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.
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24 West 40 Street, New York 18, New York
D. Van Nostrand Company, Ltd,
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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. 618542
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
11 Introduction . 1
12 Classifications of Radar Systems 2
13 Installation Environment 3
14 Functional Characteristics of Radar Systems 4
15 The Modulation of Radar Signals 16
16 Operating Carrier Frequency 26
17 The Airborne Radar Design Problem 27
18 The Systems Approach to Airborne Radar Design ... 30
19 Systems Environments 35
110 Weapons System Models 36
111 The Basic Statistical Character of Weapons System Models 39
112 Construction and Manipulation of Weapons System Models 41
113 Summary 44
2 THE DEVELOPMENT OF WEAPONS SYSTEM
REQUIREMENTS
21 Introduction to the Problem 46
22 Formulating the System Study Plan 48
23 Aircraft Carrier Task Force Weapons System 50
24 The Target Complex 55
25 The Operational Requirement 57
26 The System Concept 57
27 The System Study Plan 5^
28 Model Parameters 60
29 System Effectiveness Models 61
210 Preliminary Design of the Airborne Early Warning System 67
211 AEW System Logic and Fixed Elements 70
212 AEW Detection Range Requirements 73
213 AEW Target Resolution Requirements 75
xiii
xiv CONTENTS
214 Interrelations of the AEW System, the CIC System, the In
terceptor System, and the Tactical Problem 79
215 Accuracy of the Provisional AEW System 80
216 InformationHandling Capacity of the Provisional AEW
System 84
217 Velocity and Heading Estimates 85
218 AEW Radar Beamwidth as Dictated by the Tactical Problem 89
219 Factors Affecting HeightFinding Radar Requirements . . 92
220 Summary of AEW System Requirements 96
221 Evaluation of Tentative Design Parameters with Respect to
the Tactical Problem 98
222 Interceptor System Study Model 100
223 Probability of Reliable Operation 103
224 Probability of Viewing Target— Vectoring Probability . . 104
225 Analysis of the Vectoring Phase of Interceptor System Op
eration 106
226 AT Radar Requirements Dictated by Vectoring Considera
tions Ill
227 Analysis of the Conversion Problem 116
228 Lockon Range and LookAngle Requirements Dictated by
the Conversion Problem 130
229 AI Radar Requirements Imposed by Missile Guidance Con
siderations 135
230 Summary of AI Requirements 136
231 Summary 137
3 THE CALCULATION OF RADAR DETECTION
PROBABILITY AND ANGULAR RESOLUTION
31 General Remarks 138
32 The Radar Range Equation 138
33 The Calculation of Detection Probability for a Pulse Radar 141
34 The Effect of Scanning and the Cumulative Probability of
Detection 156
35 The Calculation of Detection Probability for a Pulsed
Doppler Radar 162
36 Factors Affecting Angular Resolution 168
4 REFLECTION AND TRANSMISSION
OF RADIO WAVES
41 Introduction • 174
42 Reflection of Radar Waves 175
43 Effect of Polarization on Reflection 179
44 Modulation of Reflected Signal by Target Motion . . . 180
CONTENTS XV
45 Reflection of Plane Waves from the Ground 181
46 Effect of Earth's Curvature 190
47 Radar Cross Sections of Aircraft 192
48 Amplitude, Angle, and Range Noise 198
49 Prediction of Target Radar Characteristics 208
410 Sea Return 211
411 Sea Return in a Doppler System 217
412 Ground Return 219
413 Altitude Return 222
414 Solutions to the Clutter Problem 224
415 Attenuation in the Atmosphere 227
416 Attenuation and Backscattering by Precipitation . . . 230
417 Attenuation by Propellant Gases 231
418 Refraction Effects in the Atmosphere 233
5 TECHNIQUES FOR SIGNAL
AND NOISE ANALYSIS
51 Introduction 238
52 Fourier Analysis . 238
53 Impulse Functions 243
54 Random Noise Processes 245
55 The Power Density Spectrum 248
56 Nonlinear and TimeDependent Operations 253
57 Narrow Band Noise 258
58 An Application to the Evaluation of Angle Tracking Noise 264
59 An Application to the Analysis of an MTI System . . . 269
510 An Application to the Analysis of a Matched Filter Radar . 272
511 Application to the Determination of a Signal's Time of
Arrival 281
6 GENERIC TYPES OF RADAR SYSTEMS
AND TECHNIQUES
61 Introduction 292
62 Basic Principles 293
63 Monopulse Angle Tracking Techniques 300
64 Correlation and Storage Radar Techniques 305
65 FM/CW Radar Systems 311
66 PulsedDoppler Radar Systems 320
67 HighResolution Radar Systems 333
68 Infrared Systems 338
7 THE RADAR RECEIVER
71 General Design Principles 347
72 The Interdependence of Receiver Components .... 352
XVI CONTENTS
73 Receiver Noise Figure 353
74 LowNoise Figure Devices for RF Amplification .... 356
75 Mixers 357
76 Coupling to the Mixer 361
77 IF Amplifier Design 362
78 Considerations of IF Preamplifier Design 368
79 Overall Amplifier Gain 373
710 Gain Variation and Gain Setting 375
711 Bandwidth and Dynamic Response 375
712 Sneak Circuits 377
713 Considerations Relating to AGC Design 379
714 Problems at HighInput Power Levels 380
715 The Second Detector (Envelope Detector) 382
716 Gating Circuits 386
717 Pulse Stretching 387
718 Connecting the Receiver to the Related Regulating and
Tracking Circuits 388
719 Angle Demodulation 389
720 Some Problems in the Measurement of Receiver Character
istics 390
8 REGULATORY CIRCUITS
81 The Need for Regulatory Circuits 394
82 External and Internal Noise Inputs to the Radar System . 395
83 Automatic Frequency Control 401
84 Variation of Transmitter Frequency with Environmental
Conditions 402
85 Magnetron Pulling . . . . ' 403
86 Static and Dynamic Accuracy Requirements 405
87 ContinuousCorrection AFC 407
88 LimitActivated AFC 412
89 The Influences of Local Oscillator Characteristics . . . 413
810 Relation to Receiver IF Characteristics 414
811 Discriminator Design 414
812 Instantaneous AFC 415
813 Problems of Frequency Search and Acquisition .... 416
814 Automatic Gain Control 416
815 Linear Analysis of AGC Loops 419
816 Static Regulation Requirements of AGC Loops .... 420
817 Dynamic Regulation Requirements of AGC Loops . . . 422
818 AGC Transfer Characteristic Design Considerations . . . 423
819 The Modulation Transmission Requirement 424
820 Design of an AGC Transfer Function 425
CONTENTS xvii
821 The IF Amplifier Control Characteristic 427
822 The Angle Measurement Stabilization Problem .... 429
823 AI Radar Angle Stabilization 433
824 Aircraft Motions 433
825 Stabilization Requirements 439
826 Search Pattern Stabilization 440
827 Search Stabilization Equations 440
828 Static and Dynamic Control Loop Errors 442
829 Search Loop Mechanization 448
830 Stabilization During Track 452
831 Possible System Configurations 453
832 Accuracy Requirements on the Angle Track Stabilization
Loop 457
833 Dynamic Stability Requirements on Angle Track Stabiliza
tion 464
834 Stabilization Loop Mechanization 468
9 AUTOMATIC TRACKING CIRCUITS
91 General Problems of Automatic Tracking 471
92 Automatic Angle Tracking 474
93 External Inputs: Undesired and Desired 475
94 Requirements in Angle Tracking Accuracy 479
95 Angle Tracking System Organization 480
96 Tracking Loop Design 485
97 Angle Tracking Loop Rate Errors 486
98 Angle Tracking Loop Position Errors 489
99 Angle Tracking Loop Mechanization 492
910 Introduction to Range and Velocity Tracking 498
911 x'\utomatic Range Tracking 498
912 Servo System Transfer Function Relationship to Input Time
Function for a Range Tracking System 502
913 Range Tracking Design Example 505
914 Practical Design Considerations 508
10 ANTENNAS AND RF COMPONENTS
101 Antennas: Introduction to Radar Antennas 512
102 Some Fundamental Concepts Useful in the Development of
Radar Antenna Requirements 513
103 The Paraboloidal Reflector as a Radar Tracking Antenna . 515
104 System Requirements for Radar Antennas 518
105 Pattern Simulation as a Link Between System Requirements
and Antenna Characteristics 520
xviii CONTENTS
106 Several Anomalous Effects in Antennas for Tracking Systems 523
107 The Linear Array as a Fan Beam Antenna for Surveillance 524
108 TwoArm Spiral Antennas 528
109 Radomes 531
1010 Introduction to Transmission Lines and Modes of Propaga
tion 535
1011 Types of Transmission Lines and Modes of Propagation . . 536
1012 Standing Waves and Impedance Matching 540
1013 Broadband System Design 543
1014 Pressurization 545
1015 Miscellaneous Microwave Components 546
1016 Microwave Ferrite Devices and Their Application . . . 557
1017 Microwave Dielectric, Magnetic, and Absorbent Materials . 565
1018 The Duplexing Problem 566
1019 Duplexing Schemes 567
1020 Special Problems of Coherent Systems 573
1021 SolidState Amplifiers 574
11 THE GENERATION OF MICROWAVE POWER
111 The Magnetron 580
112 The Klystron 590
113 Traveling Wave Tubes for High Power 597
114 Modulation Techniques for BeamType Amplifiers . . . 599
115 A Typical Radar System Employing a HighGain Amplifier 601
116 Backward Wave Oscillators — Carcinotrons 602
117 The Platinotron 603
12 DISPLAY SYSTEM DESIGN PROBLEMS
121 Introduction 607
122 Uses of Display Information 608
123 Types of Displays 613
124 Types of Input Information 619
125 The Cathode Ray Tube 621
126 Important Characteristics of ElectricaltoLight Transducers 627
127 Important Characteristics of the Human Operator . 634
128 Development of Requirements for a Display System . . . 651
129 Special Display Devices 655
1210 Special Displays 673
13 MECHANICAL DESIGN AND PACKAGING
131 The Influence of Environment on Design 680
132 Military Specifications 682
133 Temperature 683
CONTENTS xix
134 Solar Radiation 692
135 Nuclear Radiation 692
136 Vibration and Shock 694
137 Acoustic Noise 704
138 Acceleration 707
139 Moisture 708
1310 Static Electricity and Explosion 710
1311 Pressure 711
1312 Maintenance and Installation 712
1313 Transportation and Supply 714
1314 Potential Growth 715
1315 ReUabihty 715
14 AIRBORNE NAVIGATION AND GROUND
SURVEILLANCE RADAR SYSTEMS
141 Introduction to Doppler Navigation Systems 726
142 Basic Principles of Doppler Radar Navigation .... 728
143 System Considerations 733
144 Major Characteristics and Components of a Doppler Radar
Navigation System 736
145 Doppler Navigation System Errors Caused by Interactions
with the Ground and Water 746
146 Modifying the Radar Range Equation for the Doppler Navi
gation Problem 749
147 Low Altitude Performance and the "Altitude Hole" . . . 752
148 Doppler Navigation System Performance Data .... 755
149 Introduction to Weather Radar 759
1410 Meteorological Effects at Microwave Frequencies . . . 760
1411 Designing Airborne Radar Systems Explicitly for Weather
Mapping 764
1412 Modifying the Radar Range Equation for the Weather
Problem 764
1413 Relative Importance of Design Variables in Airborne Weather
Radar 766
1414 Design Features 769
1415 Introduction to Active Airborne Ground Mapping Systems 772
1416 Basic Principles 772
1417 System Considerations 774
1418 Major Characteristics and Components 778
1419 Modifying the Radar Range Equation for the Active Ground
Mapping Problem 782
1420 Resolution Limits in Ground Mapping Systems .... 784
XX CONTENTS
1421 Future Possibilities in Airborne Active Ground Mapping
Systems 787
1422 Introduction to Infrared Reconnaissance 787
1423 Basic Principles Concerning IR Ground Mapping . . 788
1424 System Considerations 791
1425 Major Systems Features 798
1426 New Developments 801
Index 805
J. POVEJSIL • P. WATERMAN
CHAPTER 1
ELEMENTS OF THE AIRBORNE RADAR
SYSTEMS DESIGN PROBLEM
11 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.
12 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)
14] 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) conicalscan pulse radar (4) operating at X Band.
13 INSTALLATION ENVIRONMENT
The most common types of radar system installations are:
1. Groundbased 3. Airborne (piloted aircraft)
2. Shipbased 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.
14 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. 11. In this
,» Search Radar
Scan Pattern
ar Search and Detection.
example, RF (radiofrequency) 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. 11.
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.
14] 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.
11, 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 identificationfriendorfoe
(IFF) systems fall into this category: e.g., in Fig. 12 a presumably friendly
Fig. 12 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 (12)
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 15.
14] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 7
Angular bearing of the target is measured by utilizing a directive beam
like that shown in Fig. 11. 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 twodimensional case dis
played in Fig. 13 illustrates the basic principle.
The relative velocity of the target,
Vtr can be represented by two com
ponents — one parallel to the lineof
sight, Vtrp, and the other normal to
the lineofsight, Vtru These quan
tities may in turn be expressed
Line • of • Sight
to Target
Vt
V,
(R) = R (13)
Target Velocity
Relative to Target
R(f)
(14)
where R = range rate along the line
ofsight and 4> = space angular rota
tion of the lineofsight.
Rangerate 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 lineofsight is measured by
an angularrate 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. 13 Relative Target Motion: Two
Dimensional Case.
8 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM
platform. This capability is essential to the solution of the firecontrol
problem.
An analysis of the twodimensional firecontrol problem (Fig. 14)
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 (15)
L=Lead Angle
= Line of sight Angular Rate
Fig. 14 AirtoAir FighterBomber Duel FireControl Problem in TwoDimen
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 firecontrol 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 (16)
Rftn = VTRjf = R4>tf (17)
Similarly, relative weapon range may be expressed
14] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS
Rfwp = ^(/f cos L
Rfwn = y^tf sin L
Equating components, we obtain the basic firecontrol equations
VqIj cos L = R — Rtf (time of flight equation)
_R^
(lead angle equation).
9
(18)
(19)
(110)
(111)
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. 15.
In the first method, Fig. 15A, 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. 15 Radar Mapping: (a) ForwardLook System, Variant of the Plan Position
System, (b) SideLook 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 (ForwardLook System).
10
ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM
In the second method, Fig. 15B, 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. 16,
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 127), thereby improving the contrast in cases
where a relationship similar to Fig. 16 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 sightline
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 FREQUENCYf
Fig. 16 Utilizing the Change of Target
Characteristics with Frequency to En
hance Mapping.
14] 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 higherfrequency
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, landwater 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. 1415.
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. 17).
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 (112)
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. 17 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 lineofsight because of the inherent
14] 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 standby
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
"homeonjam" 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. 18
14
ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM
Fig. 18 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. 19. In this
&^
r)
Rf^ct,
L_ U '
Fig. 19 Rane;e Measurement in a Semiactive system.
case, the target receives energy from the interceptorborne 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.
14] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 15
by a rearwardlooking 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) (113)
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. 215. 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 shipbased 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 oneway transmission char
acteristic as contrasted with the twoway 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.
15 THE MODULATION OF RADAR SIGNALS
A radar system may perform a number of functions (Paragraph 14)
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. 110. 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. 110 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>). (114)
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 spacemodulated by
an antenna possessing directivity. Such a characteristic is shown in Fig.
110; 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
tudemodulated 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]. (115)
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 plusorminus 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. 111. The voltage
amplitude of each sideband in this case is onehalf that of the carrier, and
the power in each sideband is onequarter of the carrier power. Obviously
18
ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM
FREQUENCY (ANGULAR)
Fig. 111 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. 112.
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. 112 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. 113. 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/setype 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 114 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.
15]
THE MODULATION OF RADAR SIGNALS
li
^T
o o o
3 33
FREQUENCY *■
■ III
Fig. 113 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)
(116)
whose envelope has a constant value. A typical frequencymodulated
wave is shown in Fig. 114.
■Ao)
FREQUENCY ^
Fig. 114 Typical FM Spectrum for HighModulation Index (Aoj/coi > 10).
A key parameter in an FM system is the ratio
— = modulation index.
(117)
If this index is relatively high — say 10 or greater — the output spectrum
has the form shown in Fig. 114. 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. Pulsewidth modulation and pulsetime 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 continuouswave (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/). (118)
The difference in the arguments of the cosine functions of Equations 116
and 118, 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
15]
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 spacemodulation 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. 115. 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. 115 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., onetenth 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 timevarying phase shift in the 1cps 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. 116)
^=>
OAAAAAAAAA.
•«
Fig. 116 The Doppler F.ffect.
causes each frequency component of the transmitted wave that strikes
the target to be shifted by an amount
/d = {VtIc)/ (119)
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)/. (120)
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.
151
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
producttaking device, as shown in Fig. 117.
Reference
Incoming Signal
Product Signal
Fig. 117 Generic (Product) Building Block for a Radar Receiver.
Conceptually, the simplest producttaking 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. 118). The output product is a signal containing
Input
F(;co)
Output
[Output] =[f(; CO)] X [input]
Fi/co)  Fl(j}
Fig. 118 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 producttaking device is the nonlinear impedance.
A simple example of such a device is shown in Fig. 119. 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. 119 Product Obtained from a Nonlinear Impedance.
input sinewave, the output consists of only the positive halfcycles as
shown.
It is interesting to observe that this process may be put into the form
of the generic device of Fig. 118 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. 120.
Thus, the input may be defined as
:rLrLn
TT 2ir 3ir Att Stt
Fig. 120 Nonlinear Impedance as a ProductTaking 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'
(121)
(122)
(123)
The consequences of this product process are quite evident from Equation
123. Although the input contains only one frequency, the output has a
dc component, an inputfrequency 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 amplitudemodulated at a frequency aj;„, such that
15] THE MODULATION OF RADAR SIGNALS 25
A = A^{\ ■\ 771 COS co„/) (124)
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 dc term would now become
Ao . mAo n nc\
1 cos co,„/ (125)
T IT
and the fundamental frequency term would become
'y sin w/ H ^ [sin(co ( w™)/ — sin(w — w™)/] (126)
and so on for the higher harmonics.
If this (Ei X Er) product were then passed through a filter, F(ju),
which eliminated the dc term and the fundamental frequency u and all
its harmonics, the final output would be
(Ei X Er) X F(jc^) = '^ cos co„,/. (127)
TT
Thus, we observe, the frequency and the magnitude of the modulation
intelligence are recovered from the incoming wave by the producttaking
procedures. The procedures just described are often referred to as de
modulation or detection.
A third type of producttaking device closely resembles the basic model
of Fig. 117. 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 crossproduct 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 crossproduct 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
16 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. 121.
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. 121 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 powerhandling 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 higherfrequency 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.
17 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. l22a. 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. l22a
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. l22b for the case of a single radar param
l_.
Fig. l22b 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.
17]
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. Stateoftheart 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. 123. 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. 123 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.
18 THE SYSTEMS APPROACH TO AIRBORNE RADAR
DESIGN
A representation of a typical weapons system design cycle is shown in
Fig. 124. 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. 124 Weapons System Design Cycle.
or a problemsolving 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 problemsolving 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 stepbystep 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. 125. The weapons system must be operative in a given time
Desired Level
1
\
1
VObsolescen
1 Minimum \ \
J Acceptable \ \
Level \ \
\ \
a TIME (YEARS) b
Fig. 125 Operational Requirement.
period al?. 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 longrange 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, stateoftheart 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 stateoftheart 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
18] 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. 124).
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
stateoftheart 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.
124. 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 fivetotenyear 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.
19] 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
reliabilitycomplexity tradeoff 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.
19 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 groundbased 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
toair 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. 126. Each element of the system design must be com
System
Environment
Logistics
Environment
Weapons System
Integration Environment
Fig. 126 The System Environment.
patible with the requirements and limitations imposed by all of these
environments.
110 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.
110]
WEAPONS SYSTEM MODELS
Radar Range
37
Target Inputs
TTT
Fixed Elements
Mission Accomplishment
RADAR DETECTION RANGE
Fig. 127 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. 127. 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 midair 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 largescale
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.
111] 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 midair 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. Stateoftheart 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.
111 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. 128 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. 128. 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
112] 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.
112 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 seriesparallel — 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
toair intercept system might be characterized by the sequence of opera
tional functions shown in Fig. 129. 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 inputoutput device. The subject of inputoutput 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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112] 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
preestablished 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 111, 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 earlywarning detection,
identification, tracking, and target assignment.
4. InterceptorBomber 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 stepbystep 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.
113 SUMMARY
In this chapter, we have covered the general characteristics of radar
systems; the environments in which they operate; the functional capabilities
113] 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
21 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 firecontrol system for allweather
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, crossreferencing,
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
21
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. 21 Steps Leading to the Formulation of the System Study Plan.
48 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
22 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. 21. 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. 22 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 :
22] 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. 22. The operational
requirement defines a weapons system problem. By the processes described
in Paragraph 18, 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. 21) and their answers are fundamental to further progress.
For the hypothetical problem, the known and unknown elements of the
problem are displayed in Figs. 22 through 28. 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. 22). 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.
22) 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 stateoftheart 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
23 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 FireControl System.
Referring to the system study plan of Fig. 22, 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) AirtoAir 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. 23 through 26. 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 airtoair missile are known; the manner in which these two
elements combine and interrelate with the interceptor firecontrol 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. 23). This system
includes two aircraft carriers and three missilefiring 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 (520 n. mi.) for tactical reasons. Mass attacks
23]
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. 23 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 missilefiring 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 4050 n. mi. in front of the carriers. A third missilefiring crusier
52
THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
guards against sneak attacks from the rear as well as turnaround 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 24hour
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
TargetHandling 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
DeckReady
OnStation StandI
On Station
A. Shock and Vibration
B. Temperature
C. Pressure
D. Humidity
E. Space and Weight
F. Environmental Noise
Interference
Clutter
Weather
Internal
Fig. 24 Airborne Early Warning System. See Fig. 218 for Tactical Deployment.
I
23] AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM 53
(2) To provide target data for the shipbased combat information center
(CIC), which supplies vectoring information to piloted interceptors
and guidance information to shipbased missile radars.
Two combat air patrols (CAP) are maintained. Each CAP contains 6
allweather 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 airtoair guided missiles and are
required to perform the interception function at altitudes from sea level to
60,000 ft.
An optimum battlecontrol 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. 24). The basic functions of
the AEW system have been described.
The carrierbased 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. 25). The known and unknown
characteristics are defined as shown. The determination of detailed radar
requirements will require an analysis of the dynamics of the closedloop
system formed by the target, the interceptor, the pilot, and the AI radar
and firecontrol system.
The interrelationships among the aircraft system, the airtoair 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 AirtoAir
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. 25 Interceptor System Characteristics.
AirtoAir Missile System (Fig. 26). The sensitive parameters of
the airtoair missile system, are shown. The major unknown parameters
involve interrelationships of the missile with the aircraft and the radar and
firecontrol 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
2Missile Salvo  0.75
Radar and Computer Integration
Seeker Angle Slaving Signal
Seeker Range Slaving Signal
J^^
Launching Signal
T^
Seeker Acquisition Signal
Fig. 26 AirtoAir Missile System Characteristics.
24 THE TARGET COMPLEX (FIG. 27)
Target Complex. The characteristics of the target complex include
its parameters and its mission as shown in Fig. 27.
The 50nautical mile (n. mi.) airtosurface missile (ASM) carried by the
hostile aircraft requires that interception take place outside of a 50n. mi.
circle around the aircraft carriers.
The target's 2g maneuver capability will exercise an important influence
on the radar and firecontrol 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
7001000 V 100 50
RANGE FROM TARGET (n.mi.)
Target
Radar Cross Section
90°
180° 1
Fig. 27 Target Complex Characteristics.
The radar crosssection characteristics of the hypothetical target are
only generally known and are shown in Fig. 22. Paragraph 47 explains
the factors contributing to characteristics of this type. Paragraph 49
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 47 and 48). This can be an important
radar design consideration.
The target is assumed to carry a highyield 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 1000ft radius
sphere around the target. Ignoring time effects, 1000 ft thus defines the
26] 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.
25 THE OPERATIONAL REQUIREMENT: MISSION
ACCOMPLISHMENT GOALS (FIG. 28)
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.
26 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
missilearmed 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. Airtoair missile guidance
j. Missile detonation and target destruction (without selfdestruction)
58 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
Function HighAttrition Air Defense
Against Medium Bombers
Features (1) Compatibility with SurfacetoAir
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. 28 Operatioric
A 20Plane 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 surfacetoair missile
support defense system.
A diagrammatic summary representation of the overall tactical situation
is shown in Fig. 29.
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
27]
THE SYSTEM STUDY PLAN 59
Zone of CIC
Vectoring Inaccuracy
Zone of
AFCS Inaccuracy
rrv
'=^X
Search
Zone of Missile
Guidance Inaccuracy
Fig. 29 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.
27 THE SYSTEM STUDY PLAN
The known (fixed) and unknown (variable) elements of the problem have
now been defined. Referring to Fig. 22, 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 stepbystep 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
firecontrol 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.
28 MODEL PARAMETERS
The interrelations between major system parameters and the contribu
tions of each parameter to overall effectiveness may be developed through
29]
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.
29 SYSTEM EFFECTIVENESS MODELS
The operational requirement (Fig. 28) 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. 22). 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 twomissile 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. 210 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. 210 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 backup surfacetoair missile
system.
Continuing in this fashion, tradeoff 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. 211.
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
NNUMBER OF INTERCEPTIONS
Fig. 211 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.
29] SYSTEM EFFECTIVENESS MODELS 63
Also shown are the limiting effects of the fixed problem elements pre
viously outlined in Figs. 23 through 26. 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 deckready 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 deckready 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. 22).
1. Number of interceptors (A^)
2. Interceptor effectiveness (Po)
3. Earlywarning 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. 212, which shows the sequence of events in a typical
raid. The interceptor and target performance characteristics were given
in Figs. 25 and 27. 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. 212 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 3minute 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. 23), 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. 212 An Attack Diagram.
300
From Fig. 25 it is seen that an additional 3minute delay is created by
the interceptor climb and acceleration characteristics. This makes the
total effective reaction time of the decklaunched 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 surfacetoair missile zone corresponding to a 50n.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 16minute 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 decklaunched inter
ceptors), which is well within the interceptor performance capabilities as
displayed in Fig. 25.
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. 213,
where tradeoff 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
29]
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. 213 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
tradeoff of parameter values elsewhere in the problem should be examined
to exploit the potential advantage which might accrue from a range
increase.^ Conversely, a 50n.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 straightline 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
tailchase 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 3minute 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 — oneonone — to the first
20 targets. The following interceptors are assigned as backup 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 backup 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
210] 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 "tradeoff" 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. 22) is to establish the allocation of responsibility
between these two systems.
210 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 23.
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 firecontrol 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. 214. 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
ShipBased 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. 214 Plan for Analysis of AEW System Requirements (Step 2 of Master
Plan — Fig. 22).
The number of interceptors A^ which can be directed against the separate
elements (single surviving targets) of the 20plane raid during the air
battle is selected as the criterion for judging AEW system performance.
As already shown (Paragraph 29) 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. 211). 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 212.
210]
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 21.
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.
Stateoftheart and schedule limitations are not considered in this
analysis (see Paragraph 18). 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 21 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. 210 and 211
See Fig. 28
Fixed system parameters
as defined in Figs. 23
to 26
Target parameters
as defined in Fig. 27
Note. Selected parameters allow the operational requirements to be met or exceeded.
Sensitivity of the system performance to parameter changes are shown in Figs. 211 and 213.
70 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
211 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. 215 and 216.
AEW Position
Line
Fig. 215 AEW Operation Illustrating Azimuth Location and HeightFinding
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. 216 Early Warning and Vectoring System Information Flow.
211] AEW SYSTEM LOGIC AND FIXED ELEMENTS 71
Initial detection of the target is provided by the fanbeam radar. This
equipment also measures slant range to the target, R, and target azimuth
position 6 with respect to a reference direction.
The heightfinding radar is positioned in azimuth with information
obtained from the fanbeam 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. 23.
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, straightline flight paths can be represented by x and y velocity
components which also remain constant. Thus, if the position, P(/), of a
constant velocity straightline target at any time t is designated in rec
tangular coordinates, then
mlox{joy (21)
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 (22)
and the position of the target at any time r seconds later can be written
Pit + r) = m + rFit) = Ux + tF,) = joiy + tF,). (23)
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 preestablished 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 heightfinding
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. 212).
The type of fighter direction best fulfilling this requirement is collision
vectoring. Its basic principle is shown in Fig. 217. 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 14 and Fig. 14 the use of polar
coordinates is indicated.
212]
AEW DETECTION RANGE REQUIREMENTS
73
Fixed Reference Direction
(e.g. North)
Fig. 217 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
(24)
where each of the variables may be defined from the figure.
The quantities, dr, Vr, 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
lineofsight 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 fanbeam 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 225.
212 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: backup 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. 218.
Range Requirement
, Reserve \
AEW #1 / AEW \ AEW #2
^1 ^ ]^
\ (Nonradiating) ;
During Normal Operation
Guided Missile
Fig. 218
Possible AEW Aircraft Detection Range, Coverage, and Disposition
to Provide 255n.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 backup 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.
213] 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. 219. 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 WdR
Fig. 219 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. 219.
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.
213 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. 27). 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 airdefense operation.
First of all, the 6 CAP aircraft would be directed to engage the threat
elements. Simultaneously, deckready 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 deckready aircraft in
response to a 10target threat. Thus, for the first 10 minutes following
initial detection (3 minutes delay time plus 7 minutes for launching 14
deckready 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 10minute 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. 212). 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 deckready 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 deckready aircraft launchings, the number of targets
counted must increase at a minimum rate of 1 per minute until all 20 are
separately resolved.
213] 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 "breakup" 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. 212 and 218 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 14) — range, angle, and velocity along
sightline to target. Resolution between targets may be done on the basis
of any or all of these.
Fig. 220 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~Zr7r~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
12Msec Pulsewidth = l n.mi.
AEW Aircraft
To Fleet Center
Fig. 220 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 stationkeeping 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. 220. 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.
214] 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
decklaunched 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 tradeoffs 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.
214 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 29, 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 datahandling 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
215 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 213.
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 firecontrol 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 lockon 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 tradeoff relation
between vectoring error and the required AI lockon range.''
Unfortunately, because of the complex interrelations between AEW/CIC
system errors and vectoring errors, the analysis in Paragraphs 222 to 228
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 lockon 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 46 and 47 of
Merrill, Greenberg, and Helmholz, of), cit.
215]
ACCURACY OF THE PROVISIONAL AEW SYSTEM
81
Measurement Errors due to Beamwidth and Range Aperture.
Referring to Fig. 221, 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. 221 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 signaltonoise ratio
and the number of hits per beamwidth. An analysis of these relations is
given in Paragraph 51 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 = 165 n.mi. (at 75 n.mi. range)
(25)
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 25.
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 1n.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 213). 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 211 and Fig. 215) that AEW
215] 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 datahandling 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 updated 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 timedelay error.
For preliminary design of the overall AEW system, it will be assumed
that timedelay 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 timedelay 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 timedelay 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).
(27)
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). (28)
One source of error — the 1 n.mi. rms navigation error of the AEW
aircraft (see Paragraph 211) — 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. 218. 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.
216 INFORMATIONHANDLING CAPACITY OF THE
PROVISIONAL AEW SYSTEM
An important aspect of system design relates to its datahandling
capabilities. Both the data link for transmitting information between the
217] VELOCITY AND HEADING ESTIMATES 85
AEW aircraft and the CIC and the dataprocessing computer at the CIC
will have limited capacities for handling data. The maximum per channel
capacity of the data link has been specified (Fig. 23) 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 selfchecking 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.
217 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. 222.
^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 sixdigit 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. 222 Positk
CIC Location
Data Used in Computing Esti
Velocity.
lates of Target Heading and
Assume that the specified 800fps 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.
(29)
(210)
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. (211)
(yN\N2 = or n.mi. (212)
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 27 and
28). 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
217] 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
(213)
'^^ = ^7^ = — (214)
where F = true velocity of object being tracked.
For example, assume that position measurements are made on an 800fps
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 213.
With the assumed parameters of the provisional AEW system, the total
position error of a single measurement was derived to be Equation 28:
ar = 2.09 n.mi. = 2 n.mi. (215)
Thus, the standard deviations of the two components of the relative
error between two measurements are from Equations 211 and 212:
ap.Po  2 n.mi. (rms) = 12,000 ft (rms) (216)
(^NiNi = 2 n.mi. (rms) = 12,000 ft (rms). (217)
Thus the rms errors in the estimated velocity and heading are calculated
by Equations 213 and 214 to be
av = 12,000/(6) (10) = 200 fps (rms) (218)
cT^ = 12,000/(800) (6) (10) = 0.25 rad = 14.5° (rms). (219)
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 tradeoff between improved accuracy and greater informa
tionhandling 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 218 and 219 are somewhat pessimistic. In some
cases, a tradeoff 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 24) 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. 223. 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. 223 Prediction Process Employing ScantoScan 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 timedelay error discussed in
Paragraph 215.
211
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. 223). 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. 224 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. 224 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.
(220)
(221)
(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.
218
AEW RADAR BEAMWIDTH AS DICTATED BY THE
TACTICAL PROBLEM
On the basis of target resolution requirements (Paragraph 213) 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 215 to 217).
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. 220) 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
218]
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. 225 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. 225 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 800fps 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. 225 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. 225 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.
219 FACTORS AFFECTING HEIGHTFINDING RADAR
REQUIREMENTS
The heightfinding 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 fanbeam 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 heightfinding 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. Heightfinding requirements during threat evaluation.
3. Heightfinding requirements during vectoring.
4. Heightfinding requirements dictated by the need to supply early
information to the groundtoair missile system.
To meet these requirements within the limitations of the hypothesized
219] FACTORS AFFECTING HEIGHTFINDING RADAR 93
system logic, the height finder must operate with azimuth input commands
obtained from the AEW fanbeam radar.
The possible dimensions of the heightfinding 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 heightfinding
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 fanbeam azimuth search
system.
The assumed placement of the heightfinder 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. Heightfinding
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 heightfinding system must be able to detect such a change
in time for appropriate defensive measures to be taken.
Immediately following detection, the heightfinding 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 heightfinding radar coverage is ±80° or 160°, the maximum turn required to bring
a target under heightfinder 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 heightfinding
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 firecontrol system characteristics. An inspection of the missile
performance (Fig. 26) 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 firecontrol system. Following AI radar lockon, the
pilot obtains a reasonably precise measurement of relative target elevation.
This may be used by the firecontrol 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 heightfinding 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 heightfinding 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 heightfinding 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.
219] FACTORS AFFECTING HEIGHTFINDING RADAR 95
It will be assumed that the same heightfinding information rate (one
measurement on each target aircraft every 2 minutes) will be maintained
during vectoring.
Requirements Dictated by the SurfacetoAir Missile System.
Heightfinding data can be used to direct the search and tracking system
associated with the surfacetoair 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 groundtoair
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 heightfinding 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 heightfinder
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
0.5n.mi. rms interceptor vectoring heightfinding requirement into an
equivalent angle for 75 n.mi. range. This angle may be expressed
^, = ^ = 0.067 rad = 0.38° (rms). (222)
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. (223)
Thus, to meet the heightfinding 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 heightfinding 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 fanbeam 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
fanbeam azimuth accuracy (Paragraph 215).
Cn = 6n/4 degrees (rms) for a single measurement (224)
an = Qnl'^yln degrees (rms) for n measurements (225)
where n = number of measurements averaged to obtain a single estimate
9n = elevation beamwidth of heightfinding radar.
Since 6 seconds can be taken for the heightfinding 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°. (226)
Actually, techniques known as beam splitting can be employed to obtain
greater angular accuracy than is implied by Equation 226. Accordingly,
the derived result is only one of the possible solutions to the heightfinding
problem.
220 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 firecontrol 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 tradeoffs 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.
220] 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 21 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 22. The number of the paragraph which discusses
each parameter is included for convenience.
Table 22 TENTATIVE AEW RADAR PARAMETERS
Detection Range 90 per cent probability of detection at 150 n.mi. (Paragraph
212)
Number of Targets 20 hostile targets, 40 interceptors (Paragraph 213)
Threat Evaluation Range 125150 n.mi. (Paragraph 211)
Nominal Vectoring Range 75 n.mi. (Paragraph 213)
Azimuth Coverage 360° (Paragraph 211)
Elevation Coverage ..... AS° up, 18.3° down. Operation at 20,000 feet (Para
graph 218).
Range Resolution 1 n.mi. (Paragraph 213)
Angle Resolution 5° maximum (Paragraph 213)
Range Accuracy a a = 0.25 n.mi., rms (Paragraph 215)
Angular Accuracy (Ta = 125° rms (Paragraph 215)
Quantization Levels (Paragraph 215)
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 215)
Time Belay Errors Less than 10 per cent of measurement error (Para
graph 215)
98 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
Total System Position Error (Paragraph 215, Equation 28)
At 75 n.mi.
or = 2 n.mi. rms (5° beam).
At 150 n.mi.
(It = 4.2 n.mi.
HeightFinding Radar (Paragraph 219)
Threat Evaluation Range 125150 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
221 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 213 and 214 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 3minute
interval are
ayr = ^;^""" = 0.0233 n.mi./sec = 142 fps rms (227)
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 3minute 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
221] 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 3minute 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 9minute period. This process yields an error of
142 fps error 142 ^ in i . ^o on\
<rvT = I . = = —j= = 82 fps = 49 knots. (229)
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 dataprocessing 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 straightline
passing through these points (Fig. 222).
At a range of 75 n.mi., with an observation time nisc equal to 60 seconds,
this technique gives rise to an error (Equation 219) 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 straightline target:
a^T = 14.5 /V2= 10.2°. (231)
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 (232)
Velocity Measureynent Error: avr = 50 knots rms (233)
Heading Measurefnent Error: g^t = 10° rms (234)
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. (235)
As will be shown later (Paragraph 225) 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 sightline,
where (trr = aRT/yjl = 2 n.mi. (236)
aRa = cTRT/^I2 = 2 n.ml (237)
The position, velocity, and heading information is employed to vector
interceptors on a collision course with assigned targets (Paragraph 211 and
Equation 24).
222 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 firecontrol 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. 22).
222] 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. 226. 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. 226 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. 29): (1) a vectoring phase which terminates in AI radar lockon,
(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. 220 as:
Pm = probability that the twomissile salvo will kill the specified target
(already specified as 0.75)
Pc = probability that the interceptor will proceed from the point of
AI radar lockon 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
223] 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 (firecontrol
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. (238)
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 Tiein). 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 238 and write
Po = 0.50 = O.lSPcPvPr (239)
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 (240)
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 tradeoffs between
the contributions of each phase of system operation.
223 PROBABILITY OF RELIABLE OPERATION
The analysis of the tactical situation showed that the total air battle
lasted less than onehalf 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 killprobability goal
are the principal factors that determine required interceptor system
reliability during the attack operational situation.
The AI radar and firecontrol 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 twothirds 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 firecontrol 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 3hour 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.
224 PROBABILITY OF VIEWING TARGET — VECTORING
PROBABILITY
The study plan — as extracted from the master plan of Fig. 226 — is
shown in Fig. 227. 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. 226 it can be seen that several variable factors — notably
lockon range and lookangle (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
lockon. Thus, these factors are functions of the lockon range and cannot
be specified until lockon 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.
224]
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
LockOn Range
Detection Range
Search Range
Look Angle
Display
Search Doctrine
Output of Study
Fig. 227 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. 228. 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
lockon), 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
lookangle 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.
228 were along line OA, a lookangle of approximately 90° would be required
for the AI radar to "see" the target.
Fig. 228 Distribution of Interceptor
Headings Due to Vectoring Errors.
225
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 lockon 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, lookangle requirements.
Vectoring System Logic. The flow of information and allocation of
function for the vectoring system are shown in Fig. 229. This diagram
expresses the system logic outlined in Paragraph 211 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 22 and Paragraph 221.
Collision vectoring was specified to minimize the average target penetra
tion. The equation defining this vectoring method was derived as
sin Ld = {VtIVf) smd (24)
225]
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. 230. 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. 230 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 collisioncourse 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 lockon.
The diagram of Fig. 231 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 interceptortarget sight line established by the vectoring system may
differ from the true sighttarget line by an amount which can be expressed
approximately as
225] ANALYSIS OF THE VECTORING PHASE
'Space Reference
109
for
Error
Signal
Fig. 231 Vectoring Error Geometry.
A^i = ARa/R
ARa « R
(241)
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 sightline angle. From Fig. 231 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
(242)
(24)
AFt.
(243)
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^.. (244)
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. (245)
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
(246)
since the closing rate, R, may be expressed
Cf.r
R = VtCosO \ Vf cos Ld. (247)
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
(248)
where aa = iARa)/R.
The evaluation of this expression for various values of lockon range from
8 to 30 n.mi. is given in Fig. 232 for the estimates of measurement uncer
tainty derived for the AEW system (Paragraph 221). 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 collisioncourse lead angle for
this condition is Ld (Fig. 230) 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 lockons and on the ability to convert a shortrange lockon into
a successful attack.
The large magnitude of the heading errors for forwardhemisphere
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.
226] AI RADAR REQUIREMENTS DICTATED BY VECTORING 111
28
30 60 90 120 150 180
ANGLE OFF TARGET'S NOSE AT LOCKON (deg)
Fig. 232 Standard Deviation of Heading Error vs. Angle off Target's Nose at
Lockon.
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. 224. 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 tailchase distributions for collision vectoring. Thus
the tactical advantage of collision vectoring is bought at the price of
increased AI radar and vectoring system requirements.
226 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 lockon is to occur at any selected
range in the 830 n.mi. interval, the radar must be able to "look" at the
target. That is to say, the maximum lookangle of the AI radar antenna
must be sufficient to encompass the distributions of probable target angular
positions relative to the interceptor heading. {Lookangle is often referred to
as gimbal angle or train angle).
A typical situation is shown by Fig. 233. 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. 233 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. 228 and 232 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 lookangles required to ensure that 95 per cent of the targets are
within the AI radar field of view are displayed in Fig. 234. The prices of
shortrange lockons and "around the clock" attack capability are evi
denced by the large radar gimbal angles required to maintain 95 per cent
probability. When the lockon range satisfying the conversion probability
requirement is found, Fig. 234 may be used to determine the AI radar
gimbal angle dictated by vectoring considerations.
Display Requirements. From Fig. 229, 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
226] 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. 234
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. 11).
Target position uncertainty relative to the interceptor determines the
required dimensions of this volume.
For a given lockon range, the azimuth lookangle needed to accom
modate 95 per cent of the expected tactical situations is shown in Fig. 234.
Since the search and acquisition procedures precede AI radar lockon, 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. 232 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 lockon range need be known to specify a maximum gimbal angle
satisfying search, acquisition, and lockon 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. 235 to 237). For example, a 67° maximum lookangle is
required to achieve 95 per cent viewing probability at 10 n.mi. range and
75° off the target's nose (Fig. 234). For a 90 per cent probability under
the same conditions, a 60° maximum lookangle is required (Fig. 235).
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 219),
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 lockon range and may be expressed
6(7//(57.3) ,
R'l — " ^^^^
Search pattern elevation coverage
(249)
The maximum range dimension Ri of the search volume is the range at
which search begins. Its value depends on the required lockon 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. 235 M
aximum
Look Angles Required for 90 Per Cent Probability of Seeing
Assigned Target with Collision Vectoring.
226]
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. 236 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. 237 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
lockon 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 spacestabilized 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. 238. 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. 238 Elevation Search Angle Requirements for 10 n.mi. Lockon (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 230.
227 ANALYSIS OF THE CONVERSION PROBLEM
The plan for analyzing the conversion problem is shown in Fig. 239.
Analysis of the conversion phase must yield the following information
relevant to the AI radar design:
227]
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 Lockon
Target Characteristics
Interceptor Charac
teristics
Pilot Characteristics
Assumed
System
Goal
(Para. 2.22)
— System
Deficiencies
Variable Elements
Al Radar
Lockon Range
Look Angle
Stabilization
Display
Tracking Ace.
Dynamic Range
Attack Doctrine
Missile Launch
Requirements
Study Output
Fig. 239 Plan for the Analysis of Conversion Probability.
1. The minimum required AI radar lockon range
2. The firecontrol 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. 240.
Following AI radar lockon, 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. 240 Interceptor System Logic Diagram During Attack Phase.
118 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
angular velocity, and range rate along the lineofsight. 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 14 and Figs. 13 and 14). 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. 241 — utilizes the fixed parameters of the target, interceptor, and
Launch Point
fn
,MaximumG Missile Trajectory
Maximum Missile
Range Envelope
Impact Point for
StraightLine /
Target Trajectory^
^
yfl Target Position
/ at Launch
^MaximumG Target
Trajectories
Fig. 241 Graphical Determination of Launching Zones.
guided missile as defined in Figs. 25 to 27. The launching tolerances
calculated from this model represent the permissible deviations from perfect
solutions to the firecontrol 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.
227]
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
timeofflight 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
rangestoimpact 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
HeadOn
30
60
90 120 150
Beam
ANGLE OFF TARGET'S NOSE, d (deg)
180
♦
TailOn
Fig. 242 Missile Launch Zones and Launching Tolerances; 50,000Ft 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. 242.
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 selfdestruction. 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.
242).
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. 26). 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 firecontrol system. This is why the AI radar
designer must be certain the missile performance is defined for operation
in the expected tactical environment.
FireControl System Parameters — Attack Doctrine. All the
basic information needed for firecontrol system specification is now
available.
The firecontrol system must be compatible with five requirements or
limitations: (1) minimum average penetration distance; (2) "aroundthe
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 straightline course to a point
where the missiles may be fired with high kill probability. The straightline
characteristic reduces penetration, reduces intcceptor maneuver require
ments, and allows missile launching to take place at any angle off the
227]
ANALYSIS OF THE CONVERSION PROBLEM
121
target's nose. In a leadcollision 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 leadcollision
system correspond closely to the collision vectoring lead angles — a fact
which is helpful in solving the conversion problem.
Leadcollision geometry is shown in Fig. 243. Solution of the firecontrol
triangle yields
Relative Range at Impact
V^iTt,)
Missile Average Velocity
Relative to interceptor
During Time of Flight, ff AV Interceptor
T = Time to Go Until Impact
Fig. 243 LeadCollision Geometry: TwoDimensional.
R^ y^T cos 6+ VtT cos L^ V^tf cos L (250)
VtT sin d = {V,^T+ V^tf) sin L. (251)
The component of relative velocity along the lineofsight is
R ^ Frcosd  Ff cos L. (252)
The component of relative velocity perpendicular to the lineofsight is
Rd = Ft sin d  Fp sin L. (253)
By definition of a lead collision course
// = a preset constant. (254)
From the definition of missile characteristics for straightline flight
(Fig. 26)
F^ =/(Ff, altitude,//). (255)
Thus, for a fixed time of flight and known speed and altitude conditions
FrJf = Ro = constant. (256)
122 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS
Using Equations 252, 253, 256 to eliminate velocity terms in Equations
250 and 251 and rearranging terms, we obtain
R \ RT RocosL = (257)
sin L = {RT!Ro) d. (258)
The firecontrol 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 254 to 256).
The measured target inputs and the computed missile characteristics
may be substituted into Equations 257 and 258 to obtain
„ — Rm + Ro cos L^f ^ , . ... > ,, n\
Jc — ■ (computed timetogo until nnpact) {^^yj
Rm
sin Lc = I R.\[^] 6m (computed correct lead angle). (260)
Firing occurs when
// (preset). (261)
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]. (262)
Both the azimuth and elevation error signals are computed from an
expression of this form.
Equations 259 to 262 define the firecontrol and tracking problems that
are to be solved by the AI radar and firecontrol system. The precision
required of this solution is determined by the angular aiming tolerances
corresponding to the selected value of preset timeofflight.
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. 242 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.
227]
ANALYSIS OF THE CONVERSION PROBLEM
123
firing. The maximum allowable launching error for a 10second time of
flight may be plotted from the data of Fig. 242 as shown in Fig. 244. 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. 244 Maximum Allowable Launching Errors.
can be seen, headon and tailon 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.
FireControl System Error Specification. The sources of system
error can be listed as follows:
(1) AI radar measurements [Rm^ Rm, Om, Lyi)
(2) Flightdata measurements (altitude, speed)
(3) Firecontrol computation
(4) Pilotairframedisplay 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 leadcollision system specified,
the variables R, R, L, and 6 can change rapidly as the launching point is
approached (Fig. 245). ^^ 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
iIn 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 229.
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. 245 Dynamic Variation of Lead Collision Fire Control Parameters.
measurement errors affect the firing time and the steering error as calculated
in the firecontrol computer by Equations 259 through 262.
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 (263)
where B = total predictable bias error
<j == standard deviation of the total random error.
227] 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 timedependent uncer
tainties into the measurements of range and angle (see Paragraph 48).
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 127 will discuss this problem in some detail. Generally
speaking, however, if the pilot is presented with an error signal which is
bandlimited 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 (264)
where cpf = standard deviation of the pilot's flyability error
oiv = 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 headon 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 262 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. Signaltonoise 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
leadcollision course is a straight line; hence the effective bandwidth of the input guidance
signals is very low. Curvedcourse trajectories such as leadpursuit 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 (265)
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 262 and 259 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
(266)
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 (headon
and 80° off the nose).
The values of the input variables and their derivatives are shown in
Fig. 245. The error sensitivity factors for each of the assumed attack
courses are shown in Fig. 246. 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 headon 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 headon attack, the total system aiming error
must be held below 7° to ensure that the missile will hit a maneuvering
target (see Fig. 244). 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 (267)
227]
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 lockon (see Fig. 249 below).
For the headon 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 headon case. The
remaining error tolerance (5°) can be split up among the sources of random
error as shown in Table 23. It will be noted that no tolerances are given
for range and timetogo quantities; their effect on the headon 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 23
MEASUREMENT ACCURACY REQUIREMENTS
FOR HEADON 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°
227]
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 264 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 lockon) may be analyzed in
a similar fashion. For this case the maximum allowable error is about 10.7°
(80° off the nose at lockon will result in about 90° off the target's nose
at time of firing). Using the allowable errors already established for the
headon 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 24. 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 24 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
228
LOCKON RANGE AND LOOKANGLE 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 lockon range
requirements for the conversion probability of 0.825. Fig. 247 displays
Distribution of
Vectoring Headings
Perfect
Lead Collision
Course
f, Contours
10 sec
800 fps
1200 fps
RANGE (n. mi.)
Fig. 247 Interceptor System Model for Conversion Problem.
the essential elements of the problem. If lockon 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 collisioncourse lead
angle at {R, d) and the correct lead collisioncourse 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 lockon must be reduced below the allowable 2° pilot bias error prior to
reaching the missile firing range at 10 seconds timetogo. We shall assume
that the time available for reduction of the steering error at lockon 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. 247
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 lockon. (The distribution of aircraft headings relative to a
perfect collision vectoring course is defined in Fig. 232.)
The lead collisioncourse lead angle is a function of both timetogo and
aspect angle. Fig. 248 illustrates the variation in leadcollision lead angle
as a function of timetogo and aspect angle. For a given {R, d) value, the
lead collisioncourse 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. 249
for various initial values of steering error. The primary factor contributing
228]
LOCKON RANGE AND LOOKANGLE 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. 248 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. 249 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 aircraftpilot
display performance.
The analysis procedure to determine the probability of conversion is
indicated by the flow diagram shown in Fig. 250. 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. 248)
Determine Vectoring
Distribution About
Collision Lead Angle
(Fig. 232)
1 £
Establisii
Distribution of
Heading About
Collision Course
Establish Distribution
of Headings About
Lead Collision Course
(Fig. 251)
Determine
Lead Collision
Lead Angle
(Fig. 248)
r
Establish
Percentage of
Distribution Which
Can Convert
Probablity of
Successful Conversion
Determine
Tff
(Fig. 247)
^
Determine Magnitude
of Steering Error
Which Can Be Reduced
(Fig. 249 ) I
Fig. 250 Conversion Probability Analysis Plan.
Aspect angle at lockon = 60°
Lockon range = 8 n.mi.
Collisioncourse lead angle = 35°
Vectoring distribution, ae.r = 21.5°
Lead collisioncourse lead angle, L = 26.5°
T  tf = 20.5 sec
Maximum correctable steering error = 36°
Fig. 251 shows the distribution of heading errors relative to the correct
collision and leadcollision 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.
228]
LOCKON RANGE AND LOOKANGLE 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. 251 Method for Calculating Conversion Probability.
The calculation of conversion probability by this technique is approxi
mate. Certain kinematic effects such as the change of collisioncourse
leadangle with timetogo 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 lockon range and
aspect angle, culminates in curves like those in Fig. 252. Notice that as
one would expect, the headon attack provides the most stringent require
ments for lockon 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. lockon range is required to
achieve the requisite conversion probability.
The lookangle requirements are dictated by vectoring considerations,
since the collisioncourse leadangle is greater than the leadcollisioncourse
leadangle for the same range and aspect angle (see Fig. 248). From
Fig. 234 we may determine that a 10 n.mi. lockon range and a vectoring
probability requirement of 0.95 combine to dictate a lookangle 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. 252 The Probability of Conversion After Lockon.
The discerning reader will note that a tradeoff analysis could be made
between lockon range and lookangle. For example, a longer lockon range
would allow a smaller lookangle. When space in the aircraft nose is at a
premium, it may be easier to increase lockon range than to provide large
lookangles. In addition, the derived lookangle specification (67°) is
pessimistic. At the aspect angle at which lookangle is critical {d = 75°) a
lockon 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 lookangle requirement corresponding to
this vectoring probability may be read from Fig. 236 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 lockon requirements are established by the headon
attack situation, and the lookangle requirements are establised by the
beam aspect approach situation. These requirements are:
Required lockon range
Required lookangle
10 n.mi. with 90 per cent cumulative
probability
±53° in azimuth and elevation
The lookangle 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.
229] REQUIREMENTS BY MISSILE GUIDANCE CONSIDERATIONS 135
The required detection range is found by specifying a mean lockon time
and adding the range closed between the target and the interceptor during
this time. For example, the closure rate in a headon attack is 2000 fps
(0.33 n. mi. /sec). For mean lockon 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.'^
229 AI RADAR REQUIREMENTS IMPOSED BY MISSILE
GUIDANCE CONSIDERATIONS
The requirements dictated by missile guidance considerations can be
derived from Fig. 26 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. 26 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. 245 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 lockon 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 headon attack (4.4 n.mi.). Fig. 46 shows that
120 kw of peak pulse power is required to ensure seeker lockon at this range
with a 24inch 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 lockon requirements made no mention of
probability. As discussed in Paragraph 212, 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 lockon). 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 lockons
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 highaltitude (i.e. clutterfree)
operation required in this tactical application.
To assist lockon 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 ljusec (500ft) range gate, then
range errors in excess of about 150200 ft can begin to affect seeker lockon
capability. The range error specification previously derived (Table 24)
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.
230 SUMMARY OF AI REQUIREMENTS
Reliability: 90 per cent for 3hour 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 6second
lockon time
14 n.mi. (90 per cent probability) with 12second
lockon time
Lockon Range: 10 n.mi. at a closing speed of 2000 fps with 90 per cent
probability
LookAngle: ±60° in azimuth and elevation
Required Computation: Lead collision (see Equations 259 to 262)
Required Accuracies: See Table 24, Paragraph 228
Dynamic Inputs: See Fig. 245
231] 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 25
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 0.51.0 second
Frequency: X band
Power: Greater than 120 kw peak with a 24inch diameter antenna
Radar Type: Pulse
231 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
31 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, signaltonoise ratio, receiver saturation, pulsing, and
system bandwidth.
32 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
32] 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 =  — ^ (32)
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
crosssectional 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;; = , . ,„„. • (33)
■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 = ^ (34)
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
33 and 34, the received signal power will be
Received signal power = ^S' = , yn^ (35)
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 101 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 47 and some
typical examples are shown in Figs. 420, 421, and 422. The effective
cross sections of sea and ground surface reflections are discussed in Para
graphs 410 through 413. In this connection, a normahzed cross section
is defined as the radar crosssectional 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. (36)
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.
460a and b. Examples showing the variation of sigma zero with environ
mental conditions and radar frequency are given in Figs. 434 through 443.
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 1023 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 35 and
37 yields the signaltonoise ratio, S /N.
Signaltonoise ratio = S/N' = . )3pkRTR'^ ^'^'^■^
^See Paragraph 73 for a further discussion of receiver noise and the origin of this expression.
33] 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 38 for the range when the
signaltonoise 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 (39)
With this definition, the expression for the signaltonoise ratio given in
Equation 38 takes the following simple and useful form:
Signaltonoise ratio = S/N = (Ro/R)'. (310)
To provide an illustration of the use of Equation 39, 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.
33 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. 31. 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 " ■ ih,_x ^..iHr, —
Integrator
Threshold
RECEIVER
TRANSMITTER
I Pulse Rate = f ,
Pulse Length = r
Fig. 31 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 47 and 48.
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
53] 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 (311)
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 signaltonoise
ratio at the output of the predetection amplifier is S jN, as was indicated
above.
A squarelaw 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 57 is applicable and can be used to establish
the amplitude distribution and powerdensity spectrum of the video signal
plus noise. The assumption of a squarelaw 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 510 it will be shown that the linear operation which
gives the greatest signaltonoise 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 signaltonoise 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 signalplusnoise 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. 516 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 falsealarm
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 singlescan probability of detection will be
employed in Paragraph 34 to develop the multiplescan 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 tacticaluse 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 "blipscan" ratio or the "single glimpse" detection prob
ability.
5See Paragraph 212 and Fig. 219.
33]
DETECTION PROBABILITY FOR A PULSE RADAR
145
with a peak power of S. We are now interested in finding analytical
expressions for the squarelaw detector output under the conditions of
noiseonly inputs and signalplusnoise 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. 32. This approximation is based upon
TIME
Fig. 32 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. (312)
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, 1021 (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 57. 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 578 and 579 in Para
graph 57. 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 (313)
Probability density video voltage noise alone =
P.(.)=2^exp[^j. (314)
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 314. When
the signal is present, the sample is chosen from a population with the
probability density of Equation 313.
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 (315)
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) (316)
Probability density integrator output, noise alone = PNi'i) (317)
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,
RM754; and /I Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix,
RM753, The RAND Corp., Santa Monica, Calif.
33] 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 580), 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 581 or Fig. 512.
Video dc voltage = v = 1{N + S)
(318)
ideo rms ac voltage = o^ = 2^N(N \ 2S).
(319)
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, dc voltage = « = 2n{N + S) (320)
Integrator output, rms ac voltage = o„ = 2^1nN(N i 2S). (321)
The probability density function of the integrator output for noise alone
can be established by standard statistical procedures to have the following
form.
''"^"^ = Wuhlji^)"' ^~""' P22)
For the statisticsminded, 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 chisquared 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. 4652, McGrawHill Book Co.,
Inc., New York, 1950.
^See P. G. Hoel, Introduction to Mathematical Statistics, pp. 134136, 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, 10minute 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 signaltonoise 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 falsealarm 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
falsealarm time can be made on the basis of minimizing total costs. Most
commonly, though, such costs cannot be established and the falsealarm
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 signalplusnoise in the false
alarm time is called the falsealarm number and is denoted by t?. With
falsealarm times varying from seconds to hours and pulse lengths varying
from fractions of a microsecond up to milliseconds, the falsealarm 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
falsealarm number. This probability, the probability that a noise sample
lew. M. Hall, "Prediction of Pulse Radar Performance," Proc. IRE (Feb. 1956) 234231.
33]
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. (323)
V Jb
This integral has been evaluated, and the result is shown in Fig. 33 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. 33
Relation Between False Alarm Number and Bias Level with a Square
Law Detector.
SingleScan Probability of Detection. Having chosen the threshold
level on the basis of a required falsealarm time, the probability of detecting
a target Pd is found by integrating the probability density function of the
signalplusnoise sum voltage
Probability of detection = Pa
i:
S+N
{u)du.
(324)
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 falsealarm number and the number of pulses integrated. The results
of this calculation are shown in Fig. 34." In this figure Pd is plotted as a
function of the relative range RjRa whose reciprocal is equal to the fourth
root of the signaltonoise 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. 34 SingleScan Probability of Detection of a Nonfluctuating Target.
We refer to Pd as the singlescan or singleglimpse 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
blipscan 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.
331 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 33 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 falsealarm 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 falsealarm nuniber will be 100 X 10^ = 10^
Referring to Fig. 34, 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. 35 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. 35 SingleScan 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 34.
The Effect of Target Fluctuations. The discussion thus far and the
curves in Fig. 34 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 narrowband noise with a probability density
function of the same form as that in Equation 314. 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. (325)
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 signaltonoise 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. (326)
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 320 and 321
for the mean and variance of the integrated video, the approximate proba
bility of detection is
or, with an appropriate change of variable, (327)
l^A detailed discussion of the fluctuations in apparent size of aircraft is given in Paragraph
48.
i^See Fig. 424 for an example.
33]
DETECTION PROBABILITY FOR A PULSE RADAR
153
P^{
''  wS
exp ( 
■2n{N+S)
xy2)dx.
2ylnN{N+2S)
In Fig. 36, 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. 36 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 326 is
easily evaluated.
  P   ibllnN  1)
P. = (1/6')/ exp ( SIS)ds = exp ^,„ '■ (328)
Jb/2nN 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 310 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 (329)
In addition, a factor K(n,r]) is defined
K = (b/2nN  1). (330)
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. 33 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. 37,
where the K factor is plotted versus the number of pulses integrated for
representative values of the falsealarm number.
154
THE CALCULATION OF RADAR DETECTION PROBABILITY
N.
S\
\\s
^X
\^
v
2
%
\,^
Ns
"'"77=10
S
^N^
/ 7710
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
■^«. ^KN^CiC^MI
^
>sj>::sy:W
^
^">^^^
^^ " = v
^•>:
t
I.
2 5 10 20 50 100 200 500 1000
NO. OF PULSES INTEGRATED,n
Fig. 37 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 tradeoff of signaltonoise ratio with n, the number of pulses
integrated. From Fig. 37, 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. (332)
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 tradeoff between signal to noise ratio
and n, then, is simply
S/N^
(333)
Because of the rather gross approximation which had to be made to
obtain Equation 331, 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. 38 where the average detection probability as given by Equation 331
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. 38 are the exact values of Pd for n = 1.
This is the case when the approximations made introduce the greatest error.
33]
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. 38 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. 38 is
largest. The approximation will also be poorest when the false alarm
number is small. Curves are plotted in Fig. 38 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. 37 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. 38, 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. (334)
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. 38, 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. 35.
34 THE EFFECT OF SCANNING AND THE CUMULATIVE
PROBABILITY OF DETECTION
Because the beamwidth of a highgain 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 groundmapping 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.
MultipleScan 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
signaltonoise 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 (335)
If the first look occurs at the range i?i, then the ^'th look will occur at the
range
34] EFFECT OF SCANNING ON DETECTION PROBABILITY 157
R, = R^ (k  l)AR. (336)
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 331). At
each look, the average probability of detection is given by Equation 337
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 wellknown expression for
the probability of at least one success in a sequence of k trials:
Pc{Rk) = 1  n[l  P.iRd]. (338)
An additional refinement needs to be introduced. Equation 339 implicitly
assumes that the last look occurred at Rk. Actually, the last look may occur
anywhere between Rk and Rki = 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 338 and 339 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). (340)
The normalized range decrement is defined similarly:
Ap = K''\AR/Ro). (341)
With these definitions, the average singleglimpse probability of detection
takes the following form:
Pa = e"' (342)
With this form for the singleglimpse probability of detection, universal
curves of the average cumulative probability of detection have been
calculated on the basis of Equations 338 and 339. These curves are plotted
in Fig. 39.
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. 39 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 332.
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. 39, we shall continue with the
AI radar example which we have previously used in Paragraph 33 to
illustrate the calculation of the singlescan 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.
(343)
Referring to Fig. 39, 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
34]
EFFECT OF SCANNING ON DETECTION PROBABILITY
20.4
i?9o% = 0.87 X
1.3
13.7 n.mi.
159
(344)
The complete curve of cumulative probability of detection versus range is
given in Fig. 35 along comparable singlescan 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 31.
Table 31
Scan Time
n
K
Al/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 31 are plotted in Fig. 310. 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. 310 Example of Detection Range vs. Scan Time.
case is the original choice of 3 seconds. Another observation which can be
made in Fig. 310 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 213). 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. 311 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. 311 Some Possible Scan Patterns with a Pencil Beam System
constantvelocity raster scan with a flyback 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 constantvelocity
azimuth motion produces a cycloidal scan of the beam centers (Fig. 3llb).
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. 311 represent attempts to minimize the flyback or dead
time. They are not generally regarded as normal designs, but may be
required for some applications.
34]
EFFECT OF SCANNING ON DETECTION PROBABILITY
161
The Number of Pulses per Scan. In the system model adopted in
Paragraph 33, to develop an analytical method for calculating the proba
bility of detection, it was assumed that n equalamplitude 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. 311 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 33 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 345:
Twoway power pattern of antenna '^ ^eVo.ise^ (345)
where Q = angular position of the antenna
= antenna beamwidth (halfpower, oneway).
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. 312. In making this approximation, the
Fig. 312 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. (346)
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 33 for the example illustrating the calculation of the
singlescan 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.
35 THE CALCULATION OF DETECTION PROBABILITY
FOR A PULSED DOPPLER RADAR
With proper interpretation, the methods developed in Paragraphs 33
and 34 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 66,
whose functional block diagram is given in Fig. 625. 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.
SingleScan Probability of Detection. The idealized range for this
type of system is essentially given by Equation 39. This is restated in
Chapter 6 as Equation 639 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.
35] 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
639.
Detecting only the fundamental doppler signal in a filter output corre
sponds to the case of detecting a single pulse in Paragraph 33. Thus the
basic singlescan probability of detecting a fluctuating target should be
given by the expression in Equation 331 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 singlescan 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. 313, 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. 313 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,. (348)
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. 314, the
Ro=25 n. mi.
AR = 0.67 n. mi.
10 15
RANGE (n. mi.)
Fig. 314 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 feedthrough. 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.
314 at intervals of slightly less than a nautical mile represent the effects of
eclipsing.
35] 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 248 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
328 for finding the approximate average value of a single channel to deter
mine the average of the twochannel expression given in Equation 348.
Equation 328 was derived on the basis of an approximation to the curve
in Fig. 36 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" (349)
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 348, since only one term is retained:
The average value of Pd2 in each case will be of the exponential form first
given in Equation 331:
 _ 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/. (352)
Of more interest is the average value, which can be used as a smooth
replacement for oscillatory curves similar to that in Fig. 314 in many cases.
The function in Equation 351 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 351. Thus,
ave P,2  P/. (353)
On the average, then, the effect of straddling can be interpreted as a 3db
loss in signaltonoise 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 34 for determining the cumulative probability of detection
and the normalized curves in Fig. 39 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 multiplePRF method of
determining range in a highPRF pulse doppler system is described in
Paragraph 6G. 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 = falsealarm 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. 37 to be 6 db,
or i^ = 4. The basic singlescan probability of detection of a fluctuating
target is thus
p, = ,,4(ff/25)^^ /^ in n.mi. (354)
This probability has been plotted in Fig. 314 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. 38. Remembering that straddling has the
effect of doubling the effective value of A', the normalized range corre
sponding to Pd~ is defined by
35] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 167
^  '"{Is)  IT9 (•«5)
The normalized range decrement is thus
The resulting cumulative probability is also plotted in Fig 314.
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 signaltonoise 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 33, 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
33 where the sampling theorem is quoted. Similarly, the output of the
lowpass, 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. (357)
168 THE CALCULATION OF RADAR DETECTION PROBABILITY
Now in general the predetection signaltonoise ratio is proportional to
the reciprocal of the predetection bandwidth:
S/N'^'^ (358)
Also, the equivalent signaltonoise ratio of a fluctuating target is
proportional to a power of the number of video pulses integrated as in
Equation 333:
S/N ~ ny. (359)
The appropriate power y corresponds to the slopes of the curves in
Fig. 37.
The equivalent gain in signaltonoise 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. (361)
In Equation 333, 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. (362)
This approximation is often used for estimating performance where post
detection filtering is involved.
36 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 firecontrol
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, 8131818 (1951), upon which parts of this section are based.
36] 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^] ,^ ,^,
Uneway voltage pattern, circular aperture = , ^ .^ , . —  — (363)
(tt/J/a) sm p
where D = aperture diameter
X = wavelength
d = angle relative to aperture normal
Ji( ) = firstorder Bessel function.
For convenience, we represent the argument of this expression by x so that
the oneway 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 363 or (2Ji{x) jxY. This is
also equal to the oneway relative power pattern of such an antenna. This
pattern is illustrated in Fig. 315 where it is referred to as the twoway
voltage envelope generated by a scan over a single target.
The antenna beamwidth is normally defined as the width between the
halfpower points of the oneway antenna pattern. This is indicated in
Fig. 315. For a uniformly illuminated circular aperture the beamwidth is
related to the diameter and wa.elength by
Beamwidth, circular aperture = 58X/Z) degrees. (364)
The envelope of the received power on the twoway power pattern is
probably most significant for defining resolution. This is given by the
square of the envelope plotted in Fig. 315 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 oneway power pattern in Fig. 315 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, McGrawHill 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. 315 TwoWay 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 364; 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 halfpower 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. 316, the twoway 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
36]
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. 316 TwoWay 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
twoway voltage envelope is also shown in Fig. 316. 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. 316 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. 317, the twoway voltage envelope
Target 1
Target 2
39/2 9
Fig. 317 Average TwoWay 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 2tol
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 4tol or 6 db. The minimum
separating the two targets in Fig. 317 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 4tol 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. 318 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. 318 Resolution as a Function of Target Power Ratio.
Another factor which can affect resolution is the signaltonoise ratio.
The simplest way to account for this factor is to apply the already adopted
definition for resolution to the received signalplusnoise power envelope.
The deterioration of angular resolution with signaltonoise ratio which
can be determined in this manner is shown in Fig. 319.
\
V
■ —
Res
jiution
with n
noise
2 3 4 5 6 7
SIGNAL ■ TO ■ NOISE RATIO ~db
Fig. 319 Resolution of Two Equal Targets as a Function of SignaltoNoise Ratio.
36] 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 365 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©' + ^^" (365)
where 9 = antenna beamwidth
A^ = angular interval between pulses.
The antenna pattern described by Equation 363 and illustrated in Fig.
315 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
41 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) twoway propagation, and (2) backscattering 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 backscattering 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. 121). 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
42] REFLECTION OF RADAR WAVES 175
differential radial movement of a halfwavelength 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 surfacereflected 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.
42 REFLECTION OF RADAR WAVES
The radar equation for free space is derived in Chapter 3 (Equation 39).
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 freespace
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 freespace 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 backscattering 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 backscattered fieldatunitdistance to the
incident field strength. Its relation to cr is
(T = 47r/r". (42)
The radar length bears a relation to the received field strength similar to
that of radar area to received power in Equation 41. Thus, the received
field strength Er is given by
Er = IE,F' '^ (43)
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,. (44)
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,. (45)
Hence
a  A'G'. (46)
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
42] 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\^ (47)
(which is large \i A' l\'^ is large) we obtain from Equation 46
(7 = 4x(/f 7X^)2, for A'/\ » 1 (48)
One of the conditions assumed in
deriving Equation 41 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. 41. The difference in range PlaneWave 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. (49)
Assuming that the antenna may be treated as a point source, the roundtrip
phase difference between the fields reflected back to the source from these
two points is
A0 = IkLR = lKX'IXR. (410)
From Equation 43 the contribution of a differential length of the target
to the received signal in free space is
^£. = 1^ ^^2^(«+^«> i/ ' (411)
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 planewave 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" '
(413)
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 42 and 414 the effective radar area a' may be derived as
(415)
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. 42 Radar Cross Section as a Function of Range.
43] EFFECT OF POLARIZATION ON REFLECTION 179
in terms of the wavelength. Fig. 42 shows a plot of a' I a versus i?/X for
various values of 2Z,/A.
43 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 righthand and left
hand circular), we may denote an arbitrarily polarized incident wave by
the matrix
(f:)
Er = (V] (416)
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 crosspolarized component balances out in backscattering, but
otherwise it usually does not. Hence, in general, the backscattered 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
1polarization is horizontal and 2polarization 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 fieldatunitdistance 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 418: 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 418 by
^mn =47r/™„2. (419)
Then
(o"ii cri2\
C2I 0'22/
(420)
However, one could not deduce the polarization theorem above from this,
since the radar area is a scalar.
44 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 47
and 48.
Another important effect produced by target motion is the change in
frequency due to the doppler effect which was discussed in Paragraph 15.
If the radar and the target have a relative approach velocity V, and the
transmitter frequency is/o, the echo frequency is (see Equation 119)
/ = /o(l + IV I C) = /o + 2/7X0 = /o +/o. (421)
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.
45]
REFLECTION OF PLANE WAVES FROM THE GROUND
181
In ordinary (nondoppler) 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 nondoppler 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 nondoppler 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 (422)
as can be seen from Fig. 43. Hence
by Equation 421 the difference in
doppler frequency between these
two reflection points is
Af=?^=?5°^. (423)
A A
Thus the doppler frequency will be
proportional to radar frequency, to
= SD cos d
Fig. 43 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.
45 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 lowaltitude targets, heightfinding 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. 44 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 wellknown Fresnel equations^
sin d  (e  cos^ 0)1/2
sin e + {e  COS" 0)1/2
es'md  (e  cos^^)!/^
p//k^^^ (424)
Iprk'*^ (425)
€ sm f (e  cos20) 1/
where 6 = depression angle of the radar (see Fig. 44)
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' (426)
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
424 and 425 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. 45 and 46 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, McGrawHill Book Co., Inc.,
New York, 1941.
3F. E. Terman, Radio Engineers' Handbook, pp. 700709, McGrawHill Book Co., Inc.,
New York, 1943.
■•C. R. Burrows, "Radio Propagation over Plane EarthField Strenirth Curves," Bell System
T^f/^. J. 16, 4575 (1937).
5R. S. Kirby, J. C. Harman, F. M. Capps, and R. N. Jones, Effective Radio GroundConduc
tivity Measurements in the United States, National Bureau of Standards Circular 546.
45]
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. 45 Dielectric Properties of Pure, Fresh, and Sea Water.
10'
Sea Water
Tresh Water 6?S\t\ <^"
10 102 103
FREQUENCY (Mc)
Fig. 46 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. 47 through 410 show the magnitude and phase angle of the
reflection coefficient of sea water for a temperature of 10°C at several
wavelengths. Similarly, Figs. 411 and 412 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^, 269275 (1952).
184
REFLECTION AND TRANSMISSION OF RADIO WAVES
IPI
H—
^
'^
^
.
10 m
1 m
/
/^
><
<
/
/
y^
>
'
^
lOcml
J. iin
/
/
/
^
^
^
/
//
^W^
ly
/
/
\\
//
ly
V
10 20 30 40 50 60 70
90
Fig. 47 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.
45]
REFLECTION OF PLANE WAVES FROM THE GROUND
185
1234bb7 8u9 10
6°
Fig. 49 Expanded Plot of Fig. 47 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. 410 Expanded Plot of Fig. 48 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. 411 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. 412 Phase of Reflection Coefficient for Average Land (i' = 10, a
10~^ mho/m) as a Function of Depression Angle.
1.6 X
45]
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. 413, if both the direct and indirect paths are illuminated equally by
the radar antenna, the resultant field at the target is
Image
Fig. 413 Path Difference Between Direct and Indirect Paths.
E = Ea{\ + p^^^^^)
where Ed is the field due to the direct wave, and Ai?
(427)
Ih sin d is the path
difference.
The ratio EjEd, obtained from Equation 427, thus is the propagation
factor F due to the presence of the ground. In Paragraph 42, 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 427, we obtain
F = E/Ed = 1 + pe''"" ^*" '". (428)
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 428. 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,
1725 (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. (429)
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 freespace 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
freespace variation which is proportional to the inverse fourth power of
range as indicated in Equation 41.
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 424 and 425, can be approximated as
p// = 1 +2(6 l)i/2sine (432)
PK = 1 + 2e(e  ])■/ sin Q. (433)
45] REFLECTION OF PLANE WAVES FROM THE GROUND 189
Similarly, the exponential term in Equation 427 may be approximated as
^iiKh sin e ^ I _ j2Kh sin d. (434)
Substituting these approximations into Equation 428, and retaining only
the firstorder terms in sin d, we obtain
Fh = 2[(e  l)^/2 \jKh] sin d (435)
Fv = 2[e{e  l)i/2 \jKh] sin 6. (436)
Thus, for sufficiently small d, both Fh and Fv are proportional to sin 6.
But sin 9, as shown by Equation 430, is inversely proportional to range.
Hence below the first lobe it follows that F is also inversely proportional
to range:
F oc R\ (437)
Therefore the received echo power, which is given by Equation 41, will be
inversely proportional to the eighth power of the range in the region below
the first lobe:
P,i oc R^ (438)
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 fourthpower
law to an eighthpower law for a target which spans more than one lobe
will be discussed in Paragraph 410.
Equations 435 and 436 show how the resultant (oneway) 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" ^ (440)
Hence the ratio of the fields at the target with vertical and horizontal
polarization will be
Fv/Fh = 6. (441)
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 435 and 436. 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 firstorder expansions of F in powers of sin 6 are limited in their
ranges of validity. This can be seen, for example, in Fig. 47. 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.
46 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. 414.
Direct \Na\je /?
Fig. 414 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 (442a)
Ji'., = /;o  A/;2 = //2  doVla (442b)
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
418, the effect of average or "standard" atmospheric refraction may be
46] 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. (443)
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. (444)
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. (445)
Once this is solved for di, h[ and h'^ may be calculated from Equation 442
and the remainder of the geometry handled like a planeearth 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 (446)
where D is the divergence factor. This is given by
y^ajsmd
(447)
(•^r
For very small values of 6 the divergence factor causes reduction of the
effective reflection coefficient p' given by Equation 446 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 446 is restricted:
h
, . . . . (448)
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 GroundWave Field Intensity over a Finitely Con
ducting Spherical Earth," Proc. IRE 29, 623639 (1941).
192 REFLECTION AND TRANSMISSION OF RADIO WAVES
47 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 48.
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 49. 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. ^~^^ Pulsetopulse measurements were made of
both fighter and bomber categories, with propellerdriven and jetpropelled
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: B16
Amplitude Distributions and Aspect Dependence, NRL Report C346094A/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 XBand Values, NRL Report
C3460132A/52, 24 Oct. 1952.
"W. S. Ament, F. C. MacDonald, H. J. Passerini, Quantitative Measurements of Radar
Echoes from Aircraft VIII: B45, 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: F5I, NRL Memorandum Report No. 127, 4 March 1953.
47]
RADAR CROSS SECTIONS OF AIRCRAFT
193
Fig. 415 shows the cumulative amplitude distribution of a 2second
sample of echoes from the B36, plotted on socalled 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. 415 Cumulative Amplitude Distribution of B36 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 2second 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 B36 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, McGrawHill 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. 416 shows a 5second sample for the B45 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 5second sample at another aspect,
in which the aspect angle varied 4° (Fig. 417) 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. 416 Cumulative Amplitude Distribution of B45 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 B45 varied by more than 4° gave a satisfactory fit to the
Rayleigh distribution.
Fig, 418 shows a set of distributions for the F51 singleengine 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. 417 Cumulative Amplitude Distribution of B45 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
pulsetopulse photographs shown in Fig. 419. 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 headon 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. 420 to 422 show
the results for the B36, B45, and F51, 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 B36 (which has a rather flat fuselage) as
shown by the 9380Mc plot in Fig. 420 (broadside data for 1250 and 2810
Mc were saturated and so are absent from this figure). The F51 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. 418 Amplitude Distribution of F51 Echo, Showing Effect of Propeller
Reflection.
The data in Figs. 420 to 422 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 41. The B36 and
B45 averages are roughly independent of frequency, but the F51 average
a increases approximately proportional to frequency.
E 41
COMPARISON OF B36, B45, AND FSl
AVERAGE RADAR AREAS
Frequency,
Mc
Average a (in db > Im"^)
Excluding Broadside Region
Average cr (in db > \m)
Including Broadside Region
B36 B45 F51
B36 B45 F51
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.
47]
EFFECT OF EARTH'S CURVATURE
1250 MC/S
197
lSEC
2810 MC/S
9 380 MC/S
Fig. 419 PuIsetoPulse Records of F51 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
B36
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°
B36
2810 Mc
Median a Vs.
Aspect
•
'^ •
T
k
U
i
•
•
•□
x=3"Ele
A = 4°
ation
•T
°
t D
» »
•
u *
•
o5°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
▼ •
"?
° '□
B36
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. 420 Plot of Median Echo of B36 Averaged over 5° of Azimuth.
48 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 1114, 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 — ClosedLoop 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.
48]
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. 421 Plot of Median Echo of B45 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
F51
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. 422 Plot of Median Echo of F51 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 47, e.g. propeller modulation. A clear example of this
is shown in Figure 423, 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. 423 Spectrum of Amplitude Noise of SNB (TwoEngine Transport) Aircraft
in an Approach Run, Showing Spectral Lines Due to Propeller Modulation (= 9400
Mc).
an approach run of an SNB (twoengine 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 F51, as illustrated in Fig. 419.^''
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. 424 Spectrum of Amplitude Noise of B45.
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 F51, 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 44.
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 straightline flight and on the
duration of the observation (that is, on the length of sample).
Figure 424 shows a spectrum^^ of the amplitude noise of the B45 for
which the observation time was 5 seconds. The voltagetime plots from
which the spectrum was prepared are shown in Fig. 425. The 9380Mc
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. 425 VoltageTime Plots of B45 Fxho, Showing LowFrequency 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.
48] AMPLITUDE, ANGLE, AND RANGE NOISE 203
spectrum in Fig. 424 shows two peaks, not quite resolved completely, at 5
and 6.6 cps. In Fig. 425 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 1250Mc plots. From an
examination of the drawings of the B45, 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 62) 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. 426
'BoresightAxis illustrates this model. Consider the
following type of tracking system.
A dualfeed antenna produces lobes
Fig. 426 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 squarelaw 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^'^ (449)
and by the lower lobe
El = Ea{\  ge) + Eb{\ + ge)e^\ (450)
The difference channel voltage is then
En = k{\Eu\'  \El\') (451)
= 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). (452)
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. (454)
(t)'
(456)
Comparing this with Equation 453 we see that the apparent reflection
center of the dualreflector 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 456, 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 — • (457)
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. 427, 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. 427 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. 428 Angle Noise Samples for R4D.
'9R. H. Delano, "A Theory of Target Glint or Angular Scintillation in Radar Tracking,
Proc. IRE^l, 17781784 (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 lineofsight 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. 429 Spectra of Angle Noise for SNB Aircraft.
48] 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. 428. 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. 429 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'~ (458)
which corresponds to the transfer characteristic of a singlesection RC
lowpass 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 lowangle 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. 413). 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 lockon 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. 430 shows typical time plots of range noise for
several classes of target. Fig. 431 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^ii^
m
ift
ELAPSED TIME
SNB
SNB
SNB
Fig. 430 Sample Time Function Plots of Range Noise from the (a) PB4y (Four
Engine Bomber) at 180° Target Angle, (b) ANB (TwoEngine 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. 431. 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.
49 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
49]
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. 431 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 47 and 48, 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 CrossSections, XV, University of Michigan, Engi
neering Research Institute, Report 22601T, 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
44, and the spectrum can be determined. This calculation can be expedited
by the use of a target simulator. ^^ As an example, Fig. 432 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. 432 Spectral Distribution of Angle Noise from a B17 (FourEngine Bomb
er) Aircraft (Actual Measurement).
Jki
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. 433 Spectra! Distribution of Simulated Angle Noise from a B17 Target
Simulator.
measured angle noise spectrum of a B17, while Fig. 433 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.
410] SEA RETURN 211
410 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, (459)
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) (460a)
A = i?2$e/sin d {6 large) (460b)
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. 481581, 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 backscattering 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 45, 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
460a above, so that the reflection
Log(R\/4/i)
Fig. 434 Composite Plot of SeaClutter
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. 434 shows a composite plot
of seaclutter 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. 435.
^'•M. Katzin, "Back Scattering From the Sea Surface," IRE Cofitrn/ion Record, 3, (1),
7277 (1955).
25M. Katzin, On the Mechanisms of Radar Sea Clutter, Proc. IRE 45, 4454 (1957).
410] SEA RETURN 213
As was shown in Paragraph 45,
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^^. 435 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 41 and 459:
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 (462a)
F=6{Rt/R)\ R> Rt (462b)
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/\ (463)
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 (464)
in which //i/iu is the cresttotrough 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. 436 shows an example of this. Spikiness is explainable by the com
214 REFLECTION AND TRANSMISSION OF RADIO WAVES
Bm$i^> ■
Fig. 436 Expanded AScope 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 Targetlike Echoes in Sea Clutter, NRL Report
4902, March 19, 1957.
410]
SEA RETURN
215
The variation of o*^ with depression angle is a function of wind speed.
Fig. 437 shows measurements on vertical and horizontal polarization at
24 cm as reported by MacDonald,^^ while Figs. 438 and 439 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. 437 Sea Clutter, 1250 Mc: Solid Line = Transmitted and Received Vertical
Polarization. Dotted Line = Transmitted and Received Horizontal Polarization.
20
10
SlO
b
20
30
40
1520
Knots
Fig. 438
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 — ^ /
1520 \ / /
 Knots. \ n
"2025 \ \ ///' /
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. 439
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), 2932 (1956).
2*C. R. Grant and B. S. Yaplee, "BackScattering from Water and Land at Centimeter and
Millimeter Wavelengths, Proc. IRE 45, 976982 (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 backscatter most effectively are those with perimeters of about a
halfwavelength. The backscattering 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 backscattering 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 upwinddownwind 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 1.259.4 kMc, and gave the formula for a" upwind
at small depression angles,
(7« = (2.6 X \0'W^i'\' (465)
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 9.435 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, 838850 (1954).
3"J. C. Wiltse, S. P. Schlesinger, and C. M. Johnson, "BackScattering Characteristics of
the Sea in the Region from 10 to 50 kmc, Proc. IRE 45, 220228 (1957).
411] SEA RETURN IN A DOPPLER SYSTEM 217
may vary somewhat from time to time, depending on the condition of
the sea surface.
411 SEA RETURN IN A DOPPLER SYSTEM
The doppler shift due to relative motion of radar and target was discussed
in Paragraph 44, and the echo frequency due to a transmitted frequency/o
was given as
/=/o + 2F/X (466)
where V is the lineofsight 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 466 corre
spondingly may be written as
/=/o +/.+/.. (467)
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 (468)
/i = lVrl\. (469)
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. 440,
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)
(470)
= cos ^o{[l  (A0)V2] cos 00 ± A0 sin0o}.
218 REFLECTION AND TRANSMISSION OF RADIO WAVES
Fig. 440 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 Xband pulse radar
(9400 Mc/sec) with a horizontal beamwidth of 3° (A</> = 1.5°), and an
aircraft speed of 200 knots. Then from Equation 469, /i = 6.44 kc. At
grazing depression angles along the ground track {do = 0), the induced
doppler spectrum has a halfpower width of 0.000343/i = 2.2 cps, while at
45° to the ground track the halfpower 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 469) 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.
412] GROUND RETURN 219
Frequency spectrums were obtained for 15second samples of recorded
data (corresponding to 3750 ft along the sea surface), and also frequency,
Bscope 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 halfpower points of 2 to 3 knots (60100 cps at X band). The
corresponding B display was generally smooth on upwind and downwind
edges for all ranges. Fig. 441 (a) shows a sample of the A and B displays
for a low sea condition (wave height 2 ft, wind 9 knots). The 3db 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. 441 (b) shows a sample
of the A and B displays for a medium sea (wave height 5 ft, wind 16 knots).
Here the 3db 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.
412 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 14.
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 height2.0ft.
3db bandwidth82cps
WIND
9 knots
80°
SAMPLE NO. 140
Sept. 22,1954
Altitude2500ft.
Range 15,080 yds.
Depression3.l7°
tieightS.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. 441 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 nndtipktimearoioid echo (MTAE). This
clutter is therefore important for ground targets whose range is given by
R„ = R.^nR^rs (471)
412] 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
secondtimearound 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. 442 shows their results for a treecovered 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
_
XU
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. 443 Comparison of a^ for Green
Fig. 442 o^ for a TreeCovered Terrain. Grass and Dry Grass.
Fig. 443 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 1520 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
416). 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 manmade structures are often too strong to be fully explained
in terms of their size, and that a certain amount of cornerreflector 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 targetseeking missile
application.
413 ALTITUDE RETURN
In Paragraphs 410 to 412, we have discussed the backscattering
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. 100101, McGrawHill Book
Co., Inc., New York, 1947.
I
413]
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. 437, 438, 439, and 443).
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 460) 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. 442, 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 460b. 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. 437 and 438.)
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 pulselength limited,
and the received power of the alti
tude line will vary as the inverse
cube of altitude in accordance with Equation 460a (see Fig. 444).
20
16
/
K
/
/
/
/
I
t
V
1QV'.
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. 444 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 areaextensive,
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. ^^
414 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 410 and 411.
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. 437 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 46.
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 (nondoppler) 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 NearVertical Incidence,
Proc. IRE ^5, 228238 (1957).
414] 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. 434, and
will be given by Equation 461 :
UtYR'
(472)
In this we may insert the values of A and F given by Equations 460a and
462, respectively. The horizontal beamwidth <!> and the gain of the radar
antenna may be expressed by
$ = y^ (473)
G = ^^ (474)
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 465 for a^,
(T« = ^oA (475)
then Equation 461 becomes for the received clutter power
^c  (^,^3) (476)
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 462 and 464. 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 41
If the target is a surface target of uniform section and height Ht, then F^ is
to be replaced by F of Equation 462, with its transition range, Rt, given by
Equation 463
Rt^^^ (478)
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'
(479)
in which h = kb'^/i4T)K
Plots of Equation 476 for a specific sea condition and of Equation 479
will then be as shown in Fig. 445. From this, it follows that the range scale
Fig. 445 Target and Clutter Power Relations vs. Range.
in which the targettoclutter ratio may
REGION 1
REGION 2
REGION 3:
kahLR
kahLR
(480)
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 targettoclutter 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. 445 that the largest targettoclutter 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,,)
415] 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
445. 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. 445 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, huntandkill). 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 nondoppler radar. Doppler radar offers
the additional possibility of increasing the targettoclutter ratio by
exploiting differences in the target and clutter spectrums. In order to
achieve a gain in targettoclutter 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 411 (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 1015
knots, for the 3° beamwidth assumed. Smaller beamwidths would reduce
this figure proportionately.
In principle it is possible to improve the targettoclutter 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 66, below.
415 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
416.
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. 446 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. 446 Atmospheric Attenuation Due to Oxygen.
level and at 4 km. Fig. 447 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. 646664.
415]
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. 447
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), 1218 (1957).
^''H. H. Theissing and P. J. Caplan, "Atmospheric Attenuation of Solar Millimeter Radia
tion," J. App. Phys. 27, 538543 (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
lOo.2adbR (4_81)
where adt is the oneway attenuation in db per unit distance.
416 ATTENUATION AND BACKSCATTERING 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 backscattering 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 backscattering
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 backscattering is proportional
to D^. Hence the attenuation through small rain drops is proportional to
the total liquid water content, but the backscattering is proportional to
SD^. Thus the larger drops are much more effective in backscattering than
the smaller ones.
Because of the dispersion of water in the microwave region (see Para
graph 415) 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. 671692 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, 522545 (1954).
417]
ATTENUATION BY PROPELLANT GASES
231
Calculations of attenuation and backscattering (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. 448 and 449.
The total radar area is found by
multiplying the value found in Fig.
449 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. 448 The Variation of Attenuation
with Wavelength for Various Rainfall
Rates.
417 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, 452460 (1943).
42A. S. Locke fEd.), Guidance, pp. 118124, 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
13

\
0.3 0.5 1 3
X(cm)
Fig. 449 The Variation of Radar Cross Section of Actual RainFilled 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 419).
The ion density varies greatly with the type of fuel. Furthermore, the
ion density is influenced markedly by small quantities of lowionization
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 acidaniline 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 highenergy fuels.
■•^H. F. Calcote, "Mechanisms for the Formation of Ions in Flames," Cotnbiistion and Flame
1, 385403 (1957).
^''F. P. Adler, "Measurement of the Conductivity of a Jet Flame," J. AppL Phvs. 25,
903906 (1954).
418] 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 5060 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.
418 REFRACTION EFFECTS IN THE ATMOSPHERE
In computing the power received from a target by means of the radar
Equation 41, allowance was made for a process other than freespace
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
threedimensional, 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 earthflattening 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^ + ^^^ (482)
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 482
f = (: + ^)x.O' (483)
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
 =  + % (484)
a, a dh
In temperate climates an average value oi dn jdh is about —\/{4a). Hence
from Equation 484
ae = \a (485)
which is the socalled "fourthirds 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 onetoone 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). (486)
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 multihop 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.
411
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. 450. Curve (a) is the standard
Fig. 450 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 487, 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. 450d.
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. 450c.
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 tradewind
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 487, 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 5075 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 oneway path,
which would mean a 30db 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 AirtoAir and AirtoGround Experimental Data, Final Report Part III,
Contract AF33(038)U)91, School of Electrical Engineering, Cornell University, 10 Dec. 1951.
418] 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
51 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. ^~^
52 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) McGrawHill 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
52] 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 welldefined 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 (51)
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.. (52)
TT.
The functions /(/) and F{co) are often regarded as constituting a Fourier
transform pair which are mutually related by Equations 51 and 52. With
this terminology, Equation 51 is said to transform /(/) into the frequency
domain, while the operation indicated in Equation 52 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. (53)
The spectrum is easily calculated:
F{.^) = f (.0 e^^dt = — ^ (54)
In this case, the spectrum is complex. Upon performing the inverse
operation indicated by Equation 52, the exponeni.ial function given by
Equation 53 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 51 :
!F(co)2 = F(co)F*(co)
= / f{ti)e^'^'^dtA f{t2)ei'''^dt^ (55)
= 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. (56)
The righthand side of Equation 56 is of exactly the same form as
Equation 51; 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 52 is applicable, and <p{t) can be expressed by
<p{r) = l_J(t + r)/(/) dt = ^l_^ \F(o:)\' .^^ do:. (57)
When T is set equal to zero, the following important special case is obtained.
^(0) = /_y'w^^ = ^/_^ \F(^)\' d^ (58)
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).
52] FOURIER ANALYSIS 241
Continuing with the example adopted in Equation 53, 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 (59)
^(0) = i
Also, the absolute value of the spectrum is easily obtained from Equation
54 in this case:
By virtue of the relationship indicated in Equations 56 and 57, 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 SS) 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 singlesection, lowpass, RC filter shown
in Fig. 51. 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^— (511)
R
AAAAA/ 1— ^ Transfer Function:
T
X
1+jRCc
Impulse Response= (l/RC)e
f>0
Fig. 51 RC Filter.
*See Paragraph 53 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 53 and 54 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;). (512)
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)\\ (513)
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 512 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. (514)
Substituting r = /] + /2 and dr = dti, and interchanging the order of
integration,
^o(
0;)=/ e^'^'drl fi{t>)y{r  t2)dt.. (515)
The righthand side of this expression is again in the form of Equation 51 ;
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. (516)
5See M. 1'". Gardner and J. 1"". Barnes, Transirn/s in Lineur Systems, Vol. 1, pp. 233236,
John Wiley & Sons, Inc., New York, 1942.
53] IMPULSE FUNCTIONS 243
53 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. 52. In this figure a
rectangular function of height A and ^
width 1 I A is shown centered at the pic. 52 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. 52 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. 52:
/" r (i+i/2^
/(/)6(/  t,)dt = lim A J{t)dt = /(/:). (517)
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. (518)
/.
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 52.
The constant spectrum given by Equation 518 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. 52 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 (519)
A^o.livJA /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 51 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
(520)
= 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 519,
a constant over the entire range of w was approximated by a truncated
function which approached che constant function in the limit. Other
54]
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.
53. As yf — > 00, this triangular function obviously approaches a constant.
2A 2A "
Fig. 53 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 55, where in an
example of a noise process the expression in Equations 521 turns up as part
of the power density spectrum (Equation 540).
54 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. 54 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. 54 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^
(522)
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. SS 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. 55 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.
Secondorder probability density function of a Gaussian noise process
27raVl
exp
■vi + 2p.Vi.V2 
2<tH1  p')
(523)
54] RANDOM NOISE PROCESSES 247
In this expression Xi and X2 are values of the noise process at times /i and /2,
02 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 xi 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 dc level.
For the process whose probability density is given by Equation 522, the
average value corresponding to the dc 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\ (524)
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 dc component:
^(±00)= p. (525)
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 dc term and dividing by the variance:
<p{t)  v?(°o)
P{t) = —77^ 7— T (527)
(p(0)  <p(oo)
For a Gaussian process with the joint probability density given in
Equation 528, the autocorrelation would be computed in the following
manner:
^W = o__2./l ~^j_^j_^ ''''''
exp I 2a'^(l  2) J^.vi^.V2 (528)
[
(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.
55 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 inphase 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 nonzero 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
spectrum. 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 52, 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
55] 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. (529)
Energy spectra will be given by expressions similar to Equation 56. 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. (530)
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 530. 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. (532)
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 532
gives
^(r) = 2^ / A^^^'^Voj. (533)
When T is set equal to zero in this relation
^(0) = C72 f .^2 = i / W(^) du^. (534)
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 534 can be regarded as a general
ization of Parseval's equality given in Equation 58.
At the end of Paragraph 52 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^). (535)
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 /,) (536)
Denoting the Fourier transform o( h{t) by H{co), the Fourier transform of
/„(/) is given by
F„(co) = [ h{t  t,)ei'^^dt = //(co) i; ^'•"'^. (537)
The power spectrum is simply the absolute square of F(oo) divided by the
observation time:
^\FM\^ = ^[//(c.)E.^][//*(co)2:%^'].
= ^j^{^)\'zi:^^''''"'''' (538)
ZI 1 1
55]
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
ej'^itktDdt^ ... dtn. (539)
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
(540)
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 521. The
power density spectrum over all time, then, will have the following form:
lim
^\F{o^)\'
i7(co)M7 + 2x7^5(0.)].
(541)
The singular part of this spectrum corresponds to a concentration of
power at zero frequency or the dc component. If h{t) has no such dc
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 53. An element of such a noise process
might then look like the example shown in Fig. 5G. The energy spectrum
of the exponential function has already been computed in Equation 510.
Fig.
56 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)]. (542)
The autocorrelation function, which is just the Fourier transform of the
power spectrum, has already been partially computed in Equation 59 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. 51 , 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 (544)
where e = electronic charge
/ = average current (dc)
AF = observation bandwidth.
We suppose that each function h(i — tk) in the sum in Equation 536
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 541. 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'. (545)
The mean square value of the noise currents corresponds to the integral of
the nonsingular term //(co)^7, in Equation 541. 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 lowfrequency power density of the electronic pulse
56] NONLINEAR AND TIMEDEPENDENT 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 541, yields the following expression for the mean square
noise current:
(KTp = \H{W'i2^F = leHC^FI^ = 2e/AF. (546)
Comparison with Equation 544 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.
56 NONLINEAR AND TIMEDEPENDENT OPERATIONS
In tracing signals and noise through radar systems, we find that the
operations of many components are either nonlinear or timedependent.
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 (547)
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. (549)
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 523.
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. (550)
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^ (551)
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 548 and
549:
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 55 (Fig. 56) 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 542 and 543. There will be no dc term, and
in order to have unit variance 7 = 2:
<p(t) = a'p(T) = .IH (554)
Nic) = y^, (555)
1 + CO
56] NONLINEAR AND TIMEDEPENDENT OPERATIONS 255
From Equation 550, the autocorrelation function of the y process will be
<pAt) = jT^ = 1 + 2^21^1. {SSG)
The Fourier transform of this expression gives the power spectrum of the
y process :
NyiiS) = lirdico) +
4 +
(557)
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. 57a Uniform Spectrum {x Process).
spectrum is shown in Fig. 57a. 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
\ 1kWt J ■
(558)
(559)
At the end of Paragraph 53 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. 57b and it is represented symbolically by
Ny(a>) = 2Tra^8(o^) + i<r'/fV)il  a;/47r/F), Ico] < 4w^. (560)
cr" (Impulse Strength)
Fig. 57b Triangular Spectrum (y = x"^ Process).
256 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS
An application of this result is made in Paragraph 57 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 timedependent 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 = ■'''1V2 cos a)mt cos COm (/ + t)
(561)
= (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. (562)
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 (563)
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
S6]
NONLINEAR AND TIMEDEPENDENT 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. 58. Symbolically, the output of this device can be repre
Clamped Output
nput Signal
7 H TIME
Fig. 58 Operation of a Clamping Circuit.
sented by
y{t) = x{tk), t, < t < tk + i = tk\ T. (564)
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). (565)
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. 59a. The Fourier transform of a triangular function has
Delay Time  r
Fig. 59a Autocorrelation Function of Pulse Stretcher Output with WideBand
Noise Input.
already been determined in Equation 521
spectrum of the stretched process will be
On using that result, the power
Nyic.)
,^( smcoT/2 y
V coT/2
(566)
The autocorrelation and the power spectrum of the pulse stretcher output
in this case are both shown in Fig. 59. 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 lowfrequency spectrum of width approximately 1 jT cps.
NyiOO)
6f 47r 27r 2ir Air Sir
Nondimensional Angular Frequency.cof
Fig. 59b Power Density Spectrum of Pulse Stretcher Output with WideBand
Noise Input.
57 NARROW BAND NOISE
Signals in radar systems normally have the form of a radiofrequency
carrier modulated by a lowfrequency 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.
57] 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 lowfrequency noise process by the carrier frequency. The
carrier frequency signal can be represented by either the inphase or
quadrature component, and, in general, the narrow band noise will be
composed of both components. Denoting the lowfrequency noise processes
corresponding to the inphase 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. (567)
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 568 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)] (568)
— (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 562. 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 563:
A^.('^) = ihWio:  CO.) + A^(co F COe)]. (569)
From this expression, we see that the spectrum of narrow band symmetric
noise has the same form as the lowfrequency 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 lowfrequency
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. (570)
The peak signal power is denoted by 6" = a} jl.. The noise power is denoted
by A^ = 0"^ so that the signaltonoise power ratio when the signal is present
is
SIN = ay2a\ (571)
With the narrow band noise represented as in Equation 567, the signal plus
noise has the form
Signal plus noise = {a { x) cos wj + jy sin o^J. (572)
A typical power spectrum of a cw signal plus noise is shown in Fig. 510.
I 5 6 Function Continuous
t Signal ^^ Noise
i I Spectrum i^^Spectrum
— COc ^c
Angular Frequency
Fig. 510 Power Density Spectrum of CW Signal Plus NarrowBand 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 572 in the following form:
Signal plus noise = yjia + x)^ + y'^ cos ccj +
57] NARROW BAND NOISE 261
'^ (573)
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^ (574)
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. (575)
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 (576)
dx dy ^ r dr dd.
Substituting these relations into the expression in Equation 575 and
integrating over the variable d gives
dp = Pi{r)dr = — exp
2a'
1 P'^
dr ^ / exp [ar cos e/a^dd. (577)
2xJDo
The integral in this expression can be recognized as a representation of a
zeroorder 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'). (578)
A curve showing Pi{r) for some representative values of S /N is given in
Fig. 511. In the two extremes of very small and very large values of the
signaltonoise 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. 511 Probability Density Functions of the Envelope of NarrowBand Signal
Plus Noise.
^^^^^^Vlx^^^P
^72(72 ,»1,
(579)
(580)
The first of these forms is often called a Rayleigh probability density and
corresponds to the case of noise alone. When the signaltonoise ratio is
large, the envelope has approximately a normal distribution as is indicated
by Equation 580.
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 signaltonoise 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 seconddegree 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.
57]
NARROW BAND NOISE
263
graph 56 in connection with the discussion of a squarelaw 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'^
+ .yij'2 + 4^2—2 (581)
= (^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 dc, (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. 512 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. 512 Power Density Spectra of the Square of the Envelope of a Sinusoidal
Signal Plus NarrowBand 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 dc 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.
58 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 signaltonoise
ratio. A block diagram showing the elements of the receiver composing this
angle tracking loop is given in Fig. 513. 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. 513 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 (582)
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).
58] 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 ac 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 squarelaw
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.
513. The AGC loop maintains the dc 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 lowfrequency 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 ac 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 57 (Equation 567). That is, inphase 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 signaltonoise ratio will be
Signaltonoise ratio  SIN = a^/ld"^. (583)
This is an average signaltonoise ratio because, on a short term basis, the
signal power is modulated by the ac 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.
(584)
The video envelope from the square law detector during a pulse consists of
the sum of the squares of the inphase 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. (585)
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
57 (Equation 574).
The average value of the video signal plus noise during a pulse will consist
of a dc term and the ac error signal:
Average video = ^2 ^ ^2 j^ 2(j^ [ la'ke cos (co./ + ip). (586)
The AGC loop will act to maintain the value of the dc 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 dc 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 ac error signal during a pulse becomes
/ S/N \
\\ + s/n)
AC error signal = f , , c/at ) ^^e cos (co^/ + <p). (587)
One effect of the noise is to introduce a factor depending upon the signal
tonoise ratio which attenuates the ac 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 ac error signal delivered to the
phasesensitive demodulator is essentially of the form given in Equation
587.
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 S6. 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:
(588)
€ COS ip.
58] 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 lowpass RC filter, and the power transfer function has the
following familiar form.
Angle tracking loop power transfer function = ^ ^^^ (589)
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
signaltonoise ratio. Thus we shall express K as the product of this factor
and a design bandwidth /3 achieved at high signaltonoise 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 lowfrequency 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 )'
(591)
The next problem is to determine the magnitude D of the inputpower
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 57. The spectrum
of the video noise is thus pictured in Fig. 512, and its autocorrelation
function is given by Equation 581. Dividing the noise power in the square
of the envelope as determined from these sources by the square of the dc
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. 512, 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 pulsetopulse fluctuations due to internal noise should be very nearly
independent.
The effect of the pulse stretching operation is considered next. In Para
graph 56 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 566 which was
illustrated in Fig. 59. 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
(593)
(1 + SINY
The effect of the demodulator on its input spectrum was established in
Paragraph 56 (Equation 563). 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.)]. (594)
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;
59] 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
(595)
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 595 into the relation
already derived for the mean square tracking noise (Equation 591) 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 signaltonoise ratio of 1.35 db. The decrease in tracking
noise at small signaltonoise 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.
59 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 delayline 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 dc 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 delayline cancellation unit. This unit
provides the difference of successive returns as an output, that is, returns
separated by the repetition period T. Fig. 514a shows a block diagram of
such a cancellation unit, while Figure 514b illustrates its operation. This
Input
Time
Delay
piujT
Output
Fig. 514a DelayLine Cancellation Unit.
Input
Signal HI
Delayed
Signal
i~x,.npv^^xi ^
\
'^t
/^Tx „ /IN ,, /I ,
Cancelled
Signal il^ ^^\ly '^^\iy
Fig. 514b Cancellation of Clutter Echc
sort of unit will attenuate the dc 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/. (597)
The modulating functions .v andjy are independent Gaussian noise processes
with identical spectra. The clutter spectrum is determined by the motion
59] 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^)/. (598)
The peak signal power is denoted by 6" = a} jl.
We shall assume that the signal plus clutter is rectified by a squarelaw
second detector to give the following video signal during a pulse:
Video = V = \a cos (coc + co^)/ + v cos (xsct + y sin cor/]". (599)
The video frequencies are, of course, limited by the video bandwidth.
Squaring this expression and retaining only the lowfrequency components
which will be passed by the video amplifier gives
Video = y = \{a}  a; + .V^ + lax cos cod/ — lay sin cod/). (5100)
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\ — vi
= ^{xi^  X2'' + yi^  y2^ + 2axi cos co^/i (5101)
— 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 signaltoclutter 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 signaltoclutter ratio = {S/C)v = (y+c — vl)/ v^ (5102)
Similarly, a signaltoclutter ratio is defined for the residue signal output
of the cancellation unit:
Residue signaltoclutter ratio = {S/N)r = (r+c — ^c)/ ^c' (5103)
With these definitions, a gain factor may be determined as the quotient of
these two ratios:
System gain factor = G = ^^^ (5104)
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 5100 and 5101, and the
details will not be given here. The following average values originally
determined in Paragraph 56 in connection with a discussion of a squarelaw
device are used in these calculations:
X = y = x^ = y^= xy — xy —
x^ = y^ = c, X1X2 = yiy2 = Cp{T) (5105)
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) (5108)
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 5108, which showed how a performance equation could be arrived
at by a straightforward application of the techniques for signal and noise
analysis previously developed.
510 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
510] 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 receiverfilter 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 receiverfilter as that which maximizes this
signaltonoise ratio. We shall subsequently indicate how a maximum
signaltonoise ratio gives a maximum probability of detection for a fixed
falsealarm rate and thus provides the most reliable detection in this sense.
It will turn out, interestingly enough, that the receiverfilter 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 receiverfilter
S(o}) = spectrum of s(t)
So{t) = signal output of receiverfilter
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, 940971
(1946^.
274 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS
Y(o:) = transfer function of receiverfilter
n(i) = noise input to receiverfilter
D = power density of noise input to receiverfilter
no(f) = noise output of receiverfilter
02 = noise power in output of receiverfilter
z^ = peak signaltonolse power ratio in output of receiverfilter
/o = observation time
The target echo is represented by a signal input to the receiverfilter
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). (5109)
The output waveform will, of course, be simply the inverse Fourier trans
form of So{oi) :
i/.
Output signal = r„(/) = ^ / Yico) S{oo) ^^"Wco. (5110)
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
02. 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 =^ 02 = / Dy(a))Vw. (5111)
The output signaltonoise ratio at the time /^ is denoted by 2:
Output signaltonoise ratio = 2 = So(/o)/(r~ =
kl.
y(co).V(a;)^^'Vco
■/ Dy(co)Va;
(5112)
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
510] 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.
(5113)
This expression is represented by
^m' + 25m + C ^ 0. (5114)
Because this polynomial is always greater than zero, the equation
^m' + 25m + C = 0. (5115)
cannot have distinct real roots, and its discriminant must be less than or
equal to zero:
B'JC^O. (5116)
Substituting for yf, B, and C gives the real form of Schwarz's inequality:
(/^VkW^^)' ^ /^ f{x)dx j\Kx)dx. (5117)
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. (5118)
This immediately leads to the more general form which we need. Putting
/(;c) and ^(^) in place of/{x) and g{x) in Equation 5118:
/
f(x)g(x)d>
j\f{x)\'dxj\g{x)\^dx. (5119)
Substituting the righthand side of this inequality for the numerator in
Equation 5112.
^j_^ \Sic.)\'dc.^j_^ y(a;)lVc.. (5
^J_^Dy(co)Va;
120)
The integrals involving the filter transfer function can be canceled:
2^^(^)^/_J^MVco. (5121)
276 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS
Thus the righthand side of this inequality is independent of the filter. Since
the signaltonoise ratio is never greater than the righthand 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 5112, 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)^'"'«. (5122)
A receiverfilter 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
(5123)
= sUo  /).
Denoting the input noise process by n{t) and the output noise process by
no(t) and using Equation 516 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. (5124)
In particular, the output at the observation time to is simply
soito) + rioito) = J_^ \sir) + niT)\siT)dr. (5125)
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 5121 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 signaltonoise ratio is equal to the ratio of
the received signal energy to the power density of the noise. That is, the
maximum signaltonoise 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
510] 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. 51 5a.
(5126)
Fig. 51 5a Pulse Train Signal.
Envelope = P(co)
^^
T"~^>^ (Width of spectral
teeth  27r/4fr)
.aJ \^^ L^Al'x?
^H 27r/fr
Fig. 515b Spectrum of Pulse Train Signal.
From Equation 5125, the signal component of the filter output at the
observation time will be
Filter output (signal) = So{t^ = / s{T)dj (5127)
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 5125. The noise output will be
/oo „_i
n(T)J2 P(r  kT)dT. (512S)
The noise power is determined by squaring no{to) and finding its average
value :
/°° /• "^ nl nl
/ w(ri)w(r2)S ^p{ri — kT)p(T2 — mT)dTidT2.
(5129)
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, (5130)
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. (5131)
It is apparent that the effect of the pulse integrator is to increase the signal
tonoise ratio for a single pulse by the factor n. This could, of course, be
inferred at the outset from Equation 5121, 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
nl
Pulse train spectrum = S{i^) = P(co) X) ^~''*"^ (5132)
\ sm coT/2 /
510] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 279
The energy spectrum of a typical train of short pulses is shown in Fig. 51 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
tonoise ratio. The output of the matched filter characterizes a signal
plusnoise 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. 516
Decision Bias
Probability Density
of Noise Alone
Probability Density
of Signal Plus Noise
h\ Value of Signal
False Alarm pi^g ^^^^^
Probability
Fig. 516 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 falsealarm 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
clearcut, 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 falsealarm 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 falsealarm 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 falsealarm 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 falsealarm 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
ofarrival 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 timeofarrival.
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.
511] 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 selfexcited 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 pulsetopulse frequency fluctuations, the
receiver must operate upon the envelope of the signal, and it will normally
employ a nonlinear device to generate it. In this case, the best that can
be done is to match the lowpass 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.
511 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 510, 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 510 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. 517 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. 517 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 5118 for the
signaltonoise power ratio. Schwarz's inequality can also be applied to
511] 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 signaltonoise 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 + . (5133)
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/). (5134)
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 filterreceiver by Y(o}) and the
signal spectrum by S(cci) as in Paragraph (510), the following representation
of s"o{to) can be obtained by differentiating Equation 5116 twice:
s'o'ito) = ^ / co2y(co)^(co)^^''«^a;. (5137)
Also assuming, as in Paragraph 510, that the input noise has a uniform
spectrum of density D, the power spectrum of the output noise is Dy(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)2y(co)Vco. (5138)
284 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS
The quotient in Equation 5136 giving tiie variance of the signal location
error thus has the following form:
_ ^f Dco2y(co)Vco
A/ = , /;~'° TT (5139)
(^/ c,'Yio^)S{co)e''^'o^oA
For convenience, we denote the quotient on the righthand side by ^. The
denominator of this quotient is in a form to which Schwarz's inequality,
given by Equation 5125, 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 510, where the maximum value of the signaltonoise
ratio was determined. Since the quotient ^is never less than the expression
given on the righthand 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 5139 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 5139 cancels one of the factors in the denominator,
and we have
A7^=^.,mi„= , ,.co ^ (5141)
i/.
cjo\S{u)\~dcjo
The optimum filter transfer function giving this result is
y(aj)  S*(c^)e''''o. (5142)
This is exactly the transfer function determined in Paragraph 510 (Equa
tion 5128) to give the maximum signaltonoise ratio. Thus to a first
approximation the matched filter giving the maximum signaltonoise ratio
also provides the minimum error in locating the signal in time.
The relation given by Equation 5141 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
5111
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^")""
(5143)
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 highfrequency carrier:
High frequency signal = f(t) cos {coj + (p). (5144)
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. 518 where a relatively smooth lowfrequency
AMBIGUOUS MAXIMA
DUE TO CARRIER
FREQUENCY
LOW  FREQUENCY // 1 MODULATION
SIGNAL=f(f)
HIGHFREQUENCY SIGNAL^ f(f)cos(Wcf+0)
Fig. 518 Typical HighFrequency 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 lowfrequency
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 5143. When the definition of bandwidth given
by Equation 5143 is combined with Equation 5141, 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 5129 as the greatest possible signaltonoise 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 signaltonoise
ratio:
(X7^)i/2 ^ iiBz (5145)
rms time error = 1 /(signal bandwidth) (voltage signaltonoise 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 210 to 220 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. 519. 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. 519 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. 520. The received signal is amplified and filtered by an IF
amplifier which is matched to the envelope of an individual pulse. Noise
511]
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. 520 Block Diagram of Receiver of Scanning Radar.
with a uniform power density is introduced at the input to this amplifier.
A squarelaw second detector is used to generate the video envelope of the
signal plus noise. The video signalplusnoise 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 5145 to be valid. Other system configurations are possible. A
more practical design might stretch the gated signalplusnoise 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 (halfpower points — oneway)
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/
(5146)
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
signaltonoise 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 5127:
Peak signal to noise ratio = S/N = a^/lD. (5147)
The noise at the output of the IF amplifier corresponds to a narrow band
noise process similar to those discussed in Paragraph 58. Since the IF
amplifier is matched to the envelope of the pulse signal, the autocorrela
tion function and power spectrum of the lowfrequency inphase and
quadrature components of the noise, x{t) and yif) in Equation 573, 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 587 which gives the autocorrelation
function of the output of a squarelaw second detector:
7^2 = (202(1 + S/NY + (2cr2)2[p2 + 2{S/N)p]. (587)
The first term in this expression corresponds to the video dc 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 signalto
noise ratio was large in the development of Equation 5145. It will simplify
the present analysis if we approximate Equation 587 for large S jN by
considering only the dominant dc and noise terms in that equation:
77^ = {2<rr~{S/NY + (:lcj^Y2{S/N)p, S/N»\. (5148)
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 5148, the term involving this factor gives the power
density of the noise at zero frequency during a pulse or with a cw signal.
511] 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 587 to be
Power density at zero frequency of gated video noise
^ {lcj''Yl{SlN)hd,SINy>\. (5149)
The dc component of the video corresponds to the signal which is used
to locate the target. From Equation 5148 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
5148 gives the square of the dc level during a pulse for large signalto
noise ratios. To obtain the dc 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.
(5150)
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 oneway transmission. The gain of the twoway 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') (5151)
Because we have assumed a squarelaw 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 5145, 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 signaltonoise 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^). (5152)
290 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS
The power density of the video noise is given in Equation 5149. It was
determined in Paragraph 510 that the signaltonoise 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
Signaltonoise 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. (5153)
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 5151 is as follows.
/ ^ e\ / co^0.099^ \
Spectrum of scan modulation = ( yO.lSTr • I exp I ■ 1
(5154)
The bandwidth, as defined by Equation 5143, is easily computed:
Scan modulation bandwidth = B
1/2
\ /"" /co20.i8e2\ "1
1/2
Uo^^^K 2v rJ
 2.35,/^/e 5155)
The rms error in measuring the target angle can now be estimated from
Equation 5145 as the scan rate divided by the product of the scan modu
lation bandwidth and the voltage signaltonoise ratio in the video filter
output:
rms angle error = (M^y^^ = (li^yi'' = i/Bz (5156)
= 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 5156
P. Swerling, "Maximum Angular Accuracy of a Pulsed Search Radar," Proc. IRE 44,
11401155 (1956).
511] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 291
but only about half as great. Actually, the estimate in Equation 5156 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 signaltonoise ratio out of the matched video filter as given by
Equation 5153 is 6 db, the rms angle error from Equation 5156 is
rms angle error = e/2.35^[4', z'~ = 4 (5157)
= 0.2139.
In Paragraph 215 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*
61 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.
*Paiagraphs 61, 62, 66, and 67 are by D. J. Povejsil. Paragraph 63 is by R. M. Page.
Paragraph 64 is by S. F. George. Paragraph 65 is by L. Hopkins. Paragraph 68 is by
H. Yates.
292
62] BASIC PRINCIPLES 293
Unfortunately, there is a tremendous variety of possible choices. In
terms of generally recognized system types and subtypes, there are pulse^
continuouswave {CW), pulsed doppler, monopulse, correlation ^highresolution,
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 ruleofthumb 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.
62 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 14) ?
(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 targetgenerated 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.rdenved target information
(2) The radarderived 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 radarderived target information takes the
form of a fourdimensional information matrix as shown in Fig. 61. 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. 61 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. (61)
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. (62)
Often, it is quite informative to translate a system requirement into the
form of Equations 61 and 62. 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°
62] BASIC PRINCIPLES 295
of angular coverage in 2 seconds; and an ability to separate targets with
radial velocities of from to 2000 knots in 40knot 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
twodimensional (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 = ^; (63)
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 "FMing." That is, the minimum
range resolution element is defined by Equation 63 and the rate Nr at
which the system collects pieces of range information is:
Nr = ^ft. (64)
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 fourdimensional 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. (65)
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 pulserepetition 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^ (66)
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^ (68)
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^.=/. = ^ (69)
where i^s = total solid angle scanned
^o = solid angle subtended by antenna pattern or instantaneous
field of view.
62] 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. (610)
Since the angular scanning frequency defines the minimum signal band
width, it also defines the minimum possible velocity resolution element.
Thus
Ar>^. (611)
where A/^ = velocity resolution element, and the total number of separate
unambiguous velocity elements is, from Equations 68 and 611,
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 dataprocessing
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) signaltonoise 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. 62. 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^ (613)
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. 62 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. (614)
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 64, 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 64, 65, and GG) 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,
62] BASIC PRINCIPLES 299
angles, and velocity. For such a system the information rate may be
expressed (from Equations 61, 62, 69, and 612):
iV = TV, X A^c X TVa X A^. = ^ X/s X N,. (615)
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. (616)
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. 611.
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 timesharing 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 singlechannel 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 65.
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 66 and the delayline
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) crosscorrelation 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 signaltonoise ratio of the target information is derived from various
modifications of the basic radar range equation (Equation 31). 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
enduse 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 enduse 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
lowaltitude capability" and "Doppler radars have excellent lowaltitude
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.
63 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. 513, 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
63] MONOPULSE ANGLE TRACKING TECHNIQUES 301
errors of one sign while decreasing apparent errors of the opposite sign (see
Paragraph 48). 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 socalled 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. 63. The sum signals from the two antennas merge the two patterns
into a single lobe pattern as shown in Fig. 64. The difference signals
produce the familiar null pattern with the sharp zero at the center, as shown
in Fig. dS. 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. 63 Overlapping
Individual Antenna Pat
terns of a Monopulse
Radar.
Fig. 64 Sum
of Overlapping
Patterns in a
Monopulse Ra
dar.
Fig. 65 Difference of
Overlapping Antenna
Patterns in a Monopulse
Radar.
The process of generating sum and difference signals results from inphase
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. 66 Monopulse Error Signal Curve.
63]
MONOPULSE ANGLE TRACKING TECHNIQUES
303
antenna difference pattern null is therefore the familiar error signal curve
of Fig. 66, 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. 67. A
conventional hybrid ring (see Para
graph 101 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. 64. 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 phasesensitive 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. 67 SingleCoordinate 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 onecoordinate system may be modified to operate as a
twocoordinate 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. 68 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. 68 TwoCoordinate Monopulse System.
64] CORRELATION AND STORAGE RADAR TECHNIQUES 305
Monopulse techniques are particularly useful for applications where
pulsetopulse amplitude fluctuations due to target variations or interfer
ence signals can degrade conical or sequential scanning tracking techniques.
64 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 Ascope
presentation to modern sophisticated and complex magnetic storage devices
for predetection integration, the use of storage has become increasingly
important. Today the lack of highcapacity memory, highspeed operation,
and wide dynamic range storage are perhaps major contributing factors
impeding the development of more effective longrange 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 SCR584 used a limited form of crosscorrelation
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 crosscorrelation 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 (617)
:; 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. (618)
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 (619)
306 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES
and an incomplete crosscorrelation function as
^,,{tJ) = ^jjm^it  r)dt (620)
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 619, 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 frequencydomain radar processes wherein a squarelaw 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 Ascope 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 squarelaw detection
versus crosscorrelation detection have been studied^ with the results shown
in Fig. 69. The outputversusinput mean power signaltonoise 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. 69
there is an apparent threshold in the autocorrelation and squarelaw
detection curve starting in the neighborhood of unity signaltonoise ratio.
This threshold is noted by the change from a linear to a squarelaw relation
ship between output and input sensitivity. Such a threshold does not exist
in cross correlation, where a noisefree reference is used.
CrossCorrelation 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 620 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 noisefree reference signal possessing characteristics identical
^Samuel F. George, Time Domahi Correlation Detectors vs Conventionat Frequency Domain
Detectors, NRL Report 4332, May 3, 1954.
64]
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 SIGNALTONOISE RATIO (db)
Fig. 69 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 crosscorrelation detector is shown
in Fig. 610. 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 crosscorrelation 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^{fT) = u{tT)
Multiplier
Integrator
2l j\(t) f2(tr)dt
■4>,Jr,T)
Fig. 610 Simplified CrossCorrelation Detector.
incomplete crosscorrelation 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 rangerate 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 pulsedoppler system
means coherent video or IF integration. The crosscorrelation principle is
embodied in all of the systems proposed for using rangerate information.
First, a coherent or stored noisefree reference must be available; then some
storage medium is required to permit integration; and finally some form of
very narrowband doppler filtering must be employed. Fig. 611 shows a
block diagram of a straightforward or bruteforce pulsedoppler 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), 4551 (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, 7682 (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, PGIT4, September 1954.
^Bernard D. Steinberg, Coherent Integration oj Doppler Echoes in Pulse Radar, Report
#1821121, General Atronics Corp, Aprif 1957.
64]
CORRELATION AND STORAGE RADAR TECHNIQUES
309
w
TR M — Transmitter — ► Delay
Receiver
Multiplier
Coherent
Video
f2
*'
D
D
Fig. 611 PulsedDoppler 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
scantoscan 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 delayline video integrator is illustrated in Fig. 612.
Video
T
= lnterpulse Period
Pulse
Adder
Video Delay
Line
—
Amplifier
Integrated
"
Output
Fig. 612 Simplified DelayLine Video Integrator.
The number of pulses which can be effectively integrated to improve the
signaltonoise 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 signaltonoise 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. 69 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 signaltonoise 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 crosscorrelation purposes.
Both electrostatic storage and magnetictape 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
magneticdrum storage. As advanced technical progress provides increased
dynamic range and higher frequency operation, the tremendous data
handling capacity combined with highspeed record and readout and long
duration storage make magneticdrum storage appear very desirable for
radar use.
Fig. 613 shows a system using transmittedwaveform storage to provide
a reference for the cross correlation and magneticdrum storage to reduce
TR
Transmitter
Storage &
Range Gate
Receiver
Multiplier
Magnetic Drum Doppler
Storage Output
^3
^
D
Fig. 613 Storage Radar.
the equipment multiplication required in Fig. 611. The transmitted
waveform storage unit could be envisioned to consist of multiple delays
corresponding to the n range gates of Fig. 611. 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
65] 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 rangerate accuracy and
resolution are limited only by the total observation time during which the
target doppler remains coherent.
65 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 timeshared basis.
For certain applications — such as semiactive missile guidance systems
— continuouswave (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.
614. 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. (120)
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 groundbased 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. 614 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. 614 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. 615. 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. 615 FilterBank Detection Showing Contiguous Filters.
65]
FM/CW RADAR SYSTEMS
313
The information matrix of this system may be visualized in Fig. 616.
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. 616 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 rangemeasuring 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 FMing deter
mines the maximum unambiguous range.*
Consider typically a simple sine wave FM as expressed by the trans
mitted FM /CW waveform of Equation 621 :
et = Et sin ( co«/ + y sin oo™/ j (621)
The received doppler signal er as detected by a coherent FM/CW radar is
cr = Er cos cod^ H J— sin ir/mT cos (comt + Tr/mr)  (622)
/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. 617. Depending upon range, modulation, and doppler
f,AF 
TIME
■Transmitted
V^^^^
^Measure of Range f,
I Doppler Frequency fj
TIME
Fig. 617 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. 617, the
magnitude of the resultant detected difference frequency /r is a measure of
altitude: i.e..
(623)
65] 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 622); i.e.,
fr = 2AFs\mr/mT (624)
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 shortterm frequency stability of the
transmitted energy; i.e., the coherence that exists between the transmitted
and received signals. Longterm frequency drift is generally not a problem
provided that shortterm coherency exists.
Any modulation that tends to broaden the spectrum of a wanted signal
may destroy the signaltonoise 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, shortterm stabilities in excess of 10~"^ may be
required.
The process of coherent detection of a timedelayed signal alters the
angular modulation index as has been indicated by Equation 622.
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 622, 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 622 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/,„). (625)
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 1200cps 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. (627)
A not uncommon composite noise deviation in a 100kc band B for practical
CW radars is 100 cps rms (AF), indicating about a 1 /3 cps rms/cps density
(A/)_.
Fig. 618 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.
(yS]
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. 618 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 signaltonoise 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 headon — is a region
where doppler sensing devices are most effective in detecting and tracking
targets in heavy ground clutter.
(2) Buplexedactive 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 (spacedaclive 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 ontarget illumination power density is practical
from the large parent radar. Both a spacedactive and a semiactive system
are illustrated in Fig. 619.
Missile Semiactive
Radar
Fig. 619 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 firecontrol systems
employing unguided weapons.
FM/CW Radar Performance. The detectability of the target is
determined, for a given falsealarm rate, by the signaltonoise ratio after
final detection. Neglecting the effects of clutter and transmitter modu
lations, the signaltonoise ratio may be calculated by approximate modifi
cation of the previously derived radar range formula (see Equation 39):
S/N ^
{AttYR'FKTB
(628)
65] 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 moderntype 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>^ (629)
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 lockon.
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. 615).
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 manmade
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,
manmade 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 transmitreceive antenna radiation pattern versus
angle as viewed at the receiver. A typical CW radar doppler clutter
spectrum is shown in Fig. 618. Note the complete absence of interference
at all frequencies above ownspeed dopplers; only noseaspect 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 multipletarget 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.
66 PULSEDDOPPLER RADAR SYSTEMS
Pulseddoppler 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.
66]
PULSEDDOPPLER RADAR SYSTEMS
321
and timeduplexing properties of a pulse radar. For applications requiring
heavy ground clutter rejection, a common transmitting and receiving
antenna, and accurate range measurement, a pulseddoppler type of system
represents the best known technical approach to the problem. However,
the pulseddoppler type of system also has certain drawbacks: principal
among these are limited targethandling capacity (when compared with a
pulse radar) and a high order of electronic system complexity.
Basic Principles of Operation. A simple pulseddoppler system is
shown in Fig. 620. It differs from the CW system of Fig. 614 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. 620 Simple PulsedDoppler 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
fnlA
(b)
Frequency
Fig. 621 Transmitted PulsedDoppler 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. 621 (a) which has the frequency spectrum
shown in Fig. 621 (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 pulseddoppler 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 spectralline transmission from a moving platform is shown in Fig.
622. 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. 622 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 clutterfree portion of the
frequency spectrum.
The effects of scanning and targetinduced modulations (see Paragraph
48) 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 = ^ (630)
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
66]
PULSEDDOPPLER RADAR SYSTEMS
323
Target Frequencies
n = 0,1,2 y
1 li
^^0^^.
Main Beam and
Sidelobe Clutter
Closing Target
Frequency
Fig. 623 Received Signal Frequency Spectrum for an Airborne PulsedDoppler
System, Showing Clutter Spectrum and Return from a Closing Target.
can be represented as shown in Fig. 623. This diagram illustrates one of
the basic limitations of a pulseddoppler radar system : in order to maintain
a clutterfree 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.
(631)
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 pulseddoppler systems. For example, an Xband
(3.2cm) system designed for operation in a 2000fps aircraft against
2000fps 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. 620, the output spectrum shown in Fig. 624 is produced. This
heterodyning operation causes an effect known ?is folding; i.e. each of the
II
Doppler Filter
Bandpass
f
(b)
Fig. 624 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 spectralline interference
due to folding, the minimum PRF is
fr > — rj:!!^ (when signal "folding" occurs) (632)
A ^
The resulting video signal then is passed through a bandpass filter to
remove all the zerofrequency 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 singlesideband 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
narrowband filters as was shown in Fig. 615, 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 pulseddoppler systems.
The simple pulseddoppler 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. 616. 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. (633)
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. (634)
66J
PULSEDDOPPLER 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 pulseddoppler system
must be
Pt = P/d\ (635)
and the average power must be
Pa,, = P/d. (636)
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 pulseddoppler
system is for doppler navigation radar systems. This will be described in
Chapter 14.
RangeGated PulsedDoppler Systems. The full benefits of a
pulseddoppler system can be realized by rangegating the receiver. This
technique permits range measurement with the resolution inherent in the
radar pulse width and it can also improve the signaltonoise ratio and
signaltoclutter ratio by the reciprocal of the rangegating duty cycle.
Master
Frequency
Control
Pulsed
Coherent
Transmitter
Target Doppler Bandpass
Filter Composed of Continuous
Narrow Band Filters
fj (Velocity)
Fig. 625 Gated PulsedDoppler System with Means for Range Measurement.
326 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES
A generic rangegated pulseddoppler system is shown in Fig. 625. 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
•^ (637)
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 signaltoclutter 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. (638)
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 635.
The improvement in signaltoclutter ratio represents an improvement
over and above what can be done with a CW radar system. Thus a range
gated pulseddoppler 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 pulseddoppler system is not
exactly the same as a range measurement of a pulse radar. The high PRF
that must be employed in a pulseddoppler 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\
66] PULSEDDOPPLER 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 pulseddoppler 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. (640)
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. 611). 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 64, Correlation and Storage Radar Techniques,
suggested still another means for processing pulseddoppler radar infor
mation.
Range Performance. The idealized range of a pulseddoppler system
may be calculated by the following modification of the basic radar range
equation (31):
" ~ [{4Tr)'FkTBdg\
(641)
where ds = signal duty cycle
dg = gating duty cycle [equal to (lds) 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 onefourth power expression.
^"In some cases, postdetection filtering will be employed to improve the final signaltonoise
ratio without increasing the number of doppler filters required. In such cases the effective
detection bandwidth is Bbff = V725 • Bpd as derived in Paragraph 35 (Equation 362).
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 pulseddoppler 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. 626. 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. 626 Effect of Eclipsing on PulsedDoppler BlipScan 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 blipscan
ratio is a function of idealized range as shown in the fourth figure. When
the blipscan ratio with no eclipsing is corrected for the eclipsing effect, the
last curve in Fig. 626 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 pulseddoppler system the ratio of PRF to the useful range
66]
PULSEDDOPPLER 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
blipscan curve with a "smoothed" curve.
If the pulseddoppler 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 — blipscan curve.
Pieces of Information
Range Measurements in
PulsedDoppler 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
pulseddoppler radar an information
matrix such as is shown in Fig. 627.
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 rangemeasuring pulseddoppler system.
There are several means for measuring true range: all are inconvenient
and all degrade radar performance in terms of information rate and /or
signaltonoise 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 pulseddoppler 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. 627
NrxN.xNcxNy
Nr = VTfr
N = fr/b
B = Doppler Filter
Detection Bandwith
PulsedDoppler 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.
628 TwoPRF 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. 628. 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 +
(642)
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 642 we obtain, as expres
sions for the time delay corresponding to true range,
h/rltJr2{l
(643)
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
(644)
66] PULSEDDOPPLER 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
642 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 twoPRF 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.
PulsedDoppler System Design Problems. Pulseddoppler 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 pulseddoppler system. To cut eclipsing losses to a minimum,
the transition from transmit to receive must be made as quickly as possible.
Ordinary transmitreceive (TR) tubes are not satisfactory for this appli
cation; however, ferrite circulators (see Paragraph 1016) 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 signaltonoise
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 pulseddoppler system may have hundreds or even thousands of these
narrow band filters; thus the tradeofF between filter performance and size
and weight is a vital consideration.
Angle tracking poses certain special problems in a pulseddoppler radar.
The doppler frequency as well as the range must be tracked prior to angle
lockon. 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 40cps
scanning frequency would require a doppler filter band width of at least
120 cps.
The use of monopulse angle tracking (see Paragraph 63) poses a most
difficult problem in a pulseddoppler 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.
PulsedDoppler Systems Applications. As previously mentioned,
pulseddoppler 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, pulseddoppler 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
closingrate tactical situation.
67] HIGH RESOLUTION RADAR SYSTEMS 333
Another application of pulseddoppler 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.
67 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 0.110 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 signaltoclutter ratio when the
clutter originates from area extensive targets. This is shown by Equation
460, 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 36. 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 Zonec — ^Far Zone
Fig. 629 Antenna Beamwidth Pattern.
shown in Fig. 629. As can be seen, the concept of angular beamwidth holds
only for the socalled 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. 630 where an antenna points straight down from an airborne
V^2 = \^FSinQ: f^2
2Vf sino :
X
Fig. 630 Improvement of Apparent Angular Resolution by Doppler Filtering.
l^See S. Silver, Microwave Antenna Theory and Design, Chap. 6, McGrawHill Book Co.
Inc., 1949.
67] 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. (646)
However, the filter bandwidth is limited by dwell time requirements to a
value
A/, = \/t,= VF/hQeff cps. (647)
Substituting, and solving for the minimum value of Qeff, we obtain
Qefs ^ VV2A radians. (648)
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. (649)
and the number of channels required is
n, = IKKld'' channels. (650)
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 3.2cm system with a 4ft 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.
ShortPulse 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, highpower pulse is a difficult
problem in itself. The design of the transmitting tube, the modulator, and
the TR switching all are complicated by the shortpulse operation.
Shortpulse 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. (651)
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, shortpulse
systems are limited to relatively shortrange 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 delayline 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 shortpulse 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. 631 Pulse Radar Spectrum. radar as shown in Fig. 631.
\
Frequency
67]
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
(652)
Since each piece of range information represents a range interval of fr/l,
the total unambiguous range interval is simply
Rn
'^'ik
(653)
= 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 510 and used as the basis for the
storage and correlation radar principles outlined in Paragraph 64. 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. 632
Optimum
Radar Spectrum
—^Frequency
Filter for Pulse
Be
NrBi = B
(654)
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. Longpulse, lowPRF 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}^
68 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 510, 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
longrange detection and tracking no sufficiently intense sources of infrared radiation are
available.
'^By detection we mean the manifestation of the presence of electromagnetic radiation.
68] INFRARED SYSTEMS 339
Conversely, infrared does not have the allweather 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
57 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 ac
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. 633.^^
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. 633 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. 634.
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.
68]
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. 634 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 cathoderay 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,. (655)
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 signalto
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. 635
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
68]
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. 635 The Useful IRFrequency 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. 634) a function of the halfscan 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. (656)
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. (657)
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. (658)
4x7 sm 7
and the time required to scan a radians is
4x7 sin 7
= r seconds. (659)
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 657:
^ 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
68]
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. 636 Target Tracking.
to such a value that a circular scan of, say, ° results (see Fig. 636). Then
as long as the image in space of the detector rotates around the target
without touching it (Fig. 636a) no output (error signal) will result. When
the line of sight moves and the detector then encounters the target (Fig.
636b) 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. 636c a simplified onoff 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. 636c the pulse occurs where both the down and left
346 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES
controls are activated (as in Fig. 636b) 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
ofFcourse position of the target, and this is usually a feature of actual
tracking systems.
Detection Performance. The actual capabilities achievable with
presentday 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 signaltonoise 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 2.7micron 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 24inchdiameter 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 singleengine 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 onetenth as intense a target as in tail
aspect and will therefore be detectable at less than onethird 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
71 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. 71 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. Crossmodulation 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 signaltonoise 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
71]
GENERAL DESIGN PRINCIPLES
349
1
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Filter
and
natio
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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
72. Selectivity is provided in both frequency and time. It is desirable to
provide the required selectivity at the lowlevel 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, shortpulse, lowPRF
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 signaltonoise
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 dynamicrange and crossmodulation
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 shortterm and longterm frequency
stability is important. The effect of shortterm frequency instability is
to introduce modulation on the signal in the receiver. Such modulation
degrades the output si.gnaltonoise 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 71
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 longterm
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 highPRF
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 noisenoise 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 71 (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.
72 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
73] 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 71.
73 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 shortcircuit 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 (71)
and
/;? = \kTLgLdj (72)
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 ^. (75)
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. 72. It is
F,,G,
Fig. 72 NoiseEquivalent 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. + ■
73] RECEIVER NOISE FIGURE 355
to FGkTBn, where F is the overall noise figure expressed as a power ratio.
+ . (76)
(77)
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)] (78)
J a J a
where Free is the receiver noise figure expressed as a power ratio
/m1 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
74 LOWNOISE FIGURE DEVICES FOR RF
AMPLIFICATION
The crystal mixer Is a rather fragile element, and its electrical character
istics deteriorate when largesignal 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 lownoise 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 gaincontrolled 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
75]
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 negativeresistance amplifiers appear to be a final step in attaining
receivers whose sensitivity is truly limited by external noise.
75 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. 73a. 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. 73 (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. 73b. 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 (79)
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./ (710)
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^\. (711)
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^\. (712)
75] 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/,)/. (713)
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,. (714)
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 1015). 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 magictee, shortslot hybrid or
ratrace.
Each individual crystal develops all of the intermodulation components,
but the relative phase of the signalsignal beats differs from that of the
signalL.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 signalsignal 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 signallocal oscillator products are two which affect the
performance of the mixer. These are included in the value K of Equation
712 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 voltagecurrent relationship is given by
'(
exp^l) (715)
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 dc current through the crystal. Equation
715 indicates that a given conversion loss could be obtained with a lower
dc 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
Hh'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 78 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.
76] COUPLING TO THE MIXER 361
A frequencydependent 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.
76 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 doubletuned 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, 876889 (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 shortslot 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 shortslot hybrid with a filter in both signal
and local oscillator paths.
77 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 pulsetype airborne radar receiver.
The IF amplifier is a filter amplifier, and its smallsignal transfer function
is given by
G(s) = H — — —^ ;^;^37r  . (718)
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*)  • (719)
77]
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 polezero 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. 74), an arrangement of the
C^+JW,
Transmission
Band
Z(a))
Due to cr3+jco3
Due to cTi+jcOj
■Due to cTj+jcOg
Fig. 74 SPlane and ZPlane 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
(720)
is used to obtain an exact lowpass 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 fourterminal network followed by an extremely wideband amplifier.
(The amplifier, of course, would have a particular polezero structure,
but the contribution to the selectivity in the frequency band of interest
is negligible.) Practically, however, the fourterminal network is limited
to a fourpole structure, and most commonly to a one or twopole 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. 75. When k = I and Li = L2 the network reduces to a onepole
structure as shown.
C2
= H
S'*+aS3+/3S2+ 7S+6
S
(ss,)(ssi)(ss^)(ss;)
Where H
7 =
Qj Q2
2
COj 002
Q2(lK^j'^Qi(lK2j
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. 75 Commonly Used IF Interstage Coupling Network.
77] 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 3db bandwidth of the cascaded circuits is the 1.5db 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^^] (722)
where X =/„( /// /°// )^ ^C//.)
n = number of poles in the circuit (low pass equivalent)
Bi = 3db 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^""]. (723)
The overall bandwidth of the amplifier is therefore
B = (21/  l)i/2«5i. (724)
For a given overall IF bandwidth it is desirable that the principal
selectivity occur in lowlevel 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 onepole or twopole
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 twopoles (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 singlepole coupling circuits, the overall
frequency response exhibits geometric symmetry with respect to the IF
center frequency. With twopole coupling provided by the magnetically
coupled doubletuned transformer, the response is more nearly arith
metically symmetrical. With twopoles 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 gridtoplate
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. Gridtocathode feedback
4. Inadequate decoupling circuits resulting from selfinductance 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 gridtoplate 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 highfrequency
stages which are gaincontrolled 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. 76. 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 gridto
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
77]
IF AMPLIFIER DESIGN
367
R c
Fig. 76 Typical IF Amplifier Configuration Showing How Proper Choice of C
Can SelfNeutralize 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
(725)
where gm is the peak transconductance and Cgp is the gridplate capacitance.
The effective noise bandwidth 5„ is the parameter involved in radar
performance computation. For any practical amplifier this is very nearly
the 3db bandwidth
Bn
G(co)
1 +
1 r"
: T / G(co)do)
; recei
an be 
/co C0o\ "")
(726)
where G(a)) is the power spectrum of the receiver. The normalized power
spectrum of the radar receiver usually can be given by
(727)
where coo is the midband frequency
B is the 3db bandwidth
n is the number of poles in a flat circuit
m is the number of groups of circuits.
The 3db bandwidth is
(21/  iy''"Bi,
(728)
368 THE RADAR RECEIVER
where Bi is 3db bandwidth of one network consisting of n poles. Then as
an example for three staggered pairs of onepoles
Jo (1 + .v^)
Noise bandwidth ^ Jo (1 + •V)' ^ 1 r (i)r(3  \) _
3db bandwidth (2"^ l)'/^ 0.714 r(4)r(3) " ^•^•
(729)
78 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, signalhandling
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 groundedcathode or
groundedgrid arrangement. For the ordinary AI radar with broad band
mixer, the groundedcathode 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. 77. From this arrangement of noise
generators and signal generator the noise figure as defined by Equation
74 becomes
1 + ^" + ^'^ + ''' + '''' + ^L + g. +,. + Kf + Sv. +/
= ]+^^ + ^y,
(730)
where ^ is the total susceptance appearing at the input terminals. The
additional parameters involved are defined in Fig. 77. This expression
yields the singlestage 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
noisenoise 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. 77 Equivalent Circuit of IF Amplifier Noise Sources.
as the bandwidth is reduced. The last term of Equation 730, 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 wideband 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 transconductancetoinput capacitance
ratio and small transit time.
An important consideration is the mixerIF coupling network. When gf
and ^12 are zero, minimum F is obtained for
,.yl'^ + (g. + g.
(731)
as can be proven by differentiating Equation 730. 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. 78 shows a simplified equivalent circuit for a groundedgrid stage.
The noise figure is given by
^Am + 1/ I I
l+^ + J + ^:^{Tr)i^^l (732)
1\ i h
Nsi
Fig. 78 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 groundedcathode and
groundedgrid stages is given by
e) (733)
(grounded grid) (734)
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 groundedcathode
triode stage followed by a groundedgrid stage. Such a configuration results
in a lower noise figure than can be obtained with a cascade connection of
two neutralized groundedcathode triode stages. This is because the
interstage bandwidth is obtained by virtue of the loading incident to the
78] CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 371
large input conductance of the groundedgrid 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.
Wideband 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 730 with frequency.
To minimize the variation of the grid admittance with frequency, a
doubletuned (twopole) mixerIF 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^ (735)
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 highPRF radars (such as the pulseddoppler systems described in
Chapter 6) and where very short range with high accuracy is required, the
doubletuned mixerIF 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
groundedgrid input stage is employed. The transmission bandwidth of
the mixerIF coupling is very wide because of the heavy damping caused
by the input conductance of the groundedgrid 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 731,
^^^ ^10_2_+_5(10_^ = 6.3 X 104 mho is required.
A source bandwidth (see Equation 735) of
6.3 X 104
^ = .V2 20.5X10 = ^^^^^
is obtained, assuming 1.0 MMf stray capacitance and a wideband neutraliza
tion of Cgp.
At 3Mc bandwidth the input coupling network produces an attenuation
(see Equation 722)
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 singletuned circuit of 6.45Mc
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 stepdown transformer is employed, the gain bandwidth product
could be improved and the conductance presented to the second tube output
is lowered.
Ine stepdown must be n = \/t^ = \/t7j'
79] 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) (1012) = 2.43 X 10^.
From Equation 730 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 732)
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 733, the available power gain of the first tube is
(20XI0y(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 77:
1196+ ^^^^ + ^Q^ ^1237
i.iyo^ 131 ^(131)(1.87)
^ 0.92 db.
79 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 intensitymodulated 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 lownoise IF preamplifier, the process of referring all of the
noise sources in the receiver to the input is extremely important, especially in
multipleconversion 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 signaltonoise 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 rangegated 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.
711] BANDWIDTH AND DYNAMIC RESPONSE 375
710 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 dc current through the tube. Gain can
therefore be stabilized by operating the tubes so that the dc plate current
is stabilized. This can be accomplished by means of large cathode resistors
or by operating a number of stages in series dc 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 highfrequency IF amplifiers it is
usually limited to a small amount of signalcurrent 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 dc 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 narrowband 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 30db range is required.
These gaincontrol 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.
711 BANDWIDTH AND DYNAMIC RESPONSE
A criterion sometimes employed for best signaltonoise ratio is that
signal plus noise should be filtered by a network which maximizes the peak
signaltorms noise power. The network which will accomplish this result
was determined in Paragraph 510 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 3db 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 510.
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 77. The transient response of these networks
governs the resolution and largesignal behavior. The rectified envelope of
this response corresponds to the video signal.
A typical transient response would
appear as shown in Fig. 79. At the
receiver output, a loss in sensitivity
may occur for the time /2 shown in
Fig. 79 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. 79 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. 86 (Radiation
Laboratory Series), McGrawHill Book Co., Inc., 1950.
712] 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 /
(736)
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 staggertuned, then the response given by Equation
736 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,
712 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 largesignal input. The transmission characteristics of the amplifier
in the lowfrequency region of the spectrum must also be considered.
To realize practical highgain bandpass amplifiers the power supplied to
the stages must not be derived from a commonsource impedance, since
instability will result. Fig. 710 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 dc 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. 710 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 dc 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.
713] CONSIDERATIONS RELATING TO AGC DESIGN 379
713 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 821.
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
tonoise ratio of the receiver. For example when the input signaltonoise
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
tvpical design might allow the output signaltonoise ratio to be +30 db
li inimum for +90 db input signaltonoise 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 squarelaw 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
gaincontrolled 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 signaltonoise 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 gaincontrolled stages but is delayed until the
input signaltonoise ratio is about 20 db. The AGC voltage delivered to
the cascode is selected so as to minimize the thirdorder coefficients of the
tube transfer characteristic. The effective cascode transfer characteristic is
somewhat superior to that of a single tube because of the dc 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 thirdorder
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 thirdorder 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 highperformance 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 712. 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.
714 PROBLEMS AT HIGHINPUT POWER LEVELS
In an airborne radar set strong signals are obtained from shortrange
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
714] PROBLEMS AT HIGHINPUT 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
timeselected.
In the first case, cross modulation caused by the strong signals can
deteriorate the weaksignal 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 712, 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 timecoincident 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 welldesigned 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. 711 shows the
transfer characteristic of a typical microwave mixer at largesignal 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
propellerdriven 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 84 4 8 12
SIGNAL POWER (dbm)
Fig. 711 Transfer Characteristic of a Microwave Mixer at LargeSignal Input
Levels (1N23C Crystal).
715 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. 712, together with the current voltage relations that exist under
largesignal 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. 712. 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 backbiased 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
715]
THE SECOND DETECTOR (ENVELOPE DETECTOR)
383
Video Amp.
Fig. 712 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 lowpass filter estab
lishing the video bandwidth.]
The efficiency of the diode detector is the ratio of the dc 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 60Mc 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 = ^ (737)
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
seconddetector characteristic is shown in Fig. 713. 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. 713 Transfer Characteristic of a Typical WideBand 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.
715] 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 predetection 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 (738)
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 signaltonoise 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 noisenoise intermodulation at the
detector. An approximate expression for signal suppression is
db suppression ^  7 + 20 logio {SIN)if. (739)
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, 282236
(1944), 24, 46156 (1945); "Response of a Linear Rectifier to Signal and Noise," J. Acoust.
Soc. Am. 15, 164 (1944).
386 THE RADAR RECEIVER
716 GATING CIRCUITS
Gating circuits are employed to improve the signaltonoise and signal
toclutter 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. 714. Gating circuits are
Gating
Signal
IF GATING CIRCUIT
Fig. 714 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
717] 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.
717 PULSE STRETCHING
In tracking radars it is required that the modulation signal associated
with a pulseamplitudemodulated signal be recovered. The modulated
pulse signal is
/i(/) = [1 + m cos (a;„/ + 0)] ^fljue^^ (740)
for periodic pulses of shape /(/)
where r„ = ^///W exp (^^^^V/. (741)
If the pulses are passed through a lowpass 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. Pulsestretching 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 amplitudemodulating 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. 715. 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. 715 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
(742)
^m +
Tp
When the output from the lengthener is passed through a lowpass 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 lowpass filter has
a cutoff frequency Wc, outputs are also obtained for modulation frequencies
satisfying
1 P
(743)
718 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
719]
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 rangeerror 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. 716.
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. 716 Connection of a Receiver Employed with Sequentially Lobed Antenna
to Related Circuits.
719 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 enveloperectified 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 dc
component of the signal and the lowfrequency modulation are fed back
to the IF amplifier as a gaincontrol 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 noisefree
carrier at the lobing frequency. The carrier signal is phaselocked with the
antenna lobing. This is usually accomplished by means of an ac generator
mechanically linked to the rotating antenna.
Fig. 717 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 signalto
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.
720 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 argonfilled
gaseous discharge tube will produce a standard noise power output equiva
720] PROBLEMS IN MEASUREMENT OF RECEIVER CHARACTERISTICS 391
Demodulated
Output
NoC+NoS
■O NeC±NoS
Output
NoC±NeS
Fig. 717 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 3db loss be inserted in the receiver
rather than let the output noise level change. The 3db 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 3db 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 (dc 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 highPRFdoppler 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
720] 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 signaltonoise 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 longterm 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*
81 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 15). 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
lossoftrack 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 81 and 83 through 813, and 821 are by D. J. Healey III. Paragraph 82 is
by D. D. Howard and C. F. White. Paragraphs 814 through 820 are by C. F. White and
R. S. Raven. Paragraphs 822 through 834 are by G. S. Axelby.
394
82] 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 angletracking 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.
82 EXTERNAL AND INTERNAL NOISE INPUTS TO THE
RADAR SYSTEM 1
Paragraph 47 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 closedloop 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 longtime
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 oa^p, is defined as the pulsetopulse
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 lineof
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 o6s,„, 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 orec, 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
tonoise 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
82]
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 closedloop
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 flightpath input information
are excluded from art.
Angle tracking noise, with rms value of aat, is defined as the closedloop
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 flightpath 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. 81. 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. 81 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
rootmeansquare manner. To use the diagram of Fig. 81 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. Firecontrol 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 splitvideo
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,
onehalf 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 firecontrol 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. 82. 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 80sec 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. 83. 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 highfrequency 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. 82 Range Noise Power Spectral Distributions.
10 100
RELATIVE RANGE
1000
Fig. 83 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 lowfrequency region of noise extending from zero to approximately 10
cps causes a noise modulation within the closedloop servotarget combina
tion superimposed upon the tracking error caused by flight path input
information and associated system tracking errors. The lowfrequency
band also influences angle noise as explained later. Removal of the effects
of lowfrequency amplitude noise on angle tracking by suitable AGC design
is also discussed later.
A highfrequency 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 highfrequency
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 highfrequency 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 lowfrequency amplitude noise exists
incident to narrowband or slow AGC, the angle noise power (in suitable
units) equals onehalf the square of the radius of gyration of the target
reflectivity distribution.^ When lowfrequency 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 lowfrequency band since it is
associated with major aspect changes of the target. Because bright spot
wander noise is an uncorrelated component of targetgenerated 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. 423.^ In the spectra illustrated, the analytical method
excluded lowfrequency results below 3040 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.
83] AUTOMATIC FREQUENCY CONTROL 401
The effects of the spectral energy distribution of closedloop angle noise
and the contributions of lowfrequency amplitude noise modulation of
tracking error caused by flight path input information are discussed in
Paragraph 817.
83 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. 84).
Fig. 84 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 intermediatefrequency (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.
84 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
longterm frequency changes which occur incident to the effects of tem
perature, vibration, deterioration, and the like; there are also shortterm
frequency changes which are the result of a timevarying load impedance
connected to the transmitter, and frequency modulation from the heater
supply and powersupply 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. 85 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 81.
Table 81 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 =^ 15
Temperature =^ 15
Pressure (0 to 50,000 h) altitude  2.5
Pushing ( ='=10% linevoltage variation)
5.0
Aging; ^ 10
MAGNETRON PULLING
403
20
10
20
30
1 1
TemperatureFrequency
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. 85 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 81. 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 81 indicates that when a radar set is first energized it is usual for
the openloop frequency error to be rather large. A wide pull in range is
therefore required. {Pullin is the process whereby the error in receiver
tuning frequency existing at the instant of an offfrequency input signal is
reduced by the AFC operation.)
85 MAGNETRON PULLING
The single mode equivalent circut of a magnetron is shown in Fig. 86.
The magnetron is considered as a conventional selfexcited power oscillator
with the LC tank circuit inductively coupled to the output transmission
line.
As will be discussed in Paragraph 111, loading mismatch can affect both
the frequency and power output of the magnetron. Transient variations of
the load admittance occur in scanning antennaradome 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. 86 Equivalent Circuit of a Magnetron.
rotation because of imperfect rotary joints. The nature of tliese transient
variations governs the timeresponse 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. 87.
Fig. 87 Possible Operating Condition of a 4J50 Magnetron in a Typical Airborne
Radar Set (Rieke Diagram).
86] 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. 87. 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. 87. 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 wideangle 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
amplitudemodulated 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 lowfrequency pulling and minimum amplitude modulation
of the transmitter. It will be observed, however, that these advantages
are purchased at the price of lowerthanrated power output.
In a welldesigned antennaradome 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 82 gives some typical pulling characteristics of a conical scanning
system.
Table 82 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
86 STATIC AND DYNAMIC ACCURACY REQUIREMENTS
Tuning errors in the radar receiver degrade the output signaltonoise
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. 88 shows
the loss in signaltonoise 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. 88 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 separatepulsed trans
mitting oscillator and a continuouswave 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 sampleddata 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 onetwentieth the PRF. Tuning errors
CONTINUOUSCORRECTION 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 limitactivated correction (see Paragraph 88).
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.
87 CONTINUOUSCORRECTION AFC
Continuouscorrection AFC constitutes a type of closedloop 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. 89. 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. 89 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 localoscillator frequency changes can be expressed
as
£/.
£/rf
(81)
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 dc or zero frequency gain of the AFC. (K = discrimi
nator sensitivity X dc gain of the filter X modulation sensi
tivity of the oscillator)
G{s) is the normalized transfer function of the filter.
Fig. 810 shows the control characteristics of a typical klystron oscillator.
Referring to Table 81, 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. 810 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.
810 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 signalhandling capability.
However, the effect of a 1.5db 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.
87]
CONTINUOUSCORRECTION 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. 810.
Pullin and holdin performance of the AFC are determined by super
imposing the oscillator control curves on the discriminator characteristic.
Fig. 811 shows such a curve. Pullin has been defined previously. Holdin
Oscillator
Control Curves
Oscillator
Control Curves
Fig. 811 Discriminator Characteristics.
is the maximum frequency interval over which AFC control is effective.
Referring to Fig. 811, an error /^i in receiver tuning will occur for a
frequency deviation Ja\ of the transmitter frequency. The frequency
deviation /d2 corresponds to the holdin range. An error /« 2 results from
an oscillator deviation/d2. The pullin 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. 81 1 is the characteristic appearing at the
output of G(s) in Fig. 89. For highperformance systems G(s) is designed
with a large zero frequency or dc gain. Fig. 812 shows the resultant discrim
.Oscillator Control
Characteristic
Typical Characteristic Where
DC Amplifier Follows Discriminator
Fig. 812 Discriminator Characteristics.
inator curve measured at the oscillator. By increasing the dc gain for a given
IF characteristic the pullin range is increased. In a number of systems,
however, a limited dc gain follows the IF detectors. To realize a large
pullin 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 narrowband IF discriminator can exhibit a large pullin 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 dc 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 pulselength 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 peaktopeak 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
(82)
87] CONTINUOUSCORRECTION 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 717 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 dc 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 holdin and pullin 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
88 LIMITACTIVATED AFC
Limitactivated 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. 89 with G(s) =
K /s. The difference between this AFC and the continuouscorrection 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. 813 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. 813 Characteristics of a Switched (LimitActivated) 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.
89] 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 ■ (83)
where co is the angular frequency at which the signal frequency is changing,
and / is the peak deviation of the signal frequency.
89 THE INFLUENCES OF LOCAL OSCILLATOR
CHARACTERISTICS
A typical electronic tuning characteristic of the local oscillator is shown in
Fig. 810. Note that both the power output and the frequencymodulation
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. 810, 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 limitactivated 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. 813 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 lowfrequency feedback to the
resonator of the klystron and a highfrequency 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.
810 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
tonoise ratio which can be determined from the rms error and Fig. 88.
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.
811 DISCRIMINATOR DESIGN
There are two types of discriminators employed in pulse AFC — the
phase discriminator and the staggertuned 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 staggertuned
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. 814 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 jj = ^5, J2 = ^7, s ^ = jg, and ^4 = ^7. The discriminator response is
then of the same form as the difference in the envelope response of two
staggertuned onepole 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(SS^)(SSl)(SS2)
\(SS^)(SS;)(SSe)(Ssl)(SSj)\
s(ss)(ss;)(ss,
(SS^)(SS;)(SSe)(SS;)(SSy)
Fig. 814 Frequency Discriminator (Phase Type).
812 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 AFCIF 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 samplingtype 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.
813 PROBLEMS OF FREQUENCY SEARCH AND
ACQUISITION
Table 81 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. 810. It has been noted that it is also not always feasible to
utilize a discriminatorIF characteristic which will provide such a wide
pullin 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 fixedfrequency 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 pullin
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
pullin 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.
814 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 82 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. 815. 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. 815 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 steadystate or dc regulation of the video voltage
2. Attenuation of amplitude fluctuations
3. Fidelity with which intelligence is transmitted
4. AGC loop stability
The steadystate 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 48, and typical spectra of this
amplitude noise for two types of aircraft are shown in Figs. 423 and 424.
In Fig. 423 the amplitude noise spectrum for a propellerdriven aircraft
illuminated by Xband radiation is shown. Very predominant propeller
modulation peaks at harmonics of about 60 cps persist to over 300 cps. In
Fig. 424, the amplitude noise spectra generated by a B45 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 smallsignal approximation to the
nonlinear loop which will be derived in the following paragraph. Adequate
815]
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 dc 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 sampleddata servos must be used and the AGC loop band
width is limited to about half the repetition rate by stability considerations.
815 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. 81 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. 816 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„) (84)
e,= {ea eo)G,{s). (85)
^B. M. Oliver, "Automatic Volume Control as a Feedback Problem," Proc. IRE, 36,
466473 (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
(86)
= 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. (87)
The approximate linear feedback loop corresponding to Equations 86
and 87 is pictured in Fig. 81 6b. The outputinput ratio for this linear
regulating loop will have the following form :
Aen 1
KiAei 1 + K2G2(sy
(88)
816 STATIC REGULATION REQUIREMENTS OF AGC
LOOPS
The transfer function represented by Equation 88 gives the smallsignal
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. 81 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 84 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 gainbias curve in decibels per
volt is a constant. Fig. 817 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. 817 Typical IF Amplifi
Control Characteristics,
20 logio Gi = ^ + Beg
(89)
816] 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
(810)
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
84.
1 ;^n
0.1155
(811)
0.1155.
For the dc 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
(812)
Substituting Equation 812 into Equation 811 yields the following expres
sion for the AGC loop gain:
K^ = — fMHf^ X [gain change (db)] (813)
^o.max ^o,min
It is apparent from this expression that with the linear gain control charac
teristic shown in Fig. 817, 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 813.
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 813, 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 813 yields
Mmm= 49.5 = 33.4 db. (814)
'^' 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.
817
DYNAMIC REGULATION REQUIREMENTS OF AGC
LOOPS
It was previously noted that amplitude noise fluctuations in the receiver
output will modulate steadystate 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. 818.^" 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, 430435 (1959).
818] AGC TRANSFER CHARACTERISTIC DESIGN CONSIDERATIONS 423
mean square tracking error is plotted versus the lag error for no AGC, a
1cps 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 fastAGC designs.
Other factors to be considered in the design of the lowfrequency 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 48). 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^ ^ , .  (815)
1 ( KiKj2\S)
The required behavior of this function and the openloop characteristic
KiGii^s) will be examined in more detail.
818 AGC TRANSFER CHARACTERISTIC DESIGN
CONSIDERATIONS
From the discussion in Paragraphs 814 through 817 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 dropoff with a —1 slope on a dbversuslog 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 closedloop
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 halfpower 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
99, 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 onehalf the repetition rate to ensure stable operation.
This is particularly important in monopulse systems where rapid AGC
action is desired.
819 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 highfrequency lag and lead terms are incorporated into the
AGC filter to provide an openloop 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 closedloop 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. 819,
tan (prr,
KiG<i
1 + K,G,
K,G,. (816)
Fig. 819 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. (817)
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 openloop 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.
820 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 816, 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 818, 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. 820 Trial Design of AGC OpenLoop 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 lowfrequency corner of the AGC filter at
4 rad/sec. The AGC loop transfer characteristic thus developed is shown
in Fig. 820.
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 819. 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 818.
Fig. 821 shows a parallelT 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. 821 ParallelT Null Network with
Symmetry Pattern m.
Voltage transfer function of
network
u'^ 1
H + //
(^)
+ 1
(818)
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. 820 in combination with the single time
constant RC filter. The value of m (Fig. 821) was selected to provide a
total attenuation of 31.9 db (as required in Paragraph 819) at 292 rad/sec,
the lowest possible modulation frequency. The highfrequency gain margin
should be checked, particularly at onehalf the PRF. An inspection of
Fig, 820 shows that a minimum gain margin of 20 db, in comparison with
the 6db requirement, is maintained at all high frequencies.
The closedloop response of the AGC loop indicates most directly the
dynamic AGC action in attenuating lowfrequency amplitude noise and
transmitting modulation frequencies. Fig. 822 shows the closedloop
response corresponding to the trial design illustrated in Fig. 820.
"C. F. White, Tmns/er Characteristics of a Bridged ParatlelT Network, NRL Report R3I67,
27 August 1947.
21]
THE IF AMPLIFIER CONTROL CHARACTERISTIC
427
10
10
20
30
40
^,
/
\
/
/
A
i
/
\
y
^^Amplitude
yPhase
:r^
j^
\
5 10 20 50 100 200
ANGULAR FREQUENCY (rad/sec)
120
100 _
<o
80 H
Q
<
20
500 1000
Fig. 822 ClosedLoop Response of AGC, Trial Design.
821 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 818 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 gridcathode
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'^)
(819)
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.
(820)
When yf and Ei are so large that the negative excursion of the signal
extends beyond cutoff, Equation 820 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 820. In the
design of a gaincontrolled 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 816, 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
822] 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 signaltonoise ratio.
In addition to grid1 control of the amplifier stages, grid3 or plate and
screen control is sometimes employed. Grid3 control provides minimum
thirdorder 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 grid1 control of remote or semiremote cutoft tubes.
822 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 linesofreasoning 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
onedimensonal representation of Fig. 823. 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 longterm 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. 823. The effects of aircraft
motion are compensated in the dataprocessing 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. 823 Basic Relationships in the Airborne Antenna Drive System.
This technique is generally applicable to fanbeam 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, sidelooking 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. 823, the space pointing direction of the antenna may be
expressed :
^TL = Ga X (antenna command) \ Gd X (aircraft disturbance inputs)
+ Gm X (maneuver commands) (821)
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. (822)
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 openloop 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. 824. 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. 824 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
(823)
AtL = GaATL,c + (1  Ga)/lA
If Ga is essentially unity over the frequency range o^ Atl,c and A a
Aa = Atl,c (824)
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 closedloop 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 lineorsight
and the antenna pointing direction. Generally, however, the stabilization
provided by the radar angle tracking control is not sufficient: a faster,
tighter inner stabilizationcontrol loop must be used to provide the neces
sary isolation from aircraft motions.
A basic tracking radar stabilization system is shown in Fig. 825. ^^ 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. 825 Basic Tracking Radar Stabilization System. Jls = sight target line;
Et = angular error signal; Jtl = stabilization feedback; G3 = ratemeasuring
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. 825. This arrangement yields the interrelations among antenna
pointing direction lineorsight 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
ofsight angle Jls and the aircraft space angle Ja, the equation in Fig. 825,
Atl
becomes
r GjG
+ G2G,
^h.s +
1 + G1G2 + G2G;
•^TL
(gttg;) '^'''
l^Variations of this system are discussed
GolG, + G3)
Paragraph 831.
.^..,
^..1 (825)
(826)
824] AIRCRAFT MOTIONS 433
Over the lowfrequency range of^LS, iGs] « Gi, and the equation becomes
Jtl = Jls + ~ (827)
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.
823 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 firecontrol 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.
824 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 firecontrol 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 attackcourse
studies and knowledge of aircraft operation.
2. By the timeresponse 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 918 in systematic
errors.
434 REGULATORY CIRCUITS
3. By the frequencyresponse 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: 12°
(c) PITCH
Pitch angle: +180°, 30°
Pitch rate : 20° /sec up to 40° /sec
Pitch frequency: 213 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 pilotaircraft steering loop.
The data of Fig. 826, 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. 826 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 0.5sec 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 trackingline 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 twogimbal
system are shown in Fig. 827.
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. 827 Angle and Angular Rate Relationships for a TwoGimbal 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. 827. This fact
"In some missile applications it is necessary to provide a third gimbal to spacestabilize the
antenna in roll. This is not considered here.
824] AIRCRAFT MOTIONS 437
is important because the rolling rates can be quite large relative to the
yawing and pitching rates (see Fig. 826).
When the sinusoids representing aircraft motions are transformed into
equivalent motions in antenna coordinates, results like those shown in
Table 83 are obtained.^^ These results are typical of what might be
Table 83 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. 826. 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
crosscoupling 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 832.
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. 6445T31, Instrumentation
Laboratory, M.I.T., May 1951.
[b) A Statistical Description of LargeScale Atmospheric Turbulence by
R. A. Summers, Report No. T55, 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 highspeed, lowaltitude 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.
825] STABILIZATION REQUIREMENTS 439
can sometimes be quite severe and in such cases their effects should be
studied as part of the systems design.
825 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 39 to 314.
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 832).
440 REGULATORY CIRCUITS
826 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 226).
827 SEARCH STABILIZATION EQUATIONS
Because the antenna motion of a twogimbal antenna does not include
roll correction directly, the aircraft roll motion must be converted into the
proper azimuth or elevation commands for the twoantenna 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> (828)
sin Aa cos Ea = sin As cos Es cos <^
f (cos As cos Es sin 6 — sin Es cos 0) sin (829)
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. 828; 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. 828 Exact Coordinate Conversion Mechanization for a TwoGimbal 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> (830)
Ja = As cos <p  (d  Es) sin (831)
These equations are shown mechanized in Fig. 829.
Elevation
Antenna
Position
Antenna Search Loops
4> = Aircraft Roll Angle \ ^ ^^^^^^^^
e = Aircraft Pitch Angle )
A=As cos</) (dEs) sin0
Fig. 829 Approximate Coordinate Conversion Mechanization.
442 REGULATORY CIRCUITS
828 STATIC AND DYNAMIC CONTROL LOOP ERRORS
Perhaps the largest errors in the searchpattern stabilization are engen
dered in the antenna control loops. These errors are reduced to satisfactory
limits by proper design of the controlloop gains and bandwidths or by
modification of the pattern command signals. Errors to be considered are:
(a) Static or steadystate errors due to aircraft motion
(i>) Static or steadystate errors due to searchpattern velocity
(c) Dynamic ierrors due to changes in searchpattern command signals
It may be considered that part of the total allowable searchpattern
errors may be allotted to the antenna control loops. For example, a 0.35°
steadystate error may be assumed and it may be divided equally between
aircraft motion and searchpattern 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
openloop 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 steadystate errors caused by
aircraft motion to 0.25°, the nature of the aircraft motion must be known.
For example. Table 83 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. 830, 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° peaktopeak or Xi =
16.8°. Thus, the required openloop 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. 830 Derivation ot Required Loop Transfer Function.
calculations can be made for other frequencies, resulting in the circled
points shown in Fig.. 831. These points define the 7nini7nu?n openloop
LowFrequency 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. 831 Search Loop: OpenLoop Transfer Function.
444 REGULATORY CIRCUITS
gain necessary to provide isolation from the expected aircraft angular
SearchPattern Velocity Errors. In addition to providing isolation
from aircraft motion, the steadystate 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 steadystate position error is the velocity divided by. the
velocity constant. ^^ The constant antenna sweep velocity is determined as
discussed in Paragraph 57 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 seci = K^. (834)
Dynamic Errors Due to SearchPattern 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 steadystate value than
does the verticalmotion transient because the vertical motion is usually a
small position step instead of a large velocity reversal. Since it is desired
to resume the steadystate 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 83.
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. 832 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^~^xJ
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. 832 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) Openloop 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 steadystate error, and a relatively large time may be
required for it to settle if aj2 in Fig. 832 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. 832 indicates approxi
mate but useful relationships between the dynamic transient, the openloop
transfer function, and the steadystate error that help determine the search
controlloop 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. 832, 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 searchloop 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 openloop 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. 832.^^
SEARCH LOOP SYNTHESIS
1. Loop gain equation:
^(^Yco, = K, = 400 (835)
C0 2\C0l/
CO, ^ /C. ^ 400 6.49 (836)
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
(837)
_ 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, PGAC1, May 1956.
^The closedloop 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 837)
Combining Equations 837 and 838
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
(838)
(839)
(840)
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. 833 Nichols Chart of Search Loop.
448
REGULATORY CIRCUITS
Nichols chart is shown in Fig. 833.^^ Note that if the corner frequency in
Fig. 832 had been a double corner, C03 would be doubled; if it had been a
triplecorner frequency or equivalent, cos would be tripled, etc. However,
Fig. 833 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.
829 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. 834. 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. 834 General Block Diagram of Search Loop (One Channel).
input signal for aircraft motion is not shown because it was discussed in
Paragraph 827 and illustrated in Fig. 828 or 829. The practical problems
involved in its construction are those of making proper resolver connections
with correct phasing of ac signals. For the exact transformation, the
signals between the six resolvers must pass through isolating amplifiers or
phaseshifting devices. The loopactuating signal, or error signal, is usually
an ac voltage proportional to the difference between the input signal Xa
and the controlled antenna position /^a as shown in Fig. 834. It is obtained
in the exact coordinate transformation from the windings of resolvers, which
are mounted on the antenna. In the approximate transformation, the ac
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. 829 and 834. Wirewound
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 + C05a)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(W4CO3a;jC02)
(C04COgC0iCO2)
(oj4 + a)5)(c0j + a;2)
SRC
>R SRC + 1
(d)
Fig. 835 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. 831 do not exist naturally,
and it is necessary to provide some form of compensation. On small,
lowpower 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. 835. 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. 831 is obtained. This is illustrated in Fig. 836. It should be
noted that networks A and especially B of Fig. 835 would be used in series
with the lowpower 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 RC Networks Shown m Fig. 835) ^
\
1.0
\
\ 3
Fig. 836 Method of Determining ControlLoop Compensating Function.
829] 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 ac
error signal to a dc signal with a demodulator.^" A peak demodulator is
often used in the sear.ch loop because it has (1) less highfrequency noise
and (2) a smaller time constant than an averaging demodulator. After the
error signal is demodulated and passed through dc networks and a power
amplifier, it is applied to the power actuator which moves the antenna.
Generally, the actuator is a twophase ac electric motor or a hydraulic
actuator controlled with an electrically operated valve. If an ac motor
is used, the dc 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 dc
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 steadystate 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 searchpattern velocities and accelerations. The output power
of the actuator must be greater than the power required to move the
antenna inertia along the searchpattern 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 ac 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 firecontrol 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.
830 STABILIZATION DURING TRACK
As was discussed in Paragraph 825, 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 onehalf
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 83. 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.
831 POSSIBLE SYSTEM CONFIGURATIONS
Several physical configurations are used to mechanize the stabilization
loops. Some of them are shown in Fig. 837.^^ 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. 837 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. 837a 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. 350353.
^^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, PGAC6,
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 selfcontained 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. 838. Such
Command
Torque \/p
Rate ^^ '^
Command
Torque
Motor
Fig. 838 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. 837b. 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 steadystate error in the integrating gyro loop is
zero" for a constant rate command while the steadystate error in the rate
gyro loop has a finite value proportional to the rate command and inversely
proportional to the dc loop gain. However, this steadystate 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 steadystate 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 firecontrol 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 firecontrol
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.
832] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 457
accurate, with low drift rates, but they are expensive and require tempera
turecompensating circuits to maintain the preset value of damping torque.
Generally, this accuracy is not needed to provide antenna stabilization, but
is needed for the firecontrol computer which uses the gyro signals to
provide accurate information about the lineofsight motion.
Aircraft Gyro Stabilization Loops. Another of the many methods
of providing antenna stabilization is shown in Fig. 837c. 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.
832 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 825 and 830) 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 targetseeking 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
ownship'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 onechannel 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 lowfrequency component of the
stabilization error must be compatible with the specification for angular
rate measurement accuracy to avoid inaccuracies in the firecontrol compu
tation.
(2) The highfrequency components of the stabilization error (greater
than 1—2 rad/sec) do not affect the firecontrol 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 127).'*^
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 227) :
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 pilotinduced 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.
832] 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 83 will be used to provide aircraft input data.
For the lowfrequency (less than 2 rad) inputs
where 0.7 Atl is the rms value of the sinusoidal inputs from Table 83, for
example, at co = 1.04, Jtl = 35/2 = 17.5°/sec and Kg < 0.00245'.
However, for the highfrequency 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 23 the value of the angular rate sensitivity factor at the time
of firing on a headon course is 14.2.
If a 0.5second 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
(844)
Using the values for Atl in Table 83, the desired Ks is given in the
following table as a function of frequency.
A plot of Ks is shown in Fig. 839. 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 lowfrequency inputs is also
indicated. The design of the stabilization loop characteristics to achieve
this attenuation is described in Paragraph 833.
460 REGULATORY CIRCUITS
Table 84 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. 839 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. 95) 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. 839.
832] 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 firecontrol 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 227
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 43.
462
REGULATORY CIRCUITS
signal fading period, it is necessary to prevent the antenna from being
moved in space away from the target direction by ownship'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. 840, this noise causes the antenna in a
Antenna should be
within the Shaded
Area 95% of the
Time After the
Radar Signal Fades
Fig. 840 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 103 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 firecontrol 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. 840 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
832] 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 dc 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. 840, 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 825 and 832, 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 onehalf at all frequencies to ensure that instability will not occur.^*
Fig. 841 Coupling Between the Air Frame and the Antenna.
G2 = amplifier, actuator, antenna; Gs = rate gyro; Gc = firecontrol 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 onetwentieth 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 onehalf, 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. 841. 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. 841 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. 841, the stabilization loop attenuation becomes
l^sl < ,^ (845)
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.
833 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. 839. 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.
833] 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. 837a as
j^ ] + G2CJ8
!^^I=^tVi i^ G2G3»1.0" (847)
The magnitude of Ks is determined as a function of frequency from one
of the criteria in Paragraph 832 and plotted as shown in Fig. 839. 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.
839 but increased by 175 per cent is shown in Fig. 842.*^ This charac
teristic need not be extended to frequency regions below 3 rad/sec as
discussed at the beginning of Paragraph 830. As shown in Fig. 842, the
magnitude of G2G3 is 250 at 3 rad/sec to make the actual gain equal that
desired. This becomes the dc gain in the rate gyro loop shown in Fig. 837a.
In an integrating gyro loop, shown in Fig. 837b, the dc gain is infinite, but
the velocity constant would be 750 sec~^ as indicated in Fig. 842.
As in the search loop design, the other characteristics of the openloop
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 828.
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. 842 Stabilization Loop Transfer Ratio Designs.
corner frequencies of Fig. 842 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
833] 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. 842. 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. 842; 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. 226241, 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.
834 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. 837. 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. 837a 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 shortcircuit the capacitors in the
compensating networks. The rate gyro is a selfcontained 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 827, 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)
834] 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. 837b 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 highfrequency
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. 837a 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. 842. The transfer function
of the compensating elements can be derived as indicated in Fig. 836.
The configuration of Fig. 837c has been described in Paragraph 831 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 firecontrol 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