and track the target. In order to arrive at 
a definite result in this example, we shall assume that the antenna controller 
is composed of a single integrator, although in a practical system the 
dynamic response of the angle tracking loop might be quite complicated. 
With this assumption, the response of the whole loop becomes the same as 
that of a low-pass RC filter, and the power transfer function has the 
following familiar form. 

Angle tracking loop power transfer function = ^ ^^^ (5-89) 

where K = gain around the tracking loop = bandwidth (rad/sec) 

As noted above, the gain K will be attenuated by a factor depending on the 
signal-to-noise ratio. Thus we shall express K as the product of this factor 
and a design bandwidth /3 achieved at high signal-to-noise ratios: 

Our primary interest in this example is to determine the response of the 
loop to internally generated noise. It will turn out that the spectrum of 
the equivalent noise input to the loop is very much broader than ^ and 
relatively flat in the low-frequency region. If we denote the power density 
of this input noise spectrum by D in angular units squared per rad/sec, 
the variance of the tracking noise will be given by 

Mean square trackmgnoise = :^j_^ ^^^^-p^, = ^ = [j^^-sJnKJ )' 

(5-91) 

The next problem is to determine the magnitude D of the input-power 
density spectrum. 



268 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

As already noted, the error is assumed zero when the spectrum of the 
video noise is determined in order to simplify the calculations. This 
corresponds to the case already considered in Paragraph 5-7. The spectrum 
of the video noise is thus pictured in Fig. 5-12, and its autocorrelation 
function is given by Equation 5-81. Dividing the noise power in the square 
of the envelope as determined from these sources by the square of the d-c 
level, to account for the effect of the AGC, gives the effective video noise 
power during a pulse: 

Video noise power (with AGC) - ^[^"V^w'^!^ ' (^-92) 

With a pulse width normally on the order of a microsecond, the width of the 
IF pass band, W cps in Fig. 5-12, must be approximately 1 Mc/sec or 
greater. The spectrum of the video noise will also be approximately of this 
width with a correlation time on the order of a microsecond. The repetition 
rate on the other hand will normally lie in the range from a few hundred to 
a thousand cps. Pulses will thus be separated by at least a millisecond, and 
the pulse-to-pulse fluctuations due to internal noise should be very nearly 
independent. 

The effect of the pulse stretching operation is considered next. In Para- 
graph 5-6 the spectrum of the output of a pulse stretcher was developed 
from an input of independent noise pulses. This is exactly the situation 
being considered in this example. Thus the spectrum of the stretched 
signal plus noise should have the form given by Equation 5-66 which was 
illustrated in Fig. 5-9. If we denote the repetition period by T, the power 
spectrum of the input to the demodulator will be of the following form: 



Noise spectrum of demodulator input 



[1 + 1{SIN)\T 
(1 + SINY 



sinjo7y2]2 
cor/2 J 

(5-93) 



(1 + SINY 

The effect of the demodulator on its input spectrum was established in 
Paragraph 5-6 (Equation 5-63). The demodulator input spectrum will be 
shifted back and forth by the demodulating frequency and multiplied by 
the factor (1/4 k''): 

Noise spectrum at demodulator output = ' ' [A(a; + wj 

+ A^(co - CO.)]. (5-94) 

The width of each component of this spectrum is approximately 1 /T cps, 
which normally might be on the order of a few hundred to a thousand cps. 
Since the bandwidth of the tracking loop will normally be only a few cps, 
only the power density in the neighborhood of zero frequency is significant; 



5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 269 

that is, the noise spectrum may be assumed to be uniform without appre- 
ciable error: 

[1 + 2(S /N)]T 
Power density of demodulator output noise = D = 7- — ; — LT\i,^ A^(cos). 

(1 + o/i\)^kl 

(5-95) 
A further simplification can often be made when the ratio of the scanning 
to the repetition frequencies is small. In this case, the factor N(cos) is 
approximately unity. For example, when the ratio of these frequencies is 
1 : 10, the value of A^(co,) is 0.97. 

Substituting the power density D given in Equation 5-95 into the relation 
already derived for the mean square tracking noise (Equation 5-91) and 
assuming that N(olIs) is unity gives the following expression for the tracking 
noise variance: 



Mean square tracking noise 



(S/N)[l + 2(.V/A^)] / 7^\ 

(1 + s/Nr \4ky' ^^"^^^ 



This expression represents the end product of our analysis of the effect of 
internally generated noise on the performance of a conically scanned angle 
tracking loop. It \& interesting that the tracking noise from this source has 
a maximum at a signal-to-noise ratio of 1.35 db. The decrease in tracking 
noise at small signal-to-noise ratios is due to the loss in loop gain and 
consequent narrowing of the loop bandwidth. When this begins to happen 
in a practical system, dynamic tracking lags usually cause an early loss of 
the target. We also note that the rms tracking noise is directly proportional 
to the square root of the repetition period and inversely proportional to the 
modulation constant of the antenna. This constant, expressed in per cent 
modulation per unit error, is itself inversely proportional to the antenna 
beamwidth. 

The analysis in this example was intended to illustrate the sort of 
considerations which are appropriate to a study of noise in a radar tracking 
loop which incorporates a variety of components — some of them nonlinear 
or time dependent. Similar analyses can be made of other types of tracking 
systems such as monopulse tracking loops, range tracking loops, and 
frequency tracking loops. The effects of externally generated random 
disturbances such as glint or amplitude noise will also be handled in a 
similar fashion. 

5-9 AN APPLICATION TO THE ANALYSIS OF AN MTI 
SYSTEM 

In this paragraph we shall make some observations on the performance 
of a radar system which provides moving target indication (MTI). This 
analysis will supply another example to illustrate the use of the mathe- 
matical techniques which have been developed. The MTI system which we 



270 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



shall consider is a noncoherent delay-line cancellation system. In such a 
system, both ground clutter and target reflections are received simul- 
taneously. The RF carrier frequency of the ground clutter is denoted 
by Wc> while that from a target moving relative to the ground will possess 
a doppler shift ud and is denoted by coc + cod. When the sum signal is 
detected, a beat is produced at the doppler frequency wd- If there is no 
target present, there is no doppler beat, and the spectrum of the detected 
video is concentrated at d-c and in the neighborhoods of harmonics of the 
pulse repetition frequency. The doppler signal can be separated from the 
clutter background by means of a delay-line cancellation unit. This unit 
provides the difference of successive returns as an output, that is, returns 
separated by the repetition period T. Fig. 5-14a shows a block diagram of 
such a cancellation unit, while Figure 5-14b illustrates its operation. This 



Input 



Time 
Delay 

p-iujT 



Output 



Fig. 5-14a Delay-Line Cancellation Unit. 



Input 

Signal HI- 
Delayed 
Signal 



i~x,.npv^^xi ^ 






-\ 



'^t 



/^Tx „ /IN ,, /I , 



Cancelled 
Signal il-^ ^^\ly '^^\iy 

Fig. 5-14b Cancellation of Clutter Echc 



sort of unit will attenuate the d-c component and all harmonics of the 
repetition frequency and in this manner cancel most of the clutter. When 
the doppler frequency lies between these harmonics, it will be transmitted 
through the cancellation unit. If by chance the doppler frequency coincides 
with one of the repetition rate harmonics, it will be canceled along with the 
clutter and produce a blind region or range of doppler frequencies to which 
the system is insensitive. Blind regions represent one of the most serious 
limitations of this type of system. 

Proceeding with the analysis, the clutter echo at a given range is repre- 
sented before detection as a narrow band noise process: 

Clutter echo = xU) cos uj + v(/) sin cor/. (5-97) 

The modulating functions .v andjy are independent Gaussian noise processes 
with identical spectra. The clutter spectrum is determined by the motion 



5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 271 

of ground objects, the scanning of the antenna, and the motion of the 
platform on which the antenna is mounted. The clutter power is denoted 
hj C = x^ = y, and the autocorrelation function of the x and y processes 
is denoted by Cp = XyX2 = yxji- 

During a pulse, the echo received from a moving target is assumed to be 
a sinusoidal signal with a doppler shift: 

Target echo = a cos (ojc + co^)/. (-5-98) 

The peak signal power is denoted by 6" = a} jl. 

We shall assume that the signal plus clutter is rectified by a square-law 
second detector to give the following video signal during a pulse: 

Video = V = \a cos (coc + co^)/ + -v cos (xsct + y sin cor/]". (5-99) 

The video frequencies are, of course, limited by the video bandwidth. 
Squaring this expression and retaining only the low-frequency components 
which will be passed by the video amplifier gives 

Video = y = \{a} -|- a;- + .V^ + lax cos cod/ — lay sin cod/). (5-100) 

The cancellation unit acts to generate the difference of video signals 
separated by a repetition period. Denoting the residue from the cancel- 
lation unit by r(/), we have 

Residue signal = r{t) = v{i) — v{t — T) = V\ — v-i 

= ^{xi^ - X2'' + yi^ - y2^ + 2axi cos co^/i (5-101) 

— 2^X2 cos 0)^/2 — 2ayi sin cod/i + 2ay2 sin cod/2). 

In order to evaluate the effect of the cancellation unit in reducing the 
clutter, it is convenient to define a video signal-to-clutter ratio. This ratio is 
defined as the difference between the video power with a signal v^j^^ 
and the video power with clutter only y^ divided by this latter quantity. 

Video signal-to-clutter ratio -= {S/C)v = (y|+c — vl)/ v^- (5-102) 

Similarly, a signal-to-clutter ratio is defined for the residue signal output 
of the cancellation unit: 

Residue signal-to-clutter ratio = {S/N)r = (r|+c — ^c)/ ^c' (5-103) 

With these definitions, a gain factor may be determined as the quotient of 
these two ratios: 

System gain factor = G = ^^^ (5-104) 

In order to evaluate this gain factor in terms of the system parameters, 
the average values of the squares of the video and residue signals must be 
calculated. This is somewhat complicated because of the large number of 



272 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

terms resulting from the squaring of Equations 5-100 and 5-101, and the 
details will not be given here. The following average values originally- 
determined in Paragraph 5-6 in connection with a discussion of a square-law 
device are used in these calculations: 

X = y = x^ = y^= xy- — x-y — 

x^ = y^ = c, X1X2 = yiy2 = Cp{T) (5-105) 



x^ = y^ = 3C2 = xi^xs^ = y.'^y^'' = C^[l + 2p2(T)l. 

The following results were determined for the video and residue signal to 
clutter ratios: 

{S/C\ = 2{S/C) [1 + i {S/C)] 

1 - p{T) coscodT] 



{S/C)r = 2(S/C) 
The gain in signal to clutter ratio will simply be 



1 - pHD I (^-'0') 



System gam factor = G = ^ _^ a)(S/C) 1 _ 2(7-) (5-108) 

This expression essentially summarizes the ability of a noncoherent MTI 
system to reduce clutter. Various interesting observations might be made 
from a study of this factor. For instance, the depth of the blind speed nulls 
at harmonics of the repetition frequency can be determined as a function of 
the normalized autocorrelation function of the clutter at the repetition 
period. The average gain over all doppler frequencies can also be found as 
a function of the same parameters. These details will not be explored here. 
The primary purpose of the example has been served by the derivation of 
Equation 5-108, which showed how a performance equation could be arrived 
at by a straightforward application of the techniques for signal and noise 
analysis previously developed. 

5-10 AN APPLICATION TO THE ANALYSIS OF A 
MATCHED FILTER RADAR 

In this paragraph, we shall consider how the mathematical techniques 
which have been developed can be applied to the derivation of optimum 
radar systems. Besides providing a good illustration of the application 
of these techniques, this example will also provide an insight into the 
important basic factors which affect system performance and set theoretical 
performance boundaries which a practical system may approach but not 
surpass. 

We shall be primarily concerned with the detection performance of radar 
systems. A fundamental problem in detecting a radar target is to distin- 



5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 273 

guish the target echo from random noise which tends to obscure it and 
render detection a matter of chance. This is the problem that we shall 
discuss in this paragraph. We shall determine the characteristics of an 
optimum receiver which will provide the most reliable detection of target 
echoes obscured by random noise. 

There are several possible approaches to this problem, depending upon 
the generality desired, the definition of most reliable detection adopted, and 
various assumptions made about the signal. We shall adopt the simplest 
possible approach, although the receiver design criterion which will be 
derived is operationally equivalent to the results of more sophisticated 
analyses in most cases. 

We suppose that in the general radar situation a signal is received as an 
echo from the target. During the process of reception, noise is added to the 
signal. The question we consider is, "What function must the receiver 
perform in order that the most reliable detection of the signal may be 
obtained?" We shall limit our study to receivers which are linear. That is, 
the effect of the receiver on the signal and noise is that of a linear filter. 
The output of the receiver-filter will consist of a filtered signal and filtered 
noise. Thus a ratio of the output signal and noise powers can be formed. 
We shall choose the optimum receiver-filter as that which maximizes this 
signal-to-noise ratio. We shall subsequently indicate how a maximum 
signal-to-noise ratio gives a maximum probability of detection for a fixed 
false-alarm rate and thus provides the most reliable detection in this sense. 
It will turn out, interestingly enough, that the receiver-filter which is 
optimum in the sense described above has a transfer function which is the 
conjugate of the target echo spectrum,^ and for this reason such a radar is 
often called a matched filter system. That is, the filter transfer function is 
matched to the target echo spectrum. We shall also demonstrate that such 
a system is equivalent to a cross correlation of the signal plus noise with an 
image of the signal waveform which is the origin of the term correlation radar 
sometimes used in reference to such systems. 

We adopt the following notation for this analysis: 

sit) = signal input to receiver-filter 

S(o}) = spectrum of s(t) 

So{t) = signal output of receiver-filter 

So{co) = spectrum of So(,t) 

^This result is sometimes called the Fourier transform criterion and is attributed to a number 
of authors: namely, D. O. North, W. W. Hansen, N. Weiner, J. H. Van Vleck, and D. Middle- 
ton. See particularly Van Vleck and Middleton, "A Theoretical Comparison of Visual, Aural, 
and Meter Reception of Pulsed Signals in the Presence of Noise," J. Appl. Phy. 17, 940-971 
(1946^. 



274 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

Y(o:) = transfer function of receiver-filter 
n(i) = noise input to receiver-filter 

D = power density of noise input to receiver-filter 
no(f) = noise output of receiver-filter 

0-2 = noise power in output of receiver-filter 

z^ = peak signal-to-nolse power ratio in output of receiver-filter 

/o = observation time 

The target echo is represented by a signal input to the receiver-filter 
denoted by s{t) with a spectrum S((a). The signal output of the filter and 
its spectrum are denoted by So(t) and «S'o(aj). The transfer function of the 
filter is represented by F(co), and the output signal spectrum is equal to the 
product of this transfer function and the input signal spectrum: 

So{c^) = FM .S'(co). (5-109) 

The output waveform will, of course, be simply the inverse Fourier trans- 
form of So{oi) : 



i/. 



Output signal = r„(/) = -^ / Yico) S{oo) ^^"Wco. (5-110) 
It J -ex, 

We choose to make our observation of the output at the time to- It is 
supposed that /o is selected so that the whole of the input signal is available 
to the filter. The signal power in the output of the filter at the observation 
time will be Sg^Uo), while the noise power in the filter output is denoted by 
0-2. The input noise is assumed to be Gaussian with a uniform or "white" 
spectrum with power density D. The output noise power will thus be 

1 f" 
Output noise power =^ 0-2 = / D|y(a))|Vw. (5-111) 

The output signal-to-noise ratio at the time /^ is denoted by 2-: 



Output signal-to-noise ratio = 2- = So-(/o)/(r~ = 



kl. 



y(co).V(a;)^^'-Vco 



■/ D|y(co)|Va; 

(5-112) 



The minimum value of this ratio can be determined by means of 
Schwarz's inequality. This can be derived in the following fashion. Suppose 
that the functions /(;c) and g{x) and the parameter n are real. Then the 



5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 275 

following quadratic function of m will always be greater than or equal to 
zero: 

J^ [m/W + ^W]Vx = m'^ j^ nx)dx + 2m y^ Ax)g{x)^x +j^ g\x)dx ^ 0. 

(5-113) 
This expression is represented by 

^m' + 25m + C ^ 0. (5-114) 

Because this polynomial is always greater than zero, the equation 

^m' + 25m + C = 0. (5-115) 

cannot have distinct real roots, and its discriminant must be less than or 
equal to zero: 

B'-JC^O. (5-116) 

Substituting for yf, B, and C gives the real form of Schwarz's inequality: 

(/^VkW^^)' ^ /^ f{x)dx j\Kx)dx. (5-117) 

The absolute value of the product of two complex numbers is always less 
than or equal to the product of their absolute values. Further, the square 
of the absolute value of the integral of a complex function is always less than 
or equal to the integral of the square of the absolute value of the integrand. 
Combining these ideas, we note that when/(;c) and g{x) are complex. 



f 



f{x)g{x)dx 



i: 



L 



\f{x)g{x)\^dx ^ \f{x)\^\g{x)\^dx. (5-118) 



This immediately leads to the more general form which we need. Putting 
|/(;c)| and |^(^)| in place of/{x) and g{x) in Equation 5-118: 



/ 



f(x)g(x)d> 



j\f{x)\'dxj\g{x)\^dx. (5-119) 



Substituting the right-hand side of this inequality for the numerator in 
Equation 5-112. 



^j_^ \Sic.)\'dc.^j_^ |y(a;)lVc.. (5- 

^J_^D|y(co)|Va; 



120) 



The integrals involving the filter transfer function can be canceled: 

2^^(^)^/_J^M|Vco. (5-121) 



276 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

Thus the right-hand side of this inequality is independent of the filter. Since 
the signal-to-noise ratio is never greater than the right-hand side of the 
expression above, and this expression does not contain y(w) at all, it must 
give the maximum value for 2^ for the optimum choice of y(co). Referring 
to Equation 5-112, it is apparent that the denominator in that equation 
will be canceled and the maximum value of z^ achieved if the filter transfer 
function is made the complex conjugate of the signal spectrum: 

y(co) = ^*(co)^-'"'«. (5-122) 

A receiver-filter which is designed on the basis of this principle, where the 
receiver transfer function is matched to the signal waveform, is often 
referred to as a matched filter system. Another general term which is also 
used in reference to such systems is correlation radar. This terminology 
originates in the observation that the ideal filtering operation is equivalent 
to a cross correlation of the signal plus noise with an image of the signal 
waveform. In order to see this, the impulse response of the matched filter 
is found by taking the inverse Fourier transform of Y{oo) : 

1 /"" 
Impulse response of matched filter = :r— / S*{ii))e '"^"'^'"^dco 

(5-123) 

= sUo - /). 

Denoting the input noise process by n{t) and the output noise process by 
no(t) and using Equation 5-16 to relate the time histories of the input and 
output signal plus noise gives for the filter output: 

Soil) + noil) = l_^ [sir) + nir)]sito - t + r)dT. (5-124) 

In particular, the output at the observation time to is simply 

soito) + rioito) = J_^ \sir) + niT)\siT)dr. (5-125) 

Thus, from this relation it is clear that the optimum receiver could consist 
of taking the cross correlation of the received signal plus noise and the pure 
signal waveform and that a matched filter receiver and a cross correlation 
receiver are equivalent. 

Going back to Equation 5-121 for a moment, we might note an interesting 
basic feature of radar systems which are theoretically optimum in the sense 
of this paragraph. The maximum signal-to-noise ratio is equal to the ratio of 
the received signal energy to the power density of the noise. That is, the 
maximum signal-to-noise ratio does not depend upon the waveform of the 
signal. This is not to say that the waveform is not important. Resolution, 
tracking accuracy, and many other system characteristics are closely related 



5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 277 

to and depend upon the wave shape of the signal. For the detection prob- 
lem, though, it is the received energy that counts. 

As a concrete illustration of a matched filter, suppose that the signal 
waveform consists of a series of n identical pulses separated by a repetition 
period T. Such a signal is of common occurrence in radar problems. 
Denoting an individual pulse by /)(/), the signal is defined by 



Pulse train - s{t) =!]/>(/- kT). 



This signal is depicted in Fig. 5-1 5a. 



(5-126) 




Fig. 5-1 5a Pulse Train Signal. 



Envelope = P(co) 



^^ 



T"~-^>^ (Width of spectral 



teeth - 27r/4fr) 



.aJ \^^ L^Al'x? 



^H 27r/fr 

Fig. 5-15b Spectrum of Pulse Train Signal. 

From Equation 5-125, the signal component of the filter output at the 
observation time will be 

Filter output (signal) = So{t^ = / s-{T)dj (5-127) 

E £V(r - kT)p{T - mT)dr 



^iLp'ir - kT)dT 



= n\ p''{j)dr. 

The effect of the correlation (or filtering) operation has been to select out 
all the available signal pulses and add them together. A device which will 
perform this addition is most often referred to as a pulse integrator, and 



278 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

almost all radar receivers whose inputs consist of a series of pulses incor- 
porate such a device in one form or another. 

The effect of the pulse integrator on the noise can also be determined 
from Equation 5-125. The noise output will be 

/oo „_i 

n(T)J2 P(r - kT)dT. (5-12S) 

The noise power is determined by squaring no{to) and finding its average 
value : 

/°° /• "^ n-l n-l 
/ w(ri)w(r2)S ^p{ri — kT)p(T2 — mT)dTidT2. 

(5-129) 

Since n(t) was assumed to have a uniform spectrum with density D, the 
average value of the product «(ri)«(T2) is an impulse function with weight 
D: 

/oo /■ oo „_!„_! 

/ D8{ti — T2)2Z ^P(tT- ~ kT)p{T2—mT)dTldT2 

= dI E jipiT2 - kT)p{T2 - 7nT)dT, (5-130) 

j -co 

= £>/ Y.pKr2- kT)dr2 
7-- 

r oo 

In evaluating the integral of the double sum, we made use of the fact that 
when the pulse functions in the integrand do not coincide {k 7^ m), their 
product is zero: 



-my 



{T)dT. (5-131) 



It is apparent that the effect of the pulse integrator is to increase the signal- 
to-noise ratio for a single pulse by the factor n. This could, of course, be 
inferred at the outset from Equation 5-121, since the signal power is 
directly proportional to n. 

The shape of the matched filter response in this case is of some interest. 
Denoting the spectrum of an individual pulse by P(co), the spectrum of the 
pulse train will be 

n-l 

Pulse train spectrum = S{i^) = P(co) X) ^~''*"^ (5-132) 



\ sm coT/2 / 



5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 279 

The energy spectrum of a typical train of short pulses is shown in Fig. 5-1 5b. 
We note that the filter primarily acts to accentuate the harmonics of the 
repetition frequency. Because of its distinctive appearance, such a device 
is often called a co7nb filter. 

Some explanation on the mechanism of the detection process itself is 
in order since the previous discussion related only to maximizing the signal- 
to-noise ratio. The output of the matched filter characterizes a signal- 
plus-noise situation by a single number So{t^ -\- njyt^. This number is a 
random variable with a normal distribution and mean Jo(0- The detection 
process will consist of a decision as to whether the observed number comes 
from a distribution with mean sj^t^ or the distribution of noise alone 
with a zero mean. This decision can be made by selecting a critical value or 
threshold and deciding for or against the existence of the signal depending 
upon whether or not the observed number exceeds the threshold. Fig. 5-16 



-Decision Bias 
Probability Density 
of Noise Alone 



Probability Density 
of Signal Plus Noise 




h\ Value of Signal 

False Alarm pi^g ^^^^^ 

Probability 

Fig. 5-16 The Use of a Decision Bias for Determining Whether Noise Alone or 
Signal Plus Noise Is Present. 

shows the probability densities of the filtered signal plus noise and noise 
alone and a decision bias b for distinguishing the two cases. Because the 
two probability densities overlap, mistakes will be made. On some occasion 
a target will be thought present when there is none, while at other times 
the signal plus noise will be thought to be noise alone. The probability 
of making an error of the first kind is equal to the crosshatched area 
under the curve of noise alone and to the right of b in the diagram. This 
probability is normally called the false-alarm probability by radar system 
designers. The shaded area under the probability density curve of signal 
plus noise and to the right of ^ is the probability of detection. The difference 
between this probability and unity is, of course, the probability of making 
an error of the second kind or not seeing a target that is actually present. 
When human operators make a detection, the situation is not nearly so 
clear-cut, but some similar mechanism must take place. The decision bias 
might be visualized as diffuse, and it will vary with operators, time, and 
other conditions. 

A basic problem is the choice of the false-alarm probability at which 
the system is to operate. Most often this operating parameter is chosen 



280 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

on subjective grounds because the data upon which to base a rational 
choice are not available. Factors which can be used to determine an 
optimum false-alarm probability are the cost of a false alarm in time and 
subsequent commitments, the gain associated with a correct decision, 
and prior probability of a target's existence. If quantitative estimates of 
these factors are available, the false-alarm probability can be chosen to 
minimize the total cost of the detection operation. Or even when the 
prior probability is not known, it is possible to operate the system at false- 
alarm rates so as to minimize the cost for the most adverse value of the 
prior probability. As noted above, however, data on detection costs and 
prior detection probabilities are known only subjectively in the majority 
of cases, and most often a rather arbitrary estimate of a desirable value 
of the false-alarm rate is made after a thorough but subjective study of 
the problem. 

At the beginning of our discussion of the optimum receiving system, it 
was assumed that the waveform of the signal was known exactly, and the 
only issue was its existence. In a practical detection situation, however, the 
signal waveform may depend upon a number of unknown parameters. 
Three such parameters which are of particular importance are the signal 
amplitude, the time of arrival of the signal, and the radio frequency of 
the carrier. The signal amplitude will vary with the range, aspect, and 
size of the target, while the time of arrival is, of course, directly proportional 
to range in a radar system. The RF carrier will vary because of the 
doppler shifts proportional to the relative target velocity. An optimum 
receiver in this case will consist of a parallel combination of optimum 
receivers for all the possible waveforms. Luckily, this does not require a 
duplication of equipment to cover the possibilities of amplitude and time- 
of-arrival variations. If the signal amplitude is changed by some factor, 
then the average value of the filter output is changed by the same factor. 
The same filter will produce the maximum value of z- for all possible signal 
amplitudes. A similar situation applies to variations in time-of-arrival. 
The optimum receiver produces its maximum output at a time T after a 
signal is received. Continuous monitoring of the receiver output, then, 
will provide an observation of the filtered signal over a continuous range of 
possibilities for the time of arrival. In order to account for variations in 
the radio frequency, however, it will in general be necessary to have 
separate receiving systems for the possible radio frequencies which may 
occur. This situation will be recognized in the design of many doppler 
systems where a bank of narrow band filters, each connected to its own 
threshold, is used to cover the possible spectrum of doppler signals. 

The situation is complicated further by the fact that some of the signal 
parameters are random variables in their own right. For example, the 
amplitudes of echoes reflected by aircraft fluctuate owing to their motions. 



5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 281 

and the radio frequency of a magnetron oscillator normally varies randomly 
from pulse to pulse by a small amount. The statistics of signal parameter 
distributions would have to be considered in a more realistic optimum 
receiver, and the result would be somewhat different from that derived 
in this paragraph. 

One should not make the mistake of thinking that great gains over 
current practice can be attained through some complicated optimizing 
scheme. Actually, most radar systems are tuned up in this respect about 
as far as they can be when consideraton is taken of limitations in the 
state of the art and fluctuations in the parameters of the input signals 
with which the systems must contend. For instance, a pulse radar employ- 
ing a self-excited magnetron oscillator is not coherent because it is not 
normally feasible to control the frequency of the power oscillator to a 
sufficient degree. Because of pulse-to-pulse frequency fluctuations, the 
receiver must operate upon the envelope of the signal, and it will normally 
employ a non-linear device to generate it. In this case, the best that can 
be done is to match the low-pass equivalent of the IF amplifier to the 
pulse envelope, and this is quite normally done as a matter of course. 
When there are a number of pulses available in an echo, some provision 
is usually made to integrate them. Most commonly, this is accomplished 
on the display where the decay time of the phosphor may be matched to 
the signal duration. The point is that insofar as is possible receivers are 
normally matched to the signal waveform, and most radar systems can 
be quite legitimately regarded as correlation or matched filter radars, 
although possibly somewhat degraded from the optimum type. 



5-11 APPLICATION TO THE DETERMINATION OF A 
SIGNAL'S TIME OF ARRIVAL 

An important function of radar systems is the measurement of a target's 
parameters such as its range, velocity, size, and location. In this paragraph 
we shall develop some characteristics of a receiver which provides optimum 
target tracking in a manner similar to that used in the preceding paragraph 
to develop the properties of matched filter receivers for optimum target 
detection. We shall restrict our analysis to the problem of measuring the 
time of arrival of the signal. Since both angle and range are measured by 
comparing the return signal with angle and range reference signals which 
are generated as functions of time, the following discussion can be applied 
to both types of tracking. As in Paragraph 5-10, the receiver will be sup- 
posed to be a linear filter, and where applicable, the notation introduced in 
that paragraph will be adopted. 

Various operational definitions might be used to fix the arrival time of a 
signal. The mean, the median, or the mode of the distribution of the signal 



282 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



in time are all quite applicable. We shall find it most convenient to adopt 
the last of these, the mode or the maximum value of the signal, as the 
primary indicator of the signal's location. This definition gives a straight- 
forward development which parallels that of Paragraph 5-10 and which to a 
first approximation leads to results in accord with more elaborate analyses. 
Even so, we must recognize that since the choice of a definition for the signal 
location is purely arbitrary, we are optimizing the tracking process only 
relative to that definition and not in an absolute sense. 

In order to use the peak value of the filtered signal plus noise as an un- 
ambiguous estimate of the signal location, we shall make several assump- 
tions about the form of the signal and signal plus noise. First, we assume 
that the signal itself either has a single maximum or that the greatest 
maximum is sufficiently larger than minor maxima to allow it to be un- 
ambiguously distinguished. Second, we assume that the primary maximum 
of the filtered signal has a finite second derivative, since we intend to locate 
it by setting the first derivative of the signal plus noise equal to zero. Third, 
the filtered signal is assumed to be enough greater than the noise that there 
are no ambiguous noise maxima in the neighborhood of the primary max- 
imum and the shift in this maximum due to the presence of the noise is 
small enough to be approximated by the first few terms in a series expansion. 
Suppositions of this kind are not unusual in parameter estimation problems, 
and equivalent assumptions and approximations almost always must be 
adopted when a specific example is worked out. 

Fig. 5-17 shows a typical example of signal plus noise in the neighborhood 
of the signal maximum and illustrates how the addition of noise acts to 



APPARENT SIGNAL 
LOCATION 



ERROR IN 
SIGNAL LOCATION 



TRUE SIGNAL 
LOCATION 




NOISE- ^ ^ 

Fig. 5-17 Generation of Signal Location Error. 



shift the maximum slightly from its former value. The magnitude of the 
shift can be determined approximately by differentiating the signal plus 
noise and setting it equal to zero. The resulting expression will be in the 
form of a quotient very similar to that given in Equation 5-118 for the 
signal-to-noise power ratio. Schwarz's inequality can also be applied to 



5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 283 

this expression, and we can determine the optimum filter for tracking which 
gives the minimum tracking error. This error will be interpreted as a 
simple relation between the signal bandwidth and the signal-to-noise ratio. 
The filtered signal and noise are denoted by So{t) and no{t) as in the 
preceding paragraph. We suppose that the maximum value of the signal 
occurs at the observation time io- We suppose further that the output signal 
at the time /o + A/ can be represented by a series expansion about the time 
to for small values of the interval A/. 

So(lo + A/) = soito) + s:'(to)Ar-/2 + -. (5-133) 

The first derivative of So{t) at /« is zero, of course, because it has a maximum 
at that time. We assume that the shift in the maximum value of the signal 
plus noise is small enough that all terms beyond the second in the expansion 
above can be neglected. The derivative of signal plus noise in the neighbor- 
hood of /o is thus given approximately by 

j^[so{t) + nom = to+At = s:'{to)M + n'^to + A/). (5-134) 

Setting this expression equal to zero and solving for A/ gives an approximate 
value for the apparent shift in signal location due to noise: 

A/= -^''(!;\^'\ (5.135) 

^o \to) 

The variance or mean square value of the signal location error is thus given 
approximately by the average of the square of this expression: 



If we denote the transfer function of the filter-receiver by Y(o}) and the 
signal spectrum by S(cci) as in Paragraph (5-10), the following representation 
of s"o{to) can be obtained by differentiating Equation 5-116 twice: 



s'o'ito) = ^ / co2y(co)^(co)^^'-'«^a;. (5-137) 



Also assuming, as in Paragraph 5-10, that the input noise has a uniform 
spectrum of density D, the power spectrum of the output noise is D|y(w)|^ 
while the power spectrum of the derivative of the output noise is Do}^\Y(p})\^. 
The integral of this last spectrum gives the mean square value or variance 
of the derivative of the output noise: 

KWP = ^/'_ Da)2|y(co)|Vco. (5-138) 



284 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

The quotient in Equation 5-136 giving tiie variance of the signal location 
error thus has the following form: 

_ ^f Dco2|y(co)|Vco 

A/- = , /;~'° TT (5-139) 



(^/ c,'Yio^)S{co)e''^'o^oA 



For convenience, we denote the quotient on the right-hand side by ^. The 
denominator of this quotient is in a form to which Schwarz's inequality, 
given by Equation 5-125, can be applied. Using this relation to split the 
denominator into two separate integrals leads to the following inequality: 

^f Dco^iy(a;)|Vco ^ 



The integrals involving the transfer function of the filter simply cancel as 
they did in Paragraph 5-10, where the maximum value of the signal-to-noise 
ratio was determined. Since the quotient ^is never less than the expression 
given on the right-hand side above, which does not contain Y(o}) at all, 
this expression must give the minimum value of ^ for the most judicious 
choice of y(co). Referring to Equation 5-139 above, it is apparent that this 
minimum value of ^ will actually be achieved if the filter transfer function 
is chosen to be the conjugate of the signal spectrum. In this case, the 
numerator in Equation 5-139 cancels one of the factors in the denominator, 
and we have 

A7^=^.,mi„= , ,.co ^ (5-141) 



i/. 



cjo-\S{u)\~dcjo 



The optimum filter transfer function giving this result is 

y(aj) - S*(c^)e-''''o. (5-142) 

This is exactly the transfer function determined in Paragraph 5-10 (Equa- 
tion 5-128) to give the maximum signal-to-noise ratio. Thus to a first 
approximation the matched filter giving the maximum signal-to-noise ratio 
also provides the minimum error in locating the signal in time. 

The relation given by Equation 5-141 above for the minimum variance of 
the error in locating the signal can be given an interesting and rather useful 
physical interpretation. We note that the denominator has the form of the 
moment of inertia of the energy spectrum of the signal. If this denominator 
is divided by the total signal energy, we obtain the square of the radius of 
gyration of the energy spectrum. Now the radius of gyration of a function 



5-111 



DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 



285 



is a measure of the width of that function. Thus the radius of gyration of 
the energy spectrum is a measure of the bandwidth of that spectrum. We 
shall adopt this definition for the bandwidth of the signal spectrum: 



Signal bandwidth (rad/sec) = B 



U 



\S(co)\"(^(ji} 



T 



{rjl !^MP^")"" 



(5-143) 



A question may arise in connection with the application of this definition 
to spectra which are not centered at zero frequency. It is clear that the 
radius of gyration of an energy spectrum which is concentrated at low 
frequencies is a good measure of the bandwidth of the spectrum. The 
radius of gyration of a spectrum whose center is displaced to some high 
frequency, though, will be very large, and it does not correspond to the 
conventional idea of bandwidth. Such a signal can be represented as a 
function, denoted by/(/), whose spectrum is concentrated at low frequen- 
cies and which is modulated by a high-frequency carrier: 

High frequency signal = f(t) cos {coj + (p). (5-144) 

This signal contains information about its time location with an accuracy 
on the order of 1 /coc- But when the ratio of the carrier frequency to the 
bandwidth of/(/) is large, this information is useless because it is ambiguous. 
This is illustrated in Fig. 5-18 where a relatively smooth low-frequency 

AMBIGUOUS MAXIMA 
DUE TO CARRIER 
FREQUENCY 
LOW - FREQUENCY // 1 MODULATION 

SIGNAL=f(f) 




HIGH-FREQUENCY SIGNAL^ f(f)cos(Wcf+0) 
Fig. 5-18 Typical High-Frequency Signal. 



signal /(/) is modulated by many carrier cycles. It can be seen in this 
figure that the carrier frequency modulation produces a number of signal 
maxima in the neighborhood of the maximum of/(/) . Because these maxima 
are all of about the same magnitude, there is no way of distinguishing one 
from another, and signal location information provided by the carrier- 
frequency modulation is ambiguous. In such a case, the low-frequency 
signal /(/) is normally regenerated by a demodulating operation, and the 



286 



TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



location of the signal is determined on the basis of/(/) alone. The effective 
bandwidth in such a case, then, is that of the spectrum of/(/), and it would 
be determined relative to the carrier frequency rather than zero frequency 
as is indicated in Equation 5-143. When the definition of bandwidth given 
by Equation 5-143 is combined with Equation 5-141, the mean square time 
error is found to be approximately equal to the ratio of D, the input noise 
power density, to the product of the square of the signal bandwidth and 
the signal energy. The ratio of the signal energy to D, however, was estab- 
lished in Equation 5-129 as the greatest possible signal-to-noise ratio an^ 
was denoted by z^. Thus we can assert that the minimum rms error in 
measuring the time of arrival of a signal is approximately equal to the re- 
ciprocal of the product of the signal bandwidth and the voltage signal-to-noise 
ratio: 

(X7^)i/2 ^ iiBz (5-145) 

rms time error = 1 /(signal bandwidth) (voltage signal-to-noise ratio). 

As an example of the application of these ideas, let us consider a pulse 
radar with a narrow antenna pattern which is scanned over the target at a 
constant angular rate. Such a system is similar to the AEW example dis- 
cussed in Paragraphs 2-10 to 2-20 and the results that we shall develop are 
applicable to the design considerations in that example. 

The video signal generated by such a system would have a form similar 
to that shown in Fig. 5-19. The time at which the envelope of the pulses 

Reference 



-.^TllrC 



•Target 



0=Beamwidth measured 
between half - power 
.Envelope points 



^^, 

i/'s = Scanner Angle 

Fig. 5-19 Receiver Voltage Pulse Train Return from a Point Target. 

reaches its peak value will be correlated with the angular position of the 
target so that the problem of locating the target in angle is essentially that 
of determining the arrival time of the signal, and the ideas and develop- 
ments of this paragraph are applicable. 

The basic functions performed by the system are indicated in the block 
diagram in Fig. 5-20. The received signal is amplified and filtered by an IF 
amplifier which is matched to the envelope of an individual pulse. Noise 



5-11] 



DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 



287 



Random 
Noise 

_L_ 



1st 
Detector 



Scanning 
Antenna 



IF Amp. 
Matclned 
To Pulse 



Square Law 

2d 

Detector 



Pulse 
Gate 



Video Filter 
Matched To 

Scan 
Modulation 



Output 

Giving Minimum 

Angular Error 



Fig. 5-20 Block Diagram of Receiver of Scanning Radar. 



with a uniform power density is introduced at the input to this amplifier. 
A square-law second detector is used to generate the video envelope of the 
signal plus noise. The video signal-plus-noise pulses are gated into a video 
filter which is matched to the scan modulation. That is, this filter is matched 
to the fundamental component of the gated video signal plus noise. All 
signals, information, and noise at the repetition frequency and its higher 
harmonics are filtered out. We assume that the number of pulses per 
beamwidth is sufficiently large that the signal spectrum about the first 
harmonic of the repetition frequency does not overlap the fundamental 
component of the signal spectrum. We also assume that the video noise is 
sufficiently uniform over the bandwidth of the scan modulation signal for 
the assumption of a constant noise spectrum under which we derived 
Equation 5-145 to be valid. Other system configurations are possible. A 
more practical design might stretch the gated signal-plus-noise pulses before 
smoothing by the scan modulation filter. Such a system, however, would 
give slightly greater angular errors than the one we have chosen to study. 

The following special notation is adopted for this example. 

a = voltage amplitude of received signal 

T = repetition period 

d = duty ratio 

■^ = antenna angle 

y^ = antenna angular rate 

9 = antenna beamwidth (half-power points — one-way) 

5 = pulse width 

n = Q l\j/T = number of pulses per beamwidth 
The signal received from the target will have the following form: 

Received signal = a (scan modulation) (pulse modulation) cos coc/- 

(5-146) 



288 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

The peak pulse power will be a'^ jl. We assume that the individual pulses 
are rectangular and of width b. The total energy in a pulse, then, is a^B jl. 
Assuming noise with a uniform power spectrum of density D, the maximum 
signal-to-noise ratio which can be obtained with a filter matched to a pulse 
is given by the ratio of the pulse energy to the power density of the noise 
as was derived in Equation 5-127: 

Peak signal to noise ratio = S/N = a^/lD. (5-147) 

The noise at the output of the IF amplifier corresponds to a narrow band 
noise process similar to those discussed in Paragraph 5-8. Since the IF 
amplifier is matched to the envelope of the pulse signal, the autocorrela- 
tion function and power spectrum of the low-frequency in-phase and 
quadrature components of the noise, x{t) and yif) in Equation 5-73, will 
be of the same form as that of a rectangular pulse. This autocorrelation is 
a triangular function of width 25 and height equal to the noise power. 

The properties of the video signal and noise after a square law detector 
can be determined from Equation 5-87 which gives the autocorrelation 
function of the output of a square-law second detector: 

7^2 = (202(1 + S/NY + (2cr2)2[p2 + 2{S/N)p]. (5-87) 

The first term in this expression corresponds to the video d-c level during a 
pulse while the term involving p and p^ corresponds to the video noise. In 
order to exclude the possibility of ambiguities incident to a noise maximum 
in the neighborhood of the signal maximum, we assumed that the signal-to- 
noise ratio was large in the development of Equation 5-145. It will simplify 
the present analysis if we approximate Equation 5-87 for large S jN by 
considering only the dominant d-c and noise terms in that equation: 

77^- = {2<rr~{S/NY + (:lcj^-Y2{S/N)p, S/N»\. (5-148) 

The normalized autocorrelation function p in this expression corresponds to 
the triangular function noted above of width ITd but of unit height. 

The shape of the video spectrum will not be exactly uniform as is re- 
quired for the developments of this paragraph to be rigorous. The total 
width of the spectrum, though, will normally be greater than the spectrum 
of the scan modulation by a factor on the order of 10^. Variation of the 
noise spectrum over the scan modulation bandwidth, then, will be quite 
small, and we are justified in approximating the noise spectrum by a 
spectrum with a uniform power density. The power density of the noise 
spectrum at zero frequency, which we shall assume to be extended to all 
frequencies, is found by integrating the autocorrelation of the noise. The 
integral of the triangular function p{t) is 5. When this value is substituted 
for p in Equation 5-148, the term involving this factor gives the power 
density of the noise at zero frequency during a pulse or with a c-w signal. 



5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 289 

Because the video signal plus noise is gated, the noise power and thus the 
noise power density will be smaller than the value during a pulse by the 
duty ratio d. Taking these factors and considerations into account, the 
effective power density is determined from Equation 5-87 to be 

Power density at zero frequency of gated video noise 

^ {lcj''Yl{SlN)hd,SINy>\. (5-149) 

The d-c component of the video corresponds to the signal which is used 
to locate the target. From Equation 5-148 it will be noted that this 
component is proportional to the received signal power. Thus the video 
signal is modulated by a scan modulation function which indicates how 
the received power fluctuates during a scan. The first term in Equation 
5-148 gives the square of the d-c level during a pulse for large signal-to- 
noise ratios. To obtain the d-c level of our gated signal, we must multiply 
the signal level during a pulse by the duty ratio d. The resulting video 
signal has the following approximate form: 

Gated video signal = (2o-^)(*S'/A^)'^(scan power modulation), S/N^ 1. 

(5-150) 

We assume that the antenna pattern of the system has a Gaussian shape 
and that the same antenna is used for both transmission and reception. 
The beamwidth of the pattern 6 is defined as the angle between the half- 
power points for one-way transmission. The gain of the two-way power 
pattern would thus be down by a factor of 4 at these points. The antenna 
angle is denoted by ^. We assume that the scan modulation is generated 
by a constant velocity scan at the rate i/'. Supposing that the signal max- 
imum occurs at the time / = 0, the scan modulation has the form 

Scan power modulation = exp (— \p^/0.1SQ^) 

= exp (- ipyyO.lSQ') (5-151) 

Because we have assumed a square-law second detector, the scan modula- 
tion of the video voltage will be proportional to this scan power modulation. 
We are now in a position to apply the result of Equation 5-145, giving the 
rms error in time of arrival, which in turn yields the rms angular error after 
multiplication by the scan rate. We first compute the signal-to-noise ratio 
in the output of the video filter matched to the scan modulation. The 
energy in the signal is given by the integral of its square: 

Signal energy at low frequencies 

= (2cr2)2(^/7V)V2 / exp ( - V'VVO.0902) dt 

= 0.53{2a'y(S/Nyd\e/i^). (5-152) 



290 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 

The power density of the video noise is given in Equation 5-149. It was 
determined in Paragraph 5-10 that the signal-to-noise ratio at the output 
of a matched filter is equal to the ratio of the signal energy to the noise 
power density, so we have 

Signal-to-noise ratio at O.S2{lcj^y{S/NYd\Q/4') 
output of video filter = z^ = {2a'^y2{S/N)d^T 

= 026S{S/N){Q/i^T) 

= 0.1GS{S/N)n. (5-153) 

The number of pulses per beamwidth given by the ratio Q jxpT has been 
denoted by n in this equation. 

The bandwidth of the signal can be determined from the spectrum of the 
scan modulation. The Fourier transform of the modulating function in 
Equation 5-151 is as follows. 

/ ^ e\ / -co^0.099^ \ 

Spectrum of scan modulation = ( -yO.lSTr • I exp I ■ 1 

(5-154) 
The bandwidth, as defined by Equation 5-143, is easily computed: 



Scan modulation bandwidth = B 





1/2 


\ /"" /-co20.i8e2\ "1 


1/2 


U-o^^^K 2v- rJ 





- 2.35,/^/e 5-155) 

The rms error in measuring the target angle can now be estimated from 
Equation 5-145 as the scan rate divided by the product of the scan modu- 
lation bandwidth and the voltage signal-to-noise ratio in the video filter 
output: 

rms angle error = (M^-y^^ = (li^yi'' = i/Bz (5-156) 



= 4^Q/13S4^^026S{S/N)n 



= O.S25e/^|(S/N)n. 

The angular error of a scanning radar has been studied in the technical 
literature^ for conditions very similar to the assumptions of this example. 
In that study the minimum possible rms angular error was found to be 
approximately proportional to an expression of the form of Equation 5-156 

-P. Swerling, "Maximum Angular Accuracy of a Pulsed Search Radar," Proc. IRE 44, 

1140-1155 (1956). 



5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 291 

but only about half as great. Actually, the estimate in Equation 5-156 is 
optimistic. The mechanism by which the maximum value is chosen can 
introduce additional errors; nor was any consideration given to the effect 
of target fluctuations which would act to increase the error. As a typical 
case, if the signal-to-noise ratio out of the matched video filter as given by 
Equation 5-153 is 6 db, the rms angle error from Equation 5-156 is 

rms angle error = e/2.35^[4', z'~ = 4 (5-157) 

= 0.2139. 

In Paragraph 2-15 it was assumed that the target angle could be determined 
in a scan to 1 /4 of a beamwidth. From the relation above, this is not an 
unreasonable assumption. 



GEORGE • L. HOPKINS • R. M. PAGE • 
D. J. POVEJSIL • H. YATES 



CHAPTER 6 



GENERIC TYPES OF RADAR SYSTEMS AND 
TECHNIQUES* 

6-1 INTRODUCTION 

The radar systems engineer is often asked to solve the following problem: 

"Given a set of performance specifications based on the tactical 
problem requirements, derive a radar system that will meet the 
specifications." 

For a variety of reasons, it is seldom possible to solve this problem in a 
straightforward fashion. Probably the most important reason is this: The 
performance specification — if properly derived — will seldom specify a 
task which simply cannot be performed by radar techniques (for example, 
the performance specification could not logically ask the radar to distinguish 
between red and blue aircraft); however, the performance specification will 
usually require the radar to perform a. group of tasks which are not logically 
consistent with any one radar system mechanization. Even when mecha- 
nization limitations are excluded from consideration, there is no such thing 
as an "ideal" radar system which can perform any group of functions in an 
optimum manner. Every conceivable type of radar system possesses a 
combination of good and bad characteristics and both must be accepted 
and rationalized in a given application. 

The usual approach is to assume a generic type of radar system which 
experience and judgment deem reasonable. The assumed system then is 
measured analytically against the overall system requirements to determine 
whether it has the inherent potential for providing an acceptable problem 
solution. This process is repeated until the best match is found between 
the performance specification and the basic laws of nature governing what 
can be done by a given radar system. 

*Pai-agraphs 6-1, 6-2, 6-6, and 6-7 are by D. J. Povejsil. Paragraph 6-3 is by R. M. Page. 
Paragraph 6-4 is by S. F. George. Paragraph 6-5 is by L. Hopkins. Paragraph 6-8 is by 
H. Yates. 

292 



6-2] BASIC PRINCIPLES 293 

Unfortunately, there is a tremendous variety of possible choices. In 
terms of generally recognized system types and subtypes, there are pulse^ 
continuous-wave {CW), pulsed doppler, monopulse, correlation ^high-resolution, 
and moving target indication {MTI) radars. Some of these types represent 
genuinely different approaches; some of them represent merely alternative 
means for performing the same job; and some of them are derivatives of 
particular system types. In each case, however, the selection of one of these 
types commits the radar system designer to a problem approach that is 
confined within uncomfortably narrow limits. The radar system designer 
must therefore have a good general knowledge of the basic system types and 
the general laws that govern their performance characteristics. Toward 
this end, this chapter will attempt to accomplish two things. 

(1) It will summarize basic radar laws in a rule-of-thumb fashion to 
provide a means for understanding the operation of any radar 
system. 

(2) It will describe the performance characteristics and limitations of 
generally recognized radar system types and will indicate their 
general areas of application. 

6-2 BASIC PRINCIPLES 

The operation of almost any radar system may be visualized and under- 
stood by asking and answering the following basic questions: 

(1) Is the system active, semiactive, or passive (see Paragraph 1-4) ? 

(2) What information is contained in the signal return from the 
assumed target complex? 

(3) What are the system sampling frequencies? 

(4) How are the radar data detected and processed in the receiving 
system ? 

(5) Where does the processed information go? 

Each of these questions may now be considered in greater detail. 

Type of Radar System. The most basic division of radar system 
types is a classification based on the origin of the target signal information. 
An active system generates the signals which are ultimately scattered back 
to the point of signal origin. A passive system is simply a receiving system 
which utilizes target-generated radiations as its signal source. A semiactive 
system employs separate transmission and receiving systems which may be 
at some distance from each other. Depending upon the degree of coupling 
between the transmitted and received signals, a semiactive system may 
resemble either an active or a passive system insofar as its basic operation 
is concerned. 



294 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

Type of Information. In any radar system problem there are two 
basic kinds of target information: 

(1) The desired ra.da.r-denved target information 

(2) The radar-derived target information that is actually obtained 
using a given system 

The latter will be exemplified by answering the last three questions posed 
at the beginning of this paragraph. 

Most commonly, the desired radar-derived target information takes the 
form of a four-dimensional information matrix as shown in Fig. 6-1. The 




Expanded View of a 
Quantized Region 

I^i Elevation 
rBeamwidth 




Azimuth 
leamwidth 



Range Resolution 
Element 



'^MAx=Max. Unambiguous Range 
i/^s = Solid Angle of Coverage 
N, =No. of Range Elements 
No =No. of Azimuth Elements 
Ne =No. of Elevation Elements 
Nv =No. of Velocity Resolution 
Elements Per Block 



Fig. 6-1 Radar System Information Matrix. 

radar is expected to detect and classify targets according to their range, 
their angular orientation (two dimensions), and their relative radial 
velocity. Depending upon the tactical problem, each dimension may be 
characterized by (1) a maximum and minimum value, and (2) a minimum 
resolution element. Thus, the total amount of information which the radar 
may gather is 

A^ = A^, X A^« X A^a X A^. elements. (6-1) 

Generally, the tactical problem sets some limit on the time taken to 
gather this information. If we define this as the lota/ scanning time, tsc, the 
required interrogation rate of the radar system is 

N = N/t,c elements/sec. (6-2) 

Often, it is quite informative to translate a system requirement into the 
form of Equations 6-1 and 6-2. For example if we consider a system which 
requires 150 n.mi. range with 0.1 n.mi. resolution; a 1° fan beam with 360° 



6-2] BASIC PRINCIPLES 295 

of angular coverage in 2 seconds; and an ability to separate targets with 
radial velocities of from to 2000 knots in 40-knot increments, we find: 

,V 150 ,^ 360 ^^ 2000 ,n r ^ -,r,2 ^ i 

A^ = -Tpj- X . ry X ^7^ = 13.5 X 102 elements/sec. 

which is a very large number even though the radar is providing only three 
dimensions — range, one angle, and velocity. This answer implies a system 
bandwidth requirement of at least 13.5 Mc even if it were possible 
(which as the reader shall soon see, it is not) to apportion this bandwidth 
between range, angle, and velocity coordinates in the desired manner. 

In the case of a passive system the information matrix is usually only 
two-dimensional (angle only) since passive systems do not ordinarily have 
range and velocity measuring capabilities.^ 

System Sampling Frequencies. The system sampling frequencies 
govern the minimum resolution element size and the total unambiguous 
measurement interval of each coordinate — range, angles, and velocities. 
In general, there are three basic sampling frequencies which are important 
in determining the character of the signal entering the receiver: (1) trans- 
mitted bandwidth, (2) transmission periodicity, and (3) angular scanning 
frequency. 

1. The transmitted bandwidth I^ft determines the rate at which the radar 
system can collect pieces of range information. It represents, in effect, the 
rate at which successive range elements of space can be interrogated. This 
principle is easily seen for the case of a pulse radar. In this case the trans- 
mitted bandwidth is the reciprocal of pulse length (A/^ = l/r). At any 
instant of time following transmission, the received pulse information 
originates from a range interval Ai? which has the width 

Ai^ = f = ^; (6-3) 

Thus every r seconds, information is received from a new range interval. 
The same principles hold whether the transmitted bandwidth is created 
by pulsing or by other means such as "FM-ing." That is, the minimum 
range resolution element is defined by Equation 6-3 and the rate Nr at 
which the system collects pieces of range information is: 

Nr = ^ft. (6-4) 

If the transmitted bandwidth is large relative to the maximum doppler 
shift and all the other sampling frequencies, then the transmitted band- 
width also defines the maximum rate at which the radar may collect all 

^A passive system designed to collect and classify radiation sources according to their 
frequency, bandwidth, polarization, and angular location can encounter bandwidth problems 
similar to those of an active three- or four-dimensional radar system. 



296 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

kinds of information, regardless of how it might be split up between range, 
angular, and velocity coordinates: 

(Total interrogation rate) N = Nr = Aft. (6-5) 

2. The transmission periodicity fr is defined as the fundamental repetition 
frequency of the radar signal. (In a pulse radar, for example, /r would be 
equal to the pulse-repetition frequency or PRF.) This quantity governs the 
radar's ability to split up information between range, angle, and velocity 
coordinates. 

The transmission periodicity defines the maximum unambiguous range 
interval as follows: 

/?.ax = 7^- (6-6) 

Target returns from greater ranges will be ambiguous because they will 
enter the receiver after the transmitter has begun another transmission 
cycle. 

The total number of separate range intervals covered by the radar is then 

The transmission periodicity also defines the maximum relative velocity 
interval that may be measured without ambiguity. This may be derived as 

(A^)rn,ax--^ (6-8) 

where (AF)r^ax = maximum relative velocity interval (cm /sec) 

X = wavelength (cm). 

For higher velocities, the total doppler spread becomes higher than the 
sampling frequency/.. Thus the sampling process will create measurement 
ambiguities regardless of how the total doppler spread is split between 
opening and closing velocities. 

3. The angular scanningjrequency is defined as the rateA^a at which signal 
information is collected from separate portions of the angular space volume. 
This quantity defines the minimum possible bandwidth of any received 
signal and it may be expressed: 

A^.=/. = ^ (6-9) 

where i^s = total solid angle scanned 

^o = solid angle subtended by antenna pattern or instantaneous 
field of view. 



6-2] BASIC PRINCIPLES 297 

The transmission periodicity Jr places an upper limit on the scanning 
frequency. If, for example, A >/r, then radar returns from the far end of 
the unambiguous range interval will not be received because the radar 
antenna will have moved to a new angular position by the time the signal 
has returned. For this case 

i?n.ax = 1^^ fs>fr. (6-10) 

Since the angular scanning frequency defines the minimum signal band- 
width, it also defines the minimum possible velocity resolution element. 
Thus 

Ar>-^. (6-11) 

where A/^ = velocity resolution element, and the total number of separate 
unambiguous velocity elements is, from Equations 6-8 and 6-11, 

N^^^^n^<-fi. (6_i2) 

For passive radar and IR systems, the scanning frequency is the basic 
system sampling frequency. The interrogation rate of such systems is 
roughly equal to the scanning bandwidth. 

Detection and Data Processing. The detection and data-processing 
system of any radar may be characterized by several basic properties: 
(1) type of detection process, i.e., coherent or noncoherent; (2) number of 
channels; (3) filtering techniques; (4) signal-to-noise ratio as a function of 
target size, range, etc. A knowledge of these properties can provide a ready 
means for estimating the performance potential of any system. Each of 
these properties will be discussed briefly; subsequent sections of this chapter 
will provide illustrations of the various possibilities for the generic systems. 

In a noncoherent detection process, no attempt is made to correlate the 
phases of the transmitted and received signals. Thus the signal returns 
from each target must be detected separately and added together after 
detection (postdetection integration) as shown in Fig. 6-2. As was shown 
in Paragraph Z-'}), this process improves the S jN ratio of the target return 
by a factor which may be expressed as 

A(^/yV) = n^ (6-13) 

where n = number of samples integrated; n ^ frifs if all the samples in 
one scan period are integrated 

7^ = 0.5 - 1.0 (0.5 for low S jN and 1.0 for high S jN). 



298 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



RF 



Signal 



Bandpass 



»^ — ► Amplifier 



(Af=Af, 



Detector 



Integration I Output 
Filter Signal* 



Noncoherent 
Reference 



NONCOHERENT INTEGRATION 



RF 



Signal 



1 



Bandpass 
Amplifier 
(Af=Af,) 




Integration 

Filter 
(Af<j>AVnj 




Detector 


Output 


Signal 



Coherent 
Reference 

COHERENT INTEGRATION 

Fig. 6-2 Noncoherent and Coherent Integration Processes. 



Thus, noncoherent detection does not make optimum use of system infor- 
mation redundancy; in fact, in the region of low S /N ra.tio, the S /N ratio 
improves approximately as the square root of the number of samples 
integrated. 

In a coherent detection process, the phase relations between the trans- 
mitted and received signals are maintained. This makes it possible to add 
successive samples before detection to obtain a direct enhancement of S jN 
ratio (predetection or coherent integration). In this case improvement in 
6" /A^ ratio may be expressed 

M^SIN)= n. (6-14) 

Thus, in a coherent system, the S jN ratio can increase linearly with 
information redundancy. This means that all other things being equal 
(average power, frequency, antenna aperture, etc.) a coherent detection 
system can obtain longer detection ranges than a noncoherent system, the 
difference between the two being particularly noticeable for high informa- 
tion redundancy. In addition, as will be shown in Paragraph 6-4, a coherent 
system can employ a more efficient detection law than a noncoherent 
system, thereby enhancing the relative detection capability of coherent 
systems even for short observation times. These characteristics coupled 
with the doppler frequency measurement ability of coherent systems (see 
Paragraphs 6-4, 6-5, and G-G) has resulted in a significant shift of develop- 
ment emphasis to coherent systems in recent years. 

The number of channels required in a radar system depends upon the 
detection bandwidth and the scanning time. The basic relationships may 
be ascertained by considering a radar which is designed to measure range, 



6-2] BASIC PRINCIPLES 299 

angles, and velocity. For such a system the information rate may be 
expressed (from Equations 6-1, 6-2, 6-9, and 6-12): 

iV = TV, X A^c X TVa X A^. = ^ X/s X N,. (6-15) 

The minimum detection bandwidth/^ that could be employed with such a 
system is of the order of the bandwidth induced by scanning /^ as previously 
mentioned. Thus the number of parallel channels needed to process all the 
radar data in minimum time is 

N//d = N/fs ^ ^ X N, = Nr X N,, number of channels. (6-16) 

This development shows that such a radar would require a separate 
channel for each range resolution element for a total of Nr range channels; 
each range channel would possess, in turn, 7V„ velocity channels. A repre- 
sentation of such a system is shown in Fig. 6-11. 

The only means for reducing the number of channels required is to 
increase the detection bandwidth or to increase the total scanning time and 
employ time-sharing of the receiving channels. A noncoherent pulse radar 
is a good example of the first approach: in this case the predetection 
bandwidth is made equal to (or greater than) Aft and only one channel 
is needed. 

A CW radar with a sweeping velocity gate is a good example of the 
second approach; in this case, the various velocity intervals are examined 
sequentially. This permits single-channel operation at the cost of increasing 
the total interrogation time by a factor equal to the number of velocity 
intervals, as will be explained in Paragraph 6-5. 

A number of means — other than the brute force approach indicated — 
exist for creating parallel information channels. Principal among these are 
the storage techniques described in Paragraph 6-6 and the delay-line 
filtering techniques described in Chapter 5. 

The filtering techniques commonly employed in radar receivers may be 
listed as follows: (1) mixing, (2) bandpass filtering, (3) gating, (4) demodu- 
lation, (5) clamping, (6) cross-correlation error detection, (7) comb filtering, 
and (8) video integration. Chapter 5 developed the basic mathematical 
theory of these techniques with illustrations taken from the example of a 
pulse radar system employing conical scan angle tracking. The basic 
principles developed for each of these operations do not change; thus the 
material developed in Chapter 5 provides a means for tracing and analyzing 
the flow of signal plus noise through any radar receiver. The generic 
systems discussed in subsequent paragraphs will provide examples of the 
various filtering and receiver sampling techniques as they are used in other 
types of systems. 



300 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

The signal-to-noise ratio of the target information is derived from various 
modifications of the basic radar range equation (Equation 3-1). As the 
examples in the following paragraphs will show, considerable care must be 
taken in the derivation of an approximate expression for 6" /TV ratio to allow 
for system losses and vagaries of the receiving system such as sweeping 
gates, filter sampling times, and postdetection filters. Three factors are 
basic in determining S jN ratio and these provide a convenient basis for 
comparing S jN performance of different systems in the same situation (i.e. 
same operating frequency, search volume, antenna size, and scan speed). 
They are (1) average power, (2) type of integration (coherent or non- 
coherent), and (3) effective integration time. 

Information Utilization. The end use of the radar information in a 
given application constitutes the reference — knowledge of which must be 
compiled to understand the operation of any given system. The end use 
requirements for a given application are derived by analyses such as those 
shown in Chapter 2. Those examples demonstrated a number of different 
end-use possibilities such as (1) display of radar information for interpre- 
tation by an operator, (2) coding and transmission to a remote location, 
(3) weapon direction computation. Other possibilities include (1) storage 
by photographic techniques, (2) correlation with information from other 
sensors such as infrared (IR) and photographic, (3) navigation computa- 
tions. 

The operation of any radar system can be judged only in terms of its 
compatability with a set of end-use requirements. This fact is often 
forgotten by people who like to categorize radar systems on an absolute 
basis. Such people originate statements such as "Pulse radars have no 
low-altitude capability" and "Doppler radars have excellent low-altitude 
capability." At best, statements such as these are partial truths; at worst, 
they are quite wrong in certain applications. The systems designer is well 
advised to avoid generalizations of this sort and analyze radar systems with 
respect to their applicability to specific problems. 



6-3 MONOPULSE ANGLE TRACKING TECHNIQUES 

Angle tracking requires measurement of two quantities in a manner that 
is effectively continuous. These quantities are magnitude and sense of angle 
tracking error. As shown in Chapter 5, Fig. 5-13, this is accomplished in 
conventional conically scanning tracking radar by purposely generating 
instantaneous tracking errors, but alternating the sign of the error, and 
averaging to zero. The method is simple and effective, but suffers errors when 
the signal fluctuates in amplitude in such a manner as to increase apparent 



6-3] MONOPULSE ANGLE TRACKING TECHNIQUES 301 

errors of one sign while decreasing apparent errors of the opposite sign (see 
Paragraph 4-8). Such a phenomenon is possible because plus errors and 
minus errors are generated alternately, not simultaneously. 

One obvious method of eliminating this source of error is to generate both 
positive and negative errors simultaneously . A straightforward technique 
for accomplishing this is to amplify the two signals from two overlapping 
antenna patterns separately and compare the two amplifier outputs. This 
technique is operable, but places severe stability requirements on the 
amplifiers, since relative drifts in amplifier gain produce changes in indi- 
cated correct tracking angle. 

An analogous method makes use of two spaced antennas in an inter- 
ferometer arrangement. Signals from the two antennas are amplified 
separately, with a common local oscillator for the two receivers, and relative 
phase is measured at intermediate frequency. In this case the phase 
stability requirement on the amplifiers is severe, since relative phase shifts 
in the two channels similarly produce changes in indicated correct tracking 
angle. 

Instability in indicated correct tracking angle may be overcome in either 
the amplitude or the phase comparison approach by connecting the two 
antennas in phase opposition before amplification, thus requiring only one 
receiving channel. Direction of arrival of signals is determined as the 
direction in which the amplifier output is near or equal to zero when increase 
of signal is produced by misaligning the antenna pattern in either direction 
from this so-called null point. This technique suffers from two objectionable 
characteristics. When there is no tracking error, there also is no signal to 
indicate presence of a target; and when there is an error signal, the sense of 
the error is not indicated. 

Monopulse radar, as its name implies, is a tracking radar that derives all 
its tracking error information from a single pulse and generates new and 
independent error information with each new pulse. In a broad sense, the 
simultaneous amplitude or phase comparison systems described above may 
be called monopulse systems. The name monopulse, however, has become 
restricted by common usage to still another method for generating both 
positive and negative errors simultaneously which overcomes the principal 
objections of the other systems. The method consists in so connecting the 
RF circuits of two antennas that both sum and difference signals are 
obtained simultaneously. The patterns of the two antennas overlap in the 
conventional way for generating tracking error information, as shown in 
Fig. 6-3. The sum signals from the two antennas merge the two patterns 
into a single lobe pattern as shown in Fig. 6-4. The difference signals 
produce the familiar null pattern with the sharp zero at the center, as shown 
in Fig. d-S. The sum and difference signals are then amplified separately 
and recombined in a product detector after amplification. 



302 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 




Fig. 6-3 Overlapping 
Individual Antenna Pat- 
terns of a Monopulse 
Radar. 




Fig. 6-4 Sum 
of Overlapping 
Patterns in a 
Monopulse Ra- 
dar. 




Fig. 6-5 Difference of 

Overlapping Antenna 

Patterns in a Monopulse 

Radar. 



The process of generating sum and difference signals results from in-phase 
connection of the two lobes for the sum pattern, and antiphase connection 
for the difference pattern. Consequently the sum and difference signals are 
mutually in phase for directions of arrival on one side of the difference 
pattern null, and in antiphase for directions of arrival on the other side of 
the null. Thus the difference signal contains within itself only angle error 
signal magnitude, while the sum signal contains the phase reference by 
which angle error sense is determined. The output of the product detector 
as a function of the direction of arrival of signal energy relative to the 







+ 








ro 










c 










op 










en 






\ 










\ 




o 


. 




\ 




UJ 


/ 




X 


_ 




/ 


Angle Off Axis 


+ 













Fig. 6-6 Monopulse Error Signal Curve. 



6-3] 



MONOPULSE ANGLE TRACKING TECHNIQUES 



303 



antenna difference pattern null is therefore the familiar error signal curve 
of Fig. 6-6, with zero signal on target, and polarity of error signals indicating 
error sense. The output of the sum amplifier provides indication of the 
presence of a target, an indication which is maximum when on target. The 
balance point representing zero angle error is not significantly affected by 
relative shifts in gain or phase between the two amplifiers. The sensitivity 
to angle error, represented by the slope of the error curve as it passes 
through zero, is influenced by relative phase shift between the two ampli- 
fiers, becoming zero at 90° relative shift. Since it is a cosine function, 
however, it is insensitive to phase shift near correct phase, a relative shift 
at 25° producing a decrease in angle error sensitivity of only 1 db. 

A system diagram illustrating the 
monopulse principle for angle error 
indication is shown in Fig. 6-7. A 
conventional hybrid ring (see Para- 
graph 10-1 5) is used for deriving sum 
and difference signals from the two 
antennas. The transmitter is con- 
nected to both antennas by suitable 
TR circuitry in the sum channel, so 
that the transmitter radiation pat- 
tern corresponds to Fig. 6-4. The 
output of the sum amplifier is recti- 
fied and applied in a conventional 
manner as a video signal to a radar A 
scope, giving indication of presence 
of target and target range. Also the 
outputs of the two amplifiers are mul- 
tiplied in the phase-sensitive or prod- 
uct detector to give an error signal 
whose sign corresponds to error sense. 
This error signal, which is a video sig- 
nal, is added to the time base signal 

and the combination applied to the indicator deflection system orthogonally 
to the output of the sum amplifier. The resulting indication presents targets 
as pips perpendicular to the time base line for targets which are in align- 
ment with the antenna difference pattern null and which "lean" forward or 
backward from the perpendicular as angle of arrival deviates to one side or 
the other from the pattern null. It is apparent that the direction in which 
the signal pip points is related to the direction of arrival of signal energy 
relative to the antenna pattern null and is independent of the amplitude of 
the signal. The length of the signal pip indicates signal amplitude. Sensi- 
tivity of the indicator to angle of arrival is a function of relative attenuation 




Fig. 6-7 Single-Coordinate Monopulse 
System. 



304 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



in the orthogonal video deflection circuits. Such an indicator has been 
called a "Pisa indicator" after the famous leaning tower. 

To accomplish the operation described it is convenient to generate 
the overlapping antenna patterns with a single aperture. This may be 
accomplished with a single parabolic reflector illuminated by two primary 
radiators symmetrically displaced laterally from the focus. 

The principle has been described for a single angle coordinate. Extension 
to two angle coordinates is not ordinarily accomplished by duplicating the 
system, but the one-coordinate system may be modified to operate as a 
two-coordinate system. It is first necessary to generate four lobes in the 
antenna pattern. This is accomplished by using a cluster of four primary 
radiators symmetrically disposed about the focus, with two up, two down, 
two right, and two left. Sum and diff^erence signals are obtained separately 
from two pairs. The two difference signals are then added to generate error 
signals in one coordinate. Sum and difference signals are then obtained 
from the two first sums. The resulting diflFerence signal is used to generate 
error signals in the other coordinate. The second or final sum signal, which 
is the sum of all four lobes, carries target amplitude and range information 
and provides a common phase reference for both coordinate error signals. 
The angle information may be utilized in a wide variety of configurations. 
Shown in Fig. 6-8 is one of the most common: an automatic angle tracking 
system such as might be employed in an AI radar or guided missile terminal 
seeker. 



L 



Antenna 



F<M 



Azimuth 

Diff. 
Channel 



Elevation 



n 




Diff. Channel 
Sum Channel 



Antenna 
Controller 



^ I 



Jg Automatic Angle ^ 

Y^ Tracking Loop M 



Azimuth 
Amp. 



Phase 
Det. 



Error 



Elevation 
Amp. 



Phase 
Det. 



Sum 
Amp. 



Range 



Fig. 6-8 Two-Coordinate Monopulse System. 



6-4] CORRELATION AND STORAGE RADAR TECHNIQUES 305 

Monopulse techniques are particularly useful for applications where 
pulse-to-pulse amplitude fluctuations due to target variations or interfer- 
ence signals can degrade conical or sequential scanning tracking techniques. 



6-4 CORRELATION AND STORAGE RADAR TECHNIQUES 

Signal storage has played a most significant role in the success of radar. 
From the earliest use of the cathode ray tube in echo ranging with A-scope 
presentation to modern sophisticated and complex magnetic storage devices 
for predetection integration, the use of storage has become increasingly 
important. Today the lack of high-capacity memory, high-speed operation, 
and wide dynamic range storage are perhaps major contributing factors 
impeding the development of more effective long-range radar. The in- 
creased emphasis on integration by storage has been brought about in part 
by the growing popularity of correlation and information theory methods 
for signal enhancement. The idea of correlation in itself is not new to radar 
— the World War II SCR-584 used a limited form of cross-correlation 
detection to separate the bearing and elevation errors. Here the correlation 
was not of the statistical nature currently in favor for signal enhancement. 
For this latter purpose, the cross-correlation device requires some form of 
storage and integration in order to fulfill its mission. Since storage can be 
considered a part of the correlation process, we will discuss the more general 
subject of correlation first. 

Correlation Processes. Two correlation techniques have appeared in 
radar during the last two decades: (1) autocorrelation, defined mathe- 
matically as 

^n(r) = lim -^jj^W^i^ - r)dt (6-17) 



-:; 2T 



where t is a time displacement (delay in the case of a radar echo), and 
(2) cross correlation, defined as 

<P,,{t) = lim ;r^ / /i(/)/2(/ - r)dt. (6-18) 

Both of these techniques are defined in the time domain and exist theoret- 
ically only in the limit as the total observation time becomes infinite. In 
practice, of course, infinite time is not available, and it becomes necessary 
to reinterpret the functions using finite limits. Let us define an incomplete 
autocorrelation function as 

^„(r,T) = ^/^/iW/i(^ - r)dt (6-19) 



306 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

and an incomplete cross-correlation function as 

^,,{tJ) = ^jjm-^it - r)dt (6-20) 

where 2T is now a finite observation or integration time. 

Autocorrelation of a limited nature, specifically for r = 0, has been used 
in conventional radar systems almost since the invention of radar. As can 
be seen from a study of Equation 6-19, the incomplete autocorrelation 
function as applied to radar for r = consists of (1) obtaining the instan- 
taneous echo power /i^(/), (2) integrating or summing for a finite time 27", 
and finally (3) dividing by the period 2T, thus forming an average power. 
These three steps can be seen to be essentially equivalent to the conven- 
tional frequency-domain radar processes wherein a square-law second 
detector converts the echo into instantaneous power and some type of 
storage provides the required averaging. In early radar sets for echo 
ranging the averaging was performed aurally by the operator or visually 
using A-scope presentation. Later the plan position indicator (PPT) used 
cathode ray tube persistence plus the operator for storage. Finally the use 
of more sophisticated video integration was adopted. 

The relative merits of autocorrelation (r = 0) and square-law detection 
versus cross-correlation detection have been studied^ with the results shown 
in Fig. 6-9. The output-versus-input mean power signal-to-noise ratios are 
plotted for a bandwidth reduction of 2 : 1 (in going from IF to video, for 
example), a practical value for echo ranging where the pulses must be 
retained. These curves apply to a single pulse where there is no integration. 
Signal enhancement resulting from the integration of pulses is discussed 
subsequently. It is interesting to note that autocorrelation can be thought 
of as comparable to postdetection bandwidth reduction, whereas cross 
correlation is comparable to predetection bandwidth reduction. In Fig. 6-9 
there is an apparent threshold in the autocorrelation and square-law 
detection curve starting in the neighborhood of unity signal-to-noise ratio. 
This threshold is noted by the change from a linear to a square-law relation- 
ship between output and input sensitivity. Such a threshold does not exist 
in cross correlation, where a noise-free reference is used. 

Cross-Correlation Radar. As soon as the signal enhancement 
capability of statistical cross correlation was recognized, applications to 
radar were considered. In order to obtain the maximum advantage from 
the process, one of the functions in Equation 6-20 must be noise free. A 
study of (pi2(r,T) reveals that if /i(/) is the delayed target echo which 
contains desired information as well as unwanted noise, then Ji{t — r) 
should be a noise-free reference signal possessing characteristics identical 

^Samuel F. George, Time Domahi Correlation Detectors vs Conventionat Frequency Domain 
Detectors, NRL Report 4332, May 3, 1954. 



6-4] 



CORRELATION AND STORAGE RADAR TECHNIQUES 

+30 



+ 20 



307 



-20 



-40 



-50 











/ / 


Cross Correlat 
Detector (Noise- 
Reference) 


on 
-ree 




/ 






V 
















/ 




















/ 


\ 








/ 


Autoc 

(r=0) a 
Law 


orrelation 
nd Square- 
Detectors 






/ 











-30 -20 -10 +10 +20 

INPUT SIGNAL-TO-NOISE RATIO (db) 
Fig. 6-9 Comparison of Autocorrelation and Cross Correlation. 



with those of the signal component in/i(/) and permitting a variable delay r 
to match the echo delay, thereby indicating range. This would be a specific 
application to the radar problem of echo ranging or the measurement of 
delay. A simplified block diagram of a cross-correlation detector is shown 
in Fig. 6-10. The output ^i2(t,T) could be used in the same manner as the 
output from the second detector of a conventional radar. 

Echo ranging as a major application of the cross-correlation principle to 
radar has been studied comprehensively by Woodward^ in the light of 
information theory. Woodward shows that in order to extract the most 
information about the exact target range from a received radar echo in 
additive Gaussian noise, the optimum receiver is one which forms the 

^P. M. Woodward, Probability and Information Theory, with Applications to Radar, McGraw- 
Hill Book Co., Inc., New York, 1953. 



308 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



Receiver 

+ n(t) 



Transmitter 
u{t) 



Storage Device 

and 
Variable Delay 

f^{f-T) = u{t-T) 



Multiplier 



Integrator 

2l j\(t) f2(t-r)dt 



■4>,Jr,T) 



Fig. 6-10 Simplified Cross-Correlation Detector. 



incomplete cross-correlation function. The problem of signal detectability 
has been very exhaustively studied"* and reported in 1954 at the MIT 
Symposium on Information Theory^. One conclusion is that for the case 
of a known signal operating through white Gaussian noise, the cross- 
correlation receiver is optimum. This result is based upon the likelihood 
ratio criterion. 

Cross correlation has become very useful in extracting the doppler 
frequency shift or range-rate information for moving targets, thus adding 
a new method to aid in target detectability as well as in more accurate 
tracking and multiple target resolution. In order to extract the doppler, 
the incoming echoes must be processed so as to permit coherent integration®. 
This is predetection integration, which in the case of a pulse-doppler system 
means coherent video or IF integration. The cross-correlation principle is 
embodied in all of the systems proposed for using range-rate information. 
First, a coherent or stored noise-free reference must be available; then some 
storage medium is required to permit integration; and finally some form of 
very narrow-band doppler filtering must be employed. Fig. 6-11 shows a 
block diagram of a straightforward or brute-force pulse-doppler system. 
Here there are n range gates with m doppler filters per gate. It is readily 
seen that a tremendous duplication of equipment is called for unless some 
storage device can be placed in the system. 

*J. Neyman and E. S. Pearson, "On the Problems of the Most Efficient Tests of Statistical 
Hypotheses," Phil. Trans. Roy. Soc. London A231, 289 (1933). 

1. L. Davis, "On Determining the Presence of Signals in Noise," Proc. Inst. Elec. Engrs. 
London 99 (III), 45-51 (1952). 

E. Reich and P. Swerling, "The Detection of a Sine Wave in Gaussian Noise," J. Appl. Phys. 
24, 289 (1953). 

R. C. Davis, "On the Detection of Sure Signals in Noise," J. Appl. Phys. 25, 76-82 (1954) 

W. W. Peterson and T. G. Birdsall, The Theory of Signal Detectability^ Electronic Defense 
Group, University of Michigan, Technical Report No. 13, July 1953. 

^Transactions of the IRE, PGIT-4, September 1954. 

^Bernard D. Steinberg, Coherent Integration oj Doppler Echoes in Pulse Radar, Report 
#182-112-1, General Atronics Corp, Aprif 1957. 



6-4] 



CORRELATION AND STORAGE RADAR TECHNIQUES 



309 



w 



TR M — Transmitter — ► Delay 



Receiver 



Multiplier 



Coherent 



Video 



f2 


-*' 


D 

D 






Fig. 6-11 Pulsed-Doppler System. 

Storage Radar. We have noted that one of the first and perhaps 
simplest of all storage mechanisms consisted of a visual observer using an 
A scope. Next, in the PPI, screen persistence performs the storage, and 
scan-to-scan integration is attained. Finally, the last form of postdetection 
integration to become significant was of the video type. Here the storage 
element could vary from a video delay line to some form of electrostatic 
storage. A simple delay-line video integrator is illustrated in Fig. 6-12. 



Video 






T 


= lnterpulse Period 








Pulse 


Adder 




Video Delay 
Line 


— 


Amplifier 


Integrated 




" 


Output 



















Fig. 6-12 Simplified Delay-Line Video Integrator. 

The number of pulses which can be effectively integrated to improve the 
signal-to-noise ratio depends upon the delay line loss, feedback circuit stabil- 
ity, distortion, and the length of time the target remains essentially at a fixed 
range. Video integration of pulses embedded in additive Gaussian noise 
at the radar input improves the signal-to-noise power ratio by the 
number of pulses added; the total improvement possible after detection, 
then, is limited by the observation or integration time permissible. 
From Fig. 6-9 it is seen that there is a detector loss for any detector 
input below threshold (S /N= 1) and that the ultimate radar sensitivity 
is obtained by predetection rather than video integration. 



310 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

In order to effect an improvement in signal-to-noise ratio by predetection 
integration, some technique must be used either to obtain signal coherence 
at IF or video, or to ensure transmitted signal recognition for cross- 
correlation purposes. The earliest methods employed coherent sources for 
transmission or coherent local oscillators in reception. As the transmitted 
signals became more complex and sophisticated, it was considered necessary 
to use storage to retain an exact replica for cross-correlation purposes. 
Both electrostatic storage and magnetic-tape storage have been developed 
and used successfully. The limited dynamic range of electrostatic storage 
combined with relatively short storage times, and the relatively slow 
accessibility of the data on magnetic tape have created an interest in 
magnetic-drum storage. As advanced technical progress provides increased 
dynamic range and higher frequency operation, the tremendous data 
handling capacity combined with high-speed record and readout and long- 
duration storage make magnetic-drum storage appear very desirable for 
radar use. 

Fig. 6-13 shows a system using transmitted-waveform storage to provide 
a reference for the cross correlation and magnetic-drum storage to reduce 



TR 



Transmitter 



Storage & 
Range Gate 






Receiver 



Multiplier 



Magnetic Drum Doppler 



Storage Output 



^3 


-^ 


D 



Fig. 6-13 Storage Radar. 



the equipment multiplication required in Fig. 6-11. The transmitted- 
waveform storage unit could be envisioned to consist of multiple delays 
corresponding to the n range gates of Fig. 6-11. There could be a separate 
track on the magnetic drum corresponding to each value of delay — i.e., to 
each range gate. For a given channel or track on the magnetic drum, the 
return echo pulses could be clipped and converted essentially to a binary 
code, which could then be painted sequentially so as to form the doppler 
signal as a modulation on the code.'' The doppler frequency could be 
explored as before by a doppler filter bank as shown. Only an elementary 



6-5] FM/CW RADAR SYSTEMS 311 

system has been illustrated here, and numerous ramifications become 
evident. 

The ultimate goal in search radar using correlation and storage tech- 
niques will be achieved when the range accuracy and resolution are limited 
only by the transmitted bandwidth, and the range-rate accuracy and 
resolution are limited only by the total observation time during which the 
target doppler remains coherent. 

6-5 FM/CW RADAR SYSTEMS 

Previous discussions have dealt primarily with pulsed radar systems — 
i.e., systems where transmission and reception occur at different times. 
Another important class of radar systems is composed of systems that 
employ continuous transmission (CW systems). In these systems, the 
transmitting and receiving systems operate simtultaneously rather than on 
a time-shared basis. 

For certain applications — such as semiactive missile guidance systems 
— continuous-wave (CW) systems can offer important advantages, 
particularly with respect to high clutter rejection for moving targets, 
and relative simplicity. 

Basic Principles of Operation. In a CW system, the transmitted 
and received signals are separated on a frequency basis rather than on a 
time basis as for a pulsed system. This is accomplished by maintaining 
phase coherence between transmission and reception — a process which 
permits the measurement of the doppler shift caused by the continual rate 
of change of phase in the radar reflection from a relatively moving object. 

The principles of operation of a simple CW system are illustrated in Fig. 
6-14. A signal at frequency/^ is transmitted. The return echo from a target 
moving with relative velocity Vr is shifted by the doppler effect to a new 
frequency /< +/d, where 

Ja = 1ft Vrlc. (1-20) 

Closing geometries shift the received frequency higher than the trans- 
mitted; opening geometries lower the frequency. 

The transmitted and received signal frequencies are mixed to recover the 
doppler frequency, which is then passed through a filter whose bandpass is 
designed to accept the doppler frequency signals from moving targets and 
suppress the return signals from fixed targets such as ground clutter. (For 
the example in the diagram, the clutter is shown at zero frequency: a 
condition that would exist for a ground-based radar.) For a moving 
airborne radar, the clutter would also possess a doppler shift relative to 
the transmitted frequency. This complicates the design of the filter; 
however, the basic principle remains the same. 



312 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 







F+ 


/. 






\' 




fo 








- Mixer 


-i 






f. 




f. 


Bandpass 


Filter 








Mixer 
Inputs 



Mixer 
Output 



Clutter 



jr 



Target 



Transmitter Signal 
Clutter Signal 
^/Target Signal 



^0 fo+fd 



Filter 
Output 



Filter ^ ^ 

Bandpass i^rget 



imin. 



(b) 



Fig. 6-14 Simple Stationary CW Doppler System: (a) Block Diagram, (b) Signal 
Frequency Relations. 

Very often, the filter is not a single filter as shown in Fig. 6-14 but rather 
a series of overlapping narrow band filters — each equipped with its own 
detector — which covers the total desired doppler frequency band as shown 
in Fig. 6-15. This permits very narrow bandwidth detection, measurement 
of target velocity, and resolution between targets with different relative 
velocities within the same antenna beamwidth. 



Filter 
Outputs 



\ rn I \ ^^ A /^^ 1^ r^ 7"^ / 

' '' - \'' / "' V' 1'' / 

II; \l \ W f^\l \i >/ / ; 



'I' *' >/ \i ^\! *' V ' iw 



1 ^ l\ 






Fig. 6-15 Filter-Bank Detection Showing Contiguous Filters. 



6-5] 



FM/CW RADAR SYSTEMS 



313 



The information matrix of this system may be visualized in Fig. 6-16. 
The maximum possible range is limited by scanning speed; that is to say, 
beyond a given range, targets cannot be seen because the dwell time on the 




N, 



Total Doppler Acceptance Band 



'N.xNgxN.xNA 



Individual Doppler Filter Bandwidth 
= Total Doppler Acceptance Band 



Fig. 6-16 CW Radar Information Matrix. 

target is less than the time required for a round trip of the radar energy. 
Since a simple CW system has no range-measuring capability, the maximum 
range is also the size of the minimum range resolution element. 

Range Measurement in CW Systems. To measure target range, 
frequency modulation (FM) of the transmitted energy is generally used in 
CW systems. The maximum deviation of the transmitted signal determines 
the range resolution obtainable, while the frequency of the FM-ing deter- 
mines the maximum unambiguous range.* 

Consider typically a simple sine wave FM as expressed by the trans- 
mitted FM /CW waveform of Equation 6-21 : 



et = Et sin ( co«/ + y- sin oo™/ j- (6-21) 

The received doppler signal er as detected by a coherent FM/CW radar is 

c-r = Er cos cod^ H J— sin ir/mT cos (comt + Tr/mr) |- (6-22) 



/m 



where Er is the peak received voltage, cod is the doppler frequency, AF is the 
peak transmitted deviation, /^ is the deviation rate, and r is the round trip 

^For a rough comparison with pulse radar the reader may consider the reciprocal of the 
bandwidth of the FM transmission to be analogous to an "effective" pulse length, while the 
FM rate is analogous to the PRF of a pulse radar. 



314 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



transit time. To determine distance, transit time can be measured by 
resolving the magnitude of the returned deviation (2AFsin irfmr) or by the 
relative phase lag of the returned modulation {irfmT) . In general, the greater 
the transmitted deviation AF, the greater is the resolvability of range. 
However, this range resolution is usually purchased at the price of in- 
creasing the minimum bandwidth of the doppler filters. 

If linear FM is used such as is provided by a sawtooth or triangular 
waveform, the Doppler frequency is merely shifted by the amount the 
transmitter has deviated during the round trip transit time. The principle 
is illustrated in Fig. 6-17. Depending upon range, modulation, and doppler 



f,-AF - 



TIME 



■Transmitted 







V^^^^ 



^Measure of Range f, 

I Doppler Frequency fj 



TIME 



Fig. 6-17 FM Range Measuring Principle. 



shift, the instantaneous received frequency will differ from the transmitted 
reference frequency. When the two are heterodyned, a frequency modula- 
tion is superimposed on the detected doppler signal. The magnitude of the 
FM deviation from the doppler is a measure of range. Care must be 
exercised in the selection of deviation values or the doppler frequency and 
range frequency may be difficult to resolve uniquely. 

In some applications such as altimetry the range frequency greatly 
exceeds the doppler values. For example, in doppler altimetry the electro- 
magnetic energy is radiated normal to the direction of flight so that no 
doppler signal results. By using triangular modulation, as in Fig. 6-17, the 
magnitude of the resultant detected difference frequency /r is a measure of 
altitude: i.e.. 



(6-23) 



6-5] FM/CW RADAR SYSTEMS 315 

where Tr = modulation period in seconds 

AF = peak transmitted deviation in cps 

h = altitude in feet 

c = velocity of propagation (984 X 10^ ft /sec). 

For more conventional radar applications using sinusoidal FM for 
ranging, the peak frequency deviation/^ on the doppler signal is a measure 
of range (see Equation 6-22); i.e., 

fr = 2AFs\mr/mT (6-24) 

or = 2Ai^7r/„/^^ 

where R = range in feet 

AF = peak transmitted deviation in cps 

fm = deviation rate in cps. 

Use can be sometimes made of the fact that the doppler frequency is 
precisely proportional to range rate. Integration of range rate can produce 
an actual measure of range provided the constant of integration can be 
determined. 

Stability Requirements. The principal theoretical advantage of CW 
radar systems derives from their ability to distinguish moving targets in the 
presence of clutter. The maintenance of this advantage in a practical design 
places strict requirements on the short-term frequency stability of the 
transmitted energy; i.e., the coherence that exists between the transmitted 
and received signals. Long-term frequency drift is generally not a problem 
provided that short-term coherency exists. 

Any modulation that tends to broaden the spectrum of a wanted signal 
may destroy the signal-to-noise ratio in the received gate. Any modulation 
of interference signals may produce sidebands which interfere with the 
resolution of desired signals. When interference signals, such as feedthrough 
or clutter, are very large, short-term stabilities in excess of 10~"^ may be 
required. 

The process of coherent detection of a time-delayed signal alters the 
angular modulation index as has been indicated by Equation 6-22. 

In a sfalic situation the RF reference phase can be adjusted to minimize 
sensitivity to angular modulations; this is impossible for a gimbaled or 
moving radar. Alternatively the radar detection reference could be time- 
delayed an amount equal to the round trip to target delay. However, this 
is usually impractical because of the continually changing delays of one or 
more targets. 



316 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

From Equation 6-22, it is possible to evaluate the extent to which the 
angular noise sidebands of a large signal will interfere with the detection of 
another signal. However, it is difficult to apply Equation 6-22 in a general 
manner since the effective modulation index is a periodic function of the 
two variables /m and t where the maximum values of/m are at «/2r (where 
« = 1, 3, 5, ... ) and the zeros are at/m = n /t (where « = 0, 1, 2, 3, ...). 
Of more usefulness is a consideration of the two extremes of the relationship 
as divided by -a cutoff frequency /^ at which the returned deviation, by 
definition, equals the transmitted deviation. Below/c, where the time delay 
is short compared to a modulation cycle, the angular indices of the detected 
and transmitted signals are related simply by 

Mfr = M/,(27rr/,„). (6-25) 

Far above /c, where the time delay is long compared with a modulation 
cycle, the two indices are, on the average, equal. 

For small indices of modulation such as are descriptive of useful trans- 
mitting tubes it is more convenient to discuss potential interference in terms 
of sideband power ratios relative to the carrier. Therefore, the ratio of the 
power in a single sideband Psb, relative to the carrier power Pc for a detected 
signal, is determined (for small indices only) by: 



n 



£f4^-i- »-j« 



Typically, for a 1200-cps power supply ripple component producing 240 cps 
of frequency deviation, a carrier to single sideband power ratio of 100 would 
exist. 

Random noise modulation may be considered to be composed of distrib- 
uted components having a mean angular excursion per cycle of A/. T+te 
composite mean deviation AF associated with a band of frequencies B cps 
wide is then 

AF = Af^fB. (6-27) 

A not uncommon composite noise deviation in a 100-kc band B for practical 
CW radars is 100 cps rms (AF), indicating about a 1 /3 cps rms/cps density 
(A/)_. 

Fig. 6-18 indicates typical signal power levels which might be present in 
a hypothetical FM /CW radar. 

FM/CW Airborne Radar Systems Applications. FM/CW doppler 
systems are most commonly employed for applications requiring high 
clutter rejection and a relatively low range information rate. AI radars, 
missile seekers, and altimeters are good examples of such applications. 



(y-S] 



FM/CW RADAR SYSTEMS 



317 



200- 



150- 



50- 



C^Speed of Electromagnetic Propagation 
Vf = Speed of Radar Platform 
V^ = Radial Target Speed 




Clutter 



'/"' 



\ 0? y^ Receiver 
^^ hE / Noise 



^ 



Density 



Ike lOkc 



2Vc 



AM Noise 
Density 



lOOkc IMc lOMc lO^Mc lO^Mc lO^Mc 

f Frequency — 

c ^ 



Fig. 6-18 Typical Signal Levels in an Airborne FM/CW Radar. 



A principal advantage of CW doppler systems is their simplicity, when 
compared with other means for obtaining high clutter rejection such as 
pulsed doppler and coherent AMTI systems. Provided that some of the 
limitations to be discussed below do not seriously limit tactical utility, an 
FM/CW system offers a lightweight and potentially reliable answer to 
many airborne radar system problems. 

The use of doppler techniques places several constraints upon the tactical 
usage of an airborne weapons system. (1) There are approach aspects where 
the target doppler frequency can be zero, or where the target doppler 
frequency falls within the clutter spectrum. These conditions lead to "blind 
regions" and regions of poor signal-to-noise ratio which must receive careful 
consideration and analysis in the overall system design. For example, 
approaches in the rear hemisphere of a target can be degraded by these 
considerations. Fortunately, the most interesting approach region from 
many tactical standpoints — forward hemisphere or head-on — is a region 
where doppler sensing devices are most effective in detecting and tracking 
targets in heavy ground clutter. 

(2) Buplexed-active operation (transmission and reception through a 
common antenna) is generally impractical in FM/CW doppler equipment 



318 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



because of excessive feedihrough of the transmitter energy directly into the 
receiver. By physically spacing the transmitting from the receiving antenna 
on a common radar vehicle (spaced-aclive operation), the isolation problem 
becomes resolvable at the cost, however, of degradation in the vehicle's 
aerodynamic profile. Semiactive operation involves transmission and 
reception on separate radar platforms, which also minimizes the feed- 
through problem. The main advantages of semiactive operation in homing 
missilery are that the transmitting hardware is deleted as an internal missile 
requirement and a greater on-target illumination power density is practical 
from the large parent radar. Both a spaced-active and a semiactive system 
are illustrated in Fig. 6-19. 




Missile Semiactive 
Radar 



Fig. 6-19 FM/CW Airborne Radar Systems. 



(3) The ranging accuracy obtainable with an FM /CW set designed for 
high ground clutter rejection may be relatively coarse — perhaps of the 
order of 1—2 n. mi. This is not usually a serious drawback for guided missile 
applications, although it is an important limitation for fire-control systems 
employing unguided weapons. 

FM/CW Radar Performance. The detectability of the target is 
determined, for a given false-alarm rate, by the signal-to-noise ratio after 
final detection. Neglecting the effects of clutter and transmitter modu- 
lations, the signal-to-noise ratio may be calculated by approximate modifi- 
cation of the previously derived radar range formula (see Equation 3-9): 



S/N ^ 



{AttYR'FKTB 



(6-28) 



6-5] FM/CW RADAR SYSTEMS 319 

where P = transmitted average CW power 
B = doppler filter bandwidth. 

The choice of a detection bandwidth B is governed by a number of 
considerations derived from the tactical problem and from the realities of 
radar design practice. 

The spectral composition of a modern-type aircraft radar reflection is sel- 
dom more than a few cycles wide when it is caused by target characteristics 
alone. A CW radar, transmitting a truly unmodulated wave, produces the 
most elementary moving target spectrum. The possibility exists, therefore, 
of detecting a radar signal as much as 190 db below a watt in about a second, 
using simple, very narrow band filters matched to the signal waveform. 
However, a number of practical and tactical considerations usually limit 
the full exploitation of this potential. 

To search the frequency spectrum of expected dopplers requires a series 
of adjacent filters or one or more individual filters scanning the spectrum. 
The minimum allowable time on target is that time required for energy to 
build up in one of a fixed bank of filters; scanning filters increases this 
minimum by the ratio of scanned to actual bandwidths. Prior information 
as to target velocity or bearing can reduce the spectrum or area to be 
searched and thus increase the probability of detection in a given situation. 

One practical constraint on exploiting very narrow bandwidths is the 
shift in the doppler spectrum caused by target maneuvers. This effect can 
cause the signal to be greatly attenuated if the target doppler transits the 
filter bandpass range before the filter has time to build up. To avoid this 
situation, the filter bandpass must satisfy the relation 

B>^ (6-29) 

A 

where a = acceleration in ft/sec^ 
X = RF wavelength in ft. 

Another usual requirement in a radar system is to yield additional radar 
target parameters. Information theory advises that to process more data, 
more bandwidth and /or time is required. To resolve target range by FM 
on CW or to resolve bearing by AM will in general increase the minimum 
radar bandwidth requirements. For example, angular scanning will broaden 
the return spectrum bandwidth by the scanning frequency modulation, 
which may be of the order of 50 cycles for a typical airborne radar. Alter- 
natively, the use of very low information rates demands an amount of time 
that may be unacceptable. 

An economically attractive technique is to employ one sweeping filter of 
a bandwidth such as to satisfy all system requirements. Conventional AFC 



320 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

methods can maintain the target signal in the velocity gate after lock-on. 
By allowing adequate time for the energy to build up in the gate,^ the 
maximum allowable sweep rate is proportional to the bandwidth squared. 
It is this "squared" sweep constraint that makes sweeping very narrow band 
filters very time consuming. As the filter bandwidth approaches the 
scanning frequency (or the reciprocal of the time on target), fixed filters 
become mandatory (see Fig. 6-15). 

From a performance standpoint, the behavior of the system in the 
presence of multiple targets is most important. Multiple radar targets 
include terrain and weather returns as well as reflections from man-made 
objects; for an airborne radar, each radar target will have a finite radial 
velocity with respect to the radar platform. Because of their physical size, 
man-made objects are usually point source targets, whereas clutter is most 
always angularly expansive. 

In the presence of a hypothetical hemisphere of homogeneously reflective 
clutter, the clutter doppler amplitude versus frequency spectrum would be 
similar to the integrated transmit-receive antenna radiation pattern versus 
angle as viewed at the receiver. A typical CW radar doppler clutter 
spectrum is shown in Fig. 6-18. Note the complete absence of interference 
at all frequencies above own-speed dopplers; only nose-aspect closing 
targets appear in this uncontaminated spectrum. In practice the clutter 
spectrum may be smeared somewhat by noise modulation on the trans- 
mitted energy. Reasonable prediction of specific clutter is possible, but 
there are a myriad of possible situations. Clutter can assume staggering 
proportions; a terrain return 100 db above the minimum detectable signal 
power is not inconceivable. Ultrahigh linearity in a receiving system is 
obviously required to avoid generating the additional interference of 
distortion products. Most radar systems are advisedly employed in a 
manner avoiding the most adverse clutter conditions. 

With some performance compromises, a practical degree of automaticity 
can be achieved in sorting multiple-target data in an FM/CW airborne 
radar. Doppler filtering can provide the necessary resolution to distinguish 
numbers of targets, but the decision as to which target is wanted may 
require the application of considerable intelligence. With experience a 
human being can reduce visual or aural doppler data satisfactorily for some 
applications. 

6-6 PULSED-DOPPLER RADAR SYSTEMS 

Pulsed-doppler radar systems represent an eflFort to combine the clutter 
rejection capabilities of doppler sensing radars with the range measurement 

^Commonly, values of from 2 to 17 time constants of the filter are used in practical appli- 
cations. 



6-6] 



PULSED-DOPPLER RADAR SYSTEMS 



321 



and time-duplexing properties of a pulse radar. For applications requiring 
heavy ground clutter rejection, a common transmitting and receiving 
antenna, and accurate range measurement, a pulsed-doppler type of system 
represents the best known technical approach to the problem. However, 
the pulsed-doppler type of system also has certain drawbacks: principal 
among these are limited target-handling capacity (when compared with a 
pulse radar) and a high order of electronic system complexity. 

Basic Principles of Operation. A simple pulsed-doppler system is 
shown in Fig. 6-20. It differs from the CW system of Fig. 6-14 only by the 



Master 

Frequency 

Control 


f 


^ 


Pulsed 
Coherent - 
Transmitter 


f^(Pulsed) 


n 


.,^l/^«^K 


f,(Pulsed) 1 








r- 




f^+f^ (Pulsed) \ 












f,+f^ (Pulsed) 






f, 


Mixer 












} 


fe^f 


^±f^+2f^± f^+ 




Bandpass 




-fd 




Fil 


er 







Fig. 6-20 Simple Pulsed-Doppler System. 

introduction of a pulsed coherent transmitter in place of a CW coherent 
transmitter; and a duplexer which turns off the receiver during a pulse 
period and isolates the receiver from the transmitter during the interval 
between pulses. A master frequency control is utilized to control the carrier 



\^T^ 



^ f^l/fo 



Vfr ^ 

(a) 



-^frr 



fn-lA 



(b) 



Frequency 



Fig. 6-21 Transmitted Pulsed-Doppler Signals. 



322 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

and pulse repetition frequencies and to provide coherent references for the 
receiver mixing processes. The transmitted signal thus consists of an RF 
pulse train as shown in Fig. 6-21 (a) which has the frequency spectrum 
shown in Fig. 6-21 (b). The width of the frequency spectrum is a function 
of pulse length; the separation between adjacent spectral lines is equal to 
the pulse repetition frequency /r. 

The operation of a pulsed-doppler system can best be visualized by 
examining the character of the return spectra from targets and clutter at 
various points in the receiving system. The target and clutter spectra for a 
single spectral-line transmission from a moving platform is shown in Fig. 
6-22. Because of sidelobes, the frequencies of the fixed clutter returns can 



Relative Target Velocity 
Velocity of Radar Aircraft 



Sidelobes 



MC^ 



'^^^IVIain Bea 
~~^,-^-Tar 


/ 

m 

get 
utter. 




^Main Beam Clutter 


Sidelobe C 


J/'' K^ Le-^larget Return 




2V,/\ 


2V,/X 


f,=2vy\ 



Fig. 6-22 Target and Clutter Spectra for a Single Spectral Line Transmission. 

vary ±2/^f/X from the transmitted frequency. The clutter possesses a high 
peak resulting from clutter return in the main beam. The position of this 
clutter peak depends upon the angle between the antenna pointing axis and 
the aircraft velocity vector. Quite obviously, if the antenna is scanning, 
the frequencies of the clutter returns will change as functions of time; in no 
case, however, can the returns from fixed clutter be doppler shifted by more 
than 1VfI\. For closing targets {Fr > Fp), the target returns will be 
shifted by IV^jX and will therefore appear in a clutter-free portion of the 
frequency spectrum. 

The effects of scanning and target-induced modulations (see Paragraph 
4-8) cause a broadening of the target spectrum. Generally, the latter effects 
are small compared with the modulation induced by scanning, so that the 
width of the target spectrum may be expressed 

^ftarort = ^ (6-30) 



where td = dwell time of the main beam on the target. 

By analogy, then, the frequency spectrum of the signal and fixed clutter 
returns for a ^«/j-^^ transmission consisting of an assembly of spectral lines 



6-6] 



PULSED-DOPPLER RADAR SYSTEMS 



323 



Target Frequencies 
n = 0,1,2 y 



1 li 



^^0^^. 



Main Beam and 
Sidelobe Clutter 



Closing Target 



Frequency 



Fig. 6-23 Received Signal Frequency Spectrum for an Airborne Pulsed-Doppler 
System, Showing Clutter Spectrum and Return from a Closing Target. 



can be represented as shown in Fig. 6-23. This diagram illustrates one of 
the basic limitations of a pulsed-doppler radar system : in order to maintain 
a clutter-free region for all closing targets of tactical interest, the spacing 
between spectral lines must be 



where Vr, 



,fr> 2{Vf^ ^..,nax)/X 

maximum closing velocity dictated by tactics. 



(6-31) 



If this spacing is not maintained, some of the closing targets will be buried 
in the clutter from adjacent spectral lines. This consideration leads to the 
use of very high PRF's in pulsed-doppler systems. For example, an X-band 
(3.2-cm) system designed for operation in a 2000-fps aircraft against 
2000-fps targets will require a minimum PRF of 112 kc — a value that is 
several orders of magnitude larger than the PWF's commonly used inpulsed 
radars. 

When the return is mixed with the coherent reference signal as shown in 
Fig. 6-20, the output spectrum shown in Fig. 6-24 is produced. This 
heterodyning operation causes an effect known ?is folding; i.e. each of the 




II 



-Doppler Filter 
Bandpass 



f 



(b) 



Fig. 6-24 Received Signal Spectrum After Heterodyning to Zero Frequency, 
Showing Effects of Spectrum Folding: (a) Video Spectrum, (b) Filtered Spectrum. 



324 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

target and clutter signals produces two symmetrical sidebands around each 
PRF line. The folding effect places a further limitation on the minimum 
allowable PRF: considering the effects of adjacent spectral-line interference 
due to folding, the minimum PRF is 

fr > — rj:!!^ (when signal "folding" occurs) (6-32) 

A ^ 

The resulting video signal then is passed through a bandpass filter to 
remove all the zero-frequency clutter components and all of the higher- 
frequency sidebands of both signal and clutter. The resulting output, then, 
is simply the doppler return associated with the central line of the trans- 
mitted spectrum. 

The process of "folding" also doubles the thermal noise which competes 
with the doppler return associated with the central line of the transmission. 
For this reason, some form of single-sideband detection process is usually 
employed in pulse doppler and CW doppler systems in preference to the 
simpler system described here. 

The doppler filter may be a single filter as shown, a bank of contiguous 
narrow-band filters as was shown in Fig. 6-15, or a single narrow band filter 
which sweeps over the total range between /d,min and/d,max- The comments 
in the preceding paragraph concerning the various filter types for CW 
systems also apply to pulsed-doppler systems. 

The simple pulsed-doppler system considered does not possess a range- 
measurement capability. Thus its information matrix and information rate 
are the same as shown for the simple CW system in Fig. 6-16. Actually the 
only reason for using a simple system of the type described is to eliminate 
the duplexing problem of a CW system and permit the use of a single 
antenna. In most other respects, this simple system is inferior to a simple 
CW system. Specifically, it possesses as deficiencies (1) greater complexity, 
and (2) less efficient use of power, since only the target power associated 
with the central spectral line is detected. 

The second point deserves further amplification. In a CW radar of 
transmitted power P, the peak and average powers are equal because the 
duty cycle is unity. All of this power is effective for detection of the target. 
However, for a pu/sed radar, the peak and average powers are related by the 
"duty cycle" — i.e., by the ratio d of "on" time to total time. Thus 

Ptd = Pare. (6-33) 

For pulsed doppler detection — as previously described — only the power 
in the central spectral line is used for detecting the target. The ratio of this 
power to the total power may be expressed 

Power effective for target doppler detection = {Pave)d. (6-34) 



6-6J 



PULSED-DOPPLER RADAR SYSTEMS 



325 



Thus, to achieve the same useful power return from the target as for a CW 
radar of average power P, the peak power of a simple pulsed-doppler system 
must be 

Pt = P/d\ (6-35) 

and the average power must be 

Pa,, = P/d. (6-36) 

The average power governs the weight of the power source and the peak 
power dictates the voltage breakdown requirements of the transmitting 
system components; thus these factors must be weighed against the 
duplexing difficulties of a CW system. 

One of the most common applications of the simple pulsed-doppler 
system is for doppler navigation radar systems. This will be described in 
Chapter 14. 

Range-Gated Pulsed-Doppler Systems. The full benefits of a 
pulsed-doppler system can be realized by range-gating the receiver. This 
technique permits range measurement with the resolution inherent in the 
radar pulse width and it can also improve the signal-to-noise ratio and 
signal-to-clutter ratio by the reciprocal of the range-gating duty cycle. 



Master 

Frequency 

Control 



Pulsed 

Coherent 

Transmitter 




Target Doppler Bandpass 

Filter Composed of Continuous 

Narrow Band Filters 



fj (Velocity) 



Fig. 6-25 Gated Pulsed-Doppler System with Means for Range Measurement. 



326 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

A generic range-gated pulsed-doppler system is shown in Fig. 6-25. In 
this system, the return signal is converted to IF and passed through an 
amplifier with a bandwidth approximately equal to the reciprocal of the 
pulse length. The IF amplifier output then is "gated" before the final 
mixing and doppler detection takes place. The width of the gate usually is 
made approximately equal to the pulse length. 

The operation of the gate is best understood by considering first a fixed 
gate which opens up the receiver at a time ti following transmission and 
closes the receiver tj seconds later at /i + xg. This action accomplishes the 
following: 

1. The only returns going into the final detection stage are those from 
ranges falling between the values 

Rn = f/2(/i + n/fr) and 

•^ (6-37) 

Rn + ^R = r/2(/i + n/fr + r) n = 0,1,2,3,-. 

Clutter originates — in the main — from area extensive targets, 
whereas the desired signal originates from point targets. Thus the 
gating will improve the signal-to-clutter ratio of a target in the gate 
by a factor which is, on the average, equal to the duty cycle of the 
gate dg^ where 

dg = rjr. (6-38) 

2. Noise enters the receiver only during the gating interval. Thus the 
average noise power is reduced by the duty cycle of the gate. 

3. Since the position of the gate is known with respect to the trans- 
mitted pulse, any target doppler detected must come from a target 
in one of the range intervals indicated by Equation 6-35. 

The improvement in signal-to-clutter ratio represents an improvement 
over and above what can be done with a CW radar system. Thus a range- 
gated pulsed-doppler system can provide greater clutter rejection than any 
other generic radar system type. The reduced receiver noise incident to 
gating tends to restore the ^ /A" ratio to the same value as would exist for a 
CW system of the same average power and bandwidth. In fact, if the gate 
width equals the pulse length, a target in the middle of the gate would 
possess the same SIN as the comparable CW system. 

The range measurement made by a gated pulsed-doppler system is not 
exactly the same as a range measurement of a pulse radar. The high PRF 
that must be employed in a pulsed-doppler system causes the unambiguous 
range interval to be relatively short compared with the maximum detection 
and tracking ranges. Since the maximum unambiguous range is 

D ^^^^^ ■ if. ^Q\ 



6-6] PULSED-DOPPLER RADAR SYSTEMS 327 

A PRF of 112 kc, as derived in the previous example, would yield an un- 
ambiguous range interval of only 0.74 n.mi. Values of this order of magni- 
tude are typical for airborne pulsed-doppler systems which are constrained 
by antenna considerations to operate in the general range of S to X band 
(10 cm to 3 cm). As a result, additional techniques — to be described 
below — must be employed to measure true range in a gated pulsed- 
doppler system. 

Range gating also levies a cost on the system; a price must be paid in 
terms of system complexity and /or information rate. The previous dis- 
cussion considered a single fixed gate. To cover the complete interpulse 
period, this gate would have to be swept. A sweeping range gate will 
increase the total required dwell time on the target tdf by the reciprocal of 
the gating duty factor dg-. 

tdt = tdf/dg. (6-40) 

where tdf = buildup time for the doppler filter. 

An alternative solution is to employ contiguous fixed range gates covering 
the entire interpulse period (see Fig. 6-11). This "brute force" solution 
requires a separate doppler filtering system for each range gate interval; 
however, in combination with fixed contiguous doppler filters it does permit 
the maximum information rate to be extracted from the system because the 
separate doppler components of each range interval are examined simul- 
taneously. Paragraph 6-4, Correlation and Storage Radar Techniques, 
suggested still another means for processing pulsed-doppler radar infor- 
mation. 

Range Performance. The idealized range of a pulsed-doppler system 
may be calculated by the following modification of the basic radar range 
equation (3-1): 



" ~ [{4Tr)'FkTBdg\ 



(6-41) 



where ds = signal duty cycle 

dg = gating duty cycle [equal to (l-ds) for an ungated system] 

B = doppler detection filter bandwidth^''. 

When "folding" occurs in the detection process, an additional factor of 2 is 
required in the denominator of the one-fourth power expression. 

^"In some cases, postdetection filtering will be employed to improve the final signal-to-noise 
ratio without increasing the number of doppler filters required. In such cases the effective 
detection bandwidth is Bbff = V725 • Bpd as derived in Paragraph 3-5 (Equation 3-62). 
In these cases, the dwell time should be matched to the bandwidth of the postdetection filter. 



328 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



Eclipsing. The relatively high duty cycle of a pulsed-doppler system 
— typical values vary from 0.5 to 0.02 — introduces a strong possibility 
that part or all of the received target pulse may arrive during a transmission 
period. Since the receiver is turned off during transmission, target infor- 
mation will be lost or "eclipsed." 

The basic problem is shown for a 0.33 duty cycle pulsed doppler radar in 
Fig. 6-26. Eclipsing causes an effective change in the duty cycle for returns 



Transmitted Pulses 



o n 



o 



Range 



1.0 



Duty Ratio Correction Factor dj = Eclipsed Duty Cycle 
djo = Normal Duty Cycle 




*- Range 



Idealized Range Correction Factor 



Ro= 7 Duty Cycle 



Probability of Detection (Single Scan) -No Eclipsing 




Fig. 6-26 Effect of Eclipsing on Pulsed-Doppler Blip-Scan Ratio. 



which overlap transmission periods. Since average power that registers in 
the doppler filters is proportional to the square of the duty cycle, the 
idealized range will vary as the square root of the duty cycle. The blip-scan 
ratio is a function of idealized range as shown in the fourth figure. When 
the blip-scan ratio with no eclipsing is corrected for the eclipsing effect, the 
last curve in Fig. 6-26 results. As can be seen, the effect of eclipsing is to 
cause "holes" in the blip scan curve in the regions of pulse overlap. In a 
practical pulsed-doppler system the ratio of PRF to the useful range 



6-6] 



PULSED-DOPPLER RADAR SYSTEMS 



329 



interval would be much higher; thus, there would be many more "holes" 
than shown in this example. For purposes of calculating the cumulative 
probability of detection, it is often convenient to approximate the notched 
blip-scan curve with a "smoothed" curve. 

If the pulsed-doppler system is operated with a fixed PRF, there will be 
certain closing velocities which could result in the target's appearing in a 
detection notch on each successive scan. For example, the interpulse period 
of the numerical example was 0.74 n.mi. If the first detection of the target 
occurred at a range corresponding to a "hole," and if the target moved a 
multiple of 0.74 n.mi. between scans, then detection would never occur. 
Eventualities such as this may be largely eliminated by a slow variation of 
the PRF which would have the effect of producing a smoothed — but 
nevertheless, degraded — blip-scan curve. 




Pieces of Information 



Range Measurements in 
Pulsed-Doppler Systems. As pre- 
viously mentioned, the range gate 
position measurement produces an 
ambiguous range indication because 
of the high repetition frequency that 
must be used in a practical pulsed- 
doppler system. The high repetition 
frequency reduces the total number 
of separate unambiguous range in- 
tervals (Nr = 1 /rfr) and gives the 
pulsed-doppler radar an information 
matrix such as is shown in Fig. 6-27. 
In almost all practical cases, it is 
desired to operate the radar against 
targets at ranges far exceeding the 
unambiguous range interval. Thus 
a means must be employed to 
circumvent the range ambiguity 
problem in a range-measuring pulsed-doppler system. 

There are several means for measuring true range: all are inconvenient 
and all degrade radar performance in terms of information rate and /or 
signal-to-noise ratio. One means for accomplishing range measurement is 
to employ the FM method used for the CW radar. The operating charac- 
teristics of this method are essentially the same as for an FM/CW radar; 
particularly, if the duty cycle of the pulsed-doppler system is relatively 
high. The range accuracy of this method is relatively poor if a narrow 
detection and tracking bandwidth is maintained. The range resolution is 
also poor because the pulse shape information is never utilized. 



Fig. 6-27 



NrxN.xNcxNy 

Nr = VTfr 
N = fr/b 

B = Doppler Filter 
Detection Bandwith 

Pulsed-Doppler Information 
Matrix. 



330 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



A = Transmitted Pulses 
I = Received Pulses 

True Range = ^- 

_ 2 X Desired Max. Range 

max- ^ 



/ \\ Al A! Al Al A! 



-f2^ 



JA. 



lA 



lA 



lA 



kA,^ 



Fig. 



6-28 Two-PRF Range Measure- 
ment. 



A second means is to employ- 
various modulations of the pulse 
repetition frequency. As an ex- 
ample, step switching of the PRF 
between several values can provide a 
ranging capability. The basic prin- 
ciple is shown in Fig. 6-28. Trans- 
mission occurs on two PRF's which 
are multiples of relatively prime 
numbers (in the example in the 
figure the numbers are 5 and 4). 
Because of ambiguities, a target at a 
true range corresponding to ttr will 
appear as a target return with time 
delays /i and /2 relative to the near- 
est transmitted pulse on each PRF, 
respectively. Thus we may write 



, ^1 



/2 + 



(6-42) 



where «i, «2 = number of unambiguous range intervals in each PRF (in 
the figure, «i = 2 and «2 = !)• 

In the example shown there are two possible relationships between «i 
and «2: 

Wl = ^2 

«i — 1 = «2 

Substituting these relationships in Equation 6-42 we obtain, as expres- 
sions for the time delay corresponding to true range, 






h/rl-tJr2-{-l 



(6-43) 



frl-/r 



If the first expression is negative, the second must be employed. Thus, 
the use of two PRF's can provide unambiguous range over a maximum 
desired ranging interval corresponding to the time delay 



/rl-/r 



(6-44) 



6-6] PULSED-DOPPLER RADAR SYSTEMS 331 

Three or more PRF's can be used to extend the unambiguous ranging 
interval further. In these cases, the data processing becomes more compli- 
cated for two PRF's; however, methods similar to those used for Equations 
6-42 to 44 may be used to derive the required relationships. The multiple 
FRF system of ranging is severely limited if more than one target return 
at the same doppler frequency is present. In a two-PRF system, two 
targets would yield four possible range values : two correct ranges and two 
"ghosts." Eclipsing also can cause difficulty, since it is quite likely that the 
target return for one of the PRF's will be eclipsed. The accuracy of this 
method of ranging is comparable to that of a pulse radar employing the 
same pulse length. 

If the "looks" at the target are taken by sequentially switching the PRF 
from one value to another, the required dwell time on the target for the 
same system bandwidth is increased by the number of PRF's employed. 
Alternative procedures such as simultaneous transmission of the PRF's or 
wider bandwidth reception could be used to keep dwell time constant. 
However, these methods will decrease the available *S'/A^ ratio for a given 
amount of total average transmitter power. In addition, simultaneous 
transmission greatly increases systems complexity in both the transmitter 
and receiver and gives rise to serious eclipsing problems because of the 
higher effective duty ratio. 

Pulsed-Doppler System Design Problems. Pulsed-doppler systems 
have the same basic problems of transmitter stability as CW systems. 
These problems are, in fact, common to any coherent system. 

Because of its high duty cycle, the duplexing problem is particularly 
difficult in a pulsed-doppler system. To cut eclipsing losses to a minimum, 
the transition from transmit to receive must be made as quickly as possible. 
Ordinary transmit-receive (TR) tubes are not satisfactory for this appli- 
cation; however, ferrite circulators (see Paragraph 10-16) and ferrite 
switches have found considerable application because of their low insertion 
losses (0.5 db) and their very rapid recovery time. 

In the receiving system, particular care must be taken to provide suffi- 
cient dynamic range to accommodate the maximum clutter amplitudes .^^ 
Linearity must be maintained over this range to avoid intermodulation 
products which spread the signal spectrum and cause loss of signal-to-noise 
ratio at the doppler filters. 

The design of the doppler filtering system — particularly, the bandpass 
characteristic and the maintenance of proper frequency spacing between 

iiln many designs, the clutter from the main beam and the altitude line is eliminated prior 
to amplification and doppler filtering. This greatly reduces the dynamic range requirements 
of subsequent stages of the receiver; however, it also makes the system completely "blind" 
at these frequencies. 



332 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

filters — is a vital design consideration. A fixed range gating, fixed filter 
bank pulsed-doppler system may have hundreds or even thousands of these 
narrow band filters; thus the trade-ofF between filter performance and size 
and weight is a vital consideration. 

Angle tracking poses certain special problems in a pulsed-doppler radar. 
The doppler frequency as well as the range must be tracked prior to angle 
lock-on. The bandwidth of the velocity loop corresponds to the width of 
the doppler filter. If conical scanning is employed, this filter must be wide 
enough to transmit the scan modulation sidebands. Actually, the doppler 
filter width should be about three times the scan rate in order to minimize 
phase and amplitude variations of the error signal. For example, a 40-cps 
scanning frequency would require a doppler filter band width of at least 
120 cps. 

The use of monopulse angle tracking (see Paragraph 6-3) poses a most 
difficult problem in a pulsed-doppler system. The sum and the difference 
signals must be handled in completely separate receiver channels — each 
with its own mixer, amplifiers, range gates, and doppler filters. In addition 
to the obvious disadvantages of weight and size, the problem of maintaining 
the proper alignment of these channels relative to each other represents a 
prodigious design problem. 

Pulsed-Doppler Systems Applications. As previously mentioned, 
pulsed-doppler systems are best employed in systems requiring substantial 
ground clutter rejection, a common transmitting and receiving antenna, 
and accurate range and /or velocity measurement. 

One other characteristic of a doppler system — either CW or pulsed- 
doppler — also has great tactical utility. This is the automaticity potential 
of such systems. Detection in such systems is inherently automatic since 
the signal is detected by the comparison of a filter output with a preset bias. 
While the same thing can be done in a pulse radar system, the problem of 
setting a bias level is enormously more difficult because of false alarms 
caused by clutter. This necessitates the use of bias levels considerably 
higher than would be dictated by thermal noise considerations. Thus the 
detection performance of an automatic pulse radar system is appreciably 
poorer than can be obtained when a human being is used as the detection 
element, since the human operator can discriminate between true targets 
and random clutter peaks so long as the clutter does not completely obscure 
the target. However, a doppler radar separates closing targets from clutter; 
thus the bias level may be set on thermal noise considerations alone. For 
this reason, as well as the others mentioned, pulsed-doppler systems are 
particularly suited for application as AI radars and guided missile active 
seekers which must find and lock on to a target buried in clutter in a high 
closing-rate tactical situation. 



6-7] HIGH RESOLUTION RADAR SYSTEMS 333 

Another application of pulsed-doppler systems — doppler navigation — ■ 
is covered in Chapter 14. In this application, precise velocity measurement 
coupled with freedom from CW radar duplexing problems make the pulsed- 
doppler system most attractive. 



6-7 HIGH RESOLUTION RADAR SYSTEMS 

Certain radar applications such as fuzing and ground mapping often 
require very fine resolution; i.e. effective radar pulse lengths of from 0.002 
to 0.2 Msec (which correspond to range resolution elements of from 1 to 100 
feet, respectively) and /or angular resolutions of the order of 0.1-10 mils. 
High resolution is also tactically useful for counting the number of separate 
targets in a given space volume. The AEW radar example of Chapter 2 
discussed this basic problem. In this case, high resolution in one dimension 
— for example range — can provide the requisite capability. Finally, high 
resolution provides a means for improving signal-to-clutter ratio when the 
clutter originates from area extensive targets. This is shown by Equation 
4-60, where the instantaneous illuminated area of ground is a direct function 
of pulse length and antenna beamwidth. 

There are a number of means for obtaining high resolution in a radar 
system. Basically, all of them are variations of the following approaches to 
the problem: 

1. Angular Resolution 2. Range Resolution 

(a) Large antenna aperture (a) Short pulse length 

(b) High frequency (b) Wide bandwidth 

(c) Beam sharpening 

(d) Doppler sensing 

Angular Resolution. This resolution problem has already been 
discussed in some detail in Paragraph 3-6. There it was shown that the 
angular resolution element of a radar system was approximately equal to a 
beamwidth where the antenna beamwidth can be expressed 

- -J radians 12 (6_45) 

a 

where X and d are the wavelength and aperture size respectively in consist- 
ent units. 

Increases in the antenna aperture d or the operating frequency (/ == 1 /X) 
will directly increase the angular resolution capability. One limitation on 
the benefits of increasing antenna aperture size is worthy of mention at this 
point. For purposes of resolution, the pattern of an antenna has the shape 

i^For a practical antenna, a value of = \.2\/d radians generally is a closer approximation 
when the effects of nonuniform illumination of the aperture are considered. 



334 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 




X/j Radians (Theoretical) 



Near Zone-c — ^Far Zone 

Fig. 6-29 Antenna Beamwidth Pattern. 



shown in Fig. 6-29. As can be seen, the concept of angular beamwidth holds 
only for the so-called far zone (Fraunhofer zone) where the range R is 
greater than d"^ l\P At closer ranges, the effective pattern width is variable, 
but as a general rule it can be considered equal to the antenna dimension. 
Thus the resolution obtainable with an antenna aperture of (S' feet cannot be 
better than d feet regardless of what is implied by the angular beamwidth 
expression. 

For a given antenna aperture and operating frequency, certain techniques 
such as monopulse and sidelobe cancellation are useful for "sharpening" the 
beam and thereby obtaining better definition (that is to say the transition 
of the signal return as the beam crosses an isolated target will be sharper). 
Improved resolution has been claimed from the use of these techniques. 
Such claims rest upon relatively shaky theoretical grounds and are based 
more upon the appearance of better resolution resulting from sharper 
definition than upon a rational repudiation of the basic laws governing the 
formation of interference patterns. 

In certain cases where prior knowledge of the target characteristics exists, 
velocity resolution may be employed to give the appearance of better angular 
resolution than one would predict from the beamwidth. Such a case is 
shown in Fig. 6-30 where an antenna points straight down from an airborne 




V^2 = \^FSinQ: f^2 



2Vf sino : 
X 



Fig. 6-30 Improvement of Apparent Angular Resolution by Doppler Filtering. 

l^See S. Silver, Microwave Antenna Theory and Design, Chap. 6, McGraw-Hill Book Co. 
Inc., 1949. 



6-7] HIGH RESOLUTION RADAR SYSTEMS 335 

platform moving with a horizontal velocity Vp. The beam illuminates two 
closely spaced fixed targets, 1 and 2; however, because of the angular 
relation, and the velocity of the radar platform, the returns from these 
targets differ slightly in frequency. Thus, narrow band filtering may be 
employed in the receiver to distinguish between the two targets. 

If a single narrow band filter of width A/d centered about the carrier 
frequency is used in the receiver, the effective beamwidth may be expressed 

e,// = ^ radians. (6-46) 

However, the filter bandwidth is limited by dwell time requirements to a 
value 

A/, = \/t,= VF/hQeff cps. (6-47) 

Substituting, and solving for the minimum value of Qeff, we obtain 

Qefs ^ VV2A radians. (6-48) 

If multiple receiver channels are used in conjunction with appropriate 
signal storage and correlation techniques, the return from each target can 
be integrated over the entire dwell time of the actual beamwidth. In this 
case, the minimum possible effective beamwidth becomes 

Qeff = X/2^e = d/lh radians. (6-49) 

and the number of channels required is 

n, = IKKld'' channels. (6-50) 

Thus, in theory at least, the resolution performance of a very long antenna 
(possibly much longer than the aircraft itself) can be obtained by coherently 
combining signals transmitted and received from various positions along 
the flight path. Quite obviously, this principle could have application to 
ground mapping radar systems. It is of some interest to note that the 
effective angular resolution of the multiple channel correlation system 
actually improves as the actual antenna beamwidth becomes larger. The 
reader should also note, however, that achievement of effective beamwidths 
approaching the minimum possible requires a radar system of enormous 
complexity. For example, a 3.2-cm system with a 4-ft antenna operating at 
10,000 ft altitude has a theoretical angular resolution limit of 0.1 mil. 
However, such a system would require the equivalent of 267 separate 
coherent receiver channels to realize this potential. 

Short-Pulse Systems. The most obvious means for obtaining high 
range resolution is to employ a short pulse length in a conventional pulse 
radar system. However, such a system has a number of important design 
problems which severely limit the usefulness of this approach. 



336 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

First of all, the generation of a short, high-power pulse is a difficult 
problem in itself. The design of the transmitting tube, the modulator, and 
the TR switching all are complicated by the short-pulse operation. 

Short-pulse operation also limits radar performance. The required 
receiver bandwidth is inversely proportional to pulse length. Thus, the 
S /N rsLt'io for a given value of peak transmitted power is directly propor- 
tional to the pulse length: 

S/N^Pt/B = Ptt. (6-51) 

Usually, peak power cannot be increased to compensate for this effect 
because of voltage breakdown limitations in the transmitter, antenna, and 
waveguide. Thus, for a given state of the art in RF components, the S jN 
performance will decrease with decreasing pulse width. Actually, because 
of the previously mentioned transmitter design problems, this decrease 
proceeds at greater than a linear rate. For these reasons, short-pulse 
systems are limited to relatively short-range operation (such as fuzes) or 
operation against targets of large cross section (ground mapping) where it is 
feasible to sacrifice *S'/A^ ratio for improved resolution. Short pulse lengths 
can also complicate certain other problems. For example, if delay line 
AMTI is employed, the tolerances on the pulse repetition frequency control 
and the delay line calibration must be held within proportionally closer 
limits. In addition, the bandwidth requirements of the delay-line elements 
are increased proportionately. 

As a result of limitations such as these, there are certain tactical applica- 
tions where no physically realizable noncoherent pulse radar system can 
provide the requisite resolution and range capabilities. To fill this gap, a 
family of radar systems has grown up during recent years which — for 
lack of any more suitable name — are called "wide bandwidth coherent 
systems." 

Wide Bandwidth Coherent Systems. From an information theory 
standpoint, the fine range resolution capability of a short-pulse system 
derives from the wide bandwidth of such a system. In fact the range 
resolution capability is a direct function of the bandwidth of the trans- 
mitted spectrum. This suggests that 
any system which employs a wide 
bandwidth has the inherent capabil- 
ity for fine range resolution. Several 
other observations — useful for in- 
venting new radar systems — may 
also be made from an examination of 
the transmitted spectrum of a pulse 
Fig. 6-31 Pulse Radar Spectrum. radar as shown in Fig. 6-31. 



\ 



-Frequency 



6-7] 



HIGH RESOLUTION RADAR SYSTEMS 



337 



First of all, the number of spectral lines contained in the transmitted 
bandwidth determines the number of individual unambiguous pieces of 
range information that the radar can collect from an angular volume 
determined by the antenna beamwidth. Thus, as previously noted. 



A^. ^ ^ = -1. 

' fr Irr 



(6-52) 



Since each piece of range information represents a range interval of fr/l, 
the total unambiguous range interval is simply 



Rn 



-'^'i-k 



(6-53) 



= Spectrum Broadening 
Introduced by Scanning 
and Target IVlodulation 



At this point it is worthwhile to recall the development of the matched 
filter principle presented in Paragraph 5-10 and used as the basis for the 
storage and correlation radar principles outlined in Paragraph 6-4. This 
principle stated that the optimum 
S IN ratio is obtained when the 
detection filter transfer function is 
the complex conjugate of the re- 
ceived signal spectrum. Thus for 
the pulse spectrum shown in Fig. 6- 

31, the optimum filter would have 
the comblike appearance of Fig. 6- 

32, where the width of each tooth of 
the comb is sufficient to pass the 
modulations produced by targets 
and scanning. The total effective 
detection bandwidth of such a filter 
may then be expressed 



r 



uu 



Fig. 6-32 



Optimum 
Radar Spectrum 



—^Frequency 
Filter for Pulse 



Be 



NrBi = B 



(6-54) 



Thus, obtainment of high range resolution with a narrow band receiving 
system is theoretically possible; in fact, for a nonscanning radar operating 
against nonfluctuating targets, the total required received bandpass 
approaches zero. The application of the matched filter concept requires a 
coherent system. 

These principles make it possible to conceive a wide variety of high- 
resolution systems which circumvent the peak power and S jN ratio 
limitations of noncoherent pulsed radar systems. In general, these systems 
have the following common characteristics: 



1. A wide transmitted bandwidth {Bt = c jlRr,^ 
minimum resolution element). 



where i?r.min = 



338 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

2. A modulating frequency /r sufficient to create the desired unam- 
biguous range interval. 

3. Some form of storage and cross correlation which attempts to 
provide an optimum match between the received signal and the 
effective receiver bandpass filtering characteristic. 

4. Usually, great complication in comparison with a noncoherent pulse 
radar. 

The following systems are indicated as possibilities for high resolution 
wide bandwidth coherent systems. 

1. Wide bandwidth FM/CW. 

2. Long-pulse, low-PRF systems where the transmitted frequency is 
FM'd during a pulse transmission period to produce the wide 
transmission bandwidth. This type of system is often called a 
matched filter radar}^ 

6-8 INFRARED SYSTEMS 

A book entitled Airborne Radar may seem a strange place to find a 
discussion of infrared techniques, but it must be remembered that the 
applications and design principles of airborne radars and of infrared 
detectors and weapon control systems are quite similar. Actually the only 
real differences between a passive^^ radar and an infrared system are (1) the 
method of detection^^ and (2) the fact that the infrared radiation emanates 
from the target itself rather than from its radars or communications 
equipment. The tuned circuits used in the detection of radio and radar 
radiations cannot presently be extended to the frequencies (3 X 10^ to 
1.5 X 10'' megacycles, or 1 to 20 microns wavelength) of that portion of the 
infrared having practical significance here. Therefore optical detectors 
must be used and these impose their own restrictions. 

An advantage of the short wavelength of infrared radiation is that 
interference patterns have correspondingly small angular relationships and 
are usually not significant in instrument design or operation. For example, 
the diameter of an infrared collector mirror depends only on the amount of 
radiation to be collected and not on the required angular accuracy, as in the 
case of a radar dish. Also interference patterns such as result in radar from 
ground reflections and cause target confusion are not encountered with 
infrared systems. Because the target is itself a source, the signal in a passive 
infrared system diminishes with range more slowly than in an active system. 

I'^Note that in Paragraph 5-10, the term matched has been used to describe a more general 
class of radar systems. 

i^Only passive infrared is considered here. Active systems do exist and are used, but for 
long-range detection and tracking no sufficiently intense sources of infrared radiation are 
available. 

'^By detection we mean the manifestation of the presence of electromagnetic radiation. 



6-8] INFRARED SYSTEMS 339 

Conversely, infrared does not have the all-weather capabilities of radar, 
its ability to penetrate haze, fog, and clouds being only slightly better than 
that of visible light. Background clutter considerations are also more 
serious, since everything in a typical tactical environment is to some degree 
a source of infrared radiation, i.e., a potential source of interference. 
Passive infrared systems — like passive radar systems — also do not possess 
the capability for measuring range in the direct and convenient manner of 
active radar systems. 

The use of infrared for detection and tracking is now new, having been 
vigorously exploited by Germany during World War II. In this country, 
where reliance was placed more heavily on the development of radar — 
with obvious beneficial results during the war — serious consideration of 
infrared systems has been more recent and stems from four facts: (1) 
modern targets are better sources of infrared radiation than their predeces- 
sors and in many cases poorer radar targets; (2) many important targets 
are encountered at high altitudes where attenuation and absorption of IR 
energy are minimized; (3) infrared is more difficult to countermeasure than 
radar, or at least the art is not so advanced; and (4) infrared technology has 
made significant advances since World War II. 

This will be a short discussion of the application of infrared to airborne 
surveillance and tracking systems. The fundamentals of infrared science 
are ably covered in an earlier book of this series {Guidance, Chapter 5, 
"Emission, Transmission, and Detection of The Infrared") and a knowledge 
gained by reading that discussion will be assumed. 

Basic Principles. Airborne infrared systems generally use mirrors 
rather than lenses. Lenses are possible but usually not practical because of 
limitations imposed by the properties of available materials (see Paragraph 
5-7 in Guidance). The infrared system is composed, then, of a mirror which 
collects radiation from the target and focuses it on the detector, a means of 
modulating the radiation striking the detector in order to produce an a-c 
signal, and a means of discriminating against spurious targets and back- 
ground radiation. Frequently, modulation and discrimination are ac- 
complished in the same process. 

Consider, as an illustration, the simplified system shown in Fig. 6-33.^^ 
Radiation from the target, background, and intervening air enters through 
a dome of transparent material (Irdome) and is focused on the detector 
after reflection from the two folding mirrors. The instantaneous field of 
view is determined by the size of the detector and the focal length of the 
main collecting mirror. Scanning is accomplished by tilting the two folding 

I'This arrangement is chosen only to illustrate the significance of the system parameters and 
not for its desirability or efficacy. It is not an example of a system in actual use, since most 
such systems are classified and cannot be discussed here. 



340 



GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 



Infrared 

Transparent 

Dome (IR Dome) 




Fig. 6-33 Typical IR System. 



mirrors by the angles a and /3 away from the perpendicular to the optic axis 
and then rotating them about the optic axis in opposite directions. If the 
mirrors turn at equal speed, the effect is to move the instantaneous field of 
view along a straight line, the length of which is determined by the two 
angles a and /3 and the intermirror distances. The velocity with which the 
line Is scanned varies sinusoidally; it is most rapid in the center of the line 
and is slowest at the ends where scan reversal occurs. If one mirror turns a 
little more slowly than the other, the line scanned ih space rotates slowly 
about its center, resulting in the rosette scan pattern shown in Fig. 6-34. 
With this pattern the surveillance capability is greatest in the center of the 
field, which is crossed on each spoke of the rosette, and diminishes toward 
the edges, a property which may or may not be desirable. 

In considering the appearance on the scope of a scene scanned by an 
infrared device, it should be remembered that while our eyes see almost 
everything by reflected light (i.e., a "semiactive" process similar to some 
types of radar), the infrared scanner sees mainly thermal radiation emitted 
by the observed objects themselves. This is particularly true if the radio- 
meter is filtered to be sensitive only to radiations of wavelength greater 
than 3 microns, since reflected or scattered sunlight beyond 3 microns is 
generally negligible compared to emitted radiation. Therefore, a "hot" 
object such as a city, or one with a high emissivity such as a cloud, will 
appear bright. The clear sky, bright blue to the eye, will appear black, 
since the air molecules do not scatter infrared as they do visible light. 



6-8] 



INFRARED SYSTEMS 



341 



,**''^ 




In 


,ar,d S.an». 


Output 


Cl«u<l> 
unci Sho 


Ground -I.V.I 
r.Un. Mo.t 1 
Sa<karo»i>d 


H«., City 
it.nia Sourc 



Fig. 6-34 Example of an IR Scan Pattern. 



The detector receives the radiation collected by the optical system and 
converts this to electrical energy, which is amplified and appropriately dis- 
played — in this case on a cathode-ray screen — or used as an error signal 
for tracking. There is radiation received from the target, from the clouds, 
haze, land, water, and sky within the scanned field, and from the atmos- 
phere lying between the detector and the target. In some instances the 



342 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

optical elements themselves, for example an imperfectly transparent dome 
heated in flight, contribute significant and undesirable radiation. The 
usable signal results from the difference between radiation collected from 
the target and all other sources. This "contrast" signal is proportional to: 



m 



V.^AE K^-7; 



^z?o-' 



SxTxd,. (6-55) 



where F^ = signal output (volts) from the detector 

A = area (cm^) of main collector mirror 

a = field of view of radiometer (radians) = width of square detector 
divided by the focal length of collector mirror 

E = efficiency (%) of windows, filters, mirrors, etc. in the optical 
system (dimensionless). This is here assumed to be independ- 
ent of the wavelength 

JxT = spectral distribution of target radiation (watts micron"^ 
steradian"^) 

JxB = spectral distribution of background and other unwanted 
radiation (watts cm~^ steradian"^ micron"^) 

6'x = spectral response (volts /watt) of detector 

Tx = spectral transmission (%) of the atmosphere between the 
target and detector (dimensionless) 

Xi,X2 = wavelength limits (microns) of system sensitivity defined by 
the optical filter or sensitivity limits of the detector. 

R = target range (cm). 

The detector will have a noise output F„ which will be a function of the 
type of detector, the bandwidth A/ of the amplification system, and the 
radiation environment of the detector. Specifying the minimum signal-to- 
noise ratio Fs jVn required for reliable detection of a target, the system noise 
defines the minimum required V^. Since 7xr, 7\b, and T^ are beyond our 
control and E is always optimized anyhow, the remaining parameters are 
chosen to give the required Vs at the desired target range. Actually Xi and 
X2 are generally determined by consideration of Jxr, 7\b, and T^. Fig. 6-35 
shows J-^T, J\B, and T^ plotted as a function of wavelength for a specific 
application: the detection of a 600° K blackbody viewed against a back- 
ground of clouds or heavy haze through 10 miles of moderately clear sea 
level air. It is plain that the best choice of wavelength limits are Xi and 
X2 = 3.3 and 4.1 microns, respectively. The properties of available de- 
tectors — sensitivity, time constant, ruggedness, reliability, etc. — may, of 



6-8] 



INFRARED SYSTEMS 



343 



100 



Ji? 40- 



Background 

Predominantly 

Scattered 

Sunlight 

Mbsent at' 

V Night 



+ 



Background 
Predominantly 
Black Body 
Radiation 
Night and^ 
Day 



U Energy Distribution 
^ From Black Body 
at 600°K 



4\ 



rx Transmission of 
10 mi. of Sea - Level 
Air Containing 6 cm 
Vapor 




Jb Cloud and Heavy 
Haze Background 

(^Energy Distribution' 
From 300°K Black 
vBody 



4 X Exaggerated 




123456789 10 

WAVELENGTH (MICRONS) 

Fig. 6-35 The Useful IR-Frequency Spectrum. 

course, force a different choice. The only really flexible parameters for the 
designer, then, are the areas of the collector mirror and of the cell. These 
are in turn influenced by the requirements of the scanning system. 

Scanning System Characteristics. The choice of a scanning system 
generally represents a compromise between the requirements of the system 
and the mechanization advantages of rotary optics (particularly for high- 
speed scanning) and fixed detector elements (which simplify cooling 
problems and maintenance of cell sensitivity). As an example of the type 
of analysis which must be performed to assess a given scanning technique, 
the rosette scanning pattern previously discussed will be analyzed to 
determine the interrelations among scan time, resolution, coverage, and 
detection element characteristics. 

The important parameters of the scanning system are: the instantaneous 
field of view a (radians) square; the whole field of view which is here (see 
Fig. 6-34) a function of the half-scan angle 7; and the time T required for 
the whole field to be covered. In order to completely cover the field the 
number of spokes in the rosette pattern is equal to the number of instan- 
taneous fields required to cover the periphery of the whole field, or 

—r—. spokes. (6-56) 

a/ sm 7 



344 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

The speed of scan is controlled by the time constant t of the detector. To 
define a practical upper limit we shall use the time required for the in- 
stantaneous field to move its own width o-, equal to t. To go faster would 
result in considerable smearing of the display pattern, with resultant loss of 
resolution. In this sinusoidal scan the velocity is not constant, but for 
simplicity we use the average value. The time required to scan one com- 
plete spoke (27 radians) is 

;= ] r \ — = n — '■ — seconds/spoke. (6-57) 

lotal no. or spokes Iirsmy 

Motor 1 then turns clockwise at — ( ) rps and motor 2 turns counter- 

11- 1 /^TT sin7 A 
clockwise at — I ~~ W ''P^- 

Since a complete spoke is 27 radians, the average scan rate is 

-. seconds/radian. (6-58) 

4x7 sm 7 

and the time required to scan a- radians is 



4x7 sin 7 



= r seconds. (6-59) 



which we accordingly equate to the time constant r of the detector. 
Consider an actual case in which we want: 

^ = ly^o = 0.0058 rad 

^ = 20° = 0.35 rad 

T = 10~^ sec 

Then the time required to scan a complete field is, from Equation 6-57: 

^ 4x7 sin yr 4x X 0.35 X 0.34 2 X lO"'^ . , .. ,^. 

^ = a-^ = mossy = ^-^ '''■ ^^-^^^ 

For many practical cases, this is obviously too long. A modern aircraft 
will have changed course considerably in this time. Therefore the instan- 
taneous field must be enlarged, the total field reduced, or a faster detector 
sought — possibly all three. Perhaps the scan mode would have to be 
abandoned in favor of a more economical one without the multiple retrace 
encountered in the center of this field — say a raster scan similar to that 
used in television. 

Target Tracking. If the system is required also to track a target, 
this could be accomplished by orienting the aircraft so that the target image 
falls in the center of the screen, setting angle a = 0, and reducing angle /3 



6-8] 



INFRARED SYSTEMS 
n-/2 7r/2 



345 





































S 












Azimuth 
Drive 


Active — >■ 


























Q 














i 

Q 


Eievatio 
Drive 




/ 


A 


^ 






Target 
Pulse 

(from 4 ( 



V2 



3/2 TT 



27r 



(c) 



Fig. 6-36 Target Tracking. 



to such a value that a circular scan of, say, |° results (see Fig. 6-36). Then 
as long as the image in space of the detector rotates around the target 
without touching it (Fig. 6-36a) no output (error signal) will result. When 
the line of sight moves and the detector then encounters the target (Fig. 
6-36b) an output pulse will result; by comparing the phase of this pulse with 
a synchronizing signal generated on the shaft of motor 2, an error signal is 
generated. By having the entire optical detection device movable in the 
aircraft and motor controlled, these error signals can be used to keep the 
device pointed at the target. 

In Fig. 6-36c a simplified on-off control is illustrated. The synchro- 
nizing pulse alternately activates and deactivates the control motors. If 
the target pulse occurs during an active period, the motor moves the optical 
system. In Fig. 6-36c the pulse occurs where both the down and left 



346 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 

controls are activated (as in Fig. 6-36b) to return the target image to the 
center of the circular scan. Smoother tracking and less tendency to hunt 
result from a system in which the error signal varies in magnitude with the 
ofF-course position of the target, and this is usually a feature of actual 
tracking systems. 

Detection Performance. The actual capabilities achievable with 
present-day infrared systems can be estimated from target intensities, 
background intensities, and detector sensitivities. As an example, consider 
a radiometer having a filter limiting the sensitivity of the radiometer to the 
region 1.7 to 2.7 microns and a lead sulfide detector at the focus of a 
collector mirror 1 ft in diameter. With a readily available detector (say an 
Eastman Kodak Ektron cell) of practical size (say 1 mm square), an output 
signal just equal to the rms noise from the detector can be achieved under 
tactical conditions when about 10~^^ watt/cm^ falls on the collector mirror 
and is focused on the detector. This would represent a signal-to-noise ratio 
of 1, here arbitrarily construed as a necessary criterion for detectability. 
If the target is the exhaust port of a typical jet engine, the irradiance 
(watts /cm^ in the 1.7 to 2.7-micron region) at the collector mirror will be 
about 400 /i?2, where R is the target range. This results from assuming the 
exhaust port to be a 24-inch-diameter blackbody of emissivity unity and to 
have a temperature of 600°C. Through a completely clear atmosphere, 
then, and with no background interference this jet exhaust port could be 
seen from a distance of 2 X 10^ cm or 125 miles, at which distance it would 
irradiate the collector mirror with the necessary 10"^^ watt/cm^. Atmos- 
pheric attenuation, which is severe in the lower atmosphere, and back- 
ground interference may under average conditions degrade this range to 
less than a third of this number. 

Further, the target we are considering, a single-engine jet aircraft, will be 
a much fainter target at any other than tail aspect where the exhaust port 
is visible. In side aspect the radiation emanates from the hot exhaust gases 
which, while of extended size and quite hot, emit only the wavelengths of 
the characteristic infrared bands of the gases. If the fuel is a hydrocarbon 
these are the bands of water vapor and carbon dioxide. Atmospheric 
attenuation is most severe in this case, since the cold water vapor and 
carbon dioxide in the intervening air path absorb most of what is emitted. 
In side aspect a jet will be less than one-tenth as intense a target as in tail 
aspect and will therefore be detectable at less than one-third the range 
realizable when looking at the exhaust port. In nose aspect, it will be 
considerably worse than this, since here most of the exhaust gases are 
hidden by the aircraft and the hot parts of the engine are not visible. 



D. J. H E A L E Y III 



CHAPTER 7 

THE RADAR RECEIVER 

7-1 GENERAL DESIGN PRINCIPLES 

The airborne radar receiver amplifies and filters the signals received by 
the radar antenna for the purpose of providing useful signals to display and 
automatic tracking devices. The receiver accepts all of the signals appear- 
ing at the antenna terminals and must filter them so as to provide maximum 
discrimination against signals which do not originate by surface reflection 
of the transmitted radar signal from certain desired targets. 

Modulation characteristics of the desired signals must be preserved in 
the filtering process. The modulation characteristics provide information 
on the number of targets, their angular position with respect to a given 
frame of reference, the distance between the radar set and the targets, 
and the velocity of the targets. 

The majority of airborne radar receivers are of the superheterodyne 
type. Ordinary pulse radar sets are usually of the single frequency con- 
version type. Doppler radar sets employ single sideband reception. The 
receivers are more complicated than in the ordinary pulse radar set and 
generally employ multiple frequency conversion in order to realize the 
required frequency selectivity. Fig. 7-1 shows a functional block diagram 
of an elementary receiver of each type. 

Performance of a radar receiver is described by the following character- 
istics: 

1. Noise figure 5. Dynamic range 

2. Sensitivity 6. Cross-modulation characteristics 

3. Selectivity characteristics 7. Tuning characteristics 

4. Gain control characteristics 8. Spurious response 

Specification of each of these characteristics depends upon the particular 
radar application. Analysis of the radar system defines the input signal 
environment, the required output signal-to-noise ratio, and required 
fidelity of modulation. Such requirements are then interpreted in terms 
of the above characteristics. Each of these characteristics is discussed 
later in this chapter and in Chapter 8. 

347 



348 



THE RADAR RECEIVER 




7-1] 



GENERAL DESIGN PRINCIPLES 



349 

















1 




















1 




















Filter 
and 
natio 




-2 E 
1 '^ 




^ 




1 




.5 


















t ^.i 




•^^ 




s 




^ 






































t 


t 








A 








▲ 












T T 


i 


1 






























^fi.i^i M- 


03 




1 
















X 










2^ 1 o V 1 


u. ^ 




y. 
















S 








>|^> C 


§6 




1 


1 




< 






' 




1 












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1 


















t 1 
























U. ,/, ^ .^ 


o 




























1^^ -§-^ 


































--iz i" 


b 






































UJ 






















, 










c 
























.2 






















E ^ 
















ao 
1 5 






-11 

aj - E 




\ 


^ 






































§5^ 

q: .E 








Q 


















^ 






Q- 
00 


o 
"5 
























- E 




















f 




























" (t 












5.i 










, 




















-■ b 
































' 






' 














<v 








_l 


1 












X 








It: 

(CO.- 


2 
2 


-►CO- 


^^ 




T^n^ 
















S 










3 


^1 


1 




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< .> < .> 












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f^ 




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a. 


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Q 




C/l 


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u, 



350 THE RADAR RECEIVER 

It is desirable that the noise figure be minimized and the sensitivity be 
maximized. This is not always feasible, as will be indicated in Paragraph 
7-2. Selectivity is provided in both frequency and time. It is desirable to 
provide the required selectivity at the low-level signal stages and prior 
to envelope detection of the signal. 

Gain control characteristics are dictated by requirements to provide 
error signals to range, speed, and angle tracking feedback mechanisms for a 
specified range of input signal power. Automatic gain control (AGC) 
systems are discussed in detail in Chapter 8. 

Dynamic range is the range of signal levels above the thermal noise 
level for which a receiver will provide a normal usable output signal. To 
reproduce faithfully the amplitude modulation on a received signal, the 
incremental gain of a receiver whose output is controlled by the average 
level of the received signal must be constant for a dynamic range on the 
order of 12 db above the average signal level. The incremental gain is the 
slope of the output /input voltage characteristic of the receiver. When an 
undesired signal appears at the receiver input which is coincident in time 
with the desired signal and nearly coincident in frequency, a much greater 
linear dynamic range or range for which incremental gain remains constant 
is required. In receivers which separate signals by frequency filtering, it is 
not unusual to require a linear dynamic range on the order of 80 db up to 
the point in the receiver at which the frequency separation of the desired 
and undesired signals occurs. On the other hand, short-pulse, low-PRF 
radar sets which separate signals by time filtering may require only a linear 
dynamic range on the order of 15 db. 

Undesired signals which occur at a different time or frequency than the 
desired signal may impart their modulation to the desired signal. This is 
called cross modulation. Such a phenomenon arises from nonlinearities in 
the receiver and is undesirable, since it degrades the output signal-to-noise 
ratio. 

A proper radar system analysis defines the signal environment and allow- 
able degradation of the receiver output signal; thus the principal factors 
governing the selection of the dynamic-range and cross-modulation 
characteristics are specified. 

Tuning characteristics are dictated by the radar transmitter. The 
receiver is designed so that it can always be tuned to the transmitter fre- 
quency. The design objective is to make the receiver tuning as accurate 
as the state of the art permits. Both short-term and long-term frequency 
stability is important. The effect of short-term frequency instability is 
to introduce modulation on the signal in the receiver. Such modulation 
degrades the output si.gnal-to-noise ratio. In a noncoherent pulse radar 
set, the tuning accuracy that can be achieved is on the order of 1 part in 
10^ Much greater stability is required in coherent radar sets. Automatic 



Table 7-1 



Receiver Characteristic 


Desirable Effect 


Effects on Other Receiver 
Characteristics 


Direct detection of the 
RF signal 


Simplicity 


Poor noise figure; poor 
rejection of spurious sig- 
nals. Nonlinear transfer 
characteristic 


Preselection (band pass 
filter between antenna and 
receiver) 


Greatly reduces spuri- 
ous signal response 


Increased noise figure due 
to insertion loss of the 
filter. Imposes long-term 
frequency stability re- 
quirements on the trans- 
mitter and preselection 
filter that otherwise would 
not be encountered 


Desensitization during 
transmitting time (TR 
switch) 


Reduces degradation of 
receiver performance 
with time by limiting the 
signal power applied to 
the microwave mixer 


Degrades performance at 
short ranges owing to the 
deionization properties of 
the gas switches that are 
employed. In high-PRF 
coherent radars, degrades 
performance in each am- 
biguous range interval im- 
mediately following a 
transmitted pulse 


High IF frequency 


Minimizes spurious sig- 
nal response; simplifies 
some tuning problems 


Results in higher IF noise 
figure. The receiver noise 
figure depends on the 
amount of noise-noise in- 
termodulation due to the 
local oscillator. If this is 
negligible, the receiver 
noise figure will generally 
be higher with the higher 
IF 


IF bandwidth on the order 
of the reciprocal of the 
transmitted pulse length 


Maximizes the peak sig- 
nal to rms thermal noise, 
thereby providing best 
detection in thermal 
noise 


Limits signal resolution; 
imposes strict require- 
ments on the tuning ac- 
curacy of the receiver 


IF bandwidth character- 
istic which enhances the 
signal sidebands greatly 
removed from the trans- 
mitted carrier frequency 
and attenuates the side- 
bands near the carrier 


Improves the detection 
capability in clutter 


Degrades the detection of 
signals in thermal noise. 
Imposes strict require- 
ments on tuning accuracy 



351 



352 



THE RADAR RECEIVER 

Table 7-1 (cont'd.) 



Receiver Characteristic 


Desirable Effect 


Effects on Other Receiver 
Characteristics 


Prevention of saturation 
in a linear receiver 


Maximum signal to 
noise ratio 


Requires an AGC loop. 
When signal contains pulse 
amplitude modulation 
that must be recovered, 
instantaneous AGC is usu- 
ally not acceptable. A 
slow AGC is demanded. 
The information rate is de- 
creased since only a se- 
lected signal may operate 
the AGC. Any other sig- 
nals that may be examined 
are modulated by the 
fluctuations of the signal 
controlling the AGC. 
Therefore multiple receiv- 
er channels are needed to 
increase the information 
rate 


Extremely narrow IF 
bandwidth 


Provides maximum sig- 
nal to thermal noise 
ratio 


Reduces the information 
rate attainable. Results 
in a loss of signal resolu- 
tion unless preceded by a 
cross correlation opera- 
tion. Wasteful of power 



frequency control (AFC) systems are required to obtain the required tuning 
accuracy. These devices are discussed in detail in Chapter 8. 

Spurious responses are outputs caused by signals at frequencies to which 
the receiver is not normally tuned. These responses are the result of in- 
adequate selectivity and nonlinear elements in the receiver, e.g. mixers. 
The receiver is designed to minimize spurious responses by properly 
selecting intermediate frequencies and mixer circuits and by providing 
the necessary selectivity. 



7-2 THE INTERDEPENDENCE OF RECEIVER COMPONENTS 

A particular receiver characteristic may be designed to give optimum 
receiver performance when the contribution of this one characteristic of 
overall performance is considered. However, many of the receiver charac- 
teristics are interdependent and therefore compromises must be made in 
the design. The compromises for a specific design are determined by the 



7-3] RECEIVER NOISE FIGURE 353 

performance requirements imposed on the radar system by the tactical 
requirements. 

Some examples of the effects that the choice of a given receiver character- 
istic has on the overall receiver performance are indicated in Table 7-1. 

7-3 RECEIVER NOISE FIGURE 

The ultimate sensitivity of a receiver is dependent upon the inherent 
noise generated in the receiver circuits. A useful measure of this noise is 
the receiver noise figure which is defined as the ratio of the actual noise 
power output of a linear receiver to the noise power output of a noiseless 
receiver of otherwise identical characteristics. 

Noise in a receiver is made up oi thermal noise., which results from thermal 
agitation of charge carriers in conductors, and shot noise, which results 
from random electron motions in vacuum tubes. These noises are charac- 
terized by a Gaussian amplitude distribution with time. Such noise sources 
are independent and uncorrelated. The average power from the various 
sources is additive, and it is therefore convenient to employ ratios involving 
power in determining noise figure. 

Consider a signal generator described by a short-circuit signal current 
source /» and an internal conductance ^s which is at an absolute temperature 
Tg. Let the generator be connected to a load gL which is at an absolute 
temperature Tl. Both gs and gL will generate fluctuation currents which 
are given by^ 

TZ' = ^kT.gJj (7-1) 

and 

/;? = \kTLgLdj (7-2) 

where ins and /„l are the rms noise currents in a frequency bandwidth 
element dj, k is Boltzmann's constant = 1.37 X 10"^^ joule /i^°. 

The available signal power from the generator is Is^ /4:gs and the available 
thermal noise power is inJ^ l^gs = kTsdf. 

The available signal power from the circuit composed of the signal 
generator and the load gi is 

4(^. + gL) 
If t is defined as Tl/Ts, the available noise power is 

7^ + ^ ^ kTsdf{gs+tgL) ,^ .. 

^{g. + ^l) g.^gL ' ^ " ^ 

^J. B. Johnson, "Thermal Agitation of Electricity in Conductors," Phys. Rev. 32 (1928). 



354 THE RADAR RECEIVER 

The noise figure is defined as 

SJN,^N^ (7.4) 

So/ No N^G ^ ' 

where F is the noise figure (a power ratio) 

SilNi is the available signal to noise ratio at the input 

So /No is the available signal to noise ratio at the output 

G is the available power gain. 

For the case of the generator connected to a load, the noise figure of the 
combination is then 

F =. I -\-t ^. (7-5) 

If both the generator and load are at the same temperature, then the 
noise figure is merely the attenuation of the signal resulting from the 
termination. 

In a radar receiver it is convenient to associate a noise figure with 
various elements and then determine the receiver noise figure resulting from 
their combination in cascade. 

Consider that a number of elements characterized by a noise figure Fj 
and an available power gain Gj are interconnected as in Fig. 7-2. It is 



F,,G, 



Fig. 7-2 Noise-Equivalent Radar Receiver. 

assumed that all the noise sources are at a temperature T (a difference in 
temperature may be included as a temperature ratio). Assume further 
that all elements are linear, and that the effective noise bandwidth of each 
element is 5„. 

The input noise is A^i = kTB„ 

The overall gain G is G1G2G3 - 

Output noise originating from the source is GkTB„. 

The additional noise at the output contributed by the first box is 

GkTBn{F\ — 1). The additional noise at the output contributed by the 

, , . GkTB„{F2 - 1) ^, r n • -k • AA 

second box is — -■ I he sum or ail noise contributions add up 

Gj 



FGkTBn 


= GkTB,, + GkTB„{F, - 1) + _ 


Gi 


Then 








^ = ^' + G. + G.G. + ■ 





7-3] RECEIVER NOISE FIGURE 355 

to FGkTBn, where F is the overall noise figure expressed as a power ratio. 

+ -. (7-6) 



(7-7) 



In the common airborne radar set, RF amplification is not employed. 
Instead, the signal is heterodyned to some intermediate frequency and 
then amplified. Microwave crystal mixers are passive nonlinear devices. 
Their noisiness is characterized by the amount of noise produced by the 
mixer compared with the noise from a resistance at the same external 
temperature. The noise is thus expressed as a temperature ratio tm- A 
mixer acts as a switch, and in terms of available power exhibits a loss at 
the conversion frequency. This loss is designated as a power ratio Lx- 
Following the previous notation the noise figure of a mixer is then tmLx. 

The noise figure of a superheterodyne radar receiver is then 

Fr.. = 1 -f- ^m - 1) + ^L,[(/. - 1) + (Fi, - 1)] (7-8) 

J a J a 

where Free is the receiver noise figure expressed as a power ratio 

/m-1 is the excess noise of the mixer 

FiF-\ is the excess IF noise figure expressed as a power ratio 

Lt is the product of the conversion loss of the mixer and loss in 
the microwave transmission circuitry between the antenna 
and mixer expressed as a power ratio 

/„ is the effective noise temperature ratio of the mixer 

T is the noise temperature of the receiver 

Ta is the temperature of the antenna. 

The noise figure is usually defined with respect to room temperature 
(about 291° K).^ A radar receiver is, however, connected to a directional 
antenna which can be represented as an equivalent generator at a tempera- 
ture less than room temperature when the antenna is directed toward 
space. In fact the equivalent antenna temperature under this condition 
may be about 4°K. 

When referred to the antenna temperature under this condition, the 
noise figure of the best airborne radar sets is on the order of 30 db. 

2Under such a definition T/Ta is unity. 



356 THE RADAR RECEIVER 

7-4 LOW-NOISE FIGURE DEVICES FOR RF 
AMPLIFICATION 

The crystal mixer Is a rather fragile element, and its electrical character- 
istics deteriorate when large-signal inputs are applied to it. Because of the 
conversion loss of the crystal mixer, the receiver noise figure is highly de- 
pendent on a low IF noise figure. From a consideration of the noise figure of 
cascaded networks it is seen that a large available power gain in the first 
network minimizes the noise contributions of the later networks. 

A low-noise RF amplifier preceding the crystal mixer can provide the 
power gain required to oflFset the loss of the mixer. Two types of RF 
amplifiers that might be employed in an airborne radar receiver are the 
traveling wave tube (TWT) amplifier and the variable parameter amplifier. 
These devices are discussed in more detail in Chapters 10 and 11. 

Noise in the traveling wave tube results from noise in the electron beam. 
Theoretically the noise can be reduced to about three times kTB. At 
present such tubes are not available for airborne radar receivers. Tubes are 
available, however, that are nearly competitive in noise figure with the 
microwave crystal mixer in the frequency range employed by the airborne 
radar set. 

A TWT will produce a saturated output under strong signal conditions 
at maximum gain, alleviating many TR switching difficulties. Since the 
tube provides gain, the noise figure of the elements which follow is not 
nearly as important as in the conventional airborne radar receiver. There- 
fore, much higher intermediate frequencies are feasible without degrading 
the noise figure. A higher IF results in fewer spurious signal outputs from 
the receiver. The tube can be gain-controlled by changing the beam current 
so that it can produce an attenuation equal to the cold loss of the tube if 
required. This is an advantage when attempting to amplify strong signals 
with minimum distortion. 

One disadvantage in the traveling (forward) wave tube results from the 
wide bandwidth. The noise spectrum is very wide and this results in more 
noise at the mixer than desired. An RF filter between the traveling wave 
tube and the mixer can, however, eliminate this condition if necessary. 
Another disadvantage is that a number of spurious signals can be generated 
in the tube, and are likely to be encountered in practice due to the wide 
RF acceptance bandwidth of the tube. The backward wave amplifier has 
a narrower bandwidth than the forward wave amplifier and may prove 
to be the most desirable type of traveling wave tube for use as an RF 
preamplifier in an airborne radar set. Traveling wave tubes may also be 
constructed with two slow wave structures to provide mixing. 

Variable parameter amplifiers — also called parametric amplifiers — are 
much simpler than the TWT amplifiers. The transmission type of amplifier 



7-5] 



MIXERS 



357 



appears to be the best suited for radar receivers, although the noise figure 
is somewhat higher than the reflection type. Many practical problems 
associated with these amplifiers, such as stabilization of the loaded ^,'s of 
the resonant circuits and regulation of pump power level, must be solved 
before these amplifiers find large use in airborne radar sets. However, 
these negative-resistance amplifiers appear to be a final step in attaining 
receivers whose sensitivity is truly limited by external noise. 

7-5 MIXERS 

The SHF (super high frequency) mixer in the majority of airborne radar 
receivers incorporates crystal diodes. Properties of the crystal mixer which 
are important to radar system operation are: 

1. The effective noise temperature 

2. The conversion loss 

3. The intermodulation components 

A crystal mixer can be represented by an equivalent circuit, as is shown 
in Fig. 7-3a. The nonlinearity of the crystal arises from the variation 



Zj Local Oscillator 
Source Impedance 



Crystal 




Image Zero Frequency 

Impedance Impedance 

(a) (b) 

Each Impedance Shown External to the Crystal 
is Zero to All Frequency Components Except 
the One to Which it Refers 

Fig. 7-3 (a) Equivalent Circuit of Crystal Mixer and (b) F-/ Characteristics of 

a Mixer. 



in the barrier resistance Rb which is a function of the voltage applied to the 
crystal. A typical transfer characteristic is shown in Fig. 7-3b. The 
spreading resistance Rs and barrier capacitance Cb are detrimental parasitic 
elements. Because of these elements, not all of the heterodyne signals' 
energy can reach the IF and image termination. 



358 THE RADAR RECEIVER 

To obtain a low conversion loss, the voltage applied to the diode by the 
local oscillator signal is very large so that there is negligible conduction 
during one half of the local oscillator cycle. The signal voltage is much 
smaller than the local oscillator voltage. 

A current flows through the IF impedance which can be described by the 
multiplication of the signal and local oscillator voltages by the transfer 
characteristic of the mixer. The transfer characteristic can be expressed as a 
power series in the applied voltage. Because of the magnitude of local 
oscillator voltage a large number of terms are required to describe the 
mixer behavior. 

Thus 

/ - E a.E- (7-9) 

n=0 

where E is the input voltage 

/ is the current flowing in mixer 

an are the coefficients of the power series describing the mixer; 
these are dependent on the local oscillator signal level 

n is an integer 0, 1, 2, 3, ... . 

Normally E consists of the sum of two voltages, the signal voltage and 
the local oscillator voltage. In general the input may be 

E=Y,Ar cos CO./ (7-10) 

r=l 

with the condition that Ai cos coi/ be the local oscillator signal and Ai )$> ^2, 

If the signal is a single frequency C02 and the IF center frequency is 
(coi — CO 2), the desired output spectrum from the mixer is the intermodula- 
tion term K cos (coi — C02)/. An expansion of the expression for the current 
in the IF impedance yields terms of the form 

Ij, = jJa.J, + ~ a,A,' + - + ^r^,2^ll^y^^ ^2„^i(-^"-^' + 

cos (coi — C02)/, r 9^\. (7-11) 

for the IF outputs incident to mixing of the signal frequencies with the local 
oscillator frequency. The term in the brackets is a constant for a particular 
value of local oscillator voltage, and the mixer thus produces an IF output 
which can be expressed as 

Itf = KA.. r9^\. (7-12) 



7-5] MIXERS 359 

When more than one signal frequency is applied to the mixer, inter- 
modulation between the signal components occurs. The output current 
caused by these intermodulation components is of the following form for 
each possible pairing of m signal components: 

cos (coy - CO/,)/. (7-13) 

where / = 2, -, 7n 

k = 1, -, m. 

Once again, the term in the brackets is constant for a particular local 
oscillator voltage, so that 

/ = AiA.K,. (7-14) 

The voltage developed by these mixer currents is prevented from affecting 
the receiver performance by the frequency selectivity of the IF networks 
when coy — cojt falls outside the IF passband. Those components that would 
ordinarily fall within the IF passband, could be eliminated by RF preselec- 
tion and proper IF frequency. However, such preselection is not always 
feasible. A balanced mixer is therefore used (see Paragraph 10-15). In 
the balanced mixer two crystals are placed at two of the ports of a micro- 
wave junction, and the signal is fed into one port and the local oscillator 
into the other port. The junction may be a magic-tee, short-slot hybrid or 
rat-race. 

Each individual crystal develops all of the intermodulation components, 
but the relative phase of the signal-signal beats differs from that of the 
signal-L.O. beats and therefore can be discriminated against in the IF 
coupling circuit. Rejection of the undesired intermodulation components 
on the order of 25 db is realized in practice. Principal factors affecting 
the rejection of signal-signal beats are the impedance match between the 
signal source and each crystal, the rectifier dynamic characteristics, and 
the balance of the IF circuit. 

Among the signal-local oscillator products are two which affect the 
performance of the mixer. These are included in the value K of Equation 
7-12 when it is determined experimentally by measuring the coi — co^ 
component from the mixer. These two products involve the generation of 
an image frequency, i.e., a signal which is separated from the desired signal 
frequency by a frequency equal to twice the IF frequency, and which is 
separated from the local oscillator frequency by the IF frequency. The 
image frequency is caused by second harmonic mixing and by an up con- 
version resulting from the IF current which flows through the mixer. 



360 THE RADAR RECEIVER 

The image frequency signal appears across the crystal and propagates 
down the waveguide toward the local oscillator and the antenna. If the 
image wave sees a match, such as would exist if it were allowed to enter 
the local oscillator channel, the energy in this signal is dissipated and energy 
that could have appeared in useful IF output is lost. Proper reflection of 
the wave can cause it to enter the mixer and arrive at the crystal in proper 
phase so that the output IF is increased. Optimum handling of the image 
can improve the noise figure about 1 or 2 db. In general, however, con- 
ventional pulse type airborne radar receivers have broad band mixers. 
The image conversion is terminated and the lowest possible noise figure 
is not obtained. 

A number of other intermodulation components involving the second 
harmonic of the local oscillator occur and can be significant when the RF 
acceptance bandwidth is great. 

The crystal diode voltage-current relationship is given by 



'( 



exp|^-l) (7-15) 



where e = electronic charge 

V = applied voltage 

K — constant depending on crystal 

T = temperature of the junction. 

Shot noise is exhibited by the crystal; the mean square fluctuation current 
is Pdf = 2eIo4f, where /« is the d-c current through the crystal. Equation 
7-15 indicates that a given conversion loss could be obtained with a lower 
d-c current by reducing the temperature and therefore producing less shot 
and granular noise. In addition to the shot noise there is a frequency- 
dependent noise. All of this noisiness of the crystal is specified by the 
crystal noise temperature ratio tx- The mixer noise temperature is /„ and 
is given by 

21 



H-h'l 



a- i'-'« 

for the broadband mixer. L is the conversion loss; U is the value specified 
by crystal manufacturers. The tm of an actual mixer may be difi^erent, 
depending on the termination of the image conversion which affects L. 
Equation 7-8 shows that a large value of Fip causes the conversion loss of 
the mixer to be the dominant parameter of the mixer contributing to the 
noise figure. In fact even for a low IF noise figure the conversion loss is 
more dominant than the noise temperature. The mixer therefore yields 
lowest noise figure when it is designed for minimum conversion loss. 



7-6] COUPLING TO THE MIXER 361 

A frequency-dependent part of tm has been observed to vary as {filfi)n, 
where/2 and/i are IF frequencies and n is between 0.5 and 1.^ Since the IF 
noise figure varies approximately as/i//2 at high IF, the IF frequency 
at which minimum Free is realized is not critical. 

The local oscillator is usually a klystron with a wide electronic tuning 
range. Such oscillators exhibit shot noise whose spectra are determined by 
the ^ of their resonators. To minimize intermodulation components be- 
tween such noise and the local oscillator signal a high IF frequency is 
desirable. The use of a balanced mixer, however, reduces this noise sig- 
nificantly. 

In any mixer design, the objective is to provide minimum tm and L. 
The conversion loss L depends on the match between the signal source at 
both the signal and image frequency, the RF signal frequency, the IF 
frequency, the crystal biasing, and the local oscillator signal level. The 
noise temperature tm is also dependent on L, the IF frequency, and local 
oscillator signal. 

7-6 COUPLING TO THE MIXER 

To obtain minimum noise figure, minimum mixer conversion loss must be 
realized. Conversion loss is defined on an available power basis; therefore 
the conversion loss does not depend on the actual IF load admittance 
connected to the mixer. The conversion loss, however, is dependent on 
the RF signal source admittance. To obtain minimum conversion loss at 
the principal beating frequency (signal frequency beating against local 
oscillator frequency) a mismatch is required between the mixer and the 
source. The input admittance of the mixer, however, depends on the IF 
conductance seen by the mixer. This in turn depends on the design of the 
first IF stage of the receiver and the network which connects it to the mixer. 

A condition frequently encountered in airborne radar receivers is that 
the IF admittance is very large incident to the use of a double-tuned trans- 
former between the mixer and first IF tube. The secondary circuit is usually 
damped only by the coil losses and circuit losses. A very large admittance 
is therefore coupled into the primary circuit near the resonant frequency 
of the secondary circuit. 

For this type of coupling between mixer and IF amplifier, an optimum 
mismatch between the signal source and the mixer is given approximately 
by 

" = zi^ (^-"' 

where p is the VSWR (voltage standing wave ratio) at the signal frequency 
and Lo is the optimum conversion loss. 

3P. D. Strum, "Some Aspects of Crystal Mixer Performance," Proc. IRE 41, 876-889 (1953). 



362 THE RADAR RECEIVER 

The source admittance can be designed on this basis when the IF cou- 
pling circuit is as specified. 

When the image frequency signal generated at the mixer is allowed to 
be radiated by the antenna or dissipated in the local oscillator source 
admittance, the value of /><, is that which is normally specified by the crystal 
manufacturer as conversion loss. 

In many airborne radar receivers a short-slot hybrid junction is employed 
in a balanced mixer. When the crystals are matched in such a mixer, all 
of the image frequency signal generated in the mixer propagates out the 
local oscillator port. Normally this port is matched, therefore the image 
signal energy is lost. 

In some radar sets a filter may precede the mixer to reduce interference 
from other radar sets. Such a filter may appear as a susceptance at the 
image frequency and reflect the image signal originating in the mixer. If 
the signal arrives in the correct phase at the mixer crystals, the performance 
is improved. The phase depends on the distance between the mixer and 
the filter. However, the distance between the mixer and the filter is also 
dependent on the mixer to IF coupling circuit, since the filter would be 
situated so as to give the optimum mismatch of the source to the mixer. 
To obtain lowest receiver noise figure, design of the RF and IF circuits 
must therefore be considered jointly, not separately. One solution to this 
problem might be the use of the short-slot hybrid with a filter in both signal 
and local oscillator paths. 

7-7 IF AMPLIFIER DESIGN 

The IF amplifier consists of a cascaded arrangement of vacuum tube 
amplifiers which employ band pass coupling networks. Frequency of 
operation is a compromise between several factors such as noise figure, 
circuit stability, spurious responses, and receiver tuning characteristics. 
Consideration of these factors usually leads to the choice of an IF frequency 
between 30 and 60 Mc in the ordinary pulse-type airborne radar receiver. 

The IF amplifier is a filter amplifier, and its small-signal transfer function 
is given by 

G(s) = H — — —^ ;^;^37-r - . (7-18) 

In this expression // is a constant depending on the number of tubes, 
their transconductance, and the capacitance values of the circuits; s is the 
complex frequency variable a -\- j(ji-, n is the number of circuits; q and m 
are determined by the network complexity. The transfer function vanishes 
when s = because of the numerator term. The function thus has a zero 
of order nq at the origin. The denominator can be factored into 

{s - s,){s - s,*){s - s,){s - S2*)(s - s,)(s - .^3*) - • (7-19) 



7-7] 



IF AMPLIFIER DESIGN 



363 



The values of s for which the denominator vanishes (j = si; s = Si* . . . 
etc.) are the zeroes of the denominator. At these values of s, the transfer 
function becomes infinite, so they are called the po/es oj the network function. 
The synthesis of an IF amplifier is facilitated by use of a potential analogy.* 
By considering each pole to represent the position of positive line charge 
normal to the complex frequency plane, and each zero to represent the 
position of a negative line charge normal to the plane, it can be shown that 
the potential measured along theyco axis resulting from the pole-zero array 
is equivalent to the logarithm of the magnitude of the normalized transfer 
function. 

When the transfer function consists of poles which are very far removed 
from the origin and near thej'co axis (as in Fig. 7-4), an arrangement of the 






C^+JW, 



Transmission 
Band 



Z(a)) 



Due to cr3+jco3 
Due to cTi+jcOj 
■Due to cTj+jcOg 



Fig. 7-4 S-Plane and Z-Plane Representations of IF Amplifier Characteristics. 



poles at each interval about a semicircle having the jw axis as diameter 
produces an approximate constant potential on the jco axis. The networks 
that are used, however, have zeroes at the origin and conjugate poles in 
the third quadrant of the plane. When the ratio of bandwidth of the overall 
receiver to the IF frequency becomes large, the contribution of these zeroes 
and poles to the transfer function in the passband region becomes significant. 
A conformal transformation 



Z = s' 



(-9 



(7-20) 



is used to obtain an exact low-pass transformation, where 

_ number of zeroes at origin of s plane 
^ number of poles in upper left s plane 

for the individual network elements employed. This transformation moves 
the zeroes to infinity in the z plane, and results in coincidence of the s plane 

*W. H. Huggins, The Potential Analogue in Network Synthesis and Analysis, Air Force, 
Cambridge Research Lab. Report, March 195L 



364 



THE RADAR RECEIVER 



pole pairs in the 2 plane, so that a single pole cluster is obtained in the 2 
plane. 

Placing the poles on a semicircle in the 2 plane produces a constant 
potential in the s plane. When the poles in the s plane are placed at the 
position given by transforming the equally spaced poles of the 2 plane to 
the s plane, a maximally flat transfer characteristic is obtained. 

The desired transfer function for the amplifier could be realized by a 
single four-terminal network followed by an extremely wide-band amplifier. 
(The amplifier, of course, would have a particular pole-zero structure, 
but the contribution to the selectivity in the frequency band of interest 
is negligible.) Practically, however, the four-terminal network is limited 
to a four-pole structure, and most commonly to a one or two-pole structure 
because of the limitation of realizable unloaded ^'s for the network in- 
ductors. The most commonly used network of the IF amplifier is shown in 
Fig. 7-5. When k = I and Li = L2 the network reduces to a one-pole 
structure as shown. 




C2 



= H- 



S'*+aS3+/3S2+ 7S+6 
S 

(s-s,)(s-si)(s-s^)(s-s;) 



Where H 






7 = 



Qj Q2 

2 

COj 002 

Q2(l-K^j'^Qi(l-K2j 



Qi Q2 



cof 


+ col 


1 


2 2 

COj CO2 



4=^ 



w,R.C, 






■'2 '2'^2'-2 

When K=l and Li= L2the Single ■ Tuned Circuit With Bifilar Coil is Obtained. In This Case 
Usually All of the Damping is Provided By R 2 -. ^ 



Then: 



H, 



= H, 



5^+ 7 S + X 
S 



7 



X 



Fig. 7-5 Commonly Used IF Interstage Coupling Network. 



7-7] IF AMPLIFIER DESIGN 365 

A parameter of great importance is the gain bandwidth product. Taking 
the simplest interstage circuit as a reference network, the gain bandwidth 
product is 

This equation shows that the quantities which determine the gain and 
bandwidth are so related that high gain can be obtained only at the ex- 
pense of reduced bandwidth. GB is determined by the tube and the circuit 
physical layout since it affects C. If two identical circuits are cascaded, 
then the 3-db bandwidth of the cascaded circuits is the 1.5-db bandwidth 
of a single circuit. For a given overall bandwidth, the individual stage 
bandwidths must be increased. With flat staggering of the circuits, this 
bandwidth shrinkage does not occur, and the GB product can be used to 
determine overall gain of the amplifier. 

The normalized attenuation characteristic of the IF coupling circuits 
employed in conventional pulse radar receivers can be expressed by the 
equation 

Attenuation (db) = 10 log to [1 + x^^] (7-22) 

where X =/„( ///- /°// )^ ^C/-/.) 

n = number of poles in the circuit (low pass equivalent) 

Bi = 3-db bandwidth 

/ = frequency at which attenuation is to be evaluated. 

If m groups of such circuits are cascaded, then the overall amplifier 
selectivity is 

Attenuation (db) = 10m logio [1 + x^""]. (7-23) 

The overall bandwidth of the amplifier is therefore 

B = (21/- - l)i/2«5i. (7-24) 

For a given overall IF bandwidth it is desirable that the principal 
selectivity occur in low-level stages. The input stage selectivity, however, 
is governed by noise figure considerations. The input stages are therefore 
designed first and then the remaining amplification and selectivity intro- 
duced. 

The selectivity of the IF amplifier is provided by one-pole or two-pole 
networks between the stages in order to realize the maximum dynamic 
range. With «-pole configurations in which the poles are distributed 
through the amplifier, the dynamic range is usually not the same at all 
frequencies within the pass band; such designs are therefore avoided. 



366 THE RADAR RECEIVER 

Groups of two-poles (staggered pairs) are frequently used. Although the 
restriction of dynamic range is not too severe, such designs are nevertheless 
inferior to synchronous stages. 

When an amplifier is built with single-pole coupling circuits, the overall 
frequency response exhibits geometric symmetry with respect to the IF 
center frequency. With two-pole coupling provided by the magnetically 
coupled double-tuned transformer, the response is more nearly arith- 
metically symmetrical. With two-poles having ^ ratios of about 3 to 1 
the usual amplifier requirements (bandwidth between 1.0 and 10.0 Mc) 
can be realized with adequate stability margin and fewer components than 
with single poles. 

One of the main difficulties encountered in the design of radar IF am- 
plifiers is accurate control of the stability margin. Pole shifting (regenera- 
tion) can occur under strong signal conditions and results in poor transient 
characteristics or modulation distortion on desired signals. 

The principal source of feedback in an IF amplifier is the grid-to-plate 
capacitance of the tube and circuit. Other feedback paths are: 

1. Coupling between input and output leads 

2. Coupling due to the chassis acting as a waveguide beyond cutoff 
frequency 

3. Grid-to-cathode feedback 

4. Inadequate decoupling circuits resulting from self-inductance of 
bypass capacitors and their connecting leads 

5. Coupling between heaters 

6. Coupling between input and output caused by ground currents. 
(The impedance of the chassis is not negligible. It is necessary that 
the output and input currents not flow through the same part of 
the chassis to ensure stable operation.) 

The grid-to-plate feedback can be partly compensated by proper circuit 
design. It is advisable that a common bypass capacitor be employed for 
the screen grid and plate return of the amplifier. Appropriate choice of 
this component then enhances the amplifier stability. In high-frequency 
stages which are gain-controlled it is also desirable that feedback be intro- 
duced in the cathode lead to stabilize the input susceptance of the tube. 
These circuits are shown in Fig. 7-6. It is further desirable that such stages 
employ vacuum tubes with separate suppressor grid terminals to minimize 
the feedback from plate to cathode. This feedback path leads to instability 
when the input susceptance of tubes having internal suppressor grid-to- 
cathode connections is to be stabilized. 

The most economical distribution of gain and selectivity in the IF 
amplifier occurs when the stages are made identical. However, this con- 
dition does not al<Vays provide the most stable operation. The latter IF 



7-7] 



IF AMPLIFIER DESIGN 



367 





R c 



Fig. 7-6 Typical IF Amplifier Configuration Showing How Proper Choice of C 
Can Self-Neutralize the Stage. (Lead Inductance Has Been Neglected.) R^Xc- 

Stages must provide adequate dynamic range and thus are operated in a 
manner which allows large peak transconductances to occur. The per- 
missible gain in these stages should therefore be less than in the early 
stages when a tube of the same type is emiployed in both places. The gain 
can be controlled by lowering the impedance of the coupling network or 
by allowing a greater bandwidth in the stages. The maximum allowable 
gain is sometimes limited to 



M 



ax gain 



0.2 



i 



:fCo 



(7-25) 



where gm is the peak transconductance and Cgp is the grid-plate capacitance. 
The effective noise bandwidth 5„ is the parameter involved in radar 
performance computation. For any practical amplifier this is very nearly 
the 3-db bandwidth 



Bn 



G(co) 



1 + 



1 r" 

: T / G(co)do) 

; recei 
an be | 

/co C0o\ "") 



(7-26) 



where G(a)) is the power spectrum of the receiver. The normalized power 
spectrum of the radar receiver usually can be given by 



(7-27) 



where coo is the midband frequency 
B is the 3-db bandwidth 
n is the number of poles in a flat circuit 
m is the number of groups of circuits. 
The 3-db bandwidth is 

(21/- - iy''"Bi, 



(7-28) 



368 THE RADAR RECEIVER 

where Bi is 3-db bandwidth of one network consisting of n poles. Then as 
an example for three staggered pairs of one-poles 



Jo (1 + .v^) 



Noise bandwidth ^ Jo (1 + •V)' ^ 1 r (i)r(3 - \) _ 
3-db bandwidth (2"^- l)'/^ 0.714 r(4)r(3) " ^•^-• 



(7-29) 



7-8 CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 

The IF amplifier is frequently divided into two units, an IF preamplifier, 
and the main IF amplifier (postamplifier). This arrangement allows the 
input stages of the IF amplifier to be physically located near the mixer. 
When long cables are used between the mixer and IF amplifier, bandwidth 
and noise figure must usually be compromised. Principal considerations 
in the design of the IF preamplifier are the noise figure, signal-handling 
capability, selectivity, and gain. 

Triode tubes are almost always required for the first two IF amplifiers. 
They are used because they exhibit less shot noise^ than pentodes. (Shot 
noise is the noise resulting from fluctuations of the currents in a vacuum 
tube.) The input IF amplifier may be used in a grounded-cathode or 
grounded-grid arrangement. For the ordinary AI radar with broad band 
mixer, the grounded-cathode amplifier is usually employed for the first 
tube. To minimize input admittance variation caused by feedback from 
grid to plate this amplifier stage is neutralized. 

The equivalent circuit representing the sources of noise associated with 
an IF amplifier stage is shown in Fig. 7-7. From this arrangement of noise 
generators and signal generator the noise figure as defined by Equation 
7-4 becomes 



1 + ^" + ^'^ + ''' + '''' + ^L + g. +,. + Kf + Sv. +/ 



= ]+^^ + ^|y, 



(7-30) 



where ^ is the total susceptance appearing at the input terminals. The 
additional parameters involved are defined in Fig. 7-7. This expression 
yields the single-stage spol noise figure of an amplifier which is defined 
as the noise figure at a specific point of input frequency. When the noise 
figure of a radar receiver is measured, a noise source is generally employed. 
If the spot noise figure is constant over the bandpass of the overall receiver, 
then the noise figure that is measured will be independent of bandwidth 

5B. J. Thompson, "Fluctuation Noise in Space Charge Limited Currents at Moderate High 
Frequencies," RCA Rev. 4, 269 (1940). 



CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 




369 



S, Ns, N„, Nt, Nf, Ni2 = current generators 

A"^ = a voltage generator 

S = signal current associated with gs 

Na = source noise, TVs = yl4KTsBgs 

(In evaluating IF noise figure Tg is usually taken as reference temperature T for 

which F is defined, or 290°K. In a radar receiver Ts is larger than T because of 

noise-noise intermodulation at mixer from local oscillator signal.) 

A^„ = noise current associated with coupling network losses, A''„ = -s^AKTnBgn 

Nt = noise current associated with grid noise of tube, Nt = ^j4KTtBgt 

^ = usually 4 or 5 

gt = grid damping due to finite transit time only 

Nf = noise current associated with feedback such as that due to cathode lead 

inductance: Nf = -yJiKyTg/; y may be between and 1 

A^i2 = noise current associated with output to input feedback conductance: Nn = 

yjiKSTgi^; d usually 1 

A'' = voltage generator representing shot noise in the tube, A^ = -yJ^KTBRn 
where R„ is the resistance that must be connected between grid and ground to 
produce the same fluctuation current in the plate circuit of the hypothetical 
noiseless tube as exists in the actual tube due to shot effect. For the triode, /?„ 
varies between 



Rn 



2.5 



and R„ 



3£ 



where gm = operating transconductance. 



Fig. 7-7 Equivalent Circuit of IF Amplifier Noise Sources. 



as the bandwidth is reduced. The last term of Equation 7-30, however, 
may cause the spot noise figure to increase for frequencies removed from 
the carrier frequency and the average value of F is then increased. To 
minimize this effect in wide-band applications, care is taken to minimize 
the increase in this term either by virtue of a small value of i? or, in some 
cases, by introducing feedback to increase gf. In all cases, IF preamplifier 
tubes are selected which have a high transconductance-to-input capacitance 
ratio and small transit time. 

An important consideration is the mixer-IF coupling network. When gf 
and ^12 are zero, minimum F is obtained for 



,.-yl'-^ + (g. + g. 



(7-31) 



as can be proven by differentiating Equation 7-30. Application of this 
equation is difficult because ^gt cannot always be easily determined. In 



370 THE RADAR RECEIVER 

practice, careful measurement of the actual input admittance of the tube 
under operating conditions and with feedback effects removed by neutral- 
ization gives an input conductance which, when employed as ^gt in the 
amplifier design, results in measured noise figures in close agreement with 
calculated values. 

Fig. 7-8 shows a simplified equivalent circuit for a grounded-grid stage. 
The noise figure is given by 



^Am + 1/ I I 



l+^ + |J + ^-:^{-T-r)|i^^l (7-32) 



1\ i h 



Nsi 



Fig. 7-8 Simplified Equivalent Circuit for a Grounded Grid Stage. A^^; in- 
cludes noise due to grid loading and network loss. Ngi is therefore taken as a noise 
current generator ^^^KTypgi, where gi is total conductance between cathode and 
grid and i/' is an effective temperature ratio. The admittance seen to the right 
of aa', is 

gli} + m) 
1 + girp 

where m is the amplification factor of the tube. In determining the overall 
noise figure of the IF amplifier, the available power gain of the amplifier 
stage must be considered. When losses in the interstage coupling network 
are not included, the available power gain for the grounded-cathode and 
grounded-grid stages is given by 

e) (7-33) 

(grounded grid) (7-34) 



where 



W^ 




(grounded cathc 


w^ 




k'^ 


-gsf'p 


" (^1 + if 


s)ig 


+ gs+ g'n.) 


, 


1 + 


M 





grr, 



The most frequently used IF preamplifier circuit in airborne radar re- 
ceivers is the cascade circuit. This circuit consists of a grounded-cathode 
triode stage followed by a grounded-grid stage. Such a configuration results 
in a lower noise figure than can be obtained with a cascade connection of 
two neutralized grounded-cathode triode stages. This is because the 
interstage bandwidth is obtained by virtue of the loading incident to the 



7-8] CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 371 

large input conductance of the grounded-grid stage. Since such a feedback 
conductance is not a noise source, less noise exists than in the case when 
such network damping is obtained by a physical resistor. 

Wide-band neutralization of the input tube is employed to stabilize the 
admittance appearing at the input grid. Such stabilization allows the widest 
bandwidth for which a low spot noise figure is obtained by minimizing the 
variation of the last term of Equation 7-30 with frequency. 

To minimize the variation of the grid admittance with frequency, a 
double-tuned (two-pole) mixer-IF coupling network is employed with the 
cascode input circuit. When this circuit is designed for a flat (Butterworth) 
response, the bandwidth is given by 

B = -4^ (7-35) 

7rV2C 

where gs is the value of source conductance required for minimum F 

B is the 3 db bandwidth 

C is the total capacitance appearing at the input of the first tube. 

The signal transmission bandwidth is slightly wider than this value because 
of the loading of the source caused by coil losses and the input conductance 
of the tube. 

In high-PRF radars (such as the pulsed-doppler systems described in 
Chapter 6) and where very short range with high accuracy is required, the 
double-tuned mixer-IF coupling network is found to introduce objectionable 
transients following the transmitter signal. These transients result from the 
nonlinear loading on the network by the mixer crystals. In such cases a 
grounded-grid input stage is employed. The transmission bandwidth of 
the mixer-IF coupling is very wide because of the heavy damping caused 
by the input conductance of the grounded-grid stage. The heavy damping 
by the tube minimizes transients resulting from the crystal mixer IF 
admittance variation when the transmitter signal is present at the mixer. 

A typical example of IF preamplifier performance is given by the 
following: 

CIRCUIT AND TUBE PARAMETERS 

g,n = 20 X 10-^ mho 

M = 44 
figt = 5.0 X 10-^ mho 
^1 = 10~^ mho 
Rn = 150 ohms 



372 THE RADAR RECEIVER 

^input tube =15 MMl 
Cgp = 1 .5 MMf 

AC =1.5 MMf due to space charge effect 

Cout = 3.0 mmF 

Cin of third = 7.0 ^iixi 
tube 

gn of input = 10~^ mho 
coil 

Problem Solution. Requirements are that the overall bandwidth 
between the mixer and third tube in which the attenuation is less than 1 db 
shall be 3 Mc, the noise figure of the amplifier shall be as low as possible, 
and a cascode circuit employing two of the tubes having the specified 
characteristics is to be used. 

For minimum F, the source conductance from Equation 7-31, 



^^^ ^10_2_+_5(10_^ = 6.3 X 10-4 mho is required. 

A source bandwidth (see Equation 7-35) of 
6.3 X 10-4 
^ = .V2 20.5X10- = ^-^^^^ 

is obtained, assuming 1.0 MMf stray capacitance and a wide-band neutraliza- 
tion of Cgp. 

At 3-Mc bandwidth the input coupling network produces an attenuation 
(see Equation 7-22) 

10 login fl +[zl??^ =0.15db. 



The network between the second and third tube can therefore exhibit 0.85 
db attenuation at a bandwidth of 3 Mc. A single-tuned circuit of 6.45-Mc 
bandwidth is adequate: 

10 log, of 1 + I T^ 1 = 0.85 db. 



'"^"O+felV"- 



Total capacitance between the second and third tube is 3.0 -f 7.0 = 10 

If a step-down transformer is employed, the gain bandwidth product 
could be improved and the conductance presented to the second tube output 
is lowered. 

Ine step-down must be n = -\/t^ = \/t7j' 



7-9] OVERALL AMPLIFIER GAIN 373 

Total capacitance at output of the second tube is then 3.0 + ^ = 6.0 mm/ 

n- 

and the conductance is 

gi - 27r(6.45)(10«) (6.0) (10-12) = 2.43 X 10-^. 
From Equation 7-30 the noise figure of the first tube is 

V 1 -L IQ"' , 5.0 X 10-^ , 150 ,, . ^ ^ _ ^, , 

^^ = ^ + 6.3 X 10- + 6.3 X 10-^ + 6.3 X 10"^ ^^'^ >< ^^ '^'^ 

1.196 = 0.78 db 

The noise figure of the second tube is (from Equation 7-32) 

P ^ J . 10-^ 5.0 X 10-^ 2.43 X 10-^ 150(0.95) 

' "^ 4.55 X 10-* ^ 4.55 X 10"* "^ 4.55 X 10-* "^ 4.55 X 10-* 



(4.65 X 10-^)2 



4.55 X 10-* = ^ = g^ip. 



= 1.74 = 2.4 db. 
From Equation 7-33, the available power gain of the first tube is 

(20XI0-y(6.3X10-)(g^^3) 

(6.5 X 10-*)2 ^^^ 

while the available power gain of second tube is approximately 
4.55 X 10- 



2.43 X 10-* 



1.87. 



Assuming 10 db for the third tube F, the preamplifier noise figure is given 
by Equation 7-7: 

1196+ ^-^^-^ + ^Q-^ ^1237 
i.iyo^ 131 ^(131)(1.87) 

^ 0.92 db. 

7-9 OVERALL AMPLIFIER GAIN 

It is necessary that signal amplitudes corresponding to the thermal noise 
level at the input of the receiver be amplified to a suitable level for detec- 
tion. The level required at the detectors depends on the use of the signal. 
For example, for signal detection on an intensity-modulated display 
providing range and azimuth coordinates, signal voltages on the order of 
50 volts are usually required. The total amplification that is required 
depends on the signal bandwidth and the receiver noise figure. The equiv- 



374 THE RADAR RECEIVER 

alent noise voltage that must be amplified may be determined by computing 
the total noise voltage at the input of each stage caused solely by the total 
grid conductance, the shot noise in the tube, and total grid admittance. 
This voltage is then referred to the input by dividing by the total gain from 
the input to the noise considered. Even though consideration is given to 
the design of a low-noise IF preamplifier, the process of referring all of the 
noise sources in the receiver to the input is extremely important, especially in 
multiple-conversion receivers such as are employed in some forms of doppler 
radar receivers. This method sometimes reveals factors such as noise on 
beating oscillator signals or noise caused by a method of selectivity dis- 
tribution that would degrade the IF signal-to-noise ratio and result in a 
sensitivity poorer than would be estimated from consideration of the IF 
preamplifier noise figure and overall receiver selectivity only. 

A typical equivalent input noise for an ordinary radar receiver having an 
overall bandwidth of 5.0 Mc is about 3.0 Mvolts rms. For signal detection 
alone, a voltage between 1 and 2 volts rms at the input to the IF envelope 
detector is satisfactory. Thus the required overall amplifier gain is on the 
order of 105 to 115 db. 

To obtain the voltages for the cathode ray tube an additional gain on the 
order of 40 db is then required. (Included in this figure is a loss of 6 to 10 db 
that is usually produced in wide bandwidth second detectors.) 

Where the envelope of the signal must be accurately demodulated, higher 

voltages may be applied to the envelope detector to recover larger negative 

peak modulation with less distortion. However, dynamic range of the 

amplifier must be exchanged for the higher operating level. In tracking 

receivers, use of a range-gated amplifier ahead of the envelope detector 

allows such an exchange to be made. The detector average output is usually 

regulated to a relatively fixed level, and noise modulation positive peaks 

have very small probability of exceeding a level more than 12 db above the 

regulated level. In a typical case of a receiver having an IF bandwidth of 

5 Mc, incremental gain can be maintained for a range of IF signal from zero 

to about 12 volts rms at the input to the IF envelope detector. Thermal 

noise can therefore be amplified to a level of about 3 volts rms at the 

detector input. It is obvious that this does not appreciably alter the IF gain 

requirements. In the case of very narrow band receivers such as are 

employed for detection and tracking of targets by means of their difference 

in doppler frequency, receiver bandwidths are on the order of several 

hundred cycles per second. Considerably more gain is therefore required 

over that encountered in conventional radar sets. For example, with a 

bandwidth of 500 cps, the required gain to the envelope detector would be 

... , 5,000,000 ,^ ^, , . , r . • , J 

^°gio — 77^7^ — = 40 db more than m the case or the conventional radar 

receiver of 5 Mc IF bandwidth. 



7-11] BANDWIDTH AND DYNAMIC RESPONSE 375 

7-10 GAIN VARIATION AND GAIN SETTING 

Gain of an amplifier stage which does not incorporate feedback is equal to 
the product of the transconductance of the vacuum tube and the transfer 
impedance of the network which the tube drives. Instability of these 
parameters results in gain variation. The effective signal transconductance 
of a tube is proportional to the d-c current through the tube. Gain can 
therefore be stabilized by operating the tubes so that the d-c plate current 
is stabilized. This can be accomplished by means of large cathode resistors 
or by operating a number of stages in series d-c connection. However, the 
first method introduces transient recovery problems and the second method 
reduces dynamic range. Application of conventional feedback stabilizing 
techniques may be employed, but in high-frequency IF amplifiers it is 
usually limited to a small amount of signal-current feedback which is 
employed to compensate for input admittance variations of the tube. 

Network transfer impedance variations are on the order of ±0.5 db; and 
when a small amount of d-c current stabilization is employed with the 
vacuum tube, the variation in signal transconductance is on the order of 
±1.0 db. 

Stage gains are limited by bandwidth requirements in the case of wide- 
band stages, and stability requirements in the case of narrow-band stages. 
In addition, restrictions are usually encountered in gain distribution 
through the receiver as a result of dynamic range requirements. Typical 
average stage gains in a receiver are between 6 and 20 db incident to these 
limitations. An amplifier providing 100 db gain might therefore require 
about 10 tubes. Since the variation in gain of each stage is on the order of 
±1.5 db, 15 db reserve gain is required in the design, and provisions for con- 
trolling the maximum gain of the amplifier over a 30-db range is required. 
These gain-control variations do not include the gain control that is 
required to accommodate target signal variations. The gain setting may 
take the form of a noise AGC loop which controls the current of several 
tubes or a manual adjustment which is periodically set. 

7-11 BANDWIDTH AND DYNAMIC RESPONSE 

A criterion sometimes employed for best signal-to-noise ratio is that 
signal plus noise should be filtered by a network which maximizes the peak 
signal-to-rms noise power. The network which will accomplish this result 
was determined in Paragraph 5-10 to be simply the conjugate of the signal 
spectrum; that is, the receiver filter should be "matched" to the signal. In 
the case of the noncoherent pulse radar, each pulse must be considered as a 
separate entity; therefore the optimum predetection filter is a bandpass 
filter shaped like the RF pulse spectrum envelope. The IF characteristics 
usually employed for maximum detection in thermal noise are reasonable 



376 THE RADAR RECEIVER 

approximations to this value when the 3-db bandwidth is approximatehy 
1.2 /(pulse length).^ 

Additional filtering can be applied to the postdetection or video signal 
when there is more than one pulse. A series of periodic pulses will have a 
spectrum consisting of a number of harmonics. The filter which is matched 
to such a signal will be tuned to these harmonics so as to amplify them and 
attenuate the intervening noise. Because of the shape of the frequency 
response of such a filter, it is sometimes called a comb filter. It is often more 
convenient to obtain the effect of a matched filter by operating in the time 
domain. The comb filter, which is appropriate for a series of pulses, can be 
simply represented by adding the pulses after they are delayed by appro- 
priate multiples of the repetition period. This operation is normally called 
pulse integration and, for search radars, is often performed by the phosphor 
of a B or PPI scope display. When the more elaborate technique of time 
domain filtering is utilized, it is sometimes referred to as signal correlation. 
A more detailed discussion of matched filters is given in Paragraph 5-10. 

In selecting a bandwidth characteristic for the receiver, three considera- 
tions must be made over and above signal to thermal noise: 

1. Adjacent channel (frequency) attenuation and discrimination 
against clutter 

2. Compatibility of transient response with required resolution 

3. Large signal operation 

The usual response characteristics that might be encountered were 
indicated in Paragraph 7-7. The transient response of these networks 
governs the resolution and large-signal behavior. The rectified envelope of 
this response corresponds to the video signal. 

A typical transient response would 
appear as shown in Fig. 7-9. At the 
receiver output, a loss in sensitivity 
may occur for the time /2 shown in 
Fig. 7-9 if the signal becomes suffi- 
ciently large that amplifier stages 
are driven into saturation. In pulse 

T7 1 n ^ ■ ] r\ ^ ^^ • .. doppler radar receivers this is a more 

i*iG. 7-9 Typical Output Transients as ^^ 

They Appear on the Rectified Envelope serious problem than in conventional 

of the IF Response to an IF Pulse Input, radar receivers. 

When all of the networks have identical transfer impedance of the form 

^ £ 

(s — Si){s — Si*) 

6See J. I. Lawson and G. E. Uhlenbeck, Threshold Signals, Vol. 24, Sec. 8-6 (Radiation 
Laboratory Series), McGraw-Hill Book Co., Inc., 1950. 




7-12] SNEAK CIRCUITS 377 

there is no overshoot when the pole frequency and the carrier frequency are 
coincident. The envelope response of such a network is given by 



eo{t) = K 



1 +f expyj - 2f exp^|-jcos (co - w,,)/ 



1 + 



\C0o CO / 



(7-36) 



where ^ is the effective circuit ^ 
Wo is the pole frequency 
CO is the carrier frequency of the step sinusoidal input. 

Note that oscillatory terms are involved when the carrier is detuned from 
bandcenter. These terms are relatively insignificant, however, for the 
amount of detuning that would normally be tolerated. When the circuits 
are not identical but are stagger-tuned, then the response given by Equation 
7-36 becomes important. If the oscillatory signal is sufficiently large, the 
output of the following stage may be blocked for a period of time in excess 
of twice the duration of the input signal. To minimize these effects, inter- 
stage bandpass networks are usually employed which are symmetric about 
the IF center frequency, 

7-12 SNEAK CIRCUITS 

When considering the dynamic response of the receiver, it is not sufficient 
to consider only the performance as a bandpass filter with saturation effects 
under large-signal input. The transmission characteristics of the amplifier 
in the low-frequency region of the spectrum must also be considered. 

To realize practical high-gain bandpass amplifiers the power supplied to 
the stages must not be derived from a common-source impedance, since 
instability will result. Fig. 7-10 shows a typical arrangement of IF stages. 
The power leads are brought into the amplifiers near the output. De- 
coupling filter elements CiCiC^RiRiLi Li are employed. The decoupling 
is designed so that a single stage will exhibit adequate gain and phase 
margin over the entire frequency spectrum when the stage is examined as 
a feedback amplifier. In particular the stability margin must be realized 
when the tubes operate at the peak transconductance values that would be 
produced by a saturating signal. 

Time domain effects must also be considered. Saturating signals cause 
the d-c currents to the various tube elements to vary. The cathode circuits 
will attempt to degenerate the effects of a saturating signal during the time 
that the signal exists. When the signal input ceases the cathode capacitor is 
charged to the value which has reduced the gain during the signal on time. 



378 



THE RADAR RECEIVER 



Gain Control Bus 




r^ 



1^2 



X^2 



1^2 



Plate 
C ' X Voltage 
Supply 



L2 Heater 

Supply 



Fig. 7-10 Typical Arrangement of IF Stages Showing Arrangement of Decoupling 
Circuits and the Feedback Paths Thereby Introduced. 

To obtain maximum receiver sensitivity the charge must be removed. This 
removal occurs with a nonlinear time constant 

C, 



+ <?^(/) 



Short time constants must be used to avoid gain modulation of desired 
signals when there are large undesired signals such as clutter appearing in 
the receiver. Grid current may also produce a similar situation, and the 
time constants must be kept short while at the same time providing 
sufficient decoupling at low frequencies and at bandpass frequencies. 

The plate circuit decoupling is perhaps more critical than the other 
circuits. With a ladder decoupling chain, the d-c path must be kept low in 
resistance so that the plate voltage is not dropped excessively. Inductors 
are therefore used as the series elements. The elements nearest the power 
input connection have the currents of several tubes flowing through them. 
When several stages are driven into saturation, each of the stages will send 
a transient input into the decoupling chain. This transient propagates 
along the chain and may result in a very complicated transient at the last 
stage which can gain modulate that stage, causing undesirable transient 
gain variations following strong pulse signal inputs. To avoid this phenom- 
enon, adequate filtering is provided between the ladder tapping point and 
the tube. 



7-13] CONSIDERATIONS RELATING TO AGC DESIGN 379 

7-13 CONSIDERATIONS RELATING TO AGC DESIGN 

The AGC of the radar may be of two types: (1) a fast AGC which 
prevents saturation of the receiver or (2) a slow AGC associated with a 
single target echo. In the radar receiver employed for tracking, AGC 
circuits of the second type are required. The IF amplifier is one of the 
limiting factors in the design of a high performance AGC. This subject 
will be discussed at greater length in Paragraph 8-21. 

In designing the IF amplifier great care must be taken to examine signal 
distribution in the amplifier as a function of the AGC voltage. The AGC 
voltage must be applied to the amplifier in a manner that will result in 
minimum signal distortion and limited degradation of the output signal- 
to-noise ratio of the receiver. For example when the input signal-to-noise 
ratio is +90 db, it is necessary to reduce the gain in early stages to minimize 
distortion, and as a result noise from latter stages becomes significant. A 
t-vpical design might allow the output signal-to-noise ratio to be +30 db 
li inimum for +90 db input signal-to-noise ratio. 

For minimum distortion of the modulation on the signal as the gain of an 
amplifier stage is varied by AGC, it is desirable that the transfer character- 
istic be a square-law when signal and gain control are applied to the control 
grid. When sharp cutoff tubes are employed for gain control, considerable 
distortion is sometimes experienced when gain control is provided for large- 
signal inputs. Restriction of the gain control to about 10 db per stage in 
these cases usually results in acceptable signal envelope reproduction. 
Output stages of the amplifier should operate with linear plate transfer 
characteristics. This allows the IF signal voltages applied to the last few 
gain-controlled stages to be small, thereby resulting in less distortion. In 
addition, wider bandwidths can then be employed in these stages, since 
filtering of the undesired spectral components of the modulated signal, 
which result from passing the signal through the nonlinear plate transfer 
characteristic required for constant incremental gain as a function of AGC 
voltage, is not required. In a typical case the gains in the IF may be 10 db 
per stage. Requiring 2 volts rms at the IF envelope detector, the minimum 
signal voltage at the third from the last stage of the amplifier would be 
0.2 volt rms if gain control is not applied to the last two stages. The 
maximum signal on the controlled stage then depends on the gain reduction 
allowed. By controlling a number of stages the maximum gain reduction 
required in any one stage can be limited to something on the order of 10 
to 20 db. It is necessary to examine the signal transmission through each 
stage for the maximum signal allowed at the input of the stage as a result 
of the distribution of the AGC control voltage. The AGC and transfer 
impedance of the stages are then arranged to provide a specified allowable 
distortion of the modulation on the signal appearing at the amplifier output. 



380 THE RADAR RECEIVER 

In the early stages of the receiver, care must be exercised in applying 
AGC. When a cascode type input amplifier is employed, relatively large 
voltages may appear at the input grid and also the third tube grid in 
receivers which must provide target tracking at very short ranges. An 
AGC voltage is therefore applied to the first tube in these cases. However, 
in order that the output signal-to-noise ratio of the receiver shall not be 
seriously degraded, this AGC is usually not applied at the same input level 
as the AGC on the other gain-controlled stages but is delayed until the 
input signal-to-noise ratio is about 20 db. The AGC voltage delivered to 
the cascode is selected so as to minimize the third-order coefficients of the 
tube transfer characteristic. The effective cascode transfer characteristic is 
somewhat superior to that of a single tube because of the d-c series connec- 
tion which allows control of the current of both tubes. Controlling the 
current of two tubes in a cascode arrangement has the advantage that the 
stability is not impaired at low gain. When only one tube is controlled, the 
grounded grid section may become unstable because of the reduced source 
conductance which drives it. A disadvantage of controlling the current of 
two tubes exists; not only does the conductance of the output of the first 
tube decrease, but the input conductance of the grounded grid section also 
decreases, thus narrowing the intercascode coupling bandwidth. 

Plate and screen grid control for AGC is attractive but reduces the 
dynamic range of the amplifier stage for large signal input. The operating 
point can be maintained at a value which minimizes the third-order 
coefficient, but signal suppression occurs when the signal peaks drive the 
control grid into cutoff and into grid current. 

Suppressor grid control is very attractive, since the third-order curvature 
can be minimized without sacrificing dynamic range. One difficulty is that 
the power dissipation of the screen grid is usually exceeded under strong 
signal conditions. 

For a high-performance system the AGC voltage will be staggered, i.e., 
the amount of AGC voltage applied to the various controlled tubes of the 
amplifier will be different. This is required to obtain minimum envelope 
distortion. The AGC decoupling circuits must be designed with the 
precautions noted in Paragraph 7-12. In particular, the transmission of the 
IF amplifier at low frequencies must not be significant — i.e., it must 
operate only as a carrier amplifier. 

7-14 PROBLEMS AT HIGH-INPUT POWER LEVELS 

In an airborne radar set strong signals are obtained from short-range 
targets, clutter, and other radar signals. Two situations occur in the 
receiver. In one case the receiver may be operating at maximum gain and 
be required to furnish output from signals having an input power of the 



7-14] PROBLEMS AT HIGH-INPUT POWER LEVELS 381 

order of magnitude of the receiver thermal noise. In the other case the 
receiver is required to furnish an output from a single signal which has been 
time-selected. 

In the first case, cross modulation caused by the strong signals can 
deteriorate the weak-signal performance; the extent to which this occurs is 
a function of the detailed receiver design. If the receiver is linear, the 
dynamic range for any particular gain setting will usually be between 10 
and 20 db. Signals more than 20 db greater than the thermal noise level can 
be expected to cause saturation in the receiver. The result of the saturation 
is a paralysis of the receiver for a certain time following the removal of the 
large signal. To minimize this effect it is necessary that attention be given 
to the circuits mentioned in Paragraph 7-12, so that a suitable transient 
characteristic is obtained from the IF amplifier. The transient should 
exhibit small overshoot and short delay time. Loss of weak signals occurs 
only when they are time-coincident with the strong signals if adequate IF 
filtering is provided. In cases where signal information is required and 
when the interference and signal occur at the same time (range), saturation 
must be prevented and the two signals separated on the basis of their 
difference in frequency spectra caused by the doppler shift. In the non- 
coherent pulse radar this is accomplished by heterodyning the weak signal 
against the strong signal at the IF second detector. 

The second case occurs when a signal is being tracked. The desired 
signal is gated and may provide range and direction signals from sidebands 
associated with each of the pulse signal sidebands. The effect of strong 
signals is to add additional sidebands at the receiver output and thereby 
cause errors in the range and direction signal. In a well-designed receiver, 
negligible intermodulation occurs when a strong signal is present which is 
not time coincident with the desired signal. 

In some instances the desired signal power level may approach the order 
of magnitude of the local oscillator signal power. Fig. 7-11 shows the 
transfer characteristic of a typical microwave mixer at large-signal levels. 
The nonlinearities of this characteristic will cause signal distortion. Inter- 
modulation components appear incident to the beating of the various signal 
components. These components are not highly significant except with some 
propeller-driven targets in which terms of the order 2wi + £02 may introduce 
more fluctuation in the final bandwidth of the system. The reduction in 
modulation percentage of the pulse signal at the fundamental modulating 
frequency results in deterioration of tracking performance, since it corre- 
sponds to a change in tracking loop gain. In many cases the signal at the 
antenna terminals is greatly distorted before it reaches the signal mixer 
because of the time varying attenuation of a gas discharge TR tube. A 
controlled TR characteristic is therefore sometimes used to advantage to 
minimize the deterioration in tracking loop performance. 



382 



THE RADAR RECEIVER 

-500t20- 




-20 -16 -12 -8-4 4 8 12 

SIGNAL POWER (dbm) 

Fig. 7-11 Transfer Characteristic of a Microwave Mixer at Large-Signal Input 

Levels (1N23C Crystal). 



7-15 THE SECOND DETECTOR (ENVELOPE DETECTOR) 

An envelope detector is employed to produce an output voltage which 
corresponds to the envelope of the IF signal. The envelope detector is 
actually a mixer in which the sidebands of the signal are heterodyned 
against the signal carrier thereby producing as one output the modulation 
that existed on the IF signal. In the ordinary noncoherent pulse radar set, 
a diode detector is frequently employed. A typical circuit is shown in 
Fig. 7-12, together with the current- voltage relations that exist under 
large-signal conditions. A pulse of IF voltage is indicated as being applied 
to the detector. A large diode current pulse flows for a short time following 
the application of the signal. Capacitor Co is a relatively low impedance at 
the IF frequency compared to i?o, and RFC is a high impedance to these 
frequency components; therefore negligible voltage appears across i?o due 
to the IF frequency components and their harmonics which appear in the 
diode current. The average value of the current pulse, however, does 
produce a voltage across i?o- This voltage builds up at a rate dependent on 
the capacitance Co + Ci + Ci and the diode resistance, and reaches an 
average value Edc as shown in Fig. 7-12. The diode only conducts when the 
instantaneous voltage applied to the diode exceeds Ex. As shown, conduc- 
tion during time ab occurs and the capacitance Co + C\ is charged at a rate 
dependent on the diode resistance and this capacitance. When the IF pulse 
ceases, the diode is back-biased and returns to the unbiased condition with 
a time constant /?o (Co + Ci + C2). [The effect of the inductance of the 
RFC on this transient is usually negligible when the product of pulse length 



7-15] 



THE SECOND DETECTOR (ENVELOPE DETECTOR) 



383 




Video Amp. 



Fig. 7-12 Typical Second Detector Circuit 



times IF frequency is greater than 50. As a result of this operation it is 
necessary that Rq (Co + Ci + C2) be considered as a low-pass filter estab- 
lishing the video bandwidth.] 

The efficiency of the diode detector is the ratio of the d-c voltage (Edc) 
to the peak carrier voltage applied to the circuit. The efficiency depends on 
the ratio of the diode resistance plus source resistance of the IF network as 
seen by the diode to the load resistor Rq. Efficiency, however, also depends 
on the ratio of the load resistance to the reactance of Co + Ci at the IF 
frequency. In practice Co is usually on the order of 10 to 20 iJifif. Smaller 
values of Co result in less voltage impressed on the diode because of the 
division of voltage between Cq and C^. Ro is then selected on the basis of 
video bandwidth requirements. A typical example is a requirement that the 
video bandwidth be 10 Mc with a network impedance as seen by the 
detector of approximately 500 ohms and a capacitance Ci of 10 finf. The 
value of Ro is then fixed by Ci and the smallest value of Co that can be 
employed. Assuming Co to be 10 nnf, Ro is required to be 796 ohms. An Ro 



384 



THE RADAR RECEIVER 



of 750 ohms would be used. The efficiency of the detector would be 0.21, 
assuming a diode resistance of 200 ohms^ and a 60-Mc IF frequency. A gain 
loss of 13.6 db is thus exhibited by the detector. This is a typical loss; the 
loss usually ranges between 6 and 15 db, depending on the video bandwidth 
and IF frequency involved. 

An important design consideration is the loading on the IF network 
produced by the detector. An approximation of this loading is given by 

R = ^ (7-37) 

where R is the IF network loading and rj is the efficiency of rectification. 
Efficiency of rectification depends on the diode resistance Rd plus the IF 
signal source resistance. Since the value of Ra depends on the voltage 
applied to the diode, the detector is nonlinear at low levels. A typical 
second-detector characteristic is shown in Fig. 7-13. Reproduction of the 
modulation on a PAM (pulse amplitude modulated) signal depends there- 



1.00 



0.10 



0.01 



0.001 

























/ 




























/ 






























/ 
























y 


/ 


/ 
























/ 


/ 




























</ 




























Jy 


/ 




























Y/ 


























/ 


/ 


/ 




rher 


mionic Diode 
Ro= 820 oh 


Type 56 

ms 


47 














/ 


f 






Cc " 


— 


lb mrr 


f 














/ 






























/ 




























/ 






























/ 




























f 





























1.0 10 

INPUT VOLTS (rms) 



100 



Fig. 7-13 Transfer Characteristic of a Typical Wide-Band Envelope Detector. 



'Determination of efficiency and input impedance is relatively complicated. Methods for 
determining these quantities may be found in K. S. Sturley, Radio Receiver Design, Vol. 1. 



7-15] THE SECOND DETECTOR (ENVELOPE DETECTOR) 385 

fore on the carrier level of the IF signal applied to the detector. With high 
percentage of modulation, the negative peak modulation is distorted 
incident to the nonlinearity of the detector at low levels. 

In receivers which provide considerable pre-detection integration (IF 
bandwidths of a few kilocycles per second) it is feasible to obtain high 
detection efficiency by use of large Rq and Co. When amplitude modulation 
on the signal must be recovered in such receivers, it is required that Ri 
and Ro satisfy the relationship 

^1 



Ro + R^ 



> m (7-38) 



where m is the highest modulation percentage that must be recovered 
without distortion. Failure to satisfy this condition results in clipping of 
the negative peaks of the modulation. 

When the signal-to-noise ratio of the IF signal is very small and the video 
bandwidth is less than the IF bandwidth, signal suppression occurs in the 
second detector.^ This is the result of noise-noise intermodulation at the 
detector. An approximate expression for signal suppression is 

db suppression ^ - 7 + 20 logio {SIN)if. (7-39) 

It is desirable to provide as much filtering as possible prior to envelope 
detection to minimize sensitivity loss caused by this signal suppression. 
However, predetection selectivity is limited by the stability of the IF filters 
and the tuning accuracy of the receiver. Some receivers, e.g. logarithmic 
receivers, do not employ a diode envelope detector but obtain the envelope 
by infinite impedance detection or plate detection in each of the IF stages. 

In monopulse receivers the IF detector which is employed to obtain 
angular error signals is usually a balanced modulator. This may take the 
form of either a phase detector or a synchronous detector. Such detectors 
ideally produce an output only when both signals are applied. The output 
is primarily dependent on one of the two signals present at the input 
(provided one signal is much larger than the other). If one of the signals, 
such as the sum signal in a monopulse receiver, is heavily filtered before 
applying it to the demodulator, significant improvement in detected S \N 
can be realized for low ^S" /A^ referred to the difference signal IF bandwidth. 
Such filtering, however, requires time selection of the sum signal before it is 
applied to the detector. Such a scheme is, in effect, a carrier reconditioning 
and exaltation method of detection and, of course, reduces the information 
rate of the radar. 

8S. O. Rice, "Mathematical Analysis of Random Noise," Bell System Tech. J. 23, 282-236 
(1944), 24, 46-156 (1945); "Response of a Linear Rectifier to Signal and Noise," J. Acoust. 
Soc. Am. 15, 164 (1944). 



386 THE RADAR RECEIVER 

7-16 GATING CIRCUITS 

Gating circuits are employed to improve the signal-to-noise and signal- 
to-clutter ratios at the output of the receiver. A gating circuit consists of a 
modulator to which the signal and the gating signal are applied. In most 
applications the only output desired is the intermodulation between gating 
signal and desired signal. To accomplish this, balanced modulators are 
required. At video frequencies, such circuits are difficult to realize, the 
dynamic range usually being small. At IF frequencies such circuits are 
more easily provided, and dynamic ranges greater than 50 db are common. 
The choice between the IF and video gating depends on the nature of the 
signals to be encountered by the radar receiver. Typical gating circuits for 
video and IF applications are shown in Fig. 7-14. Gating circuits are 




Gating 
Signal 
IF GATING CIRCUIT 

Fig. 7-14 Typical Gating Circuits. 

employed having gate lengths equal to the range displayed on an indicator 
and also with lengths equal to or somewhat less than the transmitted pulse. 
When a dynamic range greater than 50 db is required from a gating 
circuit, component selection is required. This is a result of uncontrolled 
cutoff characteristics of vacuum tubes that must be utilized. When gating 
occurs in the IF amplifier, spurious signals are always encountered. These 
spurious signals occur because it is difficult to suppress completely all of the 
modulating signal (gate pulse) at the output of the gater. The gating pulse 



7-17] PULSE STRETCHING 387 

is not usually coherent with the IF signal. The higher frequency compo- 
nents of the gating signal are the signal components which cannot be 
adequately filtered. Transients caused by the modulating signal will 
generally produce outputs from the IF filter when rectangular gate pulses 
are employed. Noise modulation of the desired signal results from these 
transients. 

In typical designs the noise modulation caused by the transients is at 
least 40 db below the signal. In addition, the seriousness depends on the 
signal processing following the gating. Appropriate sampling of the gated 
signal prior to integration reduces the noise to a negligible value. In video 
gating circuits the modulating signal is coherent with the detected signal. 
Thus the noise mentioned does not occur. 

7-17 PULSE STRETCHING 

In tracking radars it is required that the modulation signal associated 
with a pulse-amplitude-modulated signal be recovered. The modulated 
pulse signal is 

/i(/) = [1 + m cos (a;„/ + 0)] ^fljue^^ (7-40) 

for periodic pulses of shape /(/) 

where r„ = ^///W exp (^^^^V/. (7-41) 

If the pulses are passed through a low-pass filter having a cutoff frequency 
below the first harmonic of the pulse, the modulation is recovered and will 
have an amplitude m{tjT) cos [w^/ + <^]. Since t jT typically may be on the 
order of xoVo this is a very inefficient process. Pulse-stretching circuits are 
therefore used to lengthen a series of pulses without changing the relative 
pulse amplitudes in order to obtain more gain in the process of recovering 
the amplitude-modulating signal. For most efficient demodulation the 
pulse is lengthened for a full period. In either case — whether a pulse is 
simply filtered or is lengthened and then filtered, time selection of the pulse 
is required prior to the lengthening to prevent cross modulation by un- 
desired pulses. 

A pulse lengthener converts the modulation function 1 + m cos (co^^ + <^) 
into a new function F{f). Two types of lengtheners are used. In one, F{t)^ 
is set at a fixed reference level prior to a signal pulse input; in the other the 
output is changed from the value measured to the new value. Typical 
circuits of these lengtheners are shown in Fig. 7-15. The lengthened pulse 
on which the desired signal is modulated is an exponential pulse. The 
decrement is small and approaches zero in many practical cases. 



388 




THE RADAR RECEIVER 




j^i:^ 



Eexn(-af) 



Fig. 7-15 Pulse Lengtheners. 



The output spectrum of the lengthener for the case where a = 0, and 
T = Tp IS given by 

\fF;\ = 1+ 2m sin ^ \Ar cos (co./ + <A) - ^1 

r/ 27r\ ^ , ^ , U)mTp~\ 



+^.s, 



[(-4:)-' 



+ <^ + 



Tp 

UmTp 



(7-42) 



^m + 



Tp 



When the output from the lengthener is passed through a low-pass filter, 
the first term becomes the only significant term in the output if the period 
of the modulating signal is much greater than Tp. If the low-pass filter has 
a cutoff frequency Wc, outputs are also obtained for modulation frequencies 
satisfying 



1 P 



(7-43) 



7-18 CONNECTING THE RECEIVER TO THE RELATED 
REGULATING AND TRACKING CIRCUITS 

The receiver must provide signals to range or speed error detectors, and 
angular error detectors. It is desired that the outputs of these detectors 
have a stable characteristic with time and with input power level to the 



7-19] 



ANGLE DEMODULATION 



389 



receiver. If the AGC demodulator is connected to the range error detector, 
and the angle demodulator connected directly to the AGC demodulator 
output, both range and angular error characteristics will be determined by 
the AGC regulation. 

It is desirable, however, that the video signals applied to these demodu- 
lators be as large as possible to minimize the bias errors resulting from 
contact potential in the demodulators. Frequently separate filtering of the 
range and angle video signals may be performed. A single AGC loop 
operating from the angle channel controls the receiver gain. To obtain a 
stable range-error detector characteristic, the video amplification between 
the input to the range detector and the AGC demodulator must then be 
stabilized by feedback. A typical arrangement is shown in Fig. 7-16. 



Local 
Osc. 



AGC 

Amplifier 

Filter 



IF 
Amplifier 



AGC Delay 
Voltage 



Envelope 
Detector 



Range Gate 



Pulse 
Lengthener 



Cathode 
Follower 



Video 
Amplifier 



Range 

Error 

Detector 



Video 
Amplifier 



Range 
"Gate 



Noise Free Signal 
at Lobing Frequency- 



Angular 

Error 

Demodulator 



A z. Error 



Angular 

Error 

Demodulator 



El. Error 



Noise Free Signal 
•at Lobing Frequency 



Fig. 7-16 Connection of a Receiver Employed with Sequentially Lobed Antenna 
to Related Circuits. 



7-19 ANGLE DEMODULATION 

The antenna tracking error signal can be considered to be proportional 
to the magnitude of the fractional modulation of the signal resulting from 
division of the difference signal by the sum signal. In monopulse radar sets 
the sum and difference signals are separated at the receiver input, whereas 
in a conical scanning radar the composite signal is passed through the 
receiver. To obtain a tracking error signal from a monopulse radar, the 
difference signal is heterodyned with the sum signal, which is effectively a 
noisy carrier signal. In some cases carrier reconditioning may be performed 



390 THE RADAR RECEIVER 

and the difference signal heterodyned against a filtered carrier signal. This 
latter operation, however, is accomplished only with sacrifice of the infor- 
mation rate. 

In a conical scanning radar the desired target is selected by range gating 
so that the other targets, which are also PAM signals, will not be demodu- 
lated. The signal is then envelope-rectified and lengthened. Lengthening 
is employed to minimize additional modulation resulting from PRF 
variation. The signal at the output of the pulse lengthener still represents 
the composite signal, i.e. the sum and the difference signal. The low- 
frequency modulation of the composite signal is caused by the scintillation 
noise of the target and is independent of the lobing frequency. Both the d-c 
component of the signal and the low-frequency modulation are fed back 
to the IF amplifier as a gain-control signal. Modulation at the lobing 
frequency, however, is not allowed to effect a gain control of the receiver. 
The signal at the output of the pulse lengthener thus contains primarily 
the sidebands about the lobing frequency which are caused by the variation 
in direction of arrival of the signal. To demodulate this signal, and provide 
control signals for the antenna servo, the signal is multiplied by a noise-free 
carrier at the lobing frequency. The carrier signal is phase-locked with the 
antenna lobing. This is usually accomplished by means of an a-c generator 
mechanically linked to the rotating antenna. 

Fig. 7-17 shows three typical demodulator circuits. In all three of these 
circuits neither the signal nor the carrier frequency appears in the output. 
The output contains only the beats between the signal and the carrier and 
certain of their harmonics. Of the three demodulator circuits shown the 
"ring modulator" is the most desirable because the modulation products 
are effectively separated in various parts of the circuit. The carrier signal 
should be as monochromatic as possible for maximum output signal-to- 
noise ratio. 

The process of pulse lengthening merely concentrates all of the noise 
appearing in the IF in a region less than the PRF. In order that the noise 
reduction provided by the antenna servo be approximately Bi/PRF, where 
Bi is the noise bandwidth of the antenna tracking loop, it is necessary that 
the demodulator provide a true product demodulation. To approach this 
performance the ring modulator is employed in conjunction with a bandpass 
filter which filters the signal applied to the demodulator. 

7-20 SOME PROBLEMS IN THE MEASUREMENT OF 
RECEIVER CHARACTERISTICS 

Noise Figure. The most practical method of making noise figure 
measurements involves the use of a dispersed signal source. An argon-filled 
gaseous discharge tube will produce a standard noise power output equiva- 



7-20] PROBLEMS IN MEASUREMENT OF RECEIVER CHARACTERISTICS 391 



Demodulated 

Output 
NoC+NoS 

■O NeC±NoS 




Output 

NoC±NeS 



Fig. 7-17 Ane;ular Error Demodulators. 



lent to a source temperature of 9775° K. Measurement of noise figure 
merely involves the measurement of the noise power required to double 
the output noise power of the receiver under test. 

A precision microwave attenuator is used to control the noise power 
applied to the receiver. The available noise power from the discharge tube 
is equivalent to a noise figure of 15.28 db referred to a temperature of 290°K. 
The noise figure is determined by merely subtracting the attenuation 
required to produce a doubling of the noise power from 15.28 db (corrections 
for spurious signal response are required). 

Several problems arise in this type of measurement. If the noise power 
output of the receiver is allowed to double, it is necessary that the receiver 
be linear at the two output conditions and that the response of the detector 
to noise be known. It is desirable that a 3-db loss be inserted in the receiver 
rather than let the output noise level change. The 3-db loss must be inserted 
at a point in the receiver which is preceded by sufficient gain that noise 
sources following the pad do not contribute to the output. The receiver also 



392 THE RADAR RECEIVER 

must be linear ahead of the pad. Frequently it is not convenient to provide 
an accurate 3-db loss in the receiver. An example is the case where the 
preamplifier and main IF amplifier are contained in a single unit. In such 
cases an arbitrary attenuation may be introduced by means of the manual 
gain control. Measurements are made with two arbitrary output levels; 
it is only necessary that the receiver have a linear transfer characteristic 
to the noise at the selected levels. The method is as follows. 

1. Observe output deflection (d-c voltmeter or milliammeter at the 
second detector) with no additional noise input. Let the deflection 
be di. The noise is incident to Nrec- 

2. Introduce the noise source and adjust the noise power (A^i) applied 
to the receiver to produce a deflection d^. The noise is incident to 

3. Insert attenuation a by means of the manual gain control so that 
the noise A^i produces the deflection di. The noise is incident to 

4. Increase the output from the noise source (A^2) to produce the 
deflection ^2. The noise is incident to (A^2 + Nrec)oi. 

From these observations, the noise figure can be determined from 

^ rec ^^ T ( T OT \ /-'TT'J 

where Ta = 290° K and T\ and T2 are noise temperatures corresponding to 

A^i and N2. 

In making a noise figure measurement with a dispersed signal source, 
difficulty is experienced with spurious responses of the receiver. In broad 
band receivers it is usual to add 3 db to the measured result to account for 
beating at the image frequency. Because of the small available power from 
the noise source it is necessary to couple directly to the antenna terminals 
of the receiver rather than through a directional coupler. As a result the 
noise figure is not usually measured with the transmitter operating in the 
case of airborne radar sets. The measurements are also correct only if the 
noise source has the same impedance as the antenna. 

Sensitivity. With the transmitter operating, additional noise may 
appear which will degrade the performance. This is particularly the case 
with high-PRFdoppler radar receiving systems. To determine the perform- 
ance in detecting and tracking small signals a sensitivity measurement is 
generally made; this is a measure of the least signal input capable of causing 
an output signal having desired characteristics. 

In the case of a radar display it is a simple matter to determine the signal 
power required to obtain a minimum discernible signal. The signal is 



7-20] PROBLEMS IN MEASUREMENT OF RECEIVER CHARACTERISTICS 393 

obtained from a standard signal generator which can provide the same 
modulation characteristics as the radar target. In a noncoherent radar the 
sensitivity is measured at various ranges. At minimum range the sensitivity- 
is usually reduced owing to the attenuation characteristics of the TR tube. 
It is sometimes convenient to define the sensitivity of a radar by an A scope 
measurement. In these cases a ^angeniiai signal measurement is made. For 
a tangential signal the signal-to-noise ratio is approximately +4 db. 

Measurement of the sensitivity of a tracking receiver requires that the 
transfer function of the loop be determined at various input signal power 
levels. The minimum signal power required to produce the full dynamic 
tracking capability of the loop is determined. Measurement involves the 
insertion of a fixed power level RF signal having the modulation character- 
istics of the radar signal, and measurement of the transfer function of the 
particular tracking loop for this fixed input signal level. More detail on the 
means for measuring the transfer functions of the regulatory and tracking 
systems will appear in the following two chapters. 

In making sensitivity measurements, accuracy is sometimes limited by 
signal leakage from the standard signal generator. Frequently it is neces- 
sary to put additional shielding around the generator, and connect a second 
precision attenuator in the line between signal generator and receiver. 

When measuring the sensitivity of very narrow band receivers such as are 
employed for doppler radar applications, it is usual to modulate the STALO 
(stable local oscillator) signal to obtain a signal source of adequate fre- 
quency stability that will remain within the narrow predetection filters. 
If the long-term stability of the STALO is reasonably good, a standard 
signal source which is crystal controlled may be used, provided the pre- 
detection bandwidth is not less than about 10~* times the RF signal input 
frequency. 



G. S. AXELBY • D. J. HEALEY III 
D. D. HOWARD • R. S. RAVEN • C. F. WHITE 



CHAPTER 8 

REGULATORY CIRCUITS* 

8-1 THE NEED FOR REGULATORY CIRCUITS 

To determine the bearing, range, and velocity of a target with high 
accuracy, three basic conditions must be fulfilled by the radar receiver and 
its associated data processing system: (1) the desired target intelligence 
components of the received signal must be faithfully reproduced at the 
output of the receiver; (2) undesired input signals which tend to reduce the 
S /N ratio of the desired target intelligence must be suppressed; (3) sources 
of noise internal to the radar must be minimized. 

The desired target intelligence appears as amplitude, phase, and fre- 
quency modulations of the received signals. The target information is 
extracted by taking a cross product between the received signal and a 
reference signal and filtering the resultant signal to remove extraneous cross 
products (see Paragraph 1-5). In a practical radar receiver, there are 
several potentially troublesome sources of degradation in these processes. 

An optimum demodulation process depends upon the accuracy with 
which the receiver can be tuned to the incoming signal. Various environ- 
mental and electrical factors will cause receiver tuning to vary or drift as a 
function of time. Receiver tuning control or automatic frequency control 
(AFC) is therefore required to reduce the effects of such variations. 

Receiver components must be operated under such conditions that the 
linear dynamic range of the receiver is very limited. Unless some form of 
automatic gain control (AGC) is utilized, signal distortion will take place in 
the receiver. For example, saturation effects will tend to erase amplitude 
modulation on the received signal; this in turn will cause poor tracking or 
loss-of-track in a conically scanning system. 

A large number of vacuum tubes must be employed in the receiver to 
amplify the noise level to the desired output level. Variations in the tube 
characteristics occur when the voltages supplied to the tubes vary. The 
desired output signals are then modulated with the undesired fluctuations 
of the power supply voltages. Thus electronic power regulation is required. 

♦Paragraphs 8-1 and 8-3 through 8-13, and 8-21 are by D. J. Healey III. Paragraph 8-2 is 
by D. D. Howard and C. F. White. Paragraphs 8-14 through 8-20 are by C. F. White and 
R. S. Raven. Paragraphs 8-22 through 8-34 are by G. S. Axelby. 

394 



8-2] p:xternal, internal noise inputs to radar system 395 

Motions of the aircraft carrying the radar set can modulate the incoming 
signal and cause loss or degradation of the target signals. Automatic space 
stabilization systems are often required to cope with this problem. 

Finally, the measurement problem is complicated by externally and 
internally generated noise. The origins of such noise and the effects of the 
noise upon range- and angle-tracking accuracies are described in the next 
paragraph. This discussion is particularly important to the subsequent 
discussion of AGC in this chapter and the angle and range tracking as 
discussed in Chapter 9. 

The remainder of this chapter will deal with the basic considerations 
governing the preliminary design of the AFC, AGC, and space stabilization 
loops. The problem of electronic power regulation is not discussed in detail 
since this is largely a matter of good electronic design practice, a topic 
beyond the scope of this volume. 

8-2 EXTERNAL AND INTERNAL NOISE INPUTS TO THE 
RADAR SYSTEM 1 

Paragraph 4-7 presented some of the basic measurements of target noise 
characteristics. This paragraph will define the noise sources in a form more 
immediately useful to the closed-loop control designer to illustrate the 
means for utilizing the measured information for design purposes. 

External Noise Inputs. Variations in the external input to the radar 
system fall into two basic categories, i.e. frequency components associated 
with motion along the target flight path and other frequency components 
normally referred to as noise. Noise includes propagation path anomalies 
and atmospheric noise (sferics) as well as noise caused by the complex 
nature of the target, random motion, and reflectivity. The emphasis here 
is on the noise associated with the target motion and reflectivity variations 
that lead to tracking errors. The various components of external radar 
noise may be defined as follows: 

Range noise, with an rms value of o-^, is defined as deviation of the range 
information content in the received echo with respect to some reference 
point on the target. The reference point may be chosen as the long-time 
average of the range information. Range noise is independent of the target 
range since its source is pulse shape distortion caused by variations in the 
vector summation of energy reflected from target surface elements. 

Amplitude noise , with an rms value of o-a^p, is defined as the pulse-to-pulse 
variation in echo amplitude caused by the vector summation of the echoes 
from the individual elements of the target. Amplitude noise, since it is 

iSee J. H. Dunn, D. D. Howard, and A. M. King, "Phenomena of Scintillation Noise in 
Radar Tracking Systems," Proc. IRE, May 1959. 



396 REGULATORY CIRCUITS 

interpreted by the radar as amplitude modulation of the mean signal level, 
is independent of range if a good automatic gain control (AGC) system is 
used. 

Angle noise, with an rms value of (Xang-, is defined as the variation in the 
apparent angle of arrival of the echo from a target relative to the line-of- 
sight to the center of reflectivity of the target. Angle noise is a function of 
the spacing of surface elements producing echoes, and the relative am- 
plitude and phase of these echoes. Since angle noise is a function of the 
linear dimensions of the target, a variation inversely proportional with 
range results as long as the target subtended angle is small compared with 
the beamwidth of the antenna. At times, incident to angle noise, the 
direction indicated by the apparent angle of arrival of the target echo may 
fall outside the target extremes. 

Bright spot wander noise, with an rms value of o-6s,„, is defined as the 
variations in the center of reflectivity of the target relative to a selected 
physical reference point on the target. The summation of angle noise plus 
bright spot wander noise is the variation in the apparent angle of arrival 
of the echo from a target relative to the selected physical reference point on 
the target. Bright spot wander noise is a function of the relative spacing of 
target reflecting elements and the amplitude of echoes from these elements. 
Like angle noise, bright spot wander noise (in angular units) varies inversely 
with range. However, the peak excursions of the center of reflectivity of the 
target cannot extend beyond the target limits. 

Internal Noise Inputs. In addition to the primary function of 
location and tracking of targets in space, radar outputs to computers utilize 
rates of change of the basic position information. Tracking smoothness 
and accuracy depend upon the manner in which the external inputs are 
processed by the radar system. Internal radar noise components may be 
categorized as follows. 

Receiver noise, with an rms value of o-rec, is defined as the variations in the 
radar tracking arising from thermal noise generated in the receiver and any 
spurious hum pickup. Receiver noise is inversely proportional to the signal- 
to-noise ratio in the receiver, and since the signal power varies inversely 
as the fourth power of the range to the target (excluding propagation 
anomalies), this effect is directly proportional to the fourth power of range. 

Servo noise, with an rms value of aser, is defined as the variations in the 
radar tracking axis caused by backlash and compliance in the gears, shafts, 
and structures of the antenna. The magnitude of servo noise is essentially 
independent of the target and is thus independent of the range. 

Tracking Noise Definitions. An optimum radar system design can 
result only from proper consideration of the nature of all the external and 
internal noise sources. One principal objective of tracking system design 



8-2] 



EXTERNAL, INTERNAL NOISE INPUTS TO RADAR SYSTEM 



397 



may be taken as minimization of tracking noise, which may be categorized 
as follows: 

Range tracking noise, with an rms value of art, is defined as the closed-loop 
tracking variations of the measured target range relative to the range to a 
fixed point on the target. Range tracking noise includes effects of the 
complex nature of the target and receiver and range servo system noise. 
Systematic range tracking errors arising from flight-path input information 
are excluded from art. 

Angle tracking noise, with rms value of aat, is defined as the closed-loop 
tracking variations of the measured target angular position relative to a 
fixed point on the target. Angle tracking noise includes effects of the 
complex nature of the target and receiver and angle servo noise. Systematic 
angle tracking errors arising from flight-path input information are excluded 
from aat. 



Range Tracking Noise. The general shape of the dispersion versus 
range for the various noise factors entering into range tracking is shown in 
Fig. 8-1. Since the various noise factors are uncorrelated, the total output 



Overall \ 


Max. 
>Joise — ^ 


Rcvr. Gam , 






Range Noise-^ 


J~ 


.. 




Servo N 


oise— ^ 

/ 


/^Rcvr. 
/ Noise 



10 100 

RELATIVE RANGE 



1000 



Fig. 8-1 Range Noise Dispersion Factors. 



noise amplitude (shown by the heavy line) representing range dispersion in 
a given tracking system is found by summing the noise components in a 
root-mean-square manner. To use the diagram of Fig. 8-1 for prediction of 
system performance, at least one point on each characteristic must be 
determined by measurements. In the case of external range noise, the 
following facts are known: 

1. Fire-control radar range information contains noise resulting from the 
finite size of practical targets. 

2. The total rms range noise, ar (in yards), may be predicted from a 
knowledge of target size and shape. Measurements made with a split-video 
error detector on a variety of single and multiple targets show an average 



398 REGULATORY CIRCUITS 

rms value- of ar = 0.8 times the estimated radius of gyration of the reflec- 
tivity distribution of the target about its center of reflectivity. These 
results relate to average noise power. By nature, wide fluctuations from 
sample to sample may be expected with the actual value dependent upon 
sample time. Examples of spectral power distributions are shown later. 

3. The range noise power spectra for a variety of aircraft targets in 
normal flight show that the significant power is below 10 cps and, in general, 
one-half the range noise power lies below 1 cps. The frequency components 
of range noise are a function of rates of target motion in yaw, pitch, and roll 
and are influenced by air turbulence, angle of view, maneuvering of the 
target, and the target type. 

4. The influence on the noise values incident to the specific type of range 
tracking system employed has not been extensively investigated, but the 
values shown are believed to be typical for fire-control design purposes 
(assuming good system engineering and performance). 

The measured spectral range noise power distributions for an SNB twin- 
engine aircraft, for two SNB aircraft, and for a PB4Y patrol bomber are 
shown in Fig. 8-2. The curves represent mean values while the upper and 
lower maximum excursions from the mean are shown by the arrowed lines. 
The analysis was based upon 80-sec samples with the indicated mean value 
for (Tr taken over the number of runs shown. The broad frequency range of 
the radar range input noise power clearly emphasizes the requirement of 
range tracking bandwidth minimization consistent with tracking error 
specifications. 

Angle Tracking Noise. The general shape of the dispersion versus 
range for the various noise factors entering into angle tracking is shown in 
Fig. 8-3. The various noise components shown are uncorrected. The 
rms total output noise for conical scanning or sequential lobing radar is 
greater^'^ than for monopulse^ (simultaneous lobe comparison) radars 
because of the high-frequency amplitude noise at the lobing frequency. For 
prediction of system performance, at least one point on each characteristic 
must be determined by measurement. In the case of external angle noise, • 
the following facts have been established. 

1. Amplitude noise is an amplitude modulation of the echo caused by the 
vector summation of echoes from the complex multielement reflecting 

2D. D. Howard and B. L. Lewis, Tracking Radar External Range Noise Measurements and 
Analysis, NRL Report 4602, August 31, 1955. 

3J. E. Meade, A. E. Hastings, and H. L. Gerwin, Noise in Tracking Radars, NRL Report 
3759, 15 November 1950. 

■•J. E. Meade, A. E. Hastings, and H. L. Gerwin, Noise in Tracking Radars, Part II: Dis- 
tribution Functions and Further Power Spectra, NRL Report 3929, 16 January 1952. 

5R. M. Page, "Monopulse Radar," paper presented at the 1957 Institute of Radio Engineers 
Convention, IRE Convention Record, Part 8, Communications and Microwaves, p. 132. 



-2] EXTERNAL, INTERNAL NOISE INPUTS TO RADAR SYSTEM 399 



g- 3 



Tail 
View 








Target 
View 


No. 
Runs 


Aver, 
(yd)- 


^ 


Side 
View^ 


Tail 
Side 
Nose 


9 
7 
16 


3.2 
2.3 
2.0- 


- 




k 


J 


. ^Nose 

" (View 




_ 




1 


^ 


1 


\ 

^ 


^ 




{ , 



(a) Three Views of a 
Single SNB Aircraft 



2 3 4 5 6 7 
FREQUENCY (cps) 



8 9 10 




(b) Two SNB Aircraft 
in Formation 



2 3 4 5 6 7 
FREQUENCY (cps) 



14 

s;i2 

^5lO 



1^ 
uj 4 
CO 



"\ Tail 


Target 


No. 


Aver. 

(Tr 


\ View 


View 


Runs 


(yd) 


Tail 


4 


4.85 


v 


Side 


5 


2.43- 




Nosel 8 


2.15 


-Tail\ 




" 


ViewA 




_ 


=^ \ 


Side View 




^ 


^^^:=?^:^-r--^ , , 


1 



(c) Three Views of a 
Single PB4Y Aircraft 



1 



234 56789 10 
FREQUEN^"' (cps) 



Fig. 8-2 Range Noise Power Spectral Distributions. 




10 100 

RELATIVE RANGE 



1000 



Fig. 8-3 Angle Noise Dispersion Factors. 



400 REGULATORY CIRCUITS 

surfaces of the target. The frequency components of amplitude noise 
causing angle tracking noise lie in two widely separated bands. 

A low-frequency region of noise extending from zero to approximately 10 
cps causes a noise modulation within the closed-loop servo-target combina- 
tion superimposed upon the tracking error caused by flight path input 
information and associated system tracking errors. The low-frequency 
band also influences angle noise as explained later. Removal of the effects 
of low-frequency amplitude noise on angle tracking by suitable AGC design 
is also discussed later. 

A high-frequency region of amplitude noise in the vicinity of the lobing 
frequency (except in monopulse radars) contributes directly to angle 
tracking noise. The angle tracking noise power arising from high-frequency 
amplitude noise is proportional to the square of the beamwidth, the 
fractional amplitude noise power modulation per cps of bandwidth, and 
the angle tracking servo bandwidth.^ The principal sources of target- 
generated high-frequency amplitude noise are propeller (power plant) 
modulation and structural vibrations of the target surface elements. 

2. Angle noise is the variation in the apparent angle of arrival of the echo 
from the target relative to the line of sight to the center of reflectivity of the 
target. It is caused by variations in the phase front of the reradiated energy 
from the multielement target. When low-frequency amplitude noise exists 
incident to narrowband or slow AGC, the angle noise power (in suitable 
units) equals one-half the square of the radius of gyration of the target 
reflectivity distribution.^ When low-frequency amplitude noise is removed 
by wideband or fast AGC, the angle noise power is approximately doubled 
with practical AGC circuitry. 

3. Bright spot wander noise results from changes in the center of target 
reflectivity principally caused by a redistribution of the significant target 
reflecting surfaces; it does not depend upon the relative phases of the echoes 
from the individual surface elements. The frequency components of bright 
spot wander noise lie almost entirely in a low-frequency band since it is 
associated with major aspect changes of the target. Because bright spot 
wander noise is an uncorrelated component of target-generated angle 
tracking noise, a complete elimination of angle noise (as defined above) does 
not reduce angle tracking noise to zero. 

Examples of the spectral energy distribution of amplitude noise were 
shown in Fig. 4-23.^ In the spectra illustrated, the analytical method 
excluded low-frequency results below 30-40 cps. 

^Ibid., p. 3. 

■'B. L. Lewis, A. J. Stecca, and D. D. Howard, The Effect of an Automatic Gain Control on 
the Tracking Performance of a Monopulse Radar, NRL Report 4796, 31 July 1956. 

^Source: D. D. Howard, from measurements made at the Naval Research Laboratory, 
Washint^ton, D. C. 



8-3] AUTOMATIC FREQUENCY CONTROL 401 

The effects of the spectral energy distribution of closed-loop angle noise 
and the contributions of low-frequency amplitude noise modulation of 
tracking error caused by flight path input information are discussed in 
Paragraph 8-17. 

8-3 AUTOMATIC FREQUENCY CONTROL 

Automatic frequency control circuits are employed as a means of over- 
coming tuning tolerance and stability problems. The operating frequency 
of the receiver is compared to a reference. An error signal, related to the 
difference between the operating frequency and the reference, is generated. 
The error signal is then applied to the system in such a manner as to reduce 
the difference to an acceptable value. 

The general problem of automatic frequency control may be visualized 
as follows (see Fig. 8-4). 




Fig. 8-4 Automatic Frequency Control in a Pulsed Radar System. 



In the case of radar employing a pulsed oscillator as the transmitter, it is 
required that the receiver be tuned to the transmitter frequency. As 
discussed in Chapter 7, this is done by mixing the incoming signal with a 
local oscillator signal. The resulting intermediate-frequency (IF) output 
then is amplified by bandpass amplifiers designed to operate at a fixed 
intermediate frequency. With such an arrangement, the receiver tuning 
depends upon the ability of the local oscillator to follow variations in the 
transmitted frequency and thereby maintain the difference frequency (IF) 
at the value for which the bandpass amplifiers were designed. The auto- 
matic system employed to accomplish the desired regulation of the IF is 
called an automatic frequency control (AFC). 



402 REGULATORY CIRCUITS 

Automatic frequency control is accomplished by applying the generated 
difference frequency to an error detector whose reference is the desired 
IF frequency. Such an error detector is a frequency discriminator. The 
frequency discriminator provides an output whose magnitude is propor- 
tional to the error and whose polarity indicates whether the IF frequency 
is above or below the reference. Since a variation in either the transmitter 
frequency or the local oscillator frequency produces an error in the IF 
frequency, these variations can be suppressed by suitable control of the local 
oscillator frequency if they do not exceed the bandwidth limitation imposed 
on the feedback control loop by the pulsed data. 

8-4 VARIATION OF TRANSMITTER FREQUENCY WITH 
ENVIRONMENTAL CONDITIONS 

There are two types of frequency instability which result from the 
environment in which the transmitter must operate. There are relatively 
long-term frequency changes which occur incident to the effects of tem- 
perature, vibration, deterioration, and the like; there are also short-term 
frequency changes which are the result of a time-varying load impedance 
connected to the transmitter, and frequency modulation from the heater- 
supply and power-supply noise. 

Since the reference in an AFC for a conventional radar set is compared 
with the difference between the transmitter frequency and the local oscil- 
lator frequency, corresponding variations in the local oscillator frequency 
occurring at the same time as the transmitter frequency variations are also 
important. 

Fig. 8-5 shows typical frequency variation of a magnetron and a klystron 
with ambient temperature. 

Some static and slow frequency differences for typical magnetrons and 
local oscillators are listed in Table 8-1. 



Table 8-1 FREQUENCY DEVIATIONS OF 
TYPICAL MAGNETRONS AND KLYSTRON LOCAL OSCILLATORS 

Maximum Diference 

Environmental Factor Frequency {Mc) 
Scatter of magnetron and 

oscillator frequencies as received from 

manufacturer =^ 50 

Warmup of radar set =^ 1-5 

Temperature =^ 15 

Pressure (0 to 50,000 h) altitude - 2.5 
Pushing ( ='=10% line-voltage variation) 



5.0 



Aging; ^ 10 



MAGNETRON PULLING 



403 



20 



-10 



-20 



-30 













1 1 
Temperature-Frequency 
Characteristic of 4J50 
IVIagnetron and \/-270 Klystron 




^Magn 


3tron 






Magne 
Puis 
PRF 


tron 1 

3 Length 0.5 juse 

1200/se 
snt 25 Amp 
)n 

5| 
nator Voltage 300 


c 


~^ 


<. 


^ 






Klystrc 
Mod 
Reso 




" 






^ 


^ 


. V -270 


















^ 


















V 


/ 







-50 



-25 



25 50 75 100 

TEMPERATURE (°C) 



125 



Fig. 8-5 Frequency Stability of a Magnetron and Klystron vs. Ambient Temper- 
ature. 



The response time of the AFC does not have to be very great to correct 
for the frequency changes listed in Table 8-1. It is only required that; the 
controlled frequency can be adjusted over the range. To obtain a wide 
tuning range, control of both the klystron cavity resonator and the reflector 
potential may be employed. In many cases only reflector control is required 
if periodic adjustment to accommodate frequency scatter caused by tube 
replacement and aging is allowed. 

Table 8-1 indicates that when a radar set is first energized it is usual for 
the open-loop frequency error to be rather large. A wide pull in range is 
therefore required. {Pull-in is the process whereby the error in receiver 
tuning frequency existing at the instant of an off-frequency input signal is 
reduced by the AFC operation.) 

8-5 MAGNETRON PULLING 

The single mode equivalent circut of a magnetron is shown in Fig. 8-6. 
The magnetron is considered as a conventional self-excited power oscillator 
with the L-C tank circuit inductively coupled to the output transmission 
line. 

As will be discussed in Paragraph 11-1, loading mismatch can affect both 
the frequency and power output of the magnetron. Transient variations of 
the load admittance occur in scanning antenna-radome configurations. In a 
conical scanning radar, load admittance variations occur with feedhorn 



404 



REGULATORY CIRCUITS 

Ideal Transformer 



Electronic 
Admittance 



R S C_L L 



Load 
Admittance 



Output 
Voltage 



Fig. 8-6 Equivalent Circuit of a Magnetron. 

rotation because of imperfect rotary joints. The nature of tliese transient 
variations governs the time-response requirements for the AFC. 

The Rieke diagram is a fundamental performance characteristic of the 
magnetron which describes the dependence of oscillator power output and 
frequency on the load. A typical Rieke diagram is shown in Fig. 8-7. 




Fig. 8-7 Possible Operating Condition of a 4J50 Magnetron in a Typical Airborne 
Radar Set (Rieke Diagram). 



8-6] STATIC AND DYNAMIC ACCURACY REQUIREMENTS 405 

Although the Rieke diagram specifies the frequency and power output for 
any load, the pulling figure of the magnetron is defined by the total fre- 
quency variation resulting from a load which produces a VSWR of 1.5 when 
it is changed through a phase of 360°. The load corresponding to this 
condition is shown as a circle in Fig. 8-7. The total frequency variation 
caused by pulling might therefore be 13 Mc. A typical conical scanning 
antenna produces relatively small phase variation. The measured phase 
variation of a typical load is plotted as sectors A, B, and C in Fig. 8-7. The 
most unfavorable position for the phase is at Sector A, for which the 
frequency may be pulled a total of approximately 4 Mc. Pulling can also 
result from a wide-angle scanning antenna looking through discontinuities 
in the radome as well as the discontinuities in phase of a rotating feedhorn 
in a conical scanning radar. Sector C is most favorable for elimination of 
transient frequency pulling caused by phase changes; however, power 
output variations are relatively large. The transmitted signal will be 
amplitude-modulated by this effect, and the resultant amplitude mod- 
ulation on the received signals introduces errors in antenna pointing. 
Accordingly, Sector B represents the most favorable alternative from the 
standpoint of low-frequency pulling and minimum amplitude modulation 
of the transmitter. It will be observed, however, that these advantages 
are purchased at the price of lower-than-rated power output. 

In a well-designed antenna-radome combination, rapid phase changes 
with the position of the antenna are not usually severe. In a conical 
scanning radar the greatest pulling effect results from the rotation of the 
feedhorn. The phase may change quite rapidly with feedhorn position, 
and the frequency of pulling is therefore high. A typical system has been 
observed to generate two phase rotations in one revolution of the feedhorn 
with some abrupt changes. The frequency variation is thus predominantly 
at frequencies greater than twice the lobing frequency of the antenna. 

Table 8-2 gives some typical pulling characteristics of a conical scanning 
system. 

Table 8-2 PULLING CHARACTERISTICS OF A 
TYPICAL CONICAL SCANNING SYSTEM 

Frequency of FM Peak Deviation {Mc) 

f\ (lobina frequency) 0.5 

2/i ^ 1.7 

3/i 0.25 

4/i 0.25 

8-6 STATIC AND DYNAMIC ACCURACY REQUIREMENTS 

Tuning errors in the radar receiver degrade the output signal-to-noise 
ratio of the radar. Maximum range performance of the radar is thus a 



406 



REGULATORY CIRCUITS 



function of the tuning accuracy of the receiver. To obtain the maximum 
performance from a radar system the IF bandwidth must be matched to 
both the received pulse and the tuning error of the receiver. Fig. 8-8 shows 
the loss in signal-to-noise ratio (sensitivity) as a function of a static tuning 
error in the receiver. The video bandwidth is considered to be very much 
larger than the IF bandwidth. 



s 1 
o 
fe 2 

UJ 
Q- 

^ 3 



CO 5 
o 



=v 


^Br=2.0 


Br=3.0 




\- 


~^\ 


/_^ 


-Br=4.0 




\ \ 


\"^ 


\ 


Br^l.O" 


\ ^ 


\ \ 


\ 




\ 


\ 


\ \ 




\ 


V 


\ 



0.5 1.0 1.5 2.0 

AfT (TUNING ERROR x PULSE LENGTH) 



Fig. 8-8 Loss in Sensitivity with Tuning Error. After R. P. Scott, Proc. IRE 
(Feb. 1948) p. 185. 



To obtain maximum system performance a compromise between AFC 
performance and bandwidth must be made to provide maximum SjN. 
Maximum performance is obtained with Bt ==1.0 when the tuning error 
is negligible. In a MOPA (master oscillator power amplifier) system it is 
feasible to realize this condition. In the case of a separate-pulsed trans- 
mitting oscillator and a continuous-wave klystron local oscillator, some 
allowance for the static error must be made. The reference is a frequency 
discriminator and associated envelope rectifiers. The reference must be 
tuned to the center frequency of the receiver IF and must be stable within 
environmental conditions encountered. With an IF frequency of 30 Mc 
the reference can be made to be accurate within ±50 kc of the IF. When 
a master oscillator, operating continuously, is employed in the radar to 
obtain the transmitted signal, much greater accuracy can be obtained by 
the use of crystal control of the IF frequency. The static accuracy that can 
be realized in the usual radar employing two oscillators is dependent on the 
accuracy of the reference and the amount of zero frequency gain it is 
practical to employ in the AFC feedback loop. 

The dynamic error characteristic must also be considered in choosing the 
bandwidth. Since the AFC of a pulsed radar set is a sampled-data feedback 
control device, the error reduction that can be achieved is dependent on the 
sampling rate. In practice, the effective bandwidth of a continuous propor- 
tional error AFC is limited to about one-twentieth the PRF. Tuning errors 



CONTINUOUS-CORRECTION AFC 



407 



caused by variations in frequency greater than this effective bandwidth are 
not significantly reduced by the AFC, and their effect on the signal energy 
at the output of the receiver must be considered. The bandwidth of the 
receiver IF is then selected so as to maximize the output S /N in the 
presence of the error resulting from those tuning errors which cannot be 
removed by the AFC. 

In some applications, transmitter pulling is often the determining factor 
in the accuracy of receiver tuning. In these cases it is not essential that the 
static error be extremely small, and a simplification of the AFC can be 
realized by the use of limit-activated correction (see Paragraph 8-8). 

In tracking applications a further requirement exists that tuning error in 
the receiver must not produce an error in tracking exceeding a specified 
amount. Two types of errors arise from the tuning error. The frequency 
error is converted to amplitude modulation by the IF characteristic. The 
additional amplitude modulation from this source produces errors in the 
direction signal of a conical scanning radar. Distortion of the pulse shape 
also occurs and may produce errors in the measurement of range. 

8-7 CONTINUOUS-CORRECTION AFC 

Continuous-correction AFC constitutes a type of closed-loop operation 
in which the error continuously tends to be minimized. The residual error 
is a function of the loop gain. 

A block diagram of a continuous AFC is shown in Fig. 8-9. The input to 
the AFC is the frequency difference between the transmitting oscillator and 



Transmitting 




W(f) 




W(f)-f2 




Oscillator 












" 












Balanced 
Mixer 


AFC IF 
Amplifier 




Frequency 
Discriminator 
















/ 












Receiver 

Local 
Oscillator 


h 




















C 


Filter 

G(S) 


e 





















Fig. 8-9 Continuous AFC. 



the receiver local oscillator. This frequency is measured by the frequency 
discriminator which is the error detector. A voltage e proportional to the 
difference between the input frequency and the crossover frequency of the 
discriminator is applied to the local oscillator through a filter G(s). The 
output of the filter is a control C which adjusts the receiver local oscillator 
to minimize e. C may be a mechanical or an electrical output or a com- 



408 



REGULATORY CIRCUITS 



bination of the two, depending on the nature of the control mechanism of 
the oscillator. 

The characteristics of an AFC in stabilizing the receiver tuning when the 
transmitter frequency or local-oscillator frequency changes can be expressed 
as 



£/. 



£/rf 



(8-1) 



1 + KG(s) 

where fei is the frequency error in the receiver IF frequency 

fdi is the frequency error that would result without the AFC 

K is the d-c or zero frequency gain of the AFC. (K = discrimi- 
nator sensitivity X d-c gain of the filter X modulation sensi- 
tivity of the oscillator) 

G{s) is the normalized transfer function of the filter. 

Fig. 8-10 shows the control characteristics of a typical klystron oscillator. 
Referring to Table 8-1, the largest tuning error that might exist in a typical 



5.0 



^ 4.0 



^ 3.0 
> 

m 2.0 

z 

LlJ 

^ 1.0 







/ 


y 


-N 


\ 






Reflector 
Sensitivity- 


/ 






\ 

XT 

\ 


—Power 
Output 




/ 


A 








\ 
\ 






/ 


\\ 






/ 


/ \ 










\ 


^ 


.^^ 


See 


ext 



















_ 
-Ih- 



-30 -20 -10 10 20 30 
CHANGE IN FREQUENCY (Mc) 

Fig. 8-10 Local Oscillator Characteri.stics. 



system is ±15 Mc incident to temperature environment, provided that an 
initial adjustment is made on the AFC whenever a tube is changed. Fig. 
8-10 shows that the power output of the local oscillator will vary about 
1.5 db and the modulation sensitivity will change by a factor of about 2 
(or 6 db) for such a frequency variation. Variations in the power output of 
the oscillator affect both receiver sensitivity and signal-handling capability. 
However, the effect of a 1.5-db change is negligible. In cases where such a 
change cannot be tolerated or in which the frequency variation between the 
uncontrolled oscillators exceeds ±15 Mc, an additional feedback loop is 
sometimes employed. This auxiliary loop measures the oscillator power 
output and adjusts the klystron cavity for maximum output. 



8-7] 



CONTINUOUS-CORRECTION AFC 



409 



Variation in modulation sensitivity limits the bandwidth that can be 
employed in the AFC loop. The loop must be designed to have adequate 
stability with both the highest and lowest gain values. To minimize 
variations in oscillator sensitivity with tuning, the amplitude of the local 
oscillator signal is usually made smaller than the amplitude of the sample 
of the transmitter frequency at the AFC mixer. The IF voltage applied to 
the discriminator is then proportional to the amplitude of the local oscillator 
output. The discriminator sensitivity therefore is reduced as the oscillator 
modulation sensitivity increases. Since loop gain involves the product of 
the discriminator sensitivity and the oscillator modulation sensitivity, the 
variation in loop gain that would exist if the IF voltage were maintained 
constant is reduced. This is shown as an effective modulation sensitivity 
reduction by the broken curve of Fig. 8-10. 

Pull-in and hold-in performance of the AFC are determined by super- 
imposing the oscillator control curves on the discriminator characteristic. 
Fig. 8-11 shows such a curve. Pull-in has been defined previously. Hold-in 



Oscillator 
Control Curves 



Oscillator 
Control Curves 




Fig. 8-11 Discriminator Characteristics. 



is the maximum frequency interval over which AFC control is effective. 
Referring to Fig. 8-11, an error /^i in receiver tuning will occur for a 
frequency deviation Ja\ of the transmitter frequency. The frequency 
deviation /d2 corresponds to the hold-in range. An error /« 2 results from 
an oscillator deviation/d2. The pull-in range corresponds tofdz- The tuning 
error can be/^s or/es'; only the point /e3 is stable, however. For all devia- 
tions less than/d3 the tuning error is a stable condition corresponding to the 
intersection of the discriminator curve and the oscillator control curve. 



410 



REGULATORY CIRCUITS 



The discriminator curve of Fig. 8-1 1 is the characteristic appearing at the 
output of G(s) in Fig. 8-9. For high-performance systems G(s) is designed 
with a large zero frequency or d-c gain. Fig. 8-12 shows the resultant discrim- 



.Oscillator Control 
Characteristic 




Typical Characteristic Where 

D-C Amplifier Follows Discriminator 



Fig. 8-12 Discriminator Characteristics. 



inator curve measured at the oscillator. By increasing the d-c gain for a given 
IF characteristic the pull-in range is increased. In a number of systems, 
however, a limited d-c gain follows the IF detectors. To realize a large 
pull-in range a frequency searcli sweep is applied to the oscillator. The 
presence of an IF signal in the AFCIF is employed to remove the sweep. 
A relatively narrow-band IF discriminator can exhibit a large pull-in range 
by this technique. 

In a typical system the control required on the oscillator may be ±50 
volts, but this voltage is usually at some bias level, e.g. —150 volts. To 
obtain maximum performance from a given loop, a d-c voltage is added to 
the output of the IF discriminator so that the control voltage is at —150 
considerable energy in modulation sidebands at the IF frequency. 

The AFC mixer of a pulse radar set is operated as a balanced mixer to 
minimize frequency tuning error caused by discriminator outputs resulting 
from the modulation spectrum of the transmitted signal. With the usual 
IF frequencies employed, narrow pulse-length transmitted signals have 
considerable energy in modulation sidebands at the IF frequency. 

A typical IF discriminator design can provide an output of 2 to 3 volts 
per megacycle with a peak-to-peak separation of the discriminator of 4 to 5 
Mc. The output from the discriminator is in the form of video pulses. If 
these pulses are fed directly into the filter the zero frequency gain required 
from the filter is 



K/eT 



(8-2) 



8-7] CONTINUOUS-CORRECTION AFC 411 

where Eo is the maximum output voltage required 

T is the interpulse period 

fe is the static error due to finite gain of the loop 

r is the pulse length 

K is the discriminator gain. 

For a static error of 50 kc, a pulse length of 1.0 Msec, and an interpulse 
period of 1000 Msec a typical gain required is 250,000. The required gain can 
be reduced by the use of a pulse lengthener following the frequency discrimi- 
nator. With a circuit like that described in Paragraph 7-17 the filter gain is 
reduced to about 250 \i T = Tp and a = 0. A gain of 250 can be provided 
with an operational amplifier. This d-c gain can be reduced if the discrimi- 
nator is designed with higher output or if a video amplifier is employed 
between the pulse lengthener and the discriminator. Although the loop can 
provide a static accuracy of 50 kc, the tuning accuracy also includes the 
inherent accuracy of the reference. With thermionic diodes in the 
discriminator circuit and with stable capacitances and inductors a reference 
accuracy on the order of 50 kc can be achieved for the assumed discrimi- 
nator characteristic. The static accuracy is then expected to be 70 kc, 
provided that means for adjusting the reference initially to the IF frequency 
of the receiver exist. 

The dynamic accuracy will depend on hold-in and pull-in requirements. 
A single lag network is usually employed; i.e., the operational amplifier is 
made an integrator. An error only appears as a signal sampled at the pulse 
repetition frequency; therefore the integrator time constant that can be 
employed depends on the allowable overshoot. At a given time /i the error 
in frequency might be /i. This error is applied as a voltage to the integrator. 
The integrator output will change at a rate determined by the RC and the 
input voltage corresponding to/i. 

An overshoot of about 50 per cent of the initial error is a reasonable 
compromise to obtain good dynamic response. The introduction of a step 
frequency error then results in an output frequency which is 50 per cent of 
the initial error but of opposite sign. The output of the AFC thus oscillates 
about the desired frequency with diminishing error. Inputs incident to 
pulling of the transmitter may occur at the lobing frequency or multiples 
thereof. The error reduction that can be accomplished by the AFC thus 
depends on the ratio of the PRF to the pulling frequency. In typical cases 
there might be 10 to 20 samples during a cycle of the pulling frequency, and 
error reductions on the order of 10 to 1 are attained when 50 per cent 
overshoot is allowed. 



412 REGULATORY CIRCUITS 

8-8 LIMIT-ACTIVATED AFC 

Limit-activated AFC constitutes a type which has a switching action 
whenever the error exceeds a predetermined value. The static error in such 
a system is never zero, but oscillates about the correct value with a constant 
peak error. The static error is determined by the limit level for the error, 
the interpulse period, and the integrator characteristic. 

Essentially the system is the same as that shown in Fig. 8-9 with G(s) = 
K /s. The difference between this AFC and the continuous-correction AFC 
is that the input to G(s) does not produce a change in polarity of the oscil- 
lator control signal until the frequency error exceeds a certain limit L. The 
output from the frequency discriminator is applied to an integrator whose 
output controls the oscillator. A frequency error/ei results in a change of 
the oscillator frequency at a rate Kfei. An error correction signal can only 
be obtained in a shortest time T after a signal occurs which causes the 
oscillator frequency to change. The quantity T is the interpulse period. 
The output frequency therefore oscillates at a frequency which is some 
fractional integer of the pulse repetition frequency. Fig. 8-13 shows the 



-1 



5.0 
2 4.0 

2 3.0 
in 

I 20 

1.0 







-1^ 




Y 


^ 


i\ 


\ ^ Boundary Lines Represent 
\V\ ^>^ Stable Oscillations. 


-l2 - 




<i 


'^ 


iim 


SNX;^^''''^ Shaded Areas are 
^^ Areas of Instability 


O C 

ui Ll 

\ / 


'k 


A 


i«r 


Peak Freq.Error = fe 


y 




w 




^■^- 
















^^1^ 



1.0 2.0 3.0 4.0 5.0 



E.O.F.= Error Oscillation 
Frequency 



Fig. 8-13 Characteristics of a Switched (Limit-Activated) Type Control Loop for 
Constant Input Conditions. 

relationship between the parameters A', T, and L. It will be observed that 
multiple modes of oscillation can occur. These result from the fact that the 
data sample may occur when the frequency error is less than L. The limit 
level in a practical system is about 50 kc minimum. 

With the continuous AFC employing a 100 per cent pulse lengthener the 
static accuracy is independent of the PRF, provided that the response time 
of the discriminator is the same for all pulse lengths. (Normally a pulse- 
length change would be associated with a PRF change.) In the limit- 
activated correction type of AFC, the static accuracy increases as the PRF 
increases. 



8-9] THE INFLUENCES OF LOCAL OSCILLATOR CHARACTERISTICS 413 

The dynamic performance of this type of AFC is determined by K and 
the peak rate of frequency change occurring in the input signal. When the 
peak rate of the input frequency change is less than K, an error reduction 
is obtained and the peak errors are of the same order of magnitude as the 
static error. When the peak rate of the input exceeds K, the error is not 
reduced. When the input to the AFC is predominantly a single frequency, 
the required value of K may be determined by 

K = o^f ■ (8-3) 

where co is the angular frequency at which the signal frequency is changing, 
and / is the peak deviation of the signal frequency. 



8-9 THE INFLUENCES OF LOCAL OSCILLATOR 
CHARACTERISTICS 

A typical electronic tuning characteristic of the local oscillator is shown in 
Fig. 8-10. Note that both the power output and the frequency-modulation 
sensitivity vary with the tuning. A variation in modulation sensitivity 
means that there will be a variation in the loop gain of the AFC. In the case 
of a continuous AFC the change in power can be employed to compensate 
(to some extent) for the variation in modulation sensitivity. However, 
perfect compensation is not attained. As a result, if a continuous AFC is 
designed with only electronic tuning capability and it must accommodate 
dynamic inputs, there is a degradation of the static and dynamic error 
characteristics at the extremes of the tuning range if the overshoot is 
selected to be 50 per cent at the middle of the tuning range. As shown in 
Fig. 8-10, making the local oscillator signal smaller than the transmitter 
signal reduces the variation in modulation sensitivity, but the variation is 
still more than 2 to 1 over the tuning range. The static error therefore 
oscillates when the receiver is tuned at the extremes of its tuning range. 
A smaller overshoot and somewhat poorer dynamic performance must be 
accepted if the tuning range is required to exceed 30 Mc >vith a fixed static 
tuning accuracy. 

In the limit-activated correction AFC there is no benefit from making 
the local oscillator signal smaller than the transmitter signal since the rate 
of correction is independent of the magnitude of the error. (A constant 
rate of control occurs whenever the error is greater than L.) The static 
error oscillates at all times and will vary over the tuning range of the 
receiver in accordance with Fig. 8-13 when the change in modulation 
sensitivity is inserted into the value for K. 

When the system is subjected to severe dynamic error requirements, a 
double loop is sometimes employed using a low-frequency feedback to the 
resonator of the klystron and a high-frequency feedback to the reflector. 



414 REGULATORY CIRCUITS 

A smaller variation in electronic modulation sensitivity is thereby obtained, 
allowing a design of 50 per cent or greater overshoot in the step response of 
the electronic tuning loop and also providing a much greater receiver tuning 
range than can be obtained with electronic tuning alone. 



8-10 RELATION TO RECEIVER IF CHARACTERISTICS 

The dynamic error in the receiver tuning and the received signal IF 
characteristic are related. The signal at the output of the IF amplifier will 
contain pulse amplitude modulation arising from the tuning error of the 
receiver. When the IF response is perfectly symmetrical about the center 
frequency and the static error is negligibly small, the modulation at the 
output will be double the frequency of the frequency modulation of the 
tuning error. 

In conical scanning radars the dominant output incident to such effects is 
thus at two or four times the lobing frequency. The response of the angle 
demodulators to these frequencies is greatly attenuated by the use of 
balanced ring demodulators, and additional noise on the direction signal 
caused by AFC characteristics is then negligible provided that saturation 
is not present. If the modulation is large there is of course a loss in signal- 
to-noise ratio which can be determined from the rms error and Fig. 8-8. 
To minimize the conversion of the tuning error to amplitude modulation 
the nose of the receiver selectivity is made as flat as possible consistent 
with the considerations discussed in Chapter 7. 

8-11 DISCRIMINATOR DESIGN 

There are two types of discriminators employed in pulse AFC — the 
phase discriminator and the stagger-tuned discriminator. The choice 
between the two depends on the details of the control circuitry. A slightly 
higher effective transfer impedance can be realized with the stagger-tuned 
circuit, but symmetry is difficult to maintain. If a video amplifier is 
employed after the discriminator but prior to the integration, then the 
phase discriminator is the more attractive choice. Fig. 8-14 shows a typical 
phase discriminator circuit, and the form of the transfer impedance. 

In designing the discriminator the network elements can be selected so 
that j-j = ^5, J-2 = -^7, s ^ = jg, and ^-4 = ^7. The discriminator response is 
then of the same form as the difference in the envelope response of two 
stagger-tuned one-pole networks. To obtain the maximum sensitivity from 
the discriminator the poles are located so that the two equivalent response 
curves cross at their point of inflection. H is inversely proportional to Ci 
and C2 and these quaatities are minimized to obtain maximum Z,. 



■12] 



INSTANTANEOUS AFC 

iC 



415 



z,= 




l^o(.)|-|gb(s)| 

i(s) 



S(S-S^)(S-Sl)(S-S2) 
\(S-S^)(S-S;)(S-Se)(S-sl)(S-Sj)\ 



s(s-s)(s-s;)(s-s, 



(S-S^)(S-S;)(S-Se)(S-S;)(S-Sy) 



Fig. 8-14 Frequency Discriminator (Phase Type). 



8-12 INSTANTANEOUS AFC 

When the dynamic inputs are so severe that continuous AFC cannot 
satisfactorily reduce the tuning error, the possibility exists of using an 
Instantaneous AFC (I AFC). I AFC is a type in which the error correction 
is completed before the pulse has ended. Extremely wide bandwidths are 
required in the AFC-IF amplifier and discriminator in order that negligible 
time delay may be obtained in these elements. The \FC must include 
a bidirectional pulse lengthener which is required to have a negligible 
decrement; the output of the lengthener is the controlled value during the 
pulse. This type of AFC can potentially provide the best dynamic perform- 
ance in a sampling-type AFC; however, there are some practical limitations. 
The discriminator measures the instantaneous frequency and there is 
negligible lag in the loop, so that if the instantaneous frequency is constant 
during the pulse the tuning error can be reduced to a value dependent on 
the gain of the loop. In most cases there are, however, intrapulse frequency 
changes. In the continuous AFC the average frequency is controlled; in 
the lAFC the controlled frequency depends on the characteristics of the 
discriminator and pulse lengthener. The controlled frequency is different 
from the average when intrapulse frequency variations are large. As a 
result the static error of an lAFC can be larger than that of a continuous 
AFC. The wide bandwidth required in the discriminator limits its output 



416 REGULATORY CIRCUITS 

SO that the reference accuracy is poorer than in a continuous AFC. Another 
lAFC problem is associated with holding the voltage precisely during the 
interpulse period. 

The lAFC circuit is not commonly employed because of its limitations 
and because the dynamic inputs can usually be reduced by proper design of 
the transmitter and associated circuits so that the continuous AFC is 
adequate. 

8-13 PROBLEMS OF FREQUENCY SEARCH AND 
ACQUISITION 

Table 8-1 indicates that in a typical case the transmitter frequency can 
vary over a greater range than can the electronic tuning of a klystron as 
shown by Fig. 8-10. It has been noted that it is also not always feasible to 
utilize a discriminator-IF characteristic which will provide such a wide 
pull-in range. To cope with this situation it is necessary that a mode of 
operation be provided which will allow the AFC to search for the trans- 
mitter frequency, acquire it, and track it. With magnetrons which are 
tunable it is necessary that two loops be provided for the AFC. A slow- 
response loop which controls the cavity resonator by thermal or mechanical 
means is sometimes employed. With fixed-frequency magnetrons, however, 
the electronic tuning range is usually adequate. In these cases, periodic 
adjustments can be made to the klystron resonator to accommodate aging 
or replacement of magnetrons. The slow frequency variations that are then 
encountered are usually well within the electronic tuning capability of the 
local oscillator. It is sometimes more economical to provide frequency 
capture within the electronic tuning range by means of a search sweep of 
the local oscillator than to provide an IF discriminator which has a pull-in 
range equal to the maximum frequency difference between the transmitter 
and local oscillator at the time that the radar set may be energized. 

A typical frequency range over which the local oscillator must search for 
the transmitter frequency is 40 Mc. The speed at which this search can 
occur depends on the bandwidth of the IF discriminator, the interpulse 
period, and the total search range. A typical discriminator might have a 
pull-in range of 10 Mc and 10 pulses required for acquisition. The maximum 
search speed is then equal to 1 jT Mc, where T is the interpulse period. 
Circuits are provided so that the search sweep signal is automatically 
removed when the transmitter frequency has been captured. 

8-14 AUTOMATIC GAIN CONTROL 

Radar targets act to modulate the amplitude of the reflected signals in 
several ways. First of all, range variations can produce variations in the 
received power of more than 100 db. Secondly, amplitude fluctuations 



M4] 



AUTOMATIC GAIN CONTROL 



417 



caused by target motion which were discussed in Paragraph 8-2 can also be 
large. These variations in signal strength can seriously interfere with 
tracking of the target unless steps are taken to protect the receiving system 
from their effects. This is particularly important in angle tracking where 
the signal amplitudes in two offset antenna lobes are compared (either 
sequentially or simultaneously) to generate an angle error. The angle error 
will normally be directly proportional to differences in the received am- 
plitudes in the two lobes, so that signal strength variations common to the 
two lobes must be removed if a usable error signal is to be obtained. 

Regulation of the received signal level is normally accomplished by an 
automatic gain control (AGC) circuit. This is a feedback loop which adjusts 
the receiver gain to maintain the average receiver output at a constant 
reference value. A simplified block diagram of an AGC loop is shown in 
Fig. 8-15. In operation, the gain of the IF amplifier presented to its input 



IF 
Input 





First 
Detector 


e. 




F 


-^ 


Second 
Detector 


— 


Video 
Amplifier 








Amplifier 


r 




/ 








AGC 
e 


Bias 

1 














AGC 
Filter 




■^+, 












L 


nola\ 



Regulated 
Video Out 



Delay Voltage 
(Reference) 



Fig. 8-15 Radar Receiver with AGC Loop. 



voltage Ci is automatically adjusted by an AGC bias Cg. This bias is 
developed as the difference of the video output voltage Co and a reference 
voltage Cd referred to as the AGC delay. The system basically acts to 
maintain the output equal to the delay. The degree to which this is done 
when the input fluctuates is determined by the AGC filter in the feedback 
path. 

An AGC loop is a nonlinear servo in that the feedback signal Cg is not 
linearly combined with the input, but acts to modify the gain with which 
the input is amplified. In a sense, an AGC loop in combination with an 
angle tracking loop is an example of an adaptive servo system in that 
the system gain is automatically adjusted to compensate for externally 
generated variations in the received signal strength. Although basically 
nonlinear, an approximate linear analysis of the system operation for small 
deviations of the input from an average operating level is very useful and 
forms the basis for system design. 



418 REGULATORY CIRCUITS 

In designing an AGC loop, particular attention should be paid to the 
following four areas of performance which are of primary importance: 

1. The steady-state or d-c regulation of the video voltage 

2. Attenuation of amplitude fluctuations 

3. Fidelity with which intelligence is transmitted 

4. AGC loop stability 

The steady-state regulation determines the degree to which the AGC loop 
compensates for slow variations in the average signal level caused by 
changes in target aspect and range. As noted above, such variations can be 
as great as 100 db. The AGC loop is often required to reduce slow variations 
of the output level to only a few decibels. For instance, in an angle tracking 
loop, the loop gain is proportional to the output signal level so that varia- 
tions in this output produce corresponding variations in loop gain. When 
the average output varies by 2 to 1, or 6 db, the angle track loop gain will 
also vary by the same factor, and this may have a serious effect on the 
overall angle track loop stability and performance. 

Fluctuations in the strength of radar echoes reflected from aircraft 
targets have been discussed in Paragraph 4-8, and typical spectra of this 
amplitude noise for two types of aircraft are shown in Figs. 4-23 and 4-24. 
In Fig. 4-23 the amplitude noise spectrum for a propeller-driven aircraft 
illuminated by X-band radiation is shown. Very predominant propeller 
modulation peaks at harmonics of about 60 cps persist to over 300 cps. In 
Fig. 4-24, the amplitude noise spectra generated by a B-45 jet bomber 
illuminated by several wavelengths are shown. With no propeller modu- 
lation, the spectra all fall off within a few cps. In general, it is desirable for 
the AGC loop to remove amplitude noise whose frequency components fall 
within the pass band of the angle tracking loop. Otherwise, modulation of 
an angular lag error by amplitude fluctuations in the receiver output can 
produce excessive angle noise. 

Besides removing noise modulation from the receiver signal, the AGC 
loop must also transmit intelligence modulatibn without appreciable 
distortion. This is a critical problem in systems which employ sequential or 
conical scan lobing to generate angle error signals. For instance, in a 
conically scanned system, the angle error is contained in the amplitude and 
phase of a sinusoidal error signal at the scan frequency which may vary 
because of poor scan rate generator regulation. Generally, the AGC loop 
must be able to transmit this signal with negligible phase shift or change in 
amplitude. 

Since the AGC circuit is a feedback loop, stability questions are im- 
portant and servomechanism design techniques are applicable. These 
techniques are applied to a linear small-signal approximation to the 
nonlinear loop which will be derived in the following paragraph. Adequate 



8-15] 



LINEAR ANALYSIS OF AGC LOOPS 



419 



stability and dynamic response of the AGC loop is often difficult to achieve 
in combination with other requirements on d-c regulation (proportional to 
AGC loop gain) and fidelity of intelligence modulation. For pulsed radars 
where fast AGC action is desired (common in monopulse systems), methods 
for analyzmg sampled-data servos must be used and the AGC loop band- 
width is limited to about half the repetition rate by stability considerations. 

8-15 LINEAR ANALYSIS OF AGC LOOPS 

Design of AGC loops is based upon a first order or linear approximation 
to the nonlinear action of the IF amplifier for small deviations from average 
operating points.^ This approximation is illustrated in Fig. 8-1 6a. The 



Output 



Delay 



Input 


Gif 


egj 




Video 
Amplifier 








Ih 
Amplifier 












Bias 












eg 


G2(s) 
AGC 
Filter 


^ r 


!. + 










" L 


■^' 



(a) 



. ^ 



AGC 
Filter 



Ki=lncremental IF Gain 



Constant 



K2=lncremental AGC Loop Gain 



, e, = Constant 



(b) 



Fig. 8-16 Linear Approximation to AGC Loop. 



upper block diagram in this figure shows the essential components of an 
AGC loop. The nonlinear relation of the IF amplifier gain to the AGC bias 
is indicated by Gi(eg). The system equations have the following forms: 

eo = e,KsG,{e„) (8-4) 

e,= {ea- eo)G,{s). (8-5) 

^B. M. Oliver, "Automatic Volume Control as a Feedback Problem," Proc. IRE, 36, 
466-473 (1948). 



420 



REGULATORY CIRCUITS 



Small deviations in the output can be related to small deviations in d and eg 
in the following manner: 



oei deo 



(8-6) 



= KiAei + K2Aeg. 

The gain factors Ki and K2 represent the incremental gain of the IF 
amplifier to the input and the incremental AGC loop gain, respectively. 
Deviations of the bias will be simply related to output deviations through 
the AGC filter: 

Ae^ = -G2{s)Aeo. (8-7) 

The approximate linear feedback loop corresponding to Equations 8-6 
and 8-7 is pictured in Fig. 8-1 6b. The output-input ratio for this linear 
regulating loop will have the following form : 

Aen 1 



KiAei 1 + K2G2(sy 



(8-8) 



8-16 STATIC REGULATION REQUIREMENTS OF AGC 
LOOPS 

The transfer function represented by Equation 8-8 gives the small-signal 
modulation transmission characteristics of the loop. If the zero frequency 

gain of the AGC filter is assumed 
unity [G2(0)- = 1], the static gain 
around the loop is K2 as is indicated 
in Fig. 8-1 6b. The static regulation 
performance is directly related to the 
loop gain K2. In order to display this 
relation, though, the gain control 
characteristic of the IF amplifier indi- 
cated in Equation 8-4 must be ex- 
amined in detail. Typically, the 
logarithm of the IF amplifier gain is 
approximately a linear function of the 
AGC bias voltage. That is, the slope 
of the gain-bias curve in decibels per 
volt is a constant. Fig. 8-17 shows a 
typical IF amplifier gain control 
Gain characteristic. Such a linear relation 
can be generally expressed in the 
following form: 



AGC BIAS VOLTAGE 

Fig. 8-17 Typical IF Amplifi 
Control Characteristics, 



20 logio Gi = ^ + Beg 



(8-9) 



8-16] STATIC REGULATION REQUIREMENTS OF AGC LOOPS 421 

The constant B in this equation gives the slope of the gain control charac- 
teristic in decibels per AGC bias volt. Differentiating with respect to eg-. 

(201ogio.)i^' = B 
Gi a eg 

(8-10) 

i^' =0.1155. 
Gi deg 

Multiplying and dividing the LHS by eiK:i allows us to express it simply as 
the ratio of the loop gain K2 and the video output <?« by utilizing Equation 
8-4. 

1 ;^n 

0.1155 

(8-11) 
0.1155. 

For the d-c or static case with G^iO) equal to unity, changes in the bias are 
directly proportional to changes in the output. Thus the slope B can be 
expressed as the ratio of the total change in gain to the change in output 
voltage : 

„ gain- change (db) gain change (db) 



1 


X 




CiK^G, 






Co 



^O.max ^fl.min ^o.max ^o.min 



(8-12) 



Substituting Equation 8-12 into Equation 8-11 yields the following expres- 
sion for the AGC loop gain: 

K^ = — fMHf^ X [gain change (db)] (8-13) 

^o.max ^o,min 

It is apparent from this expression that with the linear gain control charac- 
teristic shown in Fig. 8-17, the loop gain will vary somewhat with the video 
output eg. Normally, the video output will be well enough controlled that 
its variation can be neglected and an average value used in Equation 8-13. 
It is possible to compensate for this variation in the loop gain K2 by 
introducing a slight curvature in the gain control characteristic. Generally, 
though, uncontrolled departures from linearity with accompanying uncon- 
trolled variations in the loop gain are a much more important design factor 
to consider. 

To illustrate the use of Equation 8-13, suppose that static input varia- 
tions of 100 db must be regulated by the AGC loop to output variations of 
only ±1 db or between 0.89^<, and 1.122 eo. Substituting these numbers 
into Equation 8-13 yields 



Mmm= 49.5 = 33.4 db. (8-14) 



'^' 1.122 - 0.89 



422 



REGULATORY CIRCUITS 



Thus, with an AGC loop gain of about 50 or 34 db, input variations of 
100 db can be reduced to output variations of only ±1 db. 



8-17 



DYNAMIC REGULATION REQUIREMENTS OF AGC 
LOOPS 



It was previously noted that amplitude noise fluctuations in the receiver 
output will modulate steady-state lag errors in the angle tracking loop 
output and can thus produce excessive angle tracking noise. For this reason 
and also to minimize the possibility of saturation, the AGC loop should be 
designed to remove most of the input amplitude fluctuations, particularly 
those within the pass band of the angle tracking loop. Actually, if there 
were no systematic errors, some slight improvement in the glint noise or 
deviations in angle of arrival could be achieved with no AGC. The reason 
for this is that there is a correlation between large deviations of the apparent 
center of reflection of an aircraft target and deep amplitude fades, since both 
effects are produced by destructive interference of the reflected signals. An 
effective AGC will increase the receiver gain to compensate for fades and 
thus increase the magnitude of the glint deviations. In a practical case, 
this effect is more than balanced by the benefits of removing spurious 
modulation from the error signal. 

Typical results from a simulator study of this problem are shown in 
Fig. 8-18.^" In this case, the target noise spectrum (amplitude and angle) 
had a width of 1 cps while the tracking servo had a similar bandwidth. The 



§;'■ 



1.6 



1.2 



0.8 



0.4 




Fig. 



3L 2L L L 2L 

LAG ERROR (Units of Target Span,/.) 

■18 Effect of AGC on Angle Tracking Noise as a Function of Servo Lag 
Error. 



*"J. H. Dunn and D. D. Howard, "The Effects of Automatic Gain Control Performance on 
the Tracking Accuracy of Monopulse Radar Systems," Proc. IRE 47, 430-435 (1959). 



8-18] AGC TRANSFER CHARACTERISTIC DESIGN CONSIDERATIONS 423 

mean square tracking error is plotted versus the lag error for no AGC, a 
1-cps AGC, and a relatively fast AGC of 12 cps. With small lag errors, 
less noise is produced with no AGC because of the correlation between 
amplitude and glint noise noted above. For typical tracking conditions, 
though, in which appreciable lag errors exist, the tracking noise and no or 
slow AGC greatly exceeds that associated with fast-AGC designs. 

Other factors to be considered in the design of the low-frequency response 
of the AGC loop are the transient recovery time of the receiver from deep 
fades (as great as 60 db) which should be such that the angle error is not 
blanked for longer than the angle tracking loop response time. This can be 
achieved by providing a high enough velocity constant for the AGC loop 
and allowing sufficient dynamic range in the output. 

The amplitude noise spectrum from most aircraft targets falls off with 
frequency approximately as if it were filtered by a single section, low pass, 
RC filter (see Paragraph 4-8). In order that no particular noise frequencies 
be emphasized in the output, it is desirable, although not absolutely 
necessary, that the AGC filter match this spectrum; that is, it should fall off 
with frequency with a — 1 slope in the frequency region covering the angle 
tracking pass band. 

The significant factor in determining the quantitative effects of the AGC 
on received modulations is the transfer characteristic 

AGC transfer characteristic = :; — ; — j^ ^ , . - (8-15) 

1 -(- KiKj2\S) 

The required behavior of this function and the open-loop characteristic 
KiGii^s) will be examined in more detail. 



8-18 AGC TRANSFER CHARACTERISTIC DESIGN 
CONSIDERATIONS 

From the discussion in Paragraphs 8-14 through 8-17 of factors signifi- 
cant to the design of a radar receiver automatic gain control, several basic 
specifications emerge as AGC transfer function desiderata in sequential lobe 
comparison radars, namely: 

1. High gain at low frequencies to provide adequate static regulation. 
Some system specifications contemplate allowing only ±1 db variation 
in the modulated envelope output for a range of input signal levels of 
approximately 100 db. 

2. An initial transfer function slope of zero from dc to as high a frequency 
(approaching the angle tracking bandwidth) as possible. 

3. Gain drop-off with a —1 slope on a db-versus-log frequency plot to 
ensure, in view of the established nature of radar noise, that the receiver 
output shows no noise emphasis at any particular frequency. 



424 REGULATORY CIRCUITS 

4. Adequate AGC bandwidth (fast AGC) to ensure isolation from 
amplitude fluctuation in the received echo and minimization of closed-loop 
angle tracking noise in the practical employment of the radar system 
(particularly including recognition of situations where angle tracking errors 
will exist). Estimates of the half-power frequency of amplitude noise for 
some target tracking problems are as high as co = 10 (a gain of +20 db or 
1-^2^21 = 10 at CO = 20 may be taken as a practical design objective). 

5. Adequate AGC bandwidth to ensure suitable transient response. This 
requirement is another aspect of system demands consistent with item (4) 
above. 

6. The phase of the quantity [1 /(I + K2G2(s))] should not vary exces- 
sively over the range of angle tracking modulation frequencies surrounding 
the scan frequency to limit crosstalk effects between the azimuth and 
elevation angle tracking axes. Phase shifts of up to 5° or 10° can be allowed 
in most systems although some applications may require this phase shift 
to be maintained less than a few degrees. As will be shown in Paragraph 
9-9, with phase shifts of more than 10°, the antenna has a tendency to spiral 
or circle, and with even larger shifts, it will become unstable. This phase 
shift must often be maintained in the face of uncontrolled variations in the 
incremental loop gain K2 of as much as 10 db and uncontrolled variations 
in scanning frequency of up to ±5 per cent. 

7. In order to ensure adequate loop stability and transient response, a 
minimum gain margin of 6 db should be maintained for all possible varia- 
tions of the incremental loop gain. Similarly, a phase margin of from 40° to 
50° should be maintained. 

8. In pulsed systems, it is necessary to provide a minimum gain margin 
of at least 6 db at one-half the repetition rate to ensure stable operation. 
This is particularly important in monopulse systems where rapid AGC 
action is desired. 



8-19 THE MODULATION TRANSMISSION REQUIREMENT 

When large fluctuations in the AGC loop gain are possible or very small 
scanning frequency phase shifts are required, special design techniques must 
be used to maintain the phase of the intelligence signal being transmitted 
through the system. Two such techniques are available. In the first, 
additional high-frequency lag and lead terms are incorporated into the 
AGC filter to provide an open-loop phase shift of 180° at and near the 
lobing frequency. The phase shift of the closed loop is then zero at the 
lobing frequency and insensitive to variations in loop gain and lobing rate. 
A second approach is to attenuate K2G2(s) with a null over the required 
frequency band so that the maximum closed-loop phase </>« will be limited 
to a small value regardless of the phase of KoGiis). 



-20] 



DESIGN OF AN AGC TRANSFER FUNCTION 



425 




Using the latter approach and 
referring to Fig. 8-19, 



tan (prr, 



KiG<i 



1 + K,G, 



K,G,. (8-16) 



Fig. 8-19 The Vector [I + K^Giijo:)]. 



If, for example, it is required that 0^ 
be maintained less than 1.5°, then 
the gain at the lobing frequency- 
must comply with 

K2G2 < tan 1.5° = 0.0262 = -31.9 db. (8-17) 

If it is assumed that the lobing frequency is 50 cps, that its regulation is 
±5 per cent or ±2.5 cps, and that the angle tracking loop bandwith is 
1 cps, the open-loop attenuation should be at least 31.9 db between 46.5 
cps = 292 rad/sec and 53.5 cps = 336 rad/sec in order to maintain the 
phase shift less than 1.5° over the anticipated range of operating conditions. 

8-20 DESIGN OF AN AGC TRANSFER FUNCTION 

As a trial design, a single time constant RC filter is selected for the AGC 
filter. This will provide a — 1 slope, which was previously noted as desir- 
able. The maximum gain is chosen on the basis of the static regulation 
requirement. In the example of Paragraph 8-16, a static gain of 34 db was 
required to regulate input variations of 100 db to ±1 db in the output. 
This requirement is adopted as the loop gain in this example. 

In Paragraph 8-18, it was mentioned that a practical AGC loop design 
for a system tracking aircraft targets should have a loop gain of 20 db 




5000 



10 20 50 100 200 500 1000 2000 

ANGULAR FREQUENCY (rad/sec) 

Fig. 8-20 Trial Design of AGC Open-Loop Transfer Function. 



426 



REGULATORY CIRCUITS 



at o) = 20 in order to provide sufficient attenuation of amplitude noise 
fluctuations. This requirement, in connection with the static gain require- 
ment, fixes the location of the low-frequency corner of the AGC filter at 
4 rad/sec. The AGC loop transfer characteristic thus developed is shown 
in Fig. 8-20. 

In order to maintain the fidelity of transmitted modulation, a network 
will be introduced into the transfer function to provide a null at the lobing 
frequency as described in Paragraph 8-19. It will be supposed that the 
phase shift must be kept less than 1.5° and the lobing frequency, its 
regulation, and the angle tracking bandwidth have the values assumed 
in the example in Paragraph 8-18. 

Fig. 8-21 shows a parallel-T net- 
work which can be used to provide 
the required null.^' The transfer 
function of this network is 



mR 





— W\/ 1 

c 

1/ r 


C/m 

If 




m 


11 ] 
< 

-R < 

< 


I J 


u 

~ m + 1 ^ 
m 





Fig. 8-21 Parallel-T Null Network with 
Symmetry Pattern m. 



Voltage transfer function of 
network 

u'^ 1 



H- + // 



(-^) 



+ 1 



(8-18) 



where u = jcoRC = jco /coc 

CO = angular frequency, rad/sec 
Wc = null location, rad/sec 



m = symmetry parameter determining null sharpness. 

The effect of this network with a null of 50 cps = 314 rad /sec and a value 
of m = 0.54 is shown in Fig. 8-20 in combination with the single time 
constant RC filter. The value of m (Fig. 8-21) was selected to provide a 
total attenuation of 31.9 db (as required in Paragraph 8-19) at 292 rad/sec, 
the lowest possible modulation frequency. The high-frequency gain margin 
should be checked, particularly at one-half the PRF. An inspection of 
Fig, 8-20 shows that a minimum gain margin of 20 db, in comparison with 
the 6-db requirement, is maintained at all high frequencies. 

The closed-loop response of the AGC loop indicates most directly the 
dynamic AGC action in attenuating low-frequency amplitude noise and 
transmitting modulation frequencies. Fig. 8-22 shows the closed-loop 
response corresponding to the trial design illustrated in Fig. 8-20. 

"C. F. White, Tmns/er Characteristics of a Bridged Paratlel-T Network, NRL Report R-3I67, 
27 August 1947. 



-21] 



THE IF AMPLIFIER CONTROL CHARACTERISTIC 



427 



10 

-10 
-20 
-30 
-40 













^, 














/ 




\ 














/ 


/ 


A 


i 












/ 






\ 










y 


^^Amplitude 




y-Phase 






:r^ 


j^ 








\ 









5 10 20 50 100 200 

ANGULAR FREQUENCY (rad/sec) 



120 
100 _ 

<o 
80 H 

Q 
< 



20 



500 1000 



Fig. 8-22 Closed-Loop Response of AGC, Trial Design. 



8-21 THE IF AMPLIFIER CONTROL CHARACTERISTIC 

The AGC loop must maintain the receiver output constant so that the 
loop gain of the various tracking loops is negligibly affected by the large 
variations in input signal power that are encountered. The analysis of 
Paragraph 8-18 is based on maintaining the average value of the receiver 
output constant. In receivers which must recover modulation from a PAM 
signal to obtain an error signal, additional consideration must be given to 
the distortion of the IF signal as it passes through the amplifier. Ideally 
it is desirable that gain control be applied to IF tubes which exhibit square- 
law transfer characteristics. Under these conditions no distortion of the IF 
signal will be apparent at the demodulated output. Usually the radar 
receiver must incorporate tubes having good gain bandwidth products. 
Such tubes are invariably of the sharp cutoff variety and are likely to 
produce distortion of the IF signal with accompanying excessive variations 
in the AGC loop gain if proper precautions are not taken. 

To determine the actual distortion through an IF stage, an accurate 
description of the transfer characteristic is required. In general any 
characteristic may be expressed as a power series in ^g, the grid-cathode 
voltage. If the signal input to the IF stage is a modulated signal ei = 
A sin (xictiX + m cos Wmf) and if an AGC voltage E\ is applied to the number 
one grid of the tube, then a convenient measure of the distortion is the 
change in the effective modulation of the signal at the output of the stage. 
Thus 



1 + 



1^3^2(1 



W) 



a, + £i(2^2 + ^azEi) + la^A^l + ^m'^) 



(8-19) 



where m' is the fractional modulation at the output and <2i, ^2, and a^ are 
the first three coefficients of the power series expansion for the transfer 
characteristic. The distortion of the signal is seen to be a function of the 



428 REGULATORY CIRCUITS 

third derivative of the transconductance. In designing an amplifier it is not 
convenient to obtain an infinite series expressing the transfer characteristic. 
The published transconductance curves can, however, be employed to 
obtain a reasonable estimate of the distortion. The transconductance 
corresponding to an infinitesimal signal applied to the tube at a grid voltage 
El is determined, and then the transconductance at the positive and 
negative peak values of the IF signal is determined. From these three 
values the distortion is computed from 



m' _ 1 
tn 2 



1+^— + ^" 



2g. 



(8-20) 



When yf and Ei are so large that the negative excursion of the signal 
extends beyond cutoff, Equation 8-20 is not sufficiently accurate. However, 
it serves as an estimate of the distortion provided that the tube is cut oflF. 
Sharp cutoff tubes do not always cut off at the voltages indicated by the 
tube characteristics. The tubes are only required to exhibit less than a 
specified maximum value of plate current at cutoff bias. As a result of 
inadequately controlled cutoff characteristics ^m.mrn does not go to zero for 
large Ei and J, and the distortion in an actual amplifier is sometimes 
observed to be much greater than estimated by Equation 8-20. In the 
design of a gain-controlled amplifier employing sharp cutoff tubes the AGC 
voltage applied to the stages is therefore restricted so that the peak negative 
voltage E -\- A does not exceed cutoff. The number of stages in the IF 
amplifier must then greatly exceed the minimum number determined by 
gain, bandwidth, and stability requirements. 

A more suitable arrangement in the radar receiver involves the use of two 
or three remote cutoff pentodes in the early stages of the amplifier. Gain 
reductions of 35 per stage with negligible distortion can be obtained with 
some of the available semiremote cutoff tubes having reasonably good gain 
bandwidth products. Very little or no AGC is then applied to the remaining 
stages of the IF amplifier. 

It is desirable to limit AGC loop gain variations with the input signal 
level in order to maintain loop stability and the required dynamic perform- 
ance. AGC loop gain variations of 2 : 1 or 6 db represent a practical goal 
employed in the design of the IF amplifier. 

As was noted in Paragraph 8-16, the AGC loop gain is proportional to 
the derivative of the logarithm of the IF amplifier gain with respect to the 
AGC bias. The contribution of an individual stage is thus proportional to 
the derivative of the logarithm of the transconductance curve. Unfor- 
tunately, this quantity, like the third derivative of the transconductance 
whose importance was noted above, is not normally specified or controlled 
in tube manufacture, and large variations can occur in the cutoff region. 
The use of a greater number of remote or semiremote cutoff tubes, limiting 



8-22] THE ANGLE MEASUREMENT STABILIZATION PROBLEM 429 

the controlled gain variation to the order of 20 db per stage, will minimize 
these gain variations. When a gain control range in excess of 50 or 60 db 
must be provided, it is frequently necessary to control the first stages of the 
radar receiver. When this is done, control voltage must be provided in a 
manner that causes the least deterioration of signal-to-noise ratio. 

In addition to grid-1 control of the amplifier stages, grid-3 or plate and 
screen control is sometimes employed. Grid-3 control provides minimum 
third-order distortion, but the screen dissipation is generally excessive when 
the tubes are operated with reasonable gain bandwidth factor. As a result 
the best compromise is grid-1 control of remote or semiremote cutoft tubes. 

8-22 THE ANGLE MEASUREMENT STABILIZATION 
PROBLEM 

In airborne radars the measurement of target angular position is compli- 
cated by the angular motion of the airborne platform. This paragraph will 
discuss the general features of this problem and the approaches that are 
employed to solve it. Subsequent paragraphs will show how a specific 
stabilization prpblem — the AI radar search and track stabilization problem 
— might be approached. The techniques and lines-of-reasoning used in 
this example are typical of those which must be employed for the solution 
of any airborne radar stabilization problem. 

The essential features of the problem are illustrated by the simple 
one-dimensonal representation of Fig. 8-23. The space pointing direction 
of the antenna Atl is made up of two components: (1) the angle At of the 
antenna with respect to the aircraft and (2) the space orientation angle of 
the aircraft A a. Thus changes in the orientation of the aircraft — due 
either to maneuvering or disturbances from wind gusts, etc. — will cause 
corresponding changes in the space pointing direction of the antenna. 

From a tactical standpoint, this situation is undesirable. The line of 
sight from the radar to the target is relatively independent of radar aircraft 
orientation (neglecting long-term kinematic effects, it is completely 
independent). Thus, the effect of aircraft platform motion is to degrade 
the radar's ability to measure the target's position in space. 

The term angle stabilization refers to the family of techniques employed 
to isolate the radar measurements from the degrading influences of aircraft 
motions. These techniques fall into two general classes: (1) data stabiliza- 
tion and (2) antenna stabilization. 

Data Stabilization. With this technique, no changes are made to 
the basic control loops illustrated in Fig. 8-23. The effects of aircraft 
motion are compensated in the data-processing system by correcting the 
antenna angle measurements by appropriate functions of the measured 
platform motion. 



430 



REGULATORY CIRCUITS 



Antenna 
Command 


Antenna 






Antenna 
Drive 






Angle 
A, 


V 













A 


rcraft 






Disturbance 






Inputs 


Aircraft 


'• 




^ Orientation 


Aircraft 


Aircraft 


'+ Angle 


Maneuver 


^A 






Commands 



Space Pointing 
Direction 

At, 



Fig. 8-23 Basic Relationships in the Airborne Antenna Drive System. 



This technique is generally applicable to fan-beam AEW systems and 
other similar applications where platform motions cause measurement 
errors but do not cause loss of the target. Data stabilization finds particular 
favor where the antenna structure is so bulky as to preclude any other 
approach. 

Antenna Stabilization. For the vast majority of airborne radar 
applications — missile seekers, AI radars, side-looking radars, infrared 
systems — stabilization of the antenna itself usually is required. 

The basic objectives of such a stabilization system may be derived as 
follows. 

From Fig. 8-23, the space pointing direction of the antenna may be 
expressed : 

^TL = Ga X (antenna command) -\- Gd X (aircraft disturbance inputs) 
+ Gm X (maneuver commands) (8-21) 

where Ga, Gd, and Gm are the transfer functions of the antenna drive and 
the aircraft. 

From a tactical standpoint, the desired relationship is 

^TL = Ga X (antenna command) = j^tl desired. (8-22) 

Thus, the stabilization system must have two primary objectives: 

1. It must provide control loops which reduce the effective couplings 
(Gd and Gm) between aircraft and antenna motion. (The required amount 
of reduction is a function of the expected tactical use requirements). 

2. It must provide control means for driving the antenna to the desired 
space pointing direction. 

Antenna Stabilization During Search. During the search phase of 
radar operation, the problem is to maintain surveillance of a predetermined 
volume of space despite the perturbations caused by platform motion. 
During this phase, target data are not used for control of the antenna; 
rather the antenna is driven by open-loop command data to sweep out the 



-22] 



THE ANGLE MEASUREMENT STABILIZATION PROBLEM 



431 



desired space volume. The antenna is commanded to move in a direction 
opposite to that of the aircraft. The general means for solving the stabiliza- 
tion problem in this phase are shown in Fig. 8-24. The antenna is driven by 



A 


^ (Stabilization Feedback) 


Air 


:raft Disturbar 
Inputs 


ce 










\ 


Antenna 
Drive Ga 


i"n* 


Aircraft 


Aircraft 


J 


a/9"a 


) 


Maneuver 






'An 
/ Ant. Pointing 
\ Direction 




Commands 



An.c 



Antenna Command 




TIME-^ SPACE REF 

Fig. 8-24 Basic Search Stabilization System: Single Axis. 

a generated command function y^TL.c as shown. A feedback signal Aa 
provides stabilization by subtracting the aircraft orientation angle from 
the command angle. Thus, we may write 

^TL = Ga{/iTL,c — ^ a) + ^'i A 

(8-23) 

AtL = GaATL,c + (1 - Ga)/lA- 

If Ga is essentially unity over the frequency range o^ Atl,c and A a 

Aa = Atl,c (8-24) 

which is the desired result. 

The critical elements of such a system are seen to be: 

1. The accuracy of the angular reference which provides the feedback 
signal. 

2. The closed-loop gain and frequency response of the antenna drive 
which must be sufficient to follow the input commands Atl,c and the 
stabilization feedback signals. Generally, the dynamic response require- 
ments imposed by the command function are the most severe. 

Additional complications are introduced by the more practical problem 
of stabilization in two or three axes. While the basic principles remain the 
same, the problem geometry will involve somewhat complex angular 



432 REGULATORY CIRCUITS 

transformations. These will be discussed later in the example of a detailed 
search stabilization design for an AI radar. 

Tracking Stabilization. During tracking, information from the 
target can be employed to position the antenna. As described in Chapter 6, 
various techniques such as conical scan or monopulse can be used to create 
an error signal which indicates the amount of error between the line-or-sight 
and the antenna pointing direction. Generally, however, the stabilization 
provided by the radar angle tracking control is not sufficient: a faster, 
tighter inner stabilization-control loop must be used to provide the neces- 
sary isolation from aircraft motions. 

A basic tracking radar stabilization system is shown in Fig. 8-25. ^^ Xhe 
outer control loop represents the generation of the automatic angle tracking 

Aircraft Disturbance 
Inputs 



Antenna 
Drive 

G2 



An 



G,h- 



Aircraft 

Maneuvering 

Inputs 



^''-[^g,g,\g,g^P'\i + 



Gi Go+ Go G 



Fig. 8-25 Basic Tracking Radar Stabilization System. Jls = sight target line; 
Et = angular error signal; Jtl = stabilization feedback; G3 = rate-measuring 

device (gyro). 



error signal. The design considerations for such control loops are covered 
in Chapter 9. Stabilization is provided by measuring the space angular rate, 
Atl, of the antenna and feeding this signal into the antenna drive as shown 
in Fig. 8-25. This arrangement yields the interrelations among antenna 
pointing direction line-or-sight inputs and aircraft motions, shown. 

If the gain of the control loops, the products of G1G2 and G2G3, are much 
greater than unity for frequencies greater than those inherent in the line- 
of-sight angle Jls and the aircraft space angle Ja, the equation in Fig. 8-25, 



Atl 
becomes 



r GjG 



+ G2G, 



^h.s + 



1 + G1G2 + G2G; 



•^TL 



(gttg;) '^''' 



l^Variations of this system are discussed 



GolG, + G3) 

Paragraph 8-31. 



.^.., 



^..1 (8-25) 



(8-26) 



8-24] AIRCRAFT MOTIONS 433 

Over the low-frequency range of^LS, iGs] « |Gi|, and the equation becomes 

Jtl = Jls + ~ (8-27) 

Since G1G2 » 1 in the frequency region of interest, Jtl ~ Als as desired. 
Of course, these relationships hold only if the loop gains are high and the 
control loops are stable. These are conflicting requirements which are 
discussed in the following paragraphs where the design of the stabilization 
loops is covered in detail. 

8-23 AI RADAR ANGLE STABILIZATION 

The primary function of the AI radar control system is to detect a target 
and to provide tracking information to the interceptor fire-control system 
about the target's relative position and motion. 

The degree of space stabilization that must be provided depends on 
{a) the magnitude of the interceptor space motions during an attack and 
{b) the accuracy with which the target position and rates must be known. 
These topics are considered in more detail in subsequent sections. 

8-24 AIRCRAFT MOTIONS 

The first step in the design of the stabilization system is to obtain a 
description of the aircraft angular^^ motions that will occur in the detection 
and tracking phases of interceptor operation. The basic angular motions 
are the roll, pitch, and yaw of the aircraft. The origins of these motions 
may be outlined as follows. First of all, the aircraft must maneuver in 
accordance with the vectoring commands or the fire-control system 
commands. Superimposed on these desired maneuvering motions are the 
oscillatory motions resulting from lightly damped aircraft motions which 
are excited by the control actions and the tendency of the human pilot 
(or autopilot) to overcorrect an error. Finally angular motions will be 
excited by disturbances such as wind gusts and release of armament or 
other stores {interference motions). 

For purposes of preliminary design of the stabilization loops, these 
motions may be described in several ways: 

1. By the maximum expected roll, pitch, and yaw angles and angular 
rates and derivatives. These data can be estimated from attack-course 
studies and knowledge of aircraft operation. 

2. By the time-response characteristics of the aircraft in yaw, pitch, and 
roll incident to impulse inputs. This information can be derived from 
equations which describe aircraft dynamics. 

i^Linear motion is not considered here since stabilization control loops are pricipally con- 
cerned with angular motion. Linear motion is considered in Paragraph 9-18 in systematic 
errors. 



434 REGULATORY CIRCUITS 

3. By the frequency-response characteristics of the aircraft in yaw, pitch, 
and roll. This information can also be derived from the equations which 
describe aircraft dynamics. 

4. By an actual time response made of angular aircraft motions as an 
ideal attack course is flown. This assumes that an actual aircraft is 
available or that it can be simulated on an analogue computer and "flown" 
realistically with an autopilot or human pilot. An ideal, simplified radar 
and antenna tracking system may be assumed, but noise and approximate 
error filtering should be included in the simulation. The time responses 
may be used as follows: 

{a) A Fourier analysis of the time responses may be made. This may 
be made by conventional methods, but it usually is not as useful in 
design synthesis as the other techniques are. 

{b) Segments of the time response may be represented by sinusoids 
or parts of sinusoids of various amplitudes and frequencies. These data 
are particularly useful in the synthesis of antenna stabilization control 
loop frequency responses. 

5. By a statistical description of the aircraft motion. This is usually not 
available, and the effort required to obtain the power density spectrum is 
considerable. However, motions due to gust disturbances are better 
described statistically, as discussed in following sections. 

The most useful descriptions of aircraft motion for preliminary stabiliza- 
tions considerations are given by methods (1), (3), and (4). Typical 
numerical values of modern interceptor aircraft motion described by these 
methods are given in the following pages. 

Maximum Disturbances Incident to Aircraft Maneuvers. The 

controlled interceptor motions, as limited by the aerodynamic charac- 
teristics of the interceptor, which affect the angle tracking system design, 
are roll and roll rate, yaw and yaw rate, and pitch and pitch rate. Estimates 
of a typical interceptor's capabilities are: 

(a) ROLL 

Roll angle: +180°, -180° 

Roll rate: 80° /sec to 90° /sec 

(b) YAW 

Oscillation frequency: 3 to 6 rad/sec 
Sideslip angle: 1-2° 

(c) PITCH 

Pitch angle: +180°, -30° 

Pitch rate : 20° /sec up to 40° /sec 

Pitch frequency: 2-13 rad/sec 

Pitch oscillation angle: Pitch rate /pitch frequency = 1.5° to 10° 



-24] 



AIRCRAFT MOTIONS 



435 



The values are typical of maxima that may be encountered. Actually, 
the kinematics of most attack courses do not require maneuvers of this 
magnitude; for example, the lead collision type of attack described in 
Chapter 2 theoretically requires no maneuvering at all once the initial error 
has been corrected. Despite this fact, however, the aircraft will experience 
relatively large angular rates during an attack because of lightly damped 
oscillatory modes in the aircraft response and the marginal stability which 
characterizes the pilot-aircraft steering loop. 

The data of Fig. 8-26, which were obtained from a typical simulation pro- 
gram, illustrate the principle. In this simulation, a human pilot attempted 



Wfi'irnmn 



1(111111111111 



M M\\\U\\\\\\\\\\\\\\\ 



^ 




IIIIIIIIIIIIIIW 



^m 



Roll Angle 



m 



MilA \\\\\\\M\\\\\\\\W^ 



20 
TIME - 



Fig. 8-26 Typical Simulation Results, Showing Aircraft Motions During an 

Attack. 



to fly an attack course using information from a display which presented 
steering error and aircraft roll and pitch angles. The steering error signal 
was contaminated by radar tracking noise. Both the steering error and the 
noise were passed through a 0.5-sec filter prior to display. 



436 REGULATORY CIRCUITS 

Under these conditions a major portion of the aircraft motions took place 
at several relatively well defined frequencies. Rolling motions predomi- 
nated; these took place at frequencies between 0.5 and 3.0 rad/sec, with 
maximum rolling rate amplitudes in the range of to 20 deg/sec. Also 
evident is a yawing oscillation at a frequency of 0.5 rad /sec, and a maximum 
yawing rate amplitude of 1.2 °/sec, and a small pitching oscillation at a 
frequency of 6 rad/sec and a maximum pitching amplitude of 2° /sec. 

Sinusoidal Representation of Disturbances. This method is more 
useful in determining the stabilization control loop specifications. Specifi- 
cally, this information may be obtained from actual time responses of a 
simulated aircraft on an analog computer as was shown in the preceding 
discussion. Portions of the time responses may be approximated by sine 
waves, and the amplitudes and frequencies of the sine waves can be recorded 
for various aircraft motions from several different courses. 

To study the effect of aircraft motion on tracking-line stabilization, the 
aircraft motions are converted into motion with respect to the axes of the 
antenna gimbals. Usually, the antenna has two gimbals. ^^ The azimuth 
gimbal allows the antenna to rotate about an axis parallel to the aircraft's 
vertical axis; the elevation gimbal permits the antenna to nod up or down. 
The basic angle and angular rate relationships for such a two-gimbal 
system are shown in Fig. 8-27. 

Aircraft 
Fore -and -Aft 



Dire 



action 




Aircraft Roll, Pitch, Yaw Rates 
l^= Azimuth Gimbal Angle 
dp = Elevation Gimbal Angle 



Transformations 

Antenna Rates Due to Aircraft Angular Rates: 

Azimuth coa = oJ;< cos 6^ sin 8,+ co^ sin d^ sin d, + co, cos e, 

Elevation w^ = - co^ sin 0^ + o>y cos Q^ 

Fig. 8-27 Angle and Angular Rate Relationships for a Two-Gimbal Radar 

Antenna. 



When the antenna tracking lead angle is large, the aircraft rolling 
motions appear as azimuth and elevation disturbances as is demonstrated 
by the cox terms in the transformation relationships in Fig. 8-27. This fact 

"In some missile applications it is necessary to provide a third gimbal to space-stabilize the 
antenna in roll. This is not considered here. 



8-24] AIRCRAFT MOTIONS 437 

is important because the rolling rates can be quite large relative to the 
yawing and pitching rates (see Fig. 8-26). 

When the sinusoids representing aircraft motions are transformed into 
equivalent motions in antenna coordinates, results like those shown in 
Table 8-3 are obtained.^^ These results are typical of what might be 



Table 8-3 ANTENNA DISTURBANCE FROM 
AIRCRAFT MOTION OF FREQUENCY c^d 



A A (Vsec) 


Aa{°) 




'eak to Peak 


Peak to Peak 


oJz) (rad/sec) 


ZS 


33.6 


1.04 


30 


9.55 


3.14 


20 


6.04 


3.31 


20 


4.77 


4.19 


22 


4.48 


4.92 


5 


0.497 


10.5 


7.5 


0.715 


10.5 



obtained for lead angles of 45 deg from a large sample of the type of 
simulation data displayed in Fig. 8-26. It is assumed that the magnitude 
of this motion could disturb either channel of the tracking control loops 
at any time without further attenuation. 

It should be emphasized that in a missile system or in an autopilot- 
controlled aircraft, these motions can be calculated by considering the 
equations of aircraft motion in three dimensions as it follows a prescribed 
course, assuming that the aircraft and autopilot design are known well 
enough to be described mathematically. The resulting data can also be 
approximated by calculating the aircraft frequency response from its design 
equations; this is often done. Although the computations may be simplified 
through the use of matrix notation and block diagram representation, the 
cross-coupling between the control loops is complex and nonlinear because 
of trigonometric functions involved, and it is difficult to interpret except for 
simple cases which are discussed in the next section. The task becomes 
more difficult, if not impossible, when a human pilot is involved because the 
human transfer function is not defined to a usable degree of accuracy or 
with sufficient reliability necessary for realistic results. Therefore, when 
available, analogue simulation is the most propitious method of obtaining 
information about aircraft motion in space with or without autopilot 
control. 

i^In practice many more values should be used because the nature of these disturbances is 
essentially random and a large sample should be made to obtain representative data. Note, 
for example, that a single frequency may have different amplitudes at different times. 



438 REGULATORY CIRCUITS 

Aircraft Transfer Functions. The aircraft cannot maneuver with 
large amplitudes at high frequencies. This can be shown from simplified 
transfer functions of the aircraft relating control surface motion to aircraft 
motion. The transfer functions can be obtained from transforms of the 
differential equations which describe aircraft motion. When simplified, to 
eliminate the short term yawing oscillation term, the transfer function 
relating aircraft heading to control surface position reduces to the form 
K/S(l + Ts).^^ K and T depend upon the particular aircraft charac- 
teristics, and a frequency plot of this function should resemble the plotted 
data described in the preceding paragraph. 

The general form of the space isolation required by the radar antenna 
should be the reciprocal of the aircraft response transfer function. The 
equivalent gain factor of the isolation transfer function ultimately depends 
upon the amount of isolation needed, the equivalent gain of the aircraft, 
pilot or autopilot, course computer, and the error presentation as discussed 
in Paragraph 8-32. 

Gust Disturbances. As an aircraft flies an attack course, it is sub- 
jected to winds and turbulence or velocity fluctuations in the surrounding 
air. Turbulence disturbs the aircraft in a random manner, and its general 
effect is referred to as a gust disturbance. 

Because of their random nature, gust disturbances are best determined 
by measurement and then described by power density spectra. The data 
of typical measurements and the associated normalized power density 
spectra are presented in the following documents: 

{a) An Investigation oj the Power Spectral Density oj Atmospheric Turbu- 
lence by G. C. Clementson, Report No. 6445-T-31, Instrumentation 
Laboratory, M.I.T., May 1951. 

[b) A Statistical Description of Large-Scale Atmospheric Turbulence by 
R. A. Summers, Report No. T-55, Instrumentation Laboratory, 
M.I.T., May 1954. 

The normalized power density spectra may be applied to a specific 
aircraft by scaling both abscissa and ordinate. The effect on the tracking 
loop antenna position and rate may be then found by multiplying the gust 
power density spectrum by the square of the transfer function magnitude 
relating the disturbance to the antenna position rate in the channel corre- 
sponding to the direction that the gust disturbances were measured. The 
square root of the integral of this product is the rms value of the antenna 
motion or rate. For most tactical situations the effect of gust disturbances 
is negligible when compared with other factors and it will not be considered 
further in this text. For high-speed, low-altitude flights, however, gusts 

i^Actually, the transfer function varies in roll, pitch', and yaw. The most useful transfer 
functions are those which transform aircraft motion to antenna motion. 



8-25] STABILIZATION REQUIREMENTS 439 

can sometimes be quite severe and in such cases their effects should be 
studied as part of the systems design. 

8-25 STABILIZATION REQUIREMENTS 

The stabilization requirements vary for the search and track modes. 
In search, a lack of space stabilization would allow the search pattern to 
move with the interceptor from its preassigned space sector, and the desired 
target might not be found. 

Deviations from true space stabilization in search are produced mainly by 

(a) Aircraft angular motions^^ 

(i) Dynamic antenna control loop errors 

(c) Incorrect commands to the control loops 

(d) Inaccuracies in the vertical reference 

The total deviation from true space stabilization that can be allowed is 
related to the loss in target detection probability that it produces. ^^ Usually 
0.25° to 1.5° deviation from the ideal space stabilized pattern can be 
permitted. However, the deviation varies as a function of the search angle 
with respect to the aircraft. 

In the tracking mode, antenna space stabilization is needed for several 
reasons: 

(a) To prevent the course computer from operating on inaccurate 
antenna motions due to interceptor space motion rather than target sight- 
line motion in space. ^^ 

(^) To prevent the antenna beam from drifting off the target because of 
aircraft motion during short periods when the radar signal fades. 

(<:) To prevent system instability caused by coupling of the interceptor 
motion with its commands through resulting antenna space motions. This 
is a form of positive feedback, because as the interceptor moves the antenna 
in space, the antenna motion produces signals used by the computer to 
direct the interceptor farther in the same direction. To prevent instability 
in this positive feedback loop, it is necessary to have the loop gain ^^ always 
less than unity. This is most important in systems using an autopilot. The 
problems involved in providing the necessary isolation in the search and 
track modes are discussed in the paragraphs that follow. 

"Another source of error may be produced, especially in some missile systems, from control 
loop disturbance torques created by an unbalanced antenna undergoing large rotational space 
accelerations. However, a detailed discussion of this is beyond the scope of the text. 

i^Detection probability is discussed in Paragraphs 3-9 to 3-14. 

i^Another source of error may be produced, especially in some missile systems, from control 
loop disturbance torques created by an unbalanced antenna undergoing large linear or rota- 
tional space accelerations. However, a detailed discussion of this is beyond the scope of the 
text. 

2fThis includes antenna motion detectors, filters, computer, pilot or autopilot, aircraft 
transfer functions, and the isolation factor provided by the closed, space stabilized antenna 
control loops (See Paragraph 8-32). 



440 REGULATORY CIRCUITS 

8-26 SEARCH PATTERN STABILIZATION 

The space reference for the search pattern control loops is derived from 
a vertical gyro. The accuracy of the vertical gyro in maintaining a true 
vertical need not be extremely good; available vertical gyros provide 
sufficient accuracy for stabilizing the antenna search pattern during the 
relatively short times that a particular target is being sought. The vertical 
gyro used has two degrees of freedom and is provided with position detec- 
tors that measure the aircraft pitch and roll angles with respect to the 
vertical. Except for some missile applications, yaw angles are generally not 
measured because the aircraft yaw motions are better controlled and the 
search pattern is usually much wider in a horizontal direction (see Para- 
graph 2-26). 

8-27 SEARCH STABILIZATION EQUATIONS 

Because the antenna motion of a two-gimbal antenna does not include 
roll correction directly, the aircraft roll motion must be converted into the 
proper azimuth or elevation commands for the two-antenna control loops. 

The exact transformation of aircraft motions to antenna commands is 
rather complex as shown by the following formulae: 

sin Ea = sin As cos Es sine/) 

— (cos As cos Es sin 6 — sin Es cos 6) cos 4> (8-28) 

sin Aa cos Ea = sin As cos Es cos <^ 

-f (cos As cos Es sin 6 — sin Es cos 0) sin (8-29) 

where Ea is the elevation antenna angle command to move the antenna 
with respect to the interceptor 

Aa is the azimuth antenna angle command to move the antenna 
with respect to the interceptor 

Es is the elevation search angle with respect to space 

As is the azimuth search angle with respect to space 

</> is the aircraft roll angle in space 

d is the aircraft pitch angle in space. 

It is possible to mechanize these equations to within a few ininutes of arc^^ 
by using several resolvers, as shown in Fig. 8-28; but to simplify the 
mechanization, the transformation equations are often simplified. This 
can be done in several different ways. One of the approximations which 

2^This inaccuracy is due to components, principally the resolvers. 



-27] 



SEARCH STABILIZATION EQUATIONS 441 

(cos E5 COS A5 sin B - sin £5 cos d) 







= Aircraft Roll Angle 
: Aircraft Pitch Angle 
£(., Ar = Search Command in Space 



L 1 Resolver for 
I each Angle 



sinE^= sinAj cos £5 sin</)-(cos A5 cos Ej sin 6 - sin £5 cos 6) cos 
sinA^ cosE^=sinA5 cos £5 cos^ + icos A5 cos £5 sin 9 -sin £3 cos 6) sin <^ 

Fig. 8-28 Exact Coordinate Conversion Mechanization for a Two-Gimbal An- 
tenna Search Pattern. 

creates very little significant error (in the command signals for search angles 

within 50 deg) is expressed by the following equations: 

Ea = As sin cf) - (d - Es) cos 4> (8-30) 

Ja = As cos <p - (d - Es) sin (8-31) 

These equations are shown mechanized in Fig. 8-29. 



Elevation 

Antenna 
Position 




Antenna Search Loops 



4> = Aircraft Roll Angle \ ^ ^^^^^^^^ 
e = Aircraft Pitch Angle ) 



A=As cos</)- (d-Es) sin0 
Fig. 8-29 Approximate Coordinate Conversion Mechanization. 



442 REGULATORY CIRCUITS 

8-28 STATIC AND DYNAMIC CONTROL LOOP ERRORS 

Perhaps the largest errors in the search-pattern stabilization are engen- 
dered in the antenna control loops. These errors are reduced to satisfactory 
limits by proper design of the control-loop gains and bandwidths or by 
modification of the pattern command signals. Errors to be considered are: 
(a) Static or steady-state errors due to aircraft motion 
(i>) Static or steady-state errors due to search-pattern velocity 
(c) Dynamic ierrors due to changes in search-pattern command signals 
It may be considered that part of the total allowable search-pattern 
errors may be allotted to the antenna control loops. For example, a 0.35° 
steady-state error may be assumed and it may be divided equally between 
aircraft motion and search-pattern velocity signals; i.e., the allowable error 
contribution of each source is 0.25°.^^ 

To provide a means for translating the error requirements into a design 
specification, a generic form muse be assumed for the search stabilization 
and drive system transfer function. For the example to follow, the assumed 
open-loop transfer function will have the form 

COl < Wo < CO3. lO-->^j 



s{\ + si^^){\ + .syco,,) 



The following analysis will demonstrate how the values of Ki,, wi, 0^2, and 
C03 can be chosen to meet a set of system requirements. 

Aircraft Motion Errors. To reduce the steady-state errors caused by 
aircraft motion to 0.25°, the nature of the aircraft motion must be known. 
For example. Table 8-3 shows the amount of antenna movement that would 
take place at large lead angles if the antenna were not stabilized. The 
search stabilization loop generates position command signals which — if 
computed correctly — are equal and opposite to the disturbance caused by 
aircraft motion. However, the control loops that drive the antenna with 
respect to the aircraft have finite gain and bandpass. Thus, the actual 
position of the antenna will tend to lag the stabilization commands. As 
shown in Fig. 8-30, the amount of lag depends upon the frequency and 
magnitude of the input relative .to the open loop gain of the stabilization 
loop at the input frequency. In order that the lag be kept below 0.25° at 
all input frequencies, the minimum loop gain must be 

error specification 0.25- 

As an example, the input at 1.04 rad/sec is 33.6° peak-to-peak or Xi = 
16.8°. Thus, the required open-loop gain of/= 1.04 is 67.2. Similar 

22Since aircraft motion is independent of the search pattern, the individual errors may be 
added by taking the square root of the sum of the squares. 



443 



Xe ~: 1 Xo 

— *• Go 1 * 



Let: Xe= Loop Error 

Xj- Loop Input Command 

Xo = Loop Output 
In Search Loop, X, Contains Signals to Move Antenna and Stabilize it in Space. 
It is Desired to keep the Loop Error Xe Less Than a Particular Magnitude |Xe| 

Since 

I X; I IX, I 

Go»l 



Xi X, I 
IXJ= W — for 



iGol 



|X/| As a Function of Frequency is Known and |Go| 
As a Function of Frequency is Found as Desired 

Fig. 8-30 Derivation ot Required Loop Transfer Function. 

calculations can be made for other frequencies, resulting in the circled 
points shown in Fig.. 8-31. These points define the 7nini7nu?n open-loop 



Low-Frequency Asymptote 
For Kv=400 



1000 




0.1 



01 1 I I 1 1 m il I I 1 1 m il I I I m ill i i i ii 

1.0 10 100 

wi FREQUENCY (rad/sec) 

Fig. 8-31 Search Loop: Open-Loop Transfer Function. 



444 REGULATORY CIRCUITS 

gain necessary to provide isolation from the expected aircraft angular 



Search-Pattern Velocity Errors. In addition to providing isolation 
from aircraft motion, the steady-state error caused by the antenna scanning 
motion must also be considered. Usually the antenna motion is uniform, 
and in the azimuth or horizontal space direction it is a constant angular 
velocity. The steady-state position error is the velocity divided by. the 
velocity constant. ^^ The constant antenna sweep velocity is determined as 
discussed in Paragraph 5-7 and is usually between 75 and 150°/sec.2* 
Therefore, to keep the error below 0.25° for a 100° /sec search velocity, the 
velocity constant must be 

^^qIP"" = 400 sec-i = K^. (8-34) 

Dynamic Errors Due to Search-Pattern Velocity Changes. Some 
distortion is likely to occur at the ends of the search pattern because 
physical control loops cannot follow the rapid changes in command signal 
which are used to change the antenna velocity and position at the end of a 
horizontal sweep to another horizontal sweep. ^^ The transient at the end of 
the sweep requires a longer time to reach a small steady-state value than 
does the vertical-motion transient because the vertical motion is usually a 
small position step instead of a large velocity reversal. Since it is desired 
to resume the steady-state error in the shortest time after the sweep 
direction is changed, both azimuth and elevation control loops may be 
designed to realize the desired sweep transient. If there were no aircraft 
roll, each control channel would have different characteristics. But because 
a large roll angle transfers much of the horizontal motion to the elevation 
channel, both channels should be designed to have the characteristics of the 
azimuth channel in horizontal flight. 

2^The specification derived in this manner is somewhat pessimistic because the antenna is 
not always at the large lead angle used to obtain the data in Table 8-3. 
241 f the open loop has the form 



i(l +s/wi)il + Vcos) 



K^ is the velocity constant. 

-^The search pattern may have several forms — a horizontal Palmer scan with several 
horizontal sweeps spaced at the beam width and a diagonal return, a horizontal sweep and 
return spaced at the beamwidth, a spiral scan, a circular scan, a vertical scan, or combination 
of these types. 

^^he command changes can be made gradually to eliminate transients, but this usually 
involves a more elaborate signal generator. Even then, the ideal pattern will not necessarily 
be obtained, and more time may be required for the antenna sweep to resume the desired 
constant velocity. 



-28] 



STATIC AND DYNAMIC CONTROL LOOP ERRORS 



445 



Fig. 8-32 shows the nature of the error transient that occurs if the 
direction of the horizontal sweep changes instantaneously. An important 











Approx. Decay Transient 












Approx 
Bound 
Error 












— L*^^ — ^ 






- 












1 \ 


^^ 








Max. Time to Peak 


' V— ^ 


^^ 








(1.5<)c<2.0 ) 


1 /\^^ 








a 




r = Horizontal search sweep velocity 


1/^ \ 






X 




CO2& 0}^ are defined in (b) below 


\ 






f=,0, 


\ 


\-2 Slope "' 












^ 


N 


J 0)^ COj 


_^ ^A ,r 


-^ V,. 


Aircraft 











'"'^^" ,^n ^M 1 •'- 


Motion 






U 


LO 


"' "^^ ^ pa]i;;rv^~^-x-J 




r 


_£J r^ 


J 




(b) 


Position T t 






1 






1 1 

Pattern Aircraft 
Generator Motion 












(9,0) 












e,0 are assumed constant in (a) 
















(c) 









Fig. 8-32 Dynamic Transients, (a) Approximate position error transient for 
linear search sweep, r = horizontal search sweep velocity; 002 and coc are defined 
in design equations, (b) Open-loop transfer function. Design equations (approxi- 
mations usually accurate enough for engineering design purposes): (1) Gain: 



(2) Phase: 



180° + (/);„ 



C02\C0i/ 
TT /tT CoA /tT 0)2 \ CO^/ A 



2/3„ 



57.3 

(3) Minimum Phase atcj^: t~^ = 

2cOm 

NOTE: Substitute ( — ^ )in Equation 2 when solving for coi or a;2- 
(c) The search loop (azimuth channel). 

point is that the peak error of the transient, frequently neglected, can be 
much larger than the steady-state error, and a relatively large time may be 
required for it to settle if aj2 in Fig. 8-32 is small. Actually, there are several 
ways in which the dynamic error can be specified. For example, the time 



446 REGULATORY CIRCUITS 

required for the peak error to decay to within a certain percentage of the 
final value could be specified, a maximum time for the peak error to occur 
could be given, or the percentage distortion of the overall pattern dimen- 
sions which can be tolerated could be specified. Fig. 8-32 indicates approxi- 
mate but useful relationships between the dynamic transient, the open-loop 
transfer function, and the steady-state error that help determine the search 
control-loop design characteristics. 

For example, if the allowable distortion incident to peak dynamic error is 
to be within 10 per cent of an overall pattern sweep of 30°, the peak error 
would be 3°. From Fig. 8-32, the upper bound is about Irl ^c above the 
negative error, and the approximate peak error above zero is 2f/coc — r/i^„ 
= 3 deg; and if K, = 400 sec^^ and r is 100° /sec, 100(2K - 1 /400) = 3 
and coc = 61.6 rad/sec. This is the search-loop bandwidth. If greater 
accuracy is required, a more sophisticated pattern command would be 
necessary with special accelerating and decelerating controls — perhaps 
nonlinear control for maximum effort. This is not usually necessary, how- 
ever, to obtain a relatively constant sweep velocity. Other characteristics 
of the open-loop transfer function are found from stability considerations, 
and an optimum system can be determined directly from the three following 
equations relating the corner frequencies, peak phase margin, and the loop 
gain shown in Fig. 8-32.^^ 

SEARCH LOOP SYNTHESIS 

1. Loop gain equation: 

^(^Yco, = K, = 400 (8-35) 

C0 2\C0l/ 

CO, ^ /C. ^ 400 6.49 (8-36) 

coi coc 61. 6 

2. Phase equation (frequency response peak = 1.3,^* peak phase margin 
</)„ = 50.3° at a frequency co^ = ^c cos 0„, = 0.64coc) : 

(-180° + 50.3°) 



r-(5-„":,) + (!-S) 



57.3 

(8-37) 

_ CO,. ^ 0.69 = ^"^-"'^ + ^' 

CO 3 CO„j CO 3 

where ^m = phase angle of G at co„(. 

2''The derivations of these equations are discussed in the paper, "Synthesis of Feedback 
Control Systems with a Minimum Lead for a Specified Performance," by George S. Axelby 
in IRE Transactions in Automatic Control, PGAC-1, May 1956. 

^The closed-loop frequency response peak A/,, occurs at a frequency co„i = wc cos <^m and 
sin <^m = ^IM ,, in the optimum, minimum lead system. 



-28] STATIC AND DYNAMIC CONTROL LOOP ERRORS 

3. Minimum phase equation : 

— — ^ — ' = — (differentiate Equation 8-37) 

Combining Equations 8-37 and 8-38 

2(c02 — CO]) 2(a)2 — OJl) 



0.69 



and 



0.69 



Wm 0.64cOc 

2co,(a;2/co, - 1) 2a)i(5.49) 



447 
(8-38) 

(8-39) 
(8-40) 



0.64a;. (0.64) (61. 6) 

from which coi = 2.48 and co2 = 16.1 
O^r. = 39.4 

CO. = 61.6 

C0.3 = 114. 

These are the search loop corner frequencies, bandwidth, and maximum 
phase frequency in radians per second. A log magnitude phase diagram or 



Odb 




400 



(ili-) 



s{m*') (ii4+' 



25.0 
39.4 



1.6 wc=61.6 




Fig. 8-33 Nichols Chart of Search Loop. 



448 



REGULATORY CIRCUITS 



Nichols chart is shown in Fig. 8-33.^^ Note that if the corner frequency in 
Fig. 8-32 had been a double corner, C03 would be doubled; if it had been a 
triple-corner frequency or equivalent, cos would be tripled, etc. However, 
Fig. 8-33 would be essentially the same in the crossover frequency region. 
This example illustrates one method of synthesizing the search control loop 
directly with a minimum of the usual cut and try effort. The procedure is 
similar, even if other criteria are used to specify the search loop perform- 
ance. 

8-29 SEARCH LOOP MECHANIZATION 

Actual circuit details of search loop mechanization cannot be discussed in 
general terms because the control loop components vary with specifications, 
with the nature of the power available, and with the antenna size. How- 
ever, a few general considerations can be discussed. 

The basic components needed to mechanize each of the two search loops 
are shown in Fig. 8-34. The coordinate converter needed to correct the 



Loop Input Signal Corrected 
'for Aircraft Space Motion 




-r 



Power 
Amplifier 
Modulator 



Actuator 



I Possible 

I Tachonneter 



■E>- 



Fig. 8-34 General Block Diagram of Search Loop (One Channel). 



input signal for aircraft motion is not shown because it was discussed in 
Paragraph 8-27 and illustrated in Fig. 8-28 or 8-29. The practical problems 
involved in its construction are those of making proper resolver connections 
with correct phasing of a-c signals. For the exact transformation, the 
signals between the six resolvers must pass through isolating amplifiers or 
phase-shifting devices. The loop-actuating signal, or error signal, is usually 
an a-c voltage proportional to the difference between the input signal Xa 
and the controlled antenna position /^a as shown in Fig. 8-34. It is obtained 
in the exact coordinate transformation from the windings of resolvers, which 
are mounted on the antenna. In the approximate transformation, the a-c 
error signal is obtained from the sum of voltages from the vertical gyro roll 

29Note that the calculated maximum phase margin is about 2° greater than the desired 
design value. This discrepancy, conservative but negligible in an actual system, occurs because 
the locus was calculated exactly from the transfer function which was determined from 
approximate equations. 



!-29] 



SEARCH LOOP MECHANIZATION 



449 



resolver and a potentiometer on the antenna. The latter is not shown, but 
it is impHed in the summing symbol in Figs. 8-29 and 8-34. Wire-wound 
potentiometers may be used in each channel, but induction potentiometers 
are preferred (especially if the loop gain is high) to prevent oscillations 
between potentiometer wires. 




Note: a = l Produces 
Simplest Form 



' — ^^^H!;^ 






2 / 1 1 \ 1 

^ ^HRsCz c^rJ'^ C2C1R1/! 



( q^C,+ R3C, + R3C, l R, 

^ y R^ + Rg / (Ri+R2)CiC2Ri(i 



Design Formulas 



Ci= arbitrary '^2~'^iVa~/ 



K = (W4 + C05-a)2j(-^) 




"4 , 



Ifi 



'^^ (ciRj-'CgRj + C3C1R1R4 



e, (s+o:^){s + o^^) 



(b) 



VC3R1 + C3R2 ^ C3R4 c^rJ C1C3R1R2R4 



Cj = arbitrary 



Design Formulas 

C ^1 

3 R2(W4CO3-a;jC02) 



(C04COg-C0iCO2) 

(oj4 + a)5)(c0j + a;2) 



SRC 
>R SRC + 1 



(d) 



Fig. 8-35 RC Compensation Networks. 



450 



REGULATORY CIRCUITS 



As in all feedback control systems, the error signal is used to position the 
antenna in a direction which will decrease the error. Generally, however, 
the control system characteristics shown in Fig. 8-31 do not exist naturally, 
and it is necessary to provide some form of compensation. On small, 
low-power systems this compensation can be provided mechanically by 
adding extra inertia to the antenna inertia with fluid coupling; but with 
large antennas where space is limited and the power is relatively high, 
it is more practical to provide electrical compensation in the form of RC 
networks as shown in Fig. 8-35. Design formulae are given to illustrate how 
the network parameters are related to the corner frequencies. Generally, 
the corner frequencies are chosen so that, in combination with the corner 
frequencies of other equipment in the loop, the desired overall characteristic 
shown in Fig. 8-31 is obtained. This is illustrated in Fig. 8-36. It should be 
noted that networks A and especially B of Fig. 8-35 would be used in series 
with the low-power circuits in the forward path of the search loop, and that 



1.0 



\ 


-1 


Gd Desired Loop 






\ Transfer Function 


=\ ' 


Gi Loop Transfer 


.9 ^ 


^-y ^Function Without 
Vy/ Compensation 


1 


,E i- 


\ \ 




5i 


\ \ 


Lowest Corner Usually Due 


^ o 


-2\ \ 


/to Actuator and Load 


"1^ 




\ 
\ 


/ 


g OJ 






I 


s ^ 






\ 


5^ 






\ Gc Loop Compensating 








\ 


\ _2 / Function 








\ 


\ 1 








-r 


V \ \ 













\y 


|GcHG,HGi| 


) ^ 










\ 




Xi '\\ 




-1 \ 











r \ 


y 


Note: Compensation Function Has Slopes Other \ 


\-3 


Than Zero Where the Asymptotic Slopes of G^andX 


\ 


Go are Different (Compensation May be \ 


\ 


Realized With R-C Networks Shown m Fig. 8-35) ^ 


\ 


1.0 


\ 

\ -3 



Fig. 8-36 Method of Determining Control-Loop Compensating Function. 



8-29] SEARCH LOOP MECHANIZATION 451 

network C, possibly in combination with D, might be used in a tachometer 
feedback path around the antenna, actuator, and power ampHfier. 

To use the RC compensating networks, it is necessary to convert the a-c 
error signal to a d-c signal with a demodulator.^" A peak demodulator is 
often used in the sear.ch loop because it has (1) less high-frequency noise 
and (2) a smaller time constant than an averaging demodulator. After the 
error signal is demodulated and passed through d-c networks and a power 
amplifier, it is applied to the power actuator which moves the antenna. 
Generally, the actuator is a two-phase a-c electric motor or a hydraulic 
actuator controlled with an electrically operated valve. If an a-c motor 
is used, the d-c actuating signal from the compensating network must be 
modulated before it is applied to the motor through a power amplifier. This 
may be done electronically with vacuum tubes or with magnetic amplifiers 
which provide amplification, modulation, and power in one reliable unit. 
However, if a hydraulic actuator with a control valve is used, the d-c 
actuating signal may be used to control the valve. Some signal amplifi- 
cation may be provided with tubes or with transistors. 

There are advantages and disadvantages in both types of antenna drives. 
The electric motor is cheaper and lighter than the hydraulic actuator; it 
does not need oil lines or rotary joints with oil seals; but it does not run as 
smoothly at low speeds, it is much less efficient, and it cannot produce as 
much steady-state torque or velocity as a hydraulic actuator of the same 
size. In addition, gear trains -with troublesome backlash and friction are 
needed with electric drives, whereas they are not used with hydraulic 
actuators.^' 

Regardless of its type, an actuator must be selected which will provide 
the required search-pattern velocities and accelerations. The output power 
of the actuator must be greater than the power required to move the 
antenna inertia along the search-pattern paths in space in the presence of 
antenna friction and unbalance as well as aircraft pitch and roll motion 
which may be directly opposed to the desired antenna space motion. In 
fact, aircraft motion adds considerably to the required actuator torque, 
velocity, and power because the actuators move with respect to the airframe 
to generate a pattern in space rather than with respect to the airframe; thus 
the expected aircraft motion must be combined with the required search- 
pattern velocities and accelerations to determine the actuator performance 
characteristics. Note that the antenna inertia and unbalance in elevation 
may be less than in azimuth and the actuator may be correspondingly 

'"Simple RC networks can be directly approximated with "notch" networks or resonant 
filters for a-c signals; but for airborne systems, this is usually not practical because of the 
accuracy required and because the carrier frequency varies. 

'lit is assumed that the hydraulic actuator discussed here consists of a shaft with a vane 
enclosed in a housing through which oil may be ported to either side of the vane to produce 
a shaft rotation. Of course, 360° rotation is not possible with this type of actuator. 



452 REGULATORY CIRCUITS 

smaller. In practice, however, the actuators often have the same size for 
production economy. Finally, it should be emphasized that the control loop 
and actuators should not be designed to have a performance much greater 
than that required, not only because increased size and weight would be 
involved but also because physical limitations inherent in the gimbals and 
the antenna structures of a given size place an upper practical limit on 
linear design, ^^ and as this limit of performance is approached, the cost and 
complexity of control equipment increases rapidly. Specifically, noise is 
always present in the search control loops, although not to the extent that 
it is in other control loops associated with fire-control systems, and its 
detrimental effects become more of a problem as an attempt is made to 
increase loop performance. In addition, structural resonant frequencies in 
the antenna make it nearly impossible to construct a stable loop with a 
crossover frequency near the resonant frequencies. Thus, there is a practical 
upper limit for the control loop bandwidth which is governed by the 
antenna characteristics. Since the performance of the loop is primarily 
a function of the bandwidth, the performance itself is limited. 

8-30 STABILIZATION DURING TRACK 

As was discussed in Paragraph 8-25, the tracking antenna must be 
stabilized in space to prevent: 

{a) system errors caused by the course computer operating on informa- 
tion from coupling between the interceptor and antenna motions. 

{b) the antenna radar beam from drifting away from the target during 
brief periods of radar signal fading 

(c) instability incident to coupling between the interceptor and antenna 
motions. 

A portion of the required space stabilization is provided by the automatic 
tracking loops discussed in Chapter 9, except during periods when the radar 
signal fades. However, during normal operation, the typical tracking loop 
cannot provide effective isolation above frequencies equal to about one-half 
the track loop bandwidth, or about 3.0 rad/sec for a typical system. On 
the other hand, the interceptor may have appreciable motion at higher 
frequencies as indicated in Table 8-3. To provide the necessary space 
stabilization, a special automatic control loop is designed to move the 
antenna relative to the aircraft in a direction opposite to that of the aircraft 
motion in space. 

To provide space stabilization, the control loop must obtain antenna 
space motion information. This is obtained and converted into useful 
electrical signals with gyroscopes mounted on the antenna or on the aircraft, 

''^It is possible to devise nonlinear control loops which will provide increased performance 
in special cases. 



-31] 



POSSIBLE SYSTEM CONFIGURATIONS 



453 



and these signals are used to move the antenna in a direction opposite to 
undesired space motions. Of course, the correction signals are not exact; 
the gyros sense space velocity only, not the space position actually desired. 
However, a control loop with sufficient accuracy and speed of response can 
reduce antenna space motion to magnitudes much lower than those of the 
interceptor, and the residual space position errors are further reduced by 
the track loop which uses the target sight line as a reference. Details of 
designing these stabilization loops are discussed in the subsequent para- 
graphs. 



8-31 POSSIBLE SYSTEM CONFIGURATIONS 

Several physical configurations are used to mechanize the stabilization 
loops. Some of them are shown in Fig. 8-37.^^ Theoretically, all of them 



Rate Command w + p 



(Volts) 
From Radar 



Rate 

Command 



^ Aircraft Space Angle 
\ An 



Aft 

(Volts) 



^ Rate 
Signals 



Tracking Line Angle 



Gj Amplifier, Actuator, 
Antenna 



G3 Rate Gyro 



(a) 
Gyro Components 



(Current) 
From 
Radar 



Torque Y- 



^'* 


G4 
G5 


1 

, 1 


G2 


+ t 


A,i 


' 


'" 




Torque 


1 









Rate 
Signals 



(b) 



G4Gyro Gimbal and Detector 
G 5 Gyro Wheel 
G 6 Torque Motor 



Three Channels /Two Channels 



Ga Aircraft 

GgAircraft Gyros j^ ^^^ ^i^^ 

Q Coordinate ^ 

" Converter 
GrTachometer 



— *Gc 



1^ 



Rate 
Signals 



•K> 



(c) 



Two Channels 



Fig. 8-37 Three Stabilization Loop Configurations, (a) Rate Gyro, (b) Inte- 
grating Gyro, (c) Aircraft Gyro. 

33Another form is not illustrated. It consists of a gimbaled antenna dish which is rotated at 
high speeds to become an effective gyroscope. It is used in some missile tracking systems 
to provide stabilization. 



454 REGULATORY CIRCUITS 

are equivalent in a mathematical sense, but practically the arrangement of 
the physical equipment is entirely different, and, depending on the appli- 
cation, the mechanization can create discrepancies between the theoretical 
and realized loop performance. In fact, in many control loops the mecha- 
nization characteristics, which are incidental to the primary loop function, 
may prevent the realization of a workable system. Therefore, it is necessary 
to know as much as possible about equipment characteristics before a 
design is finalized. Unfortunately, component characteristics vary widely 
depending on the application, and it is impossible to discuss them in detail. 
However, the general principles of operation are outlined below for three 
different systems. 

Rate Gyro Stabilization Loop. The rate gyroscope^"* is a self- 
contained unit which produces an electrical signal, usually a voltage, 
proportional to a space rate about a particular axis. In the stabilization 
loop design shown in Fig. 8-37a two rate gyros are used, one to measure 
space rates about the elevation axis and the other to measure space rates 
about the azimuth axis of the antenna. Both gyros are mounted on the 
antenna dish^^ where the antenna space rates are measured directly. As 
shown in the figure, the space rate {Atl) of the antenna tracking line 
angle, Atl, is measured by gyro and converted to a voltage proportional 
to Atl- This voltage is compared with an antenna rate command voltage 
Vr and the voltage difference, the rate error Er, is used to control the 
antenna actuators through appropriate amplifiers in a direction which will 
reduce the rate error. 

Although only one channel is shown, two control loops are needed — one 
for each antenna motion — and these loops are interconnected, not only 
through the common space platform and antenna structure but through 
the gyroscopes as well, because space accelerations in one channel influence 
the gyro output voltage in the other channel to a degree which is determined 
by the gyro characteristics and the gyro orientation. However, unless the 
stabilization loop is of very large bandwidth, this effect can be made 
negligible by selecting the proper gyro and by choosing the optimum gyro 
axis mounting.^^ 

As shown in the figure, motion of the aircraft angle, Aa^ is also detected 
by the gyro and converted to a voltage, essentially a rate error, which is 
used to move the antenna at a rate in space opposite to the aircraft space 

34See Locke, Guidance, Chap. 9, pp. 350-353. 

^^Note that the effective azimuth gyro gain in volts per unit of velocity is proportional 
to the cosine of the elevation angle and that it will change during normal operation. For 
computational purposes this is often desired, but it may affect the stability of the azimuth 
stabilization loop if the elevation angle becomes large. 

^A complete discussion of this problem is given in a paper, "Analysis of Gyro Orientation," 
by Arthur Mayer in the Transactions of the Professional Group on Automatic Control, PGAC-6, 
December 1958, p. 93. 



-31] 



POSSIBLE SYSTEM CONFIGURATIONS 



455 



motion. Of course, this principle of operation is common to any stabili- 
zation loop. 

Integrating Gyro Stabilization. The integrating rate gyroscope is 
a self-contained unit which produces an electrical signal, usually a voltage, 
proportional to the time integral of the torques applied around the gyro 
gimbal axis. A schematic representation of the integrating gyro and the 
simplified equations expressing its behavior are shown in Fig. 8-38. Such 




Command 
Torque \/p 
Rate ^^ '^ 

Command 

Torque 
Motor 



Fig. 8-38 Integrating Rate Gyro Relationships. 
do = gimbal angular displacement about output axis relative to gyro case 
Bi — input angular displacement about input axis 

Ig = moment of inertia of gimbal assembly about the output (gimbal) axis 
C = viscous damping constant 
T = Koic = rate command torque 
H = IrOir = rotor angular momentum 
Ir = rotor inertia 
ojr = rotor angular rate 



DIFFERENTIAL EQUATION 






or for K = H 



= Ko 



(coi 



S{1 + T, 



Ko 



for T« 1 



a device may be used as an integral part of the antenna control loop as 
shown in Fig. 8-37b. The rate command, in the form of variable current, 
is used to produce a torque on the gimbal holding the gyro wheel, with an 
electromagnet or torque coil in the gyro. This torque is opposed by the 
gyroscopic torque of the spinning gyro wheel which is induced primarily 
by antenna space motion about a particular axis. If these torques are not 



456 REGULATORY CIRCUITS 

equal, the gyro gimbal moves at a rate determined by the amount of viscous 
damping. The motion is detected with a sensitive transducer, often a 
microsyn, and transformed into a proportional voltage. This voltage is 
amplified and applied to the antenna actuator in a direction to reduce the 
rate error. Aircraft rates are also compensated in the same way. 

A principal difference between the integrating gyro and the rate gyro 
control loop is that the steady-state error in the integrating gyro loop is 
zero" for a constant rate command while the steady-state error in the rate 
gyro loop has a finite value proportional to the rate command and inversely 
proportional to the d-c loop gain. However, this steady-state error in the 
rate loop can be reduced to a negligibly small value without great difficulty. 
The integrating gyro must have compensating networks to make it stable, 
whereas the loop with the rate gyro may not need compensating networks, 
depending on the degree of performance required. Practically, some 
compensation is usual in both forms of the stabilization loop. 

Another difference in performance between stabilization loops using the 
integrating gyro and those using the rate gyro is incident to the saturating 
characteristics of the gyro. Large steady-state rates do not usually saturate 
the integrating gyro, because the gyro gimbal is in the forward path of the 
loop and its motion is proportional to the rate error, which is small. Even 
if it should saturate, it does not change the loop performance appreciably, 
because changes in the forward gain of a feedback control loop do not affect 
the overall loop characteristics or its performance significantly. On the 
other hand, the rate gyro saturation does occur for large, steady rates and 
its measuring and performance characteristics effectively change. This 
is not desirable, nor even allowable in most cases, because the rate gyro 
must measure antenna space rates accurately for the fire-control computer, 
it must provide a stabilizing signal proportional to the antenna rate at all 
times to prevent drift, and it must have the proper transfer function to 
provide track loop stability. This may be particularly serious for systems 
using guns because the large random space rates induced in the antenna 
during periods of gun fire cause rate gyro saturation. Consequently, it is 
necessary to provide rate gyros with linear rate measuring ranges far in 
excess of those needed to measure the aircraft space rates for the fire-control 
computer. Unfortunately, a large, linear measuring range reduces the 
accuracy of the gyro in the important low rate region. However, with 
missiles for armament and jet aircraft as an antenna platform, this problem 
is not serious. 

In recent years, HIG gyros, '^^ fluid damped and hermetically sealed, have 
been commonly used in antenna stabilizing loops. They are extremely 

S'^This neglects the effects of actuator stiction which can make the loop nonlinear and 
produce a small error. 

^Hermetic Integrating Gyros developed at the Massachusetts Institute of Technology. 



8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 457 

accurate, with low drift rates, but they are expensive and require tempera- 
ture-compensating circuits to maintain the preset value of damping torque. 
Generally, this accuracy is not needed to provide antenna stabilization, but 
is needed for the fire-control computer which uses the gyro signals to 
provide accurate information about the line-of-sight motion. 

Aircraft Gyro Stabilization Loops. Another of the many methods 
of providing antenna stabilization is shown in Fig. 8-37c. Because the 
aircraft with a fire control system and tracking antenna frequently has 
autopilot gyroscopes, it is possible to use them to provide the antennas with 
stabilization signals. This reduces the total number of rate gyros needed 
in the system, but other components must be added as shown in the 
simplified figure. ^^ 

The other components are tachometer and position indicators on the 
azimuth and elevation antenna actuators and a converter to change the 
space rate signals from three aircraft coordinates to two antenna coor- 
dinates. Although the converter is not complex, the conversions must be 
made accurately, and the three aircraft gyros should be mounted near the 
antenna base to prevent discrepancies from occurring in the rate measure- 
ments which are different in various parts of the aircraft owing to structural 
flexing. 

From this discussion it is evident that several methods of providing 
antenna stabilization are available. Although the integrating gyro may 
have slight advantages over the other systems, the ultimate choice of a 
system configuration will depend upon the required accuracy, the allowable 
expense, and the permitted complexity of the application. 

8-32 ACCURACY REQUIREMENTS ON THE ANGLE TRACK 
STABILIZATION LOOP 

The stabilization accuracy requirements for the angle track stabilization 
loop are determined from the basic functions of the stabilization loops 
(Paragraphs 8-25 and 8-30) and from different criteria depending on 
whether a pilot or autopilot is used, a lead collision course or pursuit course 
is being flown, or a ballistic or target-seeking missile is being fired. The 
significance of these factors is discussed in the following paragraphs. 

Stabilization Accuracy Required to Reduce Aircraft Rate Signal 
Errors. The magnitude of the error in the antenna rate signal caused by 
own-ship's motion must be reduced to an acceptable value. This value 
depends on the nature of the system. If an autopilot is used, it must be 

^^Note that there are three aircraft gyros which do not measure the desired antenna rates 
In azimuth and elevation. Therefore the information in three channels must be converted to 
two channels using the relative rates and angles of the antenna with respect to aircraft in two 
dimensions. For simplicity the figure depicts a one-channel conversion. 



458 REGULATORY CIRCUITS 

defined from a specified system error, the allowable miss distance, or hit 
probability density.^" Actually, for an autopilot system, the criterion 
involving the magnitude of the rate error due to aircraft motion may be less 
important than a criterion concerning the amount of attenuation needed 
from the stabilization loop to prevent system instability through the 
coupling between the rate commands, autopilot, aircraft, and antenna. 

In the case of manual control of the aircraft flight path, the specification 
of the allowable stabilization error is governed by the following basic 
observations. 

(1) The stabilization error may be considered as a random error in the 
measurement of angular rate. Thus the low-frequency component of the 
stabilization error must be compatible with the specification for angular 
rate measurement accuracy to avoid inaccuracies in the fire-control compu- 
tation. 

(2) The high-frequency components of the stabilization error (greater 
than 1—2 rad/sec) do not affect the fire-control problem directly because 
the aircraft heading cannot change this rapidly. However, they do increase 
the apparent amount of noise on the steering indication, and this does 
degrade the pilot's ability to fly an accurate course (see Paragraph 12-7).'*^ 
This degradation is proportional to the rms contribution of stabilization 
error to the total apparent noise appearing on the pilot's indicator. 

An example using the basic AI radar problem presented in Chapter 2 is 
informative in illustrating how these principles might be applied. The 
applicable specification for the azimuth and elevation channels are (see 
Paragraph 2-27) : 

Allowable random error in rate measurement 0.11° /sec rms 

Allowable magnitude of indicator noise 1° rms 

Allowable filtering 0.5 second. 

For purposes of analysis, we will assume that the contributions of the 
stabilization error should be limited to the following: 

[Low frequency rate measurement] 7^ ^ ^ ^^n / 
, , ... . \ Kl< 0.03 /sec rms 

error due to stabilization errors J 

[Indicator noise due to high 1 ,. ^ ■, ^ro / 
c 1 -r • Al S 1.25 /sec rms 

frequency stabilization errors] 

''"For a detailed discussion of this problem see "Control System Optimization to Achieve 
Maximum Hit or Accuracy Probability Density" by G. S. Axelby, Wescon Record of the IRE, 
1957. 

''^The fact that the stabilization errors are actually correlated with the pilot-induced motions 
does not seem to be important (as it is for autopilot applications where the correlation results 
in degradation of system stability). Thus for analysis of manually flown systems, the rms 
stabilization error must be combined with the rms noise errors from other sources to produce 
an equivalent noise error. 



8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 459 

The problem now is to compute the stabilization loop attenuation Ks as 
a function of frequency needed to achieve these levels of performance. 
Table 8-3 will be used to provide aircraft input data. 

For the low-frequency (less than 2 rad) inputs 

where 0.7 Atl is the rms value of the sinusoidal inputs from Table 8-3, for 
example, at co = 1.04, Jtl = 35/2 = 17.5°/sec and Kg < 0.00245'. 
However, for the high-frequency inputs, the effects of filtering and error 
sensitivity as a function of angular rate must be taken into account to 
ascertain the amount of stabilization attenuation needed. The basic 
expression may be written 

(J.1Atl\S^) {deHc/oATL)^f 

where dene Id At l = sensitivity of computed error signal to angular rate 
inputs 

Gf = rate noise filter characteristic. 

From Table 2-3 the value of the angular rate sensitivity factor at the time 
of firing on a head-on course is 14.2. 

If a 0.5-second filter is used, all sinusoidal signals above 3 rad /sec passing 
through it are attenuated by a factor co//cos, where cos is any signal frequency 
and CO/ is the filter corner frequency equal to 2 rad /sec. 

Therefore, if Kl is assumed to be equal to 1.25° rms for sinusoidal 
frequencies and Gf equal to co//co£) for disturbance frequencies greater than 
3 rad /sec, 

L25 ^ 0.125 

^' ^ (0.7)(^tl)(14.2)(co//co^) ~ Wcoz>)^rL' ^^'^^^ 



However, for sinusoidal motion Atl = Atl^d and 

,, . 0.125 0.0625 
-'^'S S — -J— = —2 — 

Wf/lTL ^TL 



(8-44) 



Using the values for Atl in Table 8-3, the desired Ks is given in the 
following table as a function of frequency. 

A plot of Ks is shown in Fig. 8-39. This is the minimum attenuation that 
must be provided by the stabilization loop if rate signal errors caused by 
aircraft motion are to be maintained below acceptable levels on the pilot's 
indicator. The isolation required for the low-frequency inputs is also 
indicated. The design of the stabilization loop characteristics to achieve 
this attenuation is described in Paragraph 8-33. 



460 REGULATORY CIRCUITS 

Table 8-4 Ks AS A FUNCTION OF FREQUENCY 



^TL 


CO/) 


Ks 


9.55/2 


3.14 


0.013 


6.04/2 


3.31 


0.021 


4.77/2 


4.19 


0.026 


4.48/2 


4.92 


0.028 


0.497/2 


10.5 


0.25 


0.715/2 


10.5 


0.174 



1.0 



0.1 



0.01 





































































































































/ 




















•/ 




















/ 


















































































7 




, , , mITI 










/ 


















'} 


















/ 




















-yv. 




— 


stabilization Loop Must Have ::: 
"Attenuation Cliaracteristic 




























/ 












MM 




' 
















/ 












1 






•7^ 




1 1 1 1 


..,....nJ... 'n ' 'JJIII 






/ 






b 


yMe 

H 


surement Ac 


curac 


Ke 


q't 





10 
cOp (rad/sec) 



100 



Fig. 8-39 Desired Stabilization Loop Attenuation. 



The magnitude of the error in the antenna rate signal may be specified in 
other ways. For example, it may be determined as the allowable error in an 
input to the course computer from a study of overall system error specifi- 
cation. In this case the total rate error may be specified where the total 
rate error includes tracking loop inaccuracy as well as unwanted aircraft 
rates due to aircraft motion. The allowable total error depends on the 
overall accuracy of the system and it may vary from about 1 mil /sec to 
4 mils /sec. In Chapter 9, where tracking loop inaccuracies are considered 
(Fig. 9-5) it is shown that for a total allowable random error of 0.15° /sec 
(2.62 mils /sec /channel), the rms error component due to the aircraft motion 
could be large enough to yield the same stabilization attenuation charac- 
teristic that is shown in Fig. 8-39. 



8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 461 

Stabilization Accuracy Required to Reduce Position Errors Due 
to Aircraft Motion. The radar antenna must not deviate from its 
desired position within a specified angle incident to aircraft motion even if 
the radar signal should not be present for a short time because of temporary- 
loss of target return. In discussing the accuracy required to accomplish 
this, it will be necessary to discuss the position accuracy required and the 
temporary loss of target return which is often referred to as signal fading. 

The position accuracy needed again depends on the type of fire control 
being considered. In some missile systems where a proportional navigation 
course is flown, only antenna rates of the missile seeker are needed to 
compute the proper course; angular information is not used. Therefore, 
the antenna angle need not be maintained very accurately. It is only 
necessary to keep the antenna positioned toward the target within the 
antenna beamwidth, which may vary from 1° to 15° depending on the 
antenna size and radiating frequency. Actually, the stabilization accuracy 
needed to maintain antenna position is an order of magnitude less than that 
needed to provide the necessary attenuation of antenna motion in the rate 
signal; and therefore it is usually not used as a criterion for stabilization 
loop design for this type of application. 

In many fire-control systems, however, the antenna position is used to 
compute the course of the aircraft. From a system analysis, analytical or 
simulated, the random and bias errors in position may be specified. These 
values may range between 2 mils and 10 mils as described in Paragraph 2-27 
for a typical lead collision course. The total random error should be less 
than 2.62 mils (0.15°) rms and the bias error should be less that 9.15 rms 
(0.525°). 

Tracking loop errors contribute to the total position error, but as will be 
shown in Chapter 9, where the tracking loop errors are considered, the 
position error component due to platform motion does not require the 
attenuation needed to reduce the corresponding rate errors to the specified 
values. Usually, the allowed error in antenna position may be greater than 
that allowed in the rate signal because the rate signal, and its errors, are 
multiplied by a large multiplying constant, the error signal to angular rate 
sensitivity factor discussed in Chapter 2. Therefore, the rate error specifi- 
cation is usually more severe, and the required stabilization accuracy may 
be determined from it. 

At long ranges the target may be detected and tracking may begin, but 
the radar return signal may become weak shortly thereafter and even fade 
away completely as the target changes its aspect angle with respect to the 
interceptor.''^ After two or three seconds, if the target signal does not 
reappear, the antenna search mode is reactivated. However, during the 

*2The change in magnitude of the reflected radar signal is discussed in Paragraph 4-3. 



462 



REGULATORY CIRCUITS 



signal fading period, it is necessary to prevent the antenna from being 
moved in space away from the target direction by own-ship's motion. Here 
the stabihzation loop plays a major role in attenuating antenna space 
motion caused by aircraft motion. Usually, the attenuation needed is not 
extremely large, because when the radar error signal (which is also the 
antenna rate command) fades, it is replaced by the thermal noise of the 
radar receiver. As is shown in Fig. 8-40, this noise causes the antenna in a 




Antenna should be 
within the Shaded 
Area 95% of the 
Time After the 
Radar Signal Fades 



Fig. 8-40 Antenna Drift During Radar Signal Fade. 



wh( 



K is per cent modulation/degree tracking error 

r is time between radar pulses 
coc is track loop bandwidth, rad/sec 

/ is amount of integration in track loop 

/ is time in seconds 
Ad is antenna drift angle in degrees 



Typical Values 
50%/deg 
1 X 10-3 sec 
6 rad/sec 
> 10 : 1 
< / < 5 
0< 15° 



typical system to drift away from the desired position by as much as 2.0° 
in 3 seconds. Thus, even if the stabilization loop operated perfectly, the 
noise commands would probably drive the antenna off target. 

Actually, the positional accuracy of the antenna cannot be maintained in 
space without a radar signal to provide a space reference. Sufficient 
accuracy for fire-control computation cannot be obtained, and it is only 
necessary to prevent the antenna from being displaced from the target sight 
line by more than the antenna beamwidth, which may be between 1° and 
15°, to ensure that tracking will be resumed when the target signal strength 
increases again. Therefore, for the system of Fig. 8-40 it would be well to 
limit the duration of the fading period to a time less than 3 seconds. If the 
system is not returned to search at this time, the antenna would probably 
drift from the target sight line and prevent resumption of tracking. An 



8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 463 

alternative to returning to the search mode would be to replace the average 
level of the stabilization loop command, ordinarily the radar error signal, 
with a fixed d-c voltage to eliminate the noise. Unfortunately, this requires 
extra mechanization which may not be worth the additional space and 
complexity. However, during the fading period, it is necessary to provide 
sufficient antenna space stabilization, with the stabilization control loop, 
to keep the positional deviation incident to aircraft disturbance motion an 
order of magnitude lower than the maximum allowable deviation. There- 
fore, for the illustrative case depicted by Fig. 8-40, the peak positional error 
incident to aircraft motion should be less than 0.125°.^^ Although this is not 
as much attenuation as is required to reduce the rate signal errors, it may 
not be so in all cases. Usually, it is necessary to check the amount of 
stabilization needed for each type of error to determine which will require 
the greatest amount of attenuation. 

Stabilization Accuracy Required to Prevent System Instability. 

As mentioned in Paragraphs 8-25 and 8-32, antenna stabilization is needed, 
especially in a system with autopilot control, to prevent system instability 
incident to coupling between the aircraft space commands from the antenna 
gyros and the resulting antenna motion from the commands. Unlike the 
human pilot, the autopilot responds to all signals. Ordinarily, it does not 
learn to reject the signals generated by the coupling between the aircraft 
and antenna motions. Since this is a form of positive feedback (an undesired 
or parasitic loop) it is necessary to keep the magnitude of this loop gain less 
than one-half at all frequencies to ensure that instability will not occur.^* 




Fig. 8-41 Coupling Between the Air Frame and the Antenna. 
G2 = amplifier, actuator, antenna; Gs = rate gyro; Gc = fire-control computer; 
Ga = autopilot and aircraft; Gcl ~ GzGcGa\ Pc = computed projectile angle with 
respect to aircraft; Vr = rate command; Atl = tracking line angle; yffz, = 
tracking line rate; Aa = aircraft space angle; Otl = antenna angle with respect to 

aircraft. 

^^This is an arbitrary figure. It is one-twentieth of the antenna beamwidth to make it 
insignificant with respect to the fading error. 

'•^Actually, the degree of stability of any feedback loop depends on the magnitude and phase 
of the loop characteristic at all frequencies, but if the gain is less than one-half, a stable loop 
is assured regardless of the phase variation with frequency. 



464 REGULATORY CIRCUITS 

A simplified block diagram of this positive feedback loop is shown in 
Fig. 8-41. Since the gains of the autopilot, aircraft, and computer are 
determined by the performance specifications, only one gain element in the 
loop may be readily adjusted. This gain, or attenuation, from Aql to Atl 
in Fig. 8-41 is the required stabilization loop attenuation Ks. If the gain 
of the coupling loop is represented by KsGcl (where Gcl is the overall 
characteristic of the' computer, autopilot, and aircraft) as indicated in 
Fig. 8-41, the stabilization loop attenuation becomes 

l^sl < ,^- (8-45) 



|G, 



CL\ 



Since |Gcz,| is a function of frequency, \Ks\ may be determined as a 
frequency function. 

Unfortunately, however, Gcl is not easily determined analytically.^* It 
varies as a function of antenna elevation and azimuth angles as well as 
the aircraft roll, pitch, and yaw angles in space because it involves the 
conversion of three aircraft coordinates of motion into two antenna coordi- 
nates including the dynamics of coordinated aircraft turns. Although the 
magnitude of \Gcl\ may be estimated or approximated for various operating 
conditions, the maximum value may be most easily and accurately found 
using analog computer simulation. Of course, the magnitude of |i^s| should 
provide adequate attenuation for the largest magnitudes of Gcl that are 
likely to be encountered under normal operating conditions. In high- 
performance autopilot systems, especially missiles, the necessary stabili- 
zation loop accuracy or attenuation may be greater than that derived by 
the criteria for manually controlled aircraft. 



8-33 DYNAMIC STABILITY REQUIREMENTS ON ANGLE 
TRACK STABILIZATION 

In order to provide the necessary antenna isolation from aircraft motion, 
the stabilization loop must have an attenuation characteristic similar to 
that shown in Fig. 8-39. It must be stable, and it must provide a smooth, 
constant antenna rate for a constant voltage input. Although it might be 
desirable to have a transient response with negligible overshoot similar to 
that produced by a critically damped quadratic system, this is not always 
necessary or even possible in some cases. It is more important to achieve 
the required attenuation characteristic even with some sacrifice in stability 
margin. 

''^This function is not as complex in a cruciform missile configurat ion as it is in a conventional 
aircraft which must roil to turn. 



8-33] DYNAMIC STABILITY REQUIREMENTS 465 

Consequently, the peak overshoot to a step input may be as high as 
40 per cent.^^ Actually, this overshoot never appears in the antenna signals 
during actual operation because the input signals are never step functions 
and the output signals are heavily filtered. Therefore, the primary objective 
is to design a stabilization loop with the necessary isolation charactersitics 
and then to obtain the minimum bandwidth possible with a stability margin 
that is not seriously affected by normal component tolerances, gain changes, 
or environmental effects. Essentially, reasonable stability margin in a 
minimum lead system for a stabilization loop would be: phase margin 
greater than 40°, gain margin greater than a factor of 4 (or 12 db). How- 
ever, if precision components are used which are not greatly affected by 
environmental conditions, these margins may be reduced to: phase margin, 
35°; gain margin, 6 db. 

The attenuation characteristic Ks of the stabilization loop with respect 
to aircraft disturbances ^gl may be derived from Fig. 8-37a as 

j^ ] + G2CJ8 

!^^I=^tVi i^ G2G3»1.0" (8-47) 

The magnitude of Ks is determined as a function of frequency from one 
of the criteria in Paragraph 8-32 and plotted as shown in Fig. 8-39. The 
magnitude of 1 /G2G3 must be below these points as shown, and conse- 
quently, the magnitude of G2G3 must be greater than the reciprocal values 
of Ks. The magnitude of ^26*3 corresponding to the values of Ks in Fig. 
8-39 but increased by 175 per cent is shown in Fig. 8-42.*^ This charac- 
teristic need not be extended to frequency regions below 3 rad/sec as 
discussed at the beginning of Paragraph 8-30. As shown in Fig. 8-42, the 
magnitude of G2G3 is 250 at 3 rad/sec to make the actual gain equal that 
desired. This becomes the d-c gain in the rate gyro loop shown in Fig. 8-37a. 
In an integrating gyro loop, shown in Fig. 8-37b, the d-c gain is infinite, but 
the velocity constant would be 750 sec~^ as indicated in Fig. 8-42. 

As in the search loop design, the other characteristics of the open-loop 
transfer function, for a minimum bandwidth, are determined directly from 
stability considerations, using the following equations relating the loop 

■^^This corresponds to a maximum phase margin of about 40° or a frequency response peak 
of about 1.6 in a minimum lead system such as that discussed in Paragraph 8-28. 

4'' Actually this approximation holds very well for IG2G3I ^ 3 in most cases. 

^*The 175 per cent increase in gain is made to counteract changes in the stabilization loop 
gain. One source of gain change is the normal variation in production tolerances, and ±20 
per cent may be allowed for this. The other large gain change is in the gyro (azimuth only) 
because the azimuth gyro gain changes in proportion to the cosine of the antenna elevation 
angle when it is mounted on the antenna dish. Since the antenna may have elevation angles 
of 50°, this causes a 55 per cent gain change, a total of 175 per cent. 



466 



REGULATORY CIRCUITS 



9.55/2 



6.04/2 



4.77/2 



.48/2 



0.497/2 



0.715/2 



3.31 



4.92 



10.5 



0.013 



0.026 



0.028 



0.25 



0.174 



IG2G3I 
O 



134 



67.5 



62.5 



7.00 



250 



100 = 



-Low Frequency Characteristic 
With Intergrating Gyro 
Kv=750 

Transfer Functions 
Rate Gyro 



IG2G3IP 
A 



129 



81.9 



64.6 



60.i 



6.74 



IG2G3I 




10.0 



1 10 100 1( 

cop (rad/sec) 

Fig. 8-42 Stabilization Loop Transfer Ratio Designs. 



corner frequencies of Fig. 8-42 to the maximum phase margin and the gain 
of the loop. 

INTEGRATING GYRO LOOP 



Gain: K = 250 = 



RATE GYRO LOOP 
C0cW2 



Phase: ^^ = - ,,7 =--+ — - - 

3 /.J 2 OJm Wm CO 3 

d^rn C02 — 2a)i 2 
Maximum Phase Equations: - — = 0; ;; — = "~ 



Kv = 750 = 



C0cW2 



COi 



TT COi CO2 2cOm 

2 OJm Wm 'I'S 

2 
C03 



8-33] DYNAMIC STABILITY REQUIREMENTS 467 

For each system: co^ = coc cos </>„, (f)m = 40°, coi = 3 rad/sec. 
Solving the equations: 



29.0 rad/sec 

59.6 rad/sec 

77.8 rad/sec (12.4 cps) 

273.9 rad/sec 



0)2 = 30 A rad/sec 

Wm = 56.6 rad/sec 

coc = 74.0 rad/sec (11.8 cps) 

W3 = 263.8 rad/sec 

The derived corner frequencies are plotted in Fig. 8-42. This establishes 
the required stabilization loop transfer function. As shown in the figure, 
there is little practical difference in bandwidth between the stabilization 
stability margin and the same attenuation characteristic. 

The gain margin was not used to establish the stabilization loop transfer 
function because it is more than adequate unless resonant frequencies in 
the antenna structure are near the crossover frequency of the loop. In fact, 
if a resonant frequency is close to ooz with a damping factor less than 0.1, 
it will reduce the gain margin below 6 db without changing the phase margin 
appreciably. Since the lower frequency characteristics of the loop are 
specified as described in previous sections, it is necessary to specify that the 
antenna resonant frequencies with low damping should be above 003 by 
50 per cent or more. Thus, the lowest resonant frequency should be greater 
than 400 rad/sec or 64 cps."*^ If resonant frequenci^ below this figure 
should exist in the stabilization loop, the loop stability or the isolation 
specifications must be reduced or precision components must be used with 
closer stability margins. 

In practice, ojs need not be a double corner as shown in Fig. 8-42; but it 
should be an equivalent where C03 would be the geometric mean of two 
corner frequencies. Actually C03 should be slightly higher to allow for the 
inevitable but unknown higher corner frequencies which are characteristic 
of all physical equipment and which often reduce the phase margin at the 
crossover frequency. However, if these frequencies are known, or if they 
can be estimated, they can be included easily in the equations which are 
used to calculate the frequencies coc, Wm, C02, and ccs- It is not always possible 
to do this with more rigorous methods. In fact, it should be emphasized 
that the approximate equations give excellent results, but only if all the 
corner frequencies, up to 20 times the loop bandwidth, of the actual 
equipment are included or estimated and used in the calculation. Actually, 

^^Theoretically, resonant frequencies can be canceled by electrical networks, and this may 
be done in practice to increase stability margins. However, in practice, the cancellation 
cannot be made perfectly under all conditions, and the resulting improvement is not very large. 
For more details about resonant frequency effects on control loop performance see "Some 
Loading Effects on Servomechanism Performance," by G. S. Axelby, Aeronautical Electronic 
Digest, 1955, pp. 226-241, National Conference on Aeronautical Electronics, Dayton, Ohio, 
May 1955. 



468 REGULATORY CIRCUITS 

any control loop design, based on a mathematical analysis, should be 
verified with actual equipment tests, and the approximate equations may 
be used to give an extremely useful preliminary design even if the exact 
mathematical description of the actual equipment is not known. 



8-34 STABILIZATION LOOP MECHANIZATION 

The type and size of the antenna actuators are chosen largely from the 
search loop requirements; and to save space and weight, the same antenna 
and actuators are used in the stabilization loop. However, the remaining 
components in the loop are not identical; they are selected according to 
type of configuration desired in Fig. 8-37. Actually each of the three 
designs has different mechanization problems and a complete discussion 
of the details is beyond the scope of this text. However, a few important 
considerations, common to the designs, will be given. 

In Fig. 8-37a two rate gyros provide electrical signals proportional to 
antenna space rates about the azimuth and elevation gimbals. The signal 
from each gyro is compared with an azimuth and elevation reference voltage 
from a track loop amplifier, and the resulting difference voltage (passed 
through appropriate compensating networks) is then used to move the 
antenna in its channel through a power amplifier and actuator. Usually 
the same power amplifiers serve the search loop and the stabilization loop, 
but the compensating networks are different and relays convert the search 
loops to stabilization loops. To prevent transients during this brief tran- 
sition period, it is often necessary to short-circuit the capacitors in the 
compensating networks. The rate gyro is a self-contained unit obtainable 
in various sizes and with various degrees of accuracy. It is important that 
the rate gyros have low threshold voltages and a dynamic range large 
enough to measure the highest space rates due to vibration, antenna 
unbalance, and aircraft motion expected in normal operation. The gyros 
are mounted securely on the antenna after being carefully aligned; but 
although they are on the antenna during the search mode, they are not 
usually used to provide space rate stabilization in the search mode. As 
discussed in Paragraph 8-27, search stabilization must be derived from a 
space position reference. Therefore, the dynamic range of the gyros need 
not be so large as to measure the higher velocities that occur during the 
search loop transients. On the other hand, the gyros must be sturdy enough 
to withstand the extra forces that occur incident to rapid changes in 
direction of motion. Integrating rate gyros such as the HIG gyros with 
gimbals floating in highly viscous fluid provide considerable damping, and 
the problem is not as serious as it is with gyros for which damping is 
provided only by electrical feedback loops. This is a necessary considera- 
tion also in configuration (b); it is not important in configuration (c) 



8-34] STABILIZATION LOOP MECHANIZATION 469 

because the gyros are not on the antenna. In some cases it is necessary to 
cage the gyro gimbals during the search mode to prevent damage and to 
keep the gimbals near neutral position in readiness for the tracking mode 
of operation when it is initiated. 

The location of the gyro mounting involves a problem that is often 
overlooked. Usually the gyros cannot be mounted directly on the actuator 
output shafts because of space limitations. In fact it is desirable to have 
the gyros mounted on the antenna dish, and the necessarily light, compact 
structural members between the actuator drive shaft and the gyro base have 
compliance, mass, and resonant frequencies which limit stabilization loop 
performance. Usually there is more than one resonant frequency which 
can affect the performance of the control loop; although electrical networks 
may be used to partially cancel the detrimental effects of these frequencies, 
they are extremely complex if more than one resonant frequency is to be 
attenuated. The most economical method of avoiding this difficulty is to 
design the antenna with the resonant frequencies ten times the loop band- 
width or to limit the bandwidth and performance of the stabilization loop. 

In Fig. 8-37b an integrating gyro is used in each stabilization loop as a 
device for comparing the antenna rate with the commanded rate and for 
providing an electrical signal proportional to the difference between these 
rates. This signal is amplified and passed through appropriate compen- 
sating networks to drive the antenna through the power amplifiers and 
actuators designed for the search loops. Theoretically, any gyro could be 
used as an integrating gyro; practically, it is difficult to provide sufficient 
stability, loop gain, and damping to prevent the gyro gimbal from moving 
into its mechanical stops during antenna transient motions unless a large 
amount of viscous damping is provided at the gimbals. Therefore, HIG 
gyros are commonly used as integrating gyros although they may be used 
as rate gyros also. 

Note that in the integrating gyro configuration, antenna rates are not 
measured directly. They are assumed to be proportional to the rate loop 
command which is the track loop radar error signal. As is discussed in 
Chapter 9, the track loop has a low bandwidth and removes high-frequency 
signals incident to antenna vibration from the measured rate signal. This 
is not a great advantage, because the rate signal is actually measured 
through a low pass filter in any configuration in order to remove system 
noise. 

In Fig. 8-37a and b the amplifier is assumed to have compensating 
networks to convert the measured or calculated component characteristics 
to the derived loop transfer ratio shown in Fig. 8-42. The transfer function 
of the compensating elements can be derived as indicated in Fig. 8-36. 

The configuration of Fig. 8-37c has been described in Paragraph 8-31 and 
the design details are beyond the scope of this text. However, it should be 



470 REGULATORY CIRCUITS 

noted that the gyros are not mounted on the antenna and that the antenna 
resonant frequencies do not influence the stabilization loop design because 
relative rate motion between the antenna and aircraft is measured with 
tachometers and converted to space motion in a coordinate converter. 
Thus, the difliculty of designing an accurate stabilization loop is transferred 
to the converter. The three gyros must be more accurate than those usually 
used for autopilots, the converter must be carefully made, and the various 
input signals must be phased and combined with precision resolvers to 
provide the necessary accuracy for the fire-control computer. As indicated 
in the figure, this method of providing space stabilization does not use 
the automatic regulation function of a feedback loop. It is primarily a 
balancing arrangement in which the antenna actuators are moved as 
directed by a computed signal. In