PRINCIPLES OF.©UrDED MiSSlLE DESI© WOODS HOLE OCEANOGRAPHIC INSTITUTION LABORATORY BOOK COLLECTION AIRBORNE RADAR H g u Pi < O CQ u Pi o Pi PRINCIPLES OF GUIDED MISSILE DESIGN Editor of the Series Grayson Merrill, captain, u.s.n. (Ret. General Manager, Astrionics Division, Fairchild Engine and Airplane Corporation GUIDANCE — by Arthur S. Locke and collaborators AERODYNAMICS, PROPULSION, STRUCTURES AND DESIGN PRACTICE— 6y E. A. Bonney, M. J. Zucrow, and C. W. Besserer OPERATIONS RESEARCH, ARMAMENT, LAUNCH- ING— 6y G. Merrill, H. Goldberg, and R. H. Helmholz MISSILE ENGINEERING HANDBOOK— 6y C. W. Bes- serer DICTIONARY OF GUIDED MISSILES AND SPACE FLIGHT — Edited by Grayson Merrill SPACE FLIGHT— 6y K. A. Ehricke Vol. I: ENVIRONMENT AND CELESTIAL MECHANICS Vol. II: DYNAMICS Vol. Ill: OPERATIONS SYSTEMS PRELIMINARY DESIGN— 6y J.J. Jerger AIRBORNE RADAR— by D. J. Povejsil, R. S. Raven, and P. Waterman RANGE TESTING— 6y J.J. ScavuUo and Eric Burgess FOUNDATIONS OF SEARCH THEORY— by N. S. Potter PRINCIPLES OF INERTIAL NAVIGATION— 6y Richard Parvin PRINCIPLES OF GUIDED MISSILE DESIGN Edited by Captain Grayson Merrill, u.s.n. (Ret.) DONALD J. POVEJSIL Director, New Products Services; Formerly Manager, Weapons Systems Engineering Air Arm Division, Westinghouse Electric Corporation Pittsburgh, Pa. ROBERT S. RAVEN Advisory Engineer, Weapons Systems Engineering ... . .- ^ , , Westinghouse Electric Corporation Air Arm Division, Baltimore, Md. Ml'-. .PETER WATERMAN Head, Naval Research Laboratories, Radar Division, Washington, D. C. o AIRBORNE RADAR MARINE BIOIO<3ICAJ- UBORATORY LIBRARY WOOBS HOLE, MASS. W. H. 0. I. D. VAN NOSTRAND COMPANY, INC. PRINCETON, NEW JERSEY • TORONTO • NEW YORK • LONDON D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey {Principal office) 24 West 40 Street, New York 18, New York D. Van Nostrand Company, Ltd, 358, Kensington High Street, London, W.14, England D. Van Nostrand Company (Canada), Ltd. 25 Hollinger Road, Toronto 16, Canada Copyright © 1961, by D. VAN NOSTRAND COMPANY, Inc. Published simultaneously in Canada by D. Van Nostrand Company (Canada), Ltd. Library of Congress Catalogue Card No. 61-8542 No reproduction in any fortn of this book, in whole or in part {except for brief quotation in critical articles or reviews), may' be made without written authorization from the publishers. PRINTED IN THE UNITED STATES OF AMERICA CONTRIBUTORS P. J. Allen • Naval Research Laboratory^ Washington^ D. C. (Coauthor, Chapter 10) G. S. AxELBY • PFestinghouse Electric Corp., Air Arm Division^ Baltimore, Md. (Coauthor, Chapters 8 and 9) B. L. CoRDRY • Bendix Aviation Corp., Radio Division, Baltimore, Md. (Coauthor, Chapter 14) W. R. Fried • General Precision Laboratory , Pleasantville, N. Y. (Coauthor, Chapter 14) S. F, George • Naval Research Laboratory , Washington, D. C. (Coauthor, Chapter 6) M, Goetz • Westinghouse Electric Corp., Central Laboratories, Pittsburgh, Pa. (Coauthor, Chapter 13) D. J. Healey III • Westinghouse Ekctric Corp., Air Arm Division, Baltimore, Md. (Author, Chapter 7; Coauthor, Chapter 8) L. Hopkins • Raytheon Manufacturing Corp., Maynard Labora- tory, Maynard, Massachusetts (Coauthor, Chapter 6) D. D. Howard • Naval Research Laboratory , Washington, D. C. (Coauthor, Chapter 8) A. Kahn • Westinghouse Electric Corp., Air Arm Division, Baltimore, Md. (Coauthor, Chapter 12) M. Katzin • Electromagnetic Research Corp., Washington, D. C. (Author, Chapter 4) R. H. Laprade • Westinghouse Electric Corp., Air Arm Division, Baltimore, Md. (Coauthor, Chapter 14) T. Moreno • Varian Associates, Palo Alto, California (Author, Chapter 11) R. M. Page • Naval Research Laboratory, Washington, D. C. (Coauthor, Chapter 6) p. M. . Pan D. J. POVEJSIL R. S. Raven R. M. Sando F. Stauffer M. Taubenslag J. W. Titus P. Waterman M. S. Wheeler C. F. White H. Yates CONTRIBUTORS Westinghouse Electric Corp., Air Arm 'Division, Baltimore, Md. (Coauthor, Chapter 10) Westinghouse Electric Corp., New Products Serv- ices, Pittsburgh, Pa. (Editor and Coauthor, Chapters 1, 2, 6, and 12) Westinghouse Electric Corp., Air Arm Division, Baltimore, Md. (Editor and Author, Chapters 3 and 5; Coauthor, Chapters 8 and 12) Westinghouse Electric Corp., Air Arm Division, Baltimore, Md. (Coauthor, Chapter 13) Westinghouse Electric Corp., Air Arm Division^ Baltimore, Md. (Coauthor, Chapter 14) Aeronica Manufacturing Corp., Aerospace Divi- sion, Baltimore, Md. (Coauthor, Chapter 12) Naval Research Laboratory , Washington, D. C. (Coauthor, Chapter 13) Naval Research Laboratory, Washington, D. C. (Editor and Coauthor, Chapters 1 and 2) Westinghouse Electric Corp., Air Arm Division, Baltimore, Md. (Coauthor, Chapter 10) Naval Research Laboratory , Washington, D. C. (Coauthor, Chapters 8 and 9) Barnes Engineering Co., hifrared Division, Stam- ford, Conn. (Coauthor, Chapter 6) FOREWORD "Airborne Radar" is the eighth volume in the series Principles of Guided Missile Design. Other titles in the series are Guidance; Aerodynamics, Propulsion, Structures and Design Practice; Operations Research, Armament, Launching; Missile Engineering Handbook; Dictionary of Guided Missiles and Space Flight; Space Flight I — Environment and Celestial Mechanics; Space Flight II — Dynamics; Space Flight III — Operations; Preliminary Systejn Design; and Range Testing. The purpose of the series as a whole is to give a basis for instruction to graduate students, professional engineers, and technical officers of the armed services so that they can become well grounded in the technology of guided missiles and space flight. This book concerns itself with one of the most important systems used now in missilery and to be used soon in space flight — airborne radar. As this is written the world is commenc- ing a great search for a defense against intercontinental ballistic missiles; airborne radar will play a key role in this defense. In the interests of brevity, this book presumes considerable knowledge on the reader's part — namely, knowledge of the basic principles of elec- tronics and electromagnetic propagation and familiarity with the associ- ated language. It also presumes a knowledge of the weapons systems employing airborne radars. Certain prior issues of the series will be found especially valuable as references; these are the Dictionary oj Guided Missiles and Space Flight; Guidance; and Operations Research, Armament, Launch- ing. Criticisms and constructive suggestions are invited. With this aid and by keeping abreast of the state of the art we hope to make timely revisions to this volume. Grateful acknowledgment is made to the many authors and publishers who kindly granted permission for the use of their material and to the Department of Defense, whose cooperation made possible a meaningful text without violation of security. The opinions or assertions contained herein are the private ones of the authors and the editor and are not to be construed as official or reflecting the views of any government agency or department. Grayson Merrill Editor Wyandanch, Long Island, New York November J 960 PREFACE The basic purpose of this book is to present a balanced treatment of the airborne radar systems design problem. Primary emphasis is placed upon the interplay between radar techniques and components on one hand, and the types of weapons systems which employ airborne radars on the other. Radar design details have been eliminated for the most part except for illustrative examples which show how a design detail can exert an impor- tant influence on the operation of a complete weapons system. Although the treatment is directed at airborne radars, this volume will be found extensively applicable to surface radars as well. Since the latter enjoy a less severe environment, especially with regard to relative target motion, stability of platform, and space and weight restrictions, the prin- ciples governing their design will implicitly be covered here. Because this book attempts to bridge the gap between the abstractions of overall system design and the hard realities of hardware design, it con- tains material which will interest almost anyone involved in the study, design, or application of airborne radars. For example, although the re- ceiver designer will not learn much that is new to him about circuitry design, he can learn a great deal about how the design of a receiver should be planned for optimum benefit to the overall system. Similarly, engineers and scientists charged with responsibility for monitoring the efi^orts of airborne radar subcontractors can find this book most useful in determin- ing the type of direction they should give to subcontractors to ensure eventual compatibility of the airborne radar with the complete weapons system. A particular eflfort has been made to present facts and combinations of facts which have not enjoyed prior publication in book form. This has been done at the expense of excluding a great deal of historical and back- ground information already available in the printed literature. This book possesses close ties with two previous volumes in the series: GUIDANCE by A. S. Locke et al. and OPERATIONS RESEARCH, ARMAMENT, LAUNCHING by Grayson Merrill, Harold Goldberg, and Robert H. Helmholz. Radar techniques and problems are presented in greater detail than was possible in GUIDANCE; similarly the problem of translating operational studies into detailed airborne radar requirements is covered in greater detail than was possible in OPERATIONS RE- SEARCH. The basic theme of these earlier volumes — the importance of the systems approach — is continued in this volume. xi xii PREFACE Many of the authors have had previous association on team efforts aimed at the development and production of complex airborne radar equipments. The technical approaches presented thus represent tools forged on the anvil of experience — tools which have facilitated the solution of many- difficult problems. It is the authors' hope that succeeding generations of system designers may use these tools to their advantage in designing the even more complex systems to come. Acknowledgments In addition to the authors, there are many individuals and organizations to whom acknowledgment must be made for an active part in the writing of this book. Members of the Naval Research Laboratories, the Naval Air Development Center, the Fairchild Astrionics Division and the West- inghouse Electric Corporation assisted with suggestions, criticisms, a.nd technical readings. Deepest thanks must be extended to the government laboratories (NRL and WADC) and the corporations (Barnes Engineering, Bendix Aviation, General Precision Laboratory, Raytheon Manufacturing, Varian Associates, and Westinghouse Electric) who provided encourage- ment and assistance to the contributors. The assistance of Mr. R. G. Clanton of Westinghouse was vital to the preparation of the examples employed in Chapter 2. In addition, Mr. Clanton's many helpful suggestions and detailed reviews of the remainder of this chapter are most gratefully acknowledged. Mr. R. H. Laprade had the responsibility of reviewing all the material relating to propagation in addition to his contribution to Chapter 14. Mr. A. Stanley Higgins, Mr. Melvyn Goetz and Dr. J. F. Miner of West- inghouse rendered invaluable services in overseeing the myriad details in- volved in the editing and production of the final text. The Westinghouse Electric Corporation deserves special thanks for the assistance provided on drawings, typing, and the reproduction of the many drafts of the manu- scripts. D, J. POVEJSIL R. S. Raven P. Waterman CONTENTS List of Contributors vii Foreword ix Preface xi 1 ELEMENTS OF THE AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 1-1 Introduction . 1 1-2 Classifications of Radar Systems 2 1-3 Installation Environment 3 1-4 Functional Characteristics of Radar Systems 4 1-5 The Modulation of Radar Signals 16 1-6 Operating Carrier Frequency 26 1-7 The Airborne Radar Design Problem 27 1-8 The Systems Approach to Airborne Radar Design ... 30 1-9 Systems Environments 35 1-10 Weapons System Models 36 1-11 The Basic Statistical Character of Weapons System Models 39 1-12 Construction and Manipulation of Weapons System Models 41 1-13 Summary 44 2 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-1 Introduction to the Problem 46 2-2 Formulating the System Study Plan 48 2-3 Aircraft Carrier Task Force Weapons System 50 2-4 The Target Complex 55 2-5 The Operational Requirement 57 2-6 The System Concept 57 2-7 The System Study Plan 5^ 2-8 Model Parameters 60 2-9 System Effectiveness Models 61 2-10 Preliminary Design of the Airborne Early Warning System 67 2-11 AEW System Logic and Fixed Elements 70 2-12 AEW Detection Range Requirements 73 2-13 AEW Target Resolution Requirements 75 xiii xiv CONTENTS 2-14 Interrelations of the AEW System, the CIC System, the In- terceptor System, and the Tactical Problem 79 2-15 Accuracy of the Provisional AEW System 80 2-16 Information-Handling Capacity of the Provisional AEW System 84 2-17 Velocity and Heading Estimates 85 2-18 AEW Radar Beamwidth as Dictated by the Tactical Problem 89 2-19 Factors Affecting Height-Finding Radar Requirements . . 92 2-20 Summary of AEW System Requirements 96 2-21 Evaluation of Tentative Design Parameters with Respect to the Tactical Problem 98 2-22 Interceptor System Study Model 100 2-23 Probability of Reliable Operation 103 2-24 Probability of Viewing Target— Vectoring Probability . . 104 2-25 Analysis of the Vectoring Phase of Interceptor System Op- eration 106 2-26 AT Radar Requirements Dictated by Vectoring Considera- tions Ill 2-27 Analysis of the Conversion Problem 116 2-28 Lock-on Range and Look-Angle Requirements Dictated by the Conversion Problem 130 2-29 AI Radar Requirements Imposed by Missile Guidance Con- siderations 135 2-30 Summary of AI Requirements 136 2-31 Summary 137 3 THE CALCULATION OF RADAR DETECTION PROBABILITY AND ANGULAR RESOLUTION 3-1 General Remarks 138 3-2 The Radar Range Equation 138 3-3 The Calculation of Detection Probability for a Pulse Radar 141 3-4 The Effect of Scanning and the Cumulative Probability of Detection 156 3-5 The Calculation of Detection Probability for a Pulsed- Doppler Radar 162 3-6 Factors Affecting Angular Resolution 168 4 REFLECTION AND TRANSMISSION OF RADIO WAVES 4-1 Introduction • 174 4-2 Reflection of Radar Waves 175 4-3 Effect of Polarization on Reflection 179 4-4 Modulation of Reflected Signal by Target Motion . . . 180 CONTENTS XV 4-5 Reflection of Plane Waves from the Ground 181 4-6 Effect of Earth's Curvature 190 4-7 Radar Cross Sections of Aircraft 192 4-8 Amplitude, Angle, and Range Noise 198 4-9 Prediction of Target Radar Characteristics 208 4-10 Sea Return 211 4-11 Sea Return in a Doppler System 217 4-12 Ground Return 219 4-13 Altitude Return 222 4-14 Solutions to the Clutter Problem 224 4-15 Attenuation in the Atmosphere 227 4-16 Attenuation and Back-scattering by Precipitation . . . 230 4-17 Attenuation by Propellant Gases 231 4-18 Refraction Effects in the Atmosphere 233 5 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 5-1 Introduction 238 5-2 Fourier Analysis . 238 5-3 Impulse Functions 243 5-4 Random Noise Processes 245 5-5 The Power Density Spectrum 248 5-6 Nonlinear and Time-Dependent Operations 253 5-7 Narrow Band Noise 258 5-8 An Application to the Evaluation of Angle Tracking Noise 264 5-9 An Application to the Analysis of an MTI System . . . 269 5-10 An Application to the Analysis of a Matched Filter Radar . 272 5-11 Application to the Determination of a Signal's Time of Arrival 281 6 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 6-1 Introduction 292 6-2 Basic Principles 293 6-3 Monopulse Angle Tracking Techniques 300 6-4 Correlation and Storage Radar Techniques 305 6-5 FM/CW Radar Systems 311 6-6 Pulsed-Doppler Radar Systems 320 6-7 High-Resolution Radar Systems 333 6-8 Infrared Systems 338 7 THE RADAR RECEIVER 7-1 General Design Principles 347 7-2 The Interdependence of Receiver Components .... 352 XVI CONTENTS 7-3 Receiver Noise Figure 353 7-4 Low-Noise Figure Devices for RF Amplification .... 356 7-5 Mixers 357 7-6 Coupling to the Mixer 361 7-7 IF Amplifier Design 362 7-8 Considerations of IF Preamplifier Design 368 7-9 Overall Amplifier Gain 373 7-10 Gain Variation and Gain Setting 375 7-11 Bandwidth and Dynamic Response 375 7-12 Sneak Circuits 377 7-13 Considerations Relating to AGC Design 379 7-14 Problems at High-Input Power Levels 380 7-15 The Second Detector (Envelope Detector) 382 7-16 Gating Circuits 386 7-17 Pulse Stretching 387 7-18 Connecting the Receiver to the Related Regulating and Tracking Circuits 388 7-19 Angle Demodulation 389 7-20 Some Problems in the Measurement of Receiver Character- istics 390 8 REGULATORY CIRCUITS 8-1 The Need for Regulatory Circuits 394 8-2 External and Internal Noise Inputs to the Radar System . 395 8-3 Automatic Frequency Control 401 8-4 Variation of Transmitter Frequency with Environmental Conditions 402 8-5 Magnetron Pulling . . . . ' 403 8-6 Static and Dynamic Accuracy Requirements 405 8-7 Continuous-Correction AFC 407 8-8 Limit-Activated AFC 412 8-9 The Influences of Local Oscillator Characteristics . . . 413 8-10 Relation to Receiver IF Characteristics 414 8-11 Discriminator Design 414 8-12 Instantaneous AFC 415 8-13 Problems of Frequency Search and Acquisition .... 416 8-14 Automatic Gain Control 416 8-15 Linear Analysis of AGC Loops 419 8-16 Static Regulation Requirements of AGC Loops .... 420 8-17 Dynamic Regulation Requirements of AGC Loops . . . 422 8-18 AGC Transfer Characteristic Design Considerations . . . 423 8-19 The Modulation Transmission Requirement 424 8-20 Design of an AGC Transfer Function 425 CONTENTS xvii 8-21 The IF Amplifier Control Characteristic 427 8-22 The Angle Measurement Stabilization Problem .... 429 8-23 AI Radar Angle Stabilization 433 8-24 Aircraft Motions 433 8-25 Stabilization Requirements 439 8-26 Search Pattern Stabilization 440 8-27 Search Stabilization Equations 440 8-28 Static and Dynamic Control Loop Errors 442 8-29 Search Loop Mechanization 448 8-30 Stabilization During Track 452 8-31 Possible System Configurations 453 8-32 Accuracy Requirements on the Angle Track Stabilization Loop 457 8-33 Dynamic Stability Requirements on Angle Track Stabiliza- tion 464 8-34 Stabilization Loop Mechanization 468 9 AUTOMATIC TRACKING CIRCUITS 9-1 General Problems of Automatic Tracking 471 9-2 Automatic Angle Tracking 474 9-3 External Inputs: Undesired and Desired 475 9-4 Requirements in Angle Tracking Accuracy 479 9-5 Angle Tracking System Organization 480 9-6 Tracking Loop Design 485 9-7 Angle Tracking Loop Rate Errors 486 9-8 Angle Tracking Loop Position Errors 489 9-9 Angle Tracking Loop Mechanization 492 9-10 Introduction to Range and Velocity Tracking 498 9-11 x'\utomatic Range Tracking 498 9-12 Servo System Transfer Function Relationship to Input Time Function for a Range Tracking System 502 9-13 Range Tracking Design Example 505 9-14 Practical Design Considerations 508 10 ANTENNAS AND RF COMPONENTS 10-1 Antennas: Introduction to Radar Antennas 512 10-2 Some Fundamental Concepts Useful in the Development of Radar Antenna Requirements 513 10-3 The Paraboloidal Reflector as a Radar Tracking Antenna . 515 10-4 System Requirements for Radar Antennas 518 10-5 Pattern Simulation as a Link Between System Requirements and Antenna Characteristics 520 xviii CONTENTS 10-6 Several Anomalous Effects in Antennas for Tracking Systems 523 10-7 The Linear Array as a Fan Beam Antenna for Surveillance 524 10-8 Two-Arm Spiral Antennas 528 10-9 Radomes 531 10-10 Introduction to Transmission Lines and Modes of Propaga- tion 535 10-11 Types of Transmission Lines and Modes of Propagation . . 536 10-12 Standing Waves and Impedance Matching 540 10-13 Broadband System Design 543 10-14 Pressurization 545 10-15 Miscellaneous Microwave Components 546 10-16 Microwave Ferrite Devices and Their Application . . . 557 10-17 Microwave Dielectric, Magnetic, and Absorbent Materials . 565 10-18 The Duplexing Problem 566 10-19 Duplexing Schemes 567 10-20 Special Problems of Coherent Systems 573 10-21 Solid-State Amplifiers 574 11 THE GENERATION OF MICROWAVE POWER 11-1 The Magnetron 580 11-2 The Klystron 590 11-3 Traveling Wave Tubes for High Power 597 11-4 Modulation Techniques for Beam-Type Amplifiers . . . 599 11-5 A Typical Radar System Employing a High-Gain Amplifier 601 11-6 Backward Wave Oscillators — Carcinotrons 602 11-7 The Platinotron 603 12 DISPLAY SYSTEM DESIGN PROBLEMS 12-1 Introduction 607 12-2 Uses of Display Information 608 12-3 Types of Displays 613 12-4 Types of Input Information 619 12-5 The Cathode Ray Tube 621 12-6 Important Characteristics of Electrical-to-Light Transducers 627 12-7 Important Characteristics of the Human Operator . 634 12-8 Development of Requirements for a Display System . . . 651 12-9 Special Display Devices 655 12-10 Special Displays 673 13 MECHANICAL DESIGN AND PACKAGING 13-1 The Influence of Environment on Design 680 13-2 Military Specifications 682 13-3 Temperature 683 CONTENTS xix 13-4 Solar Radiation 692 13-5 Nuclear Radiation 692 13-6 Vibration and Shock 694 13-7 Acoustic Noise 704 13-8 Acceleration 707 13-9 Moisture 708 13-10 Static Electricity and Explosion 710 13-11 Pressure 711 13-12 Maintenance and Installation 712 13-13 Transportation and Supply 714 13-14 Potential Growth 715 13-15 ReUabihty 715 14 AIRBORNE NAVIGATION AND GROUND SURVEILLANCE RADAR SYSTEMS 14-1 Introduction to Doppler Navigation Systems 726 14-2 Basic Principles of Doppler Radar Navigation .... 728 14-3 System Considerations 733 14-4 Major Characteristics and Components of a Doppler Radar Navigation System 736 14-5 Doppler Navigation System Errors Caused by Interactions with the Ground and Water 746 14-6 Modifying the Radar Range Equation for the Doppler Navi- gation Problem 749 14-7 Low Altitude Performance and the "Altitude Hole" . . . 752 14-8 Doppler Navigation System Performance Data .... 755 14-9 Introduction to Weather Radar 759 14-10 Meteorological Effects at Microwave Frequencies . . . 760 14-11 Designing Airborne Radar Systems Explicitly for Weather Mapping 764 14-12 Modifying the Radar Range Equation for the Weather Problem 764 14-13 Relative Importance of Design Variables in Airborne Weather Radar 766 14-14 Design Features 769 14-15 Introduction to Active Airborne Ground Mapping Systems 772 14-16 Basic Principles 772 14-17 System Considerations 774 14-18 Major Characteristics and Components 778 14-19 Modifying the Radar Range Equation for the Active Ground Mapping Problem 782 14-20 Resolution Limits in Ground Mapping Systems .... 784 XX CONTENTS 14-21 Future Possibilities in Airborne Active Ground Mapping Systems 787 14-22 Introduction to Infrared Reconnaissance 787 14-23 Basic Principles Concerning IR Ground Mapping . . 788 14-24 System Considerations 791 14-25 Major Systems Features 798 14-26 New Developments 801 Index 805 J. POVEJSIL • P. WATERMAN CHAPTER 1 ELEMENTS OF THE AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 1-1 INTRODUCTION This book presents, and illustrates by examples, the basic information and procedural techniques required to plan and execute the design of an integrated airborne radar system. Basically, this design problem has three parts: (1) the development of radar system performance requirements based on the operational requirements of the overall, weapons system; (2) the development and application of specific radar techniques that will meet the performance requirements within the limitations imposed by laws of nature and the state of the art; (3) the evaluation of the proposed radar system to determine whether or not it meets the requirements of the overall weapons system. In each part of the design problem, the systems concept is employed; i.e., the airborne radar system is viewed as an integral part of a complete weapons system rather than as a separate entity. The systems concept will be developed by the case study method. A hypothetical weapons system model will be constructed. This model will then be analyzed in relation to the operational requirements in order to derive the specific characteristics of the various system environments which have an important bearing on the airborne radar system design. Those areas which sensitively affect the overall system capability will then be developed. By using the derived characteristics, it will be shown how airborne radar systems may be selected and designed to fulfill the overall system require- ments and be compatible with the system environments. As an example, the air defense system of a naval carrier task force will be considered. Two types of airborne radar systems are included in this weapons system. 1. An airborne early warning (AEW) system for alerting the air defense of a fast carrier task force. 2. An interceptor defense system, utilizing the primary information generated by the AEW system. 1 2 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM The analysis of these examples presents a basic method of approach which involves the concept of balancing various system elements — a procedure that can and should be used in the design of any airborne radar system. Perhaps the most important concept that must be grasped by the radar designer is this: A radar is usually a small but vitally important part of a dynamic system, i.e., a system whose basic characteristics and parameters are constantly changing functions of time. Because of its role as the "eyes" of the system, the dynamic performance of the radar must be related to — and to a large extent, governed by — the dynamic perform- ance required of the entire system. For this reason, the radar designer must possess the capability for understanding and analyzing the overall weapons system in addition to his specialized knowledge of the details of radar systems analysis and design. 1-2 CLASSIFICATIONS OF RADAR SYSTEMS In order to provide background for the discussion of the systems aspects of airborne radar design, the basic characteristics and uses of radar systems are described. Many of the descriptive terms commonly used in radar system technology are defined. Some of the simpler mathematical expres- sions that arise in radar work are presented. Radar is a word derived from the function performed by early radar systems — RAdixo Detection And, /hanging. The word was meant to denote systems that transmitted and received radio signals. Today the meaning of the word has been extended to include a wide variety of systems that employ microwave techniques. It encompasses systems using received energy originating in the system (active systems), systems using received energy originating at the target (passive systems), systems using received energy originating at a transmitter separate from the receiver or target (semiactive systems), and systems emitting electromagnetic radiations for various purposes (transmitting or illuminating systems). Many complex weapon systems include combinations of these basic types. For example, an electronic countermeasures system may be composed of a passive radar system that detects the presence of hostile electromagnetic radiation and utilizes this intelligence to control the action of a jamming system to combat the enemy radiation. The profusion of radar systems in use today requires that some logical means of classification be employed. One such means that has achieved general acceptance classifies a radar system according to the four character- istics: 1. Installation environment 2. Function(s) 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 3 3. Types of modulation intelligence carried on the transmitted and received radiations and the types of demodulation processes used to extract information from the received signals 4. Operating carrier frequency Reference to these four characteristics is usually made in any general qualitative description of a radar — e.g. an (1) airborne (2) intercept search and track (3) conical-scan pulse radar (4) operating at X Band. 1-3 INSTALLATION ENVIRONMENT The most common types of radar system installations are: 1. Ground-based 3. Airborne (piloted aircraft) 2. Ship-based 4. Airborne (missile) Procurement agencies, in general, have been divided into groups according to installation environment in order to simplify their diversity of interest. Such a division facilitates the proper treatment of the complex problems associated with the development and design of a radar system for a partic- ular installation environment, but does not always provide the cross fertilization of experience needed to take advantage of progress in any one particular line. 1-4 FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS Some basic functions which may be performed by radar systems are: 1. Search and detection 6. Communication 2. Identification 7. Radiation detection (Ferret) 3. Tracking 8. Illumination 4. Mapping 9. Information relay 5. Navigation 10. Jamming 11. Scientific research (e.g. radio astronomy) A given radar system may perform only one of these functions. More frequently it will perform two or more. Multimode operation is particularly characteristic of airborne radar systems where space, size, and weight limitations dictate that maximum capability and flexibility be obtained from each pound of radar equipment. The specification of the functions that must be performed by the radar systems equipments is a major product of the system study that must precede equipment design. This system study must also produce quanti- tative performance goals for each of the required functions. In cases where multimode operation is required, the system study must set up a definitive specification of primary and secondary modes. This 4 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM definition can serve as the basis for arbitrating conflicting design require- ments resulting from the multimode requirement. The specific functions performed by a radar system are outlined below in somewhat more detail. Search and Detection. An important function of a radar system is to interrogate a given volume of space for the presence (or absence) of a target of tactical interest. One very common method by which a radar system may be used to perform this function is shown in Fig. 1-1. In this ,-»- Search Radar Scan Pattern ar Search and Detection. example, RF (radio-frequency) energy is generated in the radar system (active system). This energy is focused into a highly directional beam by an antenna and propagated through space. Should there be an object of appropriate characteristics within the radar beam, a portion of the electro- magnetic energy impinging on the object will be scattered away from it. A portion of this scattered energy finds its way back to the point of trans- mission where it may be detected by a receiver. In order to extend the space coverage of the radar system, it is customary to scan a predetermined volume of space in a cyclic manner by changing the direction of propagation as indicated in Fig. 1-1. Identification. The system may be required to operate in an area where both friendly and unfriendly aircraft or targets possibly exist. A requirement will then arise to search the area and identify any targets as friend or foe (IFF). When it is performing the search and detection function, the radar system generates answers to a specific question: Is there — or is there not — a target of tactical interest within a given volume of space? The basic characteristics of a radar — or any detection device ■ — are such that both correct and incorrect answers to this question may be generated. There are, in fact, four possible sets of circumstances: 1. There is a target within the searched volume and its presence is detected by the radar. 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 5 2. There is a target within the searched volume, but for one reason or another its presence is not detected by the radar. 3. There is, in fact, no target within the searched volume and none is indicated by the radar. 4. There is no target within the searched volume; however, the presence of a target is indicated by the radar. In cases (1) and (3) the radar provides the proper answer to the question. In case (2) the radar /^z7j to provide the proper answer by failing to provide any information whatsoever. In case (4) the radar provides the wrong answer by providing spurious information. The manner in which the identification function is performed varies widely according to the type of radar and the tactical use to which it is put. In some cases, the detection and identification functions may be combined by a logical nonmechanical process which uses a suitable choice of a detection criterion and a prior knowledge of the probable target characteristics. For example, in the search and detection system, Fig. 1-1, one might specify that the appearance of a target indication on each of three successive scan cycles constitutes a detection — the assump- tion being that it is not likely that a spurious indication would be repeated on three successive scan cycles. One might further stipulate that any target thus detected shall be considered an enemy target if it is approaching at predetermined altitudes, speeds, or courses. The identification function is sometimes performed by a completely separate radar system designed specifically to accomplish some part of the identification problem. Many forms of identification-friend-or-foe (IFF) systems fall into this category: e.g., in Fig. 1-2 a presumably friendly Fig. 1-2 IFF System. aircraft is equipped with a passive receiver that detects the search radar signals. These signals are used to initiate transmission of a coded signal back to the search radar location. This coded signal is correlated with the search radar target return signal to establish and define the presence of a "friendly" aircraft. 6 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM The foregoing discussion of the search, detection, and identification func- tions points out an important characteristic that affects the performance of these functions. This characteristic is the implied uncertainty that the desired result will be obtained in a given case. The element of uncer- tainty requires a statistical approach to the problem of understanding and analyzing the detection and identification characteristics of a radar system. Tracking. A radar system may be designed with the capability of measuring the relative range, range rate, and bearing of any object which scatters microwave energy impinging on it. When a radar makes any or all of these measurements on a more or less continuous basis (depending upon whether it is also searching), it is said to be tracking the target. The tracking function can provide information for: 1. A continuous display or record of relative target position as a function of time 2. Calculation of relative target motion 3. Prediction oi future relative target position The range measurement is achieved by measuring the elapsed time between a transmitted signal and the reception of the portion of the transmitted energy that is scattered by the target back along the direction of transmission and multiplying it by a constant representative of the average propagation velocity. The radar energy is propagated at the speed of light {c = 328 yd/jusec). Thus the time required for the radar energy to travel from the transmitter to the target and back to the trans- mitter location is . = ?^. (M) c The range to the target may be expressed R = '^= I64t yards (1-2) where R = range to target in yards / = time in microseconds between transmission and reception c — propagation velocity in yards per microsecond. The closing velocity along a line from the radar to the target (range rate) can be measured by means of the frequency difference between the trans- mitted and received signals caused by the relative target motion. This doppler effect will be discussed in Paragraph 1-5. 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 7 Angular bearing of the target is measured by utilizing a directive beam like that shown in Fig. 1-1. With this arrangement a target return is obtained only when the beam is pointed in the direction of the target. Thus by measuring the angular position of the beam with respect to some reference axis when a target return is present, a measure of relative target bearing from the radar system is obtained. The accuracy of this measure- ment depends to a large extent on the parameters associated with the detailed design. The nature of this dependence and the means that may be used to improve the accura.':y of angular measurement will be developed in later portions of this book. Target motion relative to the tracking radar platform may be computed with measured range information and the time derivatives of the measured range and angle information. Analysis of the two-dimensional case dis- played in Fig. 1-3 illustrates the basic principle. The relative velocity of the target, Vtr can be represented by two com- ponents — one parallel to the line-of- sight, Vtrp, and the other normal to the line-of-sight, Vtru- These quan- tities may in turn be expressed Line • of • Sight to Target Vt V, (R) = R (1-3) Target Velocity Relative to Target R(f) (1-4) where R = range rate along the line- of-sight and 4> = space angular rota- tion of the line-of-sight. Range-rate information can be ob- tained by differentiation of the radar range measured. It can also be measured directly by doppler fre- quency shift as previously indicated. Commonly, the space angular rate of the line-of-sight is measured by an angular-rate gyroscope mounted on the antenna of a tracking radar. The relative target velocity information may be utilized in several ways. For example, this information coupled with a knowledge of the tactical situation can provide a means for identifying targets of tactical interest. In addition, the computation of the components of relative target velocity makes it possible to predict the future target position relative to the radar Fig. 1-3 Relative Target Motion: Two- Dimensional Case. 8 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM platform. This capability is essential to the solution of the fire-control problem. An analysis of the two-dimensional fire-control problem (Fig. 1-4) illustrates the basic principles. If an aircraft is armed with a weapon which is fired along the aircraft flight line, the weapon position relative to the interceptor // seconds after firing can be expressed Rf^ = V,tf (1-5) L=Lead Angle = Line of sight Angular Rate Fig. 1-4 Air-to-Air Fighter-Bomber Duel Fire-Control Problem in Two-Dimen- sional Coordinates Relative to the Weapon Firing Aircraft. where Rfw = future relative weapon position Vq = average velocity of weapon relative to fighter velocity // = weapon time of flight (i.e. time elapsed after weapon firing). The fire-control problem is solved when the future relative range of the weapon coincides with the future relative range of the target Rft, i.e. Rfw = Rft (at some value of //). The predicted future relative target range may be expressed in terms of its components relative to the line of sight Rft^ = R- VTRvtf = R- Rtf (1-6) Rftn = VTRjf = R4>tf (1-7) Similarly, relative weapon range may be expressed 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS Rfwp = ^(/f cos L Rfwn = y^tf sin L Equating components, we obtain the basic fire-control equations VqIj cos L = R — Rtf (time of flight equation) _R^ (lead angle equation). 9 (1-8) (1-9) (1-10) (1-11) Mapping. The microwave energy scattering characteristics of physical objects provide a wide range of characteristic returns. The differences between these returns make it possible to use a radar system to obtain a map of a given area and permit the interpretation of the results through an understanding of the characteristic returns. The mapping function is accomplished by "painting" (scanning) a designated area with a radar beam of appropriate characteristics. Two common means for performing this function are shown in Fig. 1-5. In the first method, Fig. 1-5A, the picture is "painted" by rotating the antenna beam around an axis perpendicular to the area to be mapped. The resulting picture is a circular map whose center, disregarding trans- FiG. 1-5 Radar Mapping: (a) Forward-Look System, Variant of the Plan Position System, (b) Side-Look System. lational motion, is the radar's position. The coordinates of the display are conveniently in terms of angle and range. The title "Plan Position" is applied to this type of map. A variant of this scheme would be a system that mapped only a sector of the circle — for example, a sector just for- ward of the radar aircraft (Forward-Look System). 10 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM In the second method, Fig. 1-5B, fixed antennas are mounted on each side of the aircraft. The motion of the aircraft with respect to the ground provides the scanning means. Thus the picture obtained by this radar is a continuous map of two strips on either side of the aircraft flight path. In each case the detail is very diflFerent than that obtainable from photo- graphs of the same terrain under conditions of good visibility. Never- theless, a considerable amount of potentially useful tactical information can be obtained from such pictures. The distinction between land and water areas is particularly striking, and prominent targets — large ships , airfields, and cities — can also be clearly distinguished. The basic capabilities of radar provide several attractive features in the performance of the mapping function. The range to the target is directly measurable. Smoke, haze, darkness, clouds, and rain do not pro- hibit taking useful radar pictures (depending on the radar parameters chosen). A camouflaged target that might be exceedingly difficult to distinguish by visual means is often readily unmasked by a radar picture. Finally, a radar picture does not necessarily have the same problems of perspective that tend to distort a visual picture. The change of target characteristics with frequency can be employed to provide increased contrast. The basic principle is illustrated in Fig. 1-6, which shows hypothetical backscattering curves for the sea and a target. If the mapping is performed at two frequencies,/i and/2, and if the returns at these frequencies are transformed into green and blue, respectively, on a visual display, then the target will appear green and the sea blue. This color transposition utilizes the human eye's ability to discern color differences (see Paragraph 12-7), thereby improving the contrast in cases where a relationship similar to Fig. 1-6 exists. By the use of the doppler (velocity discrimination) eff"ect, a mapping system may also be provided with the capability for distinguishing moving targets that have a compo- nent of velocity along the sight-line of the radar. This is known as woy- ing target indication (MTI). Another type of radar mapping does not involve the generation and transmission of microwave energy by the radar. Rather, it utilizes the fact that all bodies — as a conse- quence of their temperature and em- issivity characteristics — emit energy in the microwave spectrum. By using highly directional antenna and a receiver that is sensitive to these radia- TRANSMITTED FREQUENCY-f Fig. 1-6 Utilizing the Change of Target Characteristics with Frequency to En- hance Mapping. 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 11 tions, a given area may be mapped by scanning the area and correlating the signals received with the antenna position. This method — often referred to as microwave thermal mapping (MTR) — is similar in concept to the various forms of infrared mapping. The only difference is the frequency spectrum covered. The use of microwave frequencies sometimes alleviates the severe weather limitations of the much higher-frequency infrared spectrum. Counterbalancing this advantage is the inherently poorer resolution obtained at microwave frequencies and the vastly smaller amounts of thermal radiation energy at these lower frequencies. Navigation. The mapping capability can be used to perform a portion of the navigation function, particularly under conditions of poor visibility. Prominent land masses, land-water boundaries, and objects located in a relatively featureless background such as an aircraft carrier at sea are usually readily distinguishable — even on a radar picture obtained from a radar system not specially designed to perform the mapping function. By a proper choice of radar parameters, cloud formations that represent a potential flight hazard can readily be detected by a radar of appropriate design. Radar systems designed specially to perform this function have become standard equipment on many transport and military aircraft. A typical radar picture obtained from such a system is shown in Fig. 14-15. Information such as this represents a valuable navigational aid. It can permit the successful completion of many missions that might otherwise be aborted because of weather uncertainty. Radars designed for other purposes can provide this information as an auxiliary function. Another radar navigational aid is the radar beacon system (Fig. 1-7). In this system an airborne radar transmits microwave energy at a specified beacon frequency. When some of the energy is received by a beacon station tuned to this frequency, this energy is, in effect, amplified greatly and transmitted back to the interrogating aircraft. There is preset, fixed time delay 4 between the reception and the transmission in the beacon. Thus if the total time between interrogation of the beacon and the reception of the beacon reply is ti /xsec, the range to the beacon is R = ^{ti- 4) R = 164(/i - 4) yards (1-12) where c jl = \ propagation speed of light in yd/jusec ti = propagation transit time 4 = beacon delay time. 12 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM (b) Beacon Station at a Known Geographical Location Fig. 1-7 Radar Beacon System. The angular position of the beacon relative to the aircraft is measured by the airborne radar. Since the pilot knows his own heading in space and the geographical position of the beacon, the knowledge of relative range and bearing of the beacon permits him to determine his own geographical location. It is quite common for an airborne radar to have a beacon mode as an auxiliary function. Despite the apparent simplicity of the mode, the proper integration of this function into an airborne radar system is often difficult, particularly if early systems planning neglects to include the cooperative beacon itself. Variations of the beacon mode of operation are also quite common in guided missile applications. An airborne radar possesses an inherent capability for providing still another type of navigational information — true ground speed — achieved through the use of the doppler effect mentioned above in the discussion of the tracking function. This application will be discussed in detail in Chapter 6. Communications. The transmitted radar signal may also be used as a carrier for the transmission of communications intelligence. While such transmission is limited essentially to line-of-sight because of the inherent 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 13 nature of microwave propagation (see Chapter 4), it has a number of potential advantages: (1) high directivity, increasing the security of the communications link; (2) dual utilization of the same antenna and carrier power source; and (3) relative predictability of the transmission character- istics. Radiation Detection. The radiation detection or passive listening function that may be performed by a radar system has already been men- tioned in the preceding discussions of IFF, ECM, beacon, and com- munications systems. A passive radar system consists of only a receiving channel or channels designed to detect and — in some applications — to track microwave energy that is emitted or scattered by a separate source. Passive radars cannot measure range without auxiliary devices. There is a variety of means for obtaining range measurements from a passive system — e.g., triangulation using several passive tracking systems at different locations; but all these methods are complicated and inaccurate when compared with the convenience of range measurement in an active radar system. Several important functions may be performed by passive radar systems in addition to those already discussed. In the Ferret application, radar receivers tuned to cover a wide band of frequencies are used to detect enemy radiations, thereby providing intelligence data on the characteristics and capabilities of enemy radar systems. Such information is of great value in determining the tactics and countermeasures to be employed in subsequent operations. A variation of the above application is one in which the enemy radiation is used as a source upon which a guided missile homes — a system known colloquially as a "radar buster." Despite their simplicity of concept, such systems may present formidable systems engineering and design problems. The multiplicity of enemy signal sources, the intermittency of trans- mission from a scanning source, and the importance of having a stand-by mode of operation in the event that the enemy ceases to radiate for exten- sive periods of time, all contribute to the difficulties. A special case of the "radar buster" passive radar homing system is the "home-on-jam" system. This system might be used as an alternative mode of operation for an active radar system. When the active radar is jammed, the jamming source could be detected and tracked by the passive system. A passive radar system also forms a vital part of a semiactive guidance system. This application is discussed later in this chapter. Illumination. A common form of radar system is the semiactive system. The functional operation of such a system is shown in Fig. 1-8 14 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Fig. 1-8 Semiactive Guidance System. and is described elsewhere in greater detail.^ In this system, the target is illuminated by a source of microwave energy. A portion of this energy is scattered by the target and may be detected and tracked by a passive receiver located at some distance from the transmitting source. Semiactive systems find their greatest use in guided missile systems, where it is often desirable to retain the basic advantages of an active system without incurring the weight penalty and transmitting antenna size restrictions that would result from placement of the transmitter in the guided missile. It is possible to obtain a crude measurement of range in a semiactive system if the missile is illuminated by the same energy transmission as the target. The accuracy of this range measurement is greatest when the illuminator, missile, and target are in line as shown in Fig. 1-9. In this &^ r) Rf^ct, L_ U ' Fig. 1-9 Rane;e Measurement in a Semiactive system. case, the target receives energy from the interceptor-borne radar /o Msec following transmission. The illuminating energy is also received directly lA. S. Locke, Guidance (Principles of Guided Missile Design Series), D. Van Nostrand Co., Princeton, N. J., 1955. 1-4] FUNCTIONAL CHARACTERISTICS OF RADAR SYSTEMS 15 by a rearward-looking antenna on the missile t\ ^tsec after transmission. The missile, by measuring the time difference between these two signals, can obtain the range to the target; thus Rft = Ct2 Rfm = ctx. Since Rmt = R/t — Rfm then Rmt = c{t2 - /i) (1-13) where c = speed of propagation in yd/jusec = 328 yd/Msec. The relative velocity between the missile and the target can be obtained by analogous means, using the frequency difference between the direct and reflected signals. This frequency difference is caused by the doppler effect. Information Relay. From a systems standpoint, it is often desirable to display and utilize radar information at a different location from the point of collection of the information. Typical of such an application is the air surveillance system shown in Fig. 2-15. Data are collected by a number of airborne early warning (search radar) systems located in such a manner as to provide the required coverage. It is desirable to assemble, correlate, and assess the data at a central location (Fleet Center) in order to provide a complete picture of the tactical situation. From this analysis, instructions and data can be relayed to the operating elements. This type of operation is typical of airborne, ground, or ship-based combat information centers (CIC). Jamming. Radars may also be used to transmit microwave energy with the object of confusing or obscuring the information that other radars are attempting to gather. Jamming is of two fundamental types: (1) "brute force" and (2) deceptive. Brute force jamming attempts to obscure as completely as possible the information contained in other radar signals by overpowering these signals. Deceptive jamyning, on the other hand, endeavors to create mutations in the information contained in other radar signals to render them less useful tactically. Both types of jamming are aided by their one-way transmission char- acteristic as contrasted with the two-way transmission characteristic of active radar. This feature allows a jammer to operate successfully with a few watts of transmitted power against a radar transmitting hundreds of thousands of watts of peak power. 16 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Despite this formidable advantage, the design of a jamming radar system can be one of the most perplexing of all radar systems problems — from the points of view of both systems engineering and hardware design. This arises from the vast multitude of possibilities with which a jamming system must cope. Scientific Research. Airborne radars are frequently utilized to gather basic scientific data such as atmospheric transmission characteristics, target reflectivity, and ground reflectivity and emission characteristics. The coming space age opens up several interesting possibilities. It is very probable that the first glimpse of the surface characteristics of the planet Venus will be provided by a radar picture taken from an inter- planetary vehicle. The use of radar techniques would permit the penetra- tion of the optically opaque atmosphere which completely obscures this enigmatic planet, as well as provide a quantitative evaluation of its atmos- pheric components. This could be accomplished by measurement of the attenuation of the radar energy as a function of frequency. As will be discussed in Chapter 4, water vapor, oxygen, and carbon dioxide exhibit a marked efl^ect upon radar energy transmission characteristics at certain frequencies. Passive radar techniques (microwave thermal mapping) could be employed to ascertain the surface temperature distributions and the heat balance. This type of scientific data would be invaluable for the determination and prediction of weather conditions. 1-5 THE MODULATION OF RADAR SIGNALS A radar system may perform a number of functions (Paragraph 1-4) that involve the collection or transmission of intelligence for some defined tactical objective. The intelligence is carried by modulations of the radar microwave signal. The means used to create these modulations and the means employed to extract information from them (demodulation) form a convenient and mathematically useful way to describe and classify radar systems. As will be seen in later portions of this book, the key to the understanding and proper design of a radar system is a knowledge of the modulation proc- esses that can take place. The various processes of modulation and de- modulation are conveniently explained by the use of simple generic repre- sentations of the three basic elements of a radar system: (1) the transmitter, (2) the target, and (3) the receiving system. A simple transmitting system is shown in Fig. 1-10. It consists of a means for generating alternating current power, a means for carrying this power to an antenna, and an antenna that radiates some portion of this power into the surrounding space. Amplitude Control Frequency Control Modulator £=Acos(wf+</));: Phase Control AC Power Source K---' Lobe ^ Fig. 1-10 Simple Transmitting System. 17 The generating device may be visualized as producing a sine wave out- put of constant amplitude and frequency. E{t) = ^cos (a;o/+ <i>). (1-14) If this power is in turn applied to an antenna which radiates a portion equally in all directions (omnidirectional), we have the simplest sort of radar transmitter. We may proceed to refine the system by modulating the radiation in different ways. Space Modulation. The radiated energy may be space-modulated by an antenna possessing directivity. Such a characteristic is shown in Fig. 1-10; the radiated energy is concentrated into a lobe by means of a parabolic reflector. Three other types of modulation — amplitude ^ frequency , and phase — may be introduced by suitable operations upon the power generator. Amplitude Modulation. If the output of the transmitter is ampli- tude-modulated at an angular frequency coi with fractional modulation m^ it then has the form E{t) = Aq {\ -\- m cos coi/) cos {ui4 + (^) = A^ cos (wo/ + <^) H 2~^ cos [(wo , mA u ^ 2 OJl) / + 0] cos [(coo + wi) / + 0]. (1-15) Note that this type of modulation produces sidebands in the generated voltage; i.e., the generated voltage has frequency components which differ from the carrier angular frequency wo by plus-or-minus the modulat- ing angular frequency wi. The transmitted spectrum for the case of 100 per cent modulation {in = 1) has the form shown in Fig. 1-11. The voltage amplitude of each sideband in this case is one-half that of the carrier, and the power in each sideband is one-quarter of the carrier power. Obviously 18 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM FREQUENCY (ANGULAR) Fig. 1-11 Generated Frequency Spec- trum for 100 Per Cent Sinusoidal Am- plitude Modulation of Carrier. as m decreases the power in the side- bands decreases and becomes a lesser fraction of the carrier power. A common type of amplitude mod- ulation arises from a modulating signal of the form shown in Fig. 1-12. Essentially, this signal turns the transmitter on and off on a periodic basis. Accordingly the output is a train of pulses of the carrier fre- quency. Since this modulating signal is periodic, it may be expressed as a Fourier series with a fundamen- tal frequency equal to the pulse TIME Fig. 1-12 Pulse Modulation. repetition frequency (PRF — 1 /T^), where Tr is the time between successive pulses.^ Thus, this type of modulation gives rise to a large number of sidebands separated from the carrier frequency by multiples of the pulse repetition frequency. The amplitude spectrum of such a modulated wave is shown in Fig. 1-13. As can be seen, the pulse width r determines the amplitude of each of the sidebands. Radar systems employing the type of amplitude modulation just de- scribed are known as pu/se-type radars. Pulse radars, however, are not limited to this type of modulation, as will be described in later paragraphs. Frequency Modulation. Another major type of modulation is frequency modulation. In this case, the argument of the cosine function in Equation 1-14 is varied in such a manner as to cause the instantaneous frequency to be altered in accordance with the modulating signal. When ^Actually, the pulse amplitude modulated AF wave can be represented by a Fourier series with a fundamental frequency equal to the pulse repetition frequency only when the carrier frequency oin is an integral multiple of the PRF. 1-5] THE MODULATION OF RADAR SIGNALS li ^T o o o 3 33 FREQUENCY *■ ■ III Fig. 1-13 Amplitude Spectrum of a Pulse Train. 19 the latter is a cosine wave of angular frequency coi and the peak excursion of the modulated transmitting angular frequency is Ao), the transmitter output is £(/) .-^ cos {(joot -\ sin coi/ -j- 0) (1-16) whose envelope has a constant value. A typical frequency-modulated wave is shown in Fig. 1-14. ■Ao)- FREQUENCY ^ Fig. 1-14 Typical FM Spectrum for High-Modulation Index (Aoj/coi > 10). A key parameter in an FM system is the ratio — = modulation index. (1-17) If this index is relatively high — say 10 or greater — the output spectrum has the form shown in Fig. 1-14. As can be seen, a single modulating frequency gives rise to a large number of sidebands separated from the carrier frequency by harmonics of the modulating frequency coi. The sidebands of primary importance lie within a bandwidth Aco centered about the carrier frequency coo- 20 FXEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Various types of transmitter frequency modulations are commonly employed. Pulse-width modulation and pulse-time modulation are used to transmit information on a train of pulses. In Chapter 6 it will be seen how range can be obtained from a continuous-wave (CW) radar by fre- quency modulation of the transmitted frequency. Frequency modulation of the transmitted signal often occurs inadvert- ently owing to the characteristics of the transmitter. The magnitude of this effect must be carefully controlled by the designer. Phase Modulation. Phase modulation is similar to frequency modulation in that the instantaneous phase angle is varied from some mean value. With phase modulation by a single cosinusoid of frequency coi and phase deviation <^, the transmitter output is E{t) = A cos (coo/ + A0 cos CO i/). (1-18) The difference in the arguments of the cosine functions of Equations 1-16 and 1-18, while not important for audio systems, is important elsewhere where the waveshape must be controlled. Subcarriers. The foregoing discussion has shown that it is possible to modulate the transmitted radar signal in four basic ways • — space, amplitude, frequency, and phase. At this juncture, it is appropriate to consider just why one would want to modulate the transmitted signal. The purpose of these modulations is to create information subcarriers, i.e., an angle information subcarrier, a range information subcarrier, etc. The target information is contained in modulations of these subcarriers (and also the carrier frequency) that are created by the target itself and is derived upon return of the signal to the receiver by correlation with the transmitted subcarriers. Target Modulations. In order to understand the basic processes involved, it is now appropriate to investigate the modulations of the main carrier and its associated subcarriers that are created by the target. First of all, the amplitudes of the transmitted radar signals that are reflected back to the transmitting location are vastly reduced — perhaps by a factor of 10^" on a power basis. Moreover, the reflecting characteristics of the target are, in general, a function of frequency. Thus, the amplitudes of each carrier frequency in the reflected wave may not be modulated by equal amounts. Additional amplitude modulations are created by characteristic time variations of the target reflective characteristics. Chapter 4 will cover this phenomenon in detail. It will suffice for the moment to state that this effect introduces additional modulation which broadens each of the returned 1-5] THE MODULATION OF RADAR SIGNALS 21 sidebands to an extent depending upon the rate of target reflection char- acteristic fluctuations. The target reflection entails phase changes with reference to the trans- mitted signal incident to the finite time required for propagation of micro- wave energy to and from the target. These phase changes occur in all the frequencies of the transmitted wave. The phase changes are linear with frequency and have a proportionality constant which depends upon the distance to the target. The phase modulation that occurs in the portion of the transmitted signal that is reflected back from the target provides the basic means for measuring range to the target. Pulse radars, for example, measure the phase (or time) difference between transmitted and received pulse trains. Phase modulations of a somewhat different sort may result from the motion of the target in conjunction with the space-modulation character- istic of the radar. As an example of this process, consider a radar which scans a directional beam through an angle of 360° once each second. If there is a stationary target at an angle of Qt with respect to the reference axis, a return from the target will be obtained as the radar beam sweeps past this point. The amplitude of the return signal will have the general shape of the radar beam resulting in a return signal having the envelope shown in Fig. 1-15. Thus, the scanning process gives rise to an angle in- A. A [- — 1 sec — H 27r 47r ANGLE (rad) ^Lll sec-^ A , A 2 TIME (sec) , J\ Stationary 67r Target I A Moving 3 Target Fig. 1-15 Effect of Target Motion. formation subcarrier which has a fundamental frequency of 1 cps. The angle information is carried on the phase angle of this subcarrier funda- mental. Now let us assume that the target flies in a circle around the radar station in the same direction the beam is revolving, at a speed of 1 revolution 22 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM every 10 seconds (i.e., one-tenth of the scanning velocity). The return signals now have the form shown in the lower diagram. The effect of the target motion has been to shift the fundamental frequency of the angle information subcarrier by 10 per cent to 0.9 cps for a target moving in the same direction as the scan. This process might also be viewed as in- troducing a time-varying phase shift in the 1-cps subcarrier (phase modula- tion). How the target modulates the information subcarriers is one of the most important problems of radar design. The choice of frequency and bandwidth for the subcarrier frequency information channels is largely governed by these characteristics. One other target modulation — the doppler frequency shift mentioned in preceding paragraphs — is of fundamental importance. Motion of the target along the direction of propagation (see Fig. 1-16) ^=> OAAAAAAAAA. •« Fig. 1-16 The Doppler F.ffect. causes each frequency component of the transmitted wave that strikes the target to be shifted by an amount /d = {VtIc)/ (1-19) where Vt = the velocity of the target c = the velocity of light / = the radio frequency. When this signal is reflected or reradiated back to the radar, the total frequency shift of each component is fn = {2FtIc)/. (1-20) The frequency modulation caused by target motion is important; an entire family of radars known as doppler radars has been developed to exploit this characteristic. However, whether use is made of this char- acteristic or not, the doppler shift occurs in all signals reflected from objects that possess relative radial motion. Thus, it can be seen that the target generates a large number of am- plitude, phase, and frequency modulations of the transmitted signal. 1-51 THE MODULATION OF RADAR SIGNALS 23 These modulations create information sidebands about the carrier and subcarrier frequencies. The designer's problem is to determine how this information may be extracted from the target return signal. Extraction of Target Intelligence from Radar Signals. One thing is common to all the many techniques for extracting target information from a radar return signal. This is the concept of taking a product between the target return signal and another quantity which serves as the reference for the particular piece of information being extracted from the target return. Thus, the generic building block for a radar receiving system is a product-taking device, as shown in Fig. 1-17. Reference Incoming Signal Product Signal Fig. 1-17 Generic (Product) Building Block for a Radar Receiver. Conceptually, the simplest product-taking device is a network — or filter — composed of linear impedances which can be characterized by a transfer function F{jui). Each frequency component of the incoming signal is multiplied by the vector transfer function of the network corresponding to the frequency (see Fig. 1-18). The output product is a signal containing Input F(;co) Output [Output] =[f(; CO)] X [input] Fi/co) - Fl(j} Fig. 1-18 Impedance Products. the same frequencies as the input; however, the amplitude and phase of each frequency component may be changed with respect to the input. In this type of product device, the references are the characteristics built into the filter. The second type of product-taking device is the nonlinear impedance. A simple example of such a device is shown in Fig. 1-19. The operation of the device is such that positive inputs are faithfully reproduced at the output while negative inputs are completely suppressed. Thus, for an 24 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Input ■5 B- / Output V-/ \^ Input Fig. 1-19 Product Obtained from a Nonlinear Impedance. input sine-wave, the output consists of only the positive half-cycles as shown. It is interesting to observe that this process may be put into the form of the generic device of Fig. 1-18 merely by considering the output to be the product of the input and a reference square wave of the same frequency and phase as the input. Such a representation is shown in Fig. 1-20. Thus, the input may be defined as :rLrLn TT 2ir 3ir Att Stt Fig. 1-20 Nonlinear Impedance as a Product-Taking Device. Ei = A sin CO/. The reference signal may be expressed by a Fourier series sin Sco/- , sin 5a)/ Er H (sin co/ + + -) and the product has the Fourier series form zr w r^ A , A . lA V Ei X /t,. = - + ;r- sin a)/ ) Inixit An' (1-21) (1-22) (1-23) The consequences of this product process are quite evident from Equation 1-23. Although the input contains only one frequency, the output has a d-c component, an input-frequency component, and components at all the even harmonic frequencies of the input. Now, if the amplitude of the incoming wave A, instead of being constant as implied, were amplitude-modulated at a frequency aj;„, such that 1-5] THE MODULATION OF RADAR SIGNALS 25 A = A^{\ ■\- 771 COS co„/) (1-24) where aj„ = modulating frequency (aj^ ^ w) and m — modulation ratio {m < 1), then, each of the terms of the product (Ei X Er) would contain modulation sidebands. For example, the d-c term would now become Ao . mAo n nc\ 1 cos co,„/ (1-25) T IT and the fundamental frequency term would become '-y sin w/ H ^ [sin(co -(- w™)/ — sin(w — w™)/] (1-26) and so on for the higher harmonics. If this (Ei X Er) product were then passed through a filter, F(ju), which eliminated the d-c term and the fundamental frequency u and all its harmonics, the final output would be (Ei X Er) X F(jc^) = '-^ cos co„,/. (1-27) TT Thus, we observe, the frequency and the magnitude of the modulation intelligence are recovered from the incoming wave by the product-taking procedures. The procedures just described are often referred to as de- modulation or detection. A third type of product-taking device closely resembles the basic model of Fig. 1-17. The incoming signal is multipled by a reference signal gener- ated within the radar receiver. One form of this process is known as 7nixing or heterodyning. In this process, a cross-product is taken between the incoming signal and a locally generated signal. This process converts the microwave signal to a much lower frequency, which may be filtered and amplified by relatively simple electronic techniques. Two general forms of microwave mixing are commonly used, noncoherent mixing and coherent mixing. In coherent mixing, the phase of the locally generated signal is made to have a known relationship to the phase of the transmitted signal. This type of mixing makes it possible to detect the phase and frequency modulations introduced by target motion. The extraction of angle and range information from the received signals is almost always accomplished by a cross-product of the received intel- ligence and an internally generated reference signal. The detailed analysis of the various means for extracting target intel- ligence from radar signals — and the problems that arise in these processes — forms a major portion of this book. Chapter 3 and Chapters 5 through 9 are all concerned with various phases of these problems. 26 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM 1-6 OPERATING CARRIER FREQUENCY The operating frequencies for radar systems cover an extremely wide band, ranging from below 100 to above 10,000 Mc. This range is divided up into bands designated P, L, S, X, K, Q, V, and W as shown in Fig. 1-21. 100 80 60 50 40 30 20 I I I I I I I I \ L WAVELENGTH - Cm 10 8 6 5 4 3 I I I I I I I L 1 0.8 0.6 Mill I 0.4 0.3 J I 100 0.5 0.6 3 5 6 10 FREQUENCY - KMc 20 30 50 60 BAND DESIGNATION TV m i I in USE ALLOCATION □ Allocated to Armed Forces and Other Departments of the U.S. Government Allocated to Radio Navigation, Radio Location, and Civilian Radar. Sometimes Used By Military Equipments. Allocated to Television, Common Carriers, Domestic Public, Industrial Safety, and International Control. Military Equipments Precluded from These Bands Except at Times of National Emergency. Fig. 1-21 Operating Radar Frequency. The specific frequencies available for airborne radar systems are, in general, regulated by the Federal Communications Commission during times of comparative peace. The operating carrier frequency has a profound effect on the following characteristics of a radar system: 1. Size, weight, and power-handling capabilities of the RF com- ponents (see Chapter 11) 2. Propagation of RF energy (see Chapter 4) 3. Scattering of RF energy (see Chapter 4) 4. Doppler frequency shift from a target moving relative to the radar direction of propagation. These characteristics vary quite radically over the range of radar operating frequencies — enough, in fact, that it becomes convenient to classify a radar according to its operating carrier frequency. This method of classification is commonly used by microwave component designers ]-7] THE AIRBORNE RADAR DESIGN PROBLEM 27 because the design problems and the techniques used to solve them are strongly dependent upon the operating frequency. This method of classification is also important to the system designer because the operating frequency determines certain of the radar's reactions to its physical and tactical environment. For example, an atmosphere heavily laden with moisture is more or less opaque in some bands to the highest radar operating frequencies, whereas the transmission of the lower frequencies is little affected. In airborne applications, the smaller size of the higher-frequency radar components has favored the use of S, X, and K bands despite their limita- tions with respect to weather and moving target indication, as discussed in Chapters 4 and 6. 1-7 THE AIRBORNE RADAR DESIGN PROBLEM Preceding sections discussed general radar characteristics. The following problem is of paramount importance: How does the radar designer select and employ the right combination of these characteristics to achieve an acceptable performance level in a given weapons system application? The design problem may be divided into two basic parts, problem definition and problem solution. Problem Definition. The airborne radar design problem is defined by the weapons system application. In such applications, an airborne radar combines with other system elements — human operators and the airborne vehicle and its associated propulsion, navigation, armament, flight control, support, and data processing systems ■ — to form a closely integrated weapons system designed to perform a specific mission. To achieve a given performance level, the weapons system requires certain performance characteristics from the airborne radar. The radar designer's first task is to examine the requirements and characteristics of the complete weapons system. From this analysis, the nature of the airborne radar's contributions to overall weapons system performance (mission accomplishment) may be obtained. Typical examples of the parametric relationships developed in such a study are shown in Fig. l-22a. From such curves, the radar requirements for a desired level of mission accomplishment may be obtained. In addition, the sensitivity of mission accomplishment to changes in radar performance is displayed, thereby providing the designer knowledge of the relative importance of each performance characteristic. The derivation of such relationships must be relatively uninhibited by known limitations in the radar state of the art. That is to say, the range of values considered for each of the radar's performance capabilities need 28 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Detection Range Fig. l-22a Tracking Error Other Resolution Element Size Typical Relations Between Mission Accomplishment and Radar Performance in the Operating Environment. not bear any relation to a presently realizable radar system. The purpose of this analysis is to define the radar problem solely as it is dictated by the weapons system problem. Whether the radar problem thus defined is technically reasonable for a given era is determined in the next major step of the design process. Problem Solution. The systems analysis defined the required radar performance. Now, the designer must attempt to solve the defined problem by (1) hypothesizing a radar system of given general characteristics and (2) examining the interrelationships between radar parameters and radar performance. Typical examples of the interrelationships developed by such a study are shown in Fig. l-22b for the case of a single radar param- l_. Fig. l-22b Typical Interrelations of Radar Frequency and Radar Performance Parameters. eter — operating frequency. Similar parametric relations are derived for each radar parameter that exercises important influences on radar performance. 1-7] THE AIRBORNE RADAR DESIGN PROBLEM 29 The information thus derived is examined and correlated to find — if possible — the combinations of radar parameters which fulfill the pre- viously derived radar performance requirements. Then and only then can the designer proceed in an intelligent manner to design the radar hardware for fabrication, evaluation, and service use. Often the proper combinations cannot be found. State-of-the-art limitations, laws of nature, and other factors may conspire to prevent a successful problem solution using the assumed radar concept. In these cases, the parametric information generated for the problem definition and the problem solution provide readily avail- able means for ascertaining the most promising course of action — whether it be a change in radar concept, the initiation of a new component develop- ment, or a change in the overall weapons system concept. In extreme cases, a failure to find a radar solution may justify abandonment of a weapons system concept; in other cases an early display of seemingly irreconcilable deficiency may provide the spur for the generation of a bold new radar concept that performs as required. Summary and Discussion. Airborne radar performance usually exercises a decisive influence on overall weapons system performance. The approach to the design problem must therefore be an overall systems approach, even though the radar is only a weapons system component. The two basic steps in the design process are problem definition and problem solution as illustrated in Fig. 1-23. The first step derives the radar Weapons System Problem Radar System Requirements Radar Problem Definition Radar State of the Art and Weapons System Schedule Limitations Radar System Design Radar ^ Problem Solution Fig. 1-23 The Airborne Radar System Design Problem Approach. requirements imposed by the complete weapons system and neglects possible limitations of radar techniques. The second step is concerned with 30 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM fulfillment of the requirements considering limitations of radar state of the art, schedules, and other factors germane to the problem of achieving a useful operational capability for the overall weapons system. In actual practice, the design process is long enough and complicated enough to justify subdivision of the two basic phases described. The next paragraph will discuss the complete design cycle of a typical weapons system development with particular emphasis on the role of the radar designer as a vital part of the designers' team. 1-8 THE SYSTEMS APPROACH TO AIRBORNE RADAR DESIGN A representation of a typical weapons system design cycle is shown in Fig. 1-24. Each step represents a subdivision of the problem definition GENERATED SYSTEM CAPABILITIES Operational Requirements £3: ^-^.^- iviission Accomplisiiment System Concept Tectinical Design Objectives Technical Analysis ^T Detail Technical Requirements Mechanization Requirements Evaluation System Mechanization Requirements Equipment Development Evaluation Model Equipment Evaluation Equipment Prototype Equipment Service Equipment Service Equipment Fig. 1-24 Weapons System Design Cycle. or a problem-solving step just discussed. On the left side of the figure are displayed the sequential steps of an orderly development process from the THE SYSTEMS APPROACH TO AIRBORNE RADAR DESIGN 31 initiation of an operational requirement to the fabrication of service equipment designed to fulfill the requirement. On the right of the figure are the definitive outputs or accomplishments resulting from this develop- ment. In the middle of the figure are the evaluation processes which meas- ure the level of accomplishment attained in the problem-solving phases of the development. The outputs of these evaluations may also be used to modify succeeding phases of the development process. The diagram also indicates feedbacks from the various development phases into preceding phases. These reflect the fact that as more is learned about the system, prior concepts must be modified and expanded to ensure that the system development objectives are current and realistic. Viewed in its entirety, the indicated procedure provides a basis for playing current accomplishment against the requirement to obtain a continuous rating factor representing the generated system capability. The step-by-step processes for executing the system development plan may be summarized as follows. Operational Requirement. The overall system objective is set forth in an operational requirement . This requirement usually outlines the military task(s) which the weapons system must perform. It will also specify — or at least indicate — the level of fnission accomplishment which the system must achieve to accomplish the desired military objective. The mission accomplishment requirement often has the general form dis- played in Fig. 1-25. The weapons system must be operative in a given time Desired Level 1 \ 1 V-Obsolescen 1 Minimum \ \ J Acceptable \ \ Level \ \ \ \ a TIME (YEARS) b Fig. 1-25 Operational Requirement. period a-l?. The desired level of mission accomplishment represents the best estimate of what the military planners believe is necessary to achieve unquestioned military superiority in a given area of interest. This goal may be variable over the expected operational use cycle (as shown) by reason of anticipated introduction of new techniques by the enemy. The minimum acceptable level of mission accomplishment represents a capability which the military planners believe is still useful enough to 32 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM justify the weapons system cost. Thus, although the system design will endeavor to meet the desired goal, some degradation may be acceptable if such degradation can be shown to be unavoidable. Unexpected developments in technology or in the long-range strategic situation can cause radical changes in the operational requirement during the design cycle. For this reason, the radar designer must constantly monitor the operational requirement to ensure quick reaction to such changes. System Concept. The operational requirement defines a military problem. The next step is to define a system concept which provides bases to presume a weapons system potential capability compatible with the operational requirement. This step usually is implemented in the following way. Various possible systems are postulated. Technical military agencies examine these in the light of available or projected technical capabilities to determine which provides the best foundation for a subsequent develop- ment. Weapons system contractors may assist this study phase by pro- viding new ideas, state-of-the-art evaluations, etc.; however, the basic responsibility for decision and action invariably rests with the military. Once a decision is made on the type of system desired, the basic features of the selected system are set forth in the form of technical design objectives These comprise the performance specification of the overall weapons system and 1. The system effectiveness goal related to the operational require- ment 2. The basic system philosophy, i.e., mode of operation 3. The system environment as defined by tactics, logistics, climate, etc. 4. The characteristics of major system elements 5. The system design, development, and evaluation program 6. Fundamental state-of-the-art limitation in various portions of the system Unless he has already participated in the definition of the system concept, the radar designer's T^^rj/ task is to become familiar with these conceptual characteristics of the overall weapons system. They define the elements of his problem which are relatively fixed and with which his design must be compatible. Technical Analysis. The systems problem and its boundary con- ditions having been defined and understood, the radar designer now is ready for the next step — the construction of a weapons system model that will define the radar problem. This model is used to determine the quan- 1-8] THE SYSTEMS APPROACH TO AIRBORNE RADAR DESIGN 33 titative interrelationships of the radar and other system elements. From manipulation of these interrelationships the designer must obtain the true technical requirements of the radar necessary to attain a proper balance between mission accomplishment and the operational requirement (see Fig. 1-24). The detailed technical requirements include specification of 1. Functional capabilities 2. Radar range and angle coverage requirements 3. Information handling, transfer, and display requirements 4. Radar information accuracy requirements 5. Radar environmental requirements 6. Radar system reliability requirements 7. Radar maintenance, stowage, and handling requirements In this stage of the design, the emphasis is placed on the job the radar must do and the environment in which it must operate. As previously mentioned, this analysis should not be inhibited by the introduction of state-of-the-art radar limitations. The problem in this phase is to ascertain what requirements the radar must satisfy to allow the defined system con- cept to demonstrate an adequate system capability. This type of analysis is not popular with radar designers. Often it results in establishing technical requirements beyond the scope of known radar technology. The radar designer is forced to admit a set of require- ments he does not believe he can meet. However, the wise weapons systems contractor will demand that such an analysis be performed and demonstrated by the radar designer for two reasons: 1. It will ensure that the radar designer really understands the problem before he tries to solve it. 2. The earlier a potential source of system degradation is known, the easier it is to correct by invention or by modification of the development program. Because this step is so important to the radar designer, Chapter 2 is devoted to a detailed discussion and example of the processes involved in the derivation of technical requirements for a radar system. Mechanization Requirements. The next step in the design process is to synthesize a realizable radar system to meet the derived technical requirements within the limitations imposed by development time, state of the art, and delivery considerations. Various radar systems are designed on paper and analyzed in detail to demonstrate their performance relative 34 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM to the previously established requirements. Often, it is found that the requirements are not compatible with realizable radar systems. This discrepancy may be corrected in some cases by using the interrelation- ships of the radar and the overall system derived in the previous phase to find a different balance of radar requirements that will still permit mission accomplishment. In other cases, a known degradation in performance — relative to the initial weapons system goal — may have to be accepted. A somewhat happier situation arises when it is found that a realizable radar system can provide greater capabilities than those required by the initial weapons system concept and operational requirement. In this case, the initial system concept could be enlarged and improved or, conversely, the development objectives could be revised with a resulting economy of design. The indicated processes of evaluation and feedback are shown in Fig. 1-24. The flexibility achieved by the feedback process is the real strength of the systems approach. There are few weapons systems that cannot benefit from modification of the originally established concepts and tech- nical requirements. Circumstances change — often in a highly unpre- dictable manner — over the five-to-ten-year development period of a weapons system. That is why the radar designer must continue to treat this problem on an overall weapons system basis throughout the life of the project. Equipment Development, Evaluation, and Use. Similar com- ments concerning the value of the systems approach apply to the vitally important task of building, evaluating, and using the equipment in accord- ance with the requirements derived in the first three phases. The problems in the latter phases can be formidable. For some perverse reason the actual equipment in certain critical areas may not perform in the manner pre- dicted, particularly with respect to reliability. A vital part of the system approach is the process of rapid isolation and correction of system deficien- cies in these phases and the anticipation of potentially critical areas. Since it is often difficult to distinguish between a genuine system deficiency and a temporary bottleneck, the judgment and experience of radar systems engineers who have also participated in the requirements derivation phase is most important. There are so many problems in these phases that it is easy to concentrate effort on the wrong ones. One problem — reliability — dominates these last phases. This is the most vexing, most difficult, and most important problem in the design of a radar system. Radar systems have never been simple; in the future, their complexity may be expected to increase. The most common failure of the systems approach to the radar design problem has been the tendency to maximize system capability by specifying unnecessarily exotic radar requirements which lead to reliability problems in the mechanization phases. 1-9] SYSTEMS ENVIRONMENTS 35 In the technical requirements analysis phase, all possible ingenuity should be employed to minimize radar complexity for a given weapons system capa- bility. This is one of the most important reasons why the radar system designer must analyze the radar problem as a weapons system problem. Important performance benefits can be achieved by making the proper reliability-complexity trade-off early in the game. Summary. The systems approach to radar design implies that the radar is considered in its relation to the construction and objectives of the entire weapons system during all phases of its conception, design, construc- tion, and use. The radar systems designer must participate in all phases of the development; he must demonstrate a good understanding of the overall system and the characteristics of its other components prior to laying out a radar design. To increase the reader's understanding of the basic features of the systems approach to airborne radar design, several of the points presented will be amplified in succeeding paragraphs. These include a more precise definition of the system environment and its eflFects upon the problem, and a brief discussion of the concepts and processes involved in the construction of a weapons system model. 1-9 SYSTEMS ENVIRONMENTS In this book, the "expected tactical conditions of operation" include all of the following environments. 1. Tactical Environment — The salient elements of this environment are the speed, altitude, operating characteristics, and mission profile of the airborne portion of the weapons system; the composition, operating char- acteristics, and relative position of the ground-based portion of the weapons system; and the characteristics (speed, course, altitude, number, physical size) of the target complex. 2. Physical Enivronment — The salient elements involved are tem- perature, pressure, humidity, precipitation, fog, salt spray, wind, clouds, sand, and dust. In systems requiring a human operator, the physical environment will include factors affecting his ability to operate the system. Among these are habitability, ease of operation, length of attention span required, and the physical readiness and mental acumen required. 3. Airframe Enviromnent — The salient elements involved are volume and configuration of allotted space within the airframe, weight limitations, vibration, and shock. 4. Electronic Environment — This includes all the external sources of electromagnetic radiations and electromagnetic radiation distortions and anomalies. Examples are ground, sea, and cloud clutter; radiation from 36 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Other systems; electronic countermeasures; propagation anomalies, at- mospheric attenuation; and target radar reflective characteristics. 5. Logistics Environment — This includes all salient considerations of the parts of the weapons system that affect reliability, maintenance, han- dling, stowage, supply, replacement, and transport. 6. Weapons System Integration Environment — This is the environment formed by other systems with which the system under consideration must be compatible. An example would be the environment formed by a ground- to-air missile weapons system operating in the vicinity of a proposed interceptor weapons system. These environments should be looked upon as boundary conditions imposed upon the systems design problem. The concept is shown diagram- matically in Fig. 1-26. Each element of the system design must be com- System Environment Logistics Environment Weapons System Integration Environment Fig. 1-26 The System Environment. patible with the requirements and limitations imposed by all of these environments. 1-10 WEAPONS SYSTEM MODELS A weapons system model is a simplified representation of the actual system which can be used to predict the changes in system performance when one or more of the components which make up the system are changed. For example, a common problem in radar system design is to determine the effect of radar detection range on the performance of the overall system. Such a problem would be solved — as will be shown in Chapter 2 — by constructing a model containing radar detection range as a variable parameter. The radar detection range would be related by appropriate means to the pertinent characteristics of the overlapping environments which make up the system complex. Within the model. 1-10] WEAPONS SYSTEM MODELS Radar Range 37 Target Inputs TTT Fixed Elements Mission Accomplishment RADAR DETECTION RANGE Fig. 1-27 Diagrammatic Representation of a Model Used to Determine Sensitivity of Mission Accomplishment to Variations in Radar Detection Range. the elements of the system model interact to provide a generated mission accomplishment as shown in Fig. 1-27. From such a model, the effect of radar range upon system effectiveness could be obtained for various inputs (targets), yielding the characteristic type curves shown. It is far more convenient to conduct an experiment with a model of a system than with the system itself. This is particularly true with a military weapons system where the testing of actual hardware is enormously expensive. Moreover, system hardware usually is not available until long after the original concepts. Theoretical models are required to predict expected system performance. There are three classes of models which can be used in systems analysis work: iconic, analogue, and symbolic. Briefly, we may define the character- istics of these model types in the following ways: 1. An iconic model represents certain characteristics of a system by visual or pictorial means. 2. An analogue model replaces certain characteristics of the system it represents by analogous characteristics. 3. A symbolic model represents certain characteristics of a system by mathematical or logical expressions. 38 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM Iconic Models. An iconic model is the most literal. It "looks like" the system it represents. Iconic models can quickly portray the role that each subsystem plays in the operation of the overall system. It is therefore particularly well adapted to illustrating the qualitative aspects of system performance, such as information flow and functional characteristics of various portions of the system. The iconic model is not well adapted to the representation of dynamic characteristics of the system because it does not reveal the quantitative relationships between various elements of the system. For the same reason, it is not very useful for studying the efi^ects of changes in the system. Because of its pictorial value, most system analyses usually begin with the construction of an iconic model (block diagram) in order to establish the characteristics of the system and to provide the investigator with a realistic frame of reference for subsequent studies. This process will be illustrated by examples later in this book. Analogue Models. Analogue models are made by transforming certain properties of a system into analogous properties, the object being to transform a complicated phenomenon into a similar form that is more easily analyzed and manipulated to reveal at any early time the initial elements of system performance. For example, fluid flow through pipes can be replaced by the flow of electrical current through wires. A slightly more abstract example would derive from the problem of calculating the probability of a mid-air collision in a situation where only the laws of chance were operative — i.e., where no special equipment or techniques were used to prevent collisions. The imaginative investigator might per- ceive that this problem bears a striking similarity to the problem of calculat- ing the mean free path of a gas molecule. Having established the validity of this insight, we would then be free to make appropriate transformations between the two problems and apply the kinetic theory of gases to his problem. Unlike the iconic model, the analogue model is very effective in repre- senting dynamic situations. In addition, it is usually a relatively simple matter to investigate the efi^ects of changes in the system with an analogue model. For these reasons, analogue models form very powerful tools for the solution of complex system problems — particularly problems involving many nonlinearities. The great utility of analogue models is evidenced by the large-scale analogue computer installations that form a part of almost every major weapons system engineering organization. Symbolic Models. The symbolic model represents the components of a system and their interrelationships by mathematical or logical symbols. 1-11] STATISTICAL CHARACTER OF MODELS 39 This type of model is the most abstract. When such a model is formulated and used without incurring prohibitive mathematical complexity, it is the most useful model for obtaining quantitative answers to systems problems. Many problems may be solved by either analogue or symbolic models. Where a choice exists, it is preferable to employ the symbolic model, for it allows one to examine the effect of changes by a few steps of mathematical deduction. This process was implied in the example of the mid-air collision analogy, just cited. Here the problem was transformed into an equivalent gas dynamics analogue. However, for such a problem we should not con- struct a complex instrument and make measurements — it is far simpler to use the symbolic models already established for the kinetic behavior of gases. Further, we would gain greater insight into the basic nature of the problem in this way than would be obtained by empirical methods. The utility of symbolic models is particularly evident for problems in- volving probability concepts. Often, answers may be obtained in closed form for problems that would otherwise require many repeated tests of an analogue model. The primary disadvantage of symbolic models arises from limitations in available mathematical and computational techniques for obtaining answers from the model. State-of-the-art improvements in applied mathematics and large digital computers are relieving this problem. Despite these advances, however, there will always be a great premium on the ability to construct symbolic models that strike at the heart of a problem and eliminate nonessentials that merely increase complexity. 1-11 THE BASIC STATISTICAL CHARACTER OF WEAPONS SYSTEM MODELS The model approach consists of abstracting from a complex system certain persistent and discernible relations and using these relations to construct a system model. Frequently, owing either to the inherent nature of the process being examined or to the complex nature of the process, the relations must be expressed in a statistical form. That is to say, certain portions of the system — and as a result of this, the system itself — will not possess a unique output for a given input. Rather, the output must be expressed as a spectrum of possible events where each event has a certain probability of occurrence. Two simple examples may serve to illustrate the nature of the phenomena involved: Example 1 — Measurement Uncertainties. Measurements of time, distance, temperature, etc., always possess a certain error tolerance. For example, a large number of distance measurements made with the same 40 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM L tcAL MEASURED LENGTH Fig. 1-28 Measurement Errors Obtained in Determining Length. instrument measuring the same distance might give rise to a distribution of values about some mean value as shown in Fig. 1-28. If these measure- ments are compared with a standard, we see that two sources of error exist: (1) a calibration or bias error, and (2) a random error. The calibra- tion error — so long as it remains fixed or if its variations can be predicted — is obviously a correctable source of inaccuracy. However, the random error is just that — random. Any given measurement may be in error by an amount determined by the character — usually Gaussian — of the randomness. Measurement uncertainties are a vitally important problem in any weapons system analysis. Unlike many engineering problems, where measurements may be made with whatever degree of preciseness is neces- sary to render inconsequential the measurement error, weapons systems habitually are required to work with measurement uncertainties that exercise a profound and usually decisive influence upon their performance. Example 2 — Dice Throwing. The cast of a die is an example of a process that is, theoretically, completely predictable; however, because of the extreme complexity of the mechanisms that govern its behavior, the whole process is more easily handled by probability concepts. For example, if we knew the exact orientation of the die, its velocity, direction of motion, physical size, shape and weight distribution, condition, characteristics and orientation of the die table, etc., we could predict with certainty which of the die faces would appear on top. However, the amount of information that must be obtained and processed to arrive at this result is usually prohibitive. It is much easier — and, also, as in the case of representations of this type, less profitable — to characterize such a process by saying that for any input (legal throw) the system (balanced die cube) may produce any output from 1 to 6 with equal probability. Weapons systems contain many processes of similar brand. Ballistic trajectories are a prime example. The behavior of a human being in a 1-12] CONSTRUCTION AND MANIPULATION OF MODELS 41 control loop is another. Thus the basic parameters of the model usually take the form of distribution functions. Attention is invited to the Operations Research volume^ of this series for a detailed treatment of the theory of probability as applied to weapons system evaluation. 1-12 CONSTRUCTION AND MANIPULATION OF WEAPONS SYSTEM MODELS Before analyzing the structure of a model, let us review some of the peculiar characteristics of a weapons system. 1. A weapons system is an organization of men and equipment designed for operation and use against specific classes of enemy targets. To carry out its overall function — usually the destruction of the enemy target — it must carry out many complex subfunctions. Each functional activity converts certain quantitative inputs into outputs. The entire weapons system is merely a series — or series-parallel — arrangement of these subfunctions connected in such a manner as to permit achievement of the overall system objective. As an example of a typical organization, an air- to-air intercept system might be characterized by the sequence of opera- tional functions shown in Fig. 1-29. Also shown are the major equip- ments that are involved in the performance of each function. The input to the system is an enemy target — the output is the destruc- tion of the enemy target. Similarly, each operational function can be viewed as an input-output device. The subject of input-output relations brings us quite naturally to a consideration of another distinguishing characteristic of a weapons system. 2. A weapons system is a dynamic or time response system. Both the system inputs and outputs have time variables. This fact makes it neces- sary to treat a weapons system in terms of the time delays that it introduces between the input (enemy target) and the output (action against the enemy target). The likelihood of mission accomplishment usually is strongly dependent upon the ability of the system to respond to an input within a specified period of time. Each operational function can contribute to the overall time response characteristic. For example, a finite time is required to process target intelligence and tactical situation information for the purpose of assigning a weapon to the target. Upon being assigned, the interceptor aircraft requires a certain amount of time to take off and fly to the target location, etc. Thus, the concept of partitioning the system into subfunctions — • ^Grayson Merrill, Harold Greenberg, and Robert H. Helmholz, Operations Research, Arma- ment, Launching (Principles of Guided Missile Design Series), D. Van Nostrand Co., Inc., Princeton, N. J., 1956. 42 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM ^ I ^ E E E ^ E o p ■t^ ki to o fS^ c^ s E E 11 -f^ r; (- -t?^ <: « > •—1 i\j 00 "* 5^1 -^ n Q_ F i i t5 ^ s ".; .7^^ CO x: t; -t c o S ^ 5 0) ?> o l'^ M LO [J2 ^ Q- iZ ^ z^ CM 00 2, it, I ^ ^^ i ^ E S ^ § -K c/5 ii O ^ '^ ^ 2 ^ 't^ TO o c r '^ C o c O § W) 3 \ t op *; o o E > "5b tj n D- O Z Li. > I .-H CM 00 "^ in 'Z^ LlI 03 5 a. = 1-12] CONSTRUCTION AND MANIPULATION OF MODELS 43 each characterized by a time delay response characteristic — is basic to our modeling approach. Bearing these observations in mind, we may outline the structural composition of a weapons system model as follows: 1. Input Variables — The input variables include all the characteristics of the enemy target complex — size, number, location, speed, defense capability, etc. — that are pertinent to the operation of the weapons system. Also included are the elements of the fixed environment — the physical environment, the environment provided by other weapons sys- tems, etc. — that affect the operation of the weapons system. 2. Mission Accomplishment Goals — This is a quantitative expression of the desired system output. Usually, it derives from the operational requirement. This quantity and the input variables define the problem that the weapons system must solve. 3. System Logic — The system logic describes the system organization and the flow of information through the system; i.e. how the system oper- ates on input data, what sequence of operations takes place, what the pre-established tactical doctrine is for a given set of input variables, etc. This structural element of the mathematical model provides the means for breaking up the overall system function into logically consistent sub- functions. 4. System Configuration Parameters — These include the basic elements and characteristics of the system needed to implement the system logic — the geometry of the system, the number of weapons, the weapon character- istics, and the capabilities and characteristics of each of the system elements such as aircraft, radars, etc. 5. Model Parameters — The basic model parameters are the time delays that are defined for each of the system subfunctions on the basis of the input parameters, the system logic, and the system configuration param- eters. For reasons that were discussed in Paragraph 1-11, the time interval associated with the performance of each function must usually be expressed as a probability distribution of time delays rather than as fixed time delay. Often, range to the target is used instead of time as the means for expressing the basic parameters of the model. This is merely another way of expressing the time delay, since range and time are related through the relative velocity between the target and the weapon. Suboptimization. Most weapons systems are so complex that it is not possible to construct a single model that includes all possible parameters and variables. Instead, many different models must be constructed, each designed to explore a certain facet of the system operating and its relation 44 ELEMENTS OF AIRBORNE RADAR SYSTEMS DESIGN PROBLEM to the whole. In the example of the interceptor system previously referred to, we might have the following models. 1. System Flow Model illustrating the qualitative aspects of system operation. 2. Overall System Effectiveness Model indicating in gross terms the defense level that could be provided against a multiplane attack. 3. Early Warning Detection and Tracking Model indicating the quantitative aspects of the problem of early-warning detection, identification, tracking, and target assignment. 4. Interceptor-Bomber Duel Model, including (a) Vectoring Model, (b) Airborne Radar Detection Model, (c) Attack Phase Model, and (d) Target Destruction Model. Each model must be logically consistent with the others, to maintain a unified systems approach. Constructing these models shows the same stages as constructing the overall system, namely, 1. Define input variables. 2. Define performance criteria ("mission accomplishment goals"). 3. Outline system logic. 4. Define the configuration parameters of the part of the system being analyzed. 5. Derive the quantitative relationships on the basis of the input parameters, system logic, and configuration parameters. This process of analyzing only a small portion of the system problem at any one time is known as suboptimization. Successive suboptimizations of various portions of the system can form a step-by-step approximation which converges to the results that would be obtained if one could analyze the entire system with one model. Counterbalancing the obvious analytical advantages of the suborpti- mization technique is the fact that it gives rise to a serious bookkeeping problem. The results of each suboptimization process must be logically consistent with the rest. As we successively suboptimize various portions of the system, the assumptions of previous suboptimizing routines may be changed. We must recognize such changes and modify previous sub- optimization routines to be consistent with these changes. Summary tables, information flow diagrams, and functional block diagrams form indispensable tools for the bookkeeping process. 1-13 SUMMARY In this chapter, we have covered the general characteristics of radar systems; the environments in which they operate; the functional capabilities 1-13] SUMMARY 45 they provide; the basic means by which intelligence is carried on and extracted from microwave radiations. In addition, we have emphasized the role of the radar as a device in- timately connected with other devices to form an overall system. The requirements for the radar must be derived from a logical study of the per- formance of the complete weapons system in its expected tactical condi- tions of operation. A knowledge of broad weapons system modeling tech- niques, technical competence in the diverse aircraft and guided missiles arts, and a capacity for logical reasoning are required for this process in addition to a detailed knowledge of radar systems. The next chapter will demonstrate how the systems approach is applied to typical examples of airborne radar system design. Succeeding chapters will cover the detailed problems of radar design and their relationship to the system problem. J. POVEJSIL • P. WATERMAN CHAPTER 2 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-1 INTRODUCTION TO THE PROBLEM Assume that a radar design group is presented with the following problem. Specify, design, and build the following radar systems for use in a fast attack carrier task force environment in the time period Jgxy to I9^x. 1. Airborne Intercept (AI) radar and fire-control system for all-weather transonic interceptors, and 2. Airborne Early Warning (AEW) radar system for installation in a 35,000 lb gross weight AEW aircraft. This chapter will demonstrate how the first part of this problem — specification of radar requirements — can be solved using the general approach outlined in Chapter 1. (Later chapters will discuss the additional problems involved in mechanizing a radar system to meet a set of derived requirements.) Although a specific problem is treated, the method of approach has applicability to all radar specification problems. Particular emphasis is placed on the processes of obtaining a good understanding of the overall weapons system problem which dictates the radar requirements. Methods for formulating a system study plan and for constructing system models are illustrated by examples. The type of information which must be collected and analyzed by the radar design group prior to attempt- ing the radar design also receives attention and comment. The hypothetical problem used as a vehicle for this discussion has been greatly simplified; however, it is still a complicated problem and the reader will be required to perform the same sort of rechecking, cross-referencing, and backtracking that is required to understand and follow an actual systems analysis problem. To facilitate this process, much of the reference data is displayed on charts and diagrams which typify the pictorial (iconic) representations that should be constructed for an actual systems problem. 46 2-1 INTRODUCTION TO THE PROBLEM 47 Wherever possible, mathematical complexities have been eliminated, minimized, or referenced. However, the reader will find that knowledge of simple probability theory, operational calculus, and feedback control theory provides a key to better understanding of a weapons system study. Readers having some familiarity with the radar design problem may be somewhat surprised — even appalled — at the amount of study which must be done prior to considering the radar design problem itself. They may be tempted to say: "Surely, you do not expect a radar designer to concern himself with such matters? These are the responsibility of the agency that prepares our specification!" The authors' reply to these remarks is based on their own not always pleasant personal experiences and close observation of the experiences of many other radar system designers. Seldom, if ever, does the basic specification given to a radar designer provide the necessary input information for a successful system design. It is a starting point — nothing more. Until the radar designer understands the nature of his contribution to the solution of the overall system problem he has little chance of designing a radar system that will operate satis- factorily as part of a complete weapons system. In recognition of this fact, the weapons system contractor not only should encourage, he should require the radar designer to demonstrate his understanding of the problem by preparing a formal document embodying the type of analysis and development to be demonstrated in this chapter. This will tend to ensure that the purposes and objectives of the weapons system are not made subservient to the preconceived biases of the radar designer. What is the problem? What is known about the problem? What remains to be understood about the problem? What is the plan for action? Fig. 2-1 Steps Leading to the Formulation of the System Study Plan. 48 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-2 FORMULATING THE SYSTEM STUDY PLAN Reduced to its barest essentials, any system may be visualized as the logical process of asking and answering the sequence of four questions out- lined in Fig. 2-1. This basic pattern is repeated — again and again — throughout the course of the study. Operational Requirements (O.R.) Carrier Task Force Weapon System Model Fixed Elements Variable Elements Interceptor tJJ Subsystem Analysis Generated Capabilities and System Deficiencies Mission Accomplishments Step 1 System Analysis Target Complex Complete Accomplishment of O.R. VARIABLE ELEMENTS Target Characteristics Radar Req'ts Step 2 Al Radar and Fine Control Req'ts - Step 3 ■ Early Warning Detection Range from Fleet Center Interceptor Kill Probability No. of Interceptors Engaging Raid Fig. 2-2 Master Plan for Weapons System Analysis First, the problem is defined. For the hypothetical carrier task force air defense system, this problem definition takes the form of the following question : 2-2] FORMULATING THE SYSTEM STUDY PLAN 49 "What does the weapons system require of the airborne radars (AI and AEW) to achieve a satisfactory level of mission accomplishment?"^ The basic elements of this problem are shown in Fig. 2-2. The operational requirement defines a weapons system problem. By the processes described in Paragraph 1-8, the operational requirement leads to the establishment of the concept of a system which depends for its operation upon the charac- teristics of a number of subsystems — aircraft, missiles, detection devices, and shipboard installations, for example. As is apparent from this figure, the solution to the radar requirements problems will involve consideration of many complex characteristics and relationships external to the airborne radars. Hence, the middle two questions (Fig. 2-1) and their answers are fundamental to further progress. For the hypothetical problem, the known and unknown elements of the problem are displayed in Figs. 2-2 through 2-8. These will be discussed in greater detail in subsequent paragraphs. The plan for action will develop quite naturally from the indicated sequence of questions and answers. Elements of the problem that are not known or understood must be investigated in greater detail and related to the known elements. In some cases, adequate information may not be available on the unknown elements of the problem — or the path to under- standing may be blocked by the inherent difficulty of the problem. These cases will require that arbitrary assumptions be made in order that the analysis can proceed. When a sufficient understanding of the overall problem is obtained by analysis (or assumption), weapons system models are constructed. These models have as variable parameters the performance characteristics of the airborne radar known to be important to weapons system operation. Using the techniques of Operations Research and Systems Analysis, these models are "played" against the target inputs (Fig. 2-2). The level of system performance (Mission Accomplishment) obtained is compared with the operational requirement, thereby generating a measure of the system capabilities (or deficiencies). By such processes, mission accomplishment may be related to the radars' performance characteristics (see inset, Fig. 2-2) thereby providing a means for obtaining the true requirements of the airborne radars as dictated by the weapons system requirements. It is important to emphasize once again that the derivation of require- ments should not be aflFected by state-of-the-art considerations in radar technology. The purpose of the analysis is to define the radar problem, not to solve it. Only after this task is completed is the radar designer free to turn his attention to the job of designing and building a specific radar system to meet the requirements imposed by the weapons system problem. iThe problem of defending a carrier task, force is quite analogous to the defense of a city or important military base. 50 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-3 AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM The fixed elements of the carrier task force environment, the target complex, and the operational requirement provide the basis for the deriva- tion of radar requirements. We shall assume that all elements of the task force air defense system are fixed, except the following: 1. The Airborne Early Warning Aircraft Radar and Data Processing System, 2. The Airborne Interceptor Radar and Fire-Control System. Referring to the system study plan of Fig. 2-2, the first task is to de- termine: (1) what is known about this weapons system, and (2) what remains to be known or understood. The overall weapons system is broken down into four major operating elements: (1) Ship Weapons System, (2) AEW Aircraft System, (3) Inter- ceptor Aircraft System, and (4) Air-to-Air Missile System. The character- istics of each element — and the interrelationship between elements — which exert a sensitive influence on the overall system performance are shown in Figs. 2-3 through 2-6. The known characteristics are checked; graphical or tabular description of these appear in the same figures. The unknown (unchecked) characteristics constitute the items a knowl- edge of which must be gained from the system study. These include the characteristics of the AEW and AI radar systems; they also include important interrelationships among the various system elements, known and unknown. For example, the individual characteristics of the interceptor aircraft and the air-to-air missile are known; the manner in which these two elements combine and interrelate with the interceptor fire-control system to produce a weapons system capability is not known and must be derived by study. The list of sensitive parameters is not complete. At the outset of a system study it is not possible to name all the parameters that may be important. As more is learned about the system through study, these must be added and considered in their proper relationship with other parameters. Carrier Task Force Weapons System (Fig. 2-3). This system includes two aircraft carriers and three missile-firing cruisers in the dis- position shown. (This configuration is, of course, fictitious and is used to illustrate the possible elements of this problem. Elements of the task force not germane to this example are excluded.) Two large carriers constitute the main ofi^ensive and defensive elements. The carriers are separated (5-20 n. mi.) for tactical reasons. Mass attacks 2-3] AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM 51 Sensitive Parameters and Elements Tactical Doctrine Task Force Structure Y Air Defense Tactics >/ Aircraft Handling and Availabilityv^ Reaction Time y^ Data Collection Early Warning Range Information Rate Communications v^ Accuracy Reliability Vectorin g Accuracy Communication v^ Interceptor Effectiveness Speed v^ Task Group Disposition Likely Direction I of Raid 100 200 300 (n.mi) I = Carrier i = AEW 6 = 6 CAP o = Missile Cruisers Geometry of the Defense Zones / Missile : Zone _ CIC ■CAP- Deck Launch Interceptors «^AEW f-^lnitial I Attack I Boundary —AEW Detection Boundary 50 100 150 200 250 RANGE FROM TASK FORCE CENTER (n.mi.) Down for Maintenance and Repairs (18) Aircraft Availability and Tactics 76 60 50 40 30 20 10 CAP Guardin g Flank ( 6) CAP Engaging Raid (6") (36) Deck -Ready Launch Rate, 2 per Minute 7 Note: Tactics Require One CAP to Maintain Station During a Raid to Guard Against Attacks from Other Direction Total Available to Engage Raid (42) Communications AEW 1000 Bits Per Second Per Channel Reaction Time 3 Minutes from Instant of Early Warning Signal Fig. 2-3 Carrier Task Force Weapons System Characteristics. are assumed to occur only on the side of the task force which is exposed to enemy territory. Local defense of the task force is augmented by missile-firing cruisers flanking, and somewhat forward of each carrier. These provide a point- defense system with an altitude capability of 50,000 ft extending out to about 40-50 n. mi. in front of the carriers. A third missile-firing crusier 52 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS guards against sneak attacks from the rear as well as turn-around and reattack tactics. Early warning detection and interceptor vectoring information is pro- vided by airborne early warning aircraft (AEW). The task force is assumed to have a capability for maintaining three AEW aircraft aloft on a 24-hour basis. The primary functions of the AEW system are: (1) To provide detection of enemy aircraft at sufficient range forward of fleet center to permit interception by piloted aircraft. Sensitive Parameters and Elements Speed w^ Detection Range Altitude v^ Information Rate Endurance v Resolution Maneuver Display Reliability Measurement Accuracy Target-Handling Capacity Communications y Integration with CIC y' Aircraft Availability v^ Human Operator Characteristics Environment v^ Navigation Accuracy v Navi g ation and Communications Communication Channel Capacity 1000 Bits per sec Navigation Accuracy (Relative to CIC) 1 n.mi. rms Mission Profile ^20K - 350 Knots ■7- \ 200 Knots ■i<A~wl^y Patrol 5 lOK 50 100 RANGE FROM FLEET CENTER (n.mi.) Inte g ration with Combat Information Center (CIC) Target ^^^^^^"AEW Aircraft Availabilit y 6 Environment Deck-Ready On-Station Stand-I On Station A. Shock and Vibration B. Temperature C. Pressure D. Humidity E. Space and Weight F. Environmental Noise Interference Clutter Weather Internal Fig. 2-4 Airborne Early Warning System. See Fig. 2-18 for Tactical Deployment. I 2-3] AIRCRAFT CARRIER TASK FORCE AIR DEFENSE SYSTEM 53 (2) To provide target data for the ship-based combat information center (CIC), which supplies vectoring information to piloted interceptors and guidance information to ship-based missile radars. Two combat air patrols (CAP) are maintained. Each CAP contains 6 all-weather interceptor aircraft. In addition, interceptors may be launched at a maximum rate of 1 per minute from each of the two carriers during attack conditions. Aircraft availability limits the total number of deck- launched interceptors to 36. During an attack, only one CAP (6 aircraft) engages the raid; the other is held in reserve to guard against attacks from other directions. Thus, a maximum of 42 interceptors can be used to engage a raid. The interceptors are armed with air-to-air guided missiles and are required to perform the interception function at altitudes from sea level to 60,000 ft. An optimum battle-control and communications system is assumed. This is to say, the deployment of interceptors by CIC is such that any interceptor which enters the interceptor zone is able to make an attack so long as there are targets within the zone. As will be shown later, system performance is sensitively affected by this assumption, which represents a condition most difficult to realize in practice. Airborne Early Warning System (Fig. 2-4). The basic functions of the AEW system have been described. The carrier-based aircraft available for this purpose is assumed to be capable of housing an antenna with a maximum dimension of 12 ft in the mushroomlike appendage shown. The exact disposition of the AEW aircraft and the AEW radar and data processing requirements will be determined by study. A major unknown is the contribution of AEW target information accu- racy to interceptor effectiveness. Interceptor Aircraft System (Fig. 2-5). The known and unknown characteristics are defined as shown. The determination of detailed radar requirements will require an analysis of the dynamics of the closed-loop system formed by the target, the interceptor, the pilot, and the AI radar and fire-control system. The interrelationships among the aircraft system, the air-to-air missile system, and attack tactics are also unknown and must be analyzed as a prelude to the ascertainment of radar requirements. A major unknown, to be determined by the system study, is the con- tribution of vectoring accuracy to interceptor effectiveness. 54 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Sensitive Parameters and Elements Speed v^ Altitude v^ Range y Climb Capability -Z' Maneuver X Environment -/ Angle Coverage Vectoring Accuracy Attack Tactics Fire Control Weapon s/ Pilot Characteristics Control Dynamics Maneuverin g Characteristics Maneuvering Capabilities 2.0 g Max. Angle of Attack 6° Transient Angle of Attack 2° Transient Roll Rate 60Vsec Mission Profiles Combat Air Patrol (CAP) Mission ffefurn y^ 1200 ips 2.8 Hours at 850 fps -15 Minutes Combat at Vc = 1200 fps 100 200 300 RANGE FROM FLEET CENTER (n.mi.) Deck Launch Mission Return 100 200 RANGE FROM FLEET CENTER (n.mi.) Weapons 2 Semiactive Air-to-Air Guided Missiles Launched in a Salvo following Acquisition of Target by Missile Seekers Environmental Factors A. Shock and Vibration Catapult Launch Landing Flight and Maneuver Missile Launch B. Temperature C. Pressure D. Humidity E. Noise: Clutter Vl/eather Interference Internal F. Size and Weight Interceptor Control System Dynamics TL Tracking (\ ^^ Target Input . Ji Tracking Error Tracking Radar r LS = Line of Sight TL = Tracking Line ■^ = Interceptor Heading i/^(- = Computed Correct Heading Vp = Interceptor Velocity V^ = Target Velocity R = Range Steering Error Computer — »Q--^-»|~ Pilot | — 1\ Aircraft Aircraft Radar Interaction Fig. 2-5 Interceptor System Characteristics. Air-to-Air Missile System (Fig. 2-6). The sensitive parameters of the air-to-air missile system, are shown. The major unknown parameters involve interrelationships of the missile with the aircraft and the radar and fire-control system. THE TARGET COMPLEX 55 Sensitive Parameters and Elements Performance y^ Kill Probability v^ Guidance y^^ Radar and Computer Integration »^ Illumination Accuracy y Illumination Power v Guidance 2 4 6 8 10 SEEKER ACQUISITION RANGE (n.mi.) ILLUMINATOR TRACKING ACCURACY (deg rms) 12 3 4 5 DOWN RANGE (n.mi.) Kill Probability (Including Reliability) Single Shot - 0.50 2-Missile Salvo - 0.75 Radar and Computer Integration Seeker Angle Slaving Signal Seeker Range Slaving Signal J^^ Launching Signal T^ Seeker Acquisition Signal Fig. 2-6 Air-to-Air Missile System Characteristics. 2-4 THE TARGET COMPLEX (FIG. 2-7) Target Complex. The characteristics of the target complex include its parameters and its mission as shown in Fig. 2-7. The 50-nautical mile (n. mi.) air-to-surface missile (ASM) carried by the hostile aircraft requires that interception take place outside of a 50-n. mi. circle around the aircraft carriers. The target's 2-g maneuver capability will exercise an important influence on the radar and fire-control system design. It will be assumed for this example that the enemy aircraft is provided with the capability for op- timum timing of this maneuver. Also, it is assumed that the target does not employ electronic countermeasures (ECM). 56 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Sensitive Parameters and Elements Speed ^ Maneuver -/^ Altitude ^ Defense "^ Number v^ Radar Size Tactics v^ Weapon y^ Raid Geometr y |-* 95 n.mi. H 20 Aircraft VM\ _j L 5 n.mi. ~^ '^Spacing Direction of Raid IVlission Profile g 50 K 300 fps (2g Evasive Action) 1000 01 (\ L 700-1000 V 100 50 RANGE FROM TARGET (n.mi.) Target Radar Cross Section 90° 180° 1 Fig. 2-7 Target Complex Characteristics. The radar cross-section characteristics of the hypothetical target are only generally known and are shown in Fig. 2-2. Paragraph 4-7 explains the factors contributing to characteristics of this type. Paragraph 4-9 discusses how the target radar area may be estimated for purposes of preliminary design. The turboprop propulsion system of the enemy aircraft was chosen to introduce into the model the effects of the modulation characteristics of the reflected radar energy (Paragraphs 4-7 and 4-8). This can be an important radar design consideration. The target is assumed to carry a high-yield nuclear weapon. Destruction resulting from impact on the target aircraft is assumed to cause a detona- tion capable of producing a destructive overpressure within a 1000-ft radius sphere around the target. Ignoring time effects, 1000 ft thus defines the 2-6] THE SYSTEM CONCEPT 57 point of allowable minimum approach of the interceptor to the target aircraft. The number of target aircraft (20) and their spacing (5 n. mi.) is charac- teristic of a raid designed to present a difficult problem to the model air defense system. In an actual problem, a number of different target complexes would have to be defined in this way. The behavior of the system would be analyzed for the several inputs and the design parameters chosen on the basis of the response to all expected target complexes with emphasis on the most effective configuration. For simplicity in this example, we will confine our attention to the single problem defined; however, the sensitivity of system performance to changes in this input (i.e. target speed and number) will be examined. 2-5 THE OPERATIONAL REQUIREMENT: MISSION ACCOMPLISHMENT GOALS (FIG. 2-8) The operational requirement defines a military problem which must be solved by the combination of known and unknown weapons system elements previously described. Bases for judging the military usefulness of any system proposed as an answer to the operational requirement are also shown. 2-6 THE SYSTEM CONCEPT The operational requirement defined a weapons system problem. The procedures for solution of this problem are determined by the system con- cept or logic. Within the framework of the system elements already defined, the system logic for the interceptor system may be developed by listing the sequence of events which lead to the interception of the target by the missile-armed interceptor. The following events would normally be expected to occur in sequence: a. Early warning detection b. Identification c. Threat evaluation d. Weapon assignment e. Interceptor direction or vectoring /. AI radar search and detection g. AI radar acquisition h. Airborne weapons system tracking control and missile launching i. Air-to-air missile guidance j. Missile detonation and target destruction (without self-destruction) 58 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Function High-Attrition Air Defense Against Medium Bombers Features (1) Compatibility with Surface-to-Air Missile System (2) Compatibilty with Fleet Elements, Logistics, and Tactics (3) Compatibility with Transonic Interceptor Aircraft -- 30,000 Lb Gross Weight Mission Accomplishment Goals 20 40 60 80 100 FREQUENCY OF OCCURRENCE Mission Accomplishment Goals (See Inset Figure) (1) (2) Fig. 2-8 Operatioric A 20-Plane Raid at 50,000 ft and 800 fps shall be Employed as the Input for Judging System Performance. System Performance shall be Judged as Satisfactory if the System can be Demonstrated to Attain or Exceed the Following Performance Levels (As Indicated by the Shaded Area in the Figure) (a) 50 per cent Probability of Killing all 20 Targets (b) 90 per cent Probability of Killing 16 Targets (c) 99 per cent Probability of Killing 12 Targets All Kills are to be Accomplished at a Minimum Distance of 50 n.mi. from Fleet Center .1 Requirements, Attack Carrier Task Force Interceptor Ai Defense. k. Return to base /. Transfer of residual target elements to the surface-to-air missile support defense system. A diagrammatic summary representation of the overall tactical situation is shown in Fig. 2-9. The statistical nature of the system operation is shown by this diagram. For various reasons — interceptor availability, time limitations, system failures, system inaccuracies — a certain percentage of the interceptors fail to complete each of the successive steps required for interception. Thus, any interceptor chosen at random from the total complement has a certain probability that it will kill a target. This probability is the product of the individual probabilities that it will pass successfully through each successive stage of the interception. This line of reasoning points out the necessity for obtaining a proper balance between the performance of various elements of the system. A 2-7] THE SYSTEM STUDY PLAN 59 Zone of CIC -Vectoring Inaccuracy Zone of AFCS Inaccuracy rrv '=^X Search Zone of Missile Guidance Inaccuracy Fig. 2-9 A Tactical Situation. very low probability of success for any phase of the intercept mission can render pointless any efforts to achieve very high probabilities in other phases and thus would serve as a guide to more effective development emphasis. 2-7 THE SYSTEM STUDY PLAN The known (fixed) and unknown (variable) elements of the problem have now been defined. Referring to Fig. 2-2, it is seen that the next step is to analyze the interrelationships between the fixed and variable elements to determine the contribution of each variable element to mission accomplish- ment. From such analyses, a quantitative understanding of system operation will be obtained and — eventually — radar requirements will evolve. The unknown or variable elements may be broken into two basic cate- gories: (1) weapons system variables, and (2) subsystem variables. The primary weapons system variables are: 1. The early warning detection range measured from fleet center. 2. The number of interceptors which may engage the specified target complex 3. The effectiveness (kill probability) of each interceptor which engages the target complex 60 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS All of the other unknown parameters (subsystem variables) affect one or more of these three basic weapons system variables. This observation makes it possible to organize the study plan on a step-by-step basis as follows : Step I: Construct a model of the overall weapons system using the three primary weapons system variables as adjustable parameters. Assume values for each of these adjustable parameters and calculate the resulting system performance. Compare this performance with the desired level of mission accomplishment; use any discrepancy between the two to adjust parameter values for another tentative design. Testing of the model continues until the following information is derived. 1. All combinations of the adjustable weapons system parameter values that will allow achievement of the mission accomplishment goal. 2. The sensitivity of system performance to changes in the values of the adjustable parameters. Additional information — useful for obtaining a good understanding of the overall problem — is obtained by ascertaining the sensitivity of system performance to changes in the fixed elements. Step 2: Assume fixed values for the three weapons system variables of Step 1 that permit the system to achieve the desired level of mission accomplishm.ent. Construct a model (or models) which expresses the relationships between the adjustable (unknown) AEW parameters (beam width, information rate, radar detection range, etc.) and the assumed weapon system parameters. Test this model for various assumed combina- tions of AEW parameters. Establish acceptable combinations of AEW parameter values and the sensitivity of system performance to parameter changes. Derive a specific set of AEW requirements. Step 3: Using the values for the unknown system variables derived in Step 1 and 2, repeat Step 2 for the adjustable parameters of the interceptor weapons system. Derive a specific set of requirements for the AI radar and fire-control system. The suggested order of Steps 2 and 3 is somewhat arbitrary; a reasonable case might be made for reversing this order. As a general rule, where a choice exists, it is wise to select an order which places the most difficult subsystems first, since this will maximize the number of adjustable parameters available for its preliminary design. 2-8 MODEL PARAMETERS The interrelations between major system parameters and the contribu- tions of each parameter to overall effectiveness may be developed through 2-9] SYSTEM EFFECTIVENESS MODELS 61 the use of mathematical models. These interrelationships form the quanti- tative bases for the choice of basic system parameters. The following paragraphs will develop a number of models designed to expose some of the more important aspects of the task force air defense problems. The techniques used to develop these models are illustrative of the means by which any complex system problem may be broken down to forms that can be handled by analytical means. 2-9 SYSTEM EFFECTIVENESS MODELS The operational requirement (Fig. 2-8) specified the effectiveness of the interceptor air defense system in terms of the degree of success which must be achieved with a required reliability. For example, the probability of destroying at least 16 of the 20 targets should exceed 90 per cent. The first step of the systems analysis must determine the nature of the relationships between system eflFectiveness and the fixed and variable elements of the defense system (Fig. 2-2). The following examples demon- strate how such an analysis may be carried out. Assume that 40 interceptors may be brought to bear against the 20- target raid previously assumed as the threat (A^ = 40). Each interceptor can make only one attack with its two-missile salvo. Thus, when one attack against a target fails, another interceptor will be assigned to that target until either 40 attacks have been made or all the targets have been destroyed. 20 18 16 Q Ul d 14 \2 12 C3 ^ 10 ■---. #2^ ^^ ' ^v. \ "v N^ Specified Effectiveness from Operational Requirement ^ ( Fig. 2 - 8) Shown by Dashed Lines 1 1 1 1 \ 1 Case^l Case*'2 _ 25 Number of Interceptors = 40 20 40 60 RELIABILITY (PER CENT) 100 Fig. 2-10 System Effectiveness Operating Characteristics. 62 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS If the effectiveness of each interceptor is assumed to be P = 0.5, the system operating characteristic shown in Fig. 2-10 may be calculated by the application of simple probability theory. ^ As can be seen, the assumed parameters allow the operational requirement to be met. To obtain a complete picture, other possibilities may be assumed and analyzed in the same fashion. For example, the effectiveness level provided by 25 interceptors, each with a kill probability of 0.7, is also shown in the figure. This combination of parameter values fulfills only part of the operational requirement. The probability that more than 16 targets will be destroyed is less than that for the previous assumptions; thus, this system would impose increased requirements on the back-up surface-to-air missile system. Continuing in this fashion, trade-off curves between the number of interceptors and the interceptor kill probability can be determined for each point of the operational requirement. Such a curve is shown in Fig. 2-11. Here all the combinations of the interceptor kill probability and number of interceptions are shown which will kill at least 16 out of 20 targets with a 90 per cent reliability. \ 1 1 No. of Targets -20 > \ QJ LlI — ° Interceptors -36 Missiebaiv \ 3 3 Kill \ Prob ability ' \ V \ li V f = 1^1 \ 1 < II- oK ^1 5 a. < 1 ' lol Icvil 1 1 3 10 20 30 40 50 60 70 80 N-NUMBER OF INTERCEPTIONS Fig. 2-11 Interceptor Kill Probability vs. Number of Interceptions Required to Kill 16 or More Targets with 90 Per Cent Probability. ^Grayson Merrill, Harold Goldberg, and Robert H. Helmholz, Operations Research, Jrma- ment. Launching {Yv\nc\p\&s oi Guided Missile Design Series), D. Van Nostrand Co., Inc., 1956. Chapters 6 and 7 provide an excellent discussion of the mathematical techniques involved. 2-9] SYSTEM EFFECTIVENESS MODELS 63 Also shown are the limiting effects of the fixed problem elements pre- viously outlined in Figs. 2-3 through 2-6. For example, the missile salvo kill probability is 0.75; obviously the interceptor kill probability cannot exceed this value. If the raid is engaged by six CAP interceptors and all 36 of the deck-ready interceptors, an interceptor kill probability of 0.48 is required. If the tactics are changed to allow both CAP patrols to engage the raid in addition to the 36 deck-ready interceptors, the individual inter- ceptor kill probability required drops to 0.42. If the total complement of 66 interceptors could be used, a kill probability of only 0.3 would be required. The basic parametric relationships between the number of interceptors and system effectiveness now are established. The next phase of the sys- tems analysis must determine the relationships between the number of interceptors and the other fixed and variable elements. Completion of this phase will provide the basic parametric data which will, in turn, allow intelligent selection of the following system parameters (see Fig. 2-2). 1. Number of interceptors (A^) 2. Interceptor effectiveness (Po) 3. Early-warning range {Raew)- The number of interceptors which can be used to defend a given raid, and thus the required interceptor kill probability, is a function of initial interceptor deployment, detection ranges, reaction times, and target and interceptor speeds. These factors can be conveniently summarized in a diagram similar to Fig. 2-12, which shows the sequence of events in a typical raid. The interceptor and target performance characteristics were given in Figs. 2-5 and 2-7. We assume, as an illustrative case, that the AEW detection range is 250 n. mi. from the fleet center. Since the target has a speed of 800 fps (474 knots), it will arrive at the fleet center 32 minutes after detection. The target track is shown in Fig. 2-12 as the straight line connecting 250 n. mi. at zero time to 32 minutes at zero range. The CAP interceptors stationed 100 n.mi. from the fleet center are vectored to intercept the raid following a 3-minute time delay consumed by the process of identification, acquisition, and assignment. The track of the CAP aircraft is constructed as a line with a slope equal to the reciprocal of their speeds (1200 fps or 710 knots). We observe that the intersection of the two tracks occurs at 175 n.mi., the maximum range at which the raid can be engaged. In accordance with defined tactical doctrine (Fig. 2-3), only one combat air patrol (6 aircraft) is committed to the raid. The remaining CAP maintains its station to guard against the possibility of attacks from other directions. 64 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 40 50 100 150 200 250 DISTANCE FROM FLEET CENTER (n.mi.) Fig. 2-12 An Attack Diagram. 300 From Fig. 2-5 it is seen that an additional 3-minute delay is created by the interceptor climb and acceleration characteristics. This makes the total effective reaction time of the deck-launched aircraft equal 6 minutes. After the initial reaction, interceptors will be launched at a rate of 1 per minute from each of the two carriers until either no more interceptors are left or it is obvious that the interceptors will not be able to intercept the targets outside of the surface-to-air missile zone corresponding to a 50-n.mi. radius from the fleet center. In our example, this latter consideration is the limiting factor, and it is possible to launch only 32 interceptors from the carriers. Thus, a total of 38 attacks can be made against the raid with the assumed deployment, tactical doctrine, and equipment performance. This is close to our previously assumed case with 40 interceptors, and the required interceptor kill probability will be slightly greater than 0.5. The air battle takes place during a 16-minute time period to enemy penetration of the missile defense zone barrier. The maximum time that any interceptor must fly at Mach 1.2 is 11 minutes (for the first two deck-launched inter- ceptors), which is well within the interceptor performance capabilities as displayed in Fig. 2-5. This model may be used to examine the effect of variations in the system parameters. The results of such an analysis are shown in Fig. 2-13, where trade-off curves relating pertinent factors are given. If the early warning range is increased to 300 n.mi., 50 interceptors can engage the raid. With 50 interceptor attacks, the required interceptor kill probability will be reduced to 0;42. However, with the tactical doctrine assumed, the maximum interceptor complement available to counter an attack is limited to 42. Thus as early warning range increases, aircraft availability in this 2-9] SYSTEM EFFECTIVENESS MODELS 65 ^t3 Po=0.7 p„-0 6 ^ Y- .Ofiej^tionaL '°'1 \ , 1 0=0.3 / y / 20 40 60 80 100 120 NUMBER OF INTERCEPTORS ENGAGING TARGETS AEW^ 250 n.ml. 08 M 1.2 1.0 0.8 TARGET SPEED, M 0.6 80 £g 60 li 40 p20 200 250 AEW RANGE (n.mi.) 300 AEW 1 = 250 n.mi. A 2 4 6 8 10 12 INTERCEPTOR SPEED, M Fig. 2-13 Sensitivity of System Effectiveness to Number of Interceptors, AEW Range, and Interceptor Kill Probability Po- model becomes the limiting factor. This limitation might indicate that a trade-off of parameter values elsewhere in the problem should be examined to exploit the potential advantage which might accrue from a range increase.^ Conversely, a 50-n.mi. decrease in early warning range would require that interceptor kill probability be increased to 0.70 — a value that is almost equal to the kill probability of the missile salvo alone — in order to maintain the system effectiveness required. Increases in target velocity have much the same effect as decreases in early warning range. If the target velocity were to increase by 10 per cent, only 34 interceptions could be made. Thus, to maintain the same defense level under these conditions, interceptor kill probability would have to be raised to 0.58 or early warning range would have to be increased by about 30 n.mi. Increases in interceptor velocity have the same general effect as increases in early warning range; i.e., aircraft availability limits the useful- ness of such increases. System sensitivity to this change is relatively small, however, for extremely high interceptor speeds. The time delays defined for the model made no allowance for any time delay introduced by the vectoring process. The assumption is that the vectoring system guides each interceptor on a straight-line path to the ^For example, we might explore the possibility of using the other CAP aircraft which were assumed to maintain their stations. 66 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS earliest possible interception point. Other types of vectoring guidance — for example, a tactic whereby it is attempted to guide the interceptor on a tail-chase attack — introduce additional time delays, and these reduce the total number of interceptions which can be made with a given early warning range. System tactics also exercise other important influences. For example, a 3-minute dead time delay was assumed between early warning detection and assignment of the first interceptors to specific targets. This type of operation places a high value on the time delay. Each minute of time delay requires 8 additional miles of early warning detection range to maintain a fixed number of interceptions. The effect of this time delay would be different if the system tactics called for launching of interceptors to begin before evaluation was com- pleted. This operation, however, incurs a risk that interceptors may be launched unnecessarily. In this latter case, it would be necessary to evaluate the consequences of a false alarm as a function of threat evaluation time; i.e., the penalties of launching interceptors when the threat does not materialize following an early warning detection in terms of fuel loss, vulnerability to attacks from other directions, etc. Some of these consider- ations may seem to go a little far afield, but the answers to such questions are of great importance to the radar designer because they affect what his equipment must do. To simplify our example, we assume that no inter- ceptors are launched until evaluation of the threat is completed. As a second example of the effect of tactics, we might consider the target assignment procedure. In our example, we assume that an optimum assignment procedure could be used. That is to say, each of the 40 inter- ceptors was able to make an attack during the course of the air battle — except in the cases where all 20 targets were destroyed by less than 40 attacks. This assumption assumes a very sophisticated battle control and communications system. Another method of assignment could be as follows: the first 20 interceptors are assigned — one-on-one — to the first 20 targets. The following interceptors are assigned as back-up interceptors on the same basis — i.e. interceptor 21 to target 1 , interceptor 22 to target 2, etc. For 40 interceptions, this would mean each target could be attacked twice. In some cases, however, the target would be killed by the first interceptor thereby leaving the back-up interceptor without a target to attack, resulting in a potential inefficiency. On the other hand, two attacks may not suffice to kill the target since each attack has less than unity success probability. With these alternate tactics, a substantially greater interceptor kill probability would be required to meet the operational requirement for the case of 40 interceptors reaching the attack zone. This value has been determined to be 0.7 as compared with 0.5 when optimum target assign- 2-10] DESIGN OF AIRBORNE EARLY WARNING SYSTEM 67 ments are made. The advantages of the optimum assignment tactic are apparent. In this paragraph we have shown how the effectiveness of an interceptor system may be analyzed for an assumed mode of operation and assumed values for system parameters. We have also illustrated the concept of obtaining the "trade-off" between various system parameters and the effects of changes in system logic. The examples chosen are merely illustra- tive of the information that must be generated for an actual problem to enable the radar system designer to understand his part of the overall problem. Using the assumed or derived values for the overall system parameters, and the defined system logic, we shall now derive the requirements for the AEW and AI radar systems. The first phase of these analyses (Steps 2 and 3 of Master Plan of Fig. 2-2) is to establish the allocation of responsibility between these two systems. 2-10 PRELIMINARY DESIGN OF THE AIRBORNE EARLY WARNING SYSTEM The AEW system must contribute to the solution of the air defense problem in several ways as may be seen from the operational sequence given in Paragraph 2-3. 1 . The targets must be detected at sufficient range from task force center to permit fulfillment of the required system kill probability. 2. The targets must be identified and evaluated. This means that their identity, number, position, heading, speed, and altitude must be obtained; this information must be evaluated in terms of the implied threat to the task force; and weapons must be assigned, if necessary. This process must be completed within a delay time that is compatible with early warning detection range and the characteristics of the interceptor defense system. 3. The AEW system must provide information which can be used to vector the interceptors toward their assigned targets so that the inter- ceptors may detect and acquire the targets with their own AI radars. The type of vectoring guidance employed must be compatible with system re- sponse times permitted by the early warning detection range. The accuracy of vectoring guidance must be compatible with the input accuracy require- ments of the interceptor aircraft, AI radar, and fire-control system. 4. The AEW system must provide information that may be used for overall battle control and surveillance. The basic plan to be used for the AEW system analysis is shown in Fig. 2-14. Also shown are the interrelations between: (1) the AEW system and the overall problem, and (2) the AEW system and interceptor system. 68 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Pp (Required Kill 1 Probability/ Interceptor) Manned Interceptor AEW System Model Interceptor Effectiveness ^ Air Defense N (Achieved] ^ ^N (Required) System >. Model Requirement , A j- 1 B .i CO "oj , ,Q Fixed Elements Variable Elements Fleet Disposition System Logic Operational Doctrine Vectoring Technique AEW Aircraft Charac- teristics AEW Aircraft Deployment AEW Radar Detection Target Characteristics Range Resolution Speed Information Rate Altitude Tracking Accuracy Number Stabilization Spacing Display N = Number of Interceptors Ship-Based CIC Data Interval Vectored into Attack Evaluation Time Smoothing Time Position Delay Beamwidth Interceptor Availability Navigation Accuracy Frequency Inputs to Interceptor System Study (Step 3) Vectoring Technique AEW Svstem Vectoring Accuracy Specif ication Fig. 2-14 Plan for Analysis of AEW System Requirements (Step 2 of Master Plan — Fig. 2-2). The number of interceptors A^ which can be directed against the separate elements (single surviving targets) of the 20-plane raid during the air battle is selected as the criterion for judging AEW system performance. As already shown (Paragraph 2-9) the required number of interceptions depends upon the kill probability per interceptor. If we assume a value of 0.50, the required number of interceptions is 40 (Fig. 2-11). T\\& task force early warning range needed to meet this requirement is 255 n.mi."* This is one possible combination of parameters satisfying the operational requirement and the interceptor availability limitation. We may select this combination as a design point — keeping in mind that it is possible to trade off interceptor effectiveness and number of interceptions should subsequent analysis indicate this to be desirable. For purposes of ''More correctly, the specification of detection range should include the required minimum probability of obtaining that range, e.g. 90 per cent probability of detection when the target has closed to 255 n.mi. This point is covered in more detail in Paragraph 2-12. 2-10] DESIGN OF AIRBORNE EARLY WARNING SYSTEM 69 ready reference, the selected system parameters and the predicted system performance compared with the operational requirement are shown in Table 2-1. Now, the problem is to find the combination of variable elements which in combination with the fixed system elements will allow the desired value of A^ to be achieved. The first phase of this process is to hypothesize a specific AEW system that provides the required functional capabilities by techniques that experienced judgment deems reasonable. The specific parameters of the assumed system are then derived from the overall problem requirements. State-of-the-art and schedule limitations are not considered in this analysis (see Paragraph 1-8). The only restrictions arise from the fixed problem elements, laws of nature, and the basic nature of the assumed AEW system concept. The latter element is variable. In an actual design study, a number of possible AEW system concepts would be examined in this manner with the object of determining which provided the best solution to the system problem. We shall investigate only one possibility to illustrate the nature of the analysis problem. The AEW system selected as an example is not intended to be an optimum solution to the AEW problem presented by the hypothetical air defense system being examined — or to any other AEW problem. It is presented only to illustrate the types of problems that must be considered in any AEW system design; t\\e. form of the specification for an AEW system; and the nature of the interrelationships of AEW parameters and other system elements. Table 2-1 SUMMARY OF SYSTEMS ANALYSIS Selected System Parameters Predicted System Performance Operational Requirement Number of interceptions A^=40 Kill probability per interceptor P, = 0.5 Early warning detection range Raew = 255 n.mi. Minimum Number of Minimum Targets Probability, Killed % Minimum Number Minimum of Targets Probability, Killed % 20 55 20 ss 16 90 16 90 14 100 12 100 See Figs. 2-10 and 2-11 See Fig. 2-8 Fixed system parameters as defined in Figs. 2-3 to 2-6 Target parameters as defined in Fig. 2-7 Note. Selected parameters allow the operational requirements to be met or exceeded. Sensitivity of the system performance to parameter changes are shown in Figs. 2-11 and 2-13. 70 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-11 AEW SYSTEM LOGIC AND FIXED ELEMENTS A hypothetical AEW system that represents a possible answer to the air defense problem being considered is shown in Figs. 2-15 and 2-16. AEW Position Line Fig. 2-15 AEW Operation Illustrating Azimuth Location and Height-Finding Means and AEW Aircraft Relations to Fleet Center. Two interrelated airborne radars are employed in each AEW aircraft: (1) a fan beam which is rotated through 360°, and (2) a pencil beam which is nodded up and down past the target to measure height. Range, R Targets AEW Radar Azimuth, Height, h AEW Commun. System R,d,h AEW Position GIG Commun. System Systems '1 t AEW Display AEW Navigation System GIG Computer, Data Store and Fighter Direction Center Fighter Direction Commun. System Target Position Target Velocity Target Heading Tactical Display and Decision Tactical Doctrine Fig. 2-16 Early Warning and Vectoring System Information Flow. 2-11] AEW SYSTEM LOGIC AND FIXED ELEMENTS 71 Initial detection of the target is provided by the fan-beam radar. This equipment also measures slant range to the target, R, and target azimuth position 6 with respect to a reference direction. The height-finding radar is positioned in azimuth with information obtained from the fan-beam radar. It measures the elevation angle of the target y with respect to the horizontal. The sine of the target elevation angle, multiplied by the range R and modified by AEW aircraft altitude and an earth curvature correction, provides a measure of target altitude h. The measured target data are displayed in the AEW aircraft for monitoring purposes. The AEW system encodes the measured range, azimuth, and height information and transmits this intelligence to the CIC in the form of a digital message. AEW aircraft position — as obtained from the navigation system — is also transmitted via the digital communications link. The task force is provided with means for ascertaining AEW aircraft position relative to the combat information center (CIC) but with an error dependent upon the specific defense problem. A standard deviation of 1 n.mi. in both the rectangular coordinates is assumed for our analysis, as defined in Fig. 2-3. Several AEW aircraft are employed — the number and disposition will be derived in the succeeding paragraph. The information from all AEW aircraft is presented on a master tactical display in CIC to permit overall battle control and surveillance. Each AEW aircraft measures range and azimuth of all aircraft within its zone of surveillance. Height measurements are made only on the designated targets; the interceptors are commanded to climb to target altitude, so there is no reason (in this example) for measuring the interceptor altitude. CIC System Information Processing. The polar coordinate (R, 6) information gathered by the AEW radars is transformed into a common rectangular (cartesian) coordinate system by the CIC computer to facilitate the generation of target heading and velocity information. Rectangular coordinates have an advantage over polar coordinates because constant- velocity, straight-line flight paths can be represented by x and y velocity components which also remain constant. Thus, if the position, P(/), of a constant velocity straight-line target at any time t is designated in rec- tangular coordinates, then m-lox-{-joy (2-1) where ^, jy = target position in rectangular coordinates at time ( to, jo = unity vectors along the Xo and jo axes of the stationary rectangular coordinate system. 72 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS The instantaneous velocity of this target may be expressed as the time derivative of P(/), or P{t) = V{t) = hx +7or = "i^Vx +]fsVy = constant (2-2) and the position of the target at any time r seconds later can be written Pit + r) = m + rFit) = Ux + tF,) = joiy + tF,). (2-3) Thus, the computation of the velocity components and the prediction of future target position can be done by relatively simple means once the present position information has been transformed to rectangular co- ordinates.^ CIC Command Functions. The position and velocity information computed by the CIC is first used for purposes of assessing the threat on the basis of numbers, position, and velocity. Then, it is employed to com- pute a vectoring guidance course for each interceptor assigned to engage specific target aircraft. The guidance information is transmitted to the interceptor and displayed there by appropriate means. Overall battle control is maintained by CIC using a master tactical display in combination with a pre-established operating doctrine. The tactical doctrine — target assignment, force deployment, etc. — applicable to the threat situation is formulated by the CIC officer and is used to monitor and adjust the processing of information in CIC. The CIC computer also generates commands which are transmitted to the AEW for the purpose of designating targets for the height-finding radars. Vectoring Guidance (Fighter Direction). The type of vectoring guidance employed is dictated by the requirements of the tactical problem and should be uniquely controlled by the weapons system requirements. In the hypothetical example, a high premium was placed on the ability to bring the interceptors into a position to fire their missiles as quickly as possible. In fact, the calculation of the number of interceptors that could engage the threat (40 for 255 n.mi. AEW range) was based on the implied assumption that each interceptor flew in a straight line from fleet center to a point where it could engage its assigned target (see Fig. 2-12). The type of fighter direction best fulfilling this requirement is collision vectoring. Its basic principle is shown in Fig. 2-17. For a target at Pi traveling with velocity Ft and an interceptor at P2 traveling with velocity •''This advantage does not always lead one to choose rectangular coordinates tor the proc- essing of radar information. For example, in Paragraph 1-4 and Fig. 1-4 the use of polar coordinates is indicated. 2-12] AEW DETECTION RANGE REQUIREMENTS 73 Fixed Reference Direction (e.g. North) Fig. 2-17 Collision Vectoring Geometry. Vp, the interceptor will close with the target in the shortest possible time if the following relationship is satisfied: sin L = {Vt IVp) sin Oj (2-4) where each of the variables may be defined from the figure. The quantities, dr, V-r, and Bls are obtained from the AEW and CIC systems and transmitted to the interceptor. A computer in the interceptor uses this information along with its measurement of its own velocity Vp to calculate the proper lead angle L. Then, it adds this angle to the space line-of-sight direction, 6 is, to obtain the desired space heading of the inter- ceptor. The pilot flies the aircraft to maintain this heading and commences to search for the target with his AI radar oriented along the line of sight. Target altitude also is transmitted to the interceptor. The interceptor climbs to this altitude using his altimeter as a reference. In the hypothetical system, vectoring guidance information is transmitted at a rate equal to the scanning rate of the AEW fan-beam radars. Vectoring guidance is continued until the interceptor acquires the target with its own AI radar. The choice of this vectoring technique has a profound effect on the inter- related requirements of the interceptor AI radar and the AEW and CIC systems. A further description of the vectoring problem and the manner by which vectoring errors affect AI radar requirements is contained in Paragraph 2-25. 2-12 AEW DETECTION RANGE REQUIREMENTS The foregoing discussion has established that an early warning range of 255 n.mi. is compatible with the system effectiveness goal and the assumed fixed elements of the problem. It has been assumed that the early warning coverage need be provided only in the area of most likely attack. The operational doctrine established that a calculated risk would be taken that the defined mass attack would 74 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS not, in this case, approach the task force from the side farthest from enemy bases. Carrier deck space and AEW aircraft cycle time limitations dictate that the required coverage be provided by a maximum of 3 AEW aircraft. Another systems consideration governs the choice of detection character- istics of the early warning radars: back-up or overlapping coverage where the loss of an AEW aircraft due to enemy action or equipment failure leaves the task force undefended. The required detection range and AEW aircraft spacing for the assumed system may be analyzed by the simple geometrical model of Fig. 2-18. Range Requirement , Reserve \ AEW #1 / AEW \ AEW #2 -^1 ^ ]^ \ (Nonradiating) ; During Normal Operation Guided Missile Fig. 2-18 Possible AEW Aircraft Detection Range, Coverage, and Disposition to Provide 255-n.mi. Early Warning Range. The arrangement shown represents one possible answer to the hypothetical system requirements. This deployment shows 2 AEW aircraft, each capable of detecting enemy targets at ranges of 150 n.mi. with a 360° search sector. The two operating AEW aircraft are positioned with respect to task force center so that detection occurs at a distance of 255 n.mi. or more from task force center in the directions from which enemy raids are expected. A third AEW aircraft is positioned as shown for use as a back-up or ready replacement for either of the other two aircraft. This aircraft does not radiate during normal operation, in order to make its detection and de- struction by the enemy more difficult. 2-13] AEW TARGET RESOLUTION REQUIREMENTS 75 The configuration developed in this manner is one of many that could be developed as a possible problem solution. An actual study would exam- ine a number of such configurations. This example is chosen to illustrate some of the quantitative and qualitative aspects of the system problems that must be considered in an AEW design. The specification of detection ranges must take account of the un- certainty attending the detection process. For identical tactical situations, detection by radar will take place within a band of possible ranges, such as are shown by the distribution density function in Fig. 2-19. The proba- Probability that Target is Detected before Range Closes to R (Cumulative Probability)- f 1.0 §0.5 Q. Probability that Target is Detected between Rand R+c/R RANGE, R -W-dR Fig. 2-19 Characteristic Forms of Radar Detection Probability Distributions and Cumulative Detection Probability Curves. bility that the target will be at some time detected bejore it closes to a given range R — customarily called the cumulative probability of detection — is the integral of the distribution density function taken from i? to °o ; its usual form is also shown in Fig. 2-19. A preliminary requirements study such as we are performing generally expresses the radar detection requirement in terms of the range for 90 per cent cumulative probability of detection. Accordingly, the detection requirements for the AEW radars may be defined as: Search Sector — 360° azimuth — Sea level to 50,000 ft. Detection Range — 90 per cent cumulative probability of detection of the specified enemy targets at 150 n.mi. 2-13 AEW TARGET RESOLUTION REQUIREMENTS A primary function of the AEW system is to provide an early description of potential targets. The description might include range, bearing, eleva- tion, and number of targets. This information is employed in the threat evaluation phase and, later, to vector interceptors against specific targets. In both phases, the ability of the AEW system to resolve separate target elements is of fundamental importance. 76 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Threat assessment and tactical decision must occur within 3 minutes following initial detection at 255 n.mi. from fleet center. During this time, enough information must be obtained to allow the CIC system to compute an estimate of the position, speed, altitude, heading, and number composition of a potential target complex. The specified targets can travel about 25 n.mi. in 3 minutes; therefore this information must be gathered and processed when the target is at ranges of 125 to 150 n.mi. from the AEW aircraft in order to provide sufficient problem lead time. As a first step, the sensitivity of system performance to the resolution quality of airborne early warning information must be examined. Then the problem of providing the necessary resolution by appropriate AEW radar design parameters may be treated. The defined target complex consists of 20 targets spaced 5 n.mi. apart (see Fig. 2-7). Now, consider the following problem: Suppose that during the threat assessment phase only 10 separate targets are indicated by the AEW radar information (such a condition could be caused by in- sufficient resolution in the AEW system — i.e. a circumstance which could cause two or more targets to appear as only one target on the radar dis- play). What effect does this condition have upon overall system operation ? This question may be answered by considering the effect of this condition upon each phase of the air-defense operation. First of all, the 6 CAP aircraft would be directed to engage the threat elements. Simultaneously, deck-ready interceptors would be launched at the rate of 2 per minute. To ensure high target attrition, tactical doctrine might dictate that at least 2 interceptors be employed for every potential target. This would require launching at least 14 deck-ready aircraft in response to a 10-target threat. Thus, for the first 10 minutes following initial detection (3 minutes delay time plus 7 minutes for launching 14 deck-ready interceptors), the conduct of the air battle would be in no wise different from what would have taken place if all 20 targets had been indicated initially. During this 10-minute interval, the threat will have closed to about 175 n.mi. from fleet center. At this range the 6 CAP aircraft will engage separate elements of the raid (see Fig. 2-12). For these interceptions to be vectored successfully, at least 6 of the separate target elements must be resolved and tracked by this time. In addition, if we assume that the number of deck-ready aircraft kanched is a direct function of the number of known targets, it is necessary to begin to distinguish more than 10 objects by the time the threat has reached 175 n.mi. from fleet center (or 75 n.mi. from the AEW aircraft). In fact, to prevent delay in deck-ready aircraft launchings, the number of targets counted must increase at a minimum rate of 1 per minute until all 20 are separately resolved. 2-13] AEW TARGET RESOLUTION REQUIREMENTS 77 Thus, it is seen that the fact that only 10 of the 20 targets were resolved initially does not in itself degrade system performance. So long as resolution is sufficient to resolve additional targets faster than the interceptor launch- ing rate, system performance will not be affected for the assumed tactical doctrine.^ A further examination would disclose that as few as 5 targets could be indicated by the initial early warning information provided that the subsequent "break-up" of targets was sufficient to keep pace with deck- ready interceptor launch rate. The vectoring phase imposes additional resolution requirements. The assumed tactical doctrine requires that individual interceptors be directed against individual targets. Thus, both targets and interceptors must be separately resolved and tracked in this phase. An inspection of the tactical geometry of Figs. 2-12 and 2-18 discloses that contact between targets and interceptors will take place at ranges that are seldom greater than 75 n.mi. from an AEW aircraft. The majority of contacts will be less than 50 n.mi. from an AEW aircraft. Thus, if the AEW radar resolution and the interceptor tactics are chosen to ensure that substantially all the targets and all the interceptors can be separately resolved at ranges of 75 n.mi. or less from the AEW aircraft, little or no degradation in system performance will result if at least 5 separate targets are indicated at the early warning range (150 n.mi.). Now the foregoing tactical requirements may be translated into radar performance requirements. With a radar, it is possible to measure three quantities directly (see Paragraph 1-4) — range, angle, and velocity along sight-line to target. Resolution between targets may be done on the basis of any or all of these. Fig. 2-20 shows a particularly difficult case that could exist for the hypo- thetical threat. The target threat complex is approaching along a radial line which passes through the AEW aircraft and fleet center. The location of each threat element relative to the AEW aircraft is shown in the ex- panded view. As can be seen, the angular differences between adjacent threat elements are of the order of 4°. The range difference between ad- jacent elements varies from about 2.5 n.mi. for the extreme outer threat elements to less than 1 n.mi. for the central elements. In the case of the two center elements, the range difference is zero. The differences in radial velocity components of adjacent elements vary from about 20 fps for the outer elements to fps for the central elements. From the diagram, it is seen that an angular resolution capability of 4° or less will provide the stipulated tactical capability. However, this is not the only means for meeting the requirement. A range resolution ca- ^This analysis does not consider the possible benefits of finer resolution to the assignment procedure. These might be significant in a practical case and should be taken into account. The analysis of this problem is too complex for consideration in this example. 78 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Target Complex Broken up into Resolution Cells 5° x 1 n.mi. \~ Unresolved ,Z~Zr7-r~c^ 20 Targets Spaced 5 n.mi., Apart Resolved into 17 Elements 75 n.mi. ' / ^^ Resolution Cell I / 5° X 1 n.mi. I / I / I / I / I / I / I / ^5° Azimuth Beamwidth 12-Msec Pulsewidth = l n.mi. AEW Aircraft To Fleet Center Fig. 2-20 AEW Radar Resolution Capacity at 75 n. pability of 0.5 n.mi. would permit resolution of 19 out of the 20 targets at 75 n.mi. range. (The two center targets would appear as one if range resolution were used.) At 150 n.mi., 15 targets would be seen. Similarly, various combinations of range and angular resolution capa- bilities may be employed. For example, the diagram shows that an angular resolution capability of 5° coupled with a range resolution of 1 n.mi. will allow resolution of 18 of the 20 targets at 75 n.mi. range. At 150 n.mi., 15 separate targets will be indicated. Several other factors must be considered in a practical treatment of the resolution problem. The individual threat elements will be unable to main- tain perfect station-keeping with respect to each other. Errors in relative heading, velocity, and position will exist at any given time. This will cause the actual target positions and velocities to be distributed around the values shown in Fig. 2-20. We shall assume that these errors are small relative to the size of a resolution element for the example. However, if these errors were of the order of magnitude of a resolution element, they could cause substantial modification of the tactical resolution capability. 2-14] INTERRELATIONS OF AEW, CIC, INTERCEPTOR SYSTEMS 79 Resolution of the interceptors may be accomplished by a combination of tactical doctrine and AEW radar resolution capability. For example, the deck-launched interceptors are launched at the rate of 2 per minute. Thus, between successive pairs of interceptors there is a range difference of about 12 miles. Each pair of interceptors may be instructed to maintain a given relative spacing (e.g., 5 mi. or more). Assignment doctrine, in turn, must be adjusted to be compatible with these tactics. If these steps are taken, the AEW radar resolution capability dictated by the threat will also be adequate for resolution of the separate interceptors. For an actual AEW design problem, many combinations of range and angular resolution would be examined for a number of different threat con- figurations and approach geometries. Such analysis would serve to place upper bounds on the required resolution capability and would establish the allowable trade-offs between range and angular resolution for the particular system problem. The principles and types of reasoning used for the single case examined in this paragraph could be employed for the more compre- hensive analysis required for an actual design. In the example problem, it was seen that angular resolution capabilities of less than 5° and range resolutions of 0.5 to 1.0 n.mi. represented potentially useful ranges of values. The final choice will depend upon the influence of other functions and problems of the AEW radar system design. 2-14 INTERRELATIONS OF THE AEW SYSTEM, THE CIC SYSTEM, THE INTERCEPTOR SYSTEM, AND THE TACTICAL PROBLEM Target tracking will follow detection. The tracking information is first utilized for threat assessment; later, tracking of both targets and interceptors provides the information needed for fighter direction. Three interrelated characteristics of the AEW /CIC system are of funda- mental importance in determining the contributions of this system to overall mission accomplishment. (1) Detection range (2) Accuracy (3) Information handling capacity (number of separate tracks, etc.) In Paragraph 2-9, the detection range was found to be one of the critical factors in determining the level of mission accomplishment along with the individual interceptor effectiveness and the number of interceptors available for defense. Implicit in the analysis, however, were the assumptions that AEW /CIC system accuracy or data-handling capacity did not limit over- all system performance. We must now determine the specific characteristics that the AEW /CIC system must possess to make these assumptions valid. 80 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-15 ACCURACY OF THE PROVISIONAL AEW SYSTEM For the hypothetical problem under consideration, there are 20 targets and 40 interceptors — all of which could conceivably be in the zone of coverage of a single AEW aircraft. Thus each AEW aircraft must be capable of keeping track of 60 objects. Height measurements must be made on a maximum of 20 objects (targets only). One facet of the accuracy problem — resolution — and its relation to the overall system problem has already been discussed in Paragraph 2-13. In addition to separating the 60 objects, the AEW/CIC system must also track each object, i.e. determine its position relative to some reference coordinate system and — for each of the attacking aircraft — its heading, velocity, and altitude. As has been described, this information is utilized to direct specific interceptors on collision courses with specific targets. The required accuracy of this guidance depends upon the characteristics of the interceptor system, particulary upon the AI radar and fire-control system. The accuracy of the AEW/CIC system determines the accuracy with which the interceptors can be vectored, and the vectoring error in turn determines the required lock-on range of the AI radar. This last factor is a critical item and may be severely limited by fixed elements of the problem and use environment. Thus the AEW accuracy can only be firmly specified after a study of the vectoring problem has determined the trade-off relation between vectoring error and the required AI lock-on range.'' Unfortunately, because of the complex interrelations between AEW/CIC system errors and vectoring errors, the analysis in Paragraphs 2-22 to 2-28 cannot be made abstractly but will require, as inputs, provisional assump- tions of the AEW /CIC system design and accuracy. Thus, in this and some of the following paragraphs, we shall assume tentative values for the required AEW/CIC system accuracy and carry on our study of the preliminary design of the AEW radar on the basis of these assumptions. We should bear in mind, however, that these provisional values may lead to an un- acceptable requirement for the AI lock-on range, in which case the analysis would have to be repeated for a modified AEW/CIC system design. The AEW radar measures the relative position — azimuth and range — of a target with respect to itself once per revolution of the fan beam. The accuracy of each measurement, as it is seen in CIC, is limited by a number of factors. The most significant of these are: 1. Beamwidth 3. Data quantization 2. Range aperture (pulse length) 4. Data stabilization 5. Time delay errors ''An illustrative analysis of this kind has been carried out in Paragraphs 4-6 and 4-7 of Merrill, Greenberg, and Helmholz, of), cit. 2-15] ACCURACY OF THE PROVISIONAL AEW SYSTEM 81 Measurement Errors due to Beamwidth and Range Aperture. Referring to Fig. 2-21, if a large number of individual measurements were made on a target at point T, the measured values could be plotted as prob- ability density distributions of azimuth and range values about the point T. For any single measurement, the indicated target position might lie anywhere in the region encompassed by these distributions, e.g., the point A indicated in the figure. T = Actual Position of Target A= Measured Position of Target t= Azimuth Measurement Error of AEW Radar e„ = Range Measurement Error of AEW a^= Standard Deviation of Azimuth Measurement Errors of AEW Radar 0), = Standard Deviation of Azimuth Measurement Errors of AEW Radar R = Range from AEW to Target T AEW Radar Location Fig. 2-21 Representation of AEW Measurement Errors. It is convenient to describe the measurement errors by their standard deviations or root mean square (rms) errors. The magnitudes of these rms errors are closely related to the resolution capabilities previously dis- cussed; however, the reader should be careful not to confuse the resolution of two targets and the accuracy in tracking one. Accuracy describes the radar's ability to measure the position of a single target; as a rough approximation the standard deviation of the errors of a single measurement of target position may be considered to be about one- quarter of the resolution capability. Actually, besides being related to the beamwidth, the measurement error is a function of the signal-to-noise ratio and the number of hits per beamwidth. An analysis of these relations is given in Paragraph 5-1 1 . As an example, with a 5° azimuth beamwidth and a pulse width of 12 Msec corresponding to 1 n.mi. (i.e. resolution capabilities shown to be adequate in the preceding paragraph), the approximate ac- curacy of a position measurement for a single scan will have rms values of (5)(0.25)(150) 57.3 3.3 n.mi. (at 150 n.mi. range) (7 A = 1-65 n.mi. (at 75 n.mi. range) (2-5) 82 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS (Tff = (0.25) (164/) = (0.25) (164) (12) = 494 yd = 0.25 n.mi. (independent of range). Several factors act to make the effective errors somewhat larger than the basic errors given in Equation 2-5. Ouantization Errors. Errors may be introduced by the process of quantizing or "rounding off" measured values of range and angle. This is done both at the AEW radar and in the CIC computer in order to minimize the amount of data which must be processed and transmitted over the as- sociated data links. The process may be visualized as follows. The space around a reference point (in this case either the AEW aircraft or the CIC) is broken into "cells" of arbitrary size and shape. An object anywhere in one of the cells is as- signed a position description corresponding to the position of the center of the cell. This process introduces rms errors which are approximately one- quarter of the cell dimensions. These errors are independent of the meas- urement errors. Thus, if the quantization level (cell size) of the AEW system is chosen to be equal to the rms measurement errors (1.25° and 0.25 n.mi. in our case), the equivalent rms error in the AEW data will be in- creased by only about 4 per cent. The insensitivity of the equivalent errors to the quantization level of the foregoing example shows that coarser range quantization could be employed if desired. For example, if the space around CIC were broken into cells 1 mi. on a side, the equivalent rms range error would be increased by about 40 per cent to a value of 0.35 n.mi. when the polar data from the AEW system were transformed to rectangular coordinates in CIC. Because the angle measurement is considerably coarser than the range measurement, the 1-n.mi. CIC data quantization cells would make almost no contribution to the rms angular error data received from the AEW air- craft. Accordingly, the following quantization levels may be chosen as reasonable: AEW: Azimuth 1.25° Range 0.25 n.mi. CIC: X coordinate 1 n.mi. Y coordinate 1 n.mi. These levels are compatible with a range resolution requirement of 1 n.mi. (Paragraph 2-13). If finer range resolution were to be employed, the CIC quantization levels would have to be reduced accordingly. Stabilization Errors. Another important possible source of error is the rolling and pitching motion of the AEW aircraft due to maneuvers and wind gusts. It was required (Paragraph 2-11 and Fig. 2-15) that AEW 2-15] ACCURACY OF THE PROVISIONAL AEW SYSTEM 83 measurements be referenced to a fixed angular coordinate reference system. Errors may be induced in the azimuth measurements by aircraft motions if the AEW radar is not space stabilized. Therefore, the preliminary design must consider how AEW aircraft motions affect the AEW radar system measurements. If the effects are substantial, means must be provided for correcting the errors thus introduced. For the purpose of this analysis it will be initially assumed that stabilization errors do not degrade the angular accuracy by more than 10 per cent. The effect of this assumption may be examined in more detail when the operation of the system has been more completely analyzed and understood. Time Delay Errors. A possible source of additional position error on a moving target is the fact that time may elapse between the measurement of target position by the AEW radar and the registration and use of this information in the CIC. If the time delay is td, the amount of position error is simply e = Vtd n.mi. where V = velocity of object being tracked. Since the data-handling system must process information at least as fast as it is coming into the system, the maximum possible value of the time delay would be approximately equal to the time, tsc, for the AEW fan beam to make a 360° scan. For example, if the scan time were 6 seconds, the maximum error against an 800 fps target caused by time delay would be approximately 0.8 n.mi. For 1200 fps interceptors, this error would be 50 per cent larger or about 1.2 n.mi. Three courses of action are open to the designer with respect to this error. (1) The error may be tolerated if it does not appreciably affect system performance. (2) Scan speed and data processing speed may be increased to reduce the error magnitude to an acceptable level. (3) The position information may be up-dated by using estimates of velocity and heading of the object being tracked along with a knowledge of the time delay, to produce a term which cancels the time-delay error. For preliminary design of the overall AEW system, it will be assumed that time-delay error does not increase the total position error by more than 10 per cent. The effects of this assumption upon system operation and the detailed requirements of AEW radar can be examined when more is under- stood about the interrelationships among various parameters of the air defense system. At that time a decision can be made about the course of action to be taken to correct the time-delay error. Summary of Assumed Accuracy Characteristics. For purposes of analysis, the AEW^ radar is assumed to have the following characteristics: 84 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Beamwidth 5° Pulse length 12 /xsec Scan time 10 sec These characteristics coupled with assumed values for quantization levels in the AEW and the CIC systems and the assumed limits for stabilization errors and time-delay errors lead to the following accuracy characteristics for the provisional AEW system. a A = (3.3) X (1.04) X (1.10) X (1.10) = 4.15 n. mi.rms (at 150 n.mi. range) CA = (1.65) X (1.04) X (1.10) X (1.10) = 2.07 n.mi. rms (at 75 n.mi. range) aR = (0.25) X (1.04) X (1.41) X (1.10) = 0.41 n. mi.rms (unaffected by maneu- vers and range). (2-7) The total rms position errors may be expressed as the vector sum of the range and azimuth errors or <TT = (c7a' + (TR^y = 4.17 n. mi. (150 n.mi. range) = 2.09 n. mi. (75 n.mi. range). (2-8) One source of error — the 1 n.mi. rms navigation error of the AEW aircraft (see Paragraph 2-11) — has not been included in this analysis. This error is not significant when all of the target and interceptor tracking data come from a single AEW aircraft and when the navigation error changes vary slowly with time. When these conditions prevail, each piece of data in CIC will be biased by the same error. The relative errors between pieces of data are therefore unaffected. As we shall see, it is these relative errors that determine tracking and vectoring accuracy. The navigation error does become important for targets and interceptors which are tracked by both AEW radar aircraft. Such an overlapping zone is shown in Fig. 2-18. A large navigational error would complicate the problem of correlating data from the same target. However, since the navigation error is less than the measurement errors and less than aircraft separation distances, no great amount of difficulty can be expected for this hypothetical case. 2-16 INFORMATION-HANDLING CAPACITY OF THE PROVISIONAL AEW SYSTEM An important aspect of system design relates to its data-handling capabilities. Both the data link for transmitting information between the 2-17] VELOCITY AND HEADING ESTIMATES 85 AEW aircraft and the CIC and the data-processing computer at the CIC will have limited capacities for handling data. The maximum per channel capacity of the data link has been specified (Fig. 2-3) as 1000 bits^ per second. There will be 60 objects (40 interceptors and 20 targets) in the field of the AEW radar. To identify each object requires 6 bits as shown in the footnote. The azimuth location of each target is determined to the nearest multiple of 1.25°. This will require 9 bits per object. Range information to the nearest 0.25 n.mi. from zero to 150 n.mi. requires 10 bits per object. If we add these items and multiply by 60, we find that the amount of information needed to specify the range and azimuth of the 60 objects on a single scan of the radar is 1500 bits. In addition, the elevation of the 20 targets must be determined. The accuracy and quantization level of the target altitude has not yet been specified. Here, we shall assume that target altitude is determined to the nearest 0.25 n.mi. = 1500 ft, the same as in range. To specify a target altitude from zero to 50,000 ft, then, requires 6 bits, and all the altitude data for 20 targets comprise 120 bits. The total information load on one scan, then, is 1500 + 120 = 1620 bits. In order to incorporate self-checking codes and message redundancy in the data link to increase reliability, this figure should be about doubled. Thus, in a round figure, the data link must transmit about 3000 bits per scan to the CIC. The actual information rate will, of course, depend upon the scan time. It is generally desirable to make the scan time relatively short in order to increase the accuracy of the heading and velocity estimates. A study in Chapter 3 indicates that the cumulative detection range tends to be relatively independent of scan time, although a broad optimum may exist. Yet the scan time cannot be made indefinitely small, because of the limitations of mechanical design and the increase in the data rate. We have chosen a provisional scan time of 6 seconds for the basic AEW radar. This radar then scans at a rate of 60° /sec. The information rate which the data link must handle is 500 bits /sec. This figure is well within the capacity of the defined data link system. 2-17 VELOCITY AND HEADING ESTIMATES The position data are used in the CIC to compute estimates of target heading and velocity. This may be done in a variety of ways. One of the simplest can be illustrated with the aid of Fig. 2-22. ^A "bit" represents a binary digit, i.e. either zero or one. Transformations from decimal to binary are made in the following manner: the number 60, for example, may be expressed 60 = (1 X 25) + (1 X 24) + (1 X 23) + (1 X 22) + (0 X 2^) + (0 X 20) In binary form, the number 60 is the six-digit number formed by the multipliers of the powers of2: 60 decimal = 111100 binary. 86 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS ; Target Positions for Eacii Look" Fig. 2-22 Positk CIC Location Data Used in Computing Esti Velocity. lates of Target Heading and Assume that the specified 800-fps target is initially detected at point 1. Since the assumed AEW radar scanning time is 6 seconds, the next look at the target will occur when it reaches position 2. It will be at position 3 on the third look, and so on. At the end of seven looks, the target will be at position 7. The position measurement on each look is characterized by an assumed radial error with a standard deviation of or. This error may be broken into components parallel and normal to the target path, where the standard deviation of each component is^ ap = ot/VS n.mi. o"7v = ot/V2 n.mi. (2-9) (2-10) If the errors of each position measurement are assumed to be independ- ent, the relative errors between any two measurements have the standard deviations (TPIP2 — v(ty-(iy= ar n.mi. (2-11) (yN\N2 = or n.mi. (2-12) A very simple procedure for determining the target velocity and heading can be based on the extreme position measurements. The estimated ^Breaking the errors into equal components ignores the influence of the fact that the range and azimuth errors for a given target measurement are markedly different (Equations 2-7 and 2-8). If this factor is considered, the mathematical complexity of the problem is greatly increased; the final answer expressing the probable position errors for any randomly chosen target position relative to the CIC is not changed substantially. (TP1P2 ntsc _ ar ~ ntsc 0'N1N2 _ 0T_ l^isc ntsc 2-17] VELOCITY AND HEADING ESTIMATES 87 velocity will be the difference of the extreme measurements divided by the total time, ni^c, where n is the number of scans and 4c is the scan time. The estimated direction is simply that determined by the two extreme measure- ments. The velocity and heading errors are expressed in terms of the parallel and normal components of the relative position errors. Thus (2-13) '^^ = ^7^ = — (2-14) where F = true velocity of object being tracked. For example, assume that position measurements are made on an 800-fps target 75 n.mi. from the AEW aircraft. This range represents a likely value of the maximum range at which accurate tracking will be required for the generation of vectoring information as explained in Paragraph 2-13. With the assumed parameters of the provisional AEW system, the total position error of a single measurement was derived to be Equation 2-8: ar = 2.09 n.mi. = 2 n.mi. (2-15) Thus, the standard deviations of the two components of the relative error between two measurements are from Equations 2-11 and 2-12: ap.Po - 2 n.mi. (rms) = 12,000 ft (rms) (2-16) (^NiNi = 2 n.mi. (rms) = 12,000 ft (rms). (2-17) Thus the rms errors in the estimated velocity and heading are calculated by Equations 2-13 and 2-14 to be av = 12,000/(6) (10) = 200 fps (rms) (2-18) cT^ = 12,000/(800) (6) (10) = 0.25 rad = 14.5° (rms). (2-19) Accordingly, we see that the accuracy of the velocity and heading estimates depends upon the following factors: 1. Accuracy of each position measurement 2. Number of position measurements used for estimates of velocity and heading 3. Elapsed time between position measurements 4. Velocity of object being tracked In addition, accelerations of the object being tracked during the time interval, ntsc, also give rise to additional errors in the position and velocity estimates. 88 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS There are several means for improving the tracking accuracy. Each of these involves a trade-off between improved accuracy and greater informa- tion-handling complexity. For instance, in the simple case we have just been discussing, when only the extreme position measurements are used all the information associated with interior measurements is lost. If all the position measurements were used to determine the velocity and heading estimates, the errors would be substantially smaller. Thus, the error estimates in Equations 2-18 and 2-19 are somewhat pessimistic. In some cases, a trade-off between maneuvering and nonmaneuvering target track- ing accuracy also is required. The velocity and heading estimates may be used in several ways. First of all, this information is used to compute vectoring guidance for the interceptors. The collision vectoring equation (Equation 2-4) illustrates a typical application. Part of the vectoring problem involves prediction of the future positions of the targets and interceptors. Prediction for a single scan is also used to update the position information. An example of such a prediction process is shown in Fig. 2-23. The AEW/CIC system indicates target position as point A. The velocity and heading estimates are used to generate a track AA^ On the next scan, target position is indicated as point B. The position data are corrected to this point and a new continuous track BB^ is estimated, etc. Thus, at any time between measurements a A,6,C,D = Measured Target Positions a\b\c\d^ = Estimated Target Positions f^c Seconds After Measurements of A,6,C,D Respectively Fig. 2-23 Prediction Process Employing Scan-to-Scan Correction of Position Data. position measurement is available which accounts for the change in target position since the last measurement was made. This type of information processing (updating) greatly reduces the time-delay error discussed in Paragraph 2-15. 2-11 AEW RADAR BEAMWIDTH AS DICTATED BY PROBLEM 89 The updating process has another advantage. It produces an estimate of target position on the next scan (see Fig. 2-23). This estimate greatly facilitates the problem of maintaining the identity of a target from scan to scan because it provides a better idea of where each target is going to be the next time the AEW radar looks at it. The heading and velocity information is used to obtain these predictions. Prediction must be paid for, though, and the longer the prediction time, the larger the errors in the predicted position. Fig. 2-24 illustrates how the Error in Present Position Errors in Future Positions -• 8n.mi. — ^- Sn.mi. — A Vj = 800 f ps a, - 200 fps 2n.mi. Fig. 2-24 Growth of Position Error with Prediction Time. indeterminacy volume of the predicted position expands with the prediction time. This figure was determined on the basis of the following expressions for the future parallel and normal rms errors app and aNF in terms of prediction time T and the present position, velocity, and heading errors. (2-20) (2-21) (JPF ctnf + av'T- A&M^V^^^^ With the same target velocity and system characteristics used previously, the position error expands from 2 n.mi. rms to 4.46 n.mi. rms with a pre- diction time of 2 minutes. 2-18 AEW RADAR BEAMWIDTH AS DICTATED BY THE TACTICAL PROBLEM On the basis of target resolution requirements (Paragraph 2-13) a value of 5° was selected for the fan beamwidth of the provisional AEW radar design. Subsequent estimates of accuracy and information handling characteristics were based on this value (Paragraphs 2-15 to 2-17). 90 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS The selected value of 5° was representative of a likely allowable upper limit for the AEW radar design on the basis of resolution. The use of larger beamwidths would complicate the problem of target resolution since the angular beamwidth would then be appreciably larger than the angular separation of the targets (Fig. 2-20) at the maximum vectoring range (75 n.mi.). Since the system accuracy is almost a direct linear function of beamwidth, the estimated accuracy of the provisional AEW design represents the poorest that might be obtained from a potentially suitable AEW design. Thus the accuracy performance characteristics of the provisional AEW design will tend to place the most severe requirements on the interceptor system. If the interceptor system can be built to meet these requirements, the same interceptor system will be more than adequate for smaller values of AEW radar beamwidth. On the other hand, if the selected value of 5° makes AI radar requirements unreasonable, the maximum permissible AEW radar beamwidth may have to be reduced. The objection to a reduced beamwidth is the larger antenna which it entails and the penalty thus imposed upon the AEW aircraft. In this chapter, only the interrelationships of AEW radar beamwidth with the tactical problem are discussed. As will be seen in Chapter 3, radar beamwidth also enjoys close interrelationships with other parameters and performance characteristics of the radar system. Among these are (1) detection range, (2) information rate, (3) operating frequency, (4) antenna size, and (5) stabilization requirements. In addition, AEW radar beamwidth affects the response of the radar system to electromagnetic disturbances arising in the tactical operating environment. Enemy countermeasures, radar returns from clouds and ground, and radiations from other AEW aircraft are representative of such phenomena. Strictly speaking, the consideration of these factors should be made at the same time as the resolution and accuracy requirements studies since they are an important part of the AEW radar's relationship with the overall tactical problem. For simplicity, the discussion of these factors is deferred until Chapter 14 because a knowledge of radar techniques and propagation phenomena is necessary to make such a discussion mean- ingful. To summarize, then, AEW radar fan beamwidth is dictated by three primary tactical considerations: resolution, vectoring accuracy, and inter- action with electromagnetic disturbances. Resolution considerations have been shown to dictate a value of about 5° or less. Vectoring accuracy requirements are unknown at the present time. In order to proceed with the problem, the vectoring accuracy obtainable with a 5° beam will be used. Subsequent analysis of the AEW and AI systems will disclose whether vectoring accuracy dictates a narrower beam. Electromagnetic disturbance 2-18] AEW RADAR BEAMWIDTH AS DICTATED BY PROBLEM 91 effects are unknown. The details of this problem will be largely neglected in the development of system requirements in this chapter. Vertical Beamwidth. Vertical beamwidth also is an important factor. The AEW radar must detect and track the specified 50,000 ft altitude targets. It should also have a capability for detecting and tracking targets at all other reasonable values of altitude, since the specified threat could not be considered realistic if there were significant holes in the early warning coverage at other altitudes which could be exploited by the enemy. The characteristics of the threat determine the required vertical coverage. If it is assumed that the primary threat (Mach 0.8, 50,000 ft) could also attack from lower altitudes — for example, 10,000 to 50,000 ft — then, AEW coverage must be provided over this range of altitudes. The coverage must be sufficient that targets are not lost for appreciable periods of time. For example. Fig. 2-25 shows that vertical coverage of 45° upward and 18.3° Altitude of Primary Specified ^Threat 50,000 - Possible Altitude Range of Targets 10,000 - 50,000 ft -10 -5n.mi. 5 10 RANGE FROM AEW AIRCRAFT - n.mi. Fig. 2-25 AEW Vertical Coverage Diagram — Example. downward can create a zone 10 n.mi. in diameter where the primary target (50,000 ft, 800 fps) can be lost from view. In the worst case, this would involve loss of the 800-fps target for a period of slightly greater than 1 minute. With the assumed target spacing — 5 n.mi. — a maximum of two targets would be within this zone at any one time. By the time targets enter this zone, the estimates of their velocity and heading have been obtained quite accurately since they have been under surveillance for almost 150 n.mi. These estimates may be used to update the target position during the blind time, thereby reducing the effect of the blind zone on system performance. Moreover, the tracking of objects entering the zone is being done at very short ranges, and this greatly improves the position accuracy of the data obtained just before the target enters the zone. On these bases, it is reasonable to assume that dead zones of the order of 10 n.mi. do not sensitively affect system performance, since surveillance is lost for a relatively short time. Thus, vertical coverages 92 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS of the order of that indicated in Fig. 2-25 should be adequate for the tactical problem. One other problem related to vertical beamwidth is of great importance to the AEW problem, namely ground and sea return. Fig. 2-25 shows that the fan beam intersects the surface of the water — or land, as the case may- be — at all ranges greater than 10 n.mi. Thus reflections from the ground will compete with target signal reflections at all ranges greater than 10 n.mi. This fact requires that means be provided in the AEW system to distinguish between returns from the surface of the earth and target returns. Interactions between the radar, the target, and the ground constitute a very complicated problem. The polarization of the radar transmission, surface characteristics, and AEW and target altitudes all interrelate to produce nulls and reinforcements which influence the system capability. These factors are discussed in some detail in Chapter 4. In an actual design study, the quantitative aspects of this problem should be carefully studied and set forth at this point of the systems requirements development. The interrelations of the tactical geometry and propagation and scattering characteristics must be ascertained to define the magnitude of the problem implied by the requirements for distinguishing between ground and target returns. 2-19 FACTORS AFFECTING HEIGHT-FINDING RADAR REQUIREMENTS The height-finding radar for the example problem is positioned in the aircraft nose. It is directed to point in a given azimuth direction at a target located by the fan-beam radar. It is then nodded up and down to determine the elevation of the target with respect to a horizontal reference in the AEW aircraft. The nodding action causes the target return to vary as a function of the space (or angle) modulation characteristic (elevation) of the height- finding beam. The particular type of space modulation characteristic that is used depends upon the accuracy requirements of the height finder. The requirements of the height-finding radar are dictated primarily by the following tactical considerations: 1. The characteristics of the expected threat including possible varia- tions from the specified values. These characteristics include speed, altitude, and number of aircraft. 2. Height-finding requirements during threat evaluation. 3. Height-finding requirements during vectoring. 4. Height-finding requirements dictated by the need to supply early information to the ground-to-air missile system. To meet these requirements within the limitations of the hypothesized 2-19] FACTORS AFFECTING HEIGHT-FINDING RADAR 93 system logic, the height finder must operate with azimuth input commands obtained from the AEW fan-beam radar. The possible dimensions of the height-finding radar are limited by the dimensions of the AEW aircraft nose. For purposes of specification, it will be assumed that the aircraft in the example may accommodate an antenna with a maximum dimension of 3 ft. It is also assumed that such an antenna may be gimbaled so as to be capable of performing the height-finding function on objects within a ±80° horizontal zone around the AEW aircraft's nose. The vertical coverage of the height finder must be matched to the primary vertical pattern coverage of the fan-beam azimuth search system. The assumed placement of the height-finder places a limitation on tactical usage. To evaluate a target the AEW aircraft must be pointed within ±80° of the line of sight to the object whose height is being measured. Thus, by virtue of the assumed system logic, the AEW aircraft must maneuver to perform its mission. The required maneuver must be within the performance characteristics of the AEW aircraft. In addition, the effect of the maneuver upon the stabilization problem must be evaluated. Requirements Dictated by Threat Evaluation. Height-finding information need not be obtained at the same rate as position information for the specified threat, since its altitude does not change during the attack. In fact it need be measured only once during the specified attack. Once again, the possibility of other attack situations must be considered. If the enemy aircraft were capable of making an abrupt altitude change during the attack, the height-finding system must be able to detect such a change in time for appropriate defensive measures to be taken. Immediately following detection, the height-finding radar is required to begin measuring target altitudes for purposes of raid evaluation. Ideally, the evaluation of target altitudes should take place within the time allowed for threat evaluation (assumed to be 3 minutes in the example). If we allow an average time of 1 minute^" for the AEW aircraft turning to face the raid, a total time of 2 minutes is available to measure target altitudes lOThe AEW aircraft speed is 200 knots. At this speed, and at a bank angle of 10°, the AEW aircraft can turn at the rate of about l°/sec. where F = velocity in knots, </> = roll angle (degrees), andi/' = horizontal turning rate (°/sec). Since the height-finding radar coverage is ±80° or 160°, the maximum turn required to bring a target under height-finder surveillance is 100°. Thus, 100 seconds would be required in the worst case for a 10° bank angle. On the average less than half this time would be required, since the orientation of the AEW aircraft relative to the raid is random. Thus, the assumption of 1 minute does not imply extreme maneuvers by the AEW aircraft. 94 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS and transmit this information to CIC for decision and target assignment. Since there are 20 targets, a maximum time of 6 seconds per target is permissible. During this phase, it is sufficient to know whether the targets are high, medium, or low altitude. Requirements Dictated by Vectoring. During vectoring, inter- ceptors are vectored to the measured altitudes of the assigned targets (for the assumed system logic of the hypothetical example). A height-finding error can limit system performance in the following ways. 1. It can cause the interceptor to fly at an unnecessarily high altitude, thereby degrading speed and maneuvering capability. 2. It can cause the interceptor to approach the target with an altitude differential which its weapon (assumed to be a guided missile) cannot overcome. 3. It increases the zone of probable target positions which must be searched by the AI radar. The first limitation can be attenuated somewhat by the use of tactical doctrine based on a prior knowledge of threat characteristics. For example, if the probable threats are known to have a performance ceiling of 50,000 ft, there would be little point in directing the interceptor to fly at 60,000 ft even though the height finder indicated such an altitude. The second limitation must be related to weapon characteristics and aircraft and fire-control system characteristics. An inspection of the missile performance (Fig. 2-6) shows that the weapon can itself correct substantial altitude errors by its maneuvering capability. For a weapon traVel of 3.2 n.mi. or more, altitude errors to 2 n.mi. can be corrected if the weapon is fired horizontally. A further attenuation of the effects of altitude can be obtained from the fire-control system. Following AI radar lock-on, the pilot obtains a reasonably precise measurement of relative target elevation. This may be used by the fire-control system to point the aircraft up or down as required to eliminate an elevation error. Of course, the required climb or dive angle must be compatible with aircraft performance character- istics. The third limitation — the required AI radar search zone needed to encompass the height-finding inaccuracies — is also most important. As will be demonstrated later, the range performance of a radar system is strongly influenced by the volume it must search. On the basis of these considerations, a height-finding error of approxi- mately 0,5 n.mi. (3000 ft) standard deviation at a range of 75 n.mi. represents a reasonable first approximation to the height-finding accuracy requirement. This corresponds to a maximum error of about 1.5 n.mi. — a value which is still within the guided missile performance capabilities. 2-19] FACTORS AFFECTING HEIGHT-FINDING RADAR 95 It will be assumed that the same height-finding information rate (one measurement on each target aircraft every 2 minutes) will be maintained during vectoring. Requirements Dictated by the Surface-to-Air Missile System. Height-finding data can be used to direct the search and tracking system associated with the surface-to-air missiles to those regions of the airspace where targets are most likely. For the system of the example, such informa- tion can be most useful, since the primary target for the ground-to-air missile system is a missile launched from the hostile aircraft at a range of about 50 n.mi. The relatively smaller size of the missile makes knowledge of where to look for it most desirable. For such an operation, provision must be made for the proper transfer of data within the CIC system. It is not likely, however, that the requirements of this function are more severe than the interceptor vectoring height-finding requirements. For the purposes of the example, this will be assumed to be the case. Once again, this is an area which deserves more detailed scrutiny in an actual design study. Requirements Dictated by the Stabilization Problem. Height- finding — even when the requirements are as coarse as indicated for the hypothetical problem — involves measuring rather small angles. The assumed system logic requires that elevation angle of the target be meas- ured with respect to the horizontal plane. In addition, the height-finder must be commanded to the measured space azimuth position of a particular target. Some idea of the problem may be obtained by translating the derived 0.5-n.mi. rms interceptor vectoring height-finding requirement into an equivalent angle for 75 n.mi. range. This angle may be expressed ^, = ^ = 0.067 rad = 0.38° (rms). (2-22) This is a total error — including the accuracy of the radar, the stabiliza- tion errors, mechanization errors, and quantization errors. If the latter two errors are assumed negligible and if the stabilization and radar meas- urement rms errors {au and dhm respectively) are assumed equal, normally distributed, and independent then c^a™ = <r,, = 0.38/V2 = 0.27°rms. (2-23) Thus, to meet the height-finding requirement of 75 n.mi., the height- finding system must be stabilized to within 0.27° of true vertical. This accuracy must be maintained despite aircraft pitching or rolling motions. In addition, the azimuth beamwidth of the height-finding radar must be large enough to include the uncertainty of the azimuth fan beam. It should 96 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS not be appreciably larger than that of the fan beam or it will have difficulty resolving between adjacent targets that are resolved by the fan beam. Thus, as a first approximation, the azimuth beamwidths of the height finder and the fan-beam radar may be made approximately equal. The elevation beamwidth depends upon the required accuracy. An approximation may be obtained by using the same expression employed for fan-beam azimuth accuracy (Paragraph 2-15). Cn = 6n/4 degrees (rms) for a single measurement (2-24) an = Qnl'^yln degrees (rms) for n measurements (2-25) where n = number of measurements averaged to obtain a single estimate 9n = elevation beamwidth of height-finding radar. Since 6 seconds can be taken for the height-finding measurement, it is reasonable to assume that five to ten separate measurements could be made. If, e.g., nine measurements are made, the required beamwidth is found by a manipulation of the above equation as e„ = 4V^ c7„ = (4) (3) (0.27) = 3.2°. (2-26) Actually, techniques known as beam splitting can be employed to obtain greater angular accuracy than is implied by Equation 2-26. Accordingly, the derived result is only one of the possible solutions to the height-finding problem. 2-20 SUMMARY OF AEW SYSTEM REQUIREMENTS The preceding discussions have shown some major considerations involved in the design of a typical AEW system. Numerical examples illustrated the various interrelations and were chosen in such a manner as to be applicable to the solution of the hypothetical air defense problem we have been considering. We may now use all of this information to compile an estimate of the basic characteristics of an AEW system which represents a reasonable answer to the overall system problem. These estimated characteristics may then be employed to provide the basic input data needed to specify the AI radar and fire-control system. All during this process, we estimate — as best we can — the overall system performance to ensure that we do not depart from the mission accomplishment objectives. As already mentioned, in an actual overall systems study, we would repeat this process several times to obtain a better feeling for the trade-offs between the AEW system and the AI system. However, for all cases, the basic considerations and the method of attack on the problem would remain very much the same; only the assumed system logic and specific parameter values would undergo appreciable change. 2-20] SUMMARY OF AEW SYSTEM REQUIREMENTS 97 One important characteristic of a systems problem has been implied by the foregoing discussion; it is worth mentioning explicitly at this point to impress the reader with its importance. In a systems study designed to derive basic system requirements, it is often necessary to make arbitrary decisions on the basis of incomplete quantitative results. The system logic described in Paragraph 2-1 1 for the hypothetical AEW system is an example of such a decision. Many choices could have been made; however, in order to get on with the problem one choice had to be made and then followed to its logical conclusion. Conceivably, it would develop that this was the wrong choice, in which case we should have to repeat the entire process for a more satisfactory initial hypothesis. Keeping in mind the provisional nature of a system specification at this stage of the analysis, we may specify the basic parameters of the AEW system as in Table 2-2. The number of the paragraph which discusses each parameter is included for convenience. Table 2-2 TENTATIVE AEW RADAR PARAMETERS Detection Range 90 per cent probability of detection at 150 n.mi. (Paragraph 2-12) Number of Targets 20 hostile targets, 40 interceptors (Paragraph 2-13) Threat Evaluation Range 125-150 n.mi. (Paragraph 2-11) Nominal Vectoring Range 75 n.mi. (Paragraph 2-13) Azimuth Coverage 360° (Paragraph 2-11) Elevation Coverage ..... AS° up, 18.3° down. Operation at 20,000 feet (Para- graph 2-18). Range Resolution 1 n.mi. (Paragraph 2-13) Angle Resolution 5° maximum (Paragraph 2-13) Range Accuracy a a = 0.25 n.mi., rms (Paragraph 2-15) Angular Accuracy (Ta = 1-25° rms (Paragraph 2-15) Quantization Levels (Paragraph 2-15) AEW Azimuth 1.25° AEW Range 1 n.mi. CIC System 1.0 n.mi. Stabilization Errors Less than 10 per cent of measurement error (Para- graph 2-15) Time Belay Errors Less than 10 per cent of measurement error (Para- graph 2-15) 98 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Total System Position Error (Paragraph 2-15, Equation 2-8) At 75 n.mi. or = 2 n.mi. rms (5° beam). At 150 n.mi. (It = 4.2 n.mi. Height-Finding Radar (Paragraph 2-19) Threat Evaluation Range 125-150 n.mi. Nominal Vectoring Range IS n.mi. Azimuth Coverage ± 80° from aircraft nose Elevation Coverage 45° up, 18.3° down Beamwidth — Elevation 3.2° Height Finding Error 0.5 n.mi. rms or 3000 ft rms Beamwidth — Azimuth Approx. 5°, to match fan beamwidth Stabilization Data stabilized to within 0.27° rms of true vertical 2-21 EVALUATION OF TENTATIVE DESIGN PARAMETERS WITH RESPECT TO THE TACTICAL PROBLEM We have discussed the general problems of AEW radar design; we also have hypothesized an AEW System which provides answers to certain of these problems (detection range, resolution, target counting, etc.). Now we shall hypothesize reasonable means for processing the radar information to provide headirig and velocity information in the tactical environment of the example. The accuracy of the heading and velocity estimates obtained — coupled with the position accuracy — form inputs for the study of the interceptor system effectiveness. Three minutes (180 seconds) are available to evaluate the threat fol- lowing detection. From Equations 2-13 and 2-14 we see that the standard deviations of the heading and velocity measurements obtained by using the position measurements made at the beginning and end of the 3-minute interval are ayr = ^;^"-""- = 0.0233 n.mi./sec = 142 fps rms (2-27) 180 sec (4.2) (6080) (57.3) „ ^^^ ,. oox ""'' = (800)(180) = ^^-^ ''''' ^^-^^^ where 6080 = conversion factor between knots and fps 57.3 = conversion factor between radians and degrees. This accuracy is sufficient to provide a basis for evaluating the threat within the 3-minute period. Actually, the accuracy is somewhat better than is indicated by these figures. As already mentioned, the range resolution capability of the radar allows fifteen of the twenty targets to be resolved at 2-21] EVALUATION OF TENTATIVE DESIGN PARAMETERS 99 the range of required detection. The remaining five would appear as one or two large targets until they reached a range close enough to the AEW radar to be resolved. Tracking can begin on each of the targets indicated by the AEW radar. The average standard deviations of the raid considered as a whole would tend to approach the standard deviations of one track divided by the square root of the number of separate target tracks. By the time the targets have closed to 75 n.mi., it is possible to make further refinements in the measurements of target velocities. Two addi- tional 3-minute intervals are available for this purpose. Neglecting the decrease in position error for each interval and considering that the meas- urement made in each interval is independent of the previous measurement, the error can be reduced by the square root of 3 by averaging the three readings taken over a 9-minute period. This process yields an error of 142 fps error 142 ^ in i . ^o on\ <rvT = I . = = —j= = 82 fps = 49 knots. (2-29) VNo. of velocity measurements V3 Smoothing times consistent with this magnitude are allowable for velocity measurements because it is not reasonable to expect large changes in target velocity. A somewhat different situation attends the measurement of heading. The target can make heading changes at a maximum rate of 3° per second. Thus, it is not desirable to use long smoothing times for heading informa- tion. In fact, a major problem in the design of the data-processing system is to choose an observation time and smoothing technique for heading information that provide a satisfactory compromise between maneuvering and nonmaneuvering targets. This is a complicated problem which cannot be considered here in detail. However, the basic nature of the problem will be indicated. The development so far has considered the very simplest type of heading measurement; the target position is measured at two different times, / and f + ntsc, and the heading is determined by the direction of the straight-line passing through these points (Fig. 2-22). At a range of 75 n.mi., with an observation time nisc equal to 60 seconds, this technique gives rise to an error (Equation 2-19) in measured heading equal to (2) (6080) (57.3) ., ^o .. .r,. ""'^ = (800)(60) = ^^-^ ^^-^^^ where the constants 6080 and 57.3 have been previously defined. Now, let us assume a scan time isc of 6 seconds. The heading of the previous expression was calculated on the basis of information obtained from two scans, which we may relate to each other by calling the first scan number 1, and the second scan, occurring 60 seconds later, number 11. A 100 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS similar computation may be made using scan number 2 and scan number 12. If the errors for the two computations are independent, we may improve the heading approximation by ^2 by sim.ply averaging the two computations to yield for a straight-line target: a^T = 14.5 /V2= 10.2°. (2-31) Such improvement is obtained at the expense of increased dynamic lags when the target maneuvers. Many other smoothing schemes could be used. However, for present purposes, it is reasonable to assume that the hypothetical AEW system can provide heading information with an accuracy of the order of 10° (standard deviation). The suitability of this figure will depend upon the sensitivity of interceptor performance to this figure. From the foregoing analysis, the accuracies of the AEW radar system with which the interceptor system must be compatible are approximately Position Error: or = 2 n. mi. radial error, rms (2-32) Velocity Measureynent Error: avr = 50 knots rms (2-33) Heading Measurefnent Error: g^t = 10° rms (2-34) For the vectoring problem we are interested in the relative position inaccuracy between the interceptor and the target. The total relative radial position error, <trt, between two objects is (TRT = ^^ (TT = 2.8 n. mi. (2-35) As will be shown later (Paragraph 2-25) it is convenient to express the total relative position error in terms of two components: (1) a component (TRR along the line of sight between target and interceptor and (2) a com- ponent, (TRa^ normal to the target sight-line, where (trr = aRT/yjl = 2 n.mi. (2-36) aRa = cTRT/^I2 = 2 n.ml (2-37) The position, velocity, and heading information is employed to vector interceptors on a collision course with assigned targets (Paragraph 2-11 and Equation 2-4). 2-22 INTERCEPTOR SYSTEM STUDY MODEL The design goal for the kill probability of a single interceptor has been derived as 0.5. We shall now study the problem of specifying the require- ments of an airborne intercept (AI) radar and fire-control system that will allow the interceptor to achieve this goal within the limitations imposed by other system elements and the operational environment (Step 3, Fig. 2-2). 2-22] INTERCEPTOR SYSTEM STUDY MODEL 101 As before, the first step is to formulate a master plan for the analysis. This master plan shows the fixed and variable elements of the interceptor system problem; it must also show the method by which the problem can be handled on a step by step (suboptimization) basis without losing the relation of each step to the overall problem. Such a master plan is shown in Fig. 2-26. It is merely a variant of the plans for steps 1 and 2 showing the details of the interceptor weapons system analysis. The system effectiveness goal Pq, and the fixed elements of the system, including AEW and vectoring system characteristics, have been derived or defined in preceding analyses. These are shown in Fig. 2-26 as providing the effectiveness criteria and inputs for the interceptor system analysis. The output of the system model is Pa (achieved). The variable elements are manipulated in such a manner as to make Pa (achieved) equal Pa (required). The combinations of variable element values for which this condition is realized form the basis for the interceptor system specification. The separate steps of the interceptor system analysis can be derived from the basic system logic and a careful consideration of the factors affecting each phase of interceptor system performance. The interceptor reaching the defense zone goes through three discrete phases in attacking a target (see Fig. 2-9): (1) a vectoring phase which terminates in AI radar lock-on, (2) a tracking phase which terminates in weapon launch, and (3) a missile guidance phase which terminates in the destruction of the target. The performance in each phase of operation may be characterized by the probability that — for a given set of fixed and variable elements — the phase will be completely successful. These probabilities and the factors^^ which determine their values are shown in Fig. 2-20 as: Pm = probability that the two-missile salvo will kill the specified target (already specified as 0.75) Pc = probability that the interceptor will proceed from the point of AI radar lock-on to a point where the missile salvo may be launched with a kill probability of 0.75 P„ = probability that the vectoring system will operate to bring the interceptor to a position and orientation where it may detect, identify, and lock on the target with its own AI radar. ^^Only the most significant factors are shown for this hypothetical example. An actual analysis might include many more. However, the same basic model would be applicable and the approach to the problem — though more complicated mathematically — could be much the same as will be used for this hypothetical example. Another important fact: Often It is difficult to establish all of the vital factors affecting a given phase of system operation — some of these are products of the analysis itself. This model has considerable flexibility in that such additions can be made by simply reanalyzing the phase(s) affected. 102 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-23] PROBABILITY OF RELIABLE OPERATION 103 Throughout all of the phases of operation, an equipment failure can cause the interception to fail. To account for this, we define a fourth probability: Pr = probability that the interceptor system equipment (fire-control system, aircraft, communications, etc.) will operate satisfac- factorily until weapon impact on the target. The interceptor kill probability Po may be defined as the likelihood that the complete sequence of events will be completed successfully for any inter- ceptor operating under the expected tactical conditions. Mathematically, this statement has the form: Po (achieved) = PmXPcXP.X Pr. (2-38) Thus, the basic model is established. We will now demonstrate how quantitative models may be derived and manipulated for each phase of operation to produce specifications for the variable elements of the inter- ceptor weapons system (AI Radar, Computer, Display, and Missile Guidance Tie-in). First, we make an estimate of the expected contribution of each phase of system operation to the overall kill probability. For instance, in the hypothetical example, we may substitute specified input values in Equation 2-38 and write Po = 0.50 = O.lSPcPvPr (2-39) 0.667 = PcPvPr. Any combination of P^, P„, and Pr which yields this result will satisfy the requirement. For preliminary design purposes, we shall select one of the possible combinations to provide a criterion for the performance of each phase: Pr = 0.85 P. = 0.95 (2-40) Pc = 0.825 Note that Po = (0.85) (0.95) (0.825) = 0.50. This choice is somewhat arbitrary. In an actual analysis a number of different combinations might be assumed to establish trade-offs between the contributions of each phase of system operation. 2-23 PROBABILITY OF RELIABLE OPERATION The analysis of the tactical situation showed that the total air battle lasted less than one-half hour. The total number of interceptors in the air 104 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS during a battle is 48 — 12 combat air patrol and 36 deck launch — out of a total complement of 66. The combat air patrol interceptors maintain station for 2.8 hours. These considerations coupled with the interceptor kill-probability goal are the principal factors that determine required interceptor system reliability during the attack operational situation. The AI radar and fire-control system may be expected to be the primary contributors to interceptor system unreliability — recognizing that the guided missiles' reliability has already been included in the specified missile kill probability. On this basis, we shall assume for purposes of specification that failures in the AI radar and fire control will cause two-thirds of the aborts due to equipment failure. Since we specified the overall reliability of the interceptor system as 0.85, the reliability requirement for the AI radar and fire-control system is 0.90. A reliability requirement has little meaning unless the element of time is included. Based on the large number of CAP interceptors that must be kept continuously aloft, it is specified that the reliability requirements shall be met for any 3-hour operating period. Chapter 13 will discuss the implications of this requirement, the type of design techniques that must be employed to meet it for the defined environmental conditions, and the means for determining whether a given radar can meet such a requirement. 2-24 PROBABILITY OF VIEWING TARGET — VECTORING PROBABILITY The study plan — as extracted from the master plan of Fig. 2-26 — is shown in Fig. 2-27. The object of the study is to derive the combinations of the variable elements that will permit achievement of the assumed performance goal and to ascertain the sensitivity of vectoring probability performance to changes in the system parameters. From Fig. 2-26 it can be seen that several variable factors — notably lock-on range and look-angle (maximum gimbal angle) — are common to conversion and vectoring probability. Accordingly, we cannot develop firm requirements for these in this phase of the study. Rather, the results will be expressed as a spectrum of possibilities, all of which satisfy the viewing probability requirement. Later we shall determine the portion of these possibilities which also satisfy the conversion probability require- ments. Search for the target and its detection obviously must precede AI radar lock-on. Thus, these factors are functions of the lock-on range and cannot be specified until lock-on range is specified. AI radar search data display and stabilization and search doctrine are dictated almost entirely by vectoring phase considerations. Thus, these may be specified by the analysis of the viewing probability problem. 2-24] PROBABILITY OF VIEWING TARGET — VECTORING 105 Probability of Viewing System Study iVIodel (Vectoring Prob.) Defined by Prior Study' P, (Achieved) P, = 0.95 Assumed System Goal (Para. 2.11) Fixed Elements Vectoring Method Vectoring Accuracy Assignment Doctrine Target Characteristics Target Aspect Interceptor Char. Pilot Characteristics Variable Elements Al Radar Lock-On Range Detection Range Search Range Look Angle Display Search Doctrine Output of Study Fig. 2-27 Plan for the Study of Viewing (Vectoring) Probability. One of the fixed elements of the problem, target aspect, deserves some discussion preparatory to the systems analysis. The target aspect or angle off the target's nose at the beginning of vectoring is a function of the geometry of the attack situation. Primary emphasis is placed on forward hemisphere attacks; the first twenty interceptors are vectored into such attacks on the twenty targets. The remaining interceptors are sent to back up the first twenty. Some of these will be initially vectored to targets that are destroyed by earlier interceptors. In such cases, the interceptor will be assigned to a new target in order to utilize fully the total interceptor fire power. These attacks may require the interceptor to approach the target on the beam or from the rear hemisphere. In addition, some of the forward hemisphere attacks will be aborted before missile launching because of a failure to see the target or to make the proper conversion. In such cases, the interceptor can turn around and employ its speed advantage to attack one of the targets from the rear. These considerations indicate that all initial angles off the target's nose must be considered. The interceptor kill probability should be realized or exceeded for all possible approach angles; i.e., the interceptor should have "around the clock" capability. Of paramount importance to both the vectoring and conversion phases is the manner in which the fixed and variable problem elements combine to produce distributions of possible aircraft headings at any point in space. We may visualize this problem from Fig. 2-28. At any selected point (R,d) relative to the target, the uncertainties of the vectoring system may cause the heading of an interceptor passing through that point to assume any value within the bounds shown. The spectrum of possible headings usually 106 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS enjoys approximately a normal dis- tribution about some mean heading as indicated. If the transition from the vectoring phase to the tracking phase is made at this point (AI radar lock-on), this distribution defines the range of initial conditions for the conversion phase. In addition, for any point in space the distribution defines the likely angular positions of the target with respect to the inter- ceptor flight path. The maximum look-angle required for the AI radar is largely determined by this consid- eration coupled with the viewing probability requirement. For exam- ple, if the interceptor heading in Fig. 2-28 were along line OA, a look-angle of approximately 90° would be required for the AI radar to "see" the target. Fig. 2-28 Distribution of Interceptor Headings Due to Vectoring Errors. 2-25 ANALYSIS OF THE VECTORING PHASE OF INTERCEPTOR SYSTEM OPERATION Analysis of the vectoring phase must yield the following information: (1) The distributions of aircraft headings as functions of lock-on range and angle off the target's nose (2) The AI radar characteristics required for compatability with the operation of the vectoring phase; i.e. display requirements, stabilization requirements, look-angle requirements. Vectoring System Logic. The flow of information and allocation of function for the vectoring system are shown in Fig. 2-29. This diagram expresses the system logic outlined in Paragraph 2-11 for the assumed AEW/CIC system. System Configuration Parameters. The basic factors governing the operation of the vectoring system may be ascertained from preceding definitions of target inputs and fixed parameters and the design objectives established for the vectoring system. These factors are summarized in Table 2-2 and Paragraph 2-21. Collision vectoring was specified to minimize the average target penetra- tion. The equation defining this vectoring method was derived as sin Ld = {VtIVf) smd (2-4) 2-25] ANALYSIS OF THE VECTORING PHASE 107 x> <u n: <1> -s. u M <i> < -^ q: B -?. > TO O" E (^ 1- V cS OJ o apniiiiv jojdaojsiui §U!PB9H JO}daoj9}u| P98ds J0}d93J9}U| w E M ^ o '^ E <t5 ^ Sc^ 108 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS where Ld = the collision course lead angle for perfect collision vectoring 6 = the angle off the target's nose. A plot of the required collision course lead angle versus angle off target's nose is shown in Fig. 2-30. The vectoring system computes this angle from 50 Qo 40 ;2 30 q 10 ^^ Vp = 1200 fps V^ = 800 fps / / "\ \ / \ / \ / \ 30 60 90 120 ANGLE OFF TARGET NOSE, 6 (deg) 150 180 Fig. 2-30 Collision Course Lead Angles Versus Angle off Target's Nose. the AEW radar measurements. It transforms this lead angle into a space heading command which is transmitted to the interceptor. The pilot flies the aircraft so that the heading as measured by the aircraft compass corresponds to the vectoring system heading command. Distribution of Aircraft Headings due to Vectoring Errors. Because of errors in the vectoring system measurements, the commanded heading does not always correspond to the correct collision-course lead angle. In addition, the ability of the pilot to follow the commanded heading is limited by the resolution of his display, compass accuracy, the aircraft stability and control characteristics, and the distracting effects of the search and acquisition functions he must perform just prior to lock-on. The diagram of Fig. 2-31 may be used for an analysis of the heading error distributions. The uncertainties of the vectoring system cause errors to develop in a sequence that may be examined as follows: The interceptor-target sight line established by the vectoring system may differ from the true sight-target line by an amount which can be expressed approximately as 2-25] ANALYSIS OF THE VECTORING PHASE 'Space Reference 109 for Error Signal Fig. 2-31 Vectoring Error Geometry. A^i = ARa/R ARa « R (2-41) where ARa = component of the relative position error between target and interceptor which is normal to the sight line. The vectoring system computes a desired interceptor lead angle Lc with respect to the erroneous sight-line angle. From Fig. 2-31 the computed lead angle is dd dLn dd dLp dVr sin Ld = Vt/^f sin 6 Then, [ l^F cos Ld\ L ^^'' <^°s Ld AxPt + Fp cos Ld (2-42) (2-4) AFt. (2-43) 110 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS The commanded heading differs from the correct heading by .c = £c + 0, - io = [l + P-'^] A«, + \^^^] ^H I ^F COS Ld] L^fcosLdJ + r^^i^lA^.. (2-44) l^F cos Ln] The error signal presented to the pilot is the difference between the commanded heading and the actual heading, A^pF - ipFc - ^F. (2-45) It is assumed that the pilot follows the commanded heading with an error whose standard deviation is 5°. The total heading error with respect to the correct heading is then \_RFf cos Ld] ll^F cos LdJ [^fcosLdJ (2-46) since the closing rate, R, may be expressed Cf.r -R = VtCosO -\- Vf cos Ld. (2-47) If the vectoring errors are assumed to be independent, we may write the standard deviation of the collision course heading error as [V^^rrJ -^[FFCosLn""''') ^\Ff cos Ln''''') + "'H (2-48) where aa = iARa)/R. The evaluation of this expression for various values of lock-on range from 8 to 30 n.mi. is given in Fig. 2-32 for the estimates of measurement uncer- tainty derived for the AEW system (Paragraph 2-21). The curves may be interpreted in the following manner. For range to the target R and an angle off the target's nose 6: if the proper collision-course lead angle for this condition is Ld (Fig. 2-30) then the vectoring errors will cause the interceptor lead angles to be normally distributed about the value Ld with a standard deviation of cr^.r degrees. The magnitude of the heading error increases very rapidly as the range decreases. This will be shown to have detrimental effects on the AI radar gimbal angle requirements for short- range lock-ons and on the ability to convert a short-range lock-on into a successful attack. The large magnitude of the heading errors for forward-hemisphere attacks is characteristic of any guidance system employing "prediction". Collision vectoring is such a system; it attempts to guide the interceptor towards a point in space where the target will be at some future time. 2-26] AI RADAR REQUIREMENTS DICTATED BY VECTORING 111 28 30 60 90 120 150 180 ANGLE OFF TARGET'S NOSE AT LOCK-ON (deg) Fig. 2-32 Standard Deviation of Heading Error vs. Angle off Target's Nose at Lock-on. Prediction guidance systems require the use of velocity as well as position information. For this reason they are most sensitive to closing speed. This phenomena was indicated in Fig. 2-24. The reader might satisfy himself on this point by analyzing the errors for a pursuit vectoring system, i.e., a system where the interceptor is commanded to point at the target. This analysis would disclose that the heading error distributions for all angles are about equal to the tail-chase distributions for collision vectoring. Thus the tactical advantage of collision vectoring is bought at the price of increased AI radar and vectoring system requirements. 2-26 AI RADAR REQUIREMENTS DICTATED BY VECTORING CONSIDERATIONS The vectoring situation gives rise to several requirements that must be fulfilled by the AT radar. 112 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Look- Angle Requirements. If lock-on is to occur at any selected range in the 8-30 n.mi. interval, the radar must be able to "look" at the target. That is to say, the maximum look-angle of the AI radar antenna must be sufficient to encompass the distributions of probable target angular positions relative to the interceptor heading. {Look-angle is often referred to as gimbal angle or train angle). A typical situation is shown by Fig. 2-33. The possible positions of the target relative to the interceptor are shown as a distribution of angular positions around the lead angle Ld that would exist for perfect vectoring. Note: Lq- Collision Course Lead Angle tg=Lead Angle Limit Total Area Under LOOK ANGLE, I Fig. 2-33 Probability Density Distribution of Target Angular Positions Relative to Interceptor Heading. For any range and angle off the nose, such a figure could be formulated from the data in Figs. 2-28 and 2-32 in the preceding paragraph. The probability that the AI radar can look at the target at this range and angle is simply the area under the curve that lies between the AI radar look angle limits Lg. The look-angles required to ensure that 95 per cent of the targets are within the AI radar field of view are displayed in Fig. 2-34. The prices of short-range lock-ons and "around the clock" attack capability are evi- denced by the large radar gimbal angles required to maintain 95 per cent probability. When the lock-on range satisfying the conversion probability requirement is found, Fig. 2-34 may be used to determine the AI radar gimbal angle dictated by vectoring considerations. Display Requirements. From Fig. 2-29, we see that the vectoring system transmits required attack altitude, time to collision, and range relative to the interceptor — in addition to the heading commands already discussed — to provide tactical situation information to the pilot. All the vectoring information plus pitch and roll information must be presented on 2-26] AI RADAR REQUIREMENTS DICTATED BY VECTORING 100 o< 80 LU UJ S"^ 60 cc < LlJ O !^ 40 o o 113 ii 20 R= 8 n.mi.^ R = l n.mi.O- 5 n.mi.^X u3 ^ /^ X ^ \^R=20 n.mi. \ y \^R=25 ^R=30 n.mi. n.mi. \ \ ^ 30 60 90 120 150 ANGLE OFF TARGET'S NOSE, d (deg) 180 Fig. 2-34 Maximum Look Angles Required for 95 Per Cent Probability of Seeing Assigned Target with Collision Vectoring. an integrated display which allows the pilot to fly the aircraft in response to the vectoring commands. During the last part of the vectoring phase, the pilot must detect and acquire the target. The displays required for these functions must also be integrated with the other vectoring displays to permit proper utilization of the information. The considerations governing the design of a display system to meet such requirements are treated in Chapter 12. This is one of the most difficult design problems for any radar system; it is particularly so for an AI radar because of the limited space and multiplicity of functions the pilot must perform. Display integration, like reliability, is easier to specify than to achieve. Search Volume Requirements. Radar search is accomplished by scanning a prescribed volume of space as was shown in Chapter 1 (Fig. 1-1). Target position uncertainty relative to the interceptor determines the required dimensions of this volume. For a given lock-on range, the azimuth look-angle needed to accom- modate 95 per cent of the expected tactical situations is shown in Fig. 2-34. Since the search and acquisition procedures precede AI radar lock-on, it is necessary to ascertain whether a target that is within the field of view at a given range would also have been continuously within the field of view at greater ranges. An inspection of Fig. 2-32 shows this to be the case. The largest gimbal angles are required by the shortest ranges. Thus, only the 114 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS required lock-on range need be known to specify a maximum gimbal angle satisfying search, acquisition, and lock-on requirements. The sensitivity of viewing probability to gimbal angle may be obtained by examining similar curves for viewing probabilities of 90, 80, and 70 per cent (Figs. 2-35 to 2-37). For example, a 67° maximum look-angle is required to achieve 95 per cent viewing probability at 10 n.mi. range and 75° off the target's nose (Fig. 2-34). For a 90 per cent probability under the same conditions, a 60° maximum look-angle is required (Fig. 2-35). This heavy price suggests that a different allocation of viewing and conver- sion probabilities might yield a result nearer the optimum. The required elevation angular coverage is determined by the elevation uncertainty of the vectoring system. As already derived (Paragraph 2-19), the elevation measurement error has a standard deviation of 0.5 n.mi. Thus the probability is virtually unity that the target height is within three standard deviations (1.5 n.mi.) of the vectoring radar system measurement. At a range of 10 n.mi. an AI radar elevation coverage of 17° (0.3 radian) is required to encompass this uncertainty. This requirement varies inversely with the required lock-on range and may be expressed 6(7//(57.3) , R'l — " ^^^^ Search pattern elevation coverage (2-49) The maximum range dimension Ri of the search volume is the range at which search begins. Its value depends on the required lock-on range and 100 §1 80 Q Q LU UJ <=y^ 60 UJ O en rn 40 20 R = l R = 15n. R=8 n.rr ) n.mi.-i mi.-v Jy 'A ■->- ^^\^ ^ ^ ^ -R = 20n.tT -R = 25 n.rr -R = 30 n.rr i. ii. V y^ > 30 60 90 120 150 ANGLE OFF TARGET'S NOSE, B (deg) 180 Fig. 2-35 M aximum Look Angles Required for 90 Per Cent Probability of Seeing Assigned Target with Collision Vectoring. 2-26] AI RADAR REQUIREMENTS DICTATED BY VECTORING 115 100 CC DC UJ UJ 5S5 LlI 00 q: < LiJ C3 _l Z < CO ^ u. OO 3e 60 40 R = R=l R=l 8 n.mi.-A^ n.mi.-O 5 n.mi.-N^ \ V ^ ■--R=20n. ^R=25n. mi. mi. \ \ ^R=30n. mi. \ 30 60 90 120 150 180 ANGLE OFF TARGET'S NOSE, 6 (deg) Fig. 2-36 Maximum Look Angles Required for 80 Per Cent Probability of Seeing Assigned Target with Collision Vectoring. 100 Q< 60 R- R=l R=l n.mi.-\s 5 n.mi.n,\ ^ ^ /^ S 20 n.mi. \ V r^ \^R=25 n.mi. ^— R=30 n.mi. \ 30 60 90 120 150 ANGLE OFF TARGET'S NOSE, d (deg) 180 Fig. 2-37 Maximum Look Angles Required for 70 Per Cent Probability of Seeing Assigned Target with Collision Vectoring. the radar characteristics. For most radars, a value of two times the required lock-on range is adequate. 116 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Stabilization Requirements. The angular requirements for the search pattern were derived with the tacit assumption that the search pattern was space-stabilized in roll and pitch about the aircraft flight line. That is to say, the volume of space illuminated by the radar is independent of aircraft angles of attack and roll. This assumption results in a con- siderably smaller search pattern than would be the case if these motions were allowed to displace the search pattern. This effect is illustrated by Fig. 2-38. Search pattern stabilization also makes the radar search display 140 Q oi 100 60 40 20 N, ^Unstabilized Searc / Value Caused by 7C / Roll Angle plus 1 )° \ < Angle of Attack ^ / 1 ^Unstabiiized Search Value Caused by Angle nf Attack / Stat ilized Sec irch^ 20 40 60 80 100 120 ANGLE OFF TARGET'S NOSE (deg) 140 160 180 Fig. 2-38 Elevation Search Angle Requirements for 10 n.mi. Lock-on (Stabilized and Unstabilized Search). problem easier to solve, as will be shown in Chapter 8. For these reasons it is required that AI radar search pattern be stabilized in roll and pitch about the aircraft flight line. Summary. A summary of the AI radar requirements dictated by vectoring considerations is shown in the overall requirements summary, Paragraph 2-30. 2-27 ANALYSIS OF THE CONVERSION PROBLEM The plan for analyzing the conversion problem is shown in Fig. 2-39. Analysis of the conversion phase must yield the following information relevant to the AI radar design: 2-27] ANALYSIS OF THE CONVERSION PROBLEM 117 Defined by Prior Study" Probability of Conversion System Model P (Achieved) P, = 0.825 Fixed Elements Heading Errors at Lock-on Target Characteristics Interceptor Charac- teristics Pilot Characteristics Assumed System Goal (Para. 2.22) — System Deficiencies Variable Elements Al Radar Lock-on Range Look Angle Stabilization Display Tracking Ace. Dynamic Range Attack Doctrine Missile Launch Requirements Study Output Fig. 2-39 Plan for the Analysis of Conversion Probability. 1. The minimum required AI radar lock-on range 2. The fire-control computer requirements 3. The attack display requirements 4. The missile launching and illumination requirements 5. The radar tracking and stabilization requirements Attack Phase System Logic. The flow of information and the allocation of function during the attack phase are shown in Fig. 2-40. Following AI radar lock-on, the AI radar measures target range, lead angle. Target Input AI Radar Measured Target Info Aircraft Flight 1^— Data (Speed, Altitude, etc.) Fire Control Computer Auxiliary Signals 1 1. Error Signal Display Aircraft Control I — I Aircraft System Aircraft Heading Aircraft Heading Missile System Fig. 2-40 Interceptor System Logic Diagram During Attack Phase. 118 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS angular velocity, and range rate along the line-of-sight. This information is utilized in conjunction with aircraft flight data (speed, altitude, etc.) to compute an attack course that permits the weapons to be launched with a high kill probability (see Paragraph 1-4 and Figs. 1-3 and 1-4). Deviations between the computed attack course and the actual interceptor flight path are presented to the pilot as a steering error signal. The pilot — -or auto- pilot — flies the aircraft to reduce the steering error to within the limits required by the weapon characteristics. Guided Missile Launching Zone Parameters. The allowable launching ranges and angular error launching tolerances for the inter- ceptor's guided missile may be obtained from a graphical representation of the launching problem. This analogue model — shown schematically in Fig. 2-41 — utilizes the fixed parameters of the target, interceptor, and Launch Point fn ,Maximum-G Missile Trajectory -Maximum Missile Range Envelope Impact Point for Straight-Line / Target Trajectory^ ^ yfl Target Position / at Launch ^Maximum-G Target Trajectories Fig. 2-41 Graphical Determination of Launching Zones. guided missile as defined in Figs. 2-5 to 2-7. The launching tolerances calculated from this model represent the permissible deviations from perfect solutions to the fire-control problem. We may construct and employ this model to analyze the problem in the following manner: STEP 1 . A value of range to impact point that lies between the maximum and minimum missile downranges is chosen. A semicircle with a radius equal to this range is drawn around the impact point. STEP 2. The missile time of flight corresponding to the chosen down- range is read from the missile performance diagram. The target position at weapon launch can then be plotted P^rff units back of the impact point. 2-27] ANALYSIS OF THE CONVERSION PROBLEM 19 STEP 3. STEP 4. The target is assumed to have two possible types of trajectories during the weapon time of flight: (a) a straight line, and (b) a maneuver at the maximum permissible target aircraft load factor. These trajectories can be plotted as functions of time after weapon launch. Now we may plot the missile performance diagram on a transparent sheet, using the same scale as the target and firing circle diagram. The origin of the missile performance diagram is made to coincide with a point on the firing circle. The missile performance diagram overlay can then be rotated with respect to the target and firing circle diagram to determine the maxi- mum aiming errors that would still permit interception of the target by the guided missile. An interception is defined as any point within the missile performance contour where a missile time-of-flight line and the time marker on the target trajectory are equal. This procedure may be repeated for a number of points on the firing circle. STEP 5. The foregoing steps may be repeated for a number of assumed ranges-to-impact and for all the assumed altitude and speed conditions. Using maximum allowable aiming error as a parameter, we may plot range against angle off the target's nose 50 40 30 20 10 V 10 sec- 1 50,000 ft Alti Vf^M 1.2 V^^M 0.8 tude p7 v\>^ Isec 6 sec \ \ ^^ a12 tt rO U / \ \ ^ ^\ ^ ^ i%f b^v^ ^^^ ^ erj\ K>^J ^ / / § ^yi -—-—. , ^ ■^7 s 12 LU 10 o s 'I 4 5 S Head-On 30 60 90 120 150 Beam ANGLE OFF TARGET'S NOSE, d (deg) 180 ♦ Tail-On Fig. 2-42 Missile Launch Zones and Launching Tolerances; 50,000-Ft Altitude. 120 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS at launch. The results of such a process as applied to our fictitious problem are shown in Fig. 2-42. This analysis shows that the allowable launching tolerance varies quite widely, depending upon the launching range. The tolerances on heading at launch are quite tight for very large or very small ranges. They are comparatively liberal for intermediate ranges. For example, if missile firing occurs from 20,000 ft range at 90° off the target's nose, an error of 12° is permitted. The missile time of flight for this instance is 12 seconds. The usable minimum missile launching range is determined by the requirement that the interceptor not pass closer than 1000 ft to either the impact point or the target in order to preclude self-destruction. Using the defined maneuvering capabilities of the interceptor, the minimum launching range or breakaway barrier dictated by this requirement may be calculated by graphical techniques similar to those used for the latmChing tolerance determination. The result of such an analysis (for a nonmaneuvering target) is shown superimposed on the missile launch zone diagram (Fig. 2-42). Thus the allowable missile launching ranges and angular aiming errors — as limited by the characteristics of the target, interceptor, and guided missile and the target avoidance problem — are determined for each angle off the target. Note that the allowable angular launching tolerances are appreciably smaller than an inspection of only the missile performance diagram would indicate — 5° to 10° compared with 10° to 30° for the missile itself (Fig. 2-6). This is a typical result of a study which examines the guided missile performance in its expected tactical environment. It can be seen that the allowable launching tolerances determine the required accuracy of the AI radar and fire-control system. This is why the AI radar designer must be certain the missile performance is defined for operation in the expected tactical environment. Fire-Control System Parameters — Attack Doctrine. All the basic information needed for fire-control system specification is now available. The fire-control system must be compatible with five requirements or limitations: (1) minimum average penetration distance; (2) "around-the- clock" launching capability; (3) collision vectoring; (4) missile launching tolerances; and (5) interceptor maneuver limits. A modified form of collision guidance — known as lead collision — provides a reasonable answer. For any tactical situation this guidance system attempts to direct the interceptor on a straight-line course to a point where the missiles may be fired with high kill probability. The straight-line characteristic reduces penetration, reduces intc-ceptor maneuver require- ments, and allows missile launching to take place at any angle off the 2-27] ANALYSIS OF THE CONVERSION PROBLEM 121 target's nose. In a lead-collision system, missile launching occurs auto- matically at such a range that the missile time of flight to the impact point equals a preset constant. The value of this constant may be chosen to utilize the best characteristics within the allowable launching zones. For a given angle off the target, the lead angles required in a lead-collision system correspond closely to the collision vectoring lead angles — a fact which is helpful in solving the conversion problem. Lead-collision geometry is shown in Fig. 2-43. Solution of the fire-control triangle yields Relative Range at Impact V^iT-t,) Missile Average Velocity Relative to interceptor During Time of Flight, ff AV Interceptor T = Time to Go Until Impact Fig. 2-43 Lead-Collision Geometry: Two-Dimensional. R^ y^T cos 6+ VtT cos L^- V^tf cos L (2-50) VtT sin d = {V,^T+ V^tf) sin L. (2-51) The component of relative velocity along the line-of-sight is R ^ -Frcosd - Ff cos L. (2-52) The component of relative velocity perpendicular to the line-of-sight is Rd = Ft sin d - Fp sin L. (2-53) By definition of a lead collision course // = a preset constant. (2-54) From the definition of missile characteristics for straight-line flight (Fig. 2-6) F^ =/(Ff, altitude,//). (2-55) Thus, for a fixed time of flight and known speed and altitude conditions FrJf = Ro = constant. (2-56) 122 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Using Equations 2-52, 2-53, 2-56 to eliminate velocity terms in Equations 2-50 and 2-51 and rearranging terms, we obtain R -\- RT- RocosL = (2-57) sin L = {RT!Ro) d. (2-58) The fire-control system must solve two problems. It must provide (1) a signal for automatically firing the missiles at the correct point, and (2) an aiming error signal for the pilot or autopilot. The i\I radar measures range, range rate, lead angle, and space angular velocity of the line of sight (R^f, Rm, L:,t, ^.i/)-'' Aircraft speed and altitude may be combined with known missile performance at the preset time of flight to obtain Rq (see Equations 2-54 to 2-56). The measured target inputs and the computed missile characteristics may be substituted into Equations 2-57 and 2-58 to obtain „ — Rm + Ro cos L^f ^ , . ... > ,-, -n\ Jc — -■ (computed time-to-go until nnpact) {^-^yj Rm sin Lc = I R.\[-^] 6m (computed correct lead angle). (2-60) Firing occurs when // (preset). (2-61) A steering error signal is obtained by taking the differences between the sines of computed and measured lead angles and multiplying this difference by a sensitivity factor (Ro cos L) /(Ro -\- VfT). This factor causes the computed angular error signal to be a close approximation of the actual angular aiming error. Thus, the computed steering error is tHc = [R^^ cos Lm\Ri^ + /VTc)][sin L, - sin L.m]. (2-62) Both the azimuth and elevation error signals are computed from an expression of this form. Equations 2-59 to 2-62 define the fire-control and tracking problems that are to be solved by the AI radar and fire-control system. The precision required of this solution is determined by the angular aiming tolerances corresponding to the selected value of preset time-of-flight. For the purpose of developing a representative set of accuracy specifi- cations we shall select 10 seconds for the preset time of flight //, which corresponds to a relative displacement at impact of Ro — 6800 ft. An inspection of Fig. 2-42 shows this is a reasonable choice since firing will occur near the center of the allowable launch zone for all angles off the nose at '-The subscript M denotes a measured quantity. 2-27] ANALYSIS OF THE CONVERSION PROBLEM 123 firing. The maximum allowable launching error for a 10-second time of flight may be plotted from the data of Fig. 2-42 as shown in Fig. 2-44. As 10 <o Op: n /" N / \ / s / ^ = ^ - Missile Flight Time = 10 sec Maneuvering Target L_ 30 60 90 120 150 ANGLE OFF TARGET'S NOSE (deg) 180 Fig. 2-44 Maximum Allowable Launching Errors. can be seen, head-on and tail-on attacks impose the most severe require- ments upon overall aiming accuracy. The functions and overall accuracy required from the fire control system have now been defined. The next problem is to specify how this error is to be divided among the possible sources of error in the system. Fire-Control System Error Specification. The sources of system error can be listed as follows: (1) AI radar measurements [Rm^ Rm, Om, Lyi) (2) Flight-data measurements (altitude, speed) (3) Fire-control computation (4) Pilot-airframe-display interaction There are two general types of errors — predictable bias errors and random errors. Predictable bias errors arise from the dynamic response characteristics of the measuring device. For example, in the lead-collision system specified, the variables R, R, L, and 6 can change rapidly as the launching point is approached (Fig. 2-45). ^^ Dynamic lags in the measuring devices will cause measurement errors whose values may be predicted from a knowledge of the input parameters and the dynamics of the measuring device. The i-In this application, the system must continue to track after missile launch to provide illumination for the missile seeker. Thus, dynamic lags are also important after missile launch (r< 10 seconds); it will be noticed in this connection, that the dynamic inputs are quite severe for this case. This point will be discussed further in Paragraph 2-29. 124 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 50 ^^ 40 - 30 ■ o, 20 10 -1200 -1100 -1000 -900 -800 -700 ^ 60 t 50 o -S 40 ^ 30 o £ 20 ^ 10 :110° ^===== __ -1.0 -0.8 o -0.6 ^ X) -0.4 I -0.2 10 20 30 40 TIME TO GO (seconds) 50 60 :110 -< — 5,^^ A / ■"■ — 6 /' / / 20 30 40 TIME TO GO (seconds^ 50 -1.0 -2.0 -3.0 -4.0' -5.0 60 y^ k* y /"Range ^L^ , V^ ^ Le ad Angle y^ -S 40 III < 20 n 2 10 20 30 40 TIME TO GO (seconds) 50 60 Fig. 2-45 Dynamic Variation of Lead Collision Fire Control Parameters. measurement errors affect the firing time and the steering error as calculated in the fire-control computer by Equations 2-59 through 2-62. Because dynamic lag errors are predictable, it is theoretically possible to eliminate them entirely by suitably clever design. However, the more usual approach is to limit the magnitude of these errors to some finite value. As a general rule of thumb, it is desirable that the total value of the predict- able bias error contribution obey the following inequality: B <yj2(r (2-63) where B = total predictable bias error <j == standard deviation of the total random error. 2-27] ANALYSIS OF THE CONVERSION PROBLEM 125 and computed quantities in this expression were correct and if the pilot flew the aircraft in such a manner as to reduce the computed error to zero, then Random errors arise from several main sources. First of all are the measurement uncertainties caused by the basic limitations of the measuring device. The angular measuring accuracy of a radar, for example, is limited by beamwidth as was indicated in the discussion of AEW radar require- ments.'^ Mechanical and electrical component tolerances also contribute to errors of this type. The system noise sources also contribute to random errors. For example, the finite dimensions of a radar target introduce time-dependent uncer- tainties into the measurements of range and angle (see Paragraph 4-8). Similarly the vagaries of airflow past the aircraft may introduce random noise errors into flight data measurements. These latter would aff^ect the computation of Rq- Random aiming errors also are caused by the pilot's inability to guide the aircraft on exactly the course indicated by the displayed error infor- mation. Paragraph 12-7 will discuss this problem in some detail. Generally speaking, however, if the pilot is presented with an error signal which is band-limited to about 0.25 rad sec'^ and if the error signal, is contaminated by random noise which is bandlimited to about 1 rad /sec, then the pilot can steer the aircraft with a random error which has a standard deviation approximately equal to the standard deviation of the noise. Thus the pilot's contribution to the total aiming error may be written: (Tpf = (tn (2-64) where cpf = standard deviation of the pilot's flyability error o-iv = rms value of the noise on the error signal display. To illustrate how the error specification might be developed we shall consider two cases: (1) a head-on attack and (2) an attack which begins at an angle oflF the target's nose at launch of 80°. The method for attacking the problem can be outlined as follows. As already mentioned. Equation 2-62 is designed to provide a reasonable approximation of the actual heading error. In fact, if all of the measured i^Actually, as will be indicated in Chapter 5, the problem is a good deal more complicated than is indicated by this statement. Signal-to-noise ratio and observation time also strongly affect the angular accuracy. However, for fixed values of these latter parameters, the state- ment is substantially correct. '^The bandwidth of the error signal depends upon the type of attack trajectory flown. A lead-collision course is a straight line; hence the effective bandwidth of the input guidance signals is very low. Curved-course trajectories such as lead-pursuit have higher effective guidance signal bandwidths. Chap. 12 of the "Guidance" volume of this series presents an excellent discussion of the concept of treating a guidance trajectory in terms of its frequency spectrum. 126 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS perfect aiming would result. Practically, however, the measured and computed quantities (Ro, Lm, Tc, Lc) are not correct for reasons previously discussed. Thus the computed error eH,c differs from the actual aiming error. The contribution of each source of error to this difference may be expressed. Aen = (den,c/dx)Ax (2-65) where Aen = the steering error due to error in the quantity x den.c/dx = partial derivative of the steering error with respect to the quantity x Ax = error in the measurement of the quantity x. As an example, the sensitivity of the steering error to an error in measured lead angle may be derived from Equations 2-62 and 2-59 as: den.c/dLM = (deH.c/dLm) + {den ,c / dr ,c){dT .c / dLxj) [ Rn cos Lm 1 [" F,T+Ro\[ r RmG ■ r cos L\i : — sm Lm Rm (2-66) It should be noted that the sensitivity is a variable quantity during an attack course; it also varies from one course to another. Consequently the sensitivities must be examined for the range of attack courses. In this discussion we will confine our attention to the two courses assumed (head-on and 80° off the nose). The values of the input variables and their derivatives are shown in Fig. 2-45. The error sensitivity factors for each of the assumed attack courses are shown in Fig. 2-46. It will be noted that dynamic variations of the input quantities are greatest for the attack which terminates near the target's beam; thus, predictable bias errors arising from dynamic lags will be greatest for this course. On the other hand, the effect of errors in angular rate and lead angle is greatest for head-on attacks. This fact is particularly significant because angular rate errors tend to be the most important source of system errors. Using the foregoing error data, an error specification may be derived in the following manner. For a head-on attack, the total system aiming error must be held below 7° to ensure that the missile will hit a maneuvering target (see Fig. 2-44). For purposes of deriving a tentative specification, we may split this error among the various error sources by appropriate manipulation of the following expression: Total system error = pilot requirement + 2(6e//,c/^>^i)Axi + 2V2[(d6/,.c/a^,)<r.v,P (2-67) 2-27] ANALYSIS OF THE CONVERSION PROBLEM 127 (08S/S9P) ie/^"'? "- o 3^ 00 II II II <!> > > 1^ o -> III °^^l^ SO IX> lO ^ 00 CM o o o o o (§9p/sap) ^e/^"9e o o o o I I I I (y/§3P) °ye/^"9e (09s/§9p/§9p)V/^"9e (§9p/S9P)*^7er"5e in ■- O CU g II g ^ O II II 7i 11 >":> kT 09S/S9P/39P) '^ee/^^^e (y/§9P)'^ae/^"3e 09s/ii/S9p)^ae/^"3e 128 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS where pilot requirement = maximum allowable indicated error at firing ^{deH,cldXi)Axi = summation of predictable bias errors 2^j'E[{^eH,c /dXi)aXiY = twice the standard deviation of the total random error. We will assign a value of 2° to the pilot requirement; i.e., the pilot is required only to bring the indicated error within a value of 2° to ensure successful missile launching. This error is, in effect, treated as an allowable predictable bias error and it is desirable that its allowable value be made as large as possible because this will reduce the total time needed to reduce an initial steering error at lock-on (see Fig. 2-49 below). For the head-on case, predictable bias errors due to dynamic lags present no problem because the input quantities (R, d, Lm) are relatively constant over the entire attack course and the system is relatively insensitive to mechanization approximations used in the computation of Rq (relative range of the guided missile at impact). Thus, predictable bias errors (other than pilot bias) can be assigned a value of zero for the head-on case. The remaining error tolerance (5°) can be split up among the sources of random error as shown in Table 2-3. It will be noted that no tolerances are given for range and time-to-go quantities; their effect on the head-on attack problem is too insignificant to provide a satisfactory basis for specification. The allowable random angular errors (^m, Lm, and pilot steering) are equally divided between the azimuth and elevation channels by dividing the total allowable error by -^2. This analysis shows that the radar must Table 2-3 MEASUREMENT ACCURACY REQUIREMENTS FOR HEAD-ON ATTACKS Source Allowable of Allowable Steering Error rms Error Error Sensitivity Error Contribution Specification per Channel {=<i) {dtH.c/d.i) {deu,c/d.i) X 2(7, i (Txi (Azimuth and Elevation) Rm Rm Ro T Bm 14.2 4.25 0.157sec 0.11°/sec Lm 0.36 0.11 0.15° 0.11° Pilot = 1 2.8 1.4° 1° Total random error = 5° Pilot bias 2° Total error 7° 2-27] ANALYSIS OF THE CONVERSION PROBLEM 129 provide angular rate and angle information which has rms errors in each channel of about 2 mils /sec (0.11° /sec) and 2 mils (0.11°) respectively. Referring to Equation 2-64 and the accompanying discussion, it is also seen that the computer filtering system must be designed to limit the rms noise on the indicator to a value of about 1.0° rms in order to meet the pilot steering accuracy requirement. The other attack course (80° off the nose at lock-on) may be analyzed in a similar fashion. For this case the maximum allowable error is about 10.7° (80° off the nose at lock-on will result in about 90° off the target's nose at time of firing). Using the allowable errors already established for the head-on case, the values of the allowable predictable bias errors and the values of the random range and time errors may be established as shown in Table 2-4. It should be emphasized that this allocation can be adjusted to suit the designer's convenience, provided the total error allowance is not exceeded. Chapter 9 will present a discussion of how error specifications and dynamic input requirements derived in this manner can be used to dictate the detailed requirements of the range and angle tracking loops of the radar. Table 2-4 MEASUREMENT ACCURACY REQUIREMENTS FOR BEAM ATTACKS Allowable Random Allowable Bias Error Random Predictable Contri- Predictable Bias Source Sensi- Bias Error bution Bias Error of tivity Contri- {deH,c/d.i) Error Specification Error d^H,c/d:c.i bution X ld,i Specification CTxi Rm 0.0014 0.25 1.85 179 662 Rm 0.0086 0.25 1.85 29 108 Ro T (computation) 0.0023 0.25 1.85 109 400 I 0.30 1.85 0.30 0.925 Per Total Channel Om 10 2 3.0 0.2 0.15 0.11 Lm 0.285 0.15 0.086 0.525 0.15 0.11 Pilot = 1 2.8 2 1.4 1.0 Units: degrees, seconds, feet. S = 5.2. Vs( )^ = S.S. 130 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 2-28 LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS DICTATED BY THE CONVERSION PROBLEM The establishment of the weapon firing range as a function of target aspect angle completes the information needed to calculate lock-on range requirements for the conversion probability of 0.825. Fig. 2-47 displays Distribution of Vectoring Headings Perfect -Lead Collision Course f, Contours 10 sec 800 fps 1200 fps RANGE (n. mi.) Fig. 2-47 Interceptor System Model for Conversion Problem. the essential elements of the problem. If lock-on occurs at {R, d) the heading error that must be corrected has two components: (1) the vectoring uncertainty and (2) the difference between the correct collision-course lead angle at {R, d) and the correct lead collision-course lead angle at {R, d). We shall assume that the distribution of vectoring errors is centered about the correct collision lead angle for point (i?, d). The error which is present at lock-on must be reduced below the allowable 2° pilot bias error prior to reaching the missile firing range at 10 seconds time-to-go. We shall assume that the time available for reduction of the steering error at lock-on is equal to the time available to an interceptor passing through the point {R, 6) on a lead collision course prior to reaching the missile launch range. Fig. 2-47 illustrates the situation. Contours indicating the time from {R, d) to missile release are shown, as well as a typical heading distribution at {R, d) which will arise at lock-on. (The distribution of aircraft headings relative to a perfect collision vectoring course is defined in Fig. 2-32.) The lead collision-course lead angle is a function of both time-to-go and aspect angle. Fig. 2-48 illustrates the variation in lead-collision lead angle as a function of time-to-go and aspect angle. For a given {R, d) value, the lead collision-course lead angle always is less than the correct collision- course lead angle. The time required to reduce an initial steering error is shown in Fig. 2-49 for various initial values of steering error. The primary factor contributing 2-28] LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS 45 131 35 ^ 3 30 LlT a 1 25 20 < 15 1 Collision -.niirsfi y Lead Angles ~ 1 Y \ me to bo f=40 hf=30 -f=20 / ^ ^ '/A N 'n, Vi l/f V ^ L\i /// / \ // /F = 1200fF ^r=800fps f =10 sec s / \ \ / \ 20 40 60 80 100 120 140 160 180 TARGET ASPECT ANGLE (deg) Fig. 2-48 Lead Collision Lead Angles. 40 30 20 -\ ^/^ "/ 35° 30° 25° <~ir\° Initial Steering -C^ / — 15° / — 10° . 5° E ror \ \ ^ S^ ^ < ^ ^ ^ Allowable BiasE Pilot Irror 15 TIME (sec) 20 25 30 Fig. 2-49 Time for Example Interceptor to Reduce an Initial Steering Error. to this time is the limitation on aircraft maneuverability (2 g's). Data of this type are usually obtained from simulation studies of aircraft-pilot- display performance. The analysis procedure to determine the probability of conversion is indicated by the flow diagram shown in Fig. 2-50. As an example of how such a calculation might be made, we may consider the following case. 132 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS Choose Aspect Angle Pick a Lock ■ on Range Determine Collision Course Lead Angle (Fig. 2-48) Determine Vectoring Distribution About Collision Lead Angle (Fig. 2-32) 1 £ Establisii Distribution of Heading About Collision Course Establish Distribution of Headings About Lead Collision Course (Fig. 2-51) Determine Lead Collision Lead Angle (Fig. 248) r Establish Percentage of Distribution Which Can Convert Probablity of Successful Conversion Determine T-ff (Fig. 247) ^ Determine Magnitude of Steering Error Which Can Be Reduced (Fig. 249 ) I Fig. 2-50 Conversion Probability Analysis Plan. Aspect angle at lock-on = 60° Lock-on range = 8 n.mi. Collision-course lead angle = 35° Vectoring distribution, ae.r = 21.5° Lead collision-course lead angle, L = 26.5° T - tf = 20.5 sec Maximum correctable steering error = 36° Fig. 2-51 shows the distribution of heading errors relative to the correct collision and lead-collision courses. Since an error of 36° may be corrected, any initial heading which results in a lead angle between 62.5° and —9.5° may be converted into a successful missile launch. Thus, the probability of conversion is equal to the shaded area sh jwn, which may be determined as 88.5 per cent. 2-28] LOCK-ON RANGE AND LOOK-ANGLE REQUIREMENTS 133 Limits of Correctable Steering Error ±36° H- R=8 n.mi, e.=60° Heading Error Distribution o-=21,5' Area of Siiaded Region =Pc LEAD ANGLES 62.5 Fig. 2-51 Method for Calculating Conversion Probability. The calculation of conversion probability by this technique is approxi- mate. Certain kinematic effects such as the change of collision-course lead-angle with time-to-go and the effects of initial steering error on the ultimate attack course flown by the pilot are neglected. Evaluation of these effects requires elaborate simulation programs. In a practical case, it is usually desirable to investigate these areas by more elaborate techniques. This simplified analysis, repeated for many values of lock-on range and aspect angle, culminates in curves like those in Fig. 2-52. Notice that as one would expect, the head-on attack provides the most stringent require- ments for lock-on range. The assumed system requirement stated that the conversion probability must be at least 0.825 for any aspect angle. The corresponding viewing probability requirement was 0.95. Thus for this hypothetical system approximately 10 n.mi. lock-on range is required to achieve the requisite conversion probability. The look-angle requirements are dictated by vectoring considerations, since the collision-course lead-angle is greater than the lead-collision-course lead-angle for the same range and aspect angle (see Fig. 2-48). From Fig. 2-34 we may determine that a 10 n.mi. lock-on range and a vectoring probability requirement of 0.95 combine to dictate a look-angle capability of 67°. 134 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS 100 90 > o 70 o ^ 60 40 6=75°-* y r ^ :^ 1 0=6Os / <d -6= J / i J" 1 1 1 Probability Goal ^ — t-^-H ■ — - - — i — • — 6=3 1 '1 / L 1 // / / 1 \a 4 6 8 10 12 14 16 18 20 22 RANGE (n.mi.) Fig. 2-52 The Probability of Conversion After Lock-on. The discerning reader will note that a trade-off analysis could be made between lock-on range and look-angle. For example, a longer lock-on range would allow a smaller look-angle. When space in the aircraft nose is at a premium, it may be easier to increase lock-on range than to provide large look-angles. In addition, the derived look-angle specification (67°) is pessimistic. At the aspect angle at which look-angle is critical {d = 75°) a lock-on range of 10 n.mi. yields a conversion probability of 100 per cent. This reduces the vectoring probability requirement for this attack from 95 per cent to 78.5 per cent. The look-angle requirement corresponding to this vectoring probability may be read from Fig. 2-36 as 53°. This is quite a significant relaxation of requirements and illustrates the advantages to be gained by examining the interrelationships among the system factors. In summary, the lock-on requirements are established by the head-on attack situation, and the look-angle requirements are establised by the beam aspect approach situation. These requirements are: Required lock-on range Required look-angle 10 n.mi. with 90 per cent cumulative probability ±53° in azimuth and elevation The look-angle capability must be provided in both azimuth and eleva- tion because the aircraft will roll to angles approaching 90° during the conversion and vectoring phases. 2-29] REQUIREMENTS BY MISSILE GUIDANCE CONSIDERATIONS 135 The required detection range is found by specifying a mean lock-on time and adding the range closed between the target and the interceptor during this time. For example, the closure rate in a head-on attack is 2000 fps (0.33 n. mi. /sec). For mean lock-on times of 6 and 12 seconds the required detection ranges are therefore 12 and 14 n.mi. respectively. In each of these cases, the required cumulative probability of detection is defined as 90 per cent.'^ 2-29 AI RADAR REQUIREMENTS IMPOSED BY MISSILE GUIDANCE CONSIDERATIONS The requirements dictated by missile guidance considerations can be derived from Fig. 2-6 and the previous analysis of the tactical situation. The AI radar illuminated the target continuously during the missile flight time; the missile seeker tracks the reflected signal and homes on the target on a proportional navigation course. From Fig. 2-6 it is seen that if the AI radar tracking accuracy is better than 0.35° rms, the AI radar will not cause degradation of missile performance. The specified tracking accuracy of 0.15° rms is well within these limits. However, dynamic lag errors pose an additional complication. The data of Fig. 2-45 show that very rapid changes in angular rate and range occur near the end of missile flight (T^O). The dynamic responses of the range and angle tracking loops must be sufficient to maintain AI radar range lock-on and limit the angle lag error. The exact determination of the allowable lag error would require a more detailed study of interrelations between the missile seeker and the AI radar. However, a value of about 0.25° would represent a reasonable estimate. The maximum range to the target for which illumination must be provided is obtained on the head-on attack (4.4 n.mi.). Fig. 4-6 shows that 120 kw of peak pulse power is required to ensure seeker lock-on at this range with a 24-inch antenna. A larger antenna would reduce the power require- ment and vice versa. The frequency of the seeker (X band) and the type of seeker (pulse radar semiactive) are major factors governing the choice of AI radar frequency and type, since a separate illuminating system would have to be provided if the two were different. As will be indicated in Chapter 6, the choice of a '^The derivation of the radar detection and lock-on requirements made no mention of probability. As discussed in Paragraph 2-12, it is customary to express ranges so derived as the range at which the radar should have 90 per cent cumulative probability of detection (on lock-on). This assumption puts a safety factor into the analysis, since a radar which meets this requirement will yield a slightly better probability of conversion than a radar which always locked on at exactly 10 n.mi. This comes about because 90 per cent of the lock-ons occur at ranges greater than 10 n.mi.; the resulting improvement in conversion probability for these cases more than offsets the decreased conversion probability of the 10 per cent which occur at ranges less than 10 n.mi. 136 THE DEVELOPMENT OF WEAPONS SYSTEM REQUIREMENTS pulse radar is entirely reasonable for the high-altitude (i.e. clutter-free) operation required in this tactical application. To assist lock-on of the missile seeker, the AI radar also is required to provide range and angle slaving signals to the missile seeker. The angular accuracy is not particularly critical, since the missile seeker beamwidth is relatively wide, perhaps of the order of 10° to 12°. Range accuracy, on the other hand, can be fairly critical. If the missile seeker is assumed to operate with a pulsewidth of 0.5 /xsec (250 ft) and a l-jusec (500-ft) range gate, then range errors in excess of about 150-200 ft can begin to affect seeker lock-on capability. The range error specification previously derived (Table 2-4) dictated allowable errors of about this magnitude (bias error plus la value of random error). In a practical case, this condition would dictate a more comprehensive analysis of seeker AI radar interrelations. 2-30 SUMMARY OF AI REQUIREMENTS Reliability: 90 per cent for 3-hour operation Search Pattern: 60° azimuth; 17° elevation; Stabilized in roll and pitch Search Range: 20 n.mi. Search Display: Vectoring heading command Time to collision Attack altitude Range to target Interceptor roll and pitch AI radar target detection information Detection Range: Yl n.mi. (90 per cent probability) with 6-second lock-on time 14 n.mi. (90 per cent probability) with 12-second lock-on time Lock-on Range: 10 n.mi. at a closing speed of 2000 fps with 90 per cent probability Look-Angle: ±60° in azimuth and elevation Required Computation: Lead collision (see Equations 2-59 to 2-62) Required Accuracies: See Table 2-4, Paragraph 2-28 Dynamic Inputs: See Fig. 2-45 2-31] SUMMARY 137 Maximum Allowable Angle Tracking Lag: 0.25° during missile guid- ance phase only Stabilization: Compatible with accuracy requirements and maneu- vering characteristics listed in Table 2-5 Display: Steering error signal display Aircraft roll and pitch (see Chap. 12) Additional tactical information as shown to be necessary (see Chap. 12) Noise filtered to 1° rms Maximum signal information delay 0.5-1.0 second Frequency: X band Power: Greater than 120 kw peak with a 24-inch diameter antenna Radar Type: Pulse 2-31 SUMMARY The foregoing analyses have demonstrated the vast amount of systems analysis that must precede the design of a successful airborne radar system for a particular application. The length of this chapter is in itself testimony to the possible complexities of such analyses. The drawing together and rationalization of the important factors of an airborne radar application problem is as difficult as it is necessary to proper system design. Moreover, work of this nature should continue in parallel with the radar system design to ensure that the radar design problem is always viewed in the light of the most advanced understanding of the overall weapons system problem. Succeeding chapters of this book will break the radar design problem into its component parts, with the general objective of showing how each element of the radar — transmitter, propagation path, target, receiver, data processing, and display — may be related to the overall functions and requirements of the system. Where appropriate, the examples developed in this chapter will be employed to develop further examples. R. S. RAVEN CHAPTER 3 THE CALCULATION OF RADAR DETECTION PROBABILITY AND ANGULAR RESOLUTION 3-1 GENERAL REMARKS In establishing the preliminary design of a radar subsystem to meet overall weapons system requirements, the designer must first choose the basic radar organization or configuration. He then endeavors to select the radar parameters so as to provide the required performance with practical equipments. In order to do this rationally, he must have reliable methods for estimating the performance of hypothetical systems. In this chapter, calculations in the critical areas of detection performance and angular resolution will be discussed. The former is a particularly complicated area of analysis because of the statistical problems introduced by receiver noise and target fluctuations. The effects of multiple looks at a target and operator performance further complicate the situation. Techniques for taking these factors into account for a conventional pulse radar and a pulsed doppler radar will be developed. The definition of angular resolution and the factors which might act to degrade it will be discussed briefly. These factors include the effects of unequal target sizes, signal-to-noise ratio, receiver saturation, pulsing, and system bandwidth. 3-2 THE RADAR RANGE EQUATION A primary basis for the choice of radar system parameters is the radar range equation. In one form or another, this relation gives the power received from a radar target or the ratio of this signal power to the power of competing noise or other interference from which the signal must be distinguished. We shall briefly consider the origin of the range equation. We suppose that a radar transmitter radiates power denoted by Pt isotropically (uniformly in all directions). At a range R, then, the power density or power per unit area will be p Power density of an isotropic radiator = T~U2' (^-1) 138 3-2] THE RADAR RANGE EQUATION 139 Normally, the transmitter is not an isotropic radiator but possesses a directivity or power gain due to the influence of an antenna. The power gain on transmission is denoted by Gt and the resulting power density at the range R is P G Power density with an antenna = D = - — ^- (3-2) This power is incident upon some sort of target which reflects a portion to the receiver. The target will be characterized by an idealized or eff"ective cross-sectional area a. This area is defined to reradiate isotropically all the incident energy collected. The target cross section will be dependent on the radar frequency being used and the aspect from which the target is viewed. It is normally determined experimentally and often represents a large unknown factor in radar system calculations. By definition, the power collected by the target is Da. When this power is reradiated isotropically, the power density at the receiver, which is assumed to be located near or at the transmitter, is simply Power density at the receiver = - — ^r;; = , . ,„„. • (3-3) ■iirK- (4x)-K* The effective area of the receiving aperture is denoted by Jr- The power intercepted by the effective area of the receiving antenna is simply the product of this area and the power density. The receiving area is related to the receiving gain Gr and the wavelength X by the following relation.^ ^r = ^- (3-4) We shall assume, as is normally the case, that the same antenna is used for reception and for transmission. In this event, the receiving gain will equal the transmission gain or Gr =" Gt — G. The power received by the receiver will be simply the product of the power density at the receiver and the receiving area. Combining Equations 3-3 and 3-4, the received signal power will be Received signal power = ^S' = , yn^ (3-5) This expression represents one version of the radar range equation. It shows how the received power varies with target range and size and with the wavelength and power gain of the antenna. The received power can represent either average power or peak power, depending upon what the transmitted power Pt represents. 'See Paragraph 10-1 for a further discussion and references. 140 THE CALCULATION OF RADAR DETECTION PROBABILITY An extensive discussion of target cross section is given in Chap. 4. The radar cross section of aircraft targets is discussed in Paragraph 4-7 and some typical examples are shown in Figs. 4-20, 4-21, and 4-22. The effective cross sections of sea and ground surface reflections are discussed in Para- graphs 4-10 through 4-13. In this connection, a normahzed cross section is defined as the radar cross-sectional area per unit surface area. This quantity is denoted by o-" and is usually referred to as sigtna zero. With the illuminated surface area denoted by A, the radar cross section and sigma zero are related by ex = a'A. (3-6) The area of the resolution element on the ground is a function of the pulse length, depression angle, and antenna beamwidths and is given by Eq. 4-60a and b. Examples showing the variation of sigma zero with environ- mental conditions and radar frequency are given in Figs. 4-34 through 4-43. The radar range equation is often expressed as the ratio of the received power reflected from the target to the power of some interfering signal. Most commonly, the interfering signal is random noise generated within the receiver; it might also be ground or sea clutter, atmospheric reflections or anomalies, or some sort of jamming. Internal receiver noise is often referred to as thermal noise, not necessarily because it arises physically from electronic agitation but because in characterizing it a comparison is made with noise which does arise from this source. Normally, internal receiver noise determines the maximum range of the radar system; and even when other sources of interference predominate, it provides a useful reference point. The equivalent input noise power of a receiver is normally expressed in the following form.^ Equivalent input noise power = A^ = FkTB watts = 4 X \0--'FB watts where F = noise figure — the factor by which the equivalent input noise of the actual receiver exceeds that of an ideal reference k = 1.37 X 10-23 joule /°K = Boltzmann's constant T = absolute temperature of noise source — arbitrarily, 290° K B = equivalent rectangular bandwidth of the receiver in cycles per second. The ratio of the signal and noise powers as given by Equations 3-5 and 3-7 yields the signal-to-noise ratio, S /N. Signal-to-noise ratio = S/N' = . )3pkRTR'^ ^'^'^■^ ^See Paragraph 7-3 for a further discussion of receiver noise and the origin of this expression. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 141 This expression is also referred to as the radar range equation. The receiver bandwidth B is normally determined by the IF amplifier in pulse radar systems, although in some cases subsequent filtering or integration is interpreted as equivalent to a narrowing of the noise bandwidth. Another convention is to solve Equation 3-8 for the range when the signal-to-noise ratio is unity. This range is called the idealized radar range and will be denoted by i?o: Idealized radar range = Ro = yj ^^^yj^jpp (3-9) With this definition, the expression for the signal-to-noise ratio given in Equation 3-8 takes the following simple and useful form: Signal-to-noise ratio = S/N = (Ro/R)'. (3-10) To provide an illustration of the use of Equation 3-9, let us suppose that an airborne radar system possesses the following parameter values: Pt = peak power = 200 kw a = target cross section = 1.0 m^ G = antenna gain = 1000 = 30 db F = noise figure = 10 db X = wavelength = 3 cm — 0.03 m 5 = IF bandwidth = 1 Mc/sec It is convenient to express each parameter in decibels relative to a con- venient set of units and then simply to add these figures with appropriate signs to obtain the logarithm of the idealized range, thus: Pt = 83.0 db (milliwatts) F = -10.0 db (unity) C = 60.0 db (unity) kTB = 114.0 db (milliwatts) X2 = -30.5 db (meters^) Ro* = 183.6 db (meters^) (T = db (meters^) i?o = 45.0 db (meters) = 3.89 X (4ir)3 = -32.9 db (unity) 10* meters = 20.4 n. mi. 3-3 THE CALCULATION OF DETECTION PROBABILITY FOR A PULSE RADAR Target detection is a radar function of primary importance and a necessary preliminary to other important functions such as tracking, resolution, and discrimination. In this paragraph, we shall discuss the detection process and describe methods for estimating its reliability as a function of the radar system and target parameters. Although the develop- ments in this paragraph will relate primarily to pulse radar systems, the principles apply generally to any type of radar system used for detection. 142 THE CALCULATION OF RADAR DETECTION PROBABILITY Factors to be considered include the model assumed to represent system operation, the effect of the operator, the effect of the target's closing velocity, and the effect of fluctuations in target size. Model of System Operation. The notation which we shall adopt in this paragraph is listed below. S /N = signal to noise power ratio tsc = scan time u = video voltage at pulse inte- grator output V = video voltage at square law detector output rj = false alarm number 6 = antenna beamwidth r = pulse length a = received signal voltage (peak) fr = pulse repetition frequency A^ = noise power n = number of pulses illumi- nating a target during scan Ro = idealized range R = actual range R = range rate AR = range decrement between scans S = received signal power fpeak) The radar system model providing the basis for our analysis of the detection process is shown in Fig. 3-1. The target is assumed to be an aircraft at a range R closing on the radar system at a constant range rate R. coc = angular carrier frequency (RF or IF) cos = scan speed Target (possibly fluctuating) Range =R Range Rate =R Scanning Antenna^/\ Noise \JI Predetection Square - Law Pulse Deci Amplifier " ■ i-h,_x ^..i-H-r, — Integrator Threshold RECEIVER TRANSMITTER I Pulse Rate = f , Pulse Length = r Fig. 3-1 Radar System Model Assumed for Analysis of Detection Process. Two cases are distinguished: (1) a target of constant size and (2) a target whose size fluctuates in accord with a Rayleigh distribution as is discussed in Paragraphs 4-7 and 4-8. A pulse radar with a small duty cycle (on the order of a thousandth or less) is assumed. The target is illuminated periodically by a scanning 5-3] DETECTION PROBABILITY FOR A PULSE RADAR 143 antenna. The antenna pattern is approximated by a constant gain over the antenna beamwidth 9, and zero gain outside of this region. The received signal on a single scan will consist of n pulses. In the absence of target size fluctuations, these pulses will all be of the same size. The number of pulses is given by the product of the repetition frequency /r and the beamwidth, divided by the scan velocity co^: Number of pulses in a scan = n = frQ/cjis- (3-11) The received signal is assumed to be a pulsed sinusoid. The signal power during a pulse is denoted by S, and the internal noise power referred to the same point in the system is denoted by A^. In the case of a fluctuating target an average signal power S will be defined. The essential parts of the receiver for this analysis consist of a predetec- tion amplifier, a square law detector, a pulse integrator, and a decision threshold. The predetection amplifier is normally the intermediate frequency (IF) amplifier, and it is assumed to be matched to the envelope of the pulse shape. That is, the bandwidth of this amplifier is approximately equal to the reciprocal of the pulse length. Noise with a uniform power density is assumed to be introduced into the system at the input to this amplifier. The power spectrum of the noise at the amplifier output will thus be equal to the power transfer function of the amplifier. The peak signal-to-noise ratio at the output of the predetection amplifier is S jN, as was indicated above. A square-law detector is assumed to generate a video voltage equal to the square of the envelope of the predetection signal plus noise. In this case, the development in Paragraph 5-7 is applicable and can be used to establish the amplitude distribution and power-density spectrum of the video signal plus noise. The assumption of a square-law detector rather than a linear detector is primarily for mathematical convenience. It does not represent a serious restriction because the basic results are only slightly dependent on the detector law. The pulse integrator combines the n pulses received during a scan over the target. In Paragraph 5-10 it will be shown that the linear operation which gives the greatest signal-to-noise ratio for a signal consisting of n pulses corresponds to the addition of these pulses to form a sum signal. Accord- ingly, in order to provide the greatest possible signal-to-noise ratio at the decision threshold — and thus the greatest reliability of detection — these n pulses are assumed to be added together by a pulse integrator.^ ^In many practical sj^stems, integration is provided by the memory of the human operators or by retention of the signal on the face of the cathode ray display tube. In such cases, the integration is not a perfect summing process, and degradation in the S/N ratio is experienced. This degradation is discussed later in this chapter and also in Chapter 12, on radar displays. 144 THE CALCULATION OF RADAR DETECTION PROBABILITY The decision element in the radar system is assumed to be simply a thresh- old or bias. When the integrated video voltage exceeds this threshold, detection is said to have occurred. When this voltage fails to exceed the threshold, no detection occurs. The bias may be exceeded for one of two reasons. (1) The integrated video signal-plus-noise may exceed the bias; in this case, "target detection" takes place. (2) The integrated noise alone may exceed the bias; in this case, a"false alarm" takes place. Fig. 5-16 shows how a decision threshold or bias is used to distinguish between the dis- tribution of signal plus noise and noise alone. The selection of the threshold level thus will represent a compromise between the desire for maximum sensitivity to integrated signal plus noise and the system penalties incurred by false alarms. Method of Analysis. Using the radar system model already described and the mathematical theory presented in Chapter 5, we will trace the progress of noise and signal plus noise through the elements of the receiver. The objectives of this analysis are to derive the target detection and false-alarm probabilities as functions of S jN ratio, threshold level, and the amount of integration. The analysis will be performed for both constant and fluctuating radar targets to determine the. probability of detection on a single scan.^ Finally, the concept of single-scan probability of detection will be employed in Paragraph 3-4 to develop the multiple-scan probability of detection for a moving target. This quantity — also called the cmnulative probability of detection — is the one most directly related to system perform- ance in the tactical-use environment.^ For example, the detection ranges specified for the examples in Chapter 2 were expressed in terms of the cumulative probability of detection. From the standpoint of clear exposition, it is rather unfortunate that a true understanding of the radar detection problem is wrapped in com- plexities of statistical theory which do not convey to the practicing designer a real feel for the problem. The author has attempted to alleviate this problem by confining some of the more detailed mathematical derivation to Chapter 5; the analysis that follows herein applies some of the results of these derivations as they pertain to the assumed model. Signal Analysis. As previously mentioned, the input to the square- law detector consists of noise with a power spectrum equal to the power transfer function of the amplifier and — when a signal is present — a signal ''This quantity is often called the "blip-scan" ratio or the "single glimpse" detection prob- ability. 5See Paragraph 2-12 and Fig. 2-19. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 145 with a peak power of S. We are now interested in finding analytical expressions for the square-law detector output under the conditions of noise-only inputs and signal-plus-noise inputs. It is conventional and convenient to approximate the video voltage in the absence of the signal by a series of independent samples which are spaced at intervals equal to the reciprocal of the predetection bandwidth. Such an approximation is shown in Fig. 3-2. This approximation is based upon TIME Fig. 3-2 Representation of Continuous Video Voltage by a Sequence of Sample the famous sampling theorem which states: If a function /{i) contains no frequencies higher than W j2 cps, it is completely determined by its ordinates at a series of points spaced 1 IW seconds apart. ^ In this connection, the envelope of the noise in a predetection band of width W cps can be shown to be equivalent to a low- frequency function limited to frequencies less than W 11 cps, and it can be represented by a series of samples spaced by T = 1 IW seconds. Since the spectrum of the predetection filter is matched to the spectrum of the pulse envelope, it will have a width approximately equal to the reciprocal of the pulse length, r. In this case, the samples will be spaced by intervals equal to t. It can also be shown by an appropriate application of the material in Chapter 5 that the statistical fluctuations in these samples are independent. In this case, each sample can be considered a separate detection trial, and the input to the decision element during an observation period can be regarded as a series of independent trials for which methods of analysis are well known. For instance, if the probability of exceeding the threshold is ^, the probability of exceeding the threshold at least once in m trials is Probability of at least one success in m trials 1 - (1 - pY. (3-12) Further, the average number of trials between successes is the same as the average number of trials per success, which is equal to the reciprocal of the probability on a single trial, 1 \p. When there is no signal present so that any exceeding of the threshold represents a false alarm, the number 1 jp is called t\i& false -alarm number. ^C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE 37, 10-21 (1949). 146 THE CALCULATION OF RADAR DETECTION PROBABILITY The video signal and noise samples are statistical variables. Their probability density functions are determined in Paragraph 5-7. We denote the video voltage out of the square law detector by v. This voltage is equal to the square of the video envelope r in Equations 5-78 and 5-79 in Para- graph 5-7. Making the transformations y = r^ ^ = a} 11^ and A^ = cr^ in these equations provides the probability density functions of the video voltage for signal plus noise and noise alone. Probability density video voltage signal plus noise = P.M^) = :^ exp 1^2^ - ^j hiyj'h^Sjm (3-13) Probability density video voltage noise alone = P.(.)=2^exp[^j. (3-14) The video voltage when no signal is present is thus represented by a series of independent samples at intervals of r = 1 jW which are chosen from a statistical population with the probability density of Equation 3-14. When the signal is present, the sample is chosen from a population with the probability density of Equation 3-13. An interpretation of these expressions may be given as follows. For a given value of noise power A^ and a given value of signal power S the probability that the video voltage will have a value between v and v -]- dv may be expressed as Ps+N{v)dv. Next we examine the effects of integration. We denote the sum signal at the pulse integrator output by u. « - ^1 + ^2 + ^3+ ••• + Vn (3-15) The components of the sum Vk are independent because they are separated in time by the repetition period while the correlation time of the video voltage is approximately the pulse length r, which is on the order of micro- seconds. Probability density functions giving the distribution of the signal plus noise and noise alone of the sum signal out of the integrator are required in order to determine whether a decision threshold will be exceeded. These probability density functions are denoted by Probability density integrator output, signal plus noise = Ps-\-n{h) (3-16) Probability density integrator output, noise alone = PNi'i)- (3-17) The probability density function of the integrator output when a signal is present is quite complicated, and we will not attempt a detailed study of this function here.'' Some calculations are greatly simplified, however, by ''For such a study see J. I. Marcum, // Statistical Theory of Target Detection by Pulsed Radar, RM-754; and /I Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix, RM-753, The RAND Corp., Santa Monica, Calif. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 147 the adoption of a suitable approximation to P s+n{u). Such an approxima- tion can be based on the assumption that n(S /N) ^ I. In this case, the distribution of u is very nearly normal. This is so because when S /N ^ 1, V itself tends to be normally distributed (see Equation 5-80), while with n y> I, the distribution of the sum u tends to normality by the central limit theorem.^ The mean and standard deviation of the video voltage v can be found from Equation 5-81 or Fig. 5-12. Video d-c voltage = v = 1{N + S) (3-18) ideo rms a-c voltage = o-^ = 2^N(N -\- 2S). (3-19) The mean and standard deviation of the sum signal u will be larger by «, the number of components in the sum, and by the square root of w, respectively. Integrator output, d-c voltage = « = 2n{N + S) (3-20) Integrator output, rms a-c voltage = o-„ = 2^1nN(N -i- 2S). (3-21) The probability density function of the integrator output for noise alone can be established by standard statistical procedures to have the following form. ''"^"^ = Wuhlji^)"' ^~""'- P-22) For the statistics-minded, we may note that each variable Vk is given by the sum of squares of two independent normal variables Xk andjy/c- Thus, u will be the sum of the squares of 2n normal variates, and Pn{u) will be the probability density function of a chi-squared distribution with 2n degrees of freedom.^ The Decision Element. The threshold type of decision element assumed for this system corresponds closely to the detection operations which would be performed by an automatic system such as might be employed in the terminal seeker of a guided missile. In many important cases, however, the human operator is the decision element. It is postulated that the human operator does something very similar to the threshold type of decision element. The functions of the human operator, though, would probably deviate somewhat from those performed by an ideal decision threshold. For instance, the threshold of human operators appears to vary 8J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, pp. 46-52, McGraw-Hill Book Co., Inc., New York, 1950. ^See P. G. Hoel, Introduction to Mathematical Statistics, pp. 134-136, John Wiley & Sons, Inc., New York. 148 THE CALCULATION OF RADAR DETECTION PROBABILITY randomly from look to look because of their inability to judge accurately the signal strength or to remember exactly the threshold level. Their average threshold would also tend to increase with fatigue and inattention. The net result of these deviations generally seems to be a loss in detection efficiency of the human operator in comparison with that of a mathematical threshold. This degradation is often introduced through an "operator factor" or efficiency factor, ^q. The probability of detection obtained on the basis of some threshold assumption is simply multiplied by />(, to give the "realistic" probability of detection. Values ranging all the way from 0.05 to 0.8 have been specified for this factor at one time or another. It is quite possible that a degradation of this kind represents certain detection operations quite well where the operators become fatigued or bored. On the other hand there are many detection situations where the use of an "operator factor" is very dubious. One such situation is that of the operator of an AI radar on a vectored, 10-minute interception mission. It is somewhat ridiculous to suppose that an operator on such a mission would completely miss, say, 50 per cent of all targets no matter how brightly they are painted on his scope. Another situation where the "operator factor" concept is obviously not applicable is in connection with automatic equip- ments. Here, the detection is directly accomplished through the use of a threshold. In this chapter the "operator factor" concept will be abandoned in favor of simply introducing an operator degradation of the signal-to-noise ratio. This procedure is a standard one,^° and it leads to a somewhat simpler formulation of the cumulative probability of detection. A typical value for the degradation is given in the footnote reference as 2 db. The decision threshold is chosen to give a false-alarm probability, or probability of detecting a target when none is present, which is compatible with the cost of committing the radar or weapon system to such an alarm. When such commitment costs can be established numerically, a selection of false-alarm time can be made on the basis of minimizing total costs. Most commonly, though, such costs cannot be established and the false-alarm time is arbitrarily fixed after a thorough but subjective study of its effect on the operational performance of the system. The number of independent samples of signal-plus-noise in the false- alarm time is called the false-alarm number and is denoted by t?. With false-alarm times varying from seconds to hours and pulse lengths varying from fractions of a microsecond up to milliseconds, the false-alarm number might have approximate upper and lower bounds of 10'- to 10^. The probability of having a false alarm on a single trial is the reciprocal of the false-alarm number. This probability, the probability that a noise sample lew. M. Hall, "Prediction of Pulse Radar Performance," Proc. IRE (Feb. 1956) 234-231. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 149 will exceed a threshold ^, is given by the integral of the probability density function of the sum voltage u of noise alone. False alarm probability = - = / P]s[{u)du. (3-23) V Jb This integral has been evaluated, and the result is shown in Fig. 3-3 for a useful range of parameters. 1000 500 100 50 20 10 — =;^' aW AW >| # A f_ -^^ / ^f// / */ff/ rj^lO A / , //// 77 = 10%\ //. 7j = 10^V\ s \ \ V/Z/i '// r7=iu\0 w ------^=- 5P :/—> 7^ r/ ' / / / yj 1 / / 1 n False Alarm Number I // 't ' // L ' '' False A larm Prob. 1 III 7 / /// 11 10 20 50 100 200 500 1000 RELATIVE BIAS LEVEL, b/2 N Fig. 3-3 Relation Between False Alarm Number and Bias Level with a Square- Law Detector. Single-Scan Probability of Detection. Having chosen the threshold level on the basis of a required false-alarm time, the probability of detecting a target Pd is found by integrating the probability density function of the signal-plus-noise sum voltage Probability of detection = Pa i: S+N {u)du. (3-24) 150 THE CALCULATION OF RADAR DETECTION PROBABILITY This integral gives the probability of detection of a nonfluctuating target on a single scan. It has been evaluated numerically for a useful range of the false-alarm number and the number of pulses integrated. The results of this calculation are shown in Fig. 3-4." In this figure Pd is plotted as a function of the relative range RjRa whose reciprocal is equal to the fourth root of the signal-to-noise ratio. 0.999 0.998 0.995 0.990 0.980 0.950 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.050 0.020 0.010 0,005 0.002 0.001 n = 10 n=100 n=1000 — Ill 1 ll W \ \ 11 1! it \\\ 1 11 u v\ 11 ill u \\\ 1 11 1 1 1 111 1 \ w lu \ \ ilu l\ \ IW \ ^ \ \ u \ \ \ \ ,\ i\ 1 \ \ \ V \ \\ \ \ A \- r — l\ w \\ \y u La \ \\ \ \\ \ \ \ s \\ W A .\^ \\ 1 \ \ \'\ fyT" \ \ \^ \i' Ki :>A \ \ \ \ 4 \ k\ \ \ ,\ \ \ -m \.\\ 1 \ V\\\ \ V\ s, !MA!!A^l \ \ s Vl- \ ^ \ \ \ '°sk i-\ -^ °\ >\l \ A A ^ ^ 0.6 0.8 1.0 1.2 1.4 1.6 RELATIVE RANGE, R/Rq n=NO. OF PULSES INTEGRATED ^=FALSE ALARM NUMBER 1.8 2.0 Fig. 3-4 Single-Scan Probability of Detection of a Nonfluctuating Target. We refer to Pd as the single-scan or single-glimpse probability to dis- tinguish it from the probability of detection when multiple looks are considered. Another term which is commonly used in this connection is blip-scan ratio which refers to the fraction of the time that an operator will see a blip in a single scan over the target. '^For a wider range of such curves and also a discussion of a method for calculating them, see J. I. Marcum, op. cit. 3-31 DETECTION PROBABILITY FOR A PULSE RADAR 151 To fix ideas, we consider an example of an AI (airborne intercept) radar. The idealized range Ro has been determined in Paragraph 3-3 to be 20.4 n.mi. The radar parameters of interest are assumed as follows: Pulse length = 1 jusec Beamwidth = 4.15° Azimuth scan = 120° Scan time = 3 sec Elevation scan = 17° PRF = 500 pps The scan area is approximately 17° X 120° = 2040° squared. The beam area is approximately (■7r/4) 4.15^ = 13.6° squared. The number of beam areas within a scan area is 2040/13.6 = 150. With a scan time of 3 seconds and a PRF of 500 pps, the number of pulses received in a scan over the target is 3 X 500/150 = 10. We suppose that a false-alarm time of 100 seconds is chosen. With the IF bandwidth matched to the pulse width (approximately equal to its reciprocal) there will be 10^ independent noise samples per second, and the false-alarm nuniber will be 100 X 10^ = 10^ Referring to Fig. 3-4, the relative range at which Pd = 0.9 for « = 10 and 7] = 10^ is determined to be 0.72. The actual range giving 90 per cent probability of detection is thus 0.72 X 20.4 = 14.7 n.mi. A similar calculation gives the range corresponding to a detection probability of 10 per cent as 17.5 n.mi. The complete probability of detection curve for this example is shown in Fig. 3-5 along with the single scan and cumulative Ro AR = 20.4 n LOn. . mi. mi. ^ ■^ ^ N ^^ ""1 1 1 1 ■—Single -Scan - ^ ^ N, i Target 1 Fluctuating . Target-^ / \ Y 1 \ \ \ . I , 1— \ s. \ \ / Target N S s \ •\ \ s \ aaJ 10 15 TARGET RANGE (n.mi.) 20 25 Fig. 3-5 Single-Scan and Cumulative Probability of Detection for Text Example. probability of detection curves for a fluctuating target when the same basic radar parameters are used. The derivation of methods for the calculation of these other curves is discussed in the rest of this paragraph and in Paragraph 3-4. The Effect of Target Fluctuations. The discussion thus far and the curves in Fig. 3-4 refer only to the case where the magnitude of the received 152 THE CALCULATION OF RADAR DETECTION PROBABILITY signal is constant. The energy reflected from an aircraft in flight is not generally constant. Such an object is a complex reflector of electromagnetic waves. As it moves in flight, it vibrates and turns relative to the radar system, and various parts of the aircraft reflect signals with more or less random amplitudes and with slightly diflPerent doppler shifts. As a result, the signal reflected from the aircraft fluctuates and can be represented as a noise process. ^^ The power in the signal will be distributed similarly to the square of the envelope of narrow-band noise with a probability density function of the same form as that in Equation 3-14. This distribution is often called a Rayleigh distribution. The probability density function of the signal power S will thus have the following form : Probability density of signal power from fluctuating target = 1 P(S) = -^e-^'s. (3-25) The factor S in this expression represents the average signal power. Because the rate of turn of aircraft is relatively slow, the spectrum of the fluctuations in S is normally less than about 3 cps in width,^^ and S is reasonably well correlated over intervals of less than about 50 msec. We shall suppose that the observation time is less than this and assume that S is constant during a look, but independent from look to look. With the signal-to-noise ratio fluctuating from scan to scan, the proba- bility of detection will also fluctuate, and an average value of Pd must be calculated by weighting the various values of Pd by their probability of occurrence. Thus the average probability of det ction is of the form Average probability of detection for a fluctuating target = Pd= Pd(u,S)PiS)^S. (3-26) This integral can be evaluated approximately by making use of the previously noted Gaussian approximation to the distribution of integrated video signal plus noise. Using the values given in Equations 3-20 and 3-21 for the mean and variance of the integrated video, the approximate proba- bility of detection is or, with an appropriate change of variable, (3-27) l^A detailed discussion of the fluctuations in apparent size of aircraft is given in Paragraph 4-8. i^See Fig. 4-24 for an example. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 153 P^{ '' - wS exp ( - ■2n{N+S) xy2)dx. 2ylnN{N+2S) In Fig. 3-6, Pd{S) is plotted as a function of ^. For small values of S, PdiS) is very small, while for large S, Pd(S) is approximately unity. A transition S=b/2n -N SIGNAL POWER Fig. 3-6 Probability of Detection as a Function of the Signal Amplitude. occurs when S = b jln — N. The width of the transition region is inversely proportional to n. When n is large, then, it is reasonable to approximate Pd{S) as zero for S < b jln — A^and unity for S larger than this transition point. We shall subsequently indicate that this is a fair approximation even when n = \. With this approximation, the integral in Equation 3-26 is easily evaluated. - - P - - ibllnN - 1) P. = (1/6')/ exp (- SIS)ds = exp ^,„ '■ (3-28) Jb/2n-N o/l\ It is convenient for. some developments to work directly with the range to the target instead of the average signal to noise ratio. The expression in Equation 3-10 giving the signal to noise ratio as the fourth power of the ratio of an ideal range to the actual range provides this relation: S/N = (Ro/Ry (3-29) In addition, a factor K(n,r]) is defined K = (b/2nN - 1). (3-30) With this notation, the average detection probability can be represented in the following very simple form. Pd = ^-if(«/-Ro)\ (3_31) The K factor can be evaluated from the data in Fig. 3-3 giving the relation between the relative bias, the number of pulses integrated, and the false- alarm number. The results of such an evaluation are shown in Fig. 3-7, where the K factor is plotted versus the number of pulses integrated for representative values of the false-alarm number. 154 THE CALCULATION OF RADAR DETECTION PROBABILITY N. S\ \\s ^X \^ v 2 % \,^ Ns "'"-77=10 S ^N^ /- 77-10 s ^ s^ >v . y , „ \ N 'nP ^S^v /. ;'^ -77=10 \ X ^ ."0<V / / '' -77=10 S ^; s^NS Ss^ ' / ' _ ^ in \ s \^ nN X \^ s^ ^s , / -77 = 10 s ^ vN^2 ^ "v V "^>^ ^3>>1 \ ■^z '•J>0^^ ^ T?r sjfl^^CN ^ r^i iru-^-^^^i 1 Mill '^^ . "^^ -^ sw ■^«. -^K-N^CiC^MI -^ >sj>::sy:W ^ ^">^^^ ^^ " = v ^•>: t I. 2 5 10 20 50 100 200 500 1000 NO. OF PULSES INTEGRATED,n Fig. 3-7 The Factor K{n,r]) as a Function of n. It is of interest to note that we can infer from the slopes of the curves in this figure the trade-off of signal-to-noise ratio with n, the number of pulses integrated. From Fig. 3-7, a typical slope is about — 6 db for a factor of 10 in the number of pulses integrated. This is equivalent to a variation of K with n of the following form: A^ ^ «-«-6. (3-32) Because the average probability of detection is a function of the ratio K/(S/N), a variation in K is equivalent to an inverse variation in the average signal to noise ratio. The trade-off between signal to noise ratio and n, then, is simply S/N^ (3-33) Because of the rather gross approximation which had to be made to obtain Equation 3-31, there is a question about its range of validity. A reasonable validation of this equation is obtained by comparing it with some examples of exact calculations and observing the error. This is done in Fig. 3-8 where the average detection probability as given by Equation 3-31 has been plotted as a function of the normalized range K^'*{R/Ro). It is approximately a straight line on the normal probability coordinates used in that figure. Also plotted in Fig. 3-8 are the exact values of Pd for n = 1. This is the case when the approximations made introduce the greatest error. 3-3] DETECTION PROBABILITY FOR A PULSE RADAR 155 0.99 0.98 0.95 Q? 90 ■^ 1— 0.80 O 0.70 rr U- O.bO ■p- O O.bU h^ 0.40 1— 0,30 2= 0.10 0.02 \ V ^= In2.^-" s V •^ ^ ,^, ^ ^ ^ Exact curves \ / \ / V ^d \ 9^ App roxi Tiatior A K' \ ^ 0.2 1.6 0.6 0.8 1.0 ^ 1.2 NORMALIZED RANGE, K4 (R/Rg) Fig. 3-8 Average Detection Probability as a Function of Normalized Range. When n = I, the probability density function departs from normality to the greatest degree, and the width of the transition region in Fig. 3-8 is largest. The approximation will also be poorest when the false alarm number is small. Curves are plotted in Fig. 3-8 for the two values r{ = 10^ and 17 = 10*. These are considered small values for this parameter. With the false alarm number rj equal to 10^, the approximation is already quite good for values of Pd greater than 20 per cent. For larger values of r? or n, the approximation becomes very good.^*. It is of interest to compare the average detection probability on a fluctuating target with that obtained on a constant target of the same size. The detection probability on a nonfluctuating target was previously determined in the case of Ro = 20.4 n.mi., n = 10, and rj = 10^. For the fluctuating case, we first determine the value of K from Fig. 3-7 for these parameters. This value is found to be 4.5 db or a factor of 2.82. The fourth root of K is then 1.3. In Fig. 3-8, the normalized range for Pd = 0.9 is found to be 0.58. The actual range giving an average 90 per cent probability of detection is thus i^More detailed development of these ideas can be found in P. Swerling, "Probability of Detection for Fluctuating Targets," Research Memorandum 1217, The RAND Corporation, Santa Monica, Calif. (17 March 1954). 156 THE CALCULATION OF RADAR DETECTION PROBABILITY /^9o% = 0.58 X 20.4/1.3 = 9.1 n.mi. (3-34) This range is substantially less than the range (14.7 n.mi.) which gives Pd = 0.9 in the case of a nonfluctuating target. From Fig. 3-8, the nor- malized range giving Pd = 0.1 is 1.23, which yields an actual range corre- sponding to an average detection probability of 10 per cent of 19.3 n.mi. This value corresponds to 17.5 n.mi. in the nonfluctuating case. Thus, while the fluctuations degrade the performance at high probabilities, they enhance the performance on small and distant targets. The complete curve of detection probability versus range in this case is plotted in Fig. 3-5. 3-4 THE EFFECT OF SCANNING AND THE CUMULATIVE PROBABILITY OF DETECTION Because the beamwidth of a high-gain antenna is normally much smaller than the search area within which a target might appear, the beam must be made to scan over the area. For AEW or ground-mapping systems where the beam is narrow in only one dimension, this motion is generally very simple, either a wigwag or a complete rotation. For systems where the beam is narrow in both azimuth and elevation, the motion of the beam can become quite complex. The efi^ect of scanning is to provide multiple looks at the target, giving multiple chances for detection. In this case, it is the cumulative probability of detection which is most significantly related to the tactical use of the system. Complex scans can produce a nonuniform coverage of the scan area, with holes in the pattern and undesired modulation of the received pulse packet. Multiple-Scan Probability of Detection. In a typical detection situation, the radar will periodically scan the target and there will be a number of looks at the target when a detection can be made. Moreover, since the target will normally move during the scan time, the average signal-to-noise ratio and thus the average single glimpse probability will vary from scan to scan. This situation is conveniently described by the cumulative probability of detection. When the target is closing on the radar, the cumulative probability of detection at a given range is defined as the probability that the target is detected on or before reaching that range. We shall assume that the radar closes on the target at a constant rate — Ry and the scan time t^c is also constant. Thus, the range interval which is closed during a scan is given by Range decrement = A/^ = —Rtsc (3-35) If the first look occurs at the range i?i, then the ^'-th look will occur at the range 3-4] EFFECT OF SCANNING ON DETECTION PROBABILITY 157 R, = R^- (k - l)AR. (3-36) We shall limit our discussion to a consideration of fluctuating targets which can be handled rather generally thanks to the simplicity of the expression for the average probability of detection (Equation 3-31). At each look, the average probability of detection is given by Equation 3-37 below. PdiRi.) = ^-^'(«^/«o)\ (3.37) The cumulative probability that a target is detected at the range Rk or before is denoted by Pc(Rk) and is given by the well-known expression for the probability of at least one success in a sequence of k trials: Pc{Rk) = 1 - n[l - P.iRd]. (3-38) An additional refinement needs to be introduced. Equation 3-39 implicitly assumes that the last look occurred at Rk-. Actually, the last look may occur anywhere between Rk and Rk-i = Rk -\- AR with equal probability. That is, there will be a random phase between the antenna scan and the relative motion of the target. To take this effect into account, an average value of the cumulative probability of detection must be computed: \rJo Fortunately, the calculations shown in Equations 3-38 and 3-39 do not have to be carried out every time the cumulative probability of detection is desired. With properly normalized variables, a universally applicable series of detection curves can be derived. In order to do this, a normalized range denoted by p is defined: p = K''*(R/Ro). (3-40) The normalized range decrement is defined similarly: Ap = K''\AR/Ro). (3-41) With these definitions, the average single-glimpse probability of detection takes the following form: Pa = e-"' (3-42) With this form for the single-glimpse probability of detection, universal curves of the average cumulative probability of detection have been calculated on the basis of Equations 3-38 and 3-39. These curves are plotted in Fig. 3-9. From the appearance of these curves, it would seem desirable to make the normalized decrement Ap as small as possible in order to obtain the 158 THE CALCULATION OF RADAR DETECTION PROBABILITY >- 1.0 -J 0.9 CD ^ 0.8 ccio." 0.7 11 0.6 0.5 S Q ID , 04 "^ LlI O.S < on 0.2 > 0.1 K, ■-"^ N ■^ \ N N ^^^Ap = 0.4 ^/^Ap = 0.2 ^— An = 0.1 K \ k V \ V A A \ / -Ap = 0.05 -Ap = 0.025 \ < ^ s r \ V \ ^ \ \ \ \ \ \, \ \ I \ \ \ \ \ \ \ K V ^\ s s \^ ^ \ ^ ^ ^ 0.4 0.6 1.0 1.2 1.4 1.6 NORMALIZED RANGE, P=K4(R/Ro) Fig. 3-9 Universal Curves of Average Cumulative Probability of Detection. maximum range. This is only part of the story, though. Normally, AR and thus Ap would be made small by decreasing the scan time, which in turn is obtained by speeding up the scan. With a higher scan speed, the number of pulses returned on a scan over the target is reduced. This reduces the factor K approximately through the relation in Equation 3-32. The net result is to give an optimum value of scan speed or scan time which maximizes the range at which a given value of cumulative detection probability is obtained. With a slower scan than this optimum, the target closes too much between scans and there will be too few chances to detect it. With a faster scan, there are not enough hits per scan. The determination of this optimum scan time will be illustrated as part of the following example. To illustrate the use of the curves in Fig. 3-9, we shall continue with the AI radar example which we have previously used in Paragraph 3-3 to illustrate the calculation of the single-scan probability of detection for both constant and fluctuating targets. We assume that the target closes on the radar at 2000 ft /sec or about Mach 2. The scan time was assumed to be 3 seconds. The range decrement is 3 X 2000 = 6000 ft or 1.0 n.mi. The value of K^''^ was previously determined to be 1.3 while the idealized range is 20.4 n.mi. The normalized range decrement is thus Ap = 1.3 X 1 20.4 0.0635. (3-43) Referring to Fig. 3-9, the normalized range giving a cumulative proba- bility of detection of 90 per cent for Ap = 0.0635 is p = 0.87. The equivalent actual range is 3-4] EFFECT OF SCANNING ON DETECTION PROBABILITY 20.4 i?9o% = 0.87 X 1.3 13.7 n.mi. 159 (3-44) The complete curve of cumulative probability of detection versus range is given in Fig. 3-5 along comparable single-scan curves for both a constant and fluctuating target. Finding an Optimum Scan Time. In order to demonstrate how an optimum scan time would be determined, the calculations made above for a scan time of 3 seconds will be repeated for scan times of 1, 2, 5, and 10 seconds as well. The values of A^, K, K^'^, Ap, p^^% and R^(^% are given in Table 3-1. Table 3-1 Scan Time n K A-l/4 Ap P90% R^Q%,n.n 1 sec 3.33 6.02 1.57 0.0256 1.025 13.3 2 sec 6.67 3.63 1.38 0.0451 0.93 13.7 3 sec 10.0 2.82 1.295 0.0635 0.87 13.7 5 sec 16.67 2.04 1.197 0.0976 0.785 13.4 10 sec 33.33 1.305 1.058 0.1725 0.655 12.65 The detection ranges given in Table 3-1 are plotted in Fig. 3-10. From this figure and the table, it is apparent that the optimum scan time in this Id 14 il2 ^ - - - - o 2 n 1 2 3 456789 10 SCAN TIME (sec) Fig. 3-10 Example of Detection Range vs. Scan Time. case is the original choice of 3 seconds. Another observation which can be made in Fig. 3-10 is that the optimum is very broad, and it actually does not make a great deal of difference whether a scan time of 2 seconds or 5 seconds is selected if the rest of the system is made compatible. This kind of 160 THE CALCULATION OF RADAR DETECTION PROBABILITY situation is often true in matters of this nature and is often not generally- recognized until after elaborate studies have been made, if at all. A question which often comes up in connection with discussions of beamwidth and scanning is, why scan at all? Why not simply use a wider beam and a fixed antenna? This thought has a good deal of merit to it. The loss in gain due to the use of a wide beam can be oflFset by the integration of a much larger number of pulses, and the actual detection ranges might very well be comparable. A narrow beam, though, has other advantages which make its use desirable. One of these is that upon detection, the location of the target is known at once so that tracking can commence immediately. Further, the resolution which can only be provided by a narrow beam is often a basic tactical requirement of the system (see Paragraph 2-13). In addition, a narrow beam is often required to give sufficient accuracy during track or to provide a means for narrowing the scan area and "search- lighting" a suspected target. Types of Scans. Fig. 3-11 shows some scan patterns which have been used with pencil beam systems. The most common type of scan is a simple X. " -^ r.x -. - X. 1 1 , " .< ' . .^ w r ^ Multi ■ Bar Raster Scan (A) Two - Bar Scan with Conical Lobing (Palmer Scan) (B) Fig. 3-11 Some Possible Scan Patterns with a Pencil Beam System constant-velocity raster scan with a fly-back at the bottom of the pattern. With a large area to be covered, up to seven or eight bars might be required. Very often the basic scan is modified by a lobing motion. Conical or circular lobing may be used during track to generate angular error signals. During search the lobing may be left on, either because there is no convenient way in which to stop and start the lobing motor or because the larger equivalent beamwidth can be utilized to cut down on the number of scan bars. When this is done, the circular lobing motion combined with the constant-velocity azimuth motion produces a cycloidal scan of the beam centers (Fig. 3-llb). This type of scanning motion is often referred to as a Palmer scan because of its resemblance to a pennmanship exercise. The cardioid and spiral scans shown in Fig. 3-11 represent attempts to minimize the fly-back or dead time. They are not generally regarded as normal designs, but may be required for some applications. 3-4] EFFECT OF SCANNING ON DETECTION PROBABILITY 161 The Number of Pulses per Scan. In the system model adopted in Paragraph 3-3, to develop an analytical method for calculating the proba- bility of detection, it was assumed that n equal-amplitude pulses were received on a scan over a target. With a complex scan and realistic beam shapes, the pulses received are not all of the same amplitude; neither is it clear just what n should be in many situations. For instance, with the Palmer scan illustrated in Fig. 3-11 b, the pulses received on a single scan over a target may be grouped into several separate packets by the cyclical motion imposed on the basic scan. The grouping and number of pulses in the individual packets can then change with the location of the target in the scan pattern. In order to analyze situations of this nature correctly and in detail, extensive analytical investigations are often required. More commonly, it is quite adequate to make reasonable approximations which will allow the methods developed in Paragraph 3-3 to be applied. This is what we shall do here. We consider first the problem of estimating the effect of the antenna beamshape in a linear scan over a target. We suppose that the antenna pattern has a Gaussian shape similar to that defined in Equation 3-45: Two-way power pattern of antenna '-^ ^-eVo.ise^ (3-45) where Q = angular position of the antenna = antenna beamwidth (half-power, one-way). We wish to approximate this antenna pattern by a uniform pattern so that the results of the preceding paragraph are applicable. This type of approximation is indicated in Fig. 3-12. In making this approximation, the Fig. 3-12 Rectangular Approximation to Gaussian Beam Shape (Equal Area Approximation) . total integrated power will be maintained constant. That is, the integral of the uniform approximation will be made equal to the integral of the antenna power pattern between the effective limits of integration. This will result in an equivalent loss in signal power for pulses in the uniform pattern in comparison with pulses in the center of the more realistic pattern. This loss is referred to as the scan loss. Following current practice, we suppose that 162 THE CALCULATION OF RADAR DETECTION PROBABILITY the effective number of pulses integrated n are those contained within the antenna beamwidth. The optimum number of pulses to integrate will differ slightly from this.^^ The effective power within the antenna beamwidth will be proportional to the following integral. re/2 Total received power = / d'-^'/o.ise' ^^ == 0.686. (3-46) 7 -e/2 Thus, where the maximum power of the pulses in a Gaussian beam is unity, the equivalent power of uniform pulses is only 0.68, giving a scan loss of 1.7db.i« A second problem concerns the number of pulses integrated when the scan is complex. Where it is probable that there is a substantial non- uniformity in the pulse distribution, a pulse count should be carried out. That is, the actual number of pulses returned from typical target locations for a sample scan would be determined by counting them. More usually, it is adequate simply to use the average number of pulses per scan as was done in Paragraph 3-3 for the example illustrating the calculation of the single-scan probability of detection. The beam area was divided into the scan area to give the number of beams per scan. This number was in turn divided into the total number of pulses per scan to yield the received pulses per scan. 3-5 THE CALCULATION OF DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR With proper interpretation, the methods developed in Paragraphs 3-3 and 3-4 are applicable to a variety of types of radar systems. To illustrate how this can be carried out, we shall develop some of the details of such an application to the gated pulsed doppler radar described in Paragraph 6-6, whose functional block diagram is given in Fig. 6-25. This type of radar transmits pulses at a very high repetition rate in order to avoid doppler frequency ambiguities. The duty ratio is also considerably greater than in a conventional pulse radar. All the possible target ranges (ambiguous) are gated into separate filter banks which cover the spectrum of possible doppler frequencies. The filters respond to the fundamental component of the gated doppler signal which is received. Single-Scan Probability of Detection. The idealized range for this type of system is essentially given by Equation 3-9. This is restated in Chapter 6 as Equation 6-39 with the effects of the signal and gating duty '^L. V. Blake, "The Number of Pulses per Beamwidth in a Scanning Radar," Proc. IRE, June, 1953. '^A scan loss of L6 db was obtained by L. V. Blake in the paper cited in footnote 15. 3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 163 cycles specifically incorporated. For a gated pulse doppler system the noise is gated with the same duty factor as the signal so that dg = ds in Equation 6-39. Detecting only the fundamental doppler signal in a filter output corre- sponds to the case of detecting a single pulse in Paragraph 3-3. Thus the basic single-scan probability of detecting a fluctuating target should be given by the expression in Equation 3-31 with the factor K correspond- ing to the integration of a single pulse. In order to account for certain features of the pulsed doppler system, this basic probability must be modified somewhat. This modification is due to the straddling of a pulse by contiguous range gates and the eclipsing of part of the received pulse by the transmitted pulse. These effects act to decrease the single-scan probability of detection from its basic value. This reduction is denoted by the factor F(R). Thus the probability of detection of a fluctuating target by a pulsed doppler radar can be represented by Prf = F(/?)e-^'(«/«o)4. (3_47) The Straddling and Eclipsing Factor. Range gate straddling refers to the situation when the received signal simultaneously falls within two range channels. This situation is illustrated in Fig. 3-13, where the received n I I I I I I ^ I I I I I I I I I I Ml I 123455789 123456789 12345 I n Transmitted Pulses Received Pulses Fig. 3-13 Range Gate Straddling. pulse lies partly in channel 2 and partly in channel 3. Since the gating of only a fraction of the received pulse into a given channel is equivalent to decreasing the duty ratio by this factor and since the noise in that channel is undiminished, the signal power in the channel will be proportional to the square of the fraction of the pulse within the channel gate. Thus, if a fraction a of the received pulse falls within gate k and the fraction 1 — a falls within gate ^ + 1, the signal power in the first gate will be proportional to a^ while that in the second channel will be proportional to (1 — a)^. When half the pulse lies in each gate, there will be a loss of 6 db in each 164 THE CALCULATION OF RADAR DETECTION PROBABILITY channel. Of course, there will be two chances to detect the target. If the probability of detecting the target in the first channel is denoted by Pi and that for the second channel by P2, the probability of detecting the target in at least one of the channels will be Pd2 = 1 - (1 - Pi)(l - P,) = P,^ p,- p,p,. (3-48) The straddling factor will be periodic in range, with a period equal to the pulse length in range units. In the example illustrated in Fig. 3-14, the Ro=25 n. mi. AR = 0.67 n. mi. 10 15 RANGE (n. mi.) Fig. 3-14 Sifigle Scan and Cumulative Probabilities of Detection for a Pulsed Doppler Radar. oscillations of the detection probability with a period of about 1 /12 n.mi., which are shown in the expanded view, represent the eflfects of straddling. When the received pulse straddles the transmitted pulse, eclipsing is produced because the receiver is gated off when the transmitter is on to prevent feed-through. The effect of eclipsing is much more severe than that •of gate straddling. When the received pulse is centered on the transmitted pulse, the signal received and the resulting probability of detection become zero and produce a blind range at which the system is completely insen- sitive. These blind ranges are periodically spaced at intervals equal to the repetition period measured in range units. The nulls in the curve in Fig. 3-14 at intervals of slightly less than a nautical mile represent the effects of eclipsing. 3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 165 It is not completely correct to substitute the average values of the detection probability in each channel into Equation 2-48 when considering a fluctuating target because the signal will fluctuate similarly in the two channels. Instead, we should use the procedure previously used in Equation 3-28 for finding the approximate average value of a single channel to deter- mine the average of the two-channel expression given in Equation 3-48. Equation 3-28 was derived on the basis of an approximation to the curve in Fig. 3-6 that Pa was zero out to the value 6" = KN and unity for higher values of S. With this approximation, Pi and P2 become ^'^""^ X.SyKN/a"- (3-49) prn .^2vi~o, ^<AW/(i ~aY Adopting these approximate expressions, it is apparent that the product P1P2 is equal to Pi when a < 1 /2, and to P2 when a > 1 /2. This observation materially simplifies Equation 3-48, since only one term is retained: The average value of Pd2 in each case will be of the exponential form first given in Equation 3-31: - _ exp - K{R/R,)V/i.\ - ar-\ = {P,iy"'-'\ cc<h ^''- ~ exp - K{R/R,Y{\/a^-) = (Prf)l/«^ a > f ^^'^'^ The minimum value is attained when a = \, and the received pulse is divided equally between the two channels: min Prf2 - P/. (3-52) Of more interest is the average value, which can be used as a smooth replacement for oscillatory curves similar to that in Fig. 3-14 in many cases. The function in Equation 3-51 could be integrated by numerical means. It is more expedient, though, and a good approximation to simply use the average of 1 /a^, which = 2 in Equation 3-51. Thus, ave P,2 - P/. (3-53) On the average, then, the effect of straddling can be interpreted as a 3-db loss in signal-to-noise ratio. _ It should also be noted that Pd2 is of the same general form as Pa itself. Thus, if the effects of eclipsing can be neglected, the methods developed in Paragraph 3-4 for determining the cumulative probability of detection and the normalized curves in Fig. 3-9 are quite applicable. 166 THE CALCULATION OF RADAR DETECTION PROBABILITY An approach similar to that taken in this paragraph should be applicable to many similar problems. For instance, a multiple-PRF method of determining range in a high-PRF pulse doppler system is described in Paragraph 6-G. In order to determine range on a given scan over the target, it must be detected in all PRF's, and the return must not be eclipsed nor can there be interference with a return from another target with the same doppler shift but at a different range. Calculating the probability of measuring range in such a situation is quite complicated, but should be possible with the methods indicated. An Example. The following system parameters of a gated pulsed doppler radar are assumed to provide an illustration of the methods under discussion. Rq = idealized range = 25 n. mi. 77 = false-alarm number = 10^ T = pulse width = l^sec fr = pulse repetition rate = 100 kc/sec d = duty ratio = 0.1 n = pulses integrated = 1 R = closing rate = 0.33 n. mi. /sec /sc = scan time = 2 sec AR = range decrement = 0.67 n.mi. For 7] = 10^ and n = I, the value of K is found from Fig. 3-7 to be 6 db, or i^ = 4. The basic single-scan probability of detection of a fluctuating target is thus p, = ,,-4(ff/25)^^ /^ in n.mi. (3-54) This probability has been plotted in Fig. 3-14 as the maximum value o( Pd2- Also plotted in that figure are Pd'^ and Pd^ corresponding to the minimum and average values of Pd2- The shaded area between the minimum and maximum values of Pdi is composed of many oscillations with a period of about 1 /12 n.mi. This is illustrated in the expanded view. At intervals of slightly less than a mile, one of these oscillations deepens into a complete null due to the eclipsing to give a narrow blind region. When the effect of the eclipsing is neglected, the cumulative probability is easily determined from Fig. 3-8. Remembering that straddling has the effect of doubling the effective value of A', the normalized range corre- sponding to Pd~ is defined by 3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 167 ^ - '"{Is) - IT-9- (•«5) The normalized range decrement is thus The resulting cumulative probability is also plotted in Fig 3-14. Postdetection Filtering. It is not uncommon in pulsed doppler systems to use a predetection doppler filter which is considerably wider than the reciprocal of the observation time of the signal. Subsequent post- detection filtering is matched to the signal observation time to provide the maximum output signal-to-noise ratio. In this manner the number of doppler filters required can be materially reduced at the expense, of course, of velocity resolution. The filtering or integration is also somewhat less efficient because it is noncoherent representing an operation on the detected signal plus noise. An exact analysis of postdetection filtering is not possible in general, and we shall look for reasonable approximations. Postdetection filtering is essentially similar to video pulse integration, whose eff'ect on detection was discussed in some detail in Paragraph 3-3, and it is natural to use this approach in establishing the approximate effect of this operation. What we shall do is to derive an equivalent predetection bandwidth which provides approximately the same detection performance as the combination of pre- and postdetection filters which it represents. It is assumed that the target fluctuates from scan to scan but has a constant size during the observation time. The following notation is adopted: B = predetection bandwidth (band pass) ^ = postdetection bandwidth (low pass) B' = equivalent predetection bandwidth (band pass) n = equivalent number of signal samples integrated The output of the bandpass predetection filter can be represented by a series of samples separated by 1 /B (seconds) as was indicated in Paragraph 3-3 where the sampling theorem is quoted. Similarly, the output of the low-pass, postdetection filter can be represented by a series of samples spaced by 1/2^ (seconds). In order to provide signal integration the postdetection sampling time will be longer than that of the predetection signal. The ratio of these sampling times gives the number of predetection samples which are integrated in the postdetection filter: Equivalent number of samples integrated = n = Bjlb. (3-57) 168 THE CALCULATION OF RADAR DETECTION PROBABILITY Now in general the predetection signal-to-noise ratio is proportional to the reciprocal of the predetection bandwidth: S/N'^'^- (3-58) Also, the equivalent signal-to-noise ratio of a fluctuating target is proportional to a power of the number of video pulses integrated as in Equation 3-33: S/N ~ ny. (3-59) The appropriate power y corresponds to the slopes of the curves in Fig. 3-7. The equivalent gain in signal-to-noise ratio obtained through postdetec- tion integration can now be expressed either as the ratio of the actual and equivalent predetection bandwidths or simply as w"^: D / D \ 7 Equivalent gain in S/N ~ "m ~ ^^ = ( ^r ) • (3- 60) The equivalent predetection bandwidth thus is given approximately by B' = (2^y B'-y. (3-61) In Equation 3-33, the value of 7 was found to be 0.6. If this is stretched a point and assumed to be 0.5, the following simple expression is obtained: B' = ^|2^f. (3-62) This approximation is often used for estimating performance where post- detection filtering is involved. 3-6 FACTORS AFFECTING ANGULAR RESOLUTION In many applications, it is required that a radar system be capable of separating or distinguishing closely spaced targets. This capability is referred to as the resolution of the system. Targets may be resolved on the basis of any of their characteristics. Thus they may be distinguished in range, velocity, or angular position. This paragraph discusses angular resolution. ^^ In ground mapping, the radar's angular resolution provides a primary means of target discrimination. In AEW radar systems, the angular resolution of the system breaks up multiple target complexes into individual components to provide an estimate of the threat. In fire-control radar, the angular resolution must be sufficient to separate desired targets from interfering targets and clutter. I'^A similar discussion can be found in J. Freedman, "Resolution in Radar Systems," Proc. IRE 39, 813-1818 (1951), upon which parts of this section are based. 3-6] FACTORS AFFECTING ANGULAR RESOLUTION 169 Antenna Pattern Characteristics. Angular resolution is provided by the directive properties of the radar antenna. The greater the direc- tivity, the better the resolution. There is an enormous variety of types of microwave antennas in use today. The most widely used in airborne radar systems are those employing parabolic reflectors. The discussion will center about this type of antenna although many of the observations are applicable to a much wider class. Parabolic reflectors can be constructed whose characteristics closely approximate those of a uniformly illuminated aperture. The relative voltage pattern radiated (or received) by a uniformly illuminated circular aperture will have the following form^^ at long ranges. r^ , • , 2Ji[(7rD/X) sin^] ,^ ,^, Une-way voltage pattern, circular aperture = , ^ .^ , . — - — (3-63) (tt/J/a) sm p where D = aperture diameter X = wavelength d = angle relative to aperture normal Ji( ) = first-order Bessel function. For convenience, we represent the argument of this expression by x so that the one-way relative voltage pattern is 2Ji(;c) /x. The received voltage reflected from a point target to a uniformly illumi- nated circular aperture used both for transmission and reception will be given by the square of the function in Equation 3-63 or (2Ji{x) jxY. This is also equal to the one-way relative power pattern of such an antenna. This pattern is illustrated in Fig. 3-15 where it is referred to as the two-way voltage envelope generated by a scan over a single target. The antenna beamwidth is normally defined as the width between the half-power points of the one-way antenna pattern. This is indicated in Fig. 3-15. For a uniformly illuminated circular aperture the beamwidth is related to the diameter and wa.elength by Beamwidth, circular aperture = 58X/Z) degrees. (3-64) The envelope of the received power on the two-way power pattern is probably most significant for defining resolution. This is given by the square of the envelope plotted in Fig. 3-15 or {2]i{x) IxY. The antenna pattern and the beamwidth can be modified by illuminating the aperture in a nonuniform manner. A uniform illumination yields one of the narrowest beams, but the sidelobe level is relatively high. The sidelobes of the one-way power pattern in Fig. 3-15 are down 17.6 db from the peak. When the illumination is tapered or stronger in the center of the i«J. D. Kraus, Antennas, p. 344, McGraw-Hill Book Co., Inc., New York, 1950. 170 THE CALCULATION OF RADAR DETECTION PROBABILITY Pattern of a Uniformly Illuminated Circular Aperture 2Ji(x) -29 -30/2 -9 -0/2 0/2 30/2 20 Fig. 3-15 Two-Way Voltage Envelope Generated by a Scan over a Single Target. aperture than at the edge, the sidelobe level can be minimized, but at the expense of a wider beamwidth. In actual practice it is customary to taper the illumination so that the effective beamwidth is about 20 per cent greater than indicated by Equation 3-64; i.e., the multiplying factor becomes 70 rather than 58. Resolution Criteria. When two targets are separated sufficiently, they can be identified as two distinct targets. When they are brought together, their returns merge into a single unresolved return. There are a number of criteria for deciding just when there are two returns and when there is only one. Fundamentally, resolution should be defined relative to the discrimination abilities of the human operator in the particular system involved. In general, though, this is much too complex an approach because of the many factors aflecting human performance, and it is more convenient to adopt an arbitrary definition of resolution. In some cases, this will lead to a situation where targets which are defined to be unresolved can actually be observed as separate entities. Most of the definitions which have been suggested for angular resolution lead approximately to the same result: targets separated by about 1 beamwidth can be resolved. A beamwidth is normally defined as the width between half-power points of the main lobe. We shall adopt a very similar definition of resolution which has the con- venient virtue of yielding a resolution of 1 beamwidth for a uniformly illuminated circular aperture. We shall say that two point targets are resolved when the average minimum of the received power envelope in a scan over thejn is less than half the power from the maxiynum of the smaller of the two. This definition is illustrated in Fig. 3-16, the two-way voltage envelope received from two point targets which are just resolved. As indicated in the figure, the voltage pattern fluctuates markedly depending upon whether the returns are in phase or out of phase. When the received reflections are 3-6] FACTORS AFFECTING ANGULAR RESOLUTION , Average ^Out of Phase ,ln Phase 171 -50/2 -29 -39/2 -9 -Q/2 9/2 9 39/2 29 5Q/2 Fig. 3-16 Two-Way Voltage Envelope Generated by a Scan over Two Targets Separated by One Beamwidth. in phase, only a small notch separates the two targets — they have merged in a single return. When the received signals are out of phase, there is a sharp null midway between the two targets. An average envelope can be determined for a random phase between the two reflections. This average two-way voltage envelope is also shown in Fig. 3-16. The minimum of this average curve is 0.707 of the maxima corresponding to half of the maximum received power. Consequently, the case illustrated in Fig. 3-16 shows the envelope of two targets which are just barely resolved. These targets are separated by a single beamwidth. Thus the definition of resolution adopted conveniently yields one beamwidth for two targets of equal size. Degrading Influences. In most practical situations the resolution will be degraded somewhat by a variety of factors. One such factor is unequal strength of the targets. In Fig. 3-17, the two-way voltage envelope Target 1 Target 2 -39/2 -9 Fig. 3-17 Average Two-Way Voltage Envelope Generated by a Scan over Two Separated Targets of Unequal Size (4 : 1 Power Ratio). is shown of two targets whose maximum received voltages have a 2-to-l ratio. The radar size of these two targets is normally expressed in terms of 172 THE CALCULATION OF RADAR DETECTION PROBABILITY the ratio of their reflected powers, which is 4-to-l or 6 db. The minimum separating the two targets in Fig. 3-17 is 0.707 of the smallest maximum, so that these targets are just barely resolved. The target separation required to achieve this resolution is 1.21 beamwidths. Thus, with a 4-to-l size ratio for targets, the resolution is 21 per cent greater than for targets of equal size. This can become important when the target's size fluctuates randomly. Fig. 3-18 shows how the effective resolution angle varies with target power ratio. 1 2.0 p p — — 1 — — n n ~ — n — ~ — 1 1.8 1 1.6 REQUIR TION . — ■ — RATION RESOLU So _^ — - — -^ KS 1.0 ^ ■^ 2 4 6 8 10 12 14 16 18 20 POWER RATIO OF TARGETS —db Fig. 3-18 Resolution as a Function of Target Power Ratio. Another factor which can affect resolution is the signal-to-noise ratio. The simplest way to account for this factor is to apply the already adopted definition for resolution to the received signal-plus-noise power envelope. The deterioration of angular resolution with signal-to-noise ratio which can be determined in this manner is shown in Fig. 3-19. \ V ■ — Res jiution with n noise 2 3 4 5 6 7 SIGNAL ■ TO ■ NOISE RATIO ~db Fig. 3-19 Resolution of Two Equal Targets as a Function of Signal-to-Noise Ratio. 3-6] FACTORS AFFECTING ANGULAR RESOLUTION 173 Very large degradations of resolution can often be attributed to non- linearities in the receiving system. The dynamic range of many search radar systems is less than 10 db above the average noise level, and 20 db is rare. The apparent beamwidth when scanning a very strong target with a system which has limited dynamic range can be as great as twice the normal beamwidth. In such cases, it is quite possible for large targets to completely blank out smaller adjacent targets which might have been resolved with a linear system. Two other minor factors might be noted, the effects of pulsing and the system bandwidth. When only a limited number of pulses compose the envelope generated by a scan over the target, the exact form of the continuous envelope is somewhat indeterminate. As an extreme example, if only two pulses are received during a scan over a target, the question arises as to whether these are two pulses from a single strong target or from two weaker targets. The effect of pulsing can be regarded as a widening of the effective beamwidth. Equation 3-65 gives a simple and useful approxi- mation for the equivalent effective beamwidth in terms of the actual beamwidth and the angular interval between pulses: Effective beamwidth = V©' + ^^" (3-65) where 9 = antenna beamwidth A^ = angular interval between pulses. The antenna pattern described by Equation 3-63 and illustrated in Fig. 3-15 assumed monochromatic radiation. In some applications where very wide bandwidths are required, the antenna beamwidth will be modified. Such an application might be the use of microwave radiometers for map- ping. When there is no chromatic aberration (approximately true when a parabolic reflector is used) and the average frequency is maintained constant, the increase in beamwidth with bandwidth is small. A maximum beamwidth increase of about 5 per cent is given for a bandwidth of 15 per cent of the average frequency. M. K A TZI N CHAPTER 4 REFLECTION AND TRANSMISSION OF RADIO WAVES 4-1 INTRODUCTION In the propagation of radio waves between a transmitter and receiver, we are interested in the problems associated with power transfer between two terminals. This involves an antenna problemi at each terminal (that is, the transformation of electrical power into electromagnetic waves or vice versa) and the problem of determining how the waves propagate to the receiver. In the case of airborne radar, the receiving antenna is replaced by the target, and interest is centered in reradiation by the target in the reverse direction, back toward the transmitter. This reradiation phenomenon is usually called scattering. The radar case with which we shall be primarily concerned is a special case of scattering in which the angle between the propagation directions of incident and scattered fields is 180°. Scattering may be viewed as an antenna problem, too, for the incident field sets up in the target currents whose distribution depends on the target material and configuration and on the distribution of the incident field. If this current distribution is known, then the field reradiated by it can be determined just as though that current distribution were set up in an antenna. In propaga- tion back from the target to the radar, the scattered wave is involved with the same factors as in propagation from the radar to the target: the radar problem involves (1) two-way propagation, and (2) back-scattering by the target. Thus, in order to predict the strength of the echo received from a target it is necessary to determine the characteristics of the propagation mechanism and also the back-scattering properties of the target. The frequencies normally used for radar operation range from about 100 Mc/sec on up, or wavelengths of 3 meters down to less than 1 cm (see Fig. 1-21). Consequently most targets are many wavelengths in dimension. An antenna of corresponding size would have an extremely sharp radiation pattern, so that the target, considered as an antenna, has a correspondingly sharp scattering pattern. It follows that in general the field scattered backward is very sensitive to target orientation. Targets which move, therefore, usually give a radar echo which varies with time. Since a 174 4-2] REFLECTION OF RADAR WAVES 175 differential radial movement of a half-wavelength of the target or a portion of it is sufficient in many cases to produce a profound variation of echo amplitude, even such targets as trees, towers, and buildings, normally considered stationary, frequently give fluctuating echoes. For a given target, the rate of fluctuation usually will be proportional to radar fre- quency. The current distribution set up in the target depends on the distribution of the incident field. In many common situations, the incident field is rather uniformly distributed over the target aperture, so that the target may be considered to be illuminated by a uniform plane wave. Then the scattering characteristics of the target may be analyzed independently of the propagation factors. This is permissible in the case of most airborne or elevated targets. More generally, however, the incident field may be distributed nonuniformly over the target, because of the nature of the propagation phenomena obtaining between the transmitter and various portions of the target. A ship is an example of a target in which the incident field varies over the target aperture because of the interference between direct and surface-reflected rays, which gives a resultant amplitude that varies with height. In such cases the scattering properties of the target cannot be separated from the propagation factors, so that a specification of the target properties becomes more complicated and involves the propagation factors. This same type of complication is also involved in sea and ground return. The principal propagation factors which affect airborne radar are the following. 1. Reflection from the ground 2. Attenuation by liquid water drops in the air 3. Absorption by atmospheric gases 4. Refraction in the atmosphere This chapter will be devoted to a discussion of these factors and to a description of the characteristics of the principal radar targets of interest in airborne applications; viz., aircraft, sea return, ground return, and rain. 4-2 REFLECTION OF RADAR WAVES The radar equation for free space is derived in Chapter 3 (Equation 3-9). It may be modified to account for the effect of obstacles such as the earth's surface or an inhomogeneous atmosphere by introduction of a quantity called the propagation factor., which is the factor by which the free-space field is to be multiplied to obtain the actual field. This factor, which we denote by F, is a complex quantity, or phasor, representing the modification in amplitude and phase of the free-space field by the actual propagation 176 REFLECTION AND TRANSMISSION OF RADIO WAVES process. F may be a function of the range and other parameters of the particular situation at hand. Thus, the radar equation becomes The quantity a is variously called the radar area, radar cross section, echoing area, and back-scattering cross section. It is sometimes useful to relate a to another quantity known as the radar length, designated by /. This is a phasor which represents the ratio, in amplitude and phase, of the back-scattered field-at-unit-distance to the incident field strength. Its relation to cr is (T = 47r|/r". (4-2) The radar length bears a relation to the received field strength similar to that of radar area to received power in Equation 4-1. Thus, the received field strength Er is given by Er = IE,F-'- '-^- (4-3) where E^ = the transmitted field at unit distance (the far field extrapolated to unit distance from the transmitting antenna) K = Itv l\ = phase constant which expresses the relationship between distance and the phase angle of a transmission of wavelength X. The radar area a may be very much larger than the actual projected area of the target. This may be shown in the following way. If the target is large relative to the wavelength, then it is essentially correct to consider that it intercepts a power P' equal to the product of its projected area A' and the incident power density Wi, P' = A'JV,. (4-4) The currents set up in the target by the intercepted field will produce a far field which has a certain directive characteristic, just as if the target were an antenna with such a current distribution. Hence the target will have a directive gain which is a function of angle. If we call the directive gain in the radar direction G', then the effective power reradiated backwards will be P'G' = A'G'lVi = ctJV,. (4-5) Hence a - A'G'. (4-6) It is obvious that if G' is large, then a will be large relative to the actual projected area A' . As an example, consider a target in the form of a flat 4-2] REFLECTION OF RADAR WAVES 177 metallic sheet perpendicular to the direction of the incoming wave. If we neglect edge effects, the current density is of constant amplitude and phase throughout the sheet. Accordingly, the sheet reradiates like an antenna of aperture A' with a uniform amplitude and phase distribution. Since the gain of such an antenna is G' - 47r A'l\^ (4-7) (which is large \i A' l\'^ is large) we obtain from Equation 4-6 (7 = 4x(/f 7X^)2, for A'/\ » 1 (4-8) One of the conditions assumed in deriving Equation 4-1 is that the x\i- ^^-^ variation in range R over the target \P-^^- "" results in a negligible variation of ^^-^^ the phase of the incident field. In ^^"^ xT order to obtain a numerical estimate \" of the significance of this limitation, we may consider, as an example, an airborne search radar viewing, in free space, a rectilinear target of length 2L at a range R, as illustrated pic. 4.1 Geometry for Limitations of in Fig. 4-1. The difference in range Plane-Wave Conditions. between a point at x on the target and the nearest point of the target is Ai? = (i?2 + ;,2)l/2 _ ^ ^ y,2i2R. (4-9) Assuming that the antenna may be treated as a point source, the round-trip phase difference between the fields reflected back to the source from these two points is A0 = IkLR = l-KX-'IXR. (4-10) From Equation 4-3 the contribution of a differential length of the target to the received signal in free space is ^£. = 1^ ^-^2^(«+^«> --i/ ' (4-11) R~ where dl = differential radar length of the differential target element dx located at a distance x from the center of the target. If we denote the plane-wave radar length of the target by / and assume for simplicity that the radar length per unit length of the target is constant. 178 REFLECTION AND TRANSMISSION OF RADIO WAVES 1 then, neglecting the slight effect of variations in the — term, the total field received will be R" e-^^'^ dl ILR" ' (4-13) where u = ILKXRy^, and C(u) and S(u) are the Fresnel integrals C(u) = / cos (x2V2) dz, S(u) = / sin (7r2V2) dz. Hence the effective radar length is /' = /[C^_,^]. (4.U) From Equations 4-2 and 4-14 the effective radar area a' may be derived as (4-15) C~{u) + S%u) Thus the radar length (and radar area a) becomes a function of range, especially for targets of great width at short range, the measurements being 10^ 2 4 7 10'' 2 4 Fig. 4-2 Radar Cross Section as a Function of Range. 4-3] EFFECT OF POLARIZATION ON REFLECTION 179 in terms of the wavelength. Fig. 4-2 shows a plot of a' I a versus i?/X for various values of 2Z,/A. 4-3 EFFECT OF POLARIZATION ON REFLECTION Although the majority of radars utilize linear polarization, for certain purposes other polarizations are found to be advantageous. The use of circular polarization, for example, reduces rain clutter considerably. Since any state of polarization may be described in terms of two orthogonal polarizations (for example, horizontal and vertical, or right-hand and left- hand circular), we may denote an arbitrarily polarized incident wave by the matrix (f:) Er = (V] (4-16) in which the orthogonal components £i, E2 are complex quantities, or phasors. The radar area of a target depends on the polarization of the incident wave. A long thin (in comparison to X) wire is a good example, since its reflection is very small when the incident field is linearly polarized at right angles to the wire axis, and maximum when parallel to the axis. It is evident, therefore, that the radar area and radar length are dependent on the polarization of the incident field. For targets of complex shape, the total field strength incident at a given point of the target is the resultant of the primary field from the radar and the reradiated fields from other parts of the target. Especially in the case of targets of large size which are in part inclined to the wave front, some of the latter fields have a component of polarization orthogonal to that of the primary field. For targets of symmetrical shape (as viewed from the radar) this cross-polarized component balances out in back-scattering, but otherwise it usually does not. Hence, in general, the back-scattered field has a different polarization from the incident field. The coupling between the incident and scattered polarizations depends on the incident polariza- tion itself. As a result, the radar length is a tensor quantity, which may be written in matrix form as in which each of the components /n, etc. is a phasor. For example, if the 1-polarization is horizontal and 2-polarization is vertical, /n represents the radar length of the horizontally polarized echo from a horizontally polarized radar, /12 is the radar length of the horizontally polarized echo from a verti- cally polarized radar, etc. The reflected field-at-unit-distance is then given by 180 REFLECTION AND TRANSMISSION OF RADIO WAVES + /l2£2\ ~r '22 £,2/ By the reciprocity theorem, /21 = /i2- An interesting theorem follows from Equation 4-18: For any given target and aspect, there is a polarization of incident field which gives maximuyn echo, and another which gives zero echo. This can be seen readily as follows. By adjustments of the radar antenna system, the ratio EilE\ may be adjusted (in magnitude and phase) until the received polarization is orthogonal to that of the receiving system, so no signal will then be received from the target.^ Similarly a polarization may be chosen such that the polarization of the echo coincides with that of the receiver, so that a maximum echo will be received. The radar area a also may be written in the form of a matrix by replacing the quantities Imn in Equation 4-18 by ^mn =47r|/™„|2. (4-19) Then (o"ii cri2\ C2I 0'22/ (4-20) However, one could not deduce the polarization theorem above from this, since the radar area is a scalar. 4-4 MODULATION OF REFLECTED SIGNAL BY TARGET MOTION The radar area of a complex target such as an aircraft depends on its orientation, or aspect relative to the radar. An aircraft is subjected to roll, pitch, and yaw motions by atmospheric turbulence. In addition, it may have internal motions due to rotating propellers and surface vibrations. Its gross aspect will vary with time if the target aircraft is on a noncollision or maneuvering course. All of these factors will affect the instantaneous radar area, so that the radar echo will have corresponding time variations. Some of these effects will be considered in greater detail in Paragraphs 4-7 and 4-8. Another important effect produced by target motion is the change in frequency due to the doppler effect which was discussed in Paragraph 1-5. If the radar and the target have a relative approach velocity V, and the transmitter frequency is/o, the echo frequency is (see Equation 1-19) / = /o(l + IV I C) = /o + 2/7X0 = /o +/o. (4-21) Ut is possible to build a radar which transmits one polarization and receives, on two separate receivers, the transmitted polarization and its orthogonal. For such a system, the theorem applies to only one received polarization at a time. 4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 181 In ordinary (non-doppler) radar, this shift in frequency due to the average approach velocity of the target is not noticed in the case of a point target. For extended targets, such as rain clouds and the ground or sea, for which various portions of the target area fill all or an appreciable part of the radar beam, the approach velocity varies over the beam, so that the composite echo has a spectrum of frequencies. In a doppler radar this spectrum will be properly discernible as frequency shifts relative to the radar frequency. In a non-doppler radar, beats between the various frequencies will be pro- duced in the final detector, so that an echo spectrum will also be obtained. In the case of an aircraft, a turn, pitch, or yaw will also introduce doppler beats which are discernible in a non-doppler radar. For example, consider the effect of a turn, which imparts an angular velocity c3 of the target about its center of gravity. Two fixed points on the target a distance D apart will then have a relative radial velocity toward the radar of A/^ = coD cos (4-22) as can be seen from Fig. 4-3. Hence by Equation 4-21 the difference in doppler frequency between these two reflection points is Af=?^=?5°^. (4-23) A A Thus the doppler frequency will be proportional to radar frequency, to = SD cos d Fig. 4-3 Differential Doppler Effect Due to Turning of Target. the angular velocity of the target, and to the gross aspect of the target. These and other effects which result in fluctuations of the target echo will be discussed further in later sections. 4-5 REFLECTION OF PLANE WAVES FROM THE GROUND The reflection of radar waves from the ground or sea surface is an important factor in a number of phenomena associated with airborne radar. Among these may be cited the lobe structure which is encountered in tracking low-altitude targets, height-finding errors for such targets, and the dependence of sea and ground clutter upon polarization and depression angle. In all these cases, an understanding of the basic phenomena can be obtained from a consideration of the reflection of plane waves from a plane homogeneous surface. The reflection of a plane wave from flat ground depends on the frequency, polarization, and angle of incidence of the wave, and on the electrical properties of the ground (dielectric constant and conductivity). A wave of 182 REFLECTION AND TRANSMISSION OF RADIO WAVES complex polarization customarily is resolved into its orthogonal linearly polarized components parallel and perpendicular to the surface, which, in the case of reflection from the Fig. 4-4 Reflection at the Ground. ground, are horizontally and verti- cally polarized components, respec- tively. These components can be treated separately and recombined after determining the change in amplitude and phase of each on reflection. The reflection coefficients for horizontal and vertical polarizations are given by the well-known Fresnel equations^ sin d - (e - cos^ 0)1/2 sin e + {e - COS" 0)1/2 es'md - (e - cos^^)!/^ |p//k-^^^ (4-24) Iprk-'*^ (4-25) € sm -f (e - cos20) 1/ where 6 = depression angle of the radar (see Fig. 4-4) e = complex dielectric constant of the surface. The complex dielectric constant e is given in terms of the permittivity and conductivity of the ground k and a by 6 = --j— = e'-7V' (4-26) eo coeo where eo = permittivity of free space. Values of typical ground constants and reflection coefficients are readily available in the literature. ■'^"^ A dependence of the reflection coefficient on frequency enters Equations 4-24 and 4-25 through the dependence of e" on frequency. In addition, however, the ground "constants" k and a themselves are functions of frequency, by virtue of the dispersion of water. This dispersion takes place just in the frequency region most used for airborne radar. The resulting dispersion of ground thus depends on its water content. For airborne radar this is particularly important for water surfaces. Figs. 4-5 and 4-6 show the variation of the dielectric properties of pure, fresh, and sea water with 2See J. A. Stratton, Electromagnetic Theory, Sees. 9.4 and 9.9, McGraw-Hill Book Co., Inc., New York, 1941. 3F. E. Terman, Radio Engineers' Handbook, pp. 700-709, McGraw-Hill Book Co., Inc., New York, 1943. ■•C. R. Burrows, "Radio Propagation over Plane Earth-Field Strenirth Curves," Bell System T^f/^. J. 16, 45-75 (1937). 5R. S. Kirby, J. C. Harman, F. M. Capps, and R. N. Jones, Effective Radio Ground-Conduc- tivity Measurements in the United States, National Bureau of Standards Circular 546. 4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 120 100 80 e' 60 40 20 183 — 1 1 — Pure and Fresh Water — Sea WatPt- i \ \\> _^20°C / o°c^^ \ \ V 10 10^ 103 104 105 FREQUENCY (Mc) Fig. 4-5 Dielectric Properties of Pure, Fresh, and Sea Water. 10' Sea Water Tresh Water 6?S\t\ <^" 10 102 103 FREQUENCY (Mc) Fig. 4-6 Dielectric Properties of Pure, Fresh, and Sea Water. frequency, taken from Saxton.^ The curves for temperatures of 0° and 20°C bring out a dependence on temperature as well. Figs. 4-7 through 4-10 show the magnitude and phase angle of the reflection coefficient of sea water for a temperature of 10°C at several wavelengths. Similarly, Figs. 4-11 and 4-12 show the reflection coeffi- cients for two different types of ground. For most airborne radar work, solid ground may be treated as a pure dielectric. These figures bring out clearly the diflFerence between horizontal and vertical polarization. For horizontal polarization, there is only a slight variation in magnitude and phase of the reflection coefficient with depression angle. For vertical polarization, however, there is a marked variation, caused by a partial impedance match of the two mediums which occurs at the Brewster angle. The reflection coefficient reaches a minimum magnitude and has a phase angle of 90° at this angle (the Brewster angle itself depends on frequency). 6J. A. Saxton, "Electrical Properties of Sea Water," Wireless Engineer 2^, 269-275 (1952). 184 REFLECTION AND TRANSMISSION OF RADIO WAVES IPI H— ^ '^ ^ . 10 m 1 m- / /^ >< -< / / y^ > ' ^ lOcml J. i-in / / / ^ ^ ^ / // ^W^ ly / / \\ // ly V 10 20 30 40 50 60 70 90 Fig. 4-7 Magnitude of Reflection Coefficient for Sea Water (Temperature = 10°C) as a Function of Depression Angle. 190 180 180 160 140 120 100 80 60 40 20 1 cm 1 m 10 cm 10 m ^ _ ^ \ \ Icm 1 1 " Ik ) cm \ V u 1 m. V \\^ \ 1^ \ 10 m ■ 10 20 30 40 50 60 70 80 90 Fig. 4- J Phase of Reflection Coefficient for Sea Water (Temperature = 10°C) as a Function of Depression Angle. 4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 185 1234bb7 8u9 10 6° Fig. 4-9 Expanded Plot of Fig. 4-7 for Depression Angles Between and lO'^ 182 'l81 180 180 160 140 120 100 80 60 40 20 Im- 10 cm 10 m. -— -'^ - \^ =^ ^^ =s^ \ \ "^ '^ ^ \ \ \ ^ \ \ \ \^? ^=lc m \ \ \^ cm \ \ \ Im \ \ \ \ 1« ni \ ^ \ N \ -^ ^ =:^ ' " ^^ 12 3 4 5 6 7 6° Fig. 4-10 Expanded Plot of Fig. 4-8 for Depression Angles Between 0° and 10° 186 REFLECTION AND TRANSMISSION OF RADIO WAVES \ \ h" \ k ""'■^ -^ \ / ^ \ V / ^ \ / / \ / v / 10 20 30 40 50 60 70 Fig. 4-11 Magnitude of Reflection Coefficient for Average Land (e' = 10, a 1.6 X 10~^ mho/m as a Function of Depression Angle. 200 180 160 140 120 100 80 60 40 20 n <^H /- —10 m 1 m S3 •-^10 cm ^x= 10 m 0V ^■0 m k 10 20 30 40 50 60 70 80 90 0° Fig. 4-12 Phase of Reflection Coefficient for Average Land (i' = 10, a 10~^ mho/m) as a Function of Depression Angle. 1.6 X 4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 187 The behavior is more complicated when the ground is stratified. The cases which are important to airborne radar are that of a layer of ice on top of a water surface, and that of a layer of snow on land. Then multiple reflections can occur between the surface and subsurface boundaries, with a resulting modification of the effective reflection coefficients^; the effective reflection coefficient then becomes an oscillating function of the electrical thickness of the ice or snow covering. When the radar target is at a low altitude, a variety of phenomena are generated by the interference of direct and reflected waves. Referring to Fig. 4-13, if both the direct and indirect paths are illuminated equally by the radar antenna, the resultant field at the target is Image Fig. 4-13 Path Difference Between Direct and Indirect Paths. E = Ea{\ + p^-^-^^^) where Ed is the field due to the direct wave, and Ai? (4-27) Ih sin d is the path difference. The ratio EjEd, obtained from Equation 4-27, thus is the propagation factor F due to the presence of the ground. In Paragraph 4-2, where this factor was defined, it was pointed out that the received power from a radar target is modified by the factor \FY. Introducing the values for K and A/? in Equation 4-27, we obtain F = E/Ed = 1 + pe-''-"" ^*" '"-. (4-28) The most significant and striking phenomena resulting from the inter- ference of direct and reflected rays are the lobe structure and the polarization dependence below the first lobe. The formation of a set of lobes is easily seen from Equation 4-28. With fixed 6 and continuous increase oi h, the resultant field will pass through alternate maximums and minimums when the phase ■^J. A. Saxton, "Reflection Coefficient of Snow and Ice at V. H. F.," Wireless Engineer 27, 17-25 (1950). 188 REFLECTION AND TRANSMISSION OF RADIO WAVES lag of the reflected wave is an even or odd multiple, respectively, of 180°. The height interval between an adjacent maximum and minimum is Ml = X/4 sin d. (4-29) This succession of maximums and minimums of the resultant field gives rise to the lobe structure in the vertical coverage of the radar, which is especially- important for search radars. The location of a given maximum or minimum is different for vertical and horizontal polarization because of the phase of the reflection coefficient p. For airborne radar work with pencil beam antennas, the lobe structure usually is of importance only for targets at small depression angles, since otherwise the narrow beamwidth of the antenna would not illuminate the indirect path strongly. The lobe structure is pronounced only if the value of Ah is large compared with the vertical extent of the target. If the target covers more than one lobe, it effectively averages out the field variation over the lobe. This actually produces a net increase of gain over the free-space field acting alone, which is due to the field reflected from the surface. A similar oscillation in the propagation factor is observed with fixed radar and target altitudes and a continuously varying range as the target passes through the lobe pattern. In this case the angle 6 can be expressed as H -\- h H -\- h ^^^ ^ = ^WTUH ^ -R- ^""^^^ where H = radar altitude h = target altitude R = target range. Neglecting the change in the phase angle of the reflection coefficient p, the range interval between an adjacent maximum and minimum is where R is the mean range. Thus for a target flying at a constant height, the lobes become packed more densely as the range is decreased. The oscillations of received power caused by the lobes are superimposed on a free-space variation which is proportional to the inverse fourth power of range as indicated in Equation 4-1. The situation is somewhat different when the target lies below the first lobe. In this case, the angle Q will be small and an expansion of F in powers of sin Q can be used. To obtain this expansion, we note first that ph and pr, which are given in Equations 4-24 and 4-25, can be approximated as p// = -1 +2(6- l)-i/2sine (4-32) PK = -1 + 2e(e - ])-■/- sin Q. (4-33) 4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 189 Similarly, the exponential term in Equation 4-27 may be approximated as ^-iiKh sin e ^ I _ j2Kh sin d. (4-34) Substituting these approximations into Equation 4-28, and retaining only the first-order terms in sin d, we obtain Fh = 2[(e - l)-^/2 -\-jKh] sin d (4-35) Fv = 2[e{e - l)-i/2 -\-jKh] sin 6. (4-36) Thus, for sufficiently small d, both Fh and Fv are proportional to sin 6. But sin 9, as shown by Equation 4-30, is inversely proportional to range. Hence below the first lobe it follows that F is also inversely proportional to range: F oc R-\ (4-37) Therefore the received echo power, which is given by Equation 4-1, will be inversely proportional to the eighth power of the range in the region below the first lobe: P,i oc R-^ (4-38) This is in contrast to the inverse fourth power of the range which holds for free space. The range at which the transition occurs from a fourth-power law to an eighth-power law for a target which spans more than one lobe will be discussed in Paragraph 4-10. Equations 4-35 and 4-36 show how the resultant (one-way) field varies with height below the first lobe. Very close to the surface, where the term is small in magnitude compared with the other term in the brackets, Fh and F^ become Fy = j^^JyT. si" ^- (4-40) Hence the ratio of the fields at the target with vertical and horizontal polarization will be Fv/Fh = 6. (4-41) If the radar area of the target is the same for these two polarizations, then the ratio of the received echo powers will approach |e|^. This difference is important in the case of sea clutter. As the height is increased, the term j2Kh eventually will become large relative to the other term in the brackets in Equations 4-35 and 4-36. Then the field at the target will be approximately proportional to height and will be almost the same for either polarization. 190 REFLECTION AND TRANSMISSION OF RADIO WAVES The first-order expansions of F in powers of sin 6 are limited in their ranges of validity. This can be seen, for example, in Fig. 4-7. For vertical polarization, the range of validity is limited to angles smaller than the Brewster angle, while for horizontal polarization, the angular range is much greater. For airborne radar frequencies, the range is about d < 30° for horizontal polarization, and < 4° for vertical polarization. The results deduced above are based on the properties of plane waves. In the case of the spherical waves radiated by an antenna, there is a surface wave which should be added to the direct and reflected waves. For airborne radar frequencies, however, this generally is unimportant. 4-6 EFFECT OF EARTH'S CURVATURE The effect of the earth's curvature is twofold. First, it alters the geometry so that the path difference between the direct and reflected waves is decreased, and second, it decreases the amplitude of the reflected wave. The change from the plane to the spherical geometry is equivalent to a reduction in the heights of radar and target, as illustrated in Fig. 4-14. Direct \Na\je /? Fig. 4-14 Curved Earth Geometry. The second ef^-ect of the earth's curvature is to decrease the amplitude of the reflected wave, because the incident waves within a small range of vertical angles are spread out, or diverged^ into a larger range of vertical angles on reflection from the convex surface of the earth. For all distances encountered in airborne radar work, the reduced heights Aj', Ao "^ay be calculated from 7;; = /;i - A/;, = /;i - d,yia (4-42a) Ji'., = /;o - A/;2 = //2 - doVla (4-42b) where a is the earth's radius, and diidi) is the distance from the reflection point to the transmitting (receiving) point. As will be shown in Paragraph 4-18, the effect of average or "standard" atmospheric refraction may be 4-6] EFFECT OF EARTH'S CURVATURE 191 allowed for by increasing the earth's radius by a factor 4/3 to an effective earth's radius a.. ^ ^/2a. (4-43) With this factor, and expressing heights in feet and distances in statute miles, the height reductions due to curvature take the simple form A/7 1,2 = ^1,2-/2. (4-44) For a given value of range R, which is practically the same as the total distance d = d\-\- di measured along the earth's surface, the determination of ^1 (or d^ leads to the cubic equation Idr^ - Ud;^ - \lalh^ + ^2) - ^-] dr + la,h^d = 0. (4-45) Once this is solved for di, h[ and h'^ may be calculated from Equation 4-42 and the remainder of the geometry handled like a plane-earth problem. Since the solution of the cubic is laborious, it is usually simpler to employ a graphical solution by plotting h[ jdi and h'^ jdi versus di. The proper value of di occurs where these two quantities are equal, since this gives equal A^alues of before and after reflection. As mentioned above, reflection at a spherical surface reduces the reflec- tion coefficient from the plane earth value p to p' = pD (4-46) where D is the divergence factor. This is given by y^ajsmd (4-47) (•-^r For very small values of 6 the divergence factor causes reduction of the effective reflection coefficient p' given by Equation 4-46 to a small value. In fact, at the horizon (9 = 0) D = 0, so that there p' = 0. However, the representation of the propagation process in terms of only a direct and a reflected ray breaks down as the horizon is approached. Norton^ gives as the limit to which Equation 4-46 is restricted: h , . . . . (4-48) d\ Practically all airborne radar ranges will be within this limit as long as atmospheric refraction does not depart greatly from the standard condition. ^K. A. Norton, "The Calculation of Ground-Wave Field Intensity over a Finitely Con- ducting Spherical Earth," Proc. IRE 29, 623-639 (1941). 192 REFLECTION AND TRANSMISSION OF RADIO WAVES 4-7 RADAR CROSS SECTIONS OF AIRCRAFT Because all aircraft have dimensions large in terms of the wavelengths used in airborne radar, the radar area of an aircraft target is very sensitive to its instantaneous aspect. Because of air turbulence, the aspect is subject to statistical variations of roll, pitch, and yaw. Consequently the radar area is a statistically fluctuating quantity and it is not possible to give a single number for the radar area of such a target. The quantities of chief interest are the probability distribution of amplitudes, the aspect and frequency dependencies, and the time characteristics, or spectra, of the fluctuations. The amplitude distributions and aspect and frequency dependencies of certain aircraft will be presented in this paragraph, while the fluctuations and their effect on tracking systems will be discussed in Paragraph 4-8. A summation of the echo characteristics and their association with the physical structure and dynamic behavior of the aircraft will then be presented in Paragraph 4-9. This should make it possible to predict, with useful accuracy, the main features of the radar properties of a new or unmeasured target aircraft. An appreciation of the complicated nature of the radar area of an aircraft and its association with the physical structure of the aircraft can be obtained from some of the results of basic investigations into the properties of radar echoes from aircraft carried out by the Naval Research Laboratory, and recently made available. ^~^^ Pulse-to-pulse measurements were made of both fighter and bomber categories, with propeller-driven and jet-propelled models in each category. The measurements were made on three fre- quencies, 1250, 2810, and 9380 Mc/sec, with the radars searchlighted on the target by an optical tracker, and pulsed simultaneously. No antenna scanning was used, so that the observed fluctuations were all attributable to the target. The data were analyzed to determine amplitude distributions, median radar area versus aspect, and frequency spectrum of the amplitude fluctuations. The particular series of measurements to be discussed was made at elevation angles less than 15°. These measurements will be discussed in some detail, since comparable data have not been published before. Many of the characteristics observed can be explained in terms of physical processes, so that from these it should be possible to predict the principal characteristics to be expected in other situations. 9F. C. MacDonald, Quantitative Measurements of Radar Echoes from Aircraft III: B-16 Amplitude Distributions and Aspect Dependence, NRL Report C-3460-94A/51, 19 June 195J. low. S. Ament, M. Katzin, F. C. MacDonald, H. J. Passerini, P. L. Watkins, Quantitative Measurements of Radar Echoes from Aircraft V: Correction of X-Band Values, NRL Report C-3460-132A/52, 24 Oct. 1952. "W. S. Ament, F. C. MacDonald, H. J. Passerini, Quantitative Measurements of Radar Echoes from Aircraft VIII: B-45, NRL Memorandum Report No. 116, 28 Jan. 1953. 12W. S. Ament, F. C. MacDonald, H. J. Passerini, Quantitative Measureynents of Radar Echoes from Aircraft IX: F-5I, NRL Memorandum Report No. 127, 4 March 1953. 4-7] RADAR CROSS SECTIONS OF AIRCRAFT 193 Fig. 4-15 shows the cumulative amplitude distribution of a 2-second sample of echoes from the B-36, plotted on so-called Rayleigh coordinates. 30 25 20 15 o 10 -10 B - 36 Run 10 X '^ Range 18,400 - 19,400 yd Az. 5.9°- 6.1° El. 5.7°- 6° X >», 1 ^ "> ^ »>. "^ .X ., ^^ ki^x , Mc ^ "^ X X .^ ^^ X ; A. ^^^ \ > >. >> *> ^ •x ^53.>» ^">s. ^*^. '^^ 1 ^ 30 25 ~h 20 # 15 2 10 I ■a 5 -5 -10 0.010.11 5 10 30 50 70 80 90 95 98 99 99.5 99.9 PERCENTAGE OF THE TIME THE VALUES EXCEED THE ORDINATE Fig. 4-15 Cumulative Amplitude Distribution of B-36 Echo, Approach Aspect. The straight lines through the points represent the Rayleigh distribution. ^^ Even with such a short sample (in this case, of only 240 pulses), the fit to a Rayleigh distribution is quite good. From the data obtained, it was concluded that for a 2-second sample the echo amplitude (and thus the radar area) is Rayleigh distributed for most aspects, except at broadside aspect. At broadside the amplitudes were compressed into a rather narrow range. The Rayleigh distribution signifies that the target consists of a large number of elements whose relative phases are independent and vary randomly during the time of the observation. The number of independent elements which constitute a "large" number, however, need be only about four or five if their amplitudes are comparable. Thus the conclusion to be drawn from the B-36 amplitudes distribution is that, except at broadside, the target consists of just such a "large" number of independent scatterers, and that in 2 seconds their relative phases pass through substantially all possible combinations.^^ At broadside, however, the echo from the flat 13J. L. Lawson and G. E. Uhlenbeck, "Threshold Signals" Mass. Inst. TechnoL, Laboratory Series 24, 53, McGraw-Hill Book Co., Inc., New York, 1950. i^Practically, "all possible combinations" probably is satisfied if the phases vary over one or two times 360°. 194 REFLECTION AND TRANSMISSION OF RADIO WAVES fuselage is so large relative to the echoes from other parts of the aircraft that it predominates over them and a relatively small amplitude variation results. Fig. 4-16 shows a 5-second sample for the B-45 twin jet bomber, taken for an approach run in which the aspect varied by 2|°. Here the approach to the Rayleigh distribution is poor, the range of amplitude variation being much more compressed. However, a 5-second sample at another aspect, in which the aspect angle varied 4|° (Fig. 4-17) shows a much closer 0.01 0.5 5 20 40 60 80 90 95 p 98 99 99.5 PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE Fig. 4-16 Cumulative Amplitude Distribution of B-45 Echo, Approach Aspect; Small Range of Aspect Angle. approach to the Rayleigh distribution. From an examination of data taken over a wide range of aspects, it was concluded that samples in which the azimuth of the B-45 varied by more than 4° gave a satisfactory fit to the Rayleigh distribution. Fig, 4-18 shows a set of distributions for the F-51 single-engine propeller- driven fighter. Although the lower amplitudes follow the Rayleigh distribu- tion quite well (on 9380 Mc/sec the lower levels were lost in the noise at the range of the measurements plotted in this figure), there is a pronounced upswing at high levels above the values expected from the extension of the PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE Fig. 4-17 Cumulative Amplitude Distribution of B-45 Echo; Larger Range of Aspect Angle. Rayleigh line fitted to the lower levels. This is attributed to reflections from the propeller, which, for a rather large range of angles, are stronger than from the remainder of the aircraft. This is shown by the original pulse-to-pulse photographs shown in Fig. 4-19. Here every fifth pulse (repetition rate 120 cps) is much larger (on all three frequencies) than the intervening ones. The dominance of the propeller echo was found to be especially marked at oblique aspects of the aircraft between head-on and broadside, corresponding to the region where a portion of the blade is nearly normal to the line from the radar. To depict the gross aspect variation of cr, the median values over roughly 5° of azimuth were plotted against azimuth angle. Figs. 4-20 to 4-22 show the results for the B-36, B-45, and F-51, respectively. In averaging over an angular range of this amount, sharp peaks of the aspect dependence are largely smoothed out. In all cases, however, a prominent and rather broad maximum occurs in the neighborhood of the broadside aspect. This is especially true in the case of the B-36 (which has a rather flat fuselage) as shown by the 9380-Mc plot in Fig. 4-20 (broadside data for 1250 and 2810 Mc were saturated and so are absent from this figure). The F-51 has its broadside maximum at an azimuth of about 98°, probably owing to the tapered tail section of the fuselage. REFLECTION AND TRANSMISSION OF RADIO WAVES PERCENTAGE OF TIME THE VALUES EXCEED THE ORDINATE Fig. 4-18 Amplitude Distribution of F-51 Echo, Showing Effect of Propeller Reflection. The data in Figs. 4-20 to 4-22 also give the frequency dependencies of the radar area. A single number for the average radar area was obtained for each frequency by averaging all the values of a (in square meters) plotted in each figure. These averages are controlled by the large peak in the neighborhood of the broadside aspect. Similar numbers were obtained for all aspects measured outside of the broadside region. These latter numbers give a measure of the frequency dependencies of the aircraft for most tactical applications. The results are shown in Table 4-1. The B-36 and B-45 averages are roughly independent of frequency, but the F-51 average a increases approximately proportional to frequency. -E 4-1 COMPARISON OF B-36, B-45, AND F-Sl AVERAGE RADAR AREAS Frequency, Mc Average a (in db > Im"^) Excluding Broadside Region Average cr (in db > \m-) Including Broadside Region B-36 B-45 F-51 B-36 B-45 F-51 1250 2810 9380 13.2 16.4 10. 10.7 10.6 10.2 -0.5 3.6 6.6 « 14.9 14.5 17.3 -0.2 5.6 9.2 'Broiulside ret^ion saturated. 4-7] EFFECT OF EARTH'S CURVATURE 1250 MC/S 197 l-SEC 2810 MC/S 9 380 MC/S Fig. 4-19 PuIse-to-Pulse Records of F-51 Echo, Showing Strong Propeller Echo Every Fifth Pulse. 198 REFLECTION AND TRANSMISSION OF RADIO WAVES 30 ^S 25 I— S 20 b g o 10 15 B-36 9380 Mc u Median a Vs. Aspect d| 10 — i = o = D = 3° Elevation 1 1 • U 5° • u ^ • V 5 f o • ib°^„ n n n ^• • D n • • a • •-x- ? 9° p ° »u •£ X o '^^ ° i ' fl rf •? f a • % o 11° n i\ I • i ,^ S3 Q COL 320°330°340°350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° B-36 2810 Mc Median a Vs. Aspect • '^ • T k U i • • •□ x=3"Ele A = 4° ation •T ° t D » » • u * • o-5°H □ 3 O 3 • • »=8° -12° 320°330°340°350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°110°120°130° 25 a: 20 tZ 2 15 _b q 10 = 3°E = 4° 3 = 5°" levatic n o u= 9"Elev *=10° tion ■=1 r u I 4 * ■ □ ■ i • • • 1 • 1 a r = 8° t . u ^ ^ i J fi -J ' / ^'^ D • t ▼ • "? ° '□ B-36 1250 Mc • 3 Mec ianff Aspec Vs. 320°330°340° 350° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° Fig. 4-20 Plot of Median Echo of B-36 Averaged over 5° of Azimuth. 4-8 AMPLITUDE, ANGLE, AND RANGE NOISE^s In a tracking radar, rapid variations in target aspect can affect the smoothness of tracking and hence its accuracy. The variations in the target J^Most of the material in this Paragraph has been derived from the following NRL reports, and from references in footnotes 11-14, which can be consulted for further details: J. W. Meade, A. E. Hastings, and H. L. Gerwin, N'oise in Tracking Radars, NRL Report 3759, Nov. 15, 1950. A. E. Hastings, J. E. Meade, and H. L. Gerwin, Noise in Tracking Radars, Part II: Dis- tribution Functions and Further Power Spectra, Jan. 16, 1952. D. D. Howard and B. L. Lewis, Tracking Radar External Range Noise Measuretnents and Analysis, NRL Report 4602, Aug. 31, 1955."^ A. J. Stecca, N. V. O'Neal, and J. J. Freeman, J Target Sitnulator, NRL Report 4694, Feb. 9, 1956. A. J. Stecca and N. V. O'Neal, Target Noise Simulator -— Closed-Loop Tracking, NRL Report 4770, July 27, 1956. B. L. Lewis, A. J. Stecca, and D. D. Howard, The Effect of an Automatic Gain Control on the Tracking Performance of a Monopulse Radar, NRL Report 4796, July 31, 1956. 4-8] AMPLITUDE, ANGLE, AND RANGE NOISE 199 1 B 45 1 ■^ 7 7 = 2° El x = 3° _J evation 12bUM Median a vs 1 Aspect v>^ A = 4° • = 6° ° = 7° - ^hese Four L] » Valu Lov Lirr s are D T O O "^^' td" its Q , • ^ » 7 "" • ° ® = 12° '^\ ^ ^ i 1000 100 10 B-/ 281f 'n Mr, o9P IVled an (7 vs. \spec t 0" o o o u ^1 X ° ^ • ^. o't J / •* ^°^o □ T • '^ il^Di 5 a D 100 10 ^ u 938C c c^ u Mc vs./ \spec a o Med an a t ",o ^ „. □ o a o o ^ • A ^ i ni " ,^ s^" '(^ ^-^8 •°"^ $ ' » ^.^ ^o ° 1 0.1 1000 100 300 320 340 10 20 30 40 50 60 70 AZIIVIUTH ANGLE (deg) 90 100 120 Fig. 4-21 Plot of Median Echo of B-45 Averaged over 5° of Azimuth. characteristics are referred to as target noise (or scintillation) and have an effect similar to noise originating in the tracking system. Target noise is subdivided into amplitude noise, angle noise (or glint), and range noise. Amplitude Noise. In a sequential lobing radar, the antenna beam is scanned over a small range of angles, and comparison is made of the signals received with the beam swung to opposite sides of the boresight axis. The difference between these signals is used to drive the antenna toward a 200 REFLECTION AND TRANSMISSION OF RADIO WAVES F-51 Average of the Median cr's for Each □ Five Degrees of Azimulli a = 1250 Mc D° ^These Values arc Lower Limits = 2810 = 9380 □ D 8 a J a n nfi Ln O 30 □ o o o a o ° n D ft a A a a a o ° A i ^ A A ' a a'^ 15 'T 10 1 5 ^o o | I I t^Yt I I I °l f- l ° l °1 ^1 I L°h- I I h ^ D0_ - OAA _Oo 2 -10 -15 300 320 340 10 20 30 40 50 60 70 80 90 100 120 140 AZIMUTH ANGLE (deg) Fig. 4-22 Plot of Median Echo of F-51 Averaged over 5° of Azimuth. position where the signals are equal. Since the two samples are not received at the same instant, any change in signal amplitude during the scanning cycle, caused by target fluctuation, will lead to an angle error indication even if the antenna is pointed correctly at the target. In order to make an optimum choice of the parameters of a sequential lobing system, therefore, it is necessary to have information on the amplitude fluctuation charac- teristics of target aircraft. Some of the causes of amplitude fluctuations already have been men- tioned in Paragraph 4-7, e.g. propeller modulation. A clear example of this is shown in Figure 4-23, which is the spectrum of amplitude fluctuations in 1 ^" G " 1 4 < ' T 1 1 C s , " 1 t 5 1 A 1 U^ \ l- ^ ^ i ihVl^ " ~Mr- -M^ 4l4^ Jt s ^B ^W^M kJ v...y k , J iL ,/i ., ^ 1 Y] ^vyyw.^ ^^w/ IW^y Uk^^^v... * 1 1 1 1 100 200 250 300 350 FREQUENCY (cps) Fig. 4-23 Spectrum of Amplitude Noise of SNB (Two-Engine Transport) Aircraft in an Approach Run, Showing Spectral Lines Due to Propeller Modulation (= 9400 Mc). an approach run of an SNB (two-engine transport) taken at a radar frequency around 9400 Mc. The repetition frequency of the radar was 1000 cycles, so spectral information up to 500 cycles is derivable. Here peaks occur at multiples of about 58 cps. The remainder of the spectrum consists of a continuous band'^ whose amplitude decreases with increasing l*The small ripples or scintillations are due to incomplete smoothing in the spectrum analyzer. AMPLITUDE, ANGLE, AND RANGE NOISE 201 frequency. The propeller modulation occurs at the blade frequency (engine rps X number of propeller blades), so the fundamental modulation fre- quency should be the same for all radar frequencies. This was seen to be the case for the F-51, as illustrated in Fig. 4-19.^'' It/) I n nc f \. \ XAh, jalI ^ j '^ ^'^"^W'V-'-*--s---~ — ^ .^^ I I I I I r 2 4 6 8 10 15 20 25 35 45 60 f(cps) B - 45 RUN 10 1250 Mc/sec RANGE 13,600 YD AZIMUTH 96°20' - 98°42' ELEVATION 6°56' - 6°55' 2 4 6 8 10 15 20 25 f(cps) 35 45 B - 45 RUN 10 2810 Mc/sec RANGE 13,300 YD AZIMUTH 91° -94° ELEVATION 6° 55' 2 4 6 8 10 35 45 60 B- 45 RUN 10 9380 Mc/sec RANGE 13,600 YD AZIMUTH 96° 20' - 98° 42' ELEVATION 6° 56' - 6° 55' Fig. 4-24 Spectrum of Amplitude Noise of B-45. For a jet aircraft, the mechanical vibrations of the salient reflection surfaces would be expected to be of high frequency and noiselike in nature '■'Because the radar samples the instantaneous radar area of the target at discrete intervals r times per second (r = the pulse repetition frequency), a frequency of s cps in the spectrum could result from beats between the actual frequency and the repetition frequency. Hence the actual frequency of the target radar area spectrum may be any value given by «r ± s, where n is an integer (including 0). For the F-51, the observed propeller modulation frequency was found to correspond to w = 7 or 8. 202 REFLECTION AND TRANSMISSION OF RADIO WAVES so that a line frequency spectrum similar to the propeller modulation spectrum would not be expected. The amplitude modulation spectrum observed in such cases would be due chiefly to beats between the doppler frequencies of the salient reflection centers, as described in Paragraph 4-4. The frequency of this type of modulation, therefore, would be proportional to the radar frequency. Whether a continuous spectrum is obtained, or one with spectral lines superposed on a continuous spectrum, depends on the nature of the perturbations of the aircraft for straight-line flight and on the duration of the observation (that is, on the length of sample). Figure 4-24 shows a spectrum^^ of the amplitude noise of the B-45 for which the observation time was 5 seconds. The voltage-time plots from which the spectrum was prepared are shown in Fig. 4-25. The 9380-Mc h B ■ 45 Run 10 1250 Mc Range 13,600 ya Azimuth 96°20'- 98°42' Elevation 6''56'- 6°55' ^ i:¥ ills V Fig. 4-25 Voltage-Time Plots of B-45 Fxho, Showing Low-Frequency Modulation. i^The resolving power of the spectrum analyzer used was about L5 cps. Also, its response begins to drop off below about 6 cps. 4-8] AMPLITUDE, ANGLE, AND RANGE NOISE 203 spectrum in Fig. 4-24 shows two peaks, not quite resolved completely, at 5 and 6.6 cps. In Fig. 4-25 the presence of a fundamental period in this region can be seen clearly; and similar fundamental periods, but of progressively lower frequency, can be seen in the 2810- and 1250-Mc plots. From an examination of the drawings of the B-45, it was concluded that the observed spectrum could be explained as the doppler beats between reflections from the engine nacelle, the wing tank, and a portion of the fuselage near the tail which was broadside at this aspect, all caused by a yaw rate of 0.14° /sec. For a longer sample (i.e., longer period of observation) during which the yaw rate varied, the discrete frequencies varied with time so that the spectrum over such a time of observation was smeared out into a more or less continuous band. Whether one should be concerned about a continuous band or discrete frequencies in the design of a tracking radar depends, therefore, on the time constant of the system — in other words, on the passband of the servo loop. This problem will be discussed in more detail in Chapter 9. Angle Noise. In a simultaneous lobing system (to be discussed in more detail in Paragraph 6-2) the signals which are compared to obtain angle information arrive simultaneously; thus amplitude fluctuations of the target echo do not generate angle error signals. If the angle tracking servo loop of a simultaneous lobing system is opened and the target is tracked optically, it is found that error signals still occur. These must be caused, therefore, by wandering of the eflFective center of reflection of the target. The principle involved in the generation of angle noise may be explained in terms of a target which consists of two point reflection centers, whose relative amplitude and phase vary as the target aspect changes. Fig. 4-26 'BoresightAxis illustrates this model. Consider the following type of tracking system. A dual-feed antenna produces lobes Fig. 4-26 Physical Arrangement for on each side of the boresight axis. Illustrating the Origin of Angle Noise. the gains being equal along that axis. The voltage from each lobe is passed through an amplifier and square-law detector, and their diff"erence is used to derive angular inform^'.tion. The sum of the detector outputs is used for the AGC voltage of the receiver, so that the angular deviation of the arriving signal is determined by the difi^erence divided by the sum of the detector outputs. Let the boresight axis be directed at the center of the target (midway between A and B) and let the angle of A (and of B) to the boresight axis be d. For small d, the slope of the antenna lobes can be considered constant, and will be denoted by-^. Denoting the received RF voltages due to A alone 204 REFLECTION AND TRANSMISSION OF RADIO WAVES and B alone when ^ = by £a and Eb, and their phase difference by </>, the total RF voltage received by the upper lobe is Eu = Ea{\ + gd) + Eb{\ - ge)e^'^ (4-49) and by the lower lobe El = Ea{\ - ge) + Eb{\ + ge)e^\ (4-50) The difference channel voltage is then En = k{\Eu\' - \El\') (4-51) = AkgdiE/ - En"-) where k = amplifier gain, and the sum channel voltage is Es = k{\EuV + \ElV) = 2k[{EA' + Eb' + IEaEb cos 4>) + {gey{EA'' + En' - lE^En cos «^)] = 2^(£x- + Ej? + 2£A£yj cos 0). (4-52) The error voltage is Eu __ ^ Ea' — Eb~ ,. rn■^ Es ~ ^ Ea' + Eb' + IEaEs cos 0' ^'^"'^^ In the presence of only a single target, say at A, the error voltage would be d. (4-54) (t)' (4-56) Comparing this with Equation 4-53 we see that the apparent reflection center of the dual-reflector target lies at an angle d' to the boresight axis 1 - {Eb/EaY ,, ... 1 + {Eb/EaY + 2{Eb/Ea) cos (^ ■ ^^'^'^ Thus, d' depends on the ratio ^ 1 - {Eb/EaY ' 1 + {Eb/EaY + 1{Eb/Ea) cos 4> and therefore on the relative amplitude and phase of the two reflections. This ratio can be less than, equal to, or greater than unity in absolute value, and may be positive or negative. In other words, the apparent reflection center can lie anywhere within the target, or even completely outside it. From Equation 4-56, r = when E^a = En-, so that equal reflectors have an apparent center midway between them, regardless of their relative phase. For values of £«/£.! other than unity, the value of r depends on the relative phase, 0. The apparent reflection center lies ovitside the target, when \r\ > 1, which requires that Er Ei - cos = COS (tt -</))> 7^ or — • (4-57) r.,\ ■ rLB 4- AMPLITUDE, ANGLE, AND RANGE NOISE 205 Thus this phenomenon occurs in the region of destructive interference be- tween the two reflections, as may be seen from the circle diagram in Fig. 4-27, which is drawn for Eb > Ea- In general, a target may have a number of reflection centers whose relative amplitudes and phases vary with the instantaneous target as- pect. Delano^^ has developed the theory for a target composed of an infinite number of statistically in- dependent point sources and has determined the statistical properties of the Fig. 4-27 Vector Diagram for Two- Target Example. Angle Noise vs Time from R4D at 0° V — 1 sec — ^ Angle Noise vs Time from R4D at 90' 1 Angle Noise vs Time from R4D at 180° Fig. 4-28 Angle Noise Samples for R4D. '9R. H. Delano, "A Theory of Target Glint or Angular Scintillation in Radar Tracking, Proc. IRE^l, 1778-1784 (1953). 206 REFLECTION AND TRANSMISSION OF RADIO WAVES apparent center of reflection. For a row of reflection centers uniformly spaced along a length L perpendicular to the line-of-sight from the radar, for example, the fraction of time that a conically scanning radar points ofl" I I I I I (a) SNB (0 SNB 180 TRUE NOISE FREQUENCY SPECTRAL ENERGY DISTRIBUTION Fig. 4-29 Spectra of Angle Noise for SNB Aircraft. 4-8] AMPLITUDE, ANGLE, AND RANGE NOISE 207 the target is 0.134. For equal reflection centers uniformly spaced over a circular area, this fraction becomes 0.2. Data on angle noise have been collected at the U. S. Naval Research Laboratory in connection with investigations of tracking noise. This work has been done at a frequency of about 9400 Mc, so that it is applicable to airborne radar problems. Examples of angle fluctuations of the type discussed above are shown for an R4D in Fig. 4-28. The variations in apparent reflection center are seen to be greater than the linear dimensions of the aircraft. Also, the deviations at the 90° aspect are larger than at 0° and 180°. Fig. 4-29 shows the spectrum^" of angle noise for several runs of an SNB. The angular noise in this spectrum has been multiplied by the target range so that the spectral density is independent of range and expressed in yards per Vcps- The amplitude decreases fairly regularly with increasing fre- quency, so that the total noise power is finite. In many cases the spectrum can be fitted satisfactorily by a curve of the form A = A,{\ ^r/U)-'i'~ (4-58) which corresponds to the transfer characteristic of a single-section RC low-pass filter. Since the vertical span of an aircraft is much less than its horizontal span, angle noise of a single aircraft is much less in elevation than in azimuth. In low-angle tracking, however, reflection from the ground or sea has the effect of creating an image aircraft at an equal distance below the surface (see Fig. 4-13). If the angle between the target and its image is less than the elevation beamwidth, the two will not be resolved. Variations in the phase difference between direct and reflected rays then will cause the effective reflection center to wander between target and image or beyond them. This has been observed to be the case. As the range decreases, the angular fluctuations increase until target and image can be resolved and the tracking system locks on. However, it is possible for lock-on to occur on the image instead of the target! Multiple targets have a similar effect on azimuthal variations. Thus, multiple targets which are not resolved give rise to a much higher level of tracking noise than a single target. Range Noise. In addition to causing angle noise, fluctuations of the effective center of reflection of the target can give rise to fluctuations in range, or range noise. Fig. 4-30 shows typical time plots of range noise for several classes of target. Fig. 4-31 shows the range noise of a single SNB at ^OThe observation time included in this spectrum is about 80 seconds, so that spectral frequencies below about 3^ cps are cut off by analyzer limitations. 208 REFLECTION AND TRANSMISSION OF RADIO WAVES k\l^ht\M^^^%^ PB4Y _i lE^^^^'^^Yy^^E^. (c) 2SNB90' ^1 H ^5 fS n -||!B^i|i^ m ift ELAPSED TIME SNB SNB SNB Fig. 4-30 Sample Time Function Plots of Range Noise from the (a) PB4y (Four- Engine Bomber) at 180° Target Angle, (b) ANB (Two-Engine Transport) at 180° Target Angle, and (c) SNB pair. aspect angles of 0°, 90°, and 1 80°, and of two SNB's at °90. Both the range noise spectrum and its probability distribution are shown. The distribution of the apparent reflection center in range generally lies wholly within the target, as can be seen from the curves at the right in Fig. 4-31. In this respect, range noise differs from angle noise. The reason lies in the different methods used for error detection in angle and in range. As in the case of angle noise, multiple targets which are not resolved will give rise to much higher noise levels than a single target. 4-9 PREDICTION OF TARGET RADAR CHARACTERISTICS Quantitative measurements of radar characteristics require special instrumentation which is not widely available and is costly. Furthermore, targets of interest may not be available for measurement. For example in a problem of the type outlined in Chapter 2, it is highly unlikely that such definitive target information will exist. Hence great importance attaches to methods whereby the characteristics of interest may be calculated. As 4-9] PREDICTION OF TARGET RADAR CHARACTERISTICS 209 (a) SNB 0' (b) SNB 90° 3.3 yd-] ^ KSNBH KSNBH 1 u It <^ s :: y ii : ^■° -5; ^^ \ ^"^ " N 01 23456789 10 11 TRUE NOISE FREQUENCY SPECTRAL ENERGY DISTRIBUTION YARDS AMPLITUDE DISTRIBUTION Fig. 4-31 Sample Spectra Obtained for the SNB at Target Angles of (a) 0' (b) 90°, (c) 180°, and for (d) the SNB pair. a result of theoretical studies, ^^ and measurement programs such as those referred to in Paragraphs 4-7 and 4-8, techniques have been developed with which rather good success can be expected in predicting the radar charac- teristics of a target or target complex if their basic configurations are known. A brief discussion will be given of methods which have been used for aircraft targets. Since the dimensions of aircraft are many wavelengths for airborne radar frequencies, the methods of geometrical and physical optics are sufficiently accurate for most purposes. The principal reflections, therefore, come from surfaces which have portions parallel to the wavefront. The aircraft then can be approximated by a small number of bodies of simple shapes, for which the radar lengths can be calculated. Over a small range of angles about any given aspect, the contributions from the mdividual bodies will pass through substantially all values of relative phase, so that the average 2iSee, for example, Studies in Radar Cross-Sections, XV, University of Michigan, Engi- neering Research Institute, Report 2260-1-T, Appendix A, and further references therein. 210 REFLECTION AND TRANSMISSION OF RADIO WAVES radar area and its rms spread may be calculated quite easily. In this way, these quantities may be determined over the range of aspect angles of interest. In addition to the average areas, it is possible to predict the target noise. After having replaced the aircraft by a finite number of simple shapes at fixed locations, the doppler frequencies generated by a known variation of angular velocity can be calculated in the manner described in Paragraph 4-4, and the spectrum can be determined. This calculation can be expedited by the use of a target simulator. ^^ As an example, Fig. 4-32 shows the ll||,,U . iWk,i.. TUD 3 ' Tfll 1 i 1 '^^w^l^^^ """^'Xl^llMtj^,,. ^^ ?*?•'*•-*'*•«-'- II 1.2 4.8 6.0 7.2 FREQUENCY (cps) 8.4 9.6 10.8 Fig. 4-32 Spectral Distribution of Angle Noise from a B-17 (Four-Engine Bomb- er) Aircraft (Actual Measurement). Jk-i u- n/k i 1 ^^WE i ^3iu < y^y*^ I - ^*^^^-^-^ , : 1 1 1 1 1 1 1 1 1 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 FREQUENCY (cps) Fig. 4-33 Spectra! Distribution of Simulated Angle Noise from a B-17 Target Simulator. measured angle noise spectrum of a B-17, while Fig. 4-33 shows the simu- lated spectrum. The latter was obtained with a "target motion" composed of a random oscillation having a Gaussian distribution of velocities about zero mean, with a standard deviation of 0.3° /sec. 22See footnote 17, NRL Reports 4694, 4770, 4796. 4-10] SEA RETURN 211 4-10 SEA RETURN23 In detection or tracking of targets on or near the surface, it is necessary to be able to distinguish the target from the background clutter due to reflections from the surface itself. For example, the AEW system discussed in Chapter 2 was required to distinguish between sea return and target echoes. In order to design a radar for such an application, the mechanism of sea return and its relation to radar and tactical parameters must be understood. In operations over water, this sea return or sea clutter is caused by all elements of the surface within a resolution element of the radar. Since this surface area is a function of pulse length and antenna beamwidth, the radar area of such a target complex is a function of range and some of the radar parameters. If the area of surface illuminated by a resolution element is not too small, then the return can be considered to be from scatterers uniformly distributed over this area, or area extensive. It is then convenient to use a quantity, the radar area per unit area of sea surface, usually denoted by 0-°, which is independent of the radar parameters. Then the radar area (J is a = a^A, (4-59) where A is the physical area of a resolution element. For pulse radar with pulse length r and azimuth and elevation beamwidths 4> and 6, respectively, where A is the smaller of the two values: A = m li^ii = R^L {6 small) (4-60a) A = i?2$e/sin d {6 large) (4-60b) withL = Tf/2. The characteristics of sea return depend on a number of parameters. These are the depression angle, polarization, frequency, and the condition (or "state") of the sea. The last quantity includes the many factors which affect the contour of the surface, such as the wind (its speed, direction, and duration), swell (wave systems generated by distant storms), currents, shoals, breakers, and others. Large waves in themselves do not necessarily produce strong clutter, since a heavy swell, with little or no wind blowing, does not produce a high level of clutter. On the other hand, clutter springs up suddenly with a sudden onset of wind, even before waves of appreciabel height are built up. Thus clutter seems to be more intimately connected with the secondary wave structure due to the local wind than with the primary wave structure. Although the many factors which influence sea 2^For a thorough discussion of the World War II investigations of sea return, see the account by H. Goldstein in D. E. Kerr (Ed.), Propagation of Short Radio Waves, pp. 481-581, McGraw- Hill Book Co., Inc., New York, 1951. 212 REFLECTION AND TRANSMISSION OF RADIO WAVES clutter make the phenomenon a complicated one, by now most of these appear to be understood. The radar parameters which control sea clutter are the depression angle (6), polarization, frequency, antenna beamwidths, and pulse length. The last two have been discussed already and may be eliminated, when the return is area extensive. The other three are interrelated. Since sea return is back-scattering from the surface itself, factors which affect the illumination of the surface elements responsible for the back scattering have an important effect on o-°. Polarization, frequency, and wave height are such factors, and their effects are intertwined. Katzin^S-^ showed that a number of their effects could be explained on the basis of an illumination of the scattering elements which is the combination of direct and reflected waves, similar to that above a plane reflecting surface. As was shown in Paragraph 4-5, interference between the direct and reflected rays creates a lobe structure above the surface. Below the lowest lobe, Pr oc R~^ for a single target. For an extended target distributed in height from the surface upward, this relation still holds if the top of the target is below the first lobe. At nearer ranges, where the target subtends one or more lobes, the target in effect integrates the varying illumination over it, so that the deep ripples of the lobe pattern are smoothed out and Pr cc R"^. For pulsed radar, the illuminated area of the sea surface at small depression angles is proportional to range, in accordance with Equation 4-60a above, so that the reflection Log(R\/4/i) Fig. 4-34 Composite Plot of Sea-Clutter Power at Three Frequencies and Six Altitudes from 200 to 10,000 ft: Coor- dinates Normalized to Test Interference Mechanism. mechanism just described should give an R~^ range variation at short ranges, and an R~'' range variation at long ranges. Fig. 4-34 shows a composite plot of sea-clutter power measured with horizontal polarization at various frequencies and altitudes to test the interference mechanism. Here the measured points clearly define a re- gion where Pr oc R~^, which changes into one where Pr oc R~''. The agreement with the type of behavior just discussed lends strong support to the reflection mechanism. Pre- sumably reflection takes place froni the region ahead of the wave crests in the manner indicated in Fig. 4-35. ^'•M. Katzin, "Back Scattering From the Sea Surface," IRE Cofitrn/ion Record, 3, (1), 72-77 (1955). 25M. Katzin, On the Mechanisms of Radar Sea Clutter, Proc. IRE 45, 44-54 (1957). 4-10] SEA RETURN 213 As was shown in Paragraph 4-5, vertical polarization produces a much stronger field on and just above a reflecting surface than does horizontal polarization. Hence the scattering elements of the sea sur- face are more strongly illuminated if p^^. 4-35 Possible Geometry of Reflec- vertical polarization is used so that ted Wave from Sea Surface, sea clutter at low angles is much stronger with vertical than with horizontal polarization, assuming that the same for both polarizations. Because of the presence of reflected waves, the appropriate radar equa- tion for sea clutter is obtained from Equations 4-1 and 4-59: where F is a suitable average value of F'^. For a uniform distribution of a with height above the surface, Katzin^^ gives F = 6, R< Rt (4-62a) F=6{Rt/R)\ R> Rt (4-62b) where Rt is the transition range between the R~^ and R~^ regions. The simple plane surface reflection theory for a surface with a reflection coeffi- cient of — 1 gives Rt = 5hH/\ (4-63) where h is the radar height and H the height of the top of the target, which here is to be interpreted as the height of the wave tops above the equivalent reflecting plane. Since wave heights themselves are distributed in a statistical manner, and the location of the equivalent reflecting plane is not known, an empirical relation must be deduced from experiment. A limited amount of experimental evidence suggests the relation Rt = 2A//i/io/X (4-64) in which //i/iu is the crest-to-trough wave height exceeded by 10 per cent of the waves (a unit frequently used by oceanographers). A further consequence of the reflection interference phenomenon at very small depression angles is that the return no longer remains "area exten- sive." The appearance of the sea clutter on an A scope then breaks up into a series of discrete echoes or "spikes" which appear much like individual targets. These can persist at fixed ranges for periods of a number of seconds. Fig. 4-36 shows an example of this. Spikiness is explainable by the com- 214 REFLECTION AND TRANSMISSION OF RADIO WAVES Bm$i^> ■ Fig. 4-36 Expanded A-Scope Photographs of Sea Clutter. The Saturated Echo in the Center of the Sweep Is from a Stationary Ship. Blanking Gates Near Both Ends of the Sweep Define the Base Line. Wavelength, 3.2 cm. bined effects of destructive interference below the first lobe and a statistical variation of wave heights. Because of the statistical distribution of wave heights, there are relatively few waves which exceed the average height, and these thus appear as isolated "targets. "-*r^ ^For a summary of a series of observations of spiky clutter, see F. C. MacDonakl, Charac- teristics of Radar Sea Clutter, Part I: Persistent Target-like Echoes in Sea Clutter, NRL Report 4902, March 19, 1957. 4-10] SEA RETURN 215 The variation of o-*^ with depression angle is a function of wind speed. Fig. 4-37 shows measurements on vertical and horizontal polarization at 24 cm as reported by MacDonald,^^ while Figs. 4-38 and 4-39 show measure- -20 -30 -60 w t- ^^' -^ /'^f ^M A 1 -<it ^R x'a t /f .a >' ;.# .^ 0.1° 1.0° 10' DEPRESSION ANGLE 100^ Fig. 4-37 Sea Clutter, 1250 Mc: Solid Line = Transmitted and Received Vertical Polarization. Dotted Line = Transmitted and Received Horizontal Polarization. 20 10 S-lO b -20 -30 -40 15-20 Knots Fig. 4-38 Velocity; 20° 40° 60° 80' ANGLE OF DEPRESSION cj"" as a Function of Wind X = 8.6 mm. 90 1.25 cm Sea Clutter 10 10 - 15 )( Knots — ^ / 15-20 \ / / - Knots-. \ n "20-25 \ \ ///' / -10 Knots — V \ /V/// / \ Y\/ / -V(l J"\^' / /C-'^/^ "' / Knots -3U C^Z^-^^ - 10 / Knots/ -40 1 1 1 1 1 1 ri 1 1 20° 40° 60° 8 ANGLE OF DEPRESSION Fig. 4-39 Velocity, X as a Function of Wind 1.25 cm. ments on vertical polarization at X = 8.6 mm and 1.25 cm by Grant and Yaplee."* At small depression angles, a^ increases with wind speed, but near vertical incidence this trend is reversed and a^ decreases with increasing wind 2^F. C. MacDonald, "Correlation of Radar Sea Clutter on Vertical and Horizontal Polar- ization with Wave Height and Slope," IRE Convention Record \ (1), 29-32 (1956). 2*C. R. Grant and B. S. Yaplee, "Back-Scattering from Water and Land at Centimeter and Millimeter Wavelengths, Proc. IRE 45, 976-982 (1957). 216 REFLECTION AND TRANSMISSION OF RADIO WAVES speed. It should be noted that near vertical incidence, c7° rises to as high as + 15 db. These and other characteristics of sea clutter have been explained by a theory developed by Katzin.^^ This theory is based on scattering by the small facets of the sea surface as the basic scattering elements. At small depression angles, where none of the facets is viewed broadside, the facets which back-scatter most effectively are those with perimeters of about a half-wavelength. The back-scattering of a facet increases with its slope, so that those near the wave crests contribute most strongly, even if the illumination is constant with height. Although at small depression angles the back-scattering is at angles far removed from the facet normals, at large depression angles some of the facets are viewed broadside, so that these contribute most strongly in this region. The larger the facet the greater is the contribution. The angular dependence at large depression angles then is governed by the slope distribution of the facets. At airborne radar frequencies, the angular dependence of o-° should follow the slope distribution rather closely. This distribution is approximately Gaussian, but is more peaked and is skewed in the upwind-downwind direction. ^^ At small depression angles, the theory shows that o-" is directly propor- tional to wind speed, but at high angles it is inversely proportional to wind speed. These features of the theory seem to be in accord with available measurements. The evidence regarding the frequency dependence of o-'^ is not uniform. Katzin^^ stated that o-° (at small d) was roughly proportional to frequency in the frequency range 1.25-9.4 kMc, and gave the formula for a" upwind at small depression angles, (7« = (2.6 X \0-'W^i'\-' (4-65) where W is the wind speed in knots and X the wavelength in cm. (In this formula, the illumination factor F is included in o-".) Wiltse, Schlesinger, and Johnson^** found a" to be substantially constant in the frequency range 10—50 kMc. Grant and Yaplee-* found cr" to increase with frequency range 9.4-35 kMc, the increase being about as the square of frequency at vertical incidence and about as the first power or less at 10° depression angle. Grant and Yaplee's measurements on the different frequencies used were made on different occasions, however, so that their results on the frequency dependence are subject to wider variations due to different surface conditions. It is quite possible that the frequency dependence of a" 2^C. Cox and W. Munk, "Measurement of the Roughness of the Sea Surface from Photo- graphs of the Sun's Glitter," J. Opt. Soc. Aju. 44, 838-850 (1954). 3"J. C. Wiltse, S. P. Schlesinger, and C. M. Johnson, "Back-Scattering Characteristics of the Sea in the Region from 10 to 50 kmc, Proc. IRE 45, 220-228 (1957). 4-11] SEA RETURN IN A DOPPLER SYSTEM 217 may vary somewhat from time to time, depending on the condition of the sea surface. 4-11 SEA RETURN IN A DOPPLER SYSTEM The doppler shift due to relative motion of radar and target was discussed in Paragraph 4-4, and the echo frequency due to a transmitted frequency/o was given as /=/o + 2F/X (4-66) where V is the line-of-sight component of the approach velocity of radar and target. If both radar and target are in motion, then with respect to fixed coordinates V may be divided into two parts, one due to the radar velocity F^, the other to the target velocity Vf Equation 4-66 corre- spondingly may be written as /=/o +/.+/.. (4-67) If the angle of the target from the ground track of the radar is Xj then the doppler frequency due to the radar motion is /. = (2F./X)cosx=/icosx (4-68) /i = lVrl\. (4-69) If the target is the surface of the sea, then the angle x will vary over the portion of the surface which is illuminated by the radar, owing to the finite width of the antenna beam. Hence there will be induced by the motion of the radar a corresponding band, or spectrum, of doppler frequencies Jr. This may be called an induced doppler spectrum. Similarly, if the various portions of the surface are in relative motion, then even if the radar is stationary or the radar beam is so narrow that no appreciable variation in cos x takes place over the illuminated area, a range of doppler frequencies/^ will result from the intrinsic motion of the surface. This may be called the intrinsic doppler spectrum. The relative importance of the induced and intrinsic doppler components depends on the relative velocities and the geometry, as well as on the antenna beamwidth. Referring to Fig. 4-40, cos X = cos ^0 cos 00 where 0o is the depression angle and 0o the azimuth angle of the surface target relative to the aircraft motion. Hence for a small azimuth deviation ±A(/) from the mean value 0o, we have cos X = COS 0o(cos 00 cos A0 ± sin 0o sin A0) (4-70) = cos ^o{[l - (A0)V2] cos 00 ± A0 sin0o}. 218 REFLECTION AND TRANSMISSION OF RADIO WAVES Fig. 4-40 Geometrical Relations for Doppler Spectrum of Sea Return. Thus it is evident that the spread in cos x, and hence the width of the induced doppler spectrum, will be minimum along the ground track (00 = 0)- As an example, we consider an airborne X-band pulse radar (9400 Mc/sec) with a horizontal beamwidth of 3° (A</> = 1.5°), and an aircraft speed of 200 knots. Then from Equation 4-69, /i = 6.44 kc. At grazing depression angles along the ground track {do = 0), the induced doppler spectrum has a half-power width of 0.000343/i = 2.2 cps, while at 45° to the ground track the half-power width is 0.0037 /i = 238 cps. In principle, the induced spectrum is known from information available at the radar and thus may be compensated, in part, by appropriate (though complicated) circuitry. There still remains the intrinsic spectrum, and a knowledge of this is necessary in order to determine the capabilities and limitations of doppler radar in target detection and tracking through clutter. Measurements of the intrinsic spectrum of sea clutter have been made by the Control Systems Laboratory of the University of Illinois. ^^ These were made with a coherent airborne radar operating on a wavelength of 3.2 cm. By making measurements along the ground track, the width of the induced spectrum was made small relative to that of the measured spectrum, so that the measurements yielded the intrinsic spectrum directly. By multiplying the frequencies by X/2 (see Equation 4-69) the results were converted to a velocity spectrum. ^iThe information on the intrinsic doppler spectrum of sea clutter was furnished through the courtesy of the Control Systems Laboratory, University of Illinois. 4-12] GROUND RETURN 219 Frequency spectrums were obtained for 15-second samples of recorded data (corresponding to 3750 ft along the sea surface), and also frequency, B-scope records of the spectrum as a function of position of the illuminated patch on the sea surface (250 ft long). These will be referred to as the A display and the B display, respectively. For low sea states, the average spectrum had a Gaussian shape, and width between half-power points of 2 to 3 knots (60-100 cps at X band). The corresponding B display was generally smooth on upwind and downwind edges for all ranges. Fig. 4-41 (a) shows a sample of the A and B displays for a low sea condition (wave height 2 ft, wind 9 knots). The 3-db band- width of 82 cps in this sample corresponds to a velocity spread of 2.55 knots. As the wind increased and white caps became evident, the A display broadened asymmetrically to 5 knots or more. The B display then was broadened on the downwind edge to an extent which varied irregularly with range, but the upwind edge remained smooth. Fig. 4-41 (b) shows a sample of the A and B displays for a medium sea (wave height 5 ft, wind 16 knots). Here the 3-db bandwidth is 172 cps, corresponding to a velocity spread of 5.35 knots. These characteristics suggest that the irregular downwind broadening was due to spray filaments or patches associated with the white caps, blown off the wave crests and moving downwind more rapidly than the crests. 4-12 GROUND RETURN The applications of airborne radar over land cover an even broader field than operations over sea. As in the case of sea clutter, reflections from a land surface form a clutter background which tends to obscure the desired echo, e.g. from a target aircraft flying at low altitude. At small depression angles, ground return generally is considerably larger than sea return. Hence the problem of detecting ground targets obscured by ground clutter is correspondingly more difficult. In another type of radar application — ground mapping — the most important characteristic is the contrast obtainable between objects and their immediate surroundings as determined by the nonuniformity of the return. This characteristic governs the type of ground map which may be obtained by radar techniques, as was discussed in Paragraph 1-4. The ground return which competes with or obscures the target echo is confined to the return from ground elements at the same apparent range as the target. Such returns can be received either on the main beam or the sidelobes of the antenna pattern. A special form of sidelobe clutter — the altitude line — will be discussed in the next paragraph. In a pulsed radar the returns which arrive at times precisely separated by the interpulse period appear at the same apparent range. This gives rise to 220 REFLECTION AND TRANSMISSION OF RADIO WAVES SAMPLE NO. 262 Sept. 24,1954 Altitude -lOOOff Ronge- 13,050yds. Depression- 1.46° Wave height-2.0ft. 3db bandwidth-82cps WIND 9 knots 80° SAMPLE NO. 140 Sept. 22,1954 Altitude-2500ft. Range- 15,080 yds. Depression-3.l7° tieight-S.Oft. 3db bandwidth -I72cps A Wind 16 knots ^^J^ <iep. 2400 2500 FREQUENCY cps (b) MEDIUM SEA STATE 2200 2400 2600 2800 FREQUENCY cps Fig. 4-41 Doppler Spectrum of Sea Clutter, Showing both the Spectrum Averaging over the Sample (a) and the Spectrum vs. Time or Range. A/C velocity refers to Ground Track. a form of clutter known as nndtipk-time-aroioid echo (MTAE). This clutter is therefore important for ground targets whose range is given by R„ = R.^nR^rs (4-71) 4-12] GROUND RETURN 221 where Rg = range of ground reflector Rt = target range Rprf = range corresponding to an interpulse period. Echoes from objects in the range interval Rt + Rprf {n = 1) are known as second-time-around echoes (STAE). It is not unusual for STAE to be comparable to or stronger than the desired target echo. The range of angles for which STAE may be troublesome depends upon the geometry and radar parameters of the particular system under consideration. Obviously, a knowledge of the characteristics of ground return is of importance in this and in other applications. Some measurements of ground return at wavelengths of 0.86, 1.25, and 3.2 cm are given in a paper by Grant and YapleCj^^ who used vertical polarization. Fig. 4-42 shows their results for a tree-covered terrain with the trees in full foliage. It will be noted that a^ is very roughly independent of the angle of incidence. o-° also increases with the frequency, but even at X = 8.6 mm does not exceed — 13 db at any angle. Thus this type of terrain absorbs most of the incident energy. 20 Trees With Foliage 10 - - 1.25 cm - 8.6 mm \ -10 _ X-U -20 y^^^^^cxr -30 - 3.2 cm -40 - III 20 - Tall Weeds or Flags — Apr 1955 Green Grass / 10 - Nov 1955 Dry Grass / / _ 1 / -10 -20 -30 -40 1 1 1 1 1 1 1 1 1 1 20° 40° 60° 80° ANGLE OF DEPRESSION 20° 40° 60° 80° ANGLE OF DEPRESSION Fig. 4-43 Comparison of a^ for Green Fig. 4-42 o^ for a Tree-Covered Terrain. Grass and Dry Grass. Fig. 4-43 shows the results for ground covered with tall weeds and grass in the spring when the grass was green and the ground wet and marshy, and in the fall when the grass and ground were dry. Two effects are clearly evident from this figure. (1) There is a very large and rapid rise in o-° near vertical incidence, amounting to 15-20 db, when the ground is wet. (2) Although a" increases steadily with frequency under dry conditions, when 222 REFLECTION AND TRANSMISSION OF RADIO WAVES the ground is wet the curve for 1.25 cm falls much below the other two. Grant and Yaplee state that this behavior was always found on 1.25 cm when the ground was wet, and suggest that this anomaly may be associated with the water vapor absorption peak near this wavelength (see Paragraph 4-16). Aside from the large rise near vertical incidence, the remainder of the curve lies approximately 5 db higher when the ground is wet than when it is dry. The large increase of o-" near vertical incidence when the ground is wet can be explained as caused by patches of water which are viewed broadside, as in the case of the facets which have been proposed as the scattering elements for sea clutter. This emphasizes the importance of plane surfaces whose dimensions are comparable to or large compared with the wavelength when they are viewed broadside. Hence, in attempting to generalize on the basis of the rather meager experimental results which have been reported in the literature, this characteristic should be kept in mind. An important example of this is the case of cultural areas, especially cities. Here, in addition to the presence of large flat surfaces, such as building walls, windows, roofs, and streets, there are many possibilities for corner reflectors. Since corner reflectors have a large radar area over a wide range of angles, they have a very large effect on the radar return. For example, observations of ground painting by airborne radar^^ show that the signals from man-made structures are often too strong to be fully explained in terms of their size, and that a certain amount of corner-reflector action ("retrodirectivity") in the targets must be present. This action is present principally at long ranges (small depression angles) and is responsible for sharp contrast in the return from arrays of buildings at long ranges. At short ranges, where the depression angle is outside the range of corner- reflector action, this contrast tends to fade. These principles have to be kept in mind, for example, in estimating the effect of STAE from a city on the performance of the radar in an AI, an AEW, or a target-seeking missile application. 4-13 ALTITUDE RETURN In Paragraphs 4-10 to 4-12, we have discussed the back-scattering properties of the sea and ground in terms of the scattering parameter o-". This has been done in order that the properties could be applied to radars with a wide range of parameters. In order to determine the response for a particular radar, one needs to consider the radar parameters in connection with the scattering characteristics of the surface. One case which is of some importance is that of the altitude return in pulse radar. This is the signal received from the ground directly beneath the aircraft. On a PPI display 32L. E. Ridenour (Ed.), Radar System Engineering, Vol. 1, pp. 100-101, McGraw-Hill Book Co., Inc., New York, 1947. I 4-13] ALTITUDE RETURN 223 it gives rise to the "altitude circle," while on an A display it is referred to as the "altitude line". In many cases this return is prominent because of the marked increase of cr° which occurs for depression angles near 90° (see Figs. 4-37, 4-38, 4-39, and 4-43). To a radar altimeter the altitude return is the desired signal, while to target detection and tracking radars it is a source of interference or "clutter." Since the antennas of these two classes of radars have widely different beam patterns, the illumination of the ground as a function of angle may vary widely between different applications. A full discussion of the problem, therefore, is beyond the scope of the present treatment, so that only some of the principal factors will be discussed. The expressions (Equation 4-60) were given for the area of a resolution element on the surface. For small depression angles this area is proportional to range, while for large depression angles it is proportional to range squared. The distinction between these two in the case of the altitude line is actually a function of altitude. For example, if both the antenna beam and the pulse shapes are rectangular, and if cr'' is a slowly varying function of angle near vertical incidence (as in the case of Fig. 4-42, for example), then the illuminated area is beamwidth limited if the leading edge of the transmitted pulse passes the outer edge of the antenna beam before the trailing edge of the pulse reaches the ground. The received power of the altitude line then will vary as the inverse square of altitude in accord- ance with Equation 4-60b. Because of the inverse square relationship (as contrasted with an inverse fourth power relationship for a point target) the altitude line return can be very strong. This is particularly true for altitude line return from a flat calm sea which tends to act as a perfect reflector (see Figs. 4-37 and 4-38.) However, if the altitude or beam- width is great enough that the trail- ing edge of the pulse reaches the ground before the leading edge passes out of the antenna beam, then the return is pulse-length limited, and the received power of the alti- tude line will vary as the inverse cube of altitude in accordance with Equation 4-60a (see Fig. 4-44). 20 16 / K / / / / I t V 1--QV'. 1 K 1 1 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 L/2/i I \ \ I \ \ 1 I 0.04 0.08 0.12 0.16 0.20 0.24 0.28 r(Msec)A(1000ft) Fig. 4-44 Angular Extent of Altitude Line vs. Pulse Length. 224 REFLECTION AND TRANSMISSION OF RADIO WAVES Actually, neither the antenna beam nor the pulse shape is rectangular, and the scattering properties of the ground, even if they are area-extensive, may vary with angle, so that a continuous transition from an inverse square to an inverse cube relation takes place. More complicated situations occur when one or more large individual scatterers are located within the illumi- nated area. A more detailed discussion of this problem can be found in a paper by Moore and Williams. ^^ 4-14 SOLUTIONS TO THE CLUTTER PROBLEM Having considered the characteristics of radar targets and of sea and ground clutter, we can now examine these together in order to find the most favorable solution to the clutter problem. There is no unique solution, since the factors involved depend on the operational problem and the limitations placed on the radar parameters. A full discussion of all the considerations and possible solutions is beyond the scope of this chapter, since the problem involves the overall system design and operational philosophy. We shall restrict ourselves to a consideration of certain features of the antisubmarine warfare (ASW) problem, in order to bring out some interesting possibilities based on sea clutter characteristics discussed in Paragraphs 4-10 and 4-11. In the first place, an early decision can be made regarding the polarization of the antenna. Both theory and experiment show that sea clutter levels are much lower on horizontal polarization than on vertical polarization. From Fig. 4-37 it is seen that this can amount to 10 db or more. Hence, unless the target shows a preference for vertical polarization by more than this amount, horizontal polarization clearly is to be chosen. Furthermore the discrimination based on target height, which will be discussed below, will be achievable only with horizontal polarization. The following discussion will be based on a flat earth and will illustrate the principles involved. The modifications necessary to take into account the effect of the earth's curvature have been described in Paragraph 4-6. These will affect the answer only quantitatively and will not change the nature of the results. The primary mission of airborne radar in ASW is search; tracking is a secondary mission. The object of system design and operation is to choose the radar parameters so that the probability of detection is optimized. Inevitably practical limitations will arise which restrict the ranges of certain of the parameters. Ordinary (non-doppler) pulse radar will be considered first, and then the additional improvement due to doppler radar will be discussed briefly. 8^R. K. Moore and C. S. Williams, Jr., Radar Terrain Return at Near-Vertical Incidence, Proc. IRE ^5, 228-238 (1957). 4-14] SOLUTIONS TO THE CLUTTER PROBLEM 225 Since in search it is desirable to sweep out a large area, the problem is concerned primarily with small depression angles. The variation of received clutter power with range will then be of the form shown in Fig. 4-34, and will be given by Equation 4-61 : UtYR' (4-72) In this we may insert the values of A and F given by Equations 4-60a and 4-62, respectively. The horizontal beamwidth <!> and the gain of the radar antenna may be expressed by $ = y^ (4-73) G = ^^ (4-74) where Iw and 4 are the horizontal and vertical antenna apertures, respec- tively, and ka and h are constants of the antenna design. If we adopt the form of relation given in Equation 4-65 for a^, (T« = ^oA (4-75) then Equation 4-61 becomes for the received clutter power ^c - (^,^3) (4-76) where kc = kah^h/i^T)' Fc = 6, R< Re Fc-^6{Rc/R)\ R> Re Re = 2hHiiio/\ = transition range for clutter as in Equations 4-62 and 4-64. kc is primarily a function of local wind speed, while //i/io is dependent rather on wind history, but may be forecast with reasonably good accuracy.^* Similarly, for the power Pt of the target echo, we have from Equation 4-1 If the target is a surface target of uniform section and height Ht, then F^ is to be replaced by F of Equation 4-62, with its transition range, Rt, given by Equation 4-63 Rt-^^-^ (4-78) A 3^W. J. Pierson, Jr., G. Neumann, and R. W. James, Practical Methods for Observing and Forecasting Ocean Waves, H. O. Pub. No. 603, U.S. Navy Hydrographic Office, 1955. 226 REFLECTION AND TRANSMISSION OF RADIO WAVES Then Pt may be written as kTPlLV<TTFT Pt \'R' (4-79) in which h = kb'^/i4T)K Plots of Equation 4-76 for a specific sea condition and of Equation 4-79 will then be as shown in Fig. 4-45. From this, it follows that the range scale Fig. 4-45 Target and Clutter Power Relations vs. Range. in which the target-to-clutter ratio may REGION 1 REGION 2 REGION 3: kahLR kahLR (4-80) may be divided into three regions, be expressed as follows: Pt Pc Pt Pc Pt Pc Obviously, a large antenna width and a short pulse length will increase the target-to-clutter ratio in all three regions. Furthermore, if or is inde- pendent of frequency, then so is Pt/Pc in regions 1 and 3. The locations of the transition ranges Rt and Re can be controlled by the height h of the radar. It is evident from Fig. 4-45 that the largest target-to-clutter ratios generally will be obtained in region 3. Since region 3 is one in which destructive interference operates on both the target and clutter signals, this is a region of relatively low signal \2hHuio) arL ( SHt Y kahLR\2H,i,,)- 4-15] ATTENUATION IN THE ATMOSPHERE 227 strength. Depending on the transmitted power and other radar parameters, therefore, the useful limit of region 3 will be set by the minimum power required to produce a signal detectable above the noise. This minimum power level is indicated by the horizontal dashed line labeled P^in in Fig- 4-45. Since Pm\n depends on receiver bandwidth, effective antenna scanning rate and beamwidth, and other factors, changes which are made in /^/L in order to increase Pt /Pc will also increase P.nin- Although Pt /Pc (in regions 1 and 3) does not contain an explicit frequency factor, both Pt and Pc contain the factor X~-, and so increase with frequency. Fig. 4-45 relates to a specific target area and sea condition. Obviously, one must consider a whole family of such curves, relating to various possible combinations of interest, in order to arrive at the optimum choice of param- eters. Some of these parameters depend on the operational philosophy (e.g. barrier patrol, hunt-and-kill). In addition, the effect of the earth's curvature, which will steepen the rates of signal decrease in region 3, will have to be taken into account. The above discussion refers to non-doppler radar. Doppler radar offers the additional possibility of increasing the target-to-clutter ratio by exploiting differences in the target and clutter spectrums. In order to achieve a gain in target-to-clutter ratio, it is necessary that the target doppler frequency spectrum lie outside the range of the induced doppler spectrum of the clutter. For the example given in Paragraph 4-11 (Vr = 200 knots, A0 = 1.5°) each doppler component of the intrinsic doppler spectrum would be broadened by about 2 cps along the ground track and about 350 cps at right angles to the ground track. The corresponding effective velocity broadening would be about 0.1 and 11 knots, respectively. Thus, no significant improvement will be obtained at large angles to the ground track unless the radial component of target velocity exceeds 10-15 knots, for the 3° beamwidth assumed. Smaller beamwidths would reduce this figure proportionately. In principle it is possible to improve the target-to-clutter ratio by exploiting the difference between the widths of the received target and doppler spectrums. This requires a "velocity" filter (or a set of them). A system employing such techniques is described in Paragraph 6-6, below. 4-15 ATTENUATION IN THE ATMOSPHERE The atmosphere is almost perfectly transparent to radio waves until frequencies in the microwave region are reached. Attenuation of radio waves in the atmosphere is due to absorption by gases (oxygen and water vapor) and absorption and scattering by suspended particles (precipitation, dust). The first effect will be discussed here, and the second in Paragraph 4-16. 228 REFLECTION AND TRANSMISSION OF RADIO WAVES The theory of microwave absorption by oxygen and water vapor has been developed by Van Vleck.^^ The oxygen absorption is due to a large number of overlapping resonance lines, resulting in peaks centered at wavelengths of 5 and 2.5 mm, while water vapor has an absorption peak at 1.35 cm. Fig. 4-46 shows the theoretical attenuation due to oxygen for paths at sea 3 100 -1 1 v\ /AVELENC 3 5TH ( 0.5 \) cm 0.3 .1_ _ 2 o 1 c o i \ Sea Level " .1 M n 1 iL \ \ c \\ // V ^-j \, 4 Kilometer 'A 1 s n ^^^ _- -^ / / \ \ ^ / / y 4- ^: --^ 3- — — — — — — 3,0 3 t- 00 \ i .8 1 0,000 54 6 60,0C 8 ^ : 00 I 0( \ £ 30- 8 6,000 30,000 100,000 FREQUENCY (f) Mc Fig. 4-46 Atmospheric Attenuation Due to Oxygen. level and at 4 km. Fig. 4-47 shows the theoretical water vapor attenuation in an atmosphere containing 1 per cent water vapor. The attenuation is closely proportional to the water vapor concentration. Experimental points 35See reference of footnote 25, pp. 646-664. 4-15] ATTENUATION IN THE ATMOSPHERE 229 3 100 1( ) 5 WAVELENGTH (\) cm 3 1 0.5 0.3 1 1 1 0. 1 — * Sea Lev c 1 4 Kilometers — 4 1 c A Q / / n 1 A / / , \ ' / / \ / i nm / A / / / / ■ / / / o / ./ / / / / / 1% Water Vapc r 0001- / 3 3,00 4 6J 5 2 10,000 3 4 6{ 50,00 3 ^ D 3 > 00,( 4 6{ )00 — 3 Fig. 4-47 6,000 30,000 100,000 FREQUENCY (f) Mc Atmospheric Attenuation Due to Water Vapor. due to Tolbert and Straiton^^ show general agreement with the oxygen attenuation, but for water vapor the measured values are 2.5 to 4 times the theoretical values. Theissing and Caplan" also found that the water vapor absorption between the peaks was higher than Van Vleck's theoretical curve by a factor 2.7. The reason for this disagreement with the theory is not known. ^^C. W. Tolbert and A. W. Straiten, "Attenuation and Fluctuation of Millimeter Waves," IRE National Convention Record 5 (1), 12-18 (1957). ^''H. H. Theissing and P. J. Caplan, "Atmospheric Attenuation of Solar Millimeter Radia- tion," J. App. Phys. 27, 538-543 (1956). 230 REFLECTION AND TRANSMISSION OF RADIO WAVES In the presence of absorption, an additional factor is required in the radar equation. This factor is lO-o.2adbR (4_81) where adt is the one-way attenuation in db per unit distance. 4-16 ATTENUATION AND BACK-SCATTERING BY PRECIPITATION Solid particles suspended in or falling through the air can affect radar operation both by the attenuation to waves passing them, and by the clutter due to back-scattering from them. The attenuation is a combination of absorption by the particles and scattering out of the forward beam. The particles which are most frequently encountered are those due to precipita- tion — viz., water, snow, and ice (hail). Of these, only water absorbs strongly, so that its attenuation is caused mainly by absorption. We shall give here only some salient features of the attenuation and back-scattering by precipitation, since rather complete summaries have been given in the literature. ^^'^^ For liquid water drops, the attenuation caused by absorption is much larger than that caused by scattering. For small drops (7rD/X<5C 1), the absorption is proportional to D^ while the back-scattering is proportional to D^. Hence the attenuation through small rain drops is proportional to the total liquid water content, but the back-scattering is proportional to SD^. Thus the larger drops are much more effective in back-scattering than the smaller ones. Because of the dispersion of water in the microwave region (see Para- graph 4-15) the attenuation varies in a complicated way with frequency, and also with drop size. The total attenuation is the integrated effect of all the drops in the beam between the radar and target, and thus depends on the drop size distribution, the drop density (number of drops per unit volume), and the length of the path through the precipitation. Drop size distribution is known only imperfectly, since most measurements have been made by catching rain drops af the ground. The distributions are then usually related to the precipitation rate. These may not be the same as the distribution and drop density encountered aloft. A further complication is that the precipitation density usually is not uniform for any great distance through the precipitation region. Hence the calculations made on the basis of such measurements necessarily must be considered as only approximate estimates of the actual effects which may be experienced. ^The wartime research is summarized on pp. 671-692 of the reference of footnote 33 above. ^^K. L. S. Gunn and T. W. R. East, "The Microwave Properties of Precipitation Particles," ^uart. J. Roy. Meteorol. Soc. 80, 522-545 (1954). -4-17] ATTENUATION BY PROPELLANT GASES 231 Calculations of attenuation and back-scattering (radar area) for spherical drops have been made by Haddock"*" on the basis of the drop size distri- butions of Laws and Parsons. "^^ These are reproduced in Figs. 4-48 and 4-49. The total radar area is found by multiplying the value found in Fig. 4-49 by the volume of precipitation illuminated by a pulse length. If the entire antenna beam is filled with precipitation, then this volume is R^^QL. The curves in these figures may be extended to longer wave- lengths by assuming a dependence as X-*. Snow is a mixture of air and ice. Since the refractive index of ice is much smaller than that of water, the scattering and attenuation due to snow are less than those of a corre- sponding mass of water. However, when a snow flake begins to melt, it becomes coated with a thin film of water. The scattering and absorp- tion then become almost the same as a water particle of the same size and shape and thus increase greatly. This effect has been advanced as the explanation for the radar "bright band" observed at or near the freezing level 10.0 1.0 0.1 0.01 5 3 0.001 - // ^^ 150 mm/hr 100 - 50 25 /// / 12.5 Cloud Burst' ^' '// / 4.0 - H/ / 2.5 1.25 1 1 , Excessive Rain^ ^ lil, // / - 1 Heavy Rain^ / / / / 0.25 _ - l\ /loderate Rain-' Light Rain-' / '1 , 7 / / - _ ^ _ Drizzle-' 1 100 5 3 X(cm) Fig. 4-48 The Variation of Attenuation with Wavelength for Various Rainfall Rates. 4-17 ATTENUATION BY PROPELLANT GASES In the transmission of information between a missile and its ground control station, the flame of the propellant gases lies in or near the path between the missile antenna and the ground station antenna. Attenuation, reflection, and refraction of the radio waves by the flame then are an important factor in determining the performance of the radio channel. A discussion of the nature of this problem appears in the Guidance volume of this series. ^^ ^"F. T. Haddock, Scattering and Attenuation of Microwave Radiation Through Rain, Report of NRL Progress, June 1956. *1J. O. Laws and D. A. Parsons, "The Relation of Raindrop Size to Intensity," Trans. Am. Geophys. Union 24, 452-460 (1943). 42A. S. Locke fEd.), Guidance, pp. 118-124, D. Van Nostrand Co., Inc., Princeton, N. J., 1955. 232 REFLECTION AND TRANSMISSION OF RADIO WAVES 10^ 3 I - 2 = ^ — ~ ^ in - '_ ?^ ^ 150 mm/h, 100 Cloud Burst - 50 Excessive RainI ?5 1 -1 : X ^ ^I'X - \ 12.5 Heavy Rain - = \\ 4 Moderate Rain 2.5 -1.25 Ught Rain 0.25 Drizzle 1-3 - \ 0.3 0.5 1 3 X(cm) Fig. 4-49 The Variation of Radar Cross Section of Actual Rain-Filled Space with Wavelength, for Various Rainfall Rates. The ionization processes which render flames conducting are still not completely understood. A recent summary of the subject by Calcote^' presents the status of the understanding of these mechanisms. He cites experimental evidence from the older literature of ion concentrations of 10^^ cm~^ From the standpoint of radio wave attenuation, only the electron density is of importance, since the conductivity due to a constituent ion of a highly ionized gas is approximately inversely proportional to the mass of the ion (see, for example, Guidance,'^'^ p. 121, Equation 4-19). The ion density varies greatly with the type of fuel. Furthermore, the ion density is influenced markedly by small quantities of low-ionization potential contaminants. For example, only trace quantities of the alkali metals such as potassium and sodium are sufficient to increase greatly the ion densities. Information on the quantitative attenuations to be expected from jet flames can be pieced together from the literature. Adler'*^ measured the attenuation in acid-aniline jet flames in a waveguide and found an atten- uation of 0.033 db/m at 200 Mc. Since attenuation is approximately proportional to/^'^, this is equivalent to 0.25 db/m at X band. Adler also observed that the addition of slight amounts of sodium caused large and erratic increases in the attenuation. It is probable that much higher attenuations would occur in modern high-energy fuels. ■•^H. F. Calcote, "Mechanisms for the Formation of Ions in Flames," Cotnbiistion and Flame 1, 385-403 (1957). ^''F. P. Adler, "Measurement of the Conductivity of a Jet Flame," J. AppL Phvs. 25, 903-906 (1954). 4-18] REFRACTION EFFECTS IN THE ATMOSPHERE 233 Andrew, Axford, and Sugden^^ measured the attenuation at X band in the flame of a rifle flash. They found values in the brightest part of the flash of 0.6 db /cm. The results quoted show that the attenuation in the flames of propellant gases can be serious whenever the geometry is such that the flame is a large obstacle in the path between transmitter and receiver. For example, a flame length of 1 meter in the path could introduce an attenuation at X band in the order of 50-60 db. The eff"ects of the flame are likely to be most serious as the missile ascends into rarefied air and the size of the flame grows. This indicates that special thought should be given to the location and design of the antenna on the missile in order to avoid placing the flame directly in the propagation path. 4-18 REFRACTION EFFECTS IN THE ATMOSPHERE In computing the power received from a target by means of the radar Equation 4-1, allowance was made for a process other than free-space propagation by means of the propagation factor F. A process which can produce profound modifications is refraction in the atmosphere. The atmosphere is a nonhomogeneous dielectric because of the variation of its pressure, temperature, and humidity. The variations actually are three-dimensional, but the most pronounced refraction eflfects are caused by variations in a vertical direction. In a homogeneous atmosphere, it is convenient to plot rays as straight lines and to show the earth's surface (assumed to be smooth) as a curve. If the atmosphere is not homogeneous it is then more convenient to use the earth's surface as a frame of reference. Rays which are straight lines in space then appear as curves when referred to the earth's surface as the abscissa. This is equivalent to the situation where the earth \s,flat and the (homogeneous) atmosphere has a constant positive gradient of refractive index. This is known as the earth-flattening procedure, in which the actual refractivity of the atmosphere is replaced by a modified refractive index. The modified index is denoted by M and is determined by the equation M = {n-\+ hi a) X 10« = A^ + ^^^ (4-82) a where h = height above the earth a = radius of the earth. Its unit of measurement is called the M unit. N is called the refractivity, and is the excess of the refractive index over unity, measured in parts per *^E. R. Andrew, W. E. Axford, and T. M. Sugden, "The Measurement of Ionization in a Transient Flame, Trans. Faraday Soc. A4t, ^HA31 (1948). 234 REFLECTION AND TRANSMISSION OF RADIO WAVES million. Numerically, 10^ hja amounts to 0.048 M unit per foot. From Equation 4-82 f = (:| + ^)x.O' (4-83) It follows from this that a homogeneous atmosphere {dn jdh = 0) has an M curve with dM jdh equal to lOV'^, and that an atmosphere with a constant gradient of refractive index is equivalent to a homogeneous atmosphere of effective radius <3e, where - = - + % (4-84) a, a dh In temperate climates an average value oi dn jdh is about —\/{4a). Hence from Equation 4-84 ae = \a (4-85) which is the so-called "four-thirds earth." Such an atmosphere is known as the standard at7nosphere, and the corresponding M curve, which is a straight line of slope 0.036 M unit per foot, as the standard M curve. Actually the M curve is rarely a straight line except in a restricted height range. The M curve is useful in ray tracing, since a one-to-one correspondence exists between the change in slope of a ray over a height interval and the change in M. In fact, if represents the elevation angle, measured in mils (1 mil = 10~^ radian = 3.44 minutes of arc), at height /z where the modified index has the value M, and 0o, Mo are the corresponding quantities at a reference height h^ (such as the ground), then Q = V^o^ + 2(M- Mo). (4-86) It can be seen from this that a height interval over which M — Mo is negative will give rise to a decrease in the absolute value of the elevation angle. Also, if the M curve has a sufficiently large negative excursion (Mo — Mmin > ^0^/2), then the ray will become horizontal at a certain height, and then curve back to earth. Assuming no loss in reflection at the earth's surface, the process will be repeated over and over, and the ray will go through a succession of hops along the surface. The ray is then trapped between the earth's surface and the height at which it becomes horizontal. A region of the atmosphere within which certain rays are trapped is called an atmospheric duct. The multi-hop trajectory resembles somewhat the crisscross path between the walls in waveguide propagation; and like a waveguide, an atmospheric duct can trap only waves of frequency higher than a lower limit. For effective utilization of the duct, both the radar and the target should be within the duct. 4-11 REFRACTION EFFECTS IN THE ATMOSPHERE 235 Since 2{Mmin — M) seldom exceeds 100 M units, trapping occurs only for rays with maximum elevation angle (which may or may not occur at the ground) of the order of 10 mils, or about |°. Hence trapping is a phenomenon which occurs only in almost horizontal propagation. The refractivity of the atmosphere for radio frequencies under about 4 X 10^ Mc*'' is given by the formula 77.6/. , _ .^ ^^_g^^ A^ ^'(p + 4.81 X 10^^ where T is the absolute temperature (°K), p the total pressure, and e the partial pressure of water vapor, both in millibars. The refractivity decreases with an increase in temperature, but increases with pressure, and is especially sensitive to variations in vapor pressure. The refractivity at a given point usually fluctuates with time, so that average values are used for drawing an M curve. The principal types of M curves observed are illustrated in Fig. 4-50. Curve (a) is the standard Fig. 4-50 Various Classes of M Curves. M curve already referred to. The substandard M curve, shown in (b), is so called because the rays are refracted less than in the standard case, and it generally results in lower field strengths. Curves (c) and (d) are types associated with surface ducts. The duct extends from the surface to the height hi, the "nose" of the M curve. In (e) the value of M at the surface is less than that at the nose, so that the duct then extends from hi to hi- This is called an elevated duct. Various combinations of types can take place, such as a surface duct (0 to hi) with an elevated duct (A2 to A3) shown in (f). From Equation 4-87, situations where the temperature increases with height together with a simultaneous decrease of vapor pressure lead to a strong decrease of M with height. Such situations are favorable for duct formation. Just such conditions occur at subsidence inversions. These «See Essen and Froome, Proc. Phys. Soc. London, B64, 873 (1951). 236 REFLECTION AND TRANSMISSION OF RADIO WAVES usually give rise to elevated ducts, since inversion levels commonly occur at 5000 to 10,000 ft. In some cases subsidence inversions descend low enough to form a strong surface duct of the type shown in Fig. 4-50d. Inversions can also be produced by cooling of the ground at night through the process of radiation. In the absence of wind, the radiation inversion grows upward as the night progresses, forming a surface duct of the type shown in Fig. 4-50c. Strong surface ducts are formed when warm air from a large land mass moves out over water. The air in contact with the water is cooled and moistened. This cooling and pickup of moisture works its way upward with time by eddy diffusion. As a result, during the formative process an inclined duct usually results, which can extend 200 miles or more out to sea. Duct heights can extend up to 1000 feet or so, and hence can influence airborne radar operation. Weak surface ducts are formed over the open oceans in the trade-wind regions. Here the air is colder than the water, so that an increase of temperature with height is accompanied by a decrease of vapor pressure with height. Their effects on the refractivity thus oppose, as can be seen from Equation 4-87, but the influence of the moisture predominates. These ducts are very persistent, lasting almost all year round, incident to the persistence of the trade winds. The duct height is about 50-75 ft, so that they are not very important for airborne radar, except possibly in unusual situations. An adverse effect on airborne radar can occur when an elevated layer lies below the radar and the target. Then, in addition to a direct ray, a ray refracted by the layer can be received. At certain ranges, well within the horizon, the two rays can interfere destructively, resulting in a decrease in field. This is referred to as a radio hole. Radio holes have been observed in which the field strength falls by as much as 15 db over a one-way path, which would mean a 30-db drop for a radar path. Radio holes extend in range for 20 to 50 miles, and so can seriously decrease the range of an airborne radar. Radio holes have been shown''^ to be caused by only small departures of the M curve from a straight line. A layer in which the slope changes by as little as 10 per cent of the slope in adjoining regions can produce a radio hole. It has been estimated that layers of this kind are present at altitudes between about 5000 and 10,000 ft between 50 and 95 per cent of the time. Thus this phenomenon can have a profound effect on airborne radar. Many of the effects of the varying refractivity of the atmosphere can be deduced, and to a certain extent predicted, from climatological considera- tions. However, most of the propagation measurements which have been '^''Investigation of Air-to-Air and Air-to-Ground Experimental Data, Final Report Part III, Contract AF33(038)-U)91, School of Electrical Engineering, Cornell University, 10 Dec. 1951. 4-18] REFRACTION EFFECTS IN THE ATMOSPHERE 237 made to evaluate the effects of atmospheric refraction have been either between two ground stations, or between an aircraft and a ground station. Thus the situations which are encountered in the use of airborne radar have not been explored sufficiently to yield a quantitative understanding of the meteorological effects which may be encountered. R. S. RAV CHAPTER 5 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 5-1 INTRODUCTION The performance of radar systems can often be determined only by tracing the received signal and corrupting noise in detail through the individual system components in order to establish the cumulative effect of each operation. In this chapter, some of the mathematical methods of signal and noise analysis which are appropriate for studies of this kind will be developed and their application illustrated with several examples. These examples will include a discussion of the characteristics of signal plus noise after undergoing some common nonlinear operations, the erratic perform- ance of an angle tracking system in response to internally generated noise, the clutter cancellation which can be achieved with a moving target indicator (MTI system) and the characteristics of a matched filter radar. Noise analysis embodies a generalization of classical Fourier methods which recognizes the statistical properties of random noise. Much of this material will be presented briefly. More detailed discussions can be found in the referenced literature. ^~^ 5-2 FOURIER ANALYSIS To develop the theory and methods of noise analysis several basic ideas relating to the representation of functions in terms of their frequency components as Fourier integrals are required. This paragraph explains and illustrates the concepts of: 1. Fourier integrals or transforms and inverse transforms. 2. Energy density spectra. 3. Transfer functions and impulse responses. 'S. O. Rice, "Mathematical Analysis of Random Noise," Bell System Tech. J. 23. 2J. L. Lawson and G. E. Uhlenbeck, "Threshold Signals," Chap. 2 (Radiation Laboratory- Series) McGraw-Hill Book Co., Inc., New York, 1950. "P. M. Woodward, Prohabilily and Information Theory .vith Applications to Radar, McGraw- Hill Book Co., Inc., New York, 1953. 238 5-2] FOURIER ANALYSIS 239 We are familiar with the representation of periodic functions by Fourier series. A Fourier integral is a limiting case of such a series where the period becomes indefinitely long. The separation between components becomes indefinitely small as do their magnitudes. For properly restricted functions, however, the magnitude density possesses well-defined values and a Fourier integral exists. The restrictions on a function /(/) in order that it have a Fourier integral are that the integrals of both its square and its absolute value have finite values and that it possess only a finite number of dis- continuities in any finite interval. When these conditions are met, a function F{co) can be defined by the relation /. F{c^) = / Me-'-'^'dL (5-1) If we suppose that/(/) is a function of time, then F{oci) is the spectrum of /(/) and gives the density of its diflPerential frequency components in much the same way that a Fourier series gives the resolution of a periodic function into finite frequency components. The variable co is the angular frequency equal to 27r times the cyclical frequency. In general, F(ci;) may be complex. The time function /(/) is given by the integral of all the differential Fourier components in a manner very similar to the way in which the sum of all the components of a Fourier series represents a periodic function. Thus,/(/) can be represented in terms of F(co) by the integral m = ^_j_^ F{c.)e'-^dc.. (5-2) TT. The functions /(/) and F{co) are often regarded as constituting a Fourier transform pair which are mutually related by Equations 5-1 and 5-2. With this terminology, Equation 5-1 is said to transform /(/) into the frequency domain, while the operation indicated in Equation 5-2 constitutes the inverse transformation. The symmetry of these transforming operations is striking. As a concrete illustration of such a pair of functions, suppose that/(/) is zero for negative values of time while for positive values it is a decaying exponential: /(/) = e-', </< oo. (5-3) The spectrum is easily calculated: F{.^) = f (.-0 e-^-^dt = — ^- (5-4) In this case, the spectrum is complex. Upon performing the inverse operation indicated by Equation 5-2, the exponeni.ial function given by Equation 5-3 will again be obtained. We shall not carry out the details of 240 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS this calculation, which involves treating cu as a complex variable and integrating around a semicircular contour in the complex plane. The square of the absolute value of the spectrum is important in the development of techniques for analyzing noise processes. This is the energy density spectrum giving the distribution of signal energy with frequency. This terminology is adopted because the function /(/) will normally be a voltage or its equivalent, and its square will be proportional to power. The integral of the square of/(/), then, will be proportional to the total energy. In the development to be given below, it will be shown that (in this sense) the square of the absolute value of the spectrum of/(/) gives a resolution of the energy into frequency components. This development is obtained by manipulating a general definition of the energy density spectrum as it is derived from Equation 5-1 : !F(co)|2 = F(co)F*(co) = / f{ti)e-^'^'^dtA f{t2)ei'''^dt^ (5-5) = j_ j_ At,)/(t,)e-^"^'r'.^ dt^dt2. Making the substitution t = ti — ti and dr = dti and interchanging the order of integration I^MP = l_^ e-^'^Ur j _J{t, + T)f{t,)dt, = j e-''''<p(T)dT. (5-6) The right-hand side of Equation 5-6 is of exactly the same form as Equation 5-1; that is |F(aj)|^ is expressed as the spectrum of the function <p(t) or its Fourier transform. If ^(t) satisfies the conditions prescribed for the existence of a Fourier integral, then the inverse operation given by Equation 5-2 is applicable, and <p{t) can be expressed by <p{r) = l_J(t + r)/(/) dt = ^l_^ \F(o:)\'- .^^ do:. (5-7) When T is set equal to zero, the following important special case is obtained. ^(0) = /_y'w^^ = ^/_^ \F(-^)\' d^- (5-8) This relation is often referred to as ParsevaFs equality. It expresses the idea that the total energy of /(/) is equal to the sum of the energies of each component of the frequency representation oif{t). 5-2] FOURIER ANALYSIS 241 Continuing with the example adopted in Equation 5-3, the form of ^(r) should be easy enough to find in this case. /• oo <p{t) = / (e-^'+^^)(e-')dt, T < ^(r) = ^-m/ e-'-'dt= i^-M (5-9) ^(0) = i Also, the absolute value of the spectrum is easily obtained from Equation 5-4 in this case: By virtue of the relationship indicated in Equations 5-6 and 5-7, the functions given by the two equations above must constitute a Fourier transform pair, and the total energy in the signal is |. We consider next the effect of transmission through a linear network on the time history and spectrum of a signal. Linear networks are conveniently characterized in terms of either their impulse response or their transfer function. The impulse response, sometimes called the network weighting function, is simply the transient output of the network for a unit impulse^ at the time / = 0. The transfer function is most commonly defined as the complex ratio of the network output to an input of the form exp {iwt). These two functions are closely related. In fact, the transfer function is the Fourier transform of the impulse response. This relation is made more understandable by noting that an impulse function has a uniform spectrum (see Paragraph S-S) and so represents an input of the required form where all the frequency components occur simultaneously with differential ampli- tudes. As an example, consider the single-section, low-pass, RC filter shown in Fig. 5-1. Suppose that the driving point impedance is zero and the load impedance is very large. Then the transfer function of this network is readily recognized as Transfer function = ^/^^^c'^{l ^ = y^— (5-11) R AAAAA/ 1— ^ Transfer Function: T X 1+jRCc Impulse Response= (l/RC)e f>0 Fig. 5-1 RC Filter. *See Paragraph 5-3 for a definition of an impulse function and a discussion of its properties. 242 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS Similarly, the impulse response of the network is recognized as a decaying exponential. If the RC time constant is assumed to be unity, the impulse response and transfer function are identical with the functions given as an example in Equations 5-3 and 5-4 which make up a Fourier transform pair. We denote the transfer function of a network and the input and output spectra by Y(o3), Fi((i:), and Fdco), respectively. Since the input spectrum gives the resolution of the input into components of the form exp (jW) and the transfer function indicates how each such component is modified by transmission through the network, it is clear that the output spectrum should be given by the product of these two functions. This can be rigor- ously demonstrated.^ Fo(a)) = y(a;) F, (a;). (5-12) The relation between the input and output energy density spectra is easily found by multiplying each side of this equation by its conjugate: |Fo(a,)|2 = \Y{coy\F,ico)\\ (5-13) Thus the input and output energy density spectra are related by the absolute square of the transfer function, which might appropriately be called the energy transfer function or, if power spectra are being considered, the power transfer function. It is often convenient to express the time history of the output of a network purely in terms of the time history of the input and the impulse response of the network. This relation is easily determined by substituting for y(co) and Fi(aj) in Equation 5-12 their expressions as Fourier transforms o{ y{t) and/i(/), the filter impulse response and the input to the filter: /:/- Fo(co) = j_ j_^J'(/0/-:(/2)^-^"^''+'^>^/i^/2. (5-14) Substituting r = /] + /2 and dr = dti, and interchanging the order of integration, ^o( 0;)=/ e-^'^'drl fi{t->)y{r - t2)dt.. (5-15) The right-hand side of this expression is again in the form of Equation 5-1 ; that is, Fo{oi) is given as a Fourier transform. Thus we can formally make an inverse transformation of both sides to obtain the desired relation between the input and output time histories: /o(r) = / //(/)v(r - t)dt. (5-16) 5See M. 1'". Gardner and J. 1"". Barnes, Transirn/s in Lineur Systems, Vol. 1, pp. 233-236, John Wiley & Sons, Inc., New York, 1942. 5-3] IMPULSE FUNCTIONS 243 5-3 IMPULSE FUNCTIONS Impulse or deltajunctions (so called because they are often denoted by the symbol 5) provide a most useful mathematical device in signal and noise studies. These functions can be visualized as the limiting form of a function whose integral is unity but which is concentrated at a particular value of its argument. Specific representa- tions of impulse functions may take a number of forms. One such form is shown in Fig. 5-2. In this figure a rectangular function of height A and ^ width 1 I A is shown centered at the pic. 5-2 Representation of an Impulse point t\. As A becomes large, the Function, function becomes very highly con- centrated at the point A. For any finite value of A, though, the integral of this function will be unity and independent of A. Thus, the limit of this integral as ^^ ^ <» exists and is equal to the value of the integral. In physical problems, it is conventional to suppose that these operations are interchanged and that an impulse function denoted by hit — /i) whose integral is unity is given by the limit of the function pictured in Fig. 5-2 as A — ^ oo . This certainly seems reasonable in view of the fact that for any finite A^ no matter how large, the integral is unity. Unfortunately, though, integration over the singularity produced when A -^ ^ cannot be justified in a mathematical sense, and these operations cannot correctly be inter- changed. Thus, although we shall formally regard impulse functions con- ventionally as being infinite in height with unit integrals, there is an implicit understanding that the limiting operation must, in actuality, be carried out after the finite function has been integrated. In this connection we note that impulse functions acquire physical significance only after being integrated and do not in themselves represent the end product of any calculation. With these provisos, we proceed to a discussion of some of the properties of impulse functions. Probably their most important characteristic is their sampling property. The integral of the product of a continuous function and an impulse is simply the value of the continuous function at the location of the impulse. We can establish this relation with the aid of the represen- tation pictured in Fig. 5-2: /" r (i+i/2^ /(/)6(/ - t,)dt = lim A J{t)dt = /(/:). (5-17) A^co Jt,-i/2A Additional properties can be established by finding the Fourier transform of an impulse function : 244 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS h{t)e-^'''dt = 1. (5-18) /. Thus the spectrum of an impulse is constant or uniform. Such a spectrum is often referred to as "white noise" in view of the fact that al! frequencies are equally represented. If the spectrum of an impulse (unity) is multiplied by the transfer function of a network, the spectrum of the network output is seen to be simply the transfer function itself. Thus, formally at least, the transfer function of a network is the Fourier transform of the transient response of the network to an impulse function input as was noted in Paragraph 5-2. The constant spectrum given by Equation 5-18 does not have a finite integral and so does not properly have an inverse Fourier transform. We can, however, approximate this spectrum by one which is unity for |co| < A and zero for |aj| > A^ where A is very large but finite, and this approxi- mation will have an inverse transform. This inverse transform should have characteristics very similar to the finite impulse pictured in Fig. 5-2 and should approach an impulse function as ^^ — ^ <^ . This turns out to be true and gives us a second representation for impulse functions: ./ N 1- 1 / ,^, 7 r sin At .. .^. bit) = hm ;:- / e'"' do: = lim (5-19) A-^o.livJ-A /l^co irt This expression occurs often in signal and noise studies. Many important functions cannot be transformed from the time to the frequency domain because the Fourier integral Equation 5-1 does not converge with time. Approximations to this integral, however, can often be derived on the same basis as for the uniform spectrum. When this is done, the resulting expression often contains expressions which can be interpreted as impulse functions in the limit. For example, it was just established that the Fourier transform of a constant/(/) = 1 is an impulse, 27r6(co), which denotes concentration of the frequency spectrum at zero frequency. Similarly, a sinusoid will have a spectrum which may be derived as lim / .l-^ccj-.l COS (j^it e '"^ dt — lim d sin (co + cji).y sin (co — o:\)A -|- coi o) — aj]rf (5-20) = f/7r[5(co + coi) + 5(co — wi)]. This expression indicates a spectrum which is concentrated at the positive and negative values at the frequency of the sinusoid. In deriving the impulse function representation given in Equation 5-19, a constant over the entire range of w was approximated by a truncated function which approached che constant function in the limit. Other 5-4] RANDOM NOISE PROCESSES 245 approximations to the constant function will give different impulse function representations. A third representation of an impulse function can be obtained in this way by using the triangular approximation shown in Fig. 5-3. As yf — > 00, this triangular function obviously approaches a constant. -2A 2A " Fig. 5-3 Triangular Approximation to a Constant Spectrum. The limit of the Fourier transform of this function will give the desired representation: It is apparent that there are a variety of specific representations of impulse functions. A familiarity with the forms of the representations, so that they may be recognized when they arise during the course of an analysis, is useful. A case of this kind occurs in Paragraph 5-5, where in an example of a noise process the expression in Equations 5-21 turns up as part of the power density spectrum (Equation 5-40). 5-4 RANDOM NOISE PROCESSES In describing noise mathematically, it is useful to visualize a very large group or ensemble of noise generators with outputs x{t), x'{t), x"(t), .... The output of a specific noise generator may be any one of the en- semble functions with equal proba- bility. The totality of all possible noise functions is referred to as a random process. Such processes are described in terms of their statistical characteristics over the ensemble. Fig. 5-4 shows a few of the elements of a noise process. At any time / the mean value, the variance, or other statistical parameters can be determined. These parameters can all be derived from the probability density Junction of the process at that time which describes the distribution of values of the elements of the process. Fig. 5-4 Elements of a Noise Process 246 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS The integral of the probability density function between any two values will give the fraction of the elements of the process which lie between those values. As an example, we consider the most common type of noise process, a Gaussian process, so called because the probability density is Gaussian or normal in form : Probability density function of a Gaussian noise process V2 — exp;r-^ (5-22) This process has an average value of zero and a variance or mean square value of 0-^. Most important, the probability density is independent of time. For most of the noise processes which are of importance in engineering applications the statistical parameters are independent of time, and such processes are therefore called stationary processes. Fig. S-S shows the probability density function for a Gaussian process. Most of the elements of the process have values in the neighborhood of the exp (x2/2(t2) Fig. 5-5 Gaussian Probability Density Function. origin. Only a very few of the noise functions will be very large or very small at any particular time. If the process is stationary, the values of the component functions will have the same distribution at any time. As previously noted, Gaussian noise processes are very common in physical applications. They can be generated by the superposition of a large number of time functions with random time origins. An example is the shot noise generated in an electron tube. The random times of arrival of electrons at the plate produce the shot noise fluctuations in the plate current, which has the properties of Gaussian noise. A mathematical example of Gaussian noise is produced by the superposition of a large number of sinusoids of different frequencies and random phases. Also very useful and important is the joint probability density function of values of the process at two different times. For a stationary Gaussian process with zero mean, this joint probability density will have the following form. Second-order probability density function of a Gaussian noise process 27ra-Vl exp ■vi- + 2p.Vi.V2 - 2<tH1 - p') (5-23) 5-4] RANDOM NOISE PROCESSES 247 In this expression Xi and X2 are values of the noise process at times /i and /2, 0-2 is the variance of Xi and X2, and p is a factor indicating the degree of correlation between Xi and X2. This factor is called the normalized autocorre- lation function. It is defined in this case, where the mean is zero, as the average value of the product XiX^, divided by the average value o{ x"^ which normalizes it so that its range is from +1 to — 1. When /i and t^. are close together so that x-i and X2 have about the same values, the value of p will be close to unity, indicating a high degree of correlation. That is, when Xx is high, Xi is also very likely high; and when Xi is low, x^ is probably low. On the other hand, when /i and ti are sufficiently far apart for several oscil- lations of the noise functions to occur between them, Xi and x^ will tend to be uncorrelated and p will be close to zero. When the process is stationary, the autocorrelation function will be independent of the particular times /i and /2 and depend only upon their difference, which we denote by T = ti — /o. The significant and meaningful attributes of noise processes must be expressed as average values. The notation we shall adopt to indicate the average value of some function of the process is to simply bar that function. Thus the average value of the process itself is denoted by x- If the process represents voltages or currents, then x can be interpreted as the d-c level. For the process whose probability density is given by Equation 5-22, the average value corresponding to the d-c is zero. The mean square value of the process about the mean or the variance can similarly be regarded as the average power in a unit resistance. As previously noted, this quantity is denoted by cr^. a^ = {^x -xY -= x^ - x\ (5-24) Actually, the term power will often be used very generally to refer to the square of arbitrarily measured variables so that sometimes it cannot be identified with physical power, although the electrical terminology has been retained. As an example, suppose that an angle 6 is found to be oscillating with an amplitude A and a frequency co or 6 = A cos cot. In this case, we might say that the angle d has a power of A^ /2 although the dimensions of this quantity are certainly not watts. The average value of the product X1X2 is very significant in signal and noise studies. This quantity is called the autocorrelation function, and we shall denote it by ^(r), where r is the time difference ti — t\. For a station- ary process, X\ and Xi are uncorrelated when t is very large except for the average value or d-c component: ^(±00)= p. (5-25) When T = 0, the autocorrelation function simply equals the average value of x^. Thus, the variance of the process is given by 248 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS ^2 = ^(0) _ ^(<x>). (5_26) The normalized autocorrelation function can be defined in terms of the function ^(t) by subtracting the d-c term and dividing by the variance: <p{t) - v?(°o) P{t) = —77^ 7— T- (5-27) (p(0) - <p(oo) For a Gaussian process with the joint probability density given in Equation 5-28, the autocorrelation would be computed in the following manner: ^W = o__2./l ~^j_^j_^ '''''' exp I 2a'^(l - 2) J^.vi^.V2 (5-28) [ (t^p{t). This integral can be evaluated by completing the square in the exponent for one of the variables and transforming to a standard form. In the next paragraph, it will be shown that the autocorrelation function is very closely related to the power spectrum of the process. 5-5 THE POWER DENSITY SPECTRUM It is possible to decompose random processes into frequency components in a certain sense, and this will provide a powerful analytic technique. For instance, it was previously mentioned that a Gaussian random process could be constructed by the superposition of a large number of sinusoids of varying frequency and random phase. This sort of a process can certainly be decomposed into frequency components. Of course, the average values of the in-phase and quadrature components at a given frequency will be zero because of the introduction of a random phase angle. The power at a given frequency, though, will be independent of phase and in general have a non-zero value. Thus a frequency decomposition could be carried out on a power basis. This possibility turns out to be valid for more general random processes and leads to the useful concept of the power density s-pectrum. Physically, the power density spectrum of a noise process corre- sponds to the average power outputs of a bank of narrow filters covering the frequency range of the process. To develop this idea, consider a stationary random process x{t). Subject to the restrictions noted in Paragraph 5-2, the portions of the elements of the process between — T and T possess Fourier transform spectra. By limiting the range we can ensure that the integrals of the squares ofthe elements of the process are finite. Over an infinite range these integrals 5-5] THE POWER DENSITY SPECTRUM 249 would not be finite and Fourier transforms could not be defined. Thus we have the spectra Xt(co) : /: Xt(c^) = / x{i) e-^'^'dt. (5-29) Energy spectra will be given by expressions similar to Equation 5-6. If these energy spectra are divided by the observation time 2T, power spectra will be obtained which we denote by A^t(co): NtW) = 2^ \Xt{o:)\' = ^ i^e-^'^^drlxit + T)x{t)dt. (5-30) The range of the last integral has been denoted by R. Because the elements of the process x{t) are in effect zero for |/| > T, the limits of integration will be from - T + r to T for r > and from - T to T + r for r < 0. In either case, the total range is 2T — |r|. We are, of course, primarily interested in the statistical average of the power spectrum since only average values represent meaningful and measurable attributes of the process. To compute the average value of Nrico), we average the product x (t -\- t) x (/) in the expression for 7Vr(aj) given by Equation 5-30. The average of this product is the autocorrelation function of the process which will depend only upon the time difference r if the process is stationary: A^r(co) = ;^ e-'^^'dr / <pir)d( Letting T— ^ oo, the factor involving Tin the integrand approaches unity, and we obtain the following expression for the average power density spectrum of the process : A^ = / ^(r)^-^"Vr. (5-32) This expression gives the power density spectrum as the Fourier transform of the autocorrelation function. These two functions form a Fourier transform pair and the knowledge of one is, at least in theory, equivalent to a knowledge of the other. The inverse of the relation in Equation 5-32 gives ^(r) = 2^ / A^^^'^Voj. (5-33) When T is set equal to zero in this relation ^(0) = C72 -f .^2 = i- / W(^) du^. (5-34) 250 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS Thus, the noise power or mean square vakie is equal to the sum of the power components at all frequencies. Equation 5-34 can be regarded as a general- ization of Parseval's equality given in Equation 5-8. At the end of Paragraph 5-2 it was pointed out that the absolute square of the transfer function of a network acts as a transfer function relating the input and output energy spectra. We have just defined power density spectra as the average of the energy spectra of the elements of the process divided by the observation time to give power. Thus the same relation must hold between the input and output power density spectra, A^i(co) and A^o(co), of a noise process being transmitted through a network with a transfer function y(co): Noio:) = \Y(w)['N iic^). (5-35) We might note at this time that it is normal practice not to use a bar to indicate specifically that the power density spectrum of a noise process is an average value unless the averaging takes place explicitly in the derivation of the power spectrum. Thus, the power density spectrum of a noise process would normally be denoted by A^(co) rather than N{co)- In order to illustrate some of these ideas, we shall make up a noise process and compute its power spectrum and autocorrelation function. We suppose the process to be composed of the sum of identical functions A(/) which occur at random times. Initially, we consider only functions which originate in the finite range — T to + T. We denote the average density of these functions by y and suppose that there are ITy = n functions in the finite range of interest. Denoting the origin of the ^-th function by 4, our approximation to a random process is given by the following expression. fn{t) =i:,h{t- /,) (5-36) Denoting the Fourier transform o( h{t) by H{co), the Fourier transform of /„(/) is given by F„(co) = [ h{t - t,)e-i'^^dt = //(co) i; ^-'•"'^. (5-37) The power spectrum is simply the absolute square of F(oo) divided by the observation time: ^\FM\-^ = ^[//(c.)E.--^-][//*(co)2:%^-']. = ^j^{^)\'zi:^^''''"'-'''- (5-38) ZI 1 1 5-5] THE POWER DENSITY SPECTRUM 251 To find the average power spectrum, we must average over each variable 4 supposing it to be uniformly distributed between —T and -\-T: 2j^l^«(co)p = 22^ H{o:)\'^ . I'T CT n n e-j'^itk-tDdt^ ... dtn. (5-39) The integrals of the terms in this sum will have two forms, depending upon whether k = I or not. When k = I, the average value of each term is unity. There are n such terms. When k t^ I, the average value of each term is (sin coT/coTy. There are n(n — 1) of these terms. Thus, the average power spectrum has the following form : «(w — 1) 2T FM\' = m<^w — + 2T^ {iry ©(-7 (5-40) As T -^ 00 , we note that the term involving the factor sin^ coT is of the same form as the definition of an impulse function given by Equation 5-21. The power density spectrum over all time, then, will have the following form: lim ^\F{o^)\' |i7(co)|M7 + 2x7^5(0.)]. (5-41) The singular part of this spectrum corresponds to a concentration of power at zero frequency or the d-c component. If h{t) has no such d-c component, then //(O) will be zero and the impulse has no significance. The continuous portion of the power spectrum is seen to be proportional to the energy spectrum of A(/). The remarkable thing about this is that the form of the spectrum is independent of the average number of functions per unit time 7. As a concrete illustration, suppose that h{t) is given by the decaying exponential defined in Equation 5-3. An element of such a noise process might then look like the example shown in Fig. 5-G. The energy spectrum of the exponential function has already been computed in Equation 5-10. Fig. 5-6 Element of a Noise Process Composed of Identical Exponential Functions with Random Time Origins. 252 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS The power density spectrum of this process will thus have the following form: A» = -r^ [t + 27rT^5(co)]. (5-42) The autocorrelation function, which is just the Fourier transform of the power spectrum, has already been partially computed in Equation 5-9 and will have the form <p(t) = ^ ^-H + ^2, (5_43) We may note that since we have used an A(/) corresponding to the impulse response of the RC filter pictured in Fig. 5-1 , the noise process that has been defined can be generated approximately by short pulses occurring at random times which are modified by this filter. A physical interpretation of our model of a noise process is provided by shot noise, fluctuations in the number of electrons arriving at the plate of a vacuum tube per unit time. We shall use our model to show that the mean square fluctuation in electron current, AP, incident to the shot eflFect is given by (a7)2 = lelAF (5-44) where e = electronic charge / = average current (d-c) AF = observation bandwidth. We suppose that each function h(i — tk) in the sum in Equation 5-36 represents the arrival of one electron at the plate. In this case, the integral oi h{t) should equal the electronic charge e, and we assume this, or what is equivalent, that //(O) = e. The magnitude of both the square of the direct current and average of the square of the fluctuation or noise currents can be determined from Equation 5-41. The square of the direct current corre- sponds to the magnitude of the impulse function at zero frequency in that expression and is given by P = \HmW' = e'y'. (5-45) The mean square value of the noise currents corresponds to the integral of the nonsingular term |//(co)|^7, in Equation 5-41. We are unable to deter- mine this exacdy without knowing the form of the spectrum of a current pulse, //(co). If, however, we are interested in the output of a filter which is narrow compared to //(co) we can approximate the mean square current in the output of the narrow filter by the product of twice the filter band- width 2AF and the low-frequency power density of the electronic pulse 5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 253 power spectrum. The factor 2 is introduced to account for contributions from negative frequencies. Forming this product and substituting //e for 7 in Equation 5-41, yields the following expression for the mean square noise current: (KTp = \H{W'i2^F = leHC^FI^ = 2e/AF. (5-46) Comparison with Equation 5-44 indicates that the noise process model used does indeed give the correct expression for shot noise. The forms of the functions h{t) are not significant in this derivation as long as their spectra are wide compared with AF. Similar discussions can be made in connection with many physical phenomena which generate noise by means of some random mechanism. 5-6 NONLINEAR AND TIME-DEPENDENT OPERATIONS In tracing signals and noise through radar systems, we find that the operations of many components are either nonlinear or time-dependent. Examples of such operations are rectification by second detectors, auto- matic gain control, time and frequency discrimination, phase demodulation, and sampling or gating. In this paragraph, procedures which can be used in the analysis of such operations will be discussed briefly and illustrated with a few examples. A basic case is provided by a nonlinear device which has no energy storing capacity; that is, it is assumed to operate instantaneously. We suppose that the input to this device is a Gaussian noise process denoted by x\ the output noise process is denoted by jy. The functional relation between these processes is denoted by v=/W (5-47) The process y will be random but not in general Gaussian. The average values of_y and jy^ can be found as the weighted averages of/(x) and/^(^): y =W) = vi^/-y^^^'"'^'"^^^^ ^^'^^^ 7 = a/+y^=n^) = j^ f nx)e-^'''~^' dx. (5-49) The power spectrum of the y process can be found by first finding its autocorrelation function and then computing the Fourier transform of this function. The autocorrelation of y is the average value of the product yiy^ = f{xv)f{xi)- The average will have to be computed relative to the joint Gaussian probability density function expressed by Equation 5-23. If this probability density function is denoted by P2{xi,Xi), the autocorre- lation of jy is given by 254 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS <p{r) = 3^2 = j Axr)f(x2)P2{x,,X2)^x,dx2. (5-50) The power spectrum of y is simply the Fourier transform of ^(t). A Square Law Device. As a specific example, suppose that the nonlinear operation is provided by a square law device: y = x^ (5-51) This type of nonlinearity is often assumed to approximate the rectifying action of second detectors in radar receivers. The mean and variance ofjy are found by carrying out the operations indicated in Equations 5-48 and 5-49: y = X" = cr'^ y = .^ = 3(,4 (5.52) The autocorrelation function is found by evaluating the following integral: \ —Xi^ + 2pXiX2 — Xi"^! /c cn\ exp 1^ 2.2(1 - p2) r^'^''' ^^-^^^ = (7^1 + 2p2). This integral is evaluated by completing the square of one of the variables in the exponent and transforming to standard forms. The constant term in <p{t) corresponds to the square of the average value of y and will contribute an impulse function at zero frequency to the power spectrum of jy. In general, the squaring operation will provide a widening of the con- tinuous noise spectrum as the various frequency components beat with themselves to produce sum and diflPerence frequencies. To show this and to illustrate this type of analysis generally, suppose the x process is similar to the one defined in Paragraph 5-5 (Fig. 5-6) by a sum of exponential func- tions. For simplicity, we assume that on the average only half of the exponential functions are positive while the other half are negative, so that the average value of the x process is zero. We assume further that the variance is unity. The power spectrum and autocorrelation of the x process will be given by Equations 5-42 and 5-43. There will be no d-c term, and in order to have unit variance 7 = 2: <p(t) = a-'p(T) = .-IH (5-54) Nic) = y^, (5-55) 1 + CO- 5-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 255 From Equation 5-50, the autocorrelation function of the y process will be <pAt) = jT^ = 1 + 2^-21^1. {S-SG) The Fourier transform of this expression gives the power spectrum of the y process : NyiiS) = lirdico) + 4 + (5-57) Thus, in this case, the form of the continuous spectrum remained the same, but the bandwidth was doubled. Another case which is very common in radar applications corresponds to the assumption of a uniform spectrum of finite bandwidth for the x process. Such an assumption normally represents a simplifying approximation to the more complicated forms which actual spectra might take. Such a 2W -2ttW 2irW Fig. 5-7a Uniform Spectrum {x Process). spectrum is shown in Fig. 5-7a. The autocorrelation function corresponding to this spectrum will be ■2wW r2wW ■J -2^W in IttIVt ItvWt The autocorrelation function of the y process will now be Ysin iTrWrV- \ 1-kWt J ■ (5-58) (5-59) At the end of Paragraph 5-3 it was indicated that the Fourier transform of a triangular function is of the same form as the trigonometric term above. Thus the continuous part of the power spectrum of y will be triangular. This spectrum is pictured in Fig. 5-7b and it is represented symbolically by Ny(a>) = 2Tra^8(o^) + i<r'/fV)il - |a;|/47r/F), Ico] < 4w^. (5-60) cr" (Impulse Strength) Fig. 5-7b Triangular Spectrum (y = x"^ Process). 256 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS An application of this result is made in Paragraph 5-7 in course of a discus- sion of the effect of the second detector in a pulse radar. A Synchronous Detector. Another example which is of interest is that of a product demodulator or synchronous detector. Such a device or an approximation to such a device is a common component in many types of radar systems. It will provide an example of a time-dependent operator. In operation, a -product demodulator simply multiplies the signal or noise by a sinusoid. Thus if the input is x{t) , the output would be x(t) cos Ww/. When x(t) possesses a component at the angular frequency Wm, the dc in the output gives a measure of the phase between the input component and the refer- ence. We again assume that ;c is a Gaussian noise process with zero mean, autocorrelation ^(r), and power density spectrum 7V(co). The autocorre- lation of the output is given by yiy2 = ■'''1-V2 cos a)mt cos COm (/ + t) (5-61) = (i) <P{t) [cos OOmT + cos C0m{2t + t)]. The autocorrelation of the output evidently varies with time periodically at the angular frequency 2ajm- The spectrum of the output will likewise vary periodically. In most cases, however, the angular frequency 2aJm is outside the range of practical interest, and we can use the time average of the autocorrelation or spectrum for our purposes. On taking the time average, the periodic component disappears: 1 [T Xr) = yiy2 = hm ;p^ / yiy2dt = (Dv'W cos w^r. (5-62) Zl 2r The wavy bar is used to indicate a time average. Bearing in mind that the autocorrelation function and power density spectrum (p{t) and A^(co) of the input noise are Fourier transforms, the Fourier transform of the expression above is easily computed to give the output power density spectrum in terms of that of the input: /:.' Nyiw) = h ^(r) cos co„t^-J"Vt -I v'(t)[^-'<"-"'"' + ^-'("+"'«)]^r (5-63) A product demodulation, then, operates to shift the input power density spectrum N{o}) into sidebands about the modulating frequency aj„ and the image of the modulating frequency —ojm- S-6] NONLINEAR AND TIME-DEPENDENT OPERATIONS 257 A Clamping Circuit. Clamping circuits, sometimes called pulse stretchers or boxcar detectors, are another common component of radar systems. They also provide an example of an operation with a periodic time dependency. Such a circuit clamps the output to a sampled value of the input for a fixed period of time; at the end of this period, the output is clamped to a new value of the input. The operation of such a circuit is shown in Fig. 5-8. Symbolically, the output of this device can be repre- Clamped Output nput Signal -7 H TIME Fig. 5-8 Operation of a Clamping Circuit. sented by y{t) = x{tk), t, < t < tk + i = tk-\- T. (5-64) Clearly the autocorrelation of_y(/) is dependent upon time. As with the case of the product demodulator, however, the time average of the autocorre- lation function and power density spectrum yield results which can be used for almost all applications. To determine the average autocorrelation of y{t), consider that when the delay ti — t\ = r, used in computing the autocorrelation functions, is a multiple of the sampling interval T, the average value of the productjyiV2 of the sampled and stretched process must be the same as the average value of the product x-^Xi because at the sample points ^1 = xi and jy2 = Xi- Thus, for t = kT, <Py{kT) = ^{kT). (5-65) When the time delay is intermediate between these isolated points, say kT < t < (k -\- l)T, the autocorrelation function of jy will sometimes be (p(kT) and sometimes (p(kT -\- T) depending upon the value of /. The fraction of the time during which <py{T) takes one of the other of these values is proportional to the relative values of r — kT and {k -\- \)T — r. Thus, the average value of ^2,(t) should vary linearly between its values at the discrete points where r = kT, and it will be composed of these points connected by straight lines. A limiting case of special interest occurs when the sampling frequency is much smaller than the width of the input spectrum. In this case, the autocorrelation function of the input is narrow compared with the sampling period. That is, values of the process which are separated by more than the sampling period are very nearly independent. Since in this case 258 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS ^v{T) ~ 0, the autocorrelation of the output is very nearly a triangle as is indicated in Fig. 5-9a. The Fourier transform of a triangular function has Delay Time - r Fig. 5-9a Autocorrelation Function of Pulse Stretcher Output with Wide-Band Noise Input. already been determined in Equation 5-21 spectrum of the stretched process will be On using that result, the power Nyic.) ,^( smcoT/2 y V coT/2 (5-66) The autocorrelation and the power spectrum of the pulse stretcher output in this case are both shown in Fig. 5-9. We might note that one of the basic features of this sort of operation is to concentrate the noise in a wide input spectrum in a low-frequency spectrum of width approximately 1 jT cps. NyiOO) -6f -47r -27r 2ir Air Sir Nondimensional Angular Frequency.cof Fig. 5-9b Power Density Spectrum of Pulse Stretcher Output with Wide-Band Noise Input. 5-7 NARROW BAND NOISE Signals in radar systems normally have the form of a radio-frequency carrier modulated by a low-frequency envelope which contains the essential intelligence. Such signals are filtered and amplified by tuned circuits with bandwidths just sufficient to pass the modulation sidebands. Noise asso- ciated with signals of this form or originating in circuits designed to amplify such signals will have a narrow spectrum centered about the carrier. In this paragraph, we shall develop some of the properties of narrow band noise and signal plus noise. 5-7] NARROW BAND NOISE 259 We suppose that the noise power is concentrated in the neighborhood of a carrier frequency coc. Such a noise process can be constructed by modulating a relatively low-frequency noise process by the carrier frequency. The carrier frequency signal can be represented by either the in-phase or quadrature component, and, in general, the narrow band noise will be composed of both components. Denoting the low-frequency noise processes corresponding to the in-phase and quadrature components about the carrier by x{t) and y{t), the narrow band noise process denoted by z{t) can be represented by z{t) = x{t) cos coc/ + y{t) sin oij. (5-67) In general, x{t) and jy(/) could be correlated and also might have dissimilar features. But in most problems of practical interest they will be independ- ent and have identical spectra and other statistical characteristics. If the X and y processes did not have the same spectra and autocorrelation functions, the narrow band process would depend upon time, as is apparent in Equation 5-68 below. Requiring x and y to be independent makes the spectrum of the narrow band process symmetrical about the carrier fre- quency coc- We assume that x and y are independent and have identical spectra. The autocorrelation function of the z process is computed as follows : (Pz{t) = [xi cos ixiJi + y\ sin Wct]\[x2 cos 0)^/2 + y^ sin coo/2] = (i) XiXi [cos OOcT -j- cos C0c(2/ + t)] + (I) Jl3'2 [cos WcT — cos C0c(2/ -(- t)] + {h) ^ [sin coeT + sin co.(2/ + r)] (5-68) — (I) yiXi [sin cocT — sin coc(2/ -{■ r)] = (p{t) cos WcT where ^(t) denotes the autocorrelation function of the x and y processes. The autocorrelation function ^z(r) is of exactly the same form as that of the output of a product demodulator discussed in the preceding paragraph and given in Equation 5-62. Thus the Fourier transform of .^^(t) giving the power spectrum of the z process will be related to the spectrum of the x and y processes, A^(co), in a manner similar to that indicated in Equation 5-63: A^.('^) = ihWio: - CO.) + A^(co -F COe)]. (5-69) From this expression, we see that the spectrum of narrow band symmetric noise has the same form as the low-frequency modulating functions, but is shifted to the vicinity of the carrier frequency. In a large class of radar systems, the transmitted and received signals have the form of an RF carrier amplitude modulated by a low-frequency waveform. In the majority of these systems, the modulation consists of a 260 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS periodic pulsing of the carrier. In such systems, the signal, when it is present, is a constant amplitude sinusoid. Noise will normally be present with a spectrum centered about the carrier frequency and a width deter- mined by the amplifiers which are designed to transmit the modulation sidebands. The ratio of the bandwidth to the carrier frequency is normally very small. Thus the noise can be considered narrow band noise with the representation and characteristics described above. When a signal is present, it is assumed to be of the form Signal = a cos wj. (5-70) The peak signal power is denoted by 6" = a} jl.. The noise power is denoted by A^ = 0"^ so that the signal-to-noise power ratio when the signal is present is SIN = ay2a\ (5-71) With the narrow band noise represented as in Equation 5-67, the signal plus noise has the form Signal plus noise = {a -{- x) cos wj + jy sin o^J. (5-72) A typical power spectrum of a c-w signal plus noise is shown in Fig. 5-10. I 5 6 -Function Continuous t Signal ^^ Noise i I Spectrum i^^Spectrum — COc ^c Angular Frequency Fig. 5-10 Power Density Spectrum of CW Signal Plus Narrow-Band Noise. General operations upon radar signals to extract desired information or to transform the signals into a more useable form are often referred to as demodulation or detection operations as discussed in Chapter 1. The simplest and most common such operation consists in the generation of the envelope of a narrow band signal by means of a rectifier. In superheterodyne receivers, this operation corresponds to the action of the second detector. The envelope output of the second detector is most often referred to as the video signal since it is commonly used as an input of some sort of visual display. In the following brief analysis, we shall develop some of the more important features of video signals and noise. The envelope of narrow band signal plus noise can be exhibited by rewriting the expression in Equation 5-72 in the following form: Signal plus noise = yjia + x)^ + y'^ cos ccj + 5-7] NARROW BAND NOISE 261 '-^ -(5-73) a -^ xj Here an envelope function modulates a carrier frequency with random phase modulation depending upon x and y. We note in passing that a frequency discriminator would be sensitive to this phase modulation and that studies similar to those which we shall make of the video envelope can also be made of a discriminator output. We first determine the probability density function of the envelope which is denoted by r: Envelope = r = -yjia + x)'- -j- y^- (5-74) The random variables x and y are assumed to represent independent Gaussian noise processes with zero means and equal variances. The differential probability that they will be found in the differential area (jxdy is given by their joint probability times this differential area: 1 \ —x^ dp = Pi(x)Pi{y)dxdy = ^ — -, exp 2^2 ^^^ 2(7^ dxdy. (5-75) In order to determine the probability density function of the video en- velope, this expression will be transformed to polar coordinates and the average value for all angles found. This transformation is represented as follows: a -{- X = r cos 6 y ^ rsmd (5-76) dx dy ^ r dr dd. Substituting these relations into the expression in Equation 5-75 and integrating over the variable d gives dp = Pi{r)dr = — exp 2a' 1 P'^ dr ^ / exp [ar cos e/a^dd. (5-77) 2xJDo The integral in this expression can be recognized as a representation of a zero-order Bessel function of the first kind with imaginary argument^ denoted by loi^r/a^). The probability density function of r is thus of the following form: Pi(r) = —exp 2(7^ - \h{ar/a'). (5-78) A curve showing Pi{r) for some representative values of S /N is given in Fig. 5-11. In the two extremes of very small and very large values of the signal-to-noise ratio, Pi(r) approaches the following forms: ^J. L. Lawson and G. E. Uhlenbeck, op. cit., p. 173. 262 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 17 S 16 or CO z Q 14 5 13 CQ 12 r 2 2 P{r)- -^"^ 2<^^ '0la2^" / \ ^aV2(r2=0 / / \ / \ ^oV? 7^ = 1 / ^ // /^ N is. K f ^O- ^^^^^ — ^^=^s 1 2 3 RELATIVE VALUE OF THE ENVELOPE, r/a Fig. 5-11 Probability Density Functions of the Envelope of Narrow-Band Signal Plus Noise. ^^^^^^Vlx^^^P ^72(72 ,»1, (5-79) (5-80) The first of these forms is often called a Rayleigh probability density and corresponds to the case of noise alone. When the signal-to-noise ratio is large, the envelope has approximately a normal distribution as is indicated by Equation 5-80. As might be expected from the form of the probability density function of r, its basic statistical properties such as its autocorrelation or spectrum cannot be expressed simply in terms of elementary functions. Approximate expressions valid for either large or small values of the signal-to-noise ratio have been developed. '^ Instead of becoming involved with such approxi- mations, however, it is often either more convenient analytically or more realistic in a physical sense to assume that the second detector is a square law rectifier producing the square of the envelope rather than the envelope itself. In most problems where such an assumption is made, the variations of many phenomena with parameters of interest are relatively independent of the detector law. The statistical properties of the square of the envelope can be expressed in much simpler forms than those of the envelope itself because r^ is a simple second-degree polynomial function of a; and j'. Thus, the autocorrelation function of r^ will involve the average values of products of the form Xi^x^"^ ^.nA yi^yi^ which have already been evaluated in Para- ''Uid., Chap. 7. 5-7] NARROW BAND NOISE 263 graph 5-6 in connection with the discussion of a square-law device. Using the results of that paragraph, the autocorrelation of r- is computed as follows : ri^ra^ a'^ + la^Xx + la^xi + arx^- + arx-^- -f- a?-y ^ + a'-y-^ + 1ax\X'^ + .yi-j'2- + 4^2—2 (5-81) = (^2 + 2cr2)2 + 4<72((72p2 + ^2^) = (2^2)2(1 + ^/yV)2 + (2a2)2[p2 + 2(.S'/A^)p]. The spectrum of the video signal plus noise has three components: (a) an impulse at zero frequency representing the d-c, (b) a continuous portion of the same shape as the spectrum of the component x and y processes repre- senting beats between the signal and the noise, (c) a continuous portion somewhat wider than the spectrum of the x and y processes representing beats between various parts of the noise spectrum. Fig. 5-12 illustrates the Impulse Signal Power = a^/2 Noise Power ■=2D\N=u S/N= a2/2(7" 27r(ay4) Impulse 27rW' 47r W Signal Plus Noise Angular ^ 27rW"^ 1^ Frequency 27r (2(7^)2 Impulse / \ t 1 / t D 1 1 Noise Alone Angular Frequency Fig. 5-12 Power Density Spectra of the Square of the Envelope of a Sinusoidal Signal Plus Narrow-Band Noise. forms of the various spectra in a typical case. The spectrum of the x and y processes is assumed rectangular with bandwidth W. The density of the positive and negative portions of the narrow band spectrum is denoted by D. The d-c level is equal to twice the sum of the signal and noise powers. The portion of the continuous spectrum corresponding to (b) is rectangular, of half the width of the narrow band spectrum (considering only positive 264 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS frequencies), and has a power density equal to the product of 8 times the signal power and D. The portion of the continuous spectrum corresponding to (c) is triangular, with a width equal to that of the narrow band spectrum and with a power density at zero frequency equal to the product of 8 times the noise power and D. 5-8 AN APPLICATION TO THE EVALUATION OF ANGLE TRACKING NOISE In this paragraph, the techniques developed for tracing signals and noise through radar systems will be illustrated by a discussion of the performance of an angle tracking loop in a pulse radar as a function of the signal-to-noise ratio. A block diagram showing the elements of the receiver composing this angle tracking loop is given in Fig. 5-13. This diagram represents a pulse Scanning Reference Antenna Gimbal Antenna T . .. Demodulator ^- Pulse Stretcher ( Controller ANGLE TRACKING LOOP ) — w Mixer IF Amplifier 2d Detector Range Gate t AFC t Gating AGO Filter Pulses 1 Fig. 5-13 Block Diagram of Angle Tracking Loop Employing Conical Scanning. radar with a pencil beam which is conically scanned to generate an angular error signal. A signal received from a target which is being tracked will have the following form: Received signal = a[\ -\- ke cos (ws/ + f)] cos Wct (pulse modulation) where a = signal amplitude (5-82) k = modulation constant of the antenna e = angular error magnitude <p = angular error direction Ws = scan frequency (rad/sec) ojc = carrier frequency (RF or IF, rad/sec). 5-8] APPLICATION TO EVALUATION OF ANGLE TRACKING NOISE 265 The angular error information is contained in amplitude modulation at the scanning frequency. We shall refer to this modulation as the a-c error signal. Its amplitude is proportional to the error amplitude, while its phase gives the error direction. The RF carrier of the received signal is transformed to an intermediate frequency in the mixer or first detector. The IF amplifier then provides the necessary gain and maintains the average level of the signal at a convenient constant value in response to the feedback signal from the AGC (automatic gain control) filter. The envelope of the IF signal is developed by the second detector, which is basically a rectifier. For our purposes we shall assume the second detector to be a square-law device whose characteristics have already been discussed to some extent in the preceding paragraph. A range gate selects only pulses occurring at the proper radar time for use in deriving the angle error. The range gate is positioned by an auxiliary range tracking loop which is not shown in Fig. 5-13. The AGC loop maintains the d-c value of the video signal during a pulse at a constant value so as to preserve a fixed relation between per cent modulation at the scanning frequency and angular error, independently of the received signal strength. A pulse stretcher generates a continuous signal suitable for use in the low-frequency control circuits from the pulsed signal delivered by the range gate. The output of the pulse stretcher is delivered to a product demodulator or synchronous detector which develops a servo control signal from the a-c error. Internally generated noise arises primarily within the mixer and the first stages of the IF amplifier. The noise may be represented exactly as in Paragraph 5-7 (Equation 5-67). That is, in-phase and quadrature components at the carrier frequency are modulated by independent low- frequency noise processes which we denote by x and y. The noise power is denoted by cr^ so that the average signal-to-noise ratio will be Signal-to-noise ratio - SIN = a^/ld"^. (5-83) This is an average signal-to-noise ratio because, on a short term basis, the signal power is modulated by the a-c error signal. The signal plus noise during a pulse will be of the following form: Signal plus noise = [a{\ + ke cos (cos/ + <p)) + x] cos Wct + y sin oij. (5-84) The video envelope from the square law detector during a pulse consists of the sum of the squares of the in-phase and quadrature components: Video signal plus noise = r^ = a'^[\ -\- 2ke cos (cos/ + tp) + kh"" C0S2 (oj,/ -f if)] -\- 2ax[l + ke cos (co^/ + cp)] + ^2_^y. (5-85) 266 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS In analyzing the effects of the remaining circuits in the loop on this signal, it is convenient to make certain simplifying approximations. First, it is assumed that the fractional modulation kt is small enough that its square may be neglected. Second, when determining the spectrum of the video noise, the target is assumed centered in the beam so that ke is zero. This reduces the expression above to the case already considered in Paragraph 5-7 (Equation 5-74). The average value of the video signal plus noise during a pulse will consist of a d-c term and the a-c error signal: Average video = ^2 ^ ^2 j^ 2(j^ -[- la'ke cos (co./ + ip). (5-86) The AGC loop will act to maintain the value of the d-c part of the video at a constant level which we may conveniently assume to be unity. Thus, ideally, the effect of the AGC is to divide the video by its d-c level. We assume that the AGC loop does indeed operate in this manner, although in an actual system only an approximate quotient would be formed. This assumption is sufficiently accurate for our purposes. In this case the effective a-c error signal during a pulse becomes / S/N \ \\ + s/n) A-C error signal = f , , c/at ) ^^e cos (co^/ + <p). (5-87) One effect of the noise is to introduce a factor depending upon the signal- to-noise ratio which attenuates the a-c error at low values of this ratio. The net result of this suppression of the signal by the noise is to decrease the gain around the angle tracking loop. A pulse stretcher is used to generate a signal suitable for use in the low- frequency control circuits from the pulsed signal delivered by the range gate. The pulse stretching operation will introduce some distortion of the angular error modulation, but because the scanning frequency is normally much smaller than the pulse repetition frequency, this distortion can be neglected and the pulse stretcher assumed to generate the fundamental component of the pulsed signal. Thus the a-c error signal delivered to the phase-sensitive demodulator is essentially of the form given in Equation 5-87. We suppose the demodulator to be a simple product type consisting of a multiplication of the modulated error signal by a sinusoidal reference, (1 Ik) cos oist. The factor 1 jk is incorporated in order that the output may be equal to the angular error. The properties of such a device with noise inputs were established in Paragraph S-6. The demodulator output is filtered so that only the very low frequencies are retained (components of cog and above are eliminated) as the angular error signal. The development of the error signal in the demodulator can be represented by the following operations: (5-88) € COS ip. 5-8] APPLICATION TO EVALUATION OF ANGLE TRACKING NOISE 267 Error signal = [ SiN r "^ ^^^ ^°^ ("«/ + ip)\{\IK) cos w,t = ( S/^ \ VI + S/NJ In this expression the wavy bar indicates the time average, which eliminates the fluctuating terms. The factor cos <p indicates that the error derived is the projection of the total error on the axis represented by one of the angle tracking loops. Complete directional control of the antenna requires it to be controlled in two directions, normally azimuth and elevation. The error signal for the other loop is obtained from a demodulator with a reference sin cos/. This error represents the input to the antenna controller which moves the antenna in order to null the error and track the target. In order to arrive at a definite result in this example, we shall assume that the antenna controller is composed of a single integrator, although in a practical system the dynamic response of the angle tracking loop might be quite complicated. With this assumption, the response of the whole loop becomes the same as that of a low-pass RC filter, and the power transfer function has the following familiar form. Angle tracking loop power transfer function = ^ ^^^ (5-89) where K = gain around the tracking loop = bandwidth (rad/sec) As noted above, the gain K will be attenuated by a factor depending on the signal-to-noise ratio. Thus we shall express K as the product of this factor and a design bandwidth /3 achieved at high signal-to-noise ratios: Our primary interest in this example is to determine the response of the loop to internally generated noise. It will turn out that the spectrum of the equivalent noise input to the loop is very much broader than ^ and relatively flat in the low-frequency region. If we denote the power density of this input noise spectrum by D in angular units squared per rad/sec, the variance of the tracking noise will be given by Mean square trackmgnoise = :^j_^ ^^^^-p^, = ^ = [j^^-sJnKJ )' (5-91) The next problem is to determine the magnitude D of the input-power density spectrum. 268 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS As already noted, the error is assumed zero when the spectrum of the video noise is determined in order to simplify the calculations. This corresponds to the case already considered in Paragraph 5-7. The spectrum of the video noise is thus pictured in Fig. 5-12, and its autocorrelation function is given by Equation 5-81. Dividing the noise power in the square of the envelope as determined from these sources by the square of the d-c level, to account for the effect of the AGC, gives the effective video noise power during a pulse: Video noise power (with AGC) - ^[^"V^w'^!^ ' (^-92) With a pulse width normally on the order of a microsecond, the width of the IF pass band, W cps in Fig. 5-12, must be approximately 1 Mc/sec or greater. The spectrum of the video noise will also be approximately of this width with a correlation time on the order of a microsecond. The repetition rate on the other hand will normally lie in the range from a few hundred to a thousand cps. Pulses will thus be separated by at least a millisecond, and the pulse-to-pulse fluctuations due to internal noise should be very nearly independent. The effect of the pulse stretching operation is considered next. In Para- graph 5-6 the spectrum of the output of a pulse stretcher was developed from an input of independent noise pulses. This is exactly the situation being considered in this example. Thus the spectrum of the stretched signal plus noise should have the form given by Equation 5-66 which was illustrated in Fig. 5-9. If we denote the repetition period by T, the power spectrum of the input to the demodulator will be of the following form: Noise spectrum of demodulator input [1 + 1{SIN)\T (1 + SINY sinjo7y2]2 cor/2 J (5-93) (1 + SINY The effect of the demodulator on its input spectrum was established in Paragraph 5-6 (Equation 5-63). The demodulator input spectrum will be shifted back and forth by the demodulating frequency and multiplied by the factor (1/4 k''): Noise spectrum at demodulator output = ' ' [A(a; + wj + A^(co - CO.)]. (5-94) The width of each component of this spectrum is approximately 1 /T cps, which normally might be on the order of a few hundred to a thousand cps. Since the bandwidth of the tracking loop will normally be only a few cps, only the power density in the neighborhood of zero frequency is significant; 5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 269 that is, the noise spectrum may be assumed to be uniform without appre- ciable error: [1 + 2(S /N)]T Power density of demodulator output noise = D = 7- — ; — LT\i,^ A^(cos). (1 + o/i\)^kl (5-95) A further simplification can often be made when the ratio of the scanning to the repetition frequencies is small. In this case, the factor N(cos) is approximately unity. For example, when the ratio of these frequencies is 1 : 10, the value of A^(co,) is 0.97. Substituting the power density D given in Equation 5-95 into the relation already derived for the mean square tracking noise (Equation 5-91) and assuming that N(olIs) is unity gives the following expression for the tracking noise variance: Mean square tracking noise (S/N)[l + 2(.V/A^)] / 7^\ (1 + s/Nr \4ky' ^^"^^^ This expression represents the end product of our analysis of the effect of internally generated noise on the performance of a conically scanned angle tracking loop. It \& interesting that the tracking noise from this source has a maximum at a signal-to-noise ratio of 1.35 db. The decrease in tracking noise at small signal-to-noise ratios is due to the loss in loop gain and consequent narrowing of the loop bandwidth. When this begins to happen in a practical system, dynamic tracking lags usually cause an early loss of the target. We also note that the rms tracking noise is directly proportional to the square root of the repetition period and inversely proportional to the modulation constant of the antenna. This constant, expressed in per cent modulation per unit error, is itself inversely proportional to the antenna beamwidth. The analysis in this example was intended to illustrate the sort of considerations which are appropriate to a study of noise in a radar tracking loop which incorporates a variety of components — some of them nonlinear or time dependent. Similar analyses can be made of other types of tracking systems such as monopulse tracking loops, range tracking loops, and frequency tracking loops. The effects of externally generated random disturbances such as glint or amplitude noise will also be handled in a similar fashion. 5-9 AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM In this paragraph we shall make some observations on the performance of a radar system which provides moving target indication (MTI). This analysis will supply another example to illustrate the use of the mathe- matical techniques which have been developed. The MTI system which we 270 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS shall consider is a noncoherent delay-line cancellation system. In such a system, both ground clutter and target reflections are received simul- taneously. The RF carrier frequency of the ground clutter is denoted by Wc> while that from a target moving relative to the ground will possess a doppler shift ud and is denoted by coc + cod. When the sum signal is detected, a beat is produced at the doppler frequency wd- If there is no target present, there is no doppler beat, and the spectrum of the detected video is concentrated at d-c and in the neighborhoods of harmonics of the pulse repetition frequency. The doppler signal can be separated from the clutter background by means of a delay-line cancellation unit. This unit provides the difference of successive returns as an output, that is, returns separated by the repetition period T. Fig. 5-14a shows a block diagram of such a cancellation unit, while Figure 5-14b illustrates its operation. This Input Time Delay p-iujT Output Fig. 5-14a Delay-Line Cancellation Unit. Input Signal HI- Delayed Signal i~x,.npv^^xi ^ -\ '^t /^Tx „ /IN ,, /I , Cancelled Signal il-^ ^^\ly '^^\iy Fig. 5-14b Cancellation of Clutter Echc sort of unit will attenuate the d-c component and all harmonics of the repetition frequency and in this manner cancel most of the clutter. When the doppler frequency lies between these harmonics, it will be transmitted through the cancellation unit. If by chance the doppler frequency coincides with one of the repetition rate harmonics, it will be canceled along with the clutter and produce a blind region or range of doppler frequencies to which the system is insensitive. Blind regions represent one of the most serious limitations of this type of system. Proceeding with the analysis, the clutter echo at a given range is repre- sented before detection as a narrow band noise process: Clutter echo = xU) cos uj + v(/) sin cor/. (5-97) The modulating functions .v andjy are independent Gaussian noise processes with identical spectra. The clutter spectrum is determined by the motion 5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 271 of ground objects, the scanning of the antenna, and the motion of the platform on which the antenna is mounted. The clutter power is denoted hj C = x^ = y, and the autocorrelation function of the x and y processes is denoted by Cp = XyX2 = yxji- During a pulse, the echo received from a moving target is assumed to be a sinusoidal signal with a doppler shift: Target echo = a cos (ojc + co^)/. (-5-98) The peak signal power is denoted by 6" = a} jl. We shall assume that the signal plus clutter is rectified by a square-law second detector to give the following video signal during a pulse: Video = V = \a cos (coc + co^)/ + -v cos (xsct + y sin cor/]". (5-99) The video frequencies are, of course, limited by the video bandwidth. Squaring this expression and retaining only the low-frequency components which will be passed by the video amplifier gives Video = y = \{a} -|- a;- + .V^ + lax cos cod/ — lay sin cod/). (5-100) The cancellation unit acts to generate the difference of video signals separated by a repetition period. Denoting the residue from the cancel- lation unit by r(/), we have Residue signal = r{t) = v{i) — v{t — T) = V\ — v-i = ^{xi^ - X2'' + yi^ - y2^ + 2axi cos co^/i (5-101) — 2^X2 cos 0)^/2 — 2ayi sin cod/i + 2ay2 sin cod/2). In order to evaluate the effect of the cancellation unit in reducing the clutter, it is convenient to define a video signal-to-clutter ratio. This ratio is defined as the difference between the video power with a signal v^j^^ and the video power with clutter only y^ divided by this latter quantity. Video signal-to-clutter ratio -= {S/C)v = (y|+c — vl)/ v^- (5-102) Similarly, a signal-to-clutter ratio is defined for the residue signal output of the cancellation unit: Residue signal-to-clutter ratio = {S/N)r = (r|+c — ^c)/ ^c' (5-103) With these definitions, a gain factor may be determined as the quotient of these two ratios: System gain factor = G = ^^^ (5-104) In order to evaluate this gain factor in terms of the system parameters, the average values of the squares of the video and residue signals must be calculated. This is somewhat complicated because of the large number of 272 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS terms resulting from the squaring of Equations 5-100 and 5-101, and the details will not be given here. The following average values originally- determined in Paragraph 5-6 in connection with a discussion of a square-law device are used in these calculations: X = y = x^ = y^= xy- — x-y — x^ = y^ = c, X1X2 = yiy2 = Cp{T) (5-105) x^ = y^ = 3C2 = xi^xs^ = y.'^y^'' = C^[l + 2p2(T)l. The following results were determined for the video and residue signal to clutter ratios: {S/C\ = 2{S/C) [1 + i {S/C)] 1 - p{T) coscodT] {S/C)r = 2(S/C) The gain in signal to clutter ratio will simply be 1 - pHD I (^-'0') System gam factor = G = ^ _^ a)(S/C) 1 _ 2(7-) (5-108) This expression essentially summarizes the ability of a noncoherent MTI system to reduce clutter. Various interesting observations might be made from a study of this factor. For instance, the depth of the blind speed nulls at harmonics of the repetition frequency can be determined as a function of the normalized autocorrelation function of the clutter at the repetition period. The average gain over all doppler frequencies can also be found as a function of the same parameters. These details will not be explored here. The primary purpose of the example has been served by the derivation of Equation 5-108, which showed how a performance equation could be arrived at by a straightforward application of the techniques for signal and noise analysis previously developed. 5-10 AN APPLICATION TO THE ANALYSIS OF A MATCHED FILTER RADAR In this paragraph, we shall consider how the mathematical techniques which have been developed can be applied to the derivation of optimum radar systems. Besides providing a good illustration of the application of these techniques, this example will also provide an insight into the important basic factors which affect system performance and set theoretical performance boundaries which a practical system may approach but not surpass. We shall be primarily concerned with the detection performance of radar systems. A fundamental problem in detecting a radar target is to distin- 5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 273 guish the target echo from random noise which tends to obscure it and render detection a matter of chance. This is the problem that we shall discuss in this paragraph. We shall determine the characteristics of an optimum receiver which will provide the most reliable detection of target echoes obscured by random noise. There are several possible approaches to this problem, depending upon the generality desired, the definition of most reliable detection adopted, and various assumptions made about the signal. We shall adopt the simplest possible approach, although the receiver design criterion which will be derived is operationally equivalent to the results of more sophisticated analyses in most cases. We suppose that in the general radar situation a signal is received as an echo from the target. During the process of reception, noise is added to the signal. The question we consider is, "What function must the receiver perform in order that the most reliable detection of the signal may be obtained?" We shall limit our study to receivers which are linear. That is, the effect of the receiver on the signal and noise is that of a linear filter. The output of the receiver-filter will consist of a filtered signal and filtered noise. Thus a ratio of the output signal and noise powers can be formed. We shall choose the optimum receiver-filter as that which maximizes this signal-to-noise ratio. We shall subsequently indicate how a maximum signal-to-noise ratio gives a maximum probability of detection for a fixed false-alarm rate and thus provides the most reliable detection in this sense. It will turn out, interestingly enough, that the receiver-filter which is optimum in the sense described above has a transfer function which is the conjugate of the target echo spectrum,^ and for this reason such a radar is often called a matched filter system. That is, the filter transfer function is matched to the target echo spectrum. We shall also demonstrate that such a system is equivalent to a cross correlation of the signal plus noise with an image of the signal waveform which is the origin of the term correlation radar sometimes used in reference to such systems. We adopt the following notation for this analysis: sit) = signal input to receiver-filter S(o}) = spectrum of s(t) So{t) = signal output of receiver-filter So{co) = spectrum of So(,t) ^This result is sometimes called the Fourier transform criterion and is attributed to a number of authors: namely, D. O. North, W. W. Hansen, N. Weiner, J. H. Van Vleck, and D. Middle- ton. See particularly Van Vleck and Middleton, "A Theoretical Comparison of Visual, Aural, and Meter Reception of Pulsed Signals in the Presence of Noise," J. Appl. Phy. 17, 940-971 (1946^. 274 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS Y(o:) = transfer function of receiver-filter n(i) = noise input to receiver-filter D = power density of noise input to receiver-filter no(f) = noise output of receiver-filter 0-2 = noise power in output of receiver-filter z^ = peak signal-to-nolse power ratio in output of receiver-filter /o = observation time The target echo is represented by a signal input to the receiver-filter denoted by s{t) with a spectrum S((a). The signal output of the filter and its spectrum are denoted by So(t) and «S'o(aj). The transfer function of the filter is represented by F(co), and the output signal spectrum is equal to the product of this transfer function and the input signal spectrum: So{c^) = FM .S'(co). (5-109) The output waveform will, of course, be simply the inverse Fourier trans- form of So{oi) : i/. Output signal = r„(/) = -^ / Yico) S{oo) ^^"Wco. (5-110) It J -ex, We choose to make our observation of the output at the time to- It is supposed that /o is selected so that the whole of the input signal is available to the filter. The signal power in the output of the filter at the observation time will be Sg^Uo), while the noise power in the filter output is denoted by 0-2. The input noise is assumed to be Gaussian with a uniform or "white" spectrum with power density D. The output noise power will thus be 1 f" Output noise power =^ 0-2 = / D|y(a))|Vw. (5-111) The output signal-to-noise ratio at the time /^ is denoted by 2-: Output signal-to-noise ratio = 2- = So-(/o)/(r~ = kl. y(co).V(a;)^^'-Vco ■/ D|y(co)|Va; (5-112) The minimum value of this ratio can be determined by means of Schwarz's inequality. This can be derived in the following fashion. Suppose that the functions /(;c) and g{x) and the parameter n are real. Then the 5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 275 following quadratic function of m will always be greater than or equal to zero: J^ [m/W + ^W]Vx = m'^ j^ nx)dx + 2m y^ Ax)g{x)^x +j^ g\x)dx ^ 0. (5-113) This expression is represented by ^m' + 25m + C ^ 0. (5-114) Because this polynomial is always greater than zero, the equation ^m' + 25m + C = 0. (5-115) cannot have distinct real roots, and its discriminant must be less than or equal to zero: B'-JC^O. (5-116) Substituting for yf, B, and C gives the real form of Schwarz's inequality: (/^VkW^^)' ^ /^ f{x)dx j\Kx)dx. (5-117) The absolute value of the product of two complex numbers is always less than or equal to the product of their absolute values. Further, the square of the absolute value of the integral of a complex function is always less than or equal to the integral of the square of the absolute value of the integrand. Combining these ideas, we note that when/(;c) and g{x) are complex. f f{x)g{x)dx i: L \f{x)g{x)\^dx ^ \f{x)\^\g{x)\^dx. (5-118) This immediately leads to the more general form which we need. Putting |/(;c)| and |^(^)| in place of/{x) and g{x) in Equation 5-118: / f(x)g(x)d> j\f{x)\'dxj\g{x)\^dx. (5-119) Substituting the right-hand side of this inequality for the numerator in Equation 5-112. ^j_^ \Sic.)\'dc.^j_^ |y(a;)lVc.. (5- ^J_^D|y(co)|Va; 120) The integrals involving the filter transfer function can be canceled: 2^^(^)^/_J^M|Vco. (5-121) 276 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS Thus the right-hand side of this inequality is independent of the filter. Since the signal-to-noise ratio is never greater than the right-hand side of the expression above, and this expression does not contain y(w) at all, it must give the maximum value for 2^ for the optimum choice of y(co). Referring to Equation 5-112, it is apparent that the denominator in that equation will be canceled and the maximum value of z^ achieved if the filter transfer function is made the complex conjugate of the signal spectrum: y(co) = ^*(co)^-'"'«. (5-122) A receiver-filter which is designed on the basis of this principle, where the receiver transfer function is matched to the signal waveform, is often referred to as a matched filter system. Another general term which is also used in reference to such systems is correlation radar. This terminology originates in the observation that the ideal filtering operation is equivalent to a cross correlation of the signal plus noise with an image of the signal waveform. In order to see this, the impulse response of the matched filter is found by taking the inverse Fourier transform of Y{oo) : 1 /"" Impulse response of matched filter = :r— / S*{ii))e '"^"'^'"^dco (5-123) = sUo - /). Denoting the input noise process by n{t) and the output noise process by no(t) and using Equation 5-16 to relate the time histories of the input and output signal plus noise gives for the filter output: Soil) + noil) = l_^ [sir) + nir)]sito - t + r)dT. (5-124) In particular, the output at the observation time to is simply soito) + rioito) = J_^ \sir) + niT)\siT)dr. (5-125) Thus, from this relation it is clear that the optimum receiver could consist of taking the cross correlation of the received signal plus noise and the pure signal waveform and that a matched filter receiver and a cross correlation receiver are equivalent. Going back to Equation 5-121 for a moment, we might note an interesting basic feature of radar systems which are theoretically optimum in the sense of this paragraph. The maximum signal-to-noise ratio is equal to the ratio of the received signal energy to the power density of the noise. That is, the maximum signal-to-noise ratio does not depend upon the waveform of the signal. This is not to say that the waveform is not important. Resolution, tracking accuracy, and many other system characteristics are closely related 5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 277 to and depend upon the wave shape of the signal. For the detection prob- lem, though, it is the received energy that counts. As a concrete illustration of a matched filter, suppose that the signal waveform consists of a series of n identical pulses separated by a repetition period T. Such a signal is of common occurrence in radar problems. Denoting an individual pulse by /)(/), the signal is defined by Pulse train - s{t) =!]/>(/- kT). This signal is depicted in Fig. 5-1 5a. (5-126) Fig. 5-1 5a Pulse Train Signal. Envelope = P(co) ^^ T"~-^>^ (Width of spectral teeth - 27r/4fr) .aJ \^^ L^Al'x? ^H 27r/fr Fig. 5-15b Spectrum of Pulse Train Signal. From Equation 5-125, the signal component of the filter output at the observation time will be Filter output (signal) = So{t^ = / s-{T)dj (5-127) E £V(r - kT)p{T - mT)dr ^iLp'ir - kT)dT = n\ p''{j)dr. The effect of the correlation (or filtering) operation has been to select out all the available signal pulses and add them together. A device which will perform this addition is most often referred to as a pulse integrator, and 278 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS almost all radar receivers whose inputs consist of a series of pulses incor- porate such a device in one form or another. The effect of the pulse integrator on the noise can also be determined from Equation 5-125. The noise output will be /oo „_i n(T)J2 P(r - kT)dT. (5-12S) The noise power is determined by squaring no{to) and finding its average value : /°° /• "^ n-l n-l / w(ri)w(r2)S ^p{ri — kT)p(T2 — mT)dTidT2. (5-129) Since n(t) was assumed to have a uniform spectrum with density D, the average value of the product «(ri)«(T2) is an impulse function with weight D: /oo /■ oo „_!„_! / D8{ti — T2)2Z ^P(tT- ~ kT)p{T2—mT)dTldT2 = dI E jipiT2 - kT)p{T2 - 7nT)dT, (5-130) j -co = £>/ Y.pKr2- kT)dr2 7-- r oo In evaluating the integral of the double sum, we made use of the fact that when the pulse functions in the integrand do not coincide {k 7^ m), their product is zero: -my {T)dT. (5-131) It is apparent that the effect of the pulse integrator is to increase the signal- to-noise ratio for a single pulse by the factor n. This could, of course, be inferred at the outset from Equation 5-121, since the signal power is directly proportional to n. The shape of the matched filter response in this case is of some interest. Denoting the spectrum of an individual pulse by P(co), the spectrum of the pulse train will be n-l Pulse train spectrum = S{i^) = P(co) X) ^~''*"^ (5-132) \ sm coT/2 / 5-10] APPLICATION TO ANALYSIS OF MATCHED FILTER RADAR 279 The energy spectrum of a typical train of short pulses is shown in Fig. 5-1 5b. We note that the filter primarily acts to accentuate the harmonics of the repetition frequency. Because of its distinctive appearance, such a device is often called a co7nb filter. Some explanation on the mechanism of the detection process itself is in order since the previous discussion related only to maximizing the signal- to-noise ratio. The output of the matched filter characterizes a signal- plus-noise situation by a single number So{t^ -\- njyt^. This number is a random variable with a normal distribution and mean Jo(0- The detection process will consist of a decision as to whether the observed number comes from a distribution with mean sj^t^ or the distribution of noise alone with a zero mean. This decision can be made by selecting a critical value or threshold and deciding for or against the existence of the signal depending upon whether or not the observed number exceeds the threshold. Fig. 5-16 -Decision Bias Probability Density of Noise Alone Probability Density of Signal Plus Noise h\ Value of Signal False Alarm pi^g ^^^^^ Probability Fig. 5-16 The Use of a Decision Bias for Determining Whether Noise Alone or Signal Plus Noise Is Present. shows the probability densities of the filtered signal plus noise and noise alone and a decision bias b for distinguishing the two cases. Because the two probability densities overlap, mistakes will be made. On some occasion a target will be thought present when there is none, while at other times the signal plus noise will be thought to be noise alone. The probability of making an error of the first kind is equal to the crosshatched area under the curve of noise alone and to the right of b in the diagram. This probability is normally called the false-alarm probability by radar system designers. The shaded area under the probability density curve of signal plus noise and to the right of ^ is the probability of detection. The difference between this probability and unity is, of course, the probability of making an error of the second kind or not seeing a target that is actually present. When human operators make a detection, the situation is not nearly so clear-cut, but some similar mechanism must take place. The decision bias might be visualized as diffuse, and it will vary with operators, time, and other conditions. A basic problem is the choice of the false-alarm probability at which the system is to operate. Most often this operating parameter is chosen 280 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS on subjective grounds because the data upon which to base a rational choice are not available. Factors which can be used to determine an optimum false-alarm probability are the cost of a false alarm in time and subsequent commitments, the gain associated with a correct decision, and prior probability of a target's existence. If quantitative estimates of these factors are available, the false-alarm probability can be chosen to minimize the total cost of the detection operation. Or even when the prior probability is not known, it is possible to operate the system at false- alarm rates so as to minimize the cost for the most adverse value of the prior probability. As noted above, however, data on detection costs and prior detection probabilities are known only subjectively in the majority of cases, and most often a rather arbitrary estimate of a desirable value of the false-alarm rate is made after a thorough but subjective study of the problem. At the beginning of our discussion of the optimum receiving system, it was assumed that the waveform of the signal was known exactly, and the only issue was its existence. In a practical detection situation, however, the signal waveform may depend upon a number of unknown parameters. Three such parameters which are of particular importance are the signal amplitude, the time of arrival of the signal, and the radio frequency of the carrier. The signal amplitude will vary with the range, aspect, and size of the target, while the time of arrival is, of course, directly proportional to range in a radar system. The RF carrier will vary because of the doppler shifts proportional to the relative target velocity. An optimum receiver in this case will consist of a parallel combination of optimum receivers for all the possible waveforms. Luckily, this does not require a duplication of equipment to cover the possibilities of amplitude and time- of-arrival variations. If the signal amplitude is changed by some factor, then the average value of the filter output is changed by the same factor. The same filter will produce the maximum value of z- for all possible signal amplitudes. A similar situation applies to variations in time-of-arrival. The optimum receiver produces its maximum output at a time T after a signal is received. Continuous monitoring of the receiver output, then, will provide an observation of the filtered signal over a continuous range of possibilities for the time of arrival. In order to account for variations in the radio frequency, however, it will in general be necessary to have separate receiving systems for the possible radio frequencies which may occur. This situation will be recognized in the design of many doppler systems where a bank of narrow band filters, each connected to its own threshold, is used to cover the possible spectrum of doppler signals. The situation is complicated further by the fact that some of the signal parameters are random variables in their own right. For example, the amplitudes of echoes reflected by aircraft fluctuate owing to their motions. 5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 281 and the radio frequency of a magnetron oscillator normally varies randomly from pulse to pulse by a small amount. The statistics of signal parameter distributions would have to be considered in a more realistic optimum receiver, and the result would be somewhat different from that derived in this paragraph. One should not make the mistake of thinking that great gains over current practice can be attained through some complicated optimizing scheme. Actually, most radar systems are tuned up in this respect about as far as they can be when consideraton is taken of limitations in the state of the art and fluctuations in the parameters of the input signals with which the systems must contend. For instance, a pulse radar employ- ing a self-excited magnetron oscillator is not coherent because it is not normally feasible to control the frequency of the power oscillator to a sufficient degree. Because of pulse-to-pulse frequency fluctuations, the receiver must operate upon the envelope of the signal, and it will normally employ a non-linear device to generate it. In this case, the best that can be done is to match the low-pass equivalent of the IF amplifier to the pulse envelope, and this is quite normally done as a matter of course. When there are a number of pulses available in an echo, some provision is usually made to integrate them. Most commonly, this is accomplished on the display where the decay time of the phosphor may be matched to the signal duration. The point is that insofar as is possible receivers are normally matched to the signal waveform, and most radar systems can be quite legitimately regarded as correlation or matched filter radars, although possibly somewhat degraded from the optimum type. 5-11 APPLICATION TO THE DETERMINATION OF A SIGNAL'S TIME OF ARRIVAL An important function of radar systems is the measurement of a target's parameters such as its range, velocity, size, and location. In this paragraph we shall develop some characteristics of a receiver which provides optimum target tracking in a manner similar to that used in the preceding paragraph to develop the properties of matched filter receivers for optimum target detection. We shall restrict our analysis to the problem of measuring the time of arrival of the signal. Since both angle and range are measured by comparing the return signal with angle and range reference signals which are generated as functions of time, the following discussion can be applied to both types of tracking. As in Paragraph 5-10, the receiver will be sup- posed to be a linear filter, and where applicable, the notation introduced in that paragraph will be adopted. Various operational definitions might be used to fix the arrival time of a signal. The mean, the median, or the mode of the distribution of the signal 282 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS in time are all quite applicable. We shall find it most convenient to adopt the last of these, the mode or the maximum value of the signal, as the primary indicator of the signal's location. This definition gives a straight- forward development which parallels that of Paragraph 5-10 and which to a first approximation leads to results in accord with more elaborate analyses. Even so, we must recognize that since the choice of a definition for the signal location is purely arbitrary, we are optimizing the tracking process only relative to that definition and not in an absolute sense. In order to use the peak value of the filtered signal plus noise as an un- ambiguous estimate of the signal location, we shall make several assump- tions about the form of the signal and signal plus noise. First, we assume that the signal itself either has a single maximum or that the greatest maximum is sufficiently larger than minor maxima to allow it to be un- ambiguously distinguished. Second, we assume that the primary maximum of the filtered signal has a finite second derivative, since we intend to locate it by setting the first derivative of the signal plus noise equal to zero. Third, the filtered signal is assumed to be enough greater than the noise that there are no ambiguous noise maxima in the neighborhood of the primary max- imum and the shift in this maximum due to the presence of the noise is small enough to be approximated by the first few terms in a series expansion. Suppositions of this kind are not unusual in parameter estimation problems, and equivalent assumptions and approximations almost always must be adopted when a specific example is worked out. Fig. 5-17 shows a typical example of signal plus noise in the neighborhood of the signal maximum and illustrates how the addition of noise acts to APPARENT SIGNAL LOCATION ERROR IN SIGNAL LOCATION TRUE SIGNAL LOCATION NOISE- ^ ^ Fig. 5-17 Generation of Signal Location Error. shift the maximum slightly from its former value. The magnitude of the shift can be determined approximately by differentiating the signal plus noise and setting it equal to zero. The resulting expression will be in the form of a quotient very similar to that given in Equation 5-118 for the signal-to-noise power ratio. Schwarz's inequality can also be applied to 5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 283 this expression, and we can determine the optimum filter for tracking which gives the minimum tracking error. This error will be interpreted as a simple relation between the signal bandwidth and the signal-to-noise ratio. The filtered signal and noise are denoted by So{t) and no{t) as in the preceding paragraph. We suppose that the maximum value of the signal occurs at the observation time io- We suppose further that the output signal at the time /o + A/ can be represented by a series expansion about the time to for small values of the interval A/. So(lo + A/) = soito) + s:'(to)Ar-/2 + -. (5-133) The first derivative of So{t) at /« is zero, of course, because it has a maximum at that time. We assume that the shift in the maximum value of the signal plus noise is small enough that all terms beyond the second in the expansion above can be neglected. The derivative of signal plus noise in the neighbor- hood of /o is thus given approximately by j^[so{t) + nom = to+At = s:'{to)M + n'^to + A/). (5-134) Setting this expression equal to zero and solving for A/ gives an approximate value for the apparent shift in signal location due to noise: A/= -^''(!;\^'\ (5.135) ^o \to) The variance or mean square value of the signal location error is thus given approximately by the average of the square of this expression: If we denote the transfer function of the filter-receiver by Y(o}) and the signal spectrum by S(cci) as in Paragraph (5-10), the following representation of s"o{to) can be obtained by differentiating Equation 5-116 twice: s'o'ito) = ^ / co2y(co)^(co)^^'-'«^a;. (5-137) Also assuming, as in Paragraph 5-10, that the input noise has a uniform spectrum of density D, the power spectrum of the output noise is D|y(w)|^ while the power spectrum of the derivative of the output noise is Do}^\Y(p})\^. The integral of this last spectrum gives the mean square value or variance of the derivative of the output noise: KWP = ^/'_ Da)2|y(co)|Vco. (5-138) 284 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS The quotient in Equation 5-136 giving tiie variance of the signal location error thus has the following form: _ ^f Dco2|y(co)|Vco A/- = , /;~'° TT (5-139) (^/ c,'Yio^)S{co)e''^'o^oA For convenience, we denote the quotient on the right-hand side by ^. The denominator of this quotient is in a form to which Schwarz's inequality, given by Equation 5-125, can be applied. Using this relation to split the denominator into two separate integrals leads to the following inequality: ^f Dco^iy(a;)|Vco ^ The integrals involving the transfer function of the filter simply cancel as they did in Paragraph 5-10, where the maximum value of the signal-to-noise ratio was determined. Since the quotient ^is never less than the expression given on the right-hand side above, which does not contain Y(o}) at all, this expression must give the minimum value of ^ for the most judicious choice of y(co). Referring to Equation 5-139 above, it is apparent that this minimum value of ^ will actually be achieved if the filter transfer function is chosen to be the conjugate of the signal spectrum. In this case, the numerator in Equation 5-139 cancels one of the factors in the denominator, and we have A7^=^.,mi„= , ,.co ^ (5-141) i/. cjo-\S{u)\~dcjo The optimum filter transfer function giving this result is y(aj) - S*(c^)e-''''o. (5-142) This is exactly the transfer function determined in Paragraph 5-10 (Equa- tion 5-128) to give the maximum signal-to-noise ratio. Thus to a first approximation the matched filter giving the maximum signal-to-noise ratio also provides the minimum error in locating the signal in time. The relation given by Equation 5-141 above for the minimum variance of the error in locating the signal can be given an interesting and rather useful physical interpretation. We note that the denominator has the form of the moment of inertia of the energy spectrum of the signal. If this denominator is divided by the total signal energy, we obtain the square of the radius of gyration of the energy spectrum. Now the radius of gyration of a function 5-111 DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 285 is a measure of the width of that function. Thus the radius of gyration of the energy spectrum is a measure of the bandwidth of that spectrum. We shall adopt this definition for the bandwidth of the signal spectrum: Signal bandwidth (rad/sec) = B U \S(co)\"(^(ji} T {rjl !^MP^")"" (5-143) A question may arise in connection with the application of this definition to spectra which are not centered at zero frequency. It is clear that the radius of gyration of an energy spectrum which is concentrated at low frequencies is a good measure of the bandwidth of the spectrum. The radius of gyration of a spectrum whose center is displaced to some high frequency, though, will be very large, and it does not correspond to the conventional idea of bandwidth. Such a signal can be represented as a function, denoted by/(/), whose spectrum is concentrated at low frequen- cies and which is modulated by a high-frequency carrier: High frequency signal = f(t) cos {coj + (p). (5-144) This signal contains information about its time location with an accuracy on the order of 1 /coc- But when the ratio of the carrier frequency to the bandwidth of/(/) is large, this information is useless because it is ambiguous. This is illustrated in Fig. 5-18 where a relatively smooth low-frequency AMBIGUOUS MAXIMA DUE TO CARRIER FREQUENCY LOW - FREQUENCY // 1 MODULATION SIGNAL=f(f) HIGH-FREQUENCY SIGNAL^ f(f)cos(Wcf+0) Fig. 5-18 Typical High-Frequency Signal. signal /(/) is modulated by many carrier cycles. It can be seen in this figure that the carrier frequency modulation produces a number of signal maxima in the neighborhood of the maximum of/(/) . Because these maxima are all of about the same magnitude, there is no way of distinguishing one from another, and signal location information provided by the carrier- frequency modulation is ambiguous. In such a case, the low-frequency signal /(/) is normally regenerated by a demodulating operation, and the 286 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS location of the signal is determined on the basis of/(/) alone. The effective bandwidth in such a case, then, is that of the spectrum of/(/), and it would be determined relative to the carrier frequency rather than zero frequency as is indicated in Equation 5-143. When the definition of bandwidth given by Equation 5-143 is combined with Equation 5-141, the mean square time error is found to be approximately equal to the ratio of D, the input noise power density, to the product of the square of the signal bandwidth and the signal energy. The ratio of the signal energy to D, however, was estab- lished in Equation 5-129 as the greatest possible signal-to-noise ratio an^ was denoted by z^. Thus we can assert that the minimum rms error in measuring the time of arrival of a signal is approximately equal to the re- ciprocal of the product of the signal bandwidth and the voltage signal-to-noise ratio: (X7^)i/2 ^ iiBz (5-145) rms time error = 1 /(signal bandwidth) (voltage signal-to-noise ratio). As an example of the application of these ideas, let us consider a pulse radar with a narrow antenna pattern which is scanned over the target at a constant angular rate. Such a system is similar to the AEW example dis- cussed in Paragraphs 2-10 to 2-20 and the results that we shall develop are applicable to the design considerations in that example. The video signal generated by such a system would have a form similar to that shown in Fig. 5-19. The time at which the envelope of the pulses Reference -.^TllrC •Target 0=Beamwidth measured between half - power .Envelope points ^^, i/'s = Scanner Angle Fig. 5-19 Receiver Voltage Pulse Train Return from a Point Target. reaches its peak value will be correlated with the angular position of the target so that the problem of locating the target in angle is essentially that of determining the arrival time of the signal, and the ideas and develop- ments of this paragraph are applicable. The basic functions performed by the system are indicated in the block diagram in Fig. 5-20. The received signal is amplified and filtered by an IF amplifier which is matched to the envelope of an individual pulse. Noise 5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 287 Random Noise _L_ 1st Detector Scanning Antenna IF Amp. Matclned To Pulse Square Law 2d Detector Pulse Gate Video Filter Matched To Scan Modulation Output Giving Minimum Angular Error Fig. 5-20 Block Diagram of Receiver of Scanning Radar. with a uniform power density is introduced at the input to this amplifier. A square-law second detector is used to generate the video envelope of the signal plus noise. The video signal-plus-noise pulses are gated into a video filter which is matched to the scan modulation. That is, this filter is matched to the fundamental component of the gated video signal plus noise. All signals, information, and noise at the repetition frequency and its higher harmonics are filtered out. We assume that the number of pulses per beamwidth is sufficiently large that the signal spectrum about the first harmonic of the repetition frequency does not overlap the fundamental component of the signal spectrum. We also assume that the video noise is sufficiently uniform over the bandwidth of the scan modulation signal for the assumption of a constant noise spectrum under which we derived Equation 5-145 to be valid. Other system configurations are possible. A more practical design might stretch the gated signal-plus-noise pulses before smoothing by the scan modulation filter. Such a system, however, would give slightly greater angular errors than the one we have chosen to study. The following special notation is adopted for this example. a = voltage amplitude of received signal T = repetition period d = duty ratio ■^ = antenna angle y^ = antenna angular rate 9 = antenna beamwidth (half-power points — one-way) 5 = pulse width n = Q l\j/T = number of pulses per beamwidth The signal received from the target will have the following form: Received signal = a (scan modulation) (pulse modulation) cos coc/- (5-146) 288 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS The peak pulse power will be a'^ jl. We assume that the individual pulses are rectangular and of width b. The total energy in a pulse, then, is a^B jl. Assuming noise with a uniform power spectrum of density D, the maximum signal-to-noise ratio which can be obtained with a filter matched to a pulse is given by the ratio of the pulse energy to the power density of the noise as was derived in Equation 5-127: Peak signal to noise ratio = S/N = a^/lD. (5-147) The noise at the output of the IF amplifier corresponds to a narrow band noise process similar to those discussed in Paragraph 5-8. Since the IF amplifier is matched to the envelope of the pulse signal, the autocorrela- tion function and power spectrum of the low-frequency in-phase and quadrature components of the noise, x{t) and yif) in Equation 5-73, will be of the same form as that of a rectangular pulse. This autocorrelation is a triangular function of width 25 and height equal to the noise power. The properties of the video signal and noise after a square law detector can be determined from Equation 5-87 which gives the autocorrelation function of the output of a square-law second detector: 7^2 = (202(1 + S/NY + (2cr2)2[p2 + 2{S/N)p]. (5-87) The first term in this expression corresponds to the video d-c level during a pulse while the term involving p and p^ corresponds to the video noise. In order to exclude the possibility of ambiguities incident to a noise maximum in the neighborhood of the signal maximum, we assumed that the signal-to- noise ratio was large in the development of Equation 5-145. It will simplify the present analysis if we approximate Equation 5-87 for large S jN by considering only the dominant d-c and noise terms in that equation: 77^- = {2<rr~{S/NY + (:lcj^-Y2{S/N)p, S/N»\. (5-148) The normalized autocorrelation function p in this expression corresponds to the triangular function noted above of width ITd but of unit height. The shape of the video spectrum will not be exactly uniform as is re- quired for the developments of this paragraph to be rigorous. The total width of the spectrum, though, will normally be greater than the spectrum of the scan modulation by a factor on the order of 10^. Variation of the noise spectrum over the scan modulation bandwidth, then, will be quite small, and we are justified in approximating the noise spectrum by a spectrum with a uniform power density. The power density of the noise spectrum at zero frequency, which we shall assume to be extended to all frequencies, is found by integrating the autocorrelation of the noise. The integral of the triangular function p{t) is 5. When this value is substituted for p in Equation 5-148, the term involving this factor gives the power density of the noise at zero frequency during a pulse or with a c-w signal. 5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 289 Because the video signal plus noise is gated, the noise power and thus the noise power density will be smaller than the value during a pulse by the duty ratio d. Taking these factors and considerations into account, the effective power density is determined from Equation 5-87 to be Power density at zero frequency of gated video noise ^ {lcj''Yl{SlN)hd,SINy>\. (5-149) The d-c component of the video corresponds to the signal which is used to locate the target. From Equation 5-148 it will be noted that this component is proportional to the received signal power. Thus the video signal is modulated by a scan modulation function which indicates how the received power fluctuates during a scan. The first term in Equation 5-148 gives the square of the d-c level during a pulse for large signal-to- noise ratios. To obtain the d-c level of our gated signal, we must multiply the signal level during a pulse by the duty ratio d. The resulting video signal has the following approximate form: Gated video signal = (2o-^)(*S'/A^)'^(scan power modulation), S/N^ 1. (5-150) We assume that the antenna pattern of the system has a Gaussian shape and that the same antenna is used for both transmission and reception. The beamwidth of the pattern 6 is defined as the angle between the half- power points for one-way transmission. The gain of the two-way power pattern would thus be down by a factor of 4 at these points. The antenna angle is denoted by ^. We assume that the scan modulation is generated by a constant velocity scan at the rate i/'. Supposing that the signal max- imum occurs at the time / = 0, the scan modulation has the form Scan power modulation = exp (— \p^/0.1SQ^) = exp (- ipyyO.lSQ') (5-151) Because we have assumed a square-law second detector, the scan modula- tion of the video voltage will be proportional to this scan power modulation. We are now in a position to apply the result of Equation 5-145, giving the rms error in time of arrival, which in turn yields the rms angular error after multiplication by the scan rate. We first compute the signal-to-noise ratio in the output of the video filter matched to the scan modulation. The energy in the signal is given by the integral of its square: Signal energy at low frequencies = (2cr2)2(^/7V)V2 / exp ( - V'VVO.0902) dt = 0.53{2a'y(S/Nyd\e/i^). (5-152) 290 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS The power density of the video noise is given in Equation 5-149. It was determined in Paragraph 5-10 that the signal-to-noise ratio at the output of a matched filter is equal to the ratio of the signal energy to the noise power density, so we have Signal-to-noise ratio at O.S2{lcj^y{S/NYd\Q/4') output of video filter = z^ = {2a'^y2{S/N)d^T = 026S{S/N){Q/i^T) = 0.1GS{S/N)n. (5-153) The number of pulses per beamwidth given by the ratio Q jxpT has been denoted by n in this equation. The bandwidth of the signal can be determined from the spectrum of the scan modulation. The Fourier transform of the modulating function in Equation 5-151 is as follows. / ^ e\ / -co^0.099^ \ Spectrum of scan modulation = ( -yO.lSTr • I exp I ■ 1 (5-154) The bandwidth, as defined by Equation 5-143, is easily computed: Scan modulation bandwidth = B 1/2 \ /"" /-co20.i8e2\ "1 1/2 U-o^^^K 2v- rJ - 2.35,/^/e 5-155) The rms error in measuring the target angle can now be estimated from Equation 5-145 as the scan rate divided by the product of the scan modu- lation bandwidth and the voltage signal-to-noise ratio in the video filter output: rms angle error = (M^-y^^ = (li^yi'' = i/Bz (5-156) = 4^Q/13S4^^026S{S/N)n = O.S25e/^|(S/N)n. The angular error of a scanning radar has been studied in the technical literature^ for conditions very similar to the assumptions of this example. In that study the minimum possible rms angular error was found to be approximately proportional to an expression of the form of Equation 5-156 -P. Swerling, "Maximum Angular Accuracy of a Pulsed Search Radar," Proc. IRE 44, 1140-1155 (1956). 5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 291 but only about half as great. Actually, the estimate in Equation 5-156 is optimistic. The mechanism by which the maximum value is chosen can introduce additional errors; nor was any consideration given to the effect of target fluctuations which would act to increase the error. As a typical case, if the signal-to-noise ratio out of the matched video filter as given by Equation 5-153 is 6 db, the rms angle error from Equation 5-156 is rms angle error = e/2.35^[4', z'~ = 4 (5-157) = 0.2139. In Paragraph 2-15 it was assumed that the target angle could be determined in a scan to 1 /4 of a beamwidth. From the relation above, this is not an unreasonable assumption. GEORGE • L. HOPKINS • R. M. PAGE • D. J. POVEJSIL • H. YATES CHAPTER 6 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES* 6-1 INTRODUCTION The radar systems engineer is often asked to solve the following problem: "Given a set of performance specifications based on the tactical problem requirements, derive a radar system that will meet the specifications." For a variety of reasons, it is seldom possible to solve this problem in a straightforward fashion. Probably the most important reason is this: The performance specification — if properly derived — will seldom specify a task which simply cannot be performed by radar techniques (for example, the performance specification could not logically ask the radar to distinguish between red and blue aircraft); however, the performance specification will usually require the radar to perform a. group of tasks which are not logically consistent with any one radar system mechanization. Even when mecha- nization limitations are excluded from consideration, there is no such thing as an "ideal" radar system which can perform any group of functions in an optimum manner. Every conceivable type of radar system possesses a combination of good and bad characteristics and both must be accepted and rationalized in a given application. The usual approach is to assume a generic type of radar system which experience and judgment deem reasonable. The assumed system then is measured analytically against the overall system requirements to determine whether it has the inherent potential for providing an acceptable problem solution. This process is repeated until the best match is found between the performance specification and the basic laws of nature governing what can be done by a given radar system. *Pai-agraphs 6-1, 6-2, 6-6, and 6-7 are by D. J. Povejsil. Paragraph 6-3 is by R. M. Page. Paragraph 6-4 is by S. F. George. Paragraph 6-5 is by L. Hopkins. Paragraph 6-8 is by H. Yates. 292 6-2] BASIC PRINCIPLES 293 Unfortunately, there is a tremendous variety of possible choices. In terms of generally recognized system types and subtypes, there are pulse^ continuous-wave {CW), pulsed doppler, monopulse, correlation ^high-resolution, and moving target indication {MTI) radars. Some of these types represent genuinely different approaches; some of them represent merely alternative means for performing the same job; and some of them are derivatives of particular system types. In each case, however, the selection of one of these types commits the radar system designer to a problem approach that is confined within uncomfortably narrow limits. The radar system designer must therefore have a good general knowledge of the basic system types and the general laws that govern their performance characteristics. Toward this end, this chapter will attempt to accomplish two things. (1) It will summarize basic radar laws in a rule-of-thumb fashion to provide a means for understanding the operation of any radar system. (2) It will describe the performance characteristics and limitations of generally recognized radar system types and will indicate their general areas of application. 6-2 BASIC PRINCIPLES The operation of almost any radar system may be visualized and under- stood by asking and answering the following basic questions: (1) Is the system active, semiactive, or passive (see Paragraph 1-4) ? (2) What information is contained in the signal return from the assumed target complex? (3) What are the system sampling frequencies? (4) How are the radar data detected and processed in the receiving system ? (5) Where does the processed information go? Each of these questions may now be considered in greater detail. Type of Radar System. The most basic division of radar system types is a classification based on the origin of the target signal information. An active system generates the signals which are ultimately scattered back to the point of signal origin. A passive system is simply a receiving system which utilizes target-generated radiations as its signal source. A semiactive system employs separate transmission and receiving systems which may be at some distance from each other. Depending upon the degree of coupling between the transmitted and received signals, a semiactive system may resemble either an active or a passive system insofar as its basic operation is concerned. 294 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES Type of Information. In any radar system problem there are two basic kinds of target information: (1) The desired ra.da.r-denved target information (2) The radar-derived target information that is actually obtained using a given system The latter will be exemplified by answering the last three questions posed at the beginning of this paragraph. Most commonly, the desired radar-derived target information takes the form of a four-dimensional information matrix as shown in Fig. 6-1. The Expanded View of a Quantized Region I^i Elevation rBeamwidth Azimuth leamwidth Range Resolution Element '^MAx=Max. Unambiguous Range i/^s = Solid Angle of Coverage N, =No. of Range Elements No =No. of Azimuth Elements Ne =No. of Elevation Elements Nv =No. of Velocity Resolution Elements Per Block Fig. 6-1 Radar System Information Matrix. radar is expected to detect and classify targets according to their range, their angular orientation (two dimensions), and their relative radial velocity. Depending upon the tactical problem, each dimension may be characterized by (1) a maximum and minimum value, and (2) a minimum resolution element. Thus, the total amount of information which the radar may gather is A^ = A^, X A^« X A^a X A^. elements. (6-1) Generally, the tactical problem sets some limit on the time taken to gather this information. If we define this as the lota/ scanning time, tsc, the required interrogation rate of the radar system is N = N/t,c elements/sec. (6-2) Often, it is quite informative to translate a system requirement into the form of Equations 6-1 and 6-2. For example if we consider a system which requires 150 n.mi. range with 0.1 n.mi. resolution; a 1° fan beam with 360° 6-2] BASIC PRINCIPLES 295 of angular coverage in 2 seconds; and an ability to separate targets with radial velocities of from to 2000 knots in 40-knot increments, we find: ,V 150 ,^ 360 ^^ 2000 ,n r ^ -,r,2 ^ i A^ = -Tpj- X . ry X ^7^ = 13.5 X 102 elements/sec. which is a very large number even though the radar is providing only three dimensions — range, one angle, and velocity. This answer implies a system bandwidth requirement of at least 13.5 Mc even if it were possible (which as the reader shall soon see, it is not) to apportion this bandwidth between range, angle, and velocity coordinates in the desired manner. In the case of a passive system the information matrix is usually only two-dimensional (angle only) since passive systems do not ordinarily have range and velocity measuring capabilities.^ System Sampling Frequencies. The system sampling frequencies govern the minimum resolution element size and the total unambiguous measurement interval of each coordinate — range, angles, and velocities. In general, there are three basic sampling frequencies which are important in determining the character of the signal entering the receiver: (1) trans- mitted bandwidth, (2) transmission periodicity, and (3) angular scanning frequency. 1. The transmitted bandwidth I^ft determines the rate at which the radar system can collect pieces of range information. It represents, in effect, the rate at which successive range elements of space can be interrogated. This principle is easily seen for the case of a pulse radar. In this case the trans- mitted bandwidth is the reciprocal of pulse length (A/^ = l/r). At any instant of time following transmission, the received pulse information originates from a range interval Ai? which has the width Ai^ = f = ^; (6-3) Thus every r seconds, information is received from a new range interval. The same principles hold whether the transmitted bandwidth is created by pulsing or by other means such as "FM-ing." That is, the minimum range resolution element is defined by Equation 6-3 and the rate Nr at which the system collects pieces of range information is: Nr = ^ft. (6-4) If the transmitted bandwidth is large relative to the maximum doppler shift and all the other sampling frequencies, then the transmitted band- width also defines the maximum rate at which the radar may collect all ^A passive system designed to collect and classify radiation sources according to their frequency, bandwidth, polarization, and angular location can encounter bandwidth problems similar to those of an active three- or four-dimensional radar system. 296 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES kinds of information, regardless of how it might be split up between range, angular, and velocity coordinates: (Total interrogation rate) N = Nr = Aft. (6-5) 2. The transmission periodicity fr is defined as the fundamental repetition frequency of the radar signal. (In a pulse radar, for example, /r would be equal to the pulse-repetition frequency or PRF.) This quantity governs the radar's ability to split up information between range, angle, and velocity coordinates. The transmission periodicity defines the maximum unambiguous range interval as follows: /?.ax = 7^- (6-6) Target returns from greater ranges will be ambiguous because they will enter the receiver after the transmitter has begun another transmission cycle. The total number of separate range intervals covered by the radar is then The transmission periodicity also defines the maximum relative velocity interval that may be measured without ambiguity. This may be derived as (A^)rn,ax--^ (6-8) where (AF)r^ax = maximum relative velocity interval (cm /sec) X = wavelength (cm). For higher velocities, the total doppler spread becomes higher than the sampling frequency/.. Thus the sampling process will create measurement ambiguities regardless of how the total doppler spread is split between opening and closing velocities. 3. The angular scanningjrequency is defined as the rateA^a at which signal information is collected from separate portions of the angular space volume. This quantity defines the minimum possible bandwidth of any received signal and it may be expressed: A^.=/. = ^ (6-9) where i^s = total solid angle scanned ^o = solid angle subtended by antenna pattern or instantaneous field of view. 6-2] BASIC PRINCIPLES 297 The transmission periodicity Jr places an upper limit on the scanning frequency. If, for example, A >/r, then radar returns from the far end of the unambiguous range interval will not be received because the radar antenna will have moved to a new angular position by the time the signal has returned. For this case i?n.ax = 1^^ fs>fr. (6-10) Since the angular scanning frequency defines the minimum signal band- width, it also defines the minimum possible velocity resolution element. Thus Ar>-^. (6-11) where A/^ = velocity resolution element, and the total number of separate unambiguous velocity elements is, from Equations 6-8 and 6-11, N^^^^n^<-fi. (6_i2) For passive radar and IR systems, the scanning frequency is the basic system sampling frequency. The interrogation rate of such systems is roughly equal to the scanning bandwidth. Detection and Data Processing. The detection and data-processing system of any radar may be characterized by several basic properties: (1) type of detection process, i.e., coherent or noncoherent; (2) number of channels; (3) filtering techniques; (4) signal-to-noise ratio as a function of target size, range, etc. A knowledge of these properties can provide a ready means for estimating the performance potential of any system. Each of these properties will be discussed briefly; subsequent sections of this chapter will provide illustrations of the various possibilities for the generic systems. In a noncoherent detection process, no attempt is made to correlate the phases of the transmitted and received signals. Thus the signal returns from each target must be detected separately and added together after detection (postdetection integration) as shown in Fig. 6-2. As was shown in Paragraph Z-'}), this process improves the S jN ratio of the target return by a factor which may be expressed as A(^/yV) = n^ (6-13) where n = number of samples integrated; n ^ frifs if all the samples in one scan period are integrated 7^ = 0.5 - 1.0 (0.5 for low S jN and 1.0 for high S jN). 298 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES RF Signal Bandpass »^ — ► Amplifier (Af=Af, Detector Integration I Output Filter Signal* Noncoherent Reference NONCOHERENT INTEGRATION RF Signal 1 Bandpass Amplifier (Af=Af,) Integration Filter (Af<j>AVnj Detector Output Signal Coherent Reference COHERENT INTEGRATION Fig. 6-2 Noncoherent and Coherent Integration Processes. Thus, noncoherent detection does not make optimum use of system infor- mation redundancy; in fact, in the region of low S /N ra.tio, the S /N ratio improves approximately as the square root of the number of samples integrated. In a coherent detection process, the phase relations between the trans- mitted and received signals are maintained. This makes it possible to add successive samples before detection to obtain a direct enhancement of S jN ratio (predetection or coherent integration). In this case improvement in 6" /A^ ratio may be expressed M^SIN)= n. (6-14) Thus, in a coherent system, the S jN ratio can increase linearly with information redundancy. This means that all other things being equal (average power, frequency, antenna aperture, etc.) a coherent detection system can obtain longer detection ranges than a noncoherent system, the difference between the two being particularly noticeable for high informa- tion redundancy. In addition, as will be shown in Paragraph 6-4, a coherent system can employ a more efficient detection law than a noncoherent system, thereby enhancing the relative detection capability of coherent systems even for short observation times. These characteristics coupled with the doppler frequency measurement ability of coherent systems (see Paragraphs 6-4, 6-5, and G-G) has resulted in a significant shift of develop- ment emphasis to coherent systems in recent years. The number of channels required in a radar system depends upon the detection bandwidth and the scanning time. The basic relationships may be ascertained by considering a radar which is designed to measure range, 6-2] BASIC PRINCIPLES 299 angles, and velocity. For such a system the information rate may be expressed (from Equations 6-1, 6-2, 6-9, and 6-12): iV = TV, X A^c X TVa X A^. = ^ X/s X N,. (6-15) The minimum detection bandwidth/^ that could be employed with such a system is of the order of the bandwidth induced by scanning /^ as previously mentioned. Thus the number of parallel channels needed to process all the radar data in minimum time is N//d = N/fs ^ ^ X N, = Nr X N,, number of channels. (6-16) This development shows that such a radar would require a separate channel for each range resolution element for a total of Nr range channels; each range channel would possess, in turn, 7V„ velocity channels. A repre- sentation of such a system is shown in Fig. 6-11. The only means for reducing the number of channels required is to increase the detection bandwidth or to increase the total scanning time and employ time-sharing of the receiving channels. A noncoherent pulse radar is a good example of the first approach: in this case the predetection bandwidth is made equal to (or greater than) Aft and only one channel is needed. A CW radar with a sweeping velocity gate is a good example of the second approach; in this case, the various velocity intervals are examined sequentially. This permits single-channel operation at the cost of increasing the total interrogation time by a factor equal to the number of velocity intervals, as will be explained in Paragraph 6-5. A number of means — other than the brute force approach indicated — exist for creating parallel information channels. Principal among these are the storage techniques described in Paragraph 6-6 and the delay-line filtering techniques described in Chapter 5. The filtering techniques commonly employed in radar receivers may be listed as follows: (1) mixing, (2) bandpass filtering, (3) gating, (4) demodu- lation, (5) clamping, (6) cross-correlation error detection, (7) comb filtering, and (8) video integration. Chapter 5 developed the basic mathematical theory of these techniques with illustrations taken from the example of a pulse radar system employing conical scan angle tracking. The basic principles developed for each of these operations do not change; thus the material developed in Chapter 5 provides a means for tracing and analyzing the flow of signal plus noise through any radar receiver. The generic systems discussed in subsequent paragraphs will provide examples of the various filtering and receiver sampling techniques as they are used in other types of systems. 300 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES The signal-to-noise ratio of the target information is derived from various modifications of the basic radar range equation (Equation 3-1). As the examples in the following paragraphs will show, considerable care must be taken in the derivation of an approximate expression for 6" /TV ratio to allow for system losses and vagaries of the receiving system such as sweeping gates, filter sampling times, and postdetection filters. Three factors are basic in determining S jN ratio and these provide a convenient basis for comparing S jN performance of different systems in the same situation (i.e. same operating frequency, search volume, antenna size, and scan speed). They are (1) average power, (2) type of integration (coherent or non- coherent), and (3) effective integration time. Information Utilization. The end use of the radar information in a given application constitutes the reference — knowledge of which must be compiled to understand the operation of any given system. The end use requirements for a given application are derived by analyses such as those shown in Chapter 2. Those examples demonstrated a number of different end-use possibilities such as (1) display of radar information for interpre- tation by an operator, (2) coding and transmission to a remote location, (3) weapon direction computation. Other possibilities include (1) storage by photographic techniques, (2) correlation with information from other sensors such as infrared (IR) and photographic, (3) navigation computa- tions. The operation of any radar system can be judged only in terms of its compatability with a set of end-use requirements. This fact is often forgotten by people who like to categorize radar systems on an absolute basis. Such people originate statements such as "Pulse radars have no low-altitude capability" and "Doppler radars have excellent low-altitude capability." At best, statements such as these are partial truths; at worst, they are quite wrong in certain applications. The systems designer is well advised to avoid generalizations of this sort and analyze radar systems with respect to their applicability to specific problems. 6-3 MONOPULSE ANGLE TRACKING TECHNIQUES Angle tracking requires measurement of two quantities in a manner that is effectively continuous. These quantities are magnitude and sense of angle tracking error. As shown in Chapter 5, Fig. 5-13, this is accomplished in conventional conically scanning tracking radar by purposely generating instantaneous tracking errors, but alternating the sign of the error, and averaging to zero. The method is simple and effective, but suffers errors when the signal fluctuates in amplitude in such a manner as to increase apparent 6-3] MONOPULSE ANGLE TRACKING TECHNIQUES 301 errors of one sign while decreasing apparent errors of the opposite sign (see Paragraph 4-8). Such a phenomenon is possible because plus errors and minus errors are generated alternately, not simultaneously. One obvious method of eliminating this source of error is to generate both positive and negative errors simultaneously . A straightforward technique for accomplishing this is to amplify the two signals from two overlapping antenna patterns separately and compare the two amplifier outputs. This technique is operable, but places severe stability requirements on the amplifiers, since relative drifts in amplifier gain produce changes in indi- cated correct tracking angle. An analogous method makes use of two spaced antennas in an inter- ferometer arrangement. Signals from the two antennas are amplified separately, with a common local oscillator for the two receivers, and relative phase is measured at intermediate frequency. In this case the phase stability requirement on the amplifiers is severe, since relative phase shifts in the two channels similarly produce changes in indicated correct tracking angle. Instability in indicated correct tracking angle may be overcome in either the amplitude or the phase comparison approach by connecting the two antennas in phase opposition before amplification, thus requiring only one receiving channel. Direction of arrival of signals is determined as the direction in which the amplifier output is near or equal to zero when increase of signal is produced by misaligning the antenna pattern in either direction from this so-called null point. This technique suffers from two objectionable characteristics. When there is no tracking error, there also is no signal to indicate presence of a target; and when there is an error signal, the sense of the error is not indicated. Monopulse radar, as its name implies, is a tracking radar that derives all its tracking error information from a single pulse and generates new and independent error information with each new pulse. In a broad sense, the simultaneous amplitude or phase comparison systems described above may be called monopulse systems. The name monopulse, however, has become restricted by common usage to still another method for generating both positive and negative errors simultaneously which overcomes the principal objections of the other systems. The method consists in so connecting the RF circuits of two antennas that both sum and difference signals are obtained simultaneously. The patterns of the two antennas overlap in the conventional way for generating tracking error information, as shown in Fig. 6-3. The sum signals from the two antennas merge the two patterns into a single lobe pattern as shown in Fig. 6-4. The difference signals produce the familiar null pattern with the sharp zero at the center, as shown in Fig. d-S. The sum and difference signals are then amplified separately and recombined in a product detector after amplification. 302 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES Fig. 6-3 Overlapping Individual Antenna Pat- terns of a Monopulse Radar. Fig. 6-4 Sum of Overlapping Patterns in a Monopulse Ra- dar. Fig. 6-5 Difference of Overlapping Antenna Patterns in a Monopulse Radar. The process of generating sum and difference signals results from in-phase connection of the two lobes for the sum pattern, and antiphase connection for the difference pattern. Consequently the sum and difference signals are mutually in phase for directions of arrival on one side of the difference pattern null, and in antiphase for directions of arrival on the other side of the null. Thus the difference signal contains within itself only angle error signal magnitude, while the sum signal contains the phase reference by which angle error sense is determined. The output of the product detector as a function of the direction of arrival of signal energy relative to the + ro c op en \ \ o . \ UJ / X _ / Angle Off Axis + Fig. 6-6 Monopulse Error Signal Curve. 6-3] MONOPULSE ANGLE TRACKING TECHNIQUES 303 antenna difference pattern null is therefore the familiar error signal curve of Fig. 6-6, with zero signal on target, and polarity of error signals indicating error sense. The output of the sum amplifier provides indication of the presence of a target, an indication which is maximum when on target. The balance point representing zero angle error is not significantly affected by relative shifts in gain or phase between the two amplifiers. The sensitivity to angle error, represented by the slope of the error curve as it passes through zero, is influenced by relative phase shift between the two ampli- fiers, becoming zero at 90° relative shift. Since it is a cosine function, however, it is insensitive to phase shift near correct phase, a relative shift at 25° producing a decrease in angle error sensitivity of only 1 db. A system diagram illustrating the monopulse principle for angle error indication is shown in Fig. 6-7. A conventional hybrid ring (see Para- graph 10-1 5) is used for deriving sum and difference signals from the two antennas. The transmitter is con- nected to both antennas by suitable TR circuitry in the sum channel, so that the transmitter radiation pat- tern corresponds to Fig. 6-4. The output of the sum amplifier is recti- fied and applied in a conventional manner as a video signal to a radar A scope, giving indication of presence of target and target range. Also the outputs of the two amplifiers are mul- tiplied in the phase-sensitive or prod- uct detector to give an error signal whose sign corresponds to error sense. This error signal, which is a video sig- nal, is added to the time base signal and the combination applied to the indicator deflection system orthogonally to the output of the sum amplifier. The resulting indication presents targets as pips perpendicular to the time base line for targets which are in align- ment with the antenna difference pattern null and which "lean" forward or backward from the perpendicular as angle of arrival deviates to one side or the other from the pattern null. It is apparent that the direction in which the signal pip points is related to the direction of arrival of signal energy relative to the antenna pattern null and is independent of the amplitude of the signal. The length of the signal pip indicates signal amplitude. Sensi- tivity of the indicator to angle of arrival is a function of relative attenuation Fig. 6-7 Single-Coordinate Monopulse System. 304 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES in the orthogonal video deflection circuits. Such an indicator has been called a "Pisa indicator" after the famous leaning tower. To accomplish the operation described it is convenient to generate the overlapping antenna patterns with a single aperture. This may be accomplished with a single parabolic reflector illuminated by two primary radiators symmetrically displaced laterally from the focus. The principle has been described for a single angle coordinate. Extension to two angle coordinates is not ordinarily accomplished by duplicating the system, but the one-coordinate system may be modified to operate as a two-coordinate system. It is first necessary to generate four lobes in the antenna pattern. This is accomplished by using a cluster of four primary radiators symmetrically disposed about the focus, with two up, two down, two right, and two left. Sum and diff^erence signals are obtained separately from two pairs. The two difference signals are then added to generate error signals in one coordinate. Sum and difference signals are then obtained from the two first sums. The resulting diflFerence signal is used to generate error signals in the other coordinate. The second or final sum signal, which is the sum of all four lobes, carries target amplitude and range information and provides a common phase reference for both coordinate error signals. The angle information may be utilized in a wide variety of configurations. Shown in Fig. 6-8 is one of the most common: an automatic angle tracking system such as might be employed in an AI radar or guided missile terminal seeker. L Antenna F<M Azimuth Diff. Channel Elevation n Diff. Channel Sum Channel Antenna Controller ^ I Jg Automatic Angle ^ Y^ Tracking Loop M Azimuth Amp. Phase Det. Error Elevation Amp. Phase Det. Sum Amp. Range Fig. 6-8 Two-Coordinate Monopulse System. 6-4] CORRELATION AND STORAGE RADAR TECHNIQUES 305 Monopulse techniques are particularly useful for applications where pulse-to-pulse amplitude fluctuations due to target variations or interfer- ence signals can degrade conical or sequential scanning tracking techniques. 6-4 CORRELATION AND STORAGE RADAR TECHNIQUES Signal storage has played a most significant role in the success of radar. From the earliest use of the cathode ray tube in echo ranging with A-scope presentation to modern sophisticated and complex magnetic storage devices for predetection integration, the use of storage has become increasingly important. Today the lack of high-capacity memory, high-speed operation, and wide dynamic range storage are perhaps major contributing factors impeding the development of more effective long-range radar. The in- creased emphasis on integration by storage has been brought about in part by the growing popularity of correlation and information theory methods for signal enhancement. The idea of correlation in itself is not new to radar — the World War II SCR-584 used a limited form of cross-correlation detection to separate the bearing and elevation errors. Here the correlation was not of the statistical nature currently in favor for signal enhancement. For this latter purpose, the cross-correlation device requires some form of storage and integration in order to fulfill its mission. Since storage can be considered a part of the correlation process, we will discuss the more general subject of correlation first. Correlation Processes. Two correlation techniques have appeared in radar during the last two decades: (1) autocorrelation, defined mathe- matically as ^n(r) = lim -^jj^W^i^ - r)dt (6-17) -:; 2T where t is a time displacement (delay in the case of a radar echo), and (2) cross correlation, defined as <P,,{t) = lim ;r^ / /i(/)/2(/ - r)dt. (6-18) Both of these techniques are defined in the time domain and exist theoret- ically only in the limit as the total observation time becomes infinite. In practice, of course, infinite time is not available, and it becomes necessary to reinterpret the functions using finite limits. Let us define an incomplete autocorrelation function as ^„(r,T) = ^/^/iW/i(^ - r)dt (6-19) 306 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES and an incomplete cross-correlation function as ^,,{tJ) = ^jjm-^it - r)dt (6-20) where 2T is now a finite observation or integration time. Autocorrelation of a limited nature, specifically for r = 0, has been used in conventional radar systems almost since the invention of radar. As can be seen from a study of Equation 6-19, the incomplete autocorrelation function as applied to radar for r = consists of (1) obtaining the instan- taneous echo power /i^(/), (2) integrating or summing for a finite time 27", and finally (3) dividing by the period 2T, thus forming an average power. These three steps can be seen to be essentially equivalent to the conven- tional frequency-domain radar processes wherein a square-law second detector converts the echo into instantaneous power and some type of storage provides the required averaging. In early radar sets for echo ranging the averaging was performed aurally by the operator or visually using A-scope presentation. Later the plan position indicator (PPT) used cathode ray tube persistence plus the operator for storage. Finally the use of more sophisticated video integration was adopted. The relative merits of autocorrelation (r = 0) and square-law detection versus cross-correlation detection have been studied^ with the results shown in Fig. 6-9. The output-versus-input mean power signal-to-noise ratios are plotted for a bandwidth reduction of 2 : 1 (in going from IF to video, for example), a practical value for echo ranging where the pulses must be retained. These curves apply to a single pulse where there is no integration. Signal enhancement resulting from the integration of pulses is discussed subsequently. It is interesting to note that autocorrelation can be thought of as comparable to postdetection bandwidth reduction, whereas cross correlation is comparable to predetection bandwidth reduction. In Fig. 6-9 there is an apparent threshold in the autocorrelation and square-law detection curve starting in the neighborhood of unity signal-to-noise ratio. This threshold is noted by the change from a linear to a square-law relation- ship between output and input sensitivity. Such a threshold does not exist in cross correlation, where a noise-free reference is used. Cross-Correlation Radar. As soon as the signal enhancement capability of statistical cross correlation was recognized, applications to radar were considered. In order to obtain the maximum advantage from the process, one of the functions in Equation 6-20 must be noise free. A study of (pi2(r,T) reveals that if /i(/) is the delayed target echo which contains desired information as well as unwanted noise, then Ji{t — r) should be a noise-free reference signal possessing characteristics identical ^Samuel F. George, Time Domahi Correlation Detectors vs Conventionat Frequency Domain Detectors, NRL Report 4332, May 3, 1954. 6-4] CORRELATION AND STORAGE RADAR TECHNIQUES +30 + 20 307 -20 -40 -50 / / Cross Correlat Detector (Noise- Reference) on -ree / V / / \ / Autoc (r=0) a Law orrelation nd Square- Detectors / -30 -20 -10 +10 +20 INPUT SIGNAL-TO-NOISE RATIO (db) Fig. 6-9 Comparison of Autocorrelation and Cross Correlation. with those of the signal component in/i(/) and permitting a variable delay r to match the echo delay, thereby indicating range. This would be a specific application to the radar problem of echo ranging or the measurement of delay. A simplified block diagram of a cross-correlation detector is shown in Fig. 6-10. The output ^i2(t,T) could be used in the same manner as the output from the second detector of a conventional radar. Echo ranging as a major application of the cross-correlation principle to radar has been studied comprehensively by Woodward^ in the light of information theory. Woodward shows that in order to extract the most information about the exact target range from a received radar echo in additive Gaussian noise, the optimum receiver is one which forms the ^P. M. Woodward, Probability and Information Theory, with Applications to Radar, McGraw- Hill Book Co., Inc., New York, 1953. 308 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES Receiver + n(t) Transmitter u{t) Storage Device and Variable Delay f^{f-T) = u{t-T) Multiplier Integrator 2l j\(t) f2(t-r)dt ■4>,Jr,T) Fig. 6-10 Simplified Cross-Correlation Detector. incomplete cross-correlation function. The problem of signal detectability has been very exhaustively studied"* and reported in 1954 at the MIT Symposium on Information Theory^. One conclusion is that for the case of a known signal operating through white Gaussian noise, the cross- correlation receiver is optimum. This result is based upon the likelihood ratio criterion. Cross correlation has become very useful in extracting the doppler frequency shift or range-rate information for moving targets, thus adding a new method to aid in target detectability as well as in more accurate tracking and multiple target resolution. In order to extract the doppler, the incoming echoes must be processed so as to permit coherent integration®. This is predetection integration, which in the case of a pulse-doppler system means coherent video or IF integration. The cross-correlation principle is embodied in all of the systems proposed for using range-rate information. First, a coherent or stored noise-free reference must be available; then some storage medium is required to permit integration; and finally some form of very narrow-band doppler filtering must be employed. Fig. 6-11 shows a block diagram of a straightforward or brute-force pulse-doppler system. Here there are n range gates with m doppler filters per gate. It is readily seen that a tremendous duplication of equipment is called for unless some storage device can be placed in the system. *J. Neyman and E. S. Pearson, "On the Problems of the Most Efficient Tests of Statistical Hypotheses," Phil. Trans. Roy. Soc. London A231, 289 (1933). 1. L. Davis, "On Determining the Presence of Signals in Noise," Proc. Inst. Elec. Engrs. London 99 (III), 45-51 (1952). E. Reich and P. Swerling, "The Detection of a Sine Wave in Gaussian Noise," J. Appl. Phys. 24, 289 (1953). R. C. Davis, "On the Detection of Sure Signals in Noise," J. Appl. Phys. 25, 76-82 (1954) W. W. Peterson and T. G. Birdsall, The Theory of Signal Detectability^ Electronic Defense Group, University of Michigan, Technical Report No. 13, July 1953. ^Transactions of the IRE, PGIT-4, September 1954. ^Bernard D. Steinberg, Coherent Integration oj Doppler Echoes in Pulse Radar, Report #182-112-1, General Atronics Corp, Aprif 1957. 6-4] CORRELATION AND STORAGE RADAR TECHNIQUES 309 w TR M — Transmitter — ► Delay Receiver Multiplier Coherent Video f2 -*' D D Fig. 6-11 Pulsed-Doppler System. Storage Radar. We have noted that one of the first and perhaps simplest of all storage mechanisms consisted of a visual observer using an A scope. Next, in the PPI, screen persistence performs the storage, and scan-to-scan integration is attained. Finally, the last form of postdetection integration to become significant was of the video type. Here the storage element could vary from a video delay line to some form of electrostatic storage. A simple delay-line video integrator is illustrated in Fig. 6-12. Video T = lnterpulse Period Pulse Adder Video Delay Line — Amplifier Integrated " Output Fig. 6-12 Simplified Delay-Line Video Integrator. The number of pulses which can be effectively integrated to improve the signal-to-noise ratio depends upon the delay line loss, feedback circuit stabil- ity, distortion, and the length of time the target remains essentially at a fixed range. Video integration of pulses embedded in additive Gaussian noise at the radar input improves the signal-to-noise power ratio by the number of pulses added; the total improvement possible after detection, then, is limited by the observation or integration time permissible. From Fig. 6-9 it is seen that there is a detector loss for any detector input below threshold (S /N= 1) and that the ultimate radar sensitivity is obtained by predetection rather than video integration. 310 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES In order to effect an improvement in signal-to-noise ratio by predetection integration, some technique must be used either to obtain signal coherence at IF or video, or to ensure transmitted signal recognition for cross- correlation purposes. The earliest methods employed coherent sources for transmission or coherent local oscillators in reception. As the transmitted signals became more complex and sophisticated, it was considered necessary to use storage to retain an exact replica for cross-correlation purposes. Both electrostatic storage and magnetic-tape storage have been developed and used successfully. The limited dynamic range of electrostatic storage combined with relatively short storage times, and the relatively slow accessibility of the data on magnetic tape have created an interest in magnetic-drum storage. As advanced technical progress provides increased dynamic range and higher frequency operation, the tremendous data handling capacity combined with high-speed record and readout and long- duration storage make magnetic-drum storage appear very desirable for radar use. Fig. 6-13 shows a system using transmitted-waveform storage to provide a reference for the cross correlation and magnetic-drum storage to reduce TR Transmitter Storage & Range Gate Receiver Multiplier Magnetic Drum Doppler Storage Output ^3 -^ D Fig. 6-13 Storage Radar. the equipment multiplication required in Fig. 6-11. The transmitted- waveform storage unit could be envisioned to consist of multiple delays corresponding to the n range gates of Fig. 6-11. There could be a separate track on the magnetic drum corresponding to each value of delay — i.e., to each range gate. For a given channel or track on the magnetic drum, the return echo pulses could be clipped and converted essentially to a binary code, which could then be painted sequentially so as to form the doppler signal as a modulation on the code.'' The doppler frequency could be explored as before by a doppler filter bank as shown. Only an elementary 6-5] FM/CW RADAR SYSTEMS 311 system has been illustrated here, and numerous ramifications become evident. The ultimate goal in search radar using correlation and storage tech- niques will be achieved when the range accuracy and resolution are limited only by the transmitted bandwidth, and the range-rate accuracy and resolution are limited only by the total observation time during which the target doppler remains coherent. 6-5 FM/CW RADAR SYSTEMS Previous discussions have dealt primarily with pulsed radar systems — i.e., systems where transmission and reception occur at different times. Another important class of radar systems is composed of systems that employ continuous transmission (CW systems). In these systems, the transmitting and receiving systems operate simtultaneously rather than on a time-shared basis. For certain applications — such as semiactive missile guidance systems — continuous-wave (CW) systems can offer important advantages, particularly with respect to high clutter rejection for moving targets, and relative simplicity. Basic Principles of Operation. In a CW system, the transmitted and received signals are separated on a frequency basis rather than on a time basis as for a pulsed system. This is accomplished by maintaining phase coherence between transmission and reception — a process which permits the measurement of the doppler shift caused by the continual rate of change of phase in the radar reflection from a relatively moving object. The principles of operation of a simple CW system are illustrated in Fig. 6-14. A signal at frequency/^ is transmitted. The return echo from a target moving with relative velocity Vr is shifted by the doppler effect to a new frequency /< +/d, where Ja = 1ft Vrlc. (1-20) Closing geometries shift the received frequency higher than the trans- mitted; opening geometries lower the frequency. The transmitted and received signal frequencies are mixed to recover the doppler frequency, which is then passed through a filter whose bandpass is designed to accept the doppler frequency signals from moving targets and suppress the return signals from fixed targets such as ground clutter. (For the example in the diagram, the clutter is shown at zero frequency: a condition that would exist for a ground-based radar.) For a moving airborne radar, the clutter would also possess a doppler shift relative to the transmitted frequency. This complicates the design of the filter; however, the basic principle remains the same. 312 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES F+ /. \' fo - Mixer -i f. f. Bandpass Filter Mixer Inputs Mixer Output Clutter jr Target Transmitter Signal Clutter Signal ^/Target Signal ^0 fo+fd Filter Output Filter ^ ^ Bandpass i^rget imin. (b) Fig. 6-14 Simple Stationary CW Doppler System: (a) Block Diagram, (b) Signal Frequency Relations. Very often, the filter is not a single filter as shown in Fig. 6-14 but rather a series of overlapping narrow band filters — each equipped with its own detector — which covers the total desired doppler frequency band as shown in Fig. 6-15. This permits very narrow bandwidth detection, measurement of target velocity, and resolution between targets with different relative velocities within the same antenna beamwidth. Filter Outputs \ rn I \ ^^ A /^^ 1^ r^ 7"^ / ' '' - \'' / "' V' 1'' / II; \l \ W f^\l \i >/ / ; 'I' *' >/ \i ^\! *' V ' iw 1 ^ l\ Fig. 6-15 Filter-Bank Detection Showing Contiguous Filters. 6-5] FM/CW RADAR SYSTEMS 313 The information matrix of this system may be visualized in Fig. 6-16. The maximum possible range is limited by scanning speed; that is to say, beyond a given range, targets cannot be seen because the dwell time on the N, Total Doppler Acceptance Band 'N.xNgxN.xNA Individual Doppler Filter Bandwidth = Total Doppler Acceptance Band Fig. 6-16 CW Radar Information Matrix. target is less than the time required for a round trip of the radar energy. Since a simple CW system has no range-measuring capability, the maximum range is also the size of the minimum range resolution element. Range Measurement in CW Systems. To measure target range, frequency modulation (FM) of the transmitted energy is generally used in CW systems. The maximum deviation of the transmitted signal determines the range resolution obtainable, while the frequency of the FM-ing deter- mines the maximum unambiguous range.* Consider typically a simple sine wave FM as expressed by the trans- mitted FM /CW waveform of Equation 6-21 : et = Et sin ( co«/ + y- sin oo™/ j- (6-21) The received doppler signal er as detected by a coherent FM/CW radar is c-r = Er cos cod^ H J— sin ir/mT cos (comt + Tr/mr) |- (6-22) /m where Er is the peak received voltage, cod is the doppler frequency, AF is the peak transmitted deviation, /^ is the deviation rate, and r is the round trip ^For a rough comparison with pulse radar the reader may consider the reciprocal of the bandwidth of the FM transmission to be analogous to an "effective" pulse length, while the FM rate is analogous to the PRF of a pulse radar. 314 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES transit time. To determine distance, transit time can be measured by resolving the magnitude of the returned deviation (2AFsin irfmr) or by the relative phase lag of the returned modulation {irfmT) . In general, the greater the transmitted deviation AF, the greater is the resolvability of range. However, this range resolution is usually purchased at the price of in- creasing the minimum bandwidth of the doppler filters. If linear FM is used such as is provided by a sawtooth or triangular waveform, the Doppler frequency is merely shifted by the amount the transmitter has deviated during the round trip transit time. The principle is illustrated in Fig. 6-17. Depending upon range, modulation, and doppler f,-AF - TIME ■Transmitted V^^^^ ^Measure of Range f, I Doppler Frequency fj TIME Fig. 6-17 FM Range Measuring Principle. shift, the instantaneous received frequency will differ from the transmitted reference frequency. When the two are heterodyned, a frequency modula- tion is superimposed on the detected doppler signal. The magnitude of the FM deviation from the doppler is a measure of range. Care must be exercised in the selection of deviation values or the doppler frequency and range frequency may be difficult to resolve uniquely. In some applications such as altimetry the range frequency greatly exceeds the doppler values. For example, in doppler altimetry the electro- magnetic energy is radiated normal to the direction of flight so that no doppler signal results. By using triangular modulation, as in Fig. 6-17, the magnitude of the resultant detected difference frequency /r is a measure of altitude: i.e.. (6-23) 6-5] FM/CW RADAR SYSTEMS 315 where Tr = modulation period in seconds AF = peak transmitted deviation in cps h = altitude in feet c = velocity of propagation (984 X 10^ ft /sec). For more conventional radar applications using sinusoidal FM for ranging, the peak frequency deviation/^ on the doppler signal is a measure of range (see Equation 6-22); i.e., fr = 2AFs\mr/mT (6-24) or = 2Ai^7r/„/^^ where R = range in feet AF = peak transmitted deviation in cps fm = deviation rate in cps. Use can be sometimes made of the fact that the doppler frequency is precisely proportional to range rate. Integration of range rate can produce an actual measure of range provided the constant of integration can be determined. Stability Requirements. The principal theoretical advantage of CW radar systems derives from their ability to distinguish moving targets in the presence of clutter. The maintenance of this advantage in a practical design places strict requirements on the short-term frequency stability of the transmitted energy; i.e., the coherence that exists between the transmitted and received signals. Long-term frequency drift is generally not a problem provided that short-term coherency exists. Any modulation that tends to broaden the spectrum of a wanted signal may destroy the signal-to-noise ratio in the received gate. Any modulation of interference signals may produce sidebands which interfere with the resolution of desired signals. When interference signals, such as feedthrough or clutter, are very large, short-term stabilities in excess of 10~"^ may be required. The process of coherent detection of a time-delayed signal alters the angular modulation index as has been indicated by Equation 6-22. In a sfalic situation the RF reference phase can be adjusted to minimize sensitivity to angular modulations; this is impossible for a gimbaled or moving radar. Alternatively the radar detection reference could be time- delayed an amount equal to the round trip to target delay. However, this is usually impractical because of the continually changing delays of one or more targets. 316 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES From Equation 6-22, it is possible to evaluate the extent to which the angular noise sidebands of a large signal will interfere with the detection of another signal. However, it is difficult to apply Equation 6-22 in a general manner since the effective modulation index is a periodic function of the two variables /m and t where the maximum values of/m are at «/2r (where « = 1, 3, 5, ... ) and the zeros are at/m = n /t (where « = 0, 1, 2, 3, ...). Of more usefulness is a consideration of the two extremes of the relationship as divided by -a cutoff frequency /^ at which the returned deviation, by definition, equals the transmitted deviation. Below/c, where the time delay is short compared to a modulation cycle, the angular indices of the detected and transmitted signals are related simply by Mfr = M/,(27rr/,„). (6-25) Far above /c, where the time delay is long compared with a modulation cycle, the two indices are, on the average, equal. For small indices of modulation such as are descriptive of useful trans- mitting tubes it is more convenient to discuss potential interference in terms of sideband power ratios relative to the carrier. Therefore, the ratio of the power in a single sideband Psb, relative to the carrier power Pc for a detected signal, is determined (for small indices only) by: n £f4^-i- »-j« Typically, for a 1200-cps power supply ripple component producing 240 cps of frequency deviation, a carrier to single sideband power ratio of 100 would exist. Random noise modulation may be considered to be composed of distrib- uted components having a mean angular excursion per cycle of A/. T+te composite mean deviation AF associated with a band of frequencies B cps wide is then AF = Af^fB. (6-27) A not uncommon composite noise deviation in a 100-kc band B for practical CW radars is 100 cps rms (AF), indicating about a 1 /3 cps rms/cps density (A/)_. Fig. 6-18 indicates typical signal power levels which might be present in a hypothetical FM /CW radar. FM/CW Airborne Radar Systems Applications. FM/CW doppler systems are most commonly employed for applications requiring high clutter rejection and a relatively low range information rate. AI radars, missile seekers, and altimeters are good examples of such applications. (y-S] FM/CW RADAR SYSTEMS 317 200- 150- 50- C^Speed of Electromagnetic Propagation Vf = Speed of Radar Platform V^ = Radial Target Speed Clutter '/"' \ 0? y^ Receiver ^^ hE / Noise ^ Density Ike lOkc 2Vc AM Noise Density lOOkc IMc lOMc lO^Mc lO^Mc lO^Mc f Frequency — c ^ Fig. 6-18 Typical Signal Levels in an Airborne FM/CW Radar. A principal advantage of CW doppler systems is their simplicity, when compared with other means for obtaining high clutter rejection such as pulsed doppler and coherent AMTI systems. Provided that some of the limitations to be discussed below do not seriously limit tactical utility, an FM/CW system offers a lightweight and potentially reliable answer to many airborne radar system problems. The use of doppler techniques places several constraints upon the tactical usage of an airborne weapons system. (1) There are approach aspects where the target doppler frequency can be zero, or where the target doppler frequency falls within the clutter spectrum. These conditions lead to "blind regions" and regions of poor signal-to-noise ratio which must receive careful consideration and analysis in the overall system design. For example, approaches in the rear hemisphere of a target can be degraded by these considerations. Fortunately, the most interesting approach region from many tactical standpoints — forward hemisphere or head-on — is a region where doppler sensing devices are most effective in detecting and tracking targets in heavy ground clutter. (2) Buplexed-active operation (transmission and reception through a common antenna) is generally impractical in FM/CW doppler equipment 318 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES because of excessive feedihrough of the transmitter energy directly into the receiver. By physically spacing the transmitting from the receiving antenna on a common radar vehicle (spaced-aclive operation), the isolation problem becomes resolvable at the cost, however, of degradation in the vehicle's aerodynamic profile. Semiactive operation involves transmission and reception on separate radar platforms, which also minimizes the feed- through problem. The main advantages of semiactive operation in homing missilery are that the transmitting hardware is deleted as an internal missile requirement and a greater on-target illumination power density is practical from the large parent radar. Both a spaced-active and a semiactive system are illustrated in Fig. 6-19. Missile Semiactive Radar Fig. 6-19 FM/CW Airborne Radar Systems. (3) The ranging accuracy obtainable with an FM /CW set designed for high ground clutter rejection may be relatively coarse — perhaps of the order of 1—2 n. mi. This is not usually a serious drawback for guided missile applications, although it is an important limitation for fire-control systems employing unguided weapons. FM/CW Radar Performance. The detectability of the target is determined, for a given false-alarm rate, by the signal-to-noise ratio after final detection. Neglecting the effects of clutter and transmitter modu- lations, the signal-to-noise ratio may be calculated by approximate modifi- cation of the previously derived radar range formula (see Equation 3-9): S/N ^ {AttYR'FKTB (6-28) 6-5] FM/CW RADAR SYSTEMS 319 where P = transmitted average CW power B = doppler filter bandwidth. The choice of a detection bandwidth B is governed by a number of considerations derived from the tactical problem and from the realities of radar design practice. The spectral composition of a modern-type aircraft radar reflection is sel- dom more than a few cycles wide when it is caused by target characteristics alone. A CW radar, transmitting a truly unmodulated wave, produces the most elementary moving target spectrum. The possibility exists, therefore, of detecting a radar signal as much as 190 db below a watt in about a second, using simple, very narrow band filters matched to the signal waveform. However, a number of practical and tactical considerations usually limit the full exploitation of this potential. To search the frequency spectrum of expected dopplers requires a series of adjacent filters or one or more individual filters scanning the spectrum. The minimum allowable time on target is that time required for energy to build up in one of a fixed bank of filters; scanning filters increases this minimum by the ratio of scanned to actual bandwidths. Prior information as to target velocity or bearing can reduce the spectrum or area to be searched and thus increase the probability of detection in a given situation. One practical constraint on exploiting very narrow bandwidths is the shift in the doppler spectrum caused by target maneuvers. This effect can cause the signal to be greatly attenuated if the target doppler transits the filter bandpass range before the filter has time to build up. To avoid this situation, the filter bandpass must satisfy the relation B>^ (6-29) A where a = acceleration in ft/sec^ X = RF wavelength in ft. Another usual requirement in a radar system is to yield additional radar target parameters. Information theory advises that to process more data, more bandwidth and /or time is required. To resolve target range by FM on CW or to resolve bearing by AM will in general increase the minimum radar bandwidth requirements. For example, angular scanning will broaden the return spectrum bandwidth by the scanning frequency modulation, which may be of the order of 50 cycles for a typical airborne radar. Alter- natively, the use of very low information rates demands an amount of time that may be unacceptable. An economically attractive technique is to employ one sweeping filter of a bandwidth such as to satisfy all system requirements. Conventional AFC 320 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES methods can maintain the target signal in the velocity gate after lock-on. By allowing adequate time for the energy to build up in the gate,^ the maximum allowable sweep rate is proportional to the bandwidth squared. It is this "squared" sweep constraint that makes sweeping very narrow band filters very time consuming. As the filter bandwidth approaches the scanning frequency (or the reciprocal of the time on target), fixed filters become mandatory (see Fig. 6-15). From a performance standpoint, the behavior of the system in the presence of multiple targets is most important. Multiple radar targets include terrain and weather returns as well as reflections from man-made objects; for an airborne radar, each radar target will have a finite radial velocity with respect to the radar platform. Because of their physical size, man-made objects are usually point source targets, whereas clutter is most always angularly expansive. In the presence of a hypothetical hemisphere of homogeneously reflective clutter, the clutter doppler amplitude versus frequency spectrum would be similar to the integrated transmit-receive antenna radiation pattern versus angle as viewed at the receiver. A typical CW radar doppler clutter spectrum is shown in Fig. 6-18. Note the complete absence of interference at all frequencies above own-speed dopplers; only nose-aspect closing targets appear in this uncontaminated spectrum. In practice the clutter spectrum may be smeared somewhat by noise modulation on the trans- mitted energy. Reasonable prediction of specific clutter is possible, but there are a myriad of possible situations. Clutter can assume staggering proportions; a terrain return 100 db above the minimum detectable signal power is not inconceivable. Ultrahigh linearity in a receiving system is obviously required to avoid generating the additional interference of distortion products. Most radar systems are advisedly employed in a manner avoiding the most adverse clutter conditions. With some performance compromises, a practical degree of automaticity can be achieved in sorting multiple-target data in an FM/CW airborne radar. Doppler filtering can provide the necessary resolution to distinguish numbers of targets, but the decision as to which target is wanted may require the application of considerable intelligence. With experience a human being can reduce visual or aural doppler data satisfactorily for some applications. 6-6 PULSED-DOPPLER RADAR SYSTEMS Pulsed-doppler radar systems represent an eflFort to combine the clutter rejection capabilities of doppler sensing radars with the range measurement ^Commonly, values of from 2 to 17 time constants of the filter are used in practical appli- cations. 6-6] PULSED-DOPPLER RADAR SYSTEMS 321 and time-duplexing properties of a pulse radar. For applications requiring heavy ground clutter rejection, a common transmitting and receiving antenna, and accurate range measurement, a pulsed-doppler type of system represents the best known technical approach to the problem. However, the pulsed-doppler type of system also has certain drawbacks: principal among these are limited target-handling capacity (when compared with a pulse radar) and a high order of electronic system complexity. Basic Principles of Operation. A simple pulsed-doppler system is shown in Fig. 6-20. It differs from the CW system of Fig. 6-14 only by the Master Frequency Control f ^ Pulsed Coherent - Transmitter f^(Pulsed) n .,^l/^«^K f,(Pulsed) 1 r- f^+f^ (Pulsed) \ f,+f^ (Pulsed) f, Mixer } fe^f ^±f^+2f^± f^+ Bandpass -fd Fil er Fig. 6-20 Simple Pulsed-Doppler System. introduction of a pulsed coherent transmitter in place of a CW coherent transmitter; and a duplexer which turns off the receiver during a pulse period and isolates the receiver from the transmitter during the interval between pulses. A master frequency control is utilized to control the carrier \^T^ ^ f^l/fo Vfr ^ (a) -^frr fn-lA (b) Frequency Fig. 6-21 Transmitted Pulsed-Doppler Signals. 322 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES and pulse repetition frequencies and to provide coherent references for the receiver mixing processes. The transmitted signal thus consists of an RF pulse train as shown in Fig. 6-21 (a) which has the frequency spectrum shown in Fig. 6-21 (b). The width of the frequency spectrum is a function of pulse length; the separation between adjacent spectral lines is equal to the pulse repetition frequency /r. The operation of a pulsed-doppler system can best be visualized by examining the character of the return spectra from targets and clutter at various points in the receiving system. The target and clutter spectra for a single spectral-line transmission from a moving platform is shown in Fig. 6-22. Because of sidelobes, the frequencies of the fixed clutter returns can Relative Target Velocity Velocity of Radar Aircraft Sidelobes MC^ '^^^IVIain Bea ~~^,-^-Tar / m get utter. ^Main Beam Clutter Sidelobe C J/'' K^ Le-^larget Return 2V,/\ 2V,/X f,=2vy\ Fig. 6-22 Target and Clutter Spectra for a Single Spectral Line Transmission. vary ±2/^f/X from the transmitted frequency. The clutter possesses a high peak resulting from clutter return in the main beam. The position of this clutter peak depends upon the angle between the antenna pointing axis and the aircraft velocity vector. Quite obviously, if the antenna is scanning, the frequencies of the clutter returns will change as functions of time; in no case, however, can the returns from fixed clutter be doppler shifted by more than 1VfI\. For closing targets {Fr > Fp), the target returns will be shifted by IV^jX and will therefore appear in a clutter-free portion of the frequency spectrum. The effects of scanning and target-induced modulations (see Paragraph 4-8) cause a broadening of the target spectrum. Generally, the latter effects are small compared with the modulation induced by scanning, so that the width of the target spectrum may be expressed ^ftarort = ^ (6-30) where td = dwell time of the main beam on the target. By analogy, then, the frequency spectrum of the signal and fixed clutter returns for a ^«/j-^^ transmission consisting of an assembly of spectral lines 6-6] PULSED-DOPPLER RADAR SYSTEMS 323 Target Frequencies n = 0,1,2 y 1 li ^^0^^. Main Beam and Sidelobe Clutter Closing Target Frequency Fig. 6-23 Received Signal Frequency Spectrum for an Airborne Pulsed-Doppler System, Showing Clutter Spectrum and Return from a Closing Target. can be represented as shown in Fig. 6-23. This diagram illustrates one of the basic limitations of a pulsed-doppler radar system : in order to maintain a clutter-free region for all closing targets of tactical interest, the spacing between spectral lines must be where Vr, ,fr> 2{Vf^ ^..,nax)/X maximum closing velocity dictated by tactics. (6-31) If this spacing is not maintained, some of the closing targets will be buried in the clutter from adjacent spectral lines. This consideration leads to the use of very high PRF's in pulsed-doppler systems. For example, an X-band (3.2-cm) system designed for operation in a 2000-fps aircraft against 2000-fps targets will require a minimum PRF of 112 kc — a value that is several orders of magnitude larger than the PWF's commonly used inpulsed radars. When the return is mixed with the coherent reference signal as shown in Fig. 6-20, the output spectrum shown in Fig. 6-24 is produced. This heterodyning operation causes an effect known ?is folding; i.e. each of the II -Doppler Filter Bandpass f (b) Fig. 6-24 Received Signal Spectrum After Heterodyning to Zero Frequency, Showing Effects of Spectrum Folding: (a) Video Spectrum, (b) Filtered Spectrum. 324 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES target and clutter signals produces two symmetrical sidebands around each PRF line. The folding effect places a further limitation on the minimum allowable PRF: considering the effects of adjacent spectral-line interference due to folding, the minimum PRF is fr > — rj:!!^ (when signal "folding" occurs) (6-32) A ^ The resulting video signal then is passed through a bandpass filter to remove all the zero-frequency clutter components and all of the higher- frequency sidebands of both signal and clutter. The resulting output, then, is simply the doppler return associated with the central line of the trans- mitted spectrum. The process of "folding" also doubles the thermal noise which competes with the doppler return associated with the central line of the transmission. For this reason, some form of single-sideband detection process is usually employed in pulse doppler and CW doppler systems in preference to the simpler system described here. The doppler filter may be a single filter as shown, a bank of contiguous narrow-band filters as was shown in Fig. 6-15, or a single narrow band filter which sweeps over the total range between /d,min and/d,max- The comments in the preceding paragraph concerning the various filter types for CW systems also apply to pulsed-doppler systems. The simple pulsed-doppler system considered does not possess a range- measurement capability. Thus its information matrix and information rate are the same as shown for the simple CW system in Fig. 6-16. Actually the only reason for using a simple system of the type described is to eliminate the duplexing problem of a CW system and permit the use of a single antenna. In most other respects, this simple system is inferior to a simple CW system. Specifically, it possesses as deficiencies (1) greater complexity, and (2) less efficient use of power, since only the target power associated with the central spectral line is detected. The second point deserves further amplification. In a CW radar of transmitted power P, the peak and average powers are equal because the duty cycle is unity. All of this power is effective for detection of the target. However, for a pu/sed radar, the peak and average powers are related by the "duty cycle" — i.e., by the ratio d of "on" time to total time. Thus Ptd = Pare. (6-33) For pulsed doppler detection — as previously described — only the power in the central spectral line is used for detecting the target. The ratio of this power to the total power may be expressed Power effective for target doppler detection = {Pave)d. (6-34) 6-6J PULSED-DOPPLER RADAR SYSTEMS 325 Thus, to achieve the same useful power return from the target as for a CW radar of average power P, the peak power of a simple pulsed-doppler system must be Pt = P/d\ (6-35) and the average power must be Pa,, = P/d. (6-36) The average power governs the weight of the power source and the peak power dictates the voltage breakdown requirements of the transmitting system components; thus these factors must be weighed against the duplexing difficulties of a CW system. One of the most common applications of the simple pulsed-doppler system is for doppler navigation radar systems. This will be described in Chapter 14. Range-Gated Pulsed-Doppler Systems. The full benefits of a pulsed-doppler system can be realized by range-gating the receiver. This technique permits range measurement with the resolution inherent in the radar pulse width and it can also improve the signal-to-noise ratio and signal-to-clutter ratio by the reciprocal of the range-gating duty cycle. Master Frequency Control Pulsed Coherent Transmitter Target Doppler Bandpass Filter Composed of Continuous Narrow Band Filters fj (Velocity) Fig. 6-25 Gated Pulsed-Doppler System with Means for Range Measurement. 326 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES A generic range-gated pulsed-doppler system is shown in Fig. 6-25. In this system, the return signal is converted to IF and passed through an amplifier with a bandwidth approximately equal to the reciprocal of the pulse length. The IF amplifier output then is "gated" before the final mixing and doppler detection takes place. The width of the gate usually is made approximately equal to the pulse length. The operation of the gate is best understood by considering first a fixed gate which opens up the receiver at a time ti following transmission and closes the receiver tj seconds later at /i + xg. This action accomplishes the following: 1. The only returns going into the final detection stage are those from ranges falling between the values Rn = f/2(/i + n/fr) and •^ (6-37) Rn + ^R = r/2(/i + n/fr + r) n = 0,1,2,3,-. Clutter originates — in the main — from area extensive targets, whereas the desired signal originates from point targets. Thus the gating will improve the signal-to-clutter ratio of a target in the gate by a factor which is, on the average, equal to the duty cycle of the gate dg^ where dg = rjr. (6-38) 2. Noise enters the receiver only during the gating interval. Thus the average noise power is reduced by the duty cycle of the gate. 3. Since the position of the gate is known with respect to the trans- mitted pulse, any target doppler detected must come from a target in one of the range intervals indicated by Equation 6-35. The improvement in signal-to-clutter ratio represents an improvement over and above what can be done with a CW radar system. Thus a range- gated pulsed-doppler system can provide greater clutter rejection than any other generic radar system type. The reduced receiver noise incident to gating tends to restore the ^ /A" ratio to the same value as would exist for a CW system of the same average power and bandwidth. In fact, if the gate width equals the pulse length, a target in the middle of the gate would possess the same SIN as the comparable CW system. The range measurement made by a gated pulsed-doppler system is not exactly the same as a range measurement of a pulse radar. The high PRF that must be employed in a pulsed-doppler system causes the unambiguous range interval to be relatively short compared with the maximum detection and tracking ranges. Since the maximum unambiguous range is D ^^^^^ ■ if. ^Q\ 6-6] PULSED-DOPPLER RADAR SYSTEMS 327 A PRF of 112 kc, as derived in the previous example, would yield an un- ambiguous range interval of only 0.74 n.mi. Values of this order of magni- tude are typical for airborne pulsed-doppler systems which are constrained by antenna considerations to operate in the general range of S to X band (10 cm to 3 cm). As a result, additional techniques — to be described below — must be employed to measure true range in a gated pulsed- doppler system. Range gating also levies a cost on the system; a price must be paid in terms of system complexity and /or information rate. The previous dis- cussion considered a single fixed gate. To cover the complete interpulse period, this gate would have to be swept. A sweeping range gate will increase the total required dwell time on the target tdf by the reciprocal of the gating duty factor dg-. tdt = tdf/dg. (6-40) where tdf = buildup time for the doppler filter. An alternative solution is to employ contiguous fixed range gates covering the entire interpulse period (see Fig. 6-11). This "brute force" solution requires a separate doppler filtering system for each range gate interval; however, in combination with fixed contiguous doppler filters it does permit the maximum information rate to be extracted from the system because the separate doppler components of each range interval are examined simul- taneously. Paragraph 6-4, Correlation and Storage Radar Techniques, suggested still another means for processing pulsed-doppler radar infor- mation. Range Performance. The idealized range of a pulsed-doppler system may be calculated by the following modification of the basic radar range equation (3-1): " ~ [{4Tr)'FkTBdg\ (6-41) where ds = signal duty cycle dg = gating duty cycle [equal to (l-ds) for an ungated system] B = doppler detection filter bandwidth^''. When "folding" occurs in the detection process, an additional factor of 2 is required in the denominator of the one-fourth power expression. ^"In some cases, postdetection filtering will be employed to improve the final signal-to-noise ratio without increasing the number of doppler filters required. In such cases the effective detection bandwidth is Bbff = V725 • Bpd as derived in Paragraph 3-5 (Equation 3-62). In these cases, the dwell time should be matched to the bandwidth of the postdetection filter. 328 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES Eclipsing. The relatively high duty cycle of a pulsed-doppler system — typical values vary from 0.5 to 0.02 — introduces a strong possibility that part or all of the received target pulse may arrive during a transmission period. Since the receiver is turned off during transmission, target infor- mation will be lost or "eclipsed." The basic problem is shown for a 0.33 duty cycle pulsed doppler radar in Fig. 6-26. Eclipsing causes an effective change in the duty cycle for returns Transmitted Pulses o n o Range 1.0 Duty Ratio Correction Factor dj = Eclipsed Duty Cycle djo = Normal Duty Cycle *- Range Idealized Range Correction Factor Ro= 7 Duty Cycle Probability of Detection (Single Scan) -No Eclipsing Fig. 6-26 Effect of Eclipsing on Pulsed-Doppler Blip-Scan Ratio. which overlap transmission periods. Since average power that registers in the doppler filters is proportional to the square of the duty cycle, the idealized range will vary as the square root of the duty cycle. The blip-scan ratio is a function of idealized range as shown in the fourth figure. When the blip-scan ratio with no eclipsing is corrected for the eclipsing effect, the last curve in Fig. 6-26 results. As can be seen, the effect of eclipsing is to cause "holes" in the blip scan curve in the regions of pulse overlap. In a practical pulsed-doppler system the ratio of PRF to the useful range 6-6] PULSED-DOPPLER RADAR SYSTEMS 329 interval would be much higher; thus, there would be many more "holes" than shown in this example. For purposes of calculating the cumulative probability of detection, it is often convenient to approximate the notched blip-scan curve with a "smoothed" curve. If the pulsed-doppler system is operated with a fixed PRF, there will be certain closing velocities which could result in the target's appearing in a detection notch on each successive scan. For example, the interpulse period of the numerical example was 0.74 n.mi. If the first detection of the target occurred at a range corresponding to a "hole," and if the target moved a multiple of 0.74 n.mi. between scans, then detection would never occur. Eventualities such as this may be largely eliminated by a slow variation of the PRF which would have the effect of producing a smoothed — but nevertheless, degraded — blip-scan curve. Pieces of Information Range Measurements in Pulsed-Doppler Systems. As pre- viously mentioned, the range gate position measurement produces an ambiguous range indication because of the high repetition frequency that must be used in a practical pulsed- doppler system. The high repetition frequency reduces the total number of separate unambiguous range in- tervals (Nr = 1 /rfr) and gives the pulsed-doppler radar an information matrix such as is shown in Fig. 6-27. In almost all practical cases, it is desired to operate the radar against targets at ranges far exceeding the unambiguous range interval. Thus a means must be employed to circumvent the range ambiguity problem in a range-measuring pulsed-doppler system. There are several means for measuring true range: all are inconvenient and all degrade radar performance in terms of information rate and /or signal-to-noise ratio. One means for accomplishing range measurement is to employ the FM method used for the CW radar. The operating charac- teristics of this method are essentially the same as for an FM/CW radar; particularly, if the duty cycle of the pulsed-doppler system is relatively high. The range accuracy of this method is relatively poor if a narrow detection and tracking bandwidth is maintained. The range resolution is also poor because the pulse shape information is never utilized. Fig. 6-27 NrxN.xNcxNy Nr = VTfr N = fr/b B = Doppler Filter Detection Bandwith Pulsed-Doppler Information Matrix. 330 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES A = Transmitted Pulses I = Received Pulses True Range = ^- _ 2 X Desired Max. Range max- ^ / \\ Al A! Al Al A! -f2^ JA. lA lA lA kA,^ Fig. 6-28 Two-PRF Range Measure- ment. A second means is to employ- various modulations of the pulse repetition frequency. As an ex- ample, step switching of the PRF between several values can provide a ranging capability. The basic prin- ciple is shown in Fig. 6-28. Trans- mission occurs on two PRF's which are multiples of relatively prime numbers (in the example in the figure the numbers are 5 and 4). Because of ambiguities, a target at a true range corresponding to ttr will appear as a target return with time delays /i and /2 relative to the near- est transmitted pulse on each PRF, respectively. Thus we may write , ^1 /2 + (6-42) where «i, «2 = number of unambiguous range intervals in each PRF (in the figure, «i = 2 and «2 = !)• In the example shown there are two possible relationships between «i and «2: Wl = ^2 «i — 1 = «2 Substituting these relationships in Equation 6-42 we obtain, as expres- sions for the time delay corresponding to true range, h/rl-tJr2-{-l (6-43) frl-/r If the first expression is negative, the second must be employed. Thus, the use of two PRF's can provide unambiguous range over a maximum desired ranging interval corresponding to the time delay /rl-/r (6-44) 6-6] PULSED-DOPPLER RADAR SYSTEMS 331 Three or more PRF's can be used to extend the unambiguous ranging interval further. In these cases, the data processing becomes more compli- cated for two PRF's; however, methods similar to those used for Equations 6-42 to 44 may be used to derive the required relationships. The multiple FRF system of ranging is severely limited if more than one target return at the same doppler frequency is present. In a two-PRF system, two targets would yield four possible range values : two correct ranges and two "ghosts." Eclipsing also can cause difficulty, since it is quite likely that the target return for one of the PRF's will be eclipsed. The accuracy of this method of ranging is comparable to that of a pulse radar employing the same pulse length. If the "looks" at the target are taken by sequentially switching the PRF from one value to another, the required dwell time on the target for the same system bandwidth is increased by the number of PRF's employed. Alternative procedures such as simultaneous transmission of the PRF's or wider bandwidth reception could be used to keep dwell time constant. However, these methods will decrease the available *S'/A^ ratio for a given amount of total average transmitter power. In addition, simultaneous transmission greatly increases systems complexity in both the transmitter and receiver and gives rise to serious eclipsing problems because of the higher effective duty ratio. Pulsed-Doppler System Design Problems. Pulsed-doppler systems have the same basic problems of transmitter stability as CW systems. These problems are, in fact, common to any coherent system. Because of its high duty cycle, the duplexing problem is particularly difficult in a pulsed-doppler system. To cut eclipsing losses to a minimum, the transition from transmit to receive must be made as quickly as possible. Ordinary transmit-receive (TR) tubes are not satisfactory for this appli- cation; however, ferrite circulators (see Paragraph 10-16) and ferrite switches have found considerable application because of their low insertion losses (0.5 db) and their very rapid recovery time. In the receiving system, particular care must be taken to provide suffi- cient dynamic range to accommodate the maximum clutter amplitudes .^^ Linearity must be maintained over this range to avoid intermodulation products which spread the signal spectrum and cause loss of signal-to-noise ratio at the doppler filters. The design of the doppler filtering system — particularly, the bandpass characteristic and the maintenance of proper frequency spacing between iiln many designs, the clutter from the main beam and the altitude line is eliminated prior to amplification and doppler filtering. This greatly reduces the dynamic range requirements of subsequent stages of the receiver; however, it also makes the system completely "blind" at these frequencies. 332 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES filters — is a vital design consideration. A fixed range gating, fixed filter bank pulsed-doppler system may have hundreds or even thousands of these narrow band filters; thus the trade-ofF between filter performance and size and weight is a vital consideration. Angle tracking poses certain special problems in a pulsed-doppler radar. The doppler frequency as well as the range must be tracked prior to angle lock-on. The bandwidth of the velocity loop corresponds to the width of the doppler filter. If conical scanning is employed, this filter must be wide enough to transmit the scan modulation sidebands. Actually, the doppler filter width should be about three times the scan rate in order to minimize phase and amplitude variations of the error signal. For example, a 40-cps scanning frequency would require a doppler filter band width of at least 120 cps. The use of monopulse angle tracking (see Paragraph 6-3) poses a most difficult problem in a pulsed-doppler system. The sum and the difference signals must be handled in completely separate receiver channels — each with its own mixer, amplifiers, range gates, and doppler filters. In addition to the obvious disadvantages of weight and size, the problem of maintaining the proper alignment of these channels relative to each other represents a prodigious design problem. Pulsed-Doppler Systems Applications. As previously mentioned, pulsed-doppler systems are best employed in systems requiring substantial ground clutter rejection, a common transmitting and receiving antenna, and accurate range and /or velocity measurement. One other characteristic of a doppler system — either CW or pulsed- doppler — also has great tactical utility. This is the automaticity potential of such systems. Detection in such systems is inherently automatic since the signal is detected by the comparison of a filter output with a preset bias. While the same thing can be done in a pulse radar system, the problem of setting a bias level is enormously more difficult because of false alarms caused by clutter. This necessitates the use of bias levels considerably higher than would be dictated by thermal noise considerations. Thus the detection performance of an automatic pulse radar system is appreciably poorer than can be obtained when a human being is used as the detection element, since the human operator can discriminate between true targets and random clutter peaks so long as the clutter does not completely obscure the target. However, a doppler radar separates closing targets from clutter; thus the bias level may be set on thermal noise considerations alone. For this reason, as well as the others mentioned, pulsed-doppler systems are particularly suited for application as AI radars and guided missile active seekers which must find and lock on to a target buried in clutter in a high closing-rate tactical situation. 6-7] HIGH RESOLUTION RADAR SYSTEMS 333 Another application of pulsed-doppler systems — doppler navigation — ■ is covered in Chapter 14. In this application, precise velocity measurement coupled with freedom from CW radar duplexing problems make the pulsed- doppler system most attractive. 6-7 HIGH RESOLUTION RADAR SYSTEMS Certain radar applications such as fuzing and ground mapping often require very fine resolution; i.e. effective radar pulse lengths of from 0.002 to 0.2 Msec (which correspond to range resolution elements of from 1 to 100 feet, respectively) and /or angular resolutions of the order of 0.1-10 mils. High resolution is also tactically useful for counting the number of separate targets in a given space volume. The AEW radar example of Chapter 2 discussed this basic problem. In this case, high resolution in one dimension — for example range — can provide the requisite capability. Finally, high resolution provides a means for improving signal-to-clutter ratio when the clutter originates from area extensive targets. This is shown by Equation 4-60, where the instantaneous illuminated area of ground is a direct function of pulse length and antenna beamwidth. There are a number of means for obtaining high resolution in a radar system. Basically, all of them are variations of the following approaches to the problem: 1. Angular Resolution 2. Range Resolution (a) Large antenna aperture (a) Short pulse length (b) High frequency (b) Wide bandwidth (c) Beam sharpening (d) Doppler sensing Angular Resolution. This resolution problem has already been discussed in some detail in Paragraph 3-6. There it was shown that the angular resolution element of a radar system was approximately equal to a beamwidth where the antenna beamwidth can be expressed - -J radians 12 (6_45) a where X and d are the wavelength and aperture size respectively in consist- ent units. Increases in the antenna aperture d or the operating frequency (/ == 1 /X) will directly increase the angular resolution capability. One limitation on the benefits of increasing antenna aperture size is worthy of mention at this point. For purposes of resolution, the pattern of an antenna has the shape i^For a practical antenna, a value of = \.2\/d radians generally is a closer approximation when the effects of nonuniform illumination of the aperture are considered. 334 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES X/j Radians (Theoretical) Near Zone-c — ^Far Zone Fig. 6-29 Antenna Beamwidth Pattern. shown in Fig. 6-29. As can be seen, the concept of angular beamwidth holds only for the so-called far zone (Fraunhofer zone) where the range R is greater than d"^ l\P At closer ranges, the effective pattern width is variable, but as a general rule it can be considered equal to the antenna dimension. Thus the resolution obtainable with an antenna aperture of (S' feet cannot be better than d feet regardless of what is implied by the angular beamwidth expression. For a given antenna aperture and operating frequency, certain techniques such as monopulse and sidelobe cancellation are useful for "sharpening" the beam and thereby obtaining better definition (that is to say the transition of the signal return as the beam crosses an isolated target will be sharper). Improved resolution has been claimed from the use of these techniques. Such claims rest upon relatively shaky theoretical grounds and are based more upon the appearance of better resolution resulting from sharper definition than upon a rational repudiation of the basic laws governing the formation of interference patterns. In certain cases where prior knowledge of the target characteristics exists, velocity resolution may be employed to give the appearance of better angular resolution than one would predict from the beamwidth. Such a case is shown in Fig. 6-30 where an antenna points straight down from an airborne V^2 = \^FSinQ: f^2 2Vf sino : X Fig. 6-30 Improvement of Apparent Angular Resolution by Doppler Filtering. l^See S. Silver, Microwave Antenna Theory and Design, Chap. 6, McGraw-Hill Book Co. Inc., 1949. 6-7] HIGH RESOLUTION RADAR SYSTEMS 335 platform moving with a horizontal velocity Vp. The beam illuminates two closely spaced fixed targets, 1 and 2; however, because of the angular relation, and the velocity of the radar platform, the returns from these targets differ slightly in frequency. Thus, narrow band filtering may be employed in the receiver to distinguish between the two targets. If a single narrow band filter of width A/d centered about the carrier frequency is used in the receiver, the effective beamwidth may be expressed e,// = ^ radians. (6-46) However, the filter bandwidth is limited by dwell time requirements to a value A/, = \/t,= VF/hQeff cps. (6-47) Substituting, and solving for the minimum value of Qeff, we obtain Qefs ^ VV2A radians. (6-48) If multiple receiver channels are used in conjunction with appropriate signal storage and correlation techniques, the return from each target can be integrated over the entire dwell time of the actual beamwidth. In this case, the minimum possible effective beamwidth becomes Qeff = X/2^e = d/lh radians. (6-49) and the number of channels required is n, = IKKld'' channels. (6-50) Thus, in theory at least, the resolution performance of a very long antenna (possibly much longer than the aircraft itself) can be obtained by coherently combining signals transmitted and received from various positions along the flight path. Quite obviously, this principle could have application to ground mapping radar systems. It is of some interest to note that the effective angular resolution of the multiple channel correlation system actually improves as the actual antenna beamwidth becomes larger. The reader should also note, however, that achievement of effective beamwidths approaching the minimum possible requires a radar system of enormous complexity. For example, a 3.2-cm system with a 4-ft antenna operating at 10,000 ft altitude has a theoretical angular resolution limit of 0.1 mil. However, such a system would require the equivalent of 267 separate coherent receiver channels to realize this potential. Short-Pulse Systems. The most obvious means for obtaining high range resolution is to employ a short pulse length in a conventional pulse radar system. However, such a system has a number of important design problems which severely limit the usefulness of this approach. 336 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES First of all, the generation of a short, high-power pulse is a difficult problem in itself. The design of the transmitting tube, the modulator, and the TR switching all are complicated by the short-pulse operation. Short-pulse operation also limits radar performance. The required receiver bandwidth is inversely proportional to pulse length. Thus, the S /N rsLt'io for a given value of peak transmitted power is directly propor- tional to the pulse length: S/N^Pt/B = Ptt. (6-51) Usually, peak power cannot be increased to compensate for this effect because of voltage breakdown limitations in the transmitter, antenna, and waveguide. Thus, for a given state of the art in RF components, the S jN performance will decrease with decreasing pulse width. Actually, because of the previously mentioned transmitter design problems, this decrease proceeds at greater than a linear rate. For these reasons, short-pulse systems are limited to relatively short-range operation (such as fuzes) or operation against targets of large cross section (ground mapping) where it is feasible to sacrifice *S'/A^ ratio for improved resolution. Short pulse lengths can also complicate certain other problems. For example, if delay line AMTI is employed, the tolerances on the pulse repetition frequency control and the delay line calibration must be held within proportionally closer limits. In addition, the bandwidth requirements of the delay-line elements are increased proportionately. As a result of limitations such as these, there are certain tactical applica- tions where no physically realizable noncoherent pulse radar system can provide the requisite resolution and range capabilities. To fill this gap, a family of radar systems has grown up during recent years which — for lack of any more suitable name — are called "wide bandwidth coherent systems." Wide Bandwidth Coherent Systems. From an information theory standpoint, the fine range resolution capability of a short-pulse system derives from the wide bandwidth of such a system. In fact the range resolution capability is a direct function of the bandwidth of the trans- mitted spectrum. This suggests that any system which employs a wide bandwidth has the inherent capabil- ity for fine range resolution. Several other observations — useful for in- venting new radar systems — may also be made from an examination of the transmitted spectrum of a pulse Fig. 6-31 Pulse Radar Spectrum. radar as shown in Fig. 6-31. \ -Frequency 6-7] HIGH RESOLUTION RADAR SYSTEMS 337 First of all, the number of spectral lines contained in the transmitted bandwidth determines the number of individual unambiguous pieces of range information that the radar can collect from an angular volume determined by the antenna beamwidth. Thus, as previously noted. A^. ^ ^ = -1. ' fr Irr (6-52) Since each piece of range information represents a range interval of fr/l, the total unambiguous range interval is simply Rn -'^'i-k (6-53) = Spectrum Broadening Introduced by Scanning and Target IVlodulation At this point it is worthwhile to recall the development of the matched filter principle presented in Paragraph 5-10 and used as the basis for the storage and correlation radar principles outlined in Paragraph 6-4. This principle stated that the optimum S IN ratio is obtained when the detection filter transfer function is the complex conjugate of the re- ceived signal spectrum. Thus for the pulse spectrum shown in Fig. 6- 31, the optimum filter would have the comblike appearance of Fig. 6- 32, where the width of each tooth of the comb is sufficient to pass the modulations produced by targets and scanning. The total effective detection bandwidth of such a filter may then be expressed r uu Fig. 6-32 Optimum Radar Spectrum —^Frequency Filter for Pulse Be NrBi = B (6-54) Thus, obtainment of high range resolution with a narrow band receiving system is theoretically possible; in fact, for a nonscanning radar operating against nonfluctuating targets, the total required received bandpass approaches zero. The application of the matched filter concept requires a coherent system. These principles make it possible to conceive a wide variety of high- resolution systems which circumvent the peak power and S jN ratio limitations of noncoherent pulsed radar systems. In general, these systems have the following common characteristics: 1. A wide transmitted bandwidth {Bt = c jlRr,^ minimum resolution element). where i?r.min = 338 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES 2. A modulating frequency /r sufficient to create the desired unam- biguous range interval. 3. Some form of storage and cross correlation which attempts to provide an optimum match between the received signal and the effective receiver bandpass filtering characteristic. 4. Usually, great complication in comparison with a noncoherent pulse radar. The following systems are indicated as possibilities for high resolution wide bandwidth coherent systems. 1. Wide bandwidth FM/CW. 2. Long-pulse, low-PRF systems where the transmitted frequency is FM'd during a pulse transmission period to produce the wide transmission bandwidth. This type of system is often called a matched filter radar}^ 6-8 INFRARED SYSTEMS A book entitled Airborne Radar may seem a strange place to find a discussion of infrared techniques, but it must be remembered that the applications and design principles of airborne radars and of infrared detectors and weapon control systems are quite similar. Actually the only real differences between a passive^^ radar and an infrared system are (1) the method of detection^^ and (2) the fact that the infrared radiation emanates from the target itself rather than from its radars or communications equipment. The tuned circuits used in the detection of radio and radar radiations cannot presently be extended to the frequencies (3 X 10^ to 1.5 X 10'' megacycles, or 1 to 20 microns wavelength) of that portion of the infrared having practical significance here. Therefore optical detectors must be used and these impose their own restrictions. An advantage of the short wavelength of infrared radiation is that interference patterns have correspondingly small angular relationships and are usually not significant in instrument design or operation. For example, the diameter of an infrared collector mirror depends only on the amount of radiation to be collected and not on the required angular accuracy, as in the case of a radar dish. Also interference patterns such as result in radar from ground reflections and cause target confusion are not encountered with infrared systems. Because the target is itself a source, the signal in a passive infrared system diminishes with range more slowly than in an active system. I'^Note that in Paragraph 5-10, the term matched has been used to describe a more general class of radar systems. i^Only passive infrared is considered here. Active systems do exist and are used, but for long-range detection and tracking no sufficiently intense sources of infrared radiation are available. '^By detection we mean the manifestation of the presence of electromagnetic radiation. 6-8] INFRARED SYSTEMS 339 Conversely, infrared does not have the all-weather capabilities of radar, its ability to penetrate haze, fog, and clouds being only slightly better than that of visible light. Background clutter considerations are also more serious, since everything in a typical tactical environment is to some degree a source of infrared radiation, i.e., a potential source of interference. Passive infrared systems — like passive radar systems — also do not possess the capability for measuring range in the direct and convenient manner of active radar systems. The use of infrared for detection and tracking is now new, having been vigorously exploited by Germany during World War II. In this country, where reliance was placed more heavily on the development of radar — with obvious beneficial results during the war — serious consideration of infrared systems has been more recent and stems from four facts: (1) modern targets are better sources of infrared radiation than their predeces- sors and in many cases poorer radar targets; (2) many important targets are encountered at high altitudes where attenuation and absorption of IR energy are minimized; (3) infrared is more difficult to countermeasure than radar, or at least the art is not so advanced; and (4) infrared technology has made significant advances since World War II. This will be a short discussion of the application of infrared to airborne surveillance and tracking systems. The fundamentals of infrared science are ably covered in an earlier book of this series {Guidance, Chapter 5, "Emission, Transmission, and Detection of The Infrared") and a knowledge gained by reading that discussion will be assumed. Basic Principles. Airborne infrared systems generally use mirrors rather than lenses. Lenses are possible but usually not practical because of limitations imposed by the properties of available materials (see Paragraph 5-7 in Guidance). The infrared system is composed, then, of a mirror which collects radiation from the target and focuses it on the detector, a means of modulating the radiation striking the detector in order to produce an a-c signal, and a means of discriminating against spurious targets and back- ground radiation. Frequently, modulation and discrimination are ac- complished in the same process. Consider, as an illustration, the simplified system shown in Fig. 6-33.^^ Radiation from the target, background, and intervening air enters through a dome of transparent material (Irdome) and is focused on the detector after reflection from the two folding mirrors. The instantaneous field of view is determined by the size of the detector and the focal length of the main collecting mirror. Scanning is accomplished by tilting the two folding I'This arrangement is chosen only to illustrate the significance of the system parameters and not for its desirability or efficacy. It is not an example of a system in actual use, since most such systems are classified and cannot be discussed here. 340 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES Infrared Transparent Dome (IR Dome) Fig. 6-33 Typical IR System. mirrors by the angles a and /3 away from the perpendicular to the optic axis and then rotating them about the optic axis in opposite directions. If the mirrors turn at equal speed, the effect is to move the instantaneous field of view along a straight line, the length of which is determined by the two angles a and /3 and the intermirror distances. The velocity with which the line Is scanned varies sinusoidally; it is most rapid in the center of the line and is slowest at the ends where scan reversal occurs. If one mirror turns a little more slowly than the other, the line scanned ih space rotates slowly about its center, resulting in the rosette scan pattern shown in Fig. 6-34. With this pattern the surveillance capability is greatest in the center of the field, which is crossed on each spoke of the rosette, and diminishes toward the edges, a property which may or may not be desirable. In considering the appearance on the scope of a scene scanned by an infrared device, it should be remembered that while our eyes see almost everything by reflected light (i.e., a "semiactive" process similar to some types of radar), the infrared scanner sees mainly thermal radiation emitted by the observed objects themselves. This is particularly true if the radio- meter is filtered to be sensitive only to radiations of wavelength greater than 3 microns, since reflected or scattered sunlight beyond 3 microns is generally negligible compared to emitted radiation. Therefore, a "hot" object such as a city, or one with a high emissivity such as a cloud, will appear bright. The clear sky, bright blue to the eye, will appear black, since the air molecules do not scatter infrared as they do visible light. 6-8] INFRARED SYSTEMS 341 ,**''^ In ,ar,d S.an». Output Cl«u<l> unci Sho Ground -I.V.I r.Un. Mo.t 1 Sa<karo»i>d H«., City it.nia Sourc Fig. 6-34 Example of an IR Scan Pattern. The detector receives the radiation collected by the optical system and converts this to electrical energy, which is amplified and appropriately dis- played — in this case on a cathode-ray screen — or used as an error signal for tracking. There is radiation received from the target, from the clouds, haze, land, water, and sky within the scanned field, and from the atmos- phere lying between the detector and the target. In some instances the 342 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES optical elements themselves, for example an imperfectly transparent dome heated in flight, contribute significant and undesirable radiation. The usable signal results from the difference between radiation collected from the target and all other sources. This "contrast" signal is proportional to: m V.^AE K^-7; ^z?o-' SxTxd,. (6-55) where F^ = signal output (volts) from the detector A = area (cm^) of main collector mirror a = field of view of radiometer (radians) = width of square detector divided by the focal length of collector mirror E = efficiency (%) of windows, filters, mirrors, etc. in the optical system (dimensionless). This is here assumed to be independ- ent of the wavelength JxT = spectral distribution of target radiation (watts micron"^ steradian"^) JxB = spectral distribution of background and other unwanted radiation (watts cm~^ steradian"^ micron"^) 6'x = spectral response (volts /watt) of detector Tx = spectral transmission (%) of the atmosphere between the target and detector (dimensionless) Xi,X2 = wavelength limits (microns) of system sensitivity defined by the optical filter or sensitivity limits of the detector. R = target range (cm). The detector will have a noise output F„ which will be a function of the type of detector, the bandwidth A/ of the amplification system, and the radiation environment of the detector. Specifying the minimum signal-to- noise ratio Fs jVn required for reliable detection of a target, the system noise defines the minimum required V^. Since 7xr, 7\b, and T^ are beyond our control and E is always optimized anyhow, the remaining parameters are chosen to give the required Vs at the desired target range. Actually Xi and X2 are generally determined by consideration of Jxr, 7\b, and T^. Fig. 6-35 shows J-^T, J\B, and T^ plotted as a function of wavelength for a specific application: the detection of a 600° K blackbody viewed against a back- ground of clouds or heavy haze through 10 miles of moderately clear sea level air. It is plain that the best choice of wavelength limits are Xi and X2 = 3.3 and 4.1 microns, respectively. The properties of available de- tectors — sensitivity, time constant, ruggedness, reliability, etc. — may, of 6-8] INFRARED SYSTEMS 343 100 Ji? 40- Background Predominantly Scattered Sunlight Mbsent at' V Night + Background Predominantly Black Body Radiation Night and^ Day U Energy Distribution ^ From Black Body at 600°K 4\ rx Transmission of 10 mi. of Sea - Level Air Containing 6 cm Vapor Jb Cloud and Heavy Haze Background (^Energy Distribution' From 300°K Black vBody 4 X Exaggerated 123456789 10 WAVELENGTH (MICRONS) Fig. 6-35 The Useful IR-Frequency Spectrum. course, force a different choice. The only really flexible parameters for the designer, then, are the areas of the collector mirror and of the cell. These are in turn influenced by the requirements of the scanning system. Scanning System Characteristics. The choice of a scanning system generally represents a compromise between the requirements of the system and the mechanization advantages of rotary optics (particularly for high- speed scanning) and fixed detector elements (which simplify cooling problems and maintenance of cell sensitivity). As an example of the type of analysis which must be performed to assess a given scanning technique, the rosette scanning pattern previously discussed will be analyzed to determine the interrelations among scan time, resolution, coverage, and detection element characteristics. The important parameters of the scanning system are: the instantaneous field of view a (radians) square; the whole field of view which is here (see Fig. 6-34) a function of the half-scan angle 7; and the time T required for the whole field to be covered. In order to completely cover the field the number of spokes in the rosette pattern is equal to the number of instan- taneous fields required to cover the periphery of the whole field, or —r—. spokes. (6-56) a/ sm 7 344 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES The speed of scan is controlled by the time constant t of the detector. To define a practical upper limit we shall use the time required for the in- stantaneous field to move its own width o-, equal to t. To go faster would result in considerable smearing of the display pattern, with resultant loss of resolution. In this sinusoidal scan the velocity is not constant, but for simplicity we use the average value. The time required to scan one com- plete spoke (27 radians) is ;= ] r \ — = n — '■ — seconds/spoke. (6-57) lotal no. or spokes Iirsmy Motor 1 then turns clockwise at — ( ) rps and motor 2 turns counter- 11- 1 /^TT sin7 A clockwise at — I ~~ W ''P^- Since a complete spoke is 27 radians, the average scan rate is -. seconds/radian. (6-58) 4x7 sm 7 and the time required to scan a- radians is 4x7 sin 7 = r seconds. (6-59) which we accordingly equate to the time constant r of the detector. Consider an actual case in which we want: ^ = ly^o = 0.0058 rad ^ = 20° = 0.35 rad T = 10~^ sec Then the time required to scan a complete field is, from Equation 6-57: ^ 4x7 sin yr 4x X 0.35 X 0.34 2 X lO"'^ . , .. ,^. ^ = a-^ = mossy = ^-^ '''■ ^^-^^^ For many practical cases, this is obviously too long. A modern aircraft will have changed course considerably in this time. Therefore the instan- taneous field must be enlarged, the total field reduced, or a faster detector sought — possibly all three. Perhaps the scan mode would have to be abandoned in favor of a more economical one without the multiple retrace encountered in the center of this field — say a raster scan similar to that used in television. Target Tracking. If the system is required also to track a target, this could be accomplished by orienting the aircraft so that the target image falls in the center of the screen, setting angle a = 0, and reducing angle /3 6-8] INFRARED SYSTEMS n-/2 7r/2 345 S Azimuth Drive Active — >■ Q i Q Eievatio Drive / A ^ Target Pulse (from 4 ( V2 3/2 TT 27r (c) Fig. 6-36 Target Tracking. to such a value that a circular scan of, say, |° results (see Fig. 6-36). Then as long as the image in space of the detector rotates around the target without touching it (Fig. 6-36a) no output (error signal) will result. When the line of sight moves and the detector then encounters the target (Fig. 6-36b) an output pulse will result; by comparing the phase of this pulse with a synchronizing signal generated on the shaft of motor 2, an error signal is generated. By having the entire optical detection device movable in the aircraft and motor controlled, these error signals can be used to keep the device pointed at the target. In Fig. 6-36c a simplified on-off control is illustrated. The synchro- nizing pulse alternately activates and deactivates the control motors. If the target pulse occurs during an active period, the motor moves the optical system. In Fig. 6-36c the pulse occurs where both the down and left 346 GENERIC TYPES OF RADAR SYSTEMS AND TECHNIQUES controls are activated (as in Fig. 6-36b) to return the target image to the center of the circular scan. Smoother tracking and less tendency to hunt result from a system in which the error signal varies in magnitude with the ofF-course position of the target, and this is usually a feature of actual tracking systems. Detection Performance. The actual capabilities achievable with present-day infrared systems can be estimated from target intensities, background intensities, and detector sensitivities. As an example, consider a radiometer having a filter limiting the sensitivity of the radiometer to the region 1.7 to 2.7 microns and a lead sulfide detector at the focus of a collector mirror 1 ft in diameter. With a readily available detector (say an Eastman Kodak Ektron cell) of practical size (say 1 mm square), an output signal just equal to the rms noise from the detector can be achieved under tactical conditions when about 10~^^ watt/cm^ falls on the collector mirror and is focused on the detector. This would represent a signal-to-noise ratio of 1, here arbitrarily construed as a necessary criterion for detectability. If the target is the exhaust port of a typical jet engine, the irradiance (watts /cm^ in the 1.7 to 2.7-micron region) at the collector mirror will be about 400 /i?2, where R is the target range. This results from assuming the exhaust port to be a 24-inch-diameter blackbody of emissivity unity and to have a temperature of 600°C. Through a completely clear atmosphere, then, and with no background interference this jet exhaust port could be seen from a distance of 2 X 10^ cm or 125 miles, at which distance it would irradiate the collector mirror with the necessary 10"^^ watt/cm^. Atmos- pheric attenuation, which is severe in the lower atmosphere, and back- ground interference may under average conditions degrade this range to less than a third of this number. Further, the target we are considering, a single-engine jet aircraft, will be a much fainter target at any other than tail aspect where the exhaust port is visible. In side aspect the radiation emanates from the hot exhaust gases which, while of extended size and quite hot, emit only the wavelengths of the characteristic infrared bands of the gases. If the fuel is a hydrocarbon these are the bands of water vapor and carbon dioxide. Atmospheric attenuation is most severe in this case, since the cold water vapor and carbon dioxide in the intervening air path absorb most of what is emitted. In side aspect a jet will be less than one-tenth as intense a target as in tail aspect and will therefore be detectable at less than one-third the range realizable when looking at the exhaust port. In nose aspect, it will be considerably worse than this, since here most of the exhaust gases are hidden by the aircraft and the hot parts of the engine are not visible. D. J. H E A L E Y III CHAPTER 7 THE RADAR RECEIVER 7-1 GENERAL DESIGN PRINCIPLES The airborne radar receiver amplifies and filters the signals received by the radar antenna for the purpose of providing useful signals to display and automatic tracking devices. The receiver accepts all of the signals appear- ing at the antenna terminals and must filter them so as to provide maximum discrimination against signals which do not originate by surface reflection of the transmitted radar signal from certain desired targets. Modulation characteristics of the desired signals must be preserved in the filtering process. The modulation characteristics provide information on the number of targets, their angular position with respect to a given frame of reference, the distance between the radar set and the targets, and the velocity of the targets. The majority of airborne radar receivers are of the superheterodyne type. Ordinary pulse radar sets are usually of the single frequency con- version type. Doppler radar sets employ single sideband reception. The receivers are more complicated than in the ordinary pulse radar set and generally employ multiple frequency conversion in order to realize the required frequency selectivity. Fig. 7-1 shows a functional block diagram of an elementary receiver of each type. Performance of a radar receiver is described by the following character- istics: 1. Noise figure 5. Dynamic range 2. Sensitivity 6. Cross-modulation characteristics 3. Selectivity characteristics 7. Tuning characteristics 4. Gain control characteristics 8. Spurious response Specification of each of these characteristics depends upon the particular radar application. Analysis of the radar system defines the input signal environment, the required output signal-to-noise ratio, and required fidelity of modulation. Such requirements are then interpreted in terms of the above characteristics. Each of these characteristics is discussed later in this chapter and in Chapter 8. 347 348 THE RADAR RECEIVER 7-1] GENERAL DESIGN PRINCIPLES 349 1 1 Filter and natio -2 E 1 '^ ^ 1 .5 t ^.i •^^ s ^ t t A ▲ T T i 1 ^fi.i^i M- 03 1 X 2^ 1 o V 1 u. ^ y. S >|^> C §6 1 1 < ' 1 T 1 t 1 U. ,/, ^ .^ o 1^^ -§-^ --iz i" b UJ , c .2 E ^ ao 1 5 -11 aj - E \ ^ §5^ q: .E Q ^ Q- 00 o "5 - E f " (t 5.i , -■ b ' ' <v _l 1 X It: (CO.- 2 2 -►CO- ^^ T^n^ S 3 ^1 1 1 1 1 < .> < .> ^Q o5 \ If I I 2 f^ 1 a. , 1 O Q C/l o u, 350 THE RADAR RECEIVER It is desirable that the noise figure be minimized and the sensitivity be maximized. This is not always feasible, as will be indicated in Paragraph 7-2. Selectivity is provided in both frequency and time. It is desirable to provide the required selectivity at the low-level signal stages and prior to envelope detection of the signal. Gain control characteristics are dictated by requirements to provide error signals to range, speed, and angle tracking feedback mechanisms for a specified range of input signal power. Automatic gain control (AGC) systems are discussed in detail in Chapter 8. Dynamic range is the range of signal levels above the thermal noise level for which a receiver will provide a normal usable output signal. To reproduce faithfully the amplitude modulation on a received signal, the incremental gain of a receiver whose output is controlled by the average level of the received signal must be constant for a dynamic range on the order of 12 db above the average signal level. The incremental gain is the slope of the output /input voltage characteristic of the receiver. When an undesired signal appears at the receiver input which is coincident in time with the desired signal and nearly coincident in frequency, a much greater linear dynamic range or range for which incremental gain remains constant is required. In receivers which separate signals by frequency filtering, it is not unusual to require a linear dynamic range on the order of 80 db up to the point in the receiver at which the frequency separation of the desired and undesired signals occurs. On the other hand, short-pulse, low-PRF radar sets which separate signals by time filtering may require only a linear dynamic range on the order of 15 db. Undesired signals which occur at a different time or frequency than the desired signal may impart their modulation to the desired signal. This is called cross modulation. Such a phenomenon arises from nonlinearities in the receiver and is undesirable, since it degrades the output signal-to-noise ratio. A proper radar system analysis defines the signal environment and allow- able degradation of the receiver output signal; thus the principal factors governing the selection of the dynamic-range and cross-modulation characteristics are specified. Tuning characteristics are dictated by the radar transmitter. The receiver is designed so that it can always be tuned to the transmitter fre- quency. The design objective is to make the receiver tuning as accurate as the state of the art permits. Both short-term and long-term frequency stability is important. The effect of short-term frequency instability is to introduce modulation on the signal in the receiver. Such modulation degrades the output si.gnal-to-noise ratio. In a noncoherent pulse radar set, the tuning accuracy that can be achieved is on the order of 1 part in 10^ Much greater stability is required in coherent radar sets. Automatic Table 7-1 Receiver Characteristic Desirable Effect Effects on Other Receiver Characteristics Direct detection of the RF signal Simplicity Poor noise figure; poor rejection of spurious sig- nals. Nonlinear transfer characteristic Preselection (band pass filter between antenna and receiver) Greatly reduces spuri- ous signal response Increased noise figure due to insertion loss of the filter. Imposes long-term frequency stability re- quirements on the trans- mitter and preselection filter that otherwise would not be encountered Desensitization during transmitting time (TR switch) Reduces degradation of receiver performance with time by limiting the signal power applied to the microwave mixer Degrades performance at short ranges owing to the deionization properties of the gas switches that are employed. In high-PRF coherent radars, degrades performance in each am- biguous range interval im- mediately following a transmitted pulse High IF frequency Minimizes spurious sig- nal response; simplifies some tuning problems Results in higher IF noise figure. The receiver noise figure depends on the amount of noise-noise in- termodulation due to the local oscillator. If this is negligible, the receiver noise figure will generally be higher with the higher IF IF bandwidth on the order of the reciprocal of the transmitted pulse length Maximizes the peak sig- nal to rms thermal noise, thereby providing best detection in thermal noise Limits signal resolution; imposes strict require- ments on the tuning ac- curacy of the receiver IF bandwidth character- istic which enhances the signal sidebands greatly removed from the trans- mitted carrier frequency and attenuates the side- bands near the carrier Improves the detection capability in clutter Degrades the detection of signals in thermal noise. Imposes strict require- ments on tuning accuracy 351 352 THE RADAR RECEIVER Table 7-1 (cont'd.) Receiver Characteristic Desirable Effect Effects on Other Receiver Characteristics Prevention of saturation in a linear receiver Maximum signal to noise ratio Requires an AGC loop. When signal contains pulse amplitude modulation that must be recovered, instantaneous AGC is usu- ally not acceptable. A slow AGC is demanded. The information rate is de- creased since only a se- lected signal may operate the AGC. Any other sig- nals that may be examined are modulated by the fluctuations of the signal controlling the AGC. Therefore multiple receiv- er channels are needed to increase the information rate Extremely narrow IF bandwidth Provides maximum sig- nal to thermal noise ratio Reduces the information rate attainable. Results in a loss of signal resolu- tion unless preceded by a cross correlation opera- tion. Wasteful of power frequency control (AFC) systems are required to obtain the required tuning accuracy. These devices are discussed in detail in Chapter 8. Spurious responses are outputs caused by signals at frequencies to which the receiver is not normally tuned. These responses are the result of in- adequate selectivity and nonlinear elements in the receiver, e.g. mixers. The receiver is designed to minimize spurious responses by properly selecting intermediate frequencies and mixer circuits and by providing the necessary selectivity. 7-2 THE INTERDEPENDENCE OF RECEIVER COMPONENTS A particular receiver characteristic may be designed to give optimum receiver performance when the contribution of this one characteristic of overall performance is considered. However, many of the receiver charac- teristics are interdependent and therefore compromises must be made in the design. The compromises for a specific design are determined by the 7-3] RECEIVER NOISE FIGURE 353 performance requirements imposed on the radar system by the tactical requirements. Some examples of the effects that the choice of a given receiver character- istic has on the overall receiver performance are indicated in Table 7-1. 7-3 RECEIVER NOISE FIGURE The ultimate sensitivity of a receiver is dependent upon the inherent noise generated in the receiver circuits. A useful measure of this noise is the receiver noise figure which is defined as the ratio of the actual noise power output of a linear receiver to the noise power output of a noiseless receiver of otherwise identical characteristics. Noise in a receiver is made up oi thermal noise., which results from thermal agitation of charge carriers in conductors, and shot noise, which results from random electron motions in vacuum tubes. These noises are charac- terized by a Gaussian amplitude distribution with time. Such noise sources are independent and uncorrelated. The average power from the various sources is additive, and it is therefore convenient to employ ratios involving power in determining noise figure. Consider a signal generator described by a short-circuit signal current source /» and an internal conductance ^s which is at an absolute temperature Tg. Let the generator be connected to a load gL which is at an absolute temperature Tl. Both gs and gL will generate fluctuation currents which are given by^ TZ' = ^kT.gJj (7-1) and /;? = \kTLgLdj (7-2) where ins and /„l are the rms noise currents in a frequency bandwidth element dj, k is Boltzmann's constant = 1.37 X 10"^^ joule /i^°. The available signal power from the generator is Is^ /4:gs and the available thermal noise power is inJ^ l^gs = kTsdf. The available signal power from the circuit composed of the signal generator and the load gi is 4(^. + gL) If t is defined as Tl/Ts, the available noise power is 7^ + ^ ^ kTsdf{gs+tgL) ,^ .. ^{g. + ^l) g.^gL ' ^ " ^ ^J. B. Johnson, "Thermal Agitation of Electricity in Conductors," Phys. Rev. 32 (1928). 354 THE RADAR RECEIVER The noise figure is defined as SJN,^N^ (7.4) So/ No N^G ^ ' where F is the noise figure (a power ratio) SilNi is the available signal to noise ratio at the input So /No is the available signal to noise ratio at the output G is the available power gain. For the case of the generator connected to a load, the noise figure of the combination is then F =. I -\-t ^. (7-5) If both the generator and load are at the same temperature, then the noise figure is merely the attenuation of the signal resulting from the termination. In a radar receiver it is convenient to associate a noise figure with various elements and then determine the receiver noise figure resulting from their combination in cascade. Consider that a number of elements characterized by a noise figure Fj and an available power gain Gj are interconnected as in Fig. 7-2. It is F,,G, Fig. 7-2 Noise-Equivalent Radar Receiver. assumed that all the noise sources are at a temperature T (a difference in temperature may be included as a temperature ratio). Assume further that all elements are linear, and that the effective noise bandwidth of each element is 5„. The input noise is A^i = kTB„ The overall gain G is G1G2G3 - Output noise originating from the source is GkTB„. The additional noise at the output contributed by the first box is GkTBn{F\ — 1). The additional noise at the output contributed by the , , . GkTB„{F2 - 1) ^, r n • -k • AA second box is — -■ I he sum or ail noise contributions add up Gj FGkTBn = GkTB,, + GkTB„{F, - 1) + _ Gi Then ^ = ^' + G. + G.G. + ■ 7-3] RECEIVER NOISE FIGURE 355 to FGkTBn, where F is the overall noise figure expressed as a power ratio. + -. (7-6) (7-7) In the common airborne radar set, RF amplification is not employed. Instead, the signal is heterodyned to some intermediate frequency and then amplified. Microwave crystal mixers are passive nonlinear devices. Their noisiness is characterized by the amount of noise produced by the mixer compared with the noise from a resistance at the same external temperature. The noise is thus expressed as a temperature ratio tm- A mixer acts as a switch, and in terms of available power exhibits a loss at the conversion frequency. This loss is designated as a power ratio Lx- Following the previous notation the noise figure of a mixer is then tmLx. The noise figure of a superheterodyne radar receiver is then Fr.. = 1 -f- ^m - 1) + ^L,[(/. - 1) + (Fi, - 1)] (7-8) J a J a where Free is the receiver noise figure expressed as a power ratio /m-1 is the excess noise of the mixer FiF-\ is the excess IF noise figure expressed as a power ratio Lt is the product of the conversion loss of the mixer and loss in the microwave transmission circuitry between the antenna and mixer expressed as a power ratio /„ is the effective noise temperature ratio of the mixer T is the noise temperature of the receiver Ta is the temperature of the antenna. The noise figure is usually defined with respect to room temperature (about 291° K).^ A radar receiver is, however, connected to a directional antenna which can be represented as an equivalent generator at a tempera- ture less than room temperature when the antenna is directed toward space. In fact the equivalent antenna temperature under this condition may be about 4°K. When referred to the antenna temperature under this condition, the noise figure of the best airborne radar sets is on the order of 30 db. 2Under such a definition T/Ta is unity. 356 THE RADAR RECEIVER 7-4 LOW-NOISE FIGURE DEVICES FOR RF AMPLIFICATION The crystal mixer Is a rather fragile element, and its electrical character- istics deteriorate when large-signal inputs are applied to it. Because of the conversion loss of the crystal mixer, the receiver noise figure is highly de- pendent on a low IF noise figure. From a consideration of the noise figure of cascaded networks it is seen that a large available power gain in the first network minimizes the noise contributions of the later networks. A low-noise RF amplifier preceding the crystal mixer can provide the power gain required to oflFset the loss of the mixer. Two types of RF amplifiers that might be employed in an airborne radar receiver are the traveling wave tube (TWT) amplifier and the variable parameter amplifier. These devices are discussed in more detail in Chapters 10 and 11. Noise in the traveling wave tube results from noise in the electron beam. Theoretically the noise can be reduced to about three times kTB. At present such tubes are not available for airborne radar receivers. Tubes are available, however, that are nearly competitive in noise figure with the microwave crystal mixer in the frequency range employed by the airborne radar set. A TWT will produce a saturated output under strong signal conditions at maximum gain, alleviating many TR switching difficulties. Since the tube provides gain, the noise figure of the elements which follow is not nearly as important as in the conventional airborne radar receiver. There- fore, much higher intermediate frequencies are feasible without degrading the noise figure. A higher IF results in fewer spurious signal outputs from the receiver. The tube can be gain-controlled by changing the beam current so that it can produce an attenuation equal to the cold loss of the tube if required. This is an advantage when attempting to amplify strong signals with minimum distortion. One disadvantage in the traveling (forward) wave tube results from the wide bandwidth. The noise spectrum is very wide and this results in more noise at the mixer than desired. An RF filter between the traveling wave tube and the mixer can, however, eliminate this condition if necessary. Another disadvantage is that a number of spurious signals can be generated in the tube, and are likely to be encountered in practice due to the wide RF acceptance bandwidth of the tube. The backward wave amplifier has a narrower bandwidth than the forward wave amplifier and may prove to be the most desirable type of traveling wave tube for use as an RF preamplifier in an airborne radar set. Traveling wave tubes may also be constructed with two slow wave structures to provide mixing. Variable parameter amplifiers — also called parametric amplifiers — are much simpler than the TWT amplifiers. The transmission type of amplifier 7-5] MIXERS 357 appears to be the best suited for radar receivers, although the noise figure is somewhat higher than the reflection type. Many practical problems associated with these amplifiers, such as stabilization of the loaded ^,'s of the resonant circuits and regulation of pump power level, must be solved before these amplifiers find large use in airborne radar sets. However, these negative-resistance amplifiers appear to be a final step in attaining receivers whose sensitivity is truly limited by external noise. 7-5 MIXERS The SHF (super high frequency) mixer in the majority of airborne radar receivers incorporates crystal diodes. Properties of the crystal mixer which are important to radar system operation are: 1. The effective noise temperature 2. The conversion loss 3. The intermodulation components A crystal mixer can be represented by an equivalent circuit, as is shown in Fig. 7-3a. The nonlinearity of the crystal arises from the variation Zj Local Oscillator Source Impedance Crystal Image Zero Frequency Impedance Impedance (a) (b) Each Impedance Shown External to the Crystal is Zero to All Frequency Components Except the One to Which it Refers Fig. 7-3 (a) Equivalent Circuit of Crystal Mixer and (b) F-/ Characteristics of a Mixer. in the barrier resistance Rb which is a function of the voltage applied to the crystal. A typical transfer characteristic is shown in Fig. 7-3b. The spreading resistance Rs and barrier capacitance Cb are detrimental parasitic elements. Because of these elements, not all of the heterodyne signals' energy can reach the IF and image termination. 358 THE RADAR RECEIVER To obtain a low conversion loss, the voltage applied to the diode by the local oscillator signal is very large so that there is negligible conduction during one half of the local oscillator cycle. The signal voltage is much smaller than the local oscillator voltage. A current flows through the IF impedance which can be described by the multiplication of the signal and local oscillator voltages by the transfer characteristic of the mixer. The transfer characteristic can be expressed as a power series in the applied voltage. Because of the magnitude of local oscillator voltage a large number of terms are required to describe the mixer behavior. Thus / - E a.E- (7-9) n=0 where E is the input voltage / is the current flowing in mixer an are the coefficients of the power series describing the mixer; these are dependent on the local oscillator signal level n is an integer 0, 1, 2, 3, ... . Normally E consists of the sum of two voltages, the signal voltage and the local oscillator voltage. In general the input may be E=Y,Ar cos CO./ (7-10) r=l with the condition that Ai cos coi/ be the local oscillator signal and Ai )$> ^2, If the signal is a single frequency C02 and the IF center frequency is (coi — CO 2), the desired output spectrum from the mixer is the intermodula- tion term K cos (coi — C02)/. An expansion of the expression for the current in the IF impedance yields terms of the form Ij, = jJa.J, + ~ a,A,' + - + ^r^,2^ll^y^^ ^2„^i(-^"-^' + cos (coi — C02)/, r 9^\. (7-11) for the IF outputs incident to mixing of the signal frequencies with the local oscillator frequency. The term in the brackets is a constant for a particular value of local oscillator voltage, and the mixer thus produces an IF output which can be expressed as Itf = KA.. r9^\. (7-12) 7-5] MIXERS 359 When more than one signal frequency is applied to the mixer, inter- modulation between the signal components occurs. The output current caused by these intermodulation components is of the following form for each possible pairing of m signal components: cos (coy - CO/,)/. (7-13) where / = 2, -, 7n k = 1, -, m. Once again, the term in the brackets is constant for a particular local oscillator voltage, so that / = AiA.K,. (7-14) The voltage developed by these mixer currents is prevented from affecting the receiver performance by the frequency selectivity of the IF networks when coy — cojt falls outside the IF passband. Those components that would ordinarily fall within the IF passband, could be eliminated by RF preselec- tion and proper IF frequency. However, such preselection is not always feasible. A balanced mixer is therefore used (see Paragraph 10-15). In the balanced mixer two crystals are placed at two of the ports of a micro- wave junction, and the signal is fed into one port and the local oscillator into the other port. The junction may be a magic-tee, short-slot hybrid or rat-race. Each individual crystal develops all of the intermodulation components, but the relative phase of the signal-signal beats differs from that of the signal-L.O. beats and therefore can be discriminated against in the IF coupling circuit. Rejection of the undesired intermodulation components on the order of 25 db is realized in practice. Principal factors affecting the rejection of signal-signal beats are the impedance match between the signal source and each crystal, the rectifier dynamic characteristics, and the balance of the IF circuit. Among the signal-local oscillator products are two which affect the performance of the mixer. These are included in the value K of Equation 7-12 when it is determined experimentally by measuring the coi — co^ component from the mixer. These two products involve the generation of an image frequency, i.e., a signal which is separated from the desired signal frequency by a frequency equal to twice the IF frequency, and which is separated from the local oscillator frequency by the IF frequency. The image frequency is caused by second harmonic mixing and by an up con- version resulting from the IF current which flows through the mixer. 360 THE RADAR RECEIVER The image frequency signal appears across the crystal and propagates down the waveguide toward the local oscillator and the antenna. If the image wave sees a match, such as would exist if it were allowed to enter the local oscillator channel, the energy in this signal is dissipated and energy that could have appeared in useful IF output is lost. Proper reflection of the wave can cause it to enter the mixer and arrive at the crystal in proper phase so that the output IF is increased. Optimum handling of the image can improve the noise figure about 1 or 2 db. In general, however, con- ventional pulse type airborne radar receivers have broad band mixers. The image conversion is terminated and the lowest possible noise figure is not obtained. A number of other intermodulation components involving the second harmonic of the local oscillator occur and can be significant when the RF acceptance bandwidth is great. The crystal diode voltage-current relationship is given by '( exp|^-l) (7-15) where e = electronic charge V = applied voltage K — constant depending on crystal T = temperature of the junction. Shot noise is exhibited by the crystal; the mean square fluctuation current is Pdf = 2eIo4f, where /« is the d-c current through the crystal. Equation 7-15 indicates that a given conversion loss could be obtained with a lower d-c current by reducing the temperature and therefore producing less shot and granular noise. In addition to the shot noise there is a frequency- dependent noise. All of this noisiness of the crystal is specified by the crystal noise temperature ratio tx- The mixer noise temperature is /„ and is given by 21 H-h'l a- i'-'« for the broadband mixer. L is the conversion loss; U is the value specified by crystal manufacturers. The tm of an actual mixer may be difi^erent, depending on the termination of the image conversion which affects L. Equation 7-8 shows that a large value of Fip causes the conversion loss of the mixer to be the dominant parameter of the mixer contributing to the noise figure. In fact even for a low IF noise figure the conversion loss is more dominant than the noise temperature. The mixer therefore yields lowest noise figure when it is designed for minimum conversion loss. 7-6] COUPLING TO THE MIXER 361 A frequency-dependent part of tm has been observed to vary as {filfi)n, where/2 and/i are IF frequencies and n is between 0.5 and 1.^ Since the IF noise figure varies approximately as/i//2 at high IF, the IF frequency at which minimum Free is realized is not critical. The local oscillator is usually a klystron with a wide electronic tuning range. Such oscillators exhibit shot noise whose spectra are determined by the ^ of their resonators. To minimize intermodulation components be- tween such noise and the local oscillator signal a high IF frequency is desirable. The use of a balanced mixer, however, reduces this noise sig- nificantly. In any mixer design, the objective is to provide minimum tm and L. The conversion loss L depends on the match between the signal source at both the signal and image frequency, the RF signal frequency, the IF frequency, the crystal biasing, and the local oscillator signal level. The noise temperature tm is also dependent on L, the IF frequency, and local oscillator signal. 7-6 COUPLING TO THE MIXER To obtain minimum noise figure, minimum mixer conversion loss must be realized. Conversion loss is defined on an available power basis; therefore the conversion loss does not depend on the actual IF load admittance connected to the mixer. The conversion loss, however, is dependent on the RF signal source admittance. To obtain minimum conversion loss at the principal beating frequency (signal frequency beating against local oscillator frequency) a mismatch is required between the mixer and the source. The input admittance of the mixer, however, depends on the IF conductance seen by the mixer. This in turn depends on the design of the first IF stage of the receiver and the network which connects it to the mixer. A condition frequently encountered in airborne radar receivers is that the IF admittance is very large incident to the use of a double-tuned trans- former between the mixer and first IF tube. The secondary circuit is usually damped only by the coil losses and circuit losses. A very large admittance is therefore coupled into the primary circuit near the resonant frequency of the secondary circuit. For this type of coupling between mixer and IF amplifier, an optimum mismatch between the signal source and the mixer is given approximately by " = zi^ (^-"' where p is the VSWR (voltage standing wave ratio) at the signal frequency and Lo is the optimum conversion loss. 3P. D. Strum, "Some Aspects of Crystal Mixer Performance," Proc. IRE 41, 876-889 (1953). 362 THE RADAR RECEIVER The source admittance can be designed on this basis when the IF cou- pling circuit is as specified. When the image frequency signal generated at the mixer is allowed to be radiated by the antenna or dissipated in the local oscillator source admittance, the value of /><, is that which is normally specified by the crystal manufacturer as conversion loss. In many airborne radar receivers a short-slot hybrid junction is employed in a balanced mixer. When the crystals are matched in such a mixer, all of the image frequency signal generated in the mixer propagates out the local oscillator port. Normally this port is matched, therefore the image signal energy is lost. In some radar sets a filter may precede the mixer to reduce interference from other radar sets. Such a filter may appear as a susceptance at the image frequency and reflect the image signal originating in the mixer. If the signal arrives in the correct phase at the mixer crystals, the performance is improved. The phase depends on the distance between the mixer and the filter. However, the distance between the mixer and the filter is also dependent on the mixer to IF coupling circuit, since the filter would be situated so as to give the optimum mismatch of the source to the mixer. To obtain lowest receiver noise figure, design of the RF and IF circuits must therefore be considered jointly, not separately. One solution to this problem might be the use of the short-slot hybrid with a filter in both signal and local oscillator paths. 7-7 IF AMPLIFIER DESIGN The IF amplifier consists of a cascaded arrangement of vacuum tube amplifiers which employ band pass coupling networks. Frequency of operation is a compromise between several factors such as noise figure, circuit stability, spurious responses, and receiver tuning characteristics. Consideration of these factors usually leads to the choice of an IF frequency between 30 and 60 Mc in the ordinary pulse-type airborne radar receiver. The IF amplifier is a filter amplifier, and its small-signal transfer function is given by G(s) = H — — —^ ;^;^37-r - . (7-18) In this expression // is a constant depending on the number of tubes, their transconductance, and the capacitance values of the circuits; s is the complex frequency variable a -\- j(ji-, n is the number of circuits; q and m are determined by the network complexity. The transfer function vanishes when s = because of the numerator term. The function thus has a zero of order nq at the origin. The denominator can be factored into {s - s,){s - s,*){s - s,){s - S2*)(s - s,)(s - .^3*) - • (7-19) 7-7] IF AMPLIFIER DESIGN 363 The values of s for which the denominator vanishes (j = si; s = Si* . . . etc.) are the zeroes of the denominator. At these values of s, the transfer function becomes infinite, so they are called the po/es oj the network function. The synthesis of an IF amplifier is facilitated by use of a potential analogy.* By considering each pole to represent the position of positive line charge normal to the complex frequency plane, and each zero to represent the position of a negative line charge normal to the plane, it can be shown that the potential measured along theyco axis resulting from the pole-zero array is equivalent to the logarithm of the magnitude of the normalized transfer function. When the transfer function consists of poles which are very far removed from the origin and near thej'co axis (as in Fig. 7-4), an arrangement of the C^+JW, Transmission Band Z(a)) Due to cr3+jco3 Due to cTi+jcOj ■Due to cTj+jcOg Fig. 7-4 S-Plane and Z-Plane Representations of IF Amplifier Characteristics. poles at each interval about a semicircle having the jw axis as diameter produces an approximate constant potential on the jco axis. The networks that are used, however, have zeroes at the origin and conjugate poles in the third quadrant of the plane. When the ratio of bandwidth of the overall receiver to the IF frequency becomes large, the contribution of these zeroes and poles to the transfer function in the passband region becomes significant. A conformal transformation Z = s' (-9 (7-20) is used to obtain an exact low-pass transformation, where _ number of zeroes at origin of s plane ^ number of poles in upper left s plane for the individual network elements employed. This transformation moves the zeroes to infinity in the z plane, and results in coincidence of the s plane *W. H. Huggins, The Potential Analogue in Network Synthesis and Analysis, Air Force, Cambridge Research Lab. Report, March 195L 364 THE RADAR RECEIVER pole pairs in the 2 plane, so that a single pole cluster is obtained in the 2 plane. Placing the poles on a semicircle in the 2 plane produces a constant potential in the s plane. When the poles in the s plane are placed at the position given by transforming the equally spaced poles of the 2 plane to the s plane, a maximally flat transfer characteristic is obtained. The desired transfer function for the amplifier could be realized by a single four-terminal network followed by an extremely wide-band amplifier. (The amplifier, of course, would have a particular pole-zero structure, but the contribution to the selectivity in the frequency band of interest is negligible.) Practically, however, the four-terminal network is limited to a four-pole structure, and most commonly to a one or two-pole structure because of the limitation of realizable unloaded ^'s for the network in- ductors. The most commonly used network of the IF amplifier is shown in Fig. 7-5. When k = I and Li = L2 the network reduces to a one-pole structure as shown. C2 = H- S'*+aS3+/3S2+ 7S+6 S (s-s,)(s-si)(s-s^)(s-s;) Where H 7 = Qj Q2 2 COj 002 Q2(l-K^j'^Qi(l-K2j Qi Q2 cof + col 1 2 2 COj CO2 4=^ w,R.C, ■'2 '2'^2'-2 When K=l and Li= L2the Single ■ Tuned Circuit With Bifilar Coil is Obtained. In This Case Usually All of the Damping is Provided By R 2 -. ^ Then: H, = H, 5^+ 7 S + X S 7 X Fig. 7-5 Commonly Used IF Interstage Coupling Network. 7-7] IF AMPLIFIER DESIGN 365 A parameter of great importance is the gain bandwidth product. Taking the simplest interstage circuit as a reference network, the gain bandwidth product is This equation shows that the quantities which determine the gain and bandwidth are so related that high gain can be obtained only at the ex- pense of reduced bandwidth. GB is determined by the tube and the circuit physical layout since it affects C. If two identical circuits are cascaded, then the 3-db bandwidth of the cascaded circuits is the 1.5-db bandwidth of a single circuit. For a given overall bandwidth, the individual stage bandwidths must be increased. With flat staggering of the circuits, this bandwidth shrinkage does not occur, and the GB product can be used to determine overall gain of the amplifier. The normalized attenuation characteristic of the IF coupling circuits employed in conventional pulse radar receivers can be expressed by the equation Attenuation (db) = 10 log to [1 + x^^] (7-22) where X =/„( ///- /°// )^ ^C/-/.) n = number of poles in the circuit (low pass equivalent) Bi = 3-db bandwidth / = frequency at which attenuation is to be evaluated. If m groups of such circuits are cascaded, then the overall amplifier selectivity is Attenuation (db) = 10m logio [1 + x^""]. (7-23) The overall bandwidth of the amplifier is therefore B = (21/- - l)i/2«5i. (7-24) For a given overall IF bandwidth it is desirable that the principal selectivity occur in low-level stages. The input stage selectivity, however, is governed by noise figure considerations. The input stages are therefore designed first and then the remaining amplification and selectivity intro- duced. The selectivity of the IF amplifier is provided by one-pole or two-pole networks between the stages in order to realize the maximum dynamic range. With «-pole configurations in which the poles are distributed through the amplifier, the dynamic range is usually not the same at all frequencies within the pass band; such designs are therefore avoided. 366 THE RADAR RECEIVER Groups of two-poles (staggered pairs) are frequently used. Although the restriction of dynamic range is not too severe, such designs are nevertheless inferior to synchronous stages. When an amplifier is built with single-pole coupling circuits, the overall frequency response exhibits geometric symmetry with respect to the IF center frequency. With two-pole coupling provided by the magnetically coupled double-tuned transformer, the response is more nearly arith- metically symmetrical. With two-poles having ^ ratios of about 3 to 1 the usual amplifier requirements (bandwidth between 1.0 and 10.0 Mc) can be realized with adequate stability margin and fewer components than with single poles. One of the main difficulties encountered in the design of radar IF am- plifiers is accurate control of the stability margin. Pole shifting (regenera- tion) can occur under strong signal conditions and results in poor transient characteristics or modulation distortion on desired signals. The principal source of feedback in an IF amplifier is the grid-to-plate capacitance of the tube and circuit. Other feedback paths are: 1. Coupling between input and output leads 2. Coupling due to the chassis acting as a waveguide beyond cutoff frequency 3. Grid-to-cathode feedback 4. Inadequate decoupling circuits resulting from self-inductance of bypass capacitors and their connecting leads 5. Coupling between heaters 6. Coupling between input and output caused by ground currents. (The impedance of the chassis is not negligible. It is necessary that the output and input currents not flow through the same part of the chassis to ensure stable operation.) The grid-to-plate feedback can be partly compensated by proper circuit design. It is advisable that a common bypass capacitor be employed for the screen grid and plate return of the amplifier. Appropriate choice of this component then enhances the amplifier stability. In high-frequency stages which are gain-controlled it is also desirable that feedback be intro- duced in the cathode lead to stabilize the input susceptance of the tube. These circuits are shown in Fig. 7-6. It is further desirable that such stages employ vacuum tubes with separate suppressor grid terminals to minimize the feedback from plate to cathode. This feedback path leads to instability when the input susceptance of tubes having internal suppressor grid-to- cathode connections is to be stabilized. The most economical distribution of gain and selectivity in the IF amplifier occurs when the stages are made identical. However, this con- dition does not al<Vays provide the most stable operation. The latter IF 7-7] IF AMPLIFIER DESIGN 367 R c Fig. 7-6 Typical IF Amplifier Configuration Showing How Proper Choice of C Can Self-Neutralize the Stage. (Lead Inductance Has Been Neglected.) R^Xc- Stages must provide adequate dynamic range and thus are operated in a manner which allows large peak transconductances to occur. The per- missible gain in these stages should therefore be less than in the early stages when a tube of the same type is emiployed in both places. The gain can be controlled by lowering the impedance of the coupling network or by allowing a greater bandwidth in the stages. The maximum allowable gain is sometimes limited to M ax gain 0.2 i :fCo (7-25) where gm is the peak transconductance and Cgp is the grid-plate capacitance. The effective noise bandwidth 5„ is the parameter involved in radar performance computation. For any practical amplifier this is very nearly the 3-db bandwidth Bn G(co) 1 + 1 r" : T / G(co)do) ; recei an be | /co C0o\ "") (7-26) where G(a)) is the power spectrum of the receiver. The normalized power spectrum of the radar receiver usually can be given by (7-27) where coo is the midband frequency B is the 3-db bandwidth n is the number of poles in a flat circuit m is the number of groups of circuits. The 3-db bandwidth is (21/- - iy''"Bi, (7-28) 368 THE RADAR RECEIVER where Bi is 3-db bandwidth of one network consisting of n poles. Then as an example for three staggered pairs of one-poles Jo (1 + .v^) Noise bandwidth ^ Jo (1 + •V)' ^ 1 r (i)r(3 - \) _ 3-db bandwidth (2"^- l)'/^ 0.714 r(4)r(3) " ^•^-• (7-29) 7-8 CONSIDERATIONS OF IF PREAMPLIFIER DESIGN The IF amplifier is frequently divided into two units, an IF preamplifier, and the main IF amplifier (postamplifier). This arrangement allows the input stages of the IF amplifier to be physically located near the mixer. When long cables are used between the mixer and IF amplifier, bandwidth and noise figure must usually be compromised. Principal considerations in the design of the IF preamplifier are the noise figure, signal-handling capability, selectivity, and gain. Triode tubes are almost always required for the first two IF amplifiers. They are used because they exhibit less shot noise^ than pentodes. (Shot noise is the noise resulting from fluctuations of the currents in a vacuum tube.) The input IF amplifier may be used in a grounded-cathode or grounded-grid arrangement. For the ordinary AI radar with broad band mixer, the grounded-cathode amplifier is usually employed for the first tube. To minimize input admittance variation caused by feedback from grid to plate this amplifier stage is neutralized. The equivalent circuit representing the sources of noise associated with an IF amplifier stage is shown in Fig. 7-7. From this arrangement of noise generators and signal generator the noise figure as defined by Equation 7-4 becomes 1 + ^" + ^'^ + ''' + '''' + ^L + g. +,. + Kf + Sv. +/ = ]+^^ + ^|y, (7-30) where ^ is the total susceptance appearing at the input terminals. The additional parameters involved are defined in Fig. 7-7. This expression yields the single-stage spol noise figure of an amplifier which is defined as the noise figure at a specific point of input frequency. When the noise figure of a radar receiver is measured, a noise source is generally employed. If the spot noise figure is constant over the bandpass of the overall receiver, then the noise figure that is measured will be independent of bandwidth 5B. J. Thompson, "Fluctuation Noise in Space Charge Limited Currents at Moderate High Frequencies," RCA Rev. 4, 269 (1940). CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 369 S, Ns, N„, Nt, Nf, Ni2 = current generators A"^ = a voltage generator S = signal current associated with gs Na = source noise, TVs = yl4KTsBgs (In evaluating IF noise figure Tg is usually taken as reference temperature T for which F is defined, or 290°K. In a radar receiver Ts is larger than T because of noise-noise intermodulation at mixer from local oscillator signal.) A^„ = noise current associated with coupling network losses, A''„ = -s^AKTnBgn Nt = noise current associated with grid noise of tube, Nt = ^j4KTtBgt ^ = usually 4 or 5 gt = grid damping due to finite transit time only Nf = noise current associated with feedback such as that due to cathode lead inductance: Nf = -yJiKyTg/; y may be between and 1 A^i2 = noise current associated with output to input feedback conductance: Nn = yjiKSTgi^; d usually 1 A'' = voltage generator representing shot noise in the tube, A^ = -yJ^KTBRn where R„ is the resistance that must be connected between grid and ground to produce the same fluctuation current in the plate circuit of the hypothetical noiseless tube as exists in the actual tube due to shot effect. For the triode, /?„ varies between Rn 2.5 and R„ 3£ where gm = operating transconductance. Fig. 7-7 Equivalent Circuit of IF Amplifier Noise Sources. as the bandwidth is reduced. The last term of Equation 7-30, however, may cause the spot noise figure to increase for frequencies removed from the carrier frequency and the average value of F is then increased. To minimize this effect in wide-band applications, care is taken to minimize the increase in this term either by virtue of a small value of i? or, in some cases, by introducing feedback to increase gf. In all cases, IF preamplifier tubes are selected which have a high transconductance-to-input capacitance ratio and small transit time. An important consideration is the mixer-IF coupling network. When gf and ^12 are zero, minimum F is obtained for ,.-yl'-^ + (g. + g. (7-31) as can be proven by differentiating Equation 7-30. Application of this equation is difficult because ^gt cannot always be easily determined. In 370 THE RADAR RECEIVER practice, careful measurement of the actual input admittance of the tube under operating conditions and with feedback effects removed by neutral- ization gives an input conductance which, when employed as ^gt in the amplifier design, results in measured noise figures in close agreement with calculated values. Fig. 7-8 shows a simplified equivalent circuit for a grounded-grid stage. The noise figure is given by ^Am + 1/ I I l+^ + |J + ^-:^{-T-r)|i^^l (7-32) 1\ i h Nsi Fig. 7-8 Simplified Equivalent Circuit for a Grounded Grid Stage. A^^; in- cludes noise due to grid loading and network loss. Ngi is therefore taken as a noise current generator ^^^KTypgi, where gi is total conductance between cathode and grid and i/' is an effective temperature ratio. The admittance seen to the right of aa', is gli} + m) 1 + girp where m is the amplification factor of the tube. In determining the overall noise figure of the IF amplifier, the available power gain of the amplifier stage must be considered. When losses in the interstage coupling network are not included, the available power gain for the grounded-cathode and grounded-grid stages is given by e) (7-33) (grounded grid) (7-34) where W^ (grounded cathc w^ k'^ -gsf'p " (^1 + if s)ig + gs+ g'n.) , 1 + M grr, The most frequently used IF preamplifier circuit in airborne radar re- ceivers is the cascade circuit. This circuit consists of a grounded-cathode triode stage followed by a grounded-grid stage. Such a configuration results in a lower noise figure than can be obtained with a cascade connection of two neutralized grounded-cathode triode stages. This is because the interstage bandwidth is obtained by virtue of the loading incident to the 7-8] CONSIDERATIONS OF IF PREAMPLIFIER DESIGN 371 large input conductance of the grounded-grid stage. Since such a feedback conductance is not a noise source, less noise exists than in the case when such network damping is obtained by a physical resistor. Wide-band neutralization of the input tube is employed to stabilize the admittance appearing at the input grid. Such stabilization allows the widest bandwidth for which a low spot noise figure is obtained by minimizing the variation of the last term of Equation 7-30 with frequency. To minimize the variation of the grid admittance with frequency, a double-tuned (two-pole) mixer-IF coupling network is employed with the cascode input circuit. When this circuit is designed for a flat (Butterworth) response, the bandwidth is given by B = -4^ (7-35) 7rV2C where gs is the value of source conductance required for minimum F B is the 3 db bandwidth C is the total capacitance appearing at the input of the first tube. The signal transmission bandwidth is slightly wider than this value because of the loading of the source caused by coil losses and the input conductance of the tube. In high-PRF radars (such as the pulsed-doppler systems described in Chapter 6) and where very short range with high accuracy is required, the double-tuned mixer-IF coupling network is found to introduce objectionable transients following the transmitter signal. These transients result from the nonlinear loading on the network by the mixer crystals. In such cases a grounded-grid input stage is employed. The transmission bandwidth of the mixer-IF coupling is very wide because of the heavy damping caused by the input conductance of the grounded-grid stage. The heavy damping by the tube minimizes transients resulting from the crystal mixer IF admittance variation when the transmitter signal is present at the mixer. A typical example of IF preamplifier performance is given by the following: CIRCUIT AND TUBE PARAMETERS g,n = 20 X 10-^ mho M = 44 figt = 5.0 X 10-^ mho ^1 = 10~^ mho Rn = 150 ohms 372 THE RADAR RECEIVER ^input tube =15 MMl Cgp = 1 .5 MMf AC =1.5 MMf due to space charge effect Cout = 3.0 mmF Cin of third = 7.0 ^iixi tube gn of input = 10~^ mho coil Problem Solution. Requirements are that the overall bandwidth between the mixer and third tube in which the attenuation is less than 1 db shall be 3 Mc, the noise figure of the amplifier shall be as low as possible, and a cascode circuit employing two of the tubes having the specified characteristics is to be used. For minimum F, the source conductance from Equation 7-31, ^^^ ^10_2_+_5(10_^ = 6.3 X 10-4 mho is required. A source bandwidth (see Equation 7-35) of 6.3 X 10-4 ^ = .V2 20.5X10- = ^-^^^^ is obtained, assuming 1.0 MMf stray capacitance and a wide-band neutraliza- tion of Cgp. At 3-Mc bandwidth the input coupling network produces an attenuation (see Equation 7-22) 10 login fl +[zl??^ =0.15db. The network between the second and third tube can therefore exhibit 0.85 db attenuation at a bandwidth of 3 Mc. A single-tuned circuit of 6.45-Mc bandwidth is adequate: 10 log, of 1 + I T^ 1 = 0.85 db. '"^"O+felV"- Total capacitance between the second and third tube is 3.0 -f 7.0 = 10 If a step-down transformer is employed, the gain bandwidth product could be improved and the conductance presented to the second tube output is lowered. Ine step-down must be n = -\/t^ = \/t7j' 7-9] OVERALL AMPLIFIER GAIN 373 Total capacitance at output of the second tube is then 3.0 + ^ = 6.0 mm/ n- and the conductance is gi - 27r(6.45)(10«) (6.0) (10-12) = 2.43 X 10-^. From Equation 7-30 the noise figure of the first tube is V 1 -L IQ"' , 5.0 X 10-^ , 150 ,, . ^ ^ _ ^, , ^^ = ^ + 6.3 X 10- + 6.3 X 10-^ + 6.3 X 10"^ ^^'^ >< ^^ '^'^ 1.196 = 0.78 db The noise figure of the second tube is (from Equation 7-32) P ^ J . 10-^ 5.0 X 10-^ 2.43 X 10-^ 150(0.95) ' "^ 4.55 X 10-* ^ 4.55 X 10"* "^ 4.55 X 10-* "^ 4.55 X 10-* (4.65 X 10-^)2 4.55 X 10-* = ^ = g^ip. = 1.74 = 2.4 db. From Equation 7-33, the available power gain of the first tube is (20XI0-y(6.3X10-)(g^^3) (6.5 X 10-*)2 ^^^ while the available power gain of second tube is approximately 4.55 X 10- 2.43 X 10-* 1.87. Assuming 10 db for the third tube F, the preamplifier noise figure is given by Equation 7-7: 1196+ ^-^^-^ + ^Q-^ ^1237 i.iyo^ 131 ^(131)(1.87) ^ 0.92 db. 7-9 OVERALL AMPLIFIER GAIN It is necessary that signal amplitudes corresponding to the thermal noise level at the input of the receiver be amplified to a suitable level for detec- tion. The level required at the detectors depends on the use of the signal. For example, for signal detection on an intensity-modulated display providing range and azimuth coordinates, signal voltages on the order of 50 volts are usually required. The total amplification that is required depends on the signal bandwidth and the receiver noise figure. The equiv- 374 THE RADAR RECEIVER alent noise voltage that must be amplified may be determined by computing the total noise voltage at the input of each stage caused solely by the total grid conductance, the shot noise in the tube, and total grid admittance. This voltage is then referred to the input by dividing by the total gain from the input to the noise considered. Even though consideration is given to the design of a low-noise IF preamplifier, the process of referring all of the noise sources in the receiver to the input is extremely important, especially in multiple-conversion receivers such as are employed in some forms of doppler radar receivers. This method sometimes reveals factors such as noise on beating oscillator signals or noise caused by a method of selectivity dis- tribution that would degrade the IF signal-to-noise ratio and result in a sensitivity poorer than would be estimated from consideration of the IF preamplifier noise figure and overall receiver selectivity only. A typical equivalent input noise for an ordinary radar receiver having an overall bandwidth of 5.0 Mc is about 3.0 Mvolts rms. For signal detection alone, a voltage between 1 and 2 volts rms at the input to the IF envelope detector is satisfactory. Thus the required overall amplifier gain is on the order of 105 to 115 db. To obtain the voltages for the cathode ray tube an additional gain on the order of 40 db is then required. (Included in this figure is a loss of 6 to 10 db that is usually produced in wide bandwidth second detectors.) Where the envelope of the signal must be accurately demodulated, higher voltages may be applied to the envelope detector to recover larger negative peak modulation with less distortion. However, dynamic range of the amplifier must be exchanged for the higher operating level. In tracking receivers, use of a range-gated amplifier ahead of the envelope detector allows such an exchange to be made. The detector average output is usually regulated to a relatively fixed level, and noise modulation positive peaks have very small probability of exceeding a level more than 12 db above the regulated level. In a typical case of a receiver having an IF bandwidth of 5 Mc, incremental gain can be maintained for a range of IF signal from zero to about 12 volts rms at the input to the IF envelope detector. Thermal noise can therefore be amplified to a level of about 3 volts rms at the detector input. It is obvious that this does not appreciably alter the IF gain requirements. In the case of very narrow band receivers such as are employed for detection and tracking of targets by means of their difference in doppler frequency, receiver bandwidths are on the order of several hundred cycles per second. Considerably more gain is therefore required over that encountered in conventional radar sets. For example, with a bandwidth of 500 cps, the required gain to the envelope detector would be ... , 5,000,000 ,^ ^, , . , r . • , J ^°gio — 77^7^ — = 40 db more than m the case or the conventional radar receiver of 5 Mc IF bandwidth. 7-11] BANDWIDTH AND DYNAMIC RESPONSE 375 7-10 GAIN VARIATION AND GAIN SETTING Gain of an amplifier stage which does not incorporate feedback is equal to the product of the transconductance of the vacuum tube and the transfer impedance of the network which the tube drives. Instability of these parameters results in gain variation. The effective signal transconductance of a tube is proportional to the d-c current through the tube. Gain can therefore be stabilized by operating the tubes so that the d-c plate current is stabilized. This can be accomplished by means of large cathode resistors or by operating a number of stages in series d-c connection. However, the first method introduces transient recovery problems and the second method reduces dynamic range. Application of conventional feedback stabilizing techniques may be employed, but in high-frequency IF amplifiers it is usually limited to a small amount of signal-current feedback which is employed to compensate for input admittance variations of the tube. Network transfer impedance variations are on the order of ±0.5 db; and when a small amount of d-c current stabilization is employed with the vacuum tube, the variation in signal transconductance is on the order of ±1.0 db. Stage gains are limited by bandwidth requirements in the case of wide- band stages, and stability requirements in the case of narrow-band stages. In addition, restrictions are usually encountered in gain distribution through the receiver as a result of dynamic range requirements. Typical average stage gains in a receiver are between 6 and 20 db incident to these limitations. An amplifier providing 100 db gain might therefore require about 10 tubes. Since the variation in gain of each stage is on the order of ±1.5 db, 15 db reserve gain is required in the design, and provisions for con- trolling the maximum gain of the amplifier over a 30-db range is required. These gain-control variations do not include the gain control that is required to accommodate target signal variations. The gain setting may take the form of a noise AGC loop which controls the current of several tubes or a manual adjustment which is periodically set. 7-11 BANDWIDTH AND DYNAMIC RESPONSE A criterion sometimes employed for best signal-to-noise ratio is that signal plus noise should be filtered by a network which maximizes the peak signal-to-rms noise power. The network which will accomplish this result was determined in Paragraph 5-10 to be simply the conjugate of the signal spectrum; that is, the receiver filter should be "matched" to the signal. In the case of the noncoherent pulse radar, each pulse must be considered as a separate entity; therefore the optimum predetection filter is a bandpass filter shaped like the RF pulse spectrum envelope. The IF characteristics usually employed for maximum detection in thermal noise are reasonable 376 THE RADAR RECEIVER approximations to this value when the 3-db bandwidth is approximatehy 1.2 /(pulse length).^ Additional filtering can be applied to the postdetection or video signal when there is more than one pulse. A series of periodic pulses will have a spectrum consisting of a number of harmonics. The filter which is matched to such a signal will be tuned to these harmonics so as to amplify them and attenuate the intervening noise. Because of the shape of the frequency response of such a filter, it is sometimes called a comb filter. It is often more convenient to obtain the effect of a matched filter by operating in the time domain. The comb filter, which is appropriate for a series of pulses, can be simply represented by adding the pulses after they are delayed by appro- priate multiples of the repetition period. This operation is normally called pulse integration and, for search radars, is often performed by the phosphor of a B or PPI scope display. When the more elaborate technique of time domain filtering is utilized, it is sometimes referred to as signal correlation. A more detailed discussion of matched filters is given in Paragraph 5-10. In selecting a bandwidth characteristic for the receiver, three considera- tions must be made over and above signal to thermal noise: 1. Adjacent channel (frequency) attenuation and discrimination against clutter 2. Compatibility of transient response with required resolution 3. Large signal operation The usual response characteristics that might be encountered were indicated in Paragraph 7-7. The transient response of these networks governs the resolution and large-signal behavior. The rectified envelope of this response corresponds to the video signal. A typical transient response would appear as shown in Fig. 7-9. At the receiver output, a loss in sensitivity may occur for the time /2 shown in Fig. 7-9 if the signal becomes suffi- ciently large that amplifier stages are driven into saturation. In pulse T7 1 n ^ ■ ] r\ ^ ^^ • .. doppler radar receivers this is a more i*iG. 7-9 Typical Output Transients as ^^ They Appear on the Rectified Envelope serious problem than in conventional of the IF Response to an IF Pulse Input, radar receivers. When all of the networks have identical transfer impedance of the form ^ £ (s — Si){s — Si*) 6See J. I. Lawson and G. E. Uhlenbeck, Threshold Signals, Vol. 24, Sec. 8-6 (Radiation Laboratory Series), McGraw-Hill Book Co., Inc., 1950. 7-12] SNEAK CIRCUITS 377 there is no overshoot when the pole frequency and the carrier frequency are coincident. The envelope response of such a network is given by eo{t) = K 1 +f expyj - 2f exp^|-jcos (co - w,,)/ 1 + \C0o CO / (7-36) where ^ is the effective circuit ^ Wo is the pole frequency CO is the carrier frequency of the step sinusoidal input. Note that oscillatory terms are involved when the carrier is detuned from bandcenter. These terms are relatively insignificant, however, for the amount of detuning that would normally be tolerated. When the circuits are not identical but are stagger-tuned, then the response given by Equation 7-36 becomes important. If the oscillatory signal is sufficiently large, the output of the following stage may be blocked for a period of time in excess of twice the duration of the input signal. To minimize these effects, inter- stage bandpass networks are usually employed which are symmetric about the IF center frequency, 7-12 SNEAK CIRCUITS When considering the dynamic response of the receiver, it is not sufficient to consider only the performance as a bandpass filter with saturation effects under large-signal input. The transmission characteristics of the amplifier in the low-frequency region of the spectrum must also be considered. To realize practical high-gain bandpass amplifiers the power supplied to the stages must not be derived from a common-source impedance, since instability will result. Fig. 7-10 shows a typical arrangement of IF stages. The power leads are brought into the amplifiers near the output. De- coupling filter elements CiCiC^RiRiLi Li are employed. The decoupling is designed so that a single stage will exhibit adequate gain and phase margin over the entire frequency spectrum when the stage is examined as a feedback amplifier. In particular the stability margin must be realized when the tubes operate at the peak transconductance values that would be produced by a saturating signal. Time domain effects must also be considered. Saturating signals cause the d-c currents to the various tube elements to vary. The cathode circuits will attempt to degenerate the effects of a saturating signal during the time that the signal exists. When the signal input ceases the cathode capacitor is charged to the value which has reduced the gain during the signal on time. 378 THE RADAR RECEIVER Gain Control Bus r^ 1^2 X^2 1^2 Plate C ' X Voltage Supply L2 Heater Supply Fig. 7-10 Typical Arrangement of IF Stages Showing Arrangement of Decoupling Circuits and the Feedback Paths Thereby Introduced. To obtain maximum receiver sensitivity the charge must be removed. This removal occurs with a nonlinear time constant C, + <?^(/) Short time constants must be used to avoid gain modulation of desired signals when there are large undesired signals such as clutter appearing in the receiver. Grid current may also produce a similar situation, and the time constants must be kept short while at the same time providing sufficient decoupling at low frequencies and at bandpass frequencies. The plate circuit decoupling is perhaps more critical than the other circuits. With a ladder decoupling chain, the d-c path must be kept low in resistance so that the plate voltage is not dropped excessively. Inductors are therefore used as the series elements. The elements nearest the power input connection have the currents of several tubes flowing through them. When several stages are driven into saturation, each of the stages will send a transient input into the decoupling chain. This transient propagates along the chain and may result in a very complicated transient at the last stage which can gain modulate that stage, causing undesirable transient gain variations following strong pulse signal inputs. To avoid this phenom- enon, adequate filtering is provided between the ladder tapping point and the tube. 7-13] CONSIDERATIONS RELATING TO AGC DESIGN 379 7-13 CONSIDERATIONS RELATING TO AGC DESIGN The AGC of the radar may be of two types: (1) a fast AGC which prevents saturation of the receiver or (2) a slow AGC associated with a single target echo. In the radar receiver employed for tracking, AGC circuits of the second type are required. The IF amplifier is one of the limiting factors in the design of a high performance AGC. This subject will be discussed at greater length in Paragraph 8-21. In designing the IF amplifier great care must be taken to examine signal distribution in the amplifier as a function of the AGC voltage. The AGC voltage must be applied to the amplifier in a manner that will result in minimum signal distortion and limited degradation of the output signal- to-noise ratio of the receiver. For example when the input signal-to-noise ratio is +90 db, it is necessary to reduce the gain in early stages to minimize distortion, and as a result noise from latter stages becomes significant. A t-vpical design might allow the output signal-to-noise ratio to be +30 db li inimum for +90 db input signal-to-noise ratio. For minimum distortion of the modulation on the signal as the gain of an amplifier stage is varied by AGC, it is desirable that the transfer character- istic be a square-law when signal and gain control are applied to the control grid. When sharp cutoff tubes are employed for gain control, considerable distortion is sometimes experienced when gain control is provided for large- signal inputs. Restriction of the gain control to about 10 db per stage in these cases usually results in acceptable signal envelope reproduction. Output stages of the amplifier should operate with linear plate transfer characteristics. This allows the IF signal voltages applied to the last few gain-controlled stages to be small, thereby resulting in less distortion. In addition, wider bandwidths can then be employed in these stages, since filtering of the undesired spectral components of the modulated signal, which result from passing the signal through the nonlinear plate transfer characteristic required for constant incremental gain as a function of AGC voltage, is not required. In a typical case the gains in the IF may be 10 db per stage. Requiring 2 volts rms at the IF envelope detector, the minimum signal voltage at the third from the last stage of the amplifier would be 0.2 volt rms if gain control is not applied to the last two stages. The maximum signal on the controlled stage then depends on the gain reduction allowed. By controlling a number of stages the maximum gain reduction required in any one stage can be limited to something on the order of 10 to 20 db. It is necessary to examine the signal transmission through each stage for the maximum signal allowed at the input of the stage as a result of the distribution of the AGC control voltage. The AGC and transfer impedance of the stages are then arranged to provide a specified allowable distortion of the modulation on the signal appearing at the amplifier output. 380 THE RADAR RECEIVER In the early stages of the receiver, care must be exercised in applying AGC. When a cascode type input amplifier is employed, relatively large voltages may appear at the input grid and also the third tube grid in receivers which must provide target tracking at very short ranges. An AGC voltage is therefore applied to the first tube in these cases. However, in order that the output signal-to-noise ratio of the receiver shall not be seriously degraded, this AGC is usually not applied at the same input level as the AGC on the other gain-controlled stages but is delayed until the input signal-to-noise ratio is about 20 db. The AGC voltage delivered to the cascode is selected so as to minimize the third-order coefficients of the tube transfer characteristic. The effective cascode transfer characteristic is somewhat superior to that of a single tube because of the d-c series connec- tion which allows control of the current of both tubes. Controlling the current of two tubes in a cascode arrangement has the advantage that the stability is not impaired at low gain. When only one tube is controlled, the grounded grid section may become unstable because of the reduced source conductance which drives it. A disadvantage of controlling the current of two tubes exists; not only does the conductance of the output of the first tube decrease, but the input conductance of the grounded grid section also decreases, thus narrowing the intercascode coupling bandwidth. Plate and screen grid control for AGC is attractive but reduces the dynamic range of the amplifier stage for large signal input. The operating point can be maintained at a value which minimizes the third-order coefficient, but signal suppression occurs when the signal peaks drive the control grid into cutoff and into grid current. Suppressor grid control is very attractive, since the third-order curvature can be minimized without sacrificing dynamic range. One difficulty is that the power dissipation of the screen grid is usually exceeded under strong signal conditions. For a high-performance system the AGC voltage will be staggered, i.e., the amount of AGC voltage applied to the various controlled tubes of the amplifier will be different. This is required to obtain minimum envelope distortion. The AGC decoupling circuits must be designed with the precautions noted in Paragraph 7-12. In particular, the transmission of the IF amplifier at low frequencies must not be significant — i.e., it must operate only as a carrier amplifier. 7-14 PROBLEMS AT HIGH-INPUT POWER LEVELS In an airborne radar set strong signals are obtained from short-range targets, clutter, and other radar signals. Two situations occur in the receiver. In one case the receiver may be operating at maximum gain and be required to furnish output from signals having an input power of the 7-14] PROBLEMS AT HIGH-INPUT POWER LEVELS 381 order of magnitude of the receiver thermal noise. In the other case the receiver is required to furnish an output from a single signal which has been time-selected. In the first case, cross modulation caused by the strong signals can deteriorate the weak-signal performance; the extent to which this occurs is a function of the detailed receiver design. If the receiver is linear, the dynamic range for any particular gain setting will usually be between 10 and 20 db. Signals more than 20 db greater than the thermal noise level can be expected to cause saturation in the receiver. The result of the saturation is a paralysis of the receiver for a certain time following the removal of the large signal. To minimize this effect it is necessary that attention be given to the circuits mentioned in Paragraph 7-12, so that a suitable transient characteristic is obtained from the IF amplifier. The transient should exhibit small overshoot and short delay time. Loss of weak signals occurs only when they are time-coincident with the strong signals if adequate IF filtering is provided. In cases where signal information is required and when the interference and signal occur at the same time (range), saturation must be prevented and the two signals separated on the basis of their difference in frequency spectra caused by the doppler shift. In the non- coherent pulse radar this is accomplished by heterodyning the weak signal against the strong signal at the IF second detector. The second case occurs when a signal is being tracked. The desired signal is gated and may provide range and direction signals from sidebands associated with each of the pulse signal sidebands. The effect of strong signals is to add additional sidebands at the receiver output and thereby cause errors in the range and direction signal. In a well-designed receiver, negligible intermodulation occurs when a strong signal is present which is not time coincident with the desired signal. In some instances the desired signal power level may approach the order of magnitude of the local oscillator signal power. Fig. 7-11 shows the transfer characteristic of a typical microwave mixer at large-signal levels. The nonlinearities of this characteristic will cause signal distortion. Inter- modulation components appear incident to the beating of the various signal components. These components are not highly significant except with some propeller-driven targets in which terms of the order 2wi + £02 may introduce more fluctuation in the final bandwidth of the system. The reduction in modulation percentage of the pulse signal at the fundamental modulating frequency results in deterioration of tracking performance, since it corre- sponds to a change in tracking loop gain. In many cases the signal at the antenna terminals is greatly distorted before it reaches the signal mixer because of the time varying attenuation of a gas discharge TR tube. A controlled TR characteristic is therefore sometimes used to advantage to minimize the deterioration in tracking loop performance. 382 THE RADAR RECEIVER -500t20- -20 -16 -12 -8-4 4 8 12 SIGNAL POWER (dbm) Fig. 7-11 Transfer Characteristic of a Microwave Mixer at Large-Signal Input Levels (1N23C Crystal). 7-15 THE SECOND DETECTOR (ENVELOPE DETECTOR) An envelope detector is employed to produce an output voltage which corresponds to the envelope of the IF signal. The envelope detector is actually a mixer in which the sidebands of the signal are heterodyned against the signal carrier thereby producing as one output the modulation that existed on the IF signal. In the ordinary noncoherent pulse radar set, a diode detector is frequently employed. A typical circuit is shown in Fig. 7-12, together with the current- voltage relations that exist under large-signal conditions. A pulse of IF voltage is indicated as being applied to the detector. A large diode current pulse flows for a short time following the application of the signal. Capacitor Co is a relatively low impedance at the IF frequency compared to i?o, and RFC is a high impedance to these frequency components; therefore negligible voltage appears across i?o due to the IF frequency components and their harmonics which appear in the diode current. The average value of the current pulse, however, does produce a voltage across i?o- This voltage builds up at a rate dependent on the capacitance Co + Ci + Ci and the diode resistance, and reaches an average value Edc as shown in Fig. 7-12. The diode only conducts when the instantaneous voltage applied to the diode exceeds Ex. As shown, conduc- tion during time ab occurs and the capacitance Co + C\ is charged at a rate dependent on the diode resistance and this capacitance. When the IF pulse ceases, the diode is back-biased and returns to the unbiased condition with a time constant /?o (Co + Ci + C2). [The effect of the inductance of the RFC on this transient is usually negligible when the product of pulse length 7-15] THE SECOND DETECTOR (ENVELOPE DETECTOR) 383 Video Amp. Fig. 7-12 Typical Second Detector Circuit times IF frequency is greater than 50. As a result of this operation it is necessary that Rq (Co + Ci + C2) be considered as a low-pass filter estab- lishing the video bandwidth.] The efficiency of the diode detector is the ratio of the d-c voltage (Edc) to the peak carrier voltage applied to the circuit. The efficiency depends on the ratio of the diode resistance plus source resistance of the IF network as seen by the diode to the load resistor Rq. Efficiency, however, also depends on the ratio of the load resistance to the reactance of Co + Ci at the IF frequency. In practice Co is usually on the order of 10 to 20 iJifif. Smaller values of Co result in less voltage impressed on the diode because of the division of voltage between Cq and C^. Ro is then selected on the basis of video bandwidth requirements. A typical example is a requirement that the video bandwidth be 10 Mc with a network impedance as seen by the detector of approximately 500 ohms and a capacitance Ci of 10 finf. The value of Ro is then fixed by Ci and the smallest value of Co that can be employed. Assuming Co to be 10 nnf, Ro is required to be 796 ohms. An Ro 384 THE RADAR RECEIVER of 750 ohms would be used. The efficiency of the detector would be 0.21, assuming a diode resistance of 200 ohms^ and a 60-Mc IF frequency. A gain loss of 13.6 db is thus exhibited by the detector. This is a typical loss; the loss usually ranges between 6 and 15 db, depending on the video bandwidth and IF frequency involved. An important design consideration is the loading on the IF network produced by the detector. An approximation of this loading is given by R = ^ (7-37) where R is the IF network loading and rj is the efficiency of rectification. Efficiency of rectification depends on the diode resistance Rd plus the IF signal source resistance. Since the value of Ra depends on the voltage applied to the diode, the detector is nonlinear at low levels. A typical second-detector characteristic is shown in Fig. 7-13. Reproduction of the modulation on a PAM (pulse amplitude modulated) signal depends there- 1.00 0.10 0.01 0.001 / / / y / / / / </ Jy / Y/ / / / rher mionic Diode Ro= 820 oh Type 56 ms 47 / f Cc " — lb mrr f / / / / f 1.0 10 INPUT VOLTS (rms) 100 Fig. 7-13 Transfer Characteristic of a Typical Wide-Band Envelope Detector. 'Determination of efficiency and input impedance is relatively complicated. Methods for determining these quantities may be found in K. S. Sturley, Radio Receiver Design, Vol. 1. 7-15] THE SECOND DETECTOR (ENVELOPE DETECTOR) 385 fore on the carrier level of the IF signal applied to the detector. With high percentage of modulation, the negative peak modulation is distorted incident to the nonlinearity of the detector at low levels. In receivers which provide considerable pre-detection integration (IF bandwidths of a few kilocycles per second) it is feasible to obtain high detection efficiency by use of large Rq and Co. When amplitude modulation on the signal must be recovered in such receivers, it is required that Ri and Ro satisfy the relationship ^1 Ro + R^ > m (7-38) where m is the highest modulation percentage that must be recovered without distortion. Failure to satisfy this condition results in clipping of the negative peaks of the modulation. When the signal-to-noise ratio of the IF signal is very small and the video bandwidth is less than the IF bandwidth, signal suppression occurs in the second detector.^ This is the result of noise-noise intermodulation at the detector. An approximate expression for signal suppression is db suppression ^ - 7 + 20 logio {SIN)if. (7-39) It is desirable to provide as much filtering as possible prior to envelope detection to minimize sensitivity loss caused by this signal suppression. However, predetection selectivity is limited by the stability of the IF filters and the tuning accuracy of the receiver. Some receivers, e.g. logarithmic receivers, do not employ a diode envelope detector but obtain the envelope by infinite impedance detection or plate detection in each of the IF stages. In monopulse receivers the IF detector which is employed to obtain angular error signals is usually a balanced modulator. This may take the form of either a phase detector or a synchronous detector. Such detectors ideally produce an output only when both signals are applied. The output is primarily dependent on one of the two signals present at the input (provided one signal is much larger than the other). If one of the signals, such as the sum signal in a monopulse receiver, is heavily filtered before applying it to the demodulator, significant improvement in detected S \N can be realized for low ^S" /A^ referred to the difference signal IF bandwidth. Such filtering, however, requires time selection of the sum signal before it is applied to the detector. Such a scheme is, in effect, a carrier reconditioning and exaltation method of detection and, of course, reduces the information rate of the radar. 8S. O. Rice, "Mathematical Analysis of Random Noise," Bell System Tech. J. 23, 282-236 (1944), 24, 46-156 (1945); "Response of a Linear Rectifier to Signal and Noise," J. Acoust. Soc. Am. 15, 164 (1944). 386 THE RADAR RECEIVER 7-16 GATING CIRCUITS Gating circuits are employed to improve the signal-to-noise and signal- to-clutter ratios at the output of the receiver. A gating circuit consists of a modulator to which the signal and the gating signal are applied. In most applications the only output desired is the intermodulation between gating signal and desired signal. To accomplish this, balanced modulators are required. At video frequencies, such circuits are difficult to realize, the dynamic range usually being small. At IF frequencies such circuits are more easily provided, and dynamic ranges greater than 50 db are common. The choice between the IF and video gating depends on the nature of the signals to be encountered by the radar receiver. Typical gating circuits for video and IF applications are shown in Fig. 7-14. Gating circuits are Gating Signal IF GATING CIRCUIT Fig. 7-14 Typical Gating Circuits. employed having gate lengths equal to the range displayed on an indicator and also with lengths equal to or somewhat less than the transmitted pulse. When a dynamic range greater than 50 db is required from a gating circuit, component selection is required. This is a result of uncontrolled cutoff characteristics of vacuum tubes that must be utilized. When gating occurs in the IF amplifier, spurious signals are always encountered. These spurious signals occur because it is difficult to suppress completely all of the modulating signal (gate pulse) at the output of the gater. The gating pulse 7-17] PULSE STRETCHING 387 is not usually coherent with the IF signal. The higher frequency compo- nents of the gating signal are the signal components which cannot be adequately filtered. Transients caused by the modulating signal will generally produce outputs from the IF filter when rectangular gate pulses are employed. Noise modulation of the desired signal results from these transients. In typical designs the noise modulation caused by the transients is at least 40 db below the signal. In addition, the seriousness depends on the signal processing following the gating. Appropriate sampling of the gated signal prior to integration reduces the noise to a negligible value. In video gating circuits the modulating signal is coherent with the detected signal. Thus the noise mentioned does not occur. 7-17 PULSE STRETCHING In tracking radars it is required that the modulation signal associated with a pulse-amplitude-modulated signal be recovered. The modulated pulse signal is /i(/) = [1 + m cos (a;„/ + 0)] ^fljue^^ (7-40) for periodic pulses of shape /(/) where r„ = ^///W exp (^^^^V/. (7-41) If the pulses are passed through a low-pass filter having a cutoff frequency below the first harmonic of the pulse, the modulation is recovered and will have an amplitude m{tjT) cos [w^/ + <^]. Since t jT typically may be on the order of xoVo this is a very inefficient process. Pulse-stretching circuits are therefore used to lengthen a series of pulses without changing the relative pulse amplitudes in order to obtain more gain in the process of recovering the amplitude-modulating signal. For most efficient demodulation the pulse is lengthened for a full period. In either case — whether a pulse is simply filtered or is lengthened and then filtered, time selection of the pulse is required prior to the lengthening to prevent cross modulation by un- desired pulses. A pulse lengthener converts the modulation function 1 + m cos (co^^ + <^) into a new function F{f). Two types of lengtheners are used. In one, F{t)^ is set at a fixed reference level prior to a signal pulse input; in the other the output is changed from the value measured to the new value. Typical circuits of these lengtheners are shown in Fig. 7-15. The lengthened pulse on which the desired signal is modulated is an exponential pulse. The decrement is small and approaches zero in many practical cases. 388 THE RADAR RECEIVER j^i:^ Eexn(-af) Fig. 7-15 Pulse Lengtheners. The output spectrum of the lengthener for the case where a = 0, and T = Tp IS given by \fF;\ = 1+ 2m sin ^ \Ar cos (co./ + <A) - ^1 r/ 27r\ ^ , ^ , U)mTp~\ +^.s, [(-4:)-' + <^ + Tp UmTp (7-42) ^m + Tp When the output from the lengthener is passed through a low-pass filter, the first term becomes the only significant term in the output if the period of the modulating signal is much greater than Tp. If the low-pass filter has a cutoff frequency Wc, outputs are also obtained for modulation frequencies satisfying 1 P (7-43) 7-18 CONNECTING THE RECEIVER TO THE RELATED REGULATING AND TRACKING CIRCUITS The receiver must provide signals to range or speed error detectors, and angular error detectors. It is desired that the outputs of these detectors have a stable characteristic with time and with input power level to the 7-19] ANGLE DEMODULATION 389 receiver. If the AGC demodulator is connected to the range error detector, and the angle demodulator connected directly to the AGC demodulator output, both range and angular error characteristics will be determined by the AGC regulation. It is desirable, however, that the video signals applied to these demodu- lators be as large as possible to minimize the bias errors resulting from contact potential in the demodulators. Frequently separate filtering of the range and angle video signals may be performed. A single AGC loop operating from the angle channel controls the receiver gain. To obtain a stable range-error detector characteristic, the video amplification between the input to the range detector and the AGC demodulator must then be stabilized by feedback. A typical arrangement is shown in Fig. 7-16. Local Osc. AGC Amplifier Filter IF Amplifier AGC Delay Voltage Envelope Detector Range Gate Pulse Lengthener Cathode Follower Video Amplifier Range Error Detector Video Amplifier Range "Gate Noise Free Signal at Lobing Frequency- Angular Error Demodulator A z. Error Angular Error Demodulator El. Error Noise Free Signal •at Lobing Frequency Fig. 7-16 Connection of a Receiver Employed with Sequentially Lobed Antenna to Related Circuits. 7-19 ANGLE DEMODULATION The antenna tracking error signal can be considered to be proportional to the magnitude of the fractional modulation of the signal resulting from division of the difference signal by the sum signal. In monopulse radar sets the sum and difference signals are separated at the receiver input, whereas in a conical scanning radar the composite signal is passed through the receiver. To obtain a tracking error signal from a monopulse radar, the difference signal is heterodyned with the sum signal, which is effectively a noisy carrier signal. In some cases carrier reconditioning may be performed 390 THE RADAR RECEIVER and the difference signal heterodyned against a filtered carrier signal. This latter operation, however, is accomplished only with sacrifice of the infor- mation rate. In a conical scanning radar the desired target is selected by range gating so that the other targets, which are also PAM signals, will not be demodu- lated. The signal is then envelope-rectified and lengthened. Lengthening is employed to minimize additional modulation resulting from PRF variation. The signal at the output of the pulse lengthener still represents the composite signal, i.e. the sum and the difference signal. The low- frequency modulation of the composite signal is caused by the scintillation noise of the target and is independent of the lobing frequency. Both the d-c component of the signal and the low-frequency modulation are fed back to the IF amplifier as a gain-control signal. Modulation at the lobing frequency, however, is not allowed to effect a gain control of the receiver. The signal at the output of the pulse lengthener thus contains primarily the sidebands about the lobing frequency which are caused by the variation in direction of arrival of the signal. To demodulate this signal, and provide control signals for the antenna servo, the signal is multiplied by a noise-free carrier at the lobing frequency. The carrier signal is phase-locked with the antenna lobing. This is usually accomplished by means of an a-c generator mechanically linked to the rotating antenna. Fig. 7-17 shows three typical demodulator circuits. In all three of these circuits neither the signal nor the carrier frequency appears in the output. The output contains only the beats between the signal and the carrier and certain of their harmonics. Of the three demodulator circuits shown the "ring modulator" is the most desirable because the modulation products are effectively separated in various parts of the circuit. The carrier signal should be as monochromatic as possible for maximum output signal-to- noise ratio. The process of pulse lengthening merely concentrates all of the noise appearing in the IF in a region less than the PRF. In order that the noise reduction provided by the antenna servo be approximately Bi/PRF, where Bi is the noise bandwidth of the antenna tracking loop, it is necessary that the demodulator provide a true product demodulation. To approach this performance the ring modulator is employed in conjunction with a bandpass filter which filters the signal applied to the demodulator. 7-20 SOME PROBLEMS IN THE MEASUREMENT OF RECEIVER CHARACTERISTICS Noise Figure. The most practical method of making noise figure measurements involves the use of a dispersed signal source. An argon-filled gaseous discharge tube will produce a standard noise power output equiva- 7-20] PROBLEMS IN MEASUREMENT OF RECEIVER CHARACTERISTICS 391 Demodulated Output NoC+NoS ■O NeC±NoS Output NoC±NeS Fig. 7-17 Ane;ular Error Demodulators. lent to a source temperature of 9775° K. Measurement of noise figure merely involves the measurement of the noise power required to double the output noise power of the receiver under test. A precision microwave attenuator is used to control the noise power applied to the receiver. The available noise power from the discharge tube is equivalent to a noise figure of 15.28 db referred to a temperature of 290°K. The noise figure is determined by merely subtracting the attenuation required to produce a doubling of the noise power from 15.28 db (corrections for spurious signal response are required). Several problems arise in this type of measurement. If the noise power output of the receiver is allowed to double, it is necessary that the receiver be linear at the two output conditions and that the response of the detector to noise be known. It is desirable that a 3-db loss be inserted in the receiver rather than let the output noise level change. The 3-db loss must be inserted at a point in the receiver which is preceded by sufficient gain that noise sources following the pad do not contribute to the output. The receiver also 392 THE RADAR RECEIVER must be linear ahead of the pad. Frequently it is not convenient to provide an accurate 3-db loss in the receiver. An example is the case where the preamplifier and main IF amplifier are contained in a single unit. In such cases an arbitrary attenuation may be introduced by means of the manual gain control. Measurements are made with two arbitrary output levels; it is only necessary that the receiver have a linear transfer characteristic to the noise at the selected levels. The method is as follows. 1. Observe output deflection (d-c voltmeter or milliammeter at the second detector) with no additional noise input. Let the deflection be di. The noise is incident to Nrec- 2. Introduce the noise source and adjust the noise power (A^i) applied to the receiver to produce a deflection d^. The noise is incident to 3. Insert attenuation a by means of the manual gain control so that the noise A^i produces the deflection di. The noise is incident to 4. Increase the output from the noise source (A^2) to produce the deflection ^2. The noise is incident to (A^2 + Nrec)oi. From these observations, the noise figure can be determined from ^ rec ^^ T ( T OT \ /-'TT'J where Ta = 290° K and T\ and T2 are noise temperatures corresponding to A^i and N2. In making a noise figure measurement with a dispersed signal source, difficulty is experienced with spurious responses of the receiver. In broad band receivers it is usual to add 3 db to the measured result to account for beating at the image frequency. Because of the small available power from the noise source it is necessary to couple directly to the antenna terminals of the receiver rather than through a directional coupler. As a result the noise figure is not usually measured with the transmitter operating in the case of airborne radar sets. The measurements are also correct only if the noise source has the same impedance as the antenna. Sensitivity. With the transmitter operating, additional noise may appear which will degrade the performance. This is particularly the case with high-PRFdoppler radar receiving systems. To determine the perform- ance in detecting and tracking small signals a sensitivity measurement is generally made; this is a measure of the least signal input capable of causing an output signal having desired characteristics. In the case of a radar display it is a simple matter to determine the signal power required to obtain a minimum discernible signal. The signal is 7-20] PROBLEMS IN MEASUREMENT OF RECEIVER CHARACTERISTICS 393 obtained from a standard signal generator which can provide the same modulation characteristics as the radar target. In a noncoherent radar the sensitivity is measured at various ranges. At minimum range the sensitivity- is usually reduced owing to the attenuation characteristics of the TR tube. It is sometimes convenient to define the sensitivity of a radar by an A scope measurement. In these cases a ^angeniiai signal measurement is made. For a tangential signal the signal-to-noise ratio is approximately +4 db. Measurement of the sensitivity of a tracking receiver requires that the transfer function of the loop be determined at various input signal power levels. The minimum signal power required to produce the full dynamic tracking capability of the loop is determined. Measurement involves the insertion of a fixed power level RF signal having the modulation character- istics of the radar signal, and measurement of the transfer function of the particular tracking loop for this fixed input signal level. More detail on the means for measuring the transfer functions of the regulatory and tracking systems will appear in the following two chapters. In making sensitivity measurements, accuracy is sometimes limited by signal leakage from the standard signal generator. Frequently it is neces- sary to put additional shielding around the generator, and connect a second precision attenuator in the line between signal generator and receiver. When measuring the sensitivity of very narrow band receivers such as are employed for doppler radar applications, it is usual to modulate the STALO (stable local oscillator) signal to obtain a signal source of adequate fre- quency stability that will remain within the narrow predetection filters. If the long-term stability of the STALO is reasonably good, a standard signal source which is crystal controlled may be used, provided the pre- detection bandwidth is not less than about 10~* times the RF signal input frequency. G. S. AXELBY • D. J. HEALEY III D. D. HOWARD • R. S. RAVEN • C. F. WHITE CHAPTER 8 REGULATORY CIRCUITS* 8-1 THE NEED FOR REGULATORY CIRCUITS To determine the bearing, range, and velocity of a target with high accuracy, three basic conditions must be fulfilled by the radar receiver and its associated data processing system: (1) the desired target intelligence components of the received signal must be faithfully reproduced at the output of the receiver; (2) undesired input signals which tend to reduce the S /N ratio of the desired target intelligence must be suppressed; (3) sources of noise internal to the radar must be minimized. The desired target intelligence appears as amplitude, phase, and fre- quency modulations of the received signals. The target information is extracted by taking a cross product between the received signal and a reference signal and filtering the resultant signal to remove extraneous cross products (see Paragraph 1-5). In a practical radar receiver, there are several potentially troublesome sources of degradation in these processes. An optimum demodulation process depends upon the accuracy with which the receiver can be tuned to the incoming signal. Various environ- mental and electrical factors will cause receiver tuning to vary or drift as a function of time. Receiver tuning control or automatic frequency control (AFC) is therefore required to reduce the effects of such variations. Receiver components must be operated under such conditions that the linear dynamic range of the receiver is very limited. Unless some form of automatic gain control (AGC) is utilized, signal distortion will take place in the receiver. For example, saturation effects will tend to erase amplitude modulation on the received signal; this in turn will cause poor tracking or loss-of-track in a conically scanning system. A large number of vacuum tubes must be employed in the receiver to amplify the noise level to the desired output level. Variations in the tube characteristics occur when the voltages supplied to the tubes vary. The desired output signals are then modulated with the undesired fluctuations of the power supply voltages. Thus electronic power regulation is required. ♦Paragraphs 8-1 and 8-3 through 8-13, and 8-21 are by D. J. Healey III. Paragraph 8-2 is by D. D. Howard and C. F. White. Paragraphs 8-14 through 8-20 are by C. F. White and R. S. Raven. Paragraphs 8-22 through 8-34 are by G. S. Axelby. 394 8-2] p:xternal, internal noise inputs to radar system 395 Motions of the aircraft carrying the radar set can modulate the incoming signal and cause loss or degradation of the target signals. Automatic space stabilization systems are often required to cope with this problem. Finally, the measurement problem is complicated by externally and internally generated noise. The origins of such noise and the effects of the noise upon range- and angle-tracking accuracies are described in the next paragraph. This discussion is particularly important to the subsequent discussion of AGC in this chapter and the angle and range tracking as discussed in Chapter 9. The remainder of this chapter will deal with the basic considerations governing the preliminary design of the AFC, AGC, and space stabilization loops. The problem of electronic power regulation is not discussed in detail since this is largely a matter of good electronic design practice, a topic beyond the scope of this volume. 8-2 EXTERNAL AND INTERNAL NOISE INPUTS TO THE RADAR SYSTEM 1 Paragraph 4-7 presented some of the basic measurements of target noise characteristics. This paragraph will define the noise sources in a form more immediately useful to the closed-loop control designer to illustrate the means for utilizing the measured information for design purposes. External Noise Inputs. Variations in the external input to the radar system fall into two basic categories, i.e. frequency components associated with motion along the target flight path and other frequency components normally referred to as noise. Noise includes propagation path anomalies and atmospheric noise (sferics) as well as noise caused by the complex nature of the target, random motion, and reflectivity. The emphasis here is on the noise associated with the target motion and reflectivity variations that lead to tracking errors. The various components of external radar noise may be defined as follows: Range noise, with an rms value of o-^, is defined as deviation of the range information content in the received echo with respect to some reference point on the target. The reference point may be chosen as the long-time average of the range information. Range noise is independent of the target range since its source is pulse shape distortion caused by variations in the vector summation of energy reflected from target surface elements. Amplitude noise , with an rms value of o-a^p, is defined as the pulse-to-pulse variation in echo amplitude caused by the vector summation of the echoes from the individual elements of the target. Amplitude noise, since it is iSee J. H. Dunn, D. D. Howard, and A. M. King, "Phenomena of Scintillation Noise in Radar Tracking Systems," Proc. IRE, May 1959. 396 REGULATORY CIRCUITS interpreted by the radar as amplitude modulation of the mean signal level, is independent of range if a good automatic gain control (AGC) system is used. Angle noise, with an rms value of (Xang-, is defined as the variation in the apparent angle of arrival of the echo from a target relative to the line-of- sight to the center of reflectivity of the target. Angle noise is a function of the spacing of surface elements producing echoes, and the relative am- plitude and phase of these echoes. Since angle noise is a function of the linear dimensions of the target, a variation inversely proportional with range results as long as the target subtended angle is small compared with the beamwidth of the antenna. At times, incident to angle noise, the direction indicated by the apparent angle of arrival of the target echo may fall outside the target extremes. Bright spot wander noise, with an rms value of o-6s,„, is defined as the variations in the center of reflectivity of the target relative to a selected physical reference point on the target. The summation of angle noise plus bright spot wander noise is the variation in the apparent angle of arrival of the echo from a target relative to the selected physical reference point on the target. Bright spot wander noise is a function of the relative spacing of target reflecting elements and the amplitude of echoes from these elements. Like angle noise, bright spot wander noise (in angular units) varies inversely with range. However, the peak excursions of the center of reflectivity of the target cannot extend beyond the target limits. Internal Noise Inputs. In addition to the primary function of location and tracking of targets in space, radar outputs to computers utilize rates of change of the basic position information. Tracking smoothness and accuracy depend upon the manner in which the external inputs are processed by the radar system. Internal radar noise components may be categorized as follows. Receiver noise, with an rms value of o-rec, is defined as the variations in the radar tracking arising from thermal noise generated in the receiver and any spurious hum pickup. Receiver noise is inversely proportional to the signal- to-noise ratio in the receiver, and since the signal power varies inversely as the fourth power of the range to the target (excluding propagation anomalies), this effect is directly proportional to the fourth power of range. Servo noise, with an rms value of aser, is defined as the variations in the radar tracking axis caused by backlash and compliance in the gears, shafts, and structures of the antenna. The magnitude of servo noise is essentially independent of the target and is thus independent of the range. Tracking Noise Definitions. An optimum radar system design can result only from proper consideration of the nature of all the external and internal noise sources. One principal objective of tracking system design 8-2] EXTERNAL, INTERNAL NOISE INPUTS TO RADAR SYSTEM 397 may be taken as minimization of tracking noise, which may be categorized as follows: Range tracking noise, with an rms value of art, is defined as the closed-loop tracking variations of the measured target range relative to the range to a fixed point on the target. Range tracking noise includes effects of the complex nature of the target and receiver and range servo system noise. Systematic range tracking errors arising from flight-path input information are excluded from art. Angle tracking noise, with rms value of aat, is defined as the closed-loop tracking variations of the measured target angular position relative to a fixed point on the target. Angle tracking noise includes effects of the complex nature of the target and receiver and angle servo noise. Systematic angle tracking errors arising from flight-path input information are excluded from aat. Range Tracking Noise. The general shape of the dispersion versus range for the various noise factors entering into range tracking is shown in Fig. 8-1. Since the various noise factors are uncorrelated, the total output Overall \ Max. >Joise — ^ Rcvr. Gam , Range Noise-^ J~ .. Servo N oise— ^ / /^Rcvr. / Noise 10 100 RELATIVE RANGE 1000 Fig. 8-1 Range Noise Dispersion Factors. noise amplitude (shown by the heavy line) representing range dispersion in a given tracking system is found by summing the noise components in a root-mean-square manner. To use the diagram of Fig. 8-1 for prediction of system performance, at least one point on each characteristic must be determined by measurements. In the case of external range noise, the following facts are known: 1. Fire-control radar range information contains noise resulting from the finite size of practical targets. 2. The total rms range noise, ar (in yards), may be predicted from a knowledge of target size and shape. Measurements made with a split-video error detector on a variety of single and multiple targets show an average 398 REGULATORY CIRCUITS rms value- of ar = 0.8 times the estimated radius of gyration of the reflec- tivity distribution of the target about its center of reflectivity. These results relate to average noise power. By nature, wide fluctuations from sample to sample may be expected with the actual value dependent upon sample time. Examples of spectral power distributions are shown later. 3. The range noise power spectra for a variety of aircraft targets in normal flight show that the significant power is below 10 cps and, in general, one-half the range noise power lies below 1 cps. The frequency components of range noise are a function of rates of target motion in yaw, pitch, and roll and are influenced by air turbulence, angle of view, maneuvering of the target, and the target type. 4. The influence on the noise values incident to the specific type of range tracking system employed has not been extensively investigated, but the values shown are believed to be typical for fire-control design purposes (assuming good system engineering and performance). The measured spectral range noise power distributions for an SNB twin- engine aircraft, for two SNB aircraft, and for a PB4Y patrol bomber are shown in Fig. 8-2. The curves represent mean values while the upper and lower maximum excursions from the mean are shown by the arrowed lines. The analysis was based upon 80-sec samples with the indicated mean value for (Tr taken over the number of runs shown. The broad frequency range of the radar range input noise power clearly emphasizes the requirement of range tracking bandwidth minimization consistent with tracking error specifications. Angle Tracking Noise. The general shape of the dispersion versus range for the various noise factors entering into angle tracking is shown in Fig. 8-3. The various noise components shown are uncorrected. The rms total output noise for conical scanning or sequential lobing radar is greater^'^ than for monopulse^ (simultaneous lobe comparison) radars because of the high-frequency amplitude noise at the lobing frequency. For prediction of system performance, at least one point on each characteristic must be determined by measurement. In the case of external angle noise, • the following facts have been established. 1. Amplitude noise is an amplitude modulation of the echo caused by the vector summation of echoes from the complex multielement reflecting 2D. D. Howard and B. L. Lewis, Tracking Radar External Range Noise Measurements and Analysis, NRL Report 4602, August 31, 1955. 3J. E. Meade, A. E. Hastings, and H. L. Gerwin, Noise in Tracking Radars, NRL Report 3759, 15 November 1950. ■•J. E. Meade, A. E. Hastings, and H. L. Gerwin, Noise in Tracking Radars, Part II: Dis- tribution Functions and Further Power Spectra, NRL Report 3929, 16 January 1952. 5R. M. Page, "Monopulse Radar," paper presented at the 1957 Institute of Radio Engineers Convention, IRE Convention Record, Part 8, Communications and Microwaves, p. 132. -2] EXTERNAL, INTERNAL NOISE INPUTS TO RADAR SYSTEM 399 g- 3 Tail View Target View No. Runs Aver, (yd)- ^ Side View^ Tail Side Nose 9 7 16 3.2 2.3 2.0- - k J . ^Nose " (View _ 1 ^ 1 \ ^ ^ { , (a) Three Views of a Single SNB Aircraft 2 3 4 5 6 7 FREQUENCY (cps) 8 9 10 (b) Two SNB Aircraft in Formation 2 3 4 5 6 7 FREQUENCY (cps) 14 s;i2 ^5lO 1^ uj 4 CO "\ Tail Target No. Aver. (Tr \ View View Runs (yd) Tail 4 4.85 v Side 5 2.43- Nosel 8 2.15 -Tail\ " ViewA _ =^ \ Side View ^ ^^^:=?^:^-r--^ , , 1 (c) Three Views of a Single PB4Y Aircraft 1 234 56789 10 FREQUEN^"' (cps) Fig. 8-2 Range Noise Power Spectral Distributions. 10 100 RELATIVE RANGE 1000 Fig. 8-3 Angle Noise Dispersion Factors. 400 REGULATORY CIRCUITS surfaces of the target. The frequency components of amplitude noise causing angle tracking noise lie in two widely separated bands. A low-frequency region of noise extending from zero to approximately 10 cps causes a noise modulation within the closed-loop servo-target combina- tion superimposed upon the tracking error caused by flight path input information and associated system tracking errors. The low-frequency band also influences angle noise as explained later. Removal of the effects of low-frequency amplitude noise on angle tracking by suitable AGC design is also discussed later. A high-frequency region of amplitude noise in the vicinity of the lobing frequency (except in monopulse radars) contributes directly to angle tracking noise. The angle tracking noise power arising from high-frequency amplitude noise is proportional to the square of the beamwidth, the fractional amplitude noise power modulation per cps of bandwidth, and the angle tracking servo bandwidth.^ The principal sources of target- generated high-frequency amplitude noise are propeller (power plant) modulation and structural vibrations of the target surface elements. 2. Angle noise is the variation in the apparent angle of arrival of the echo from the target relative to the line of sight to the center of reflectivity of the target. It is caused by variations in the phase front of the reradiated energy from the multielement target. When low-frequency amplitude noise exists incident to narrowband or slow AGC, the angle noise power (in suitable units) equals one-half the square of the radius of gyration of the target reflectivity distribution.^ When low-frequency amplitude noise is removed by wideband or fast AGC, the angle noise power is approximately doubled with practical AGC circuitry. 3. Bright spot wander noise results from changes in the center of target reflectivity principally caused by a redistribution of the significant target reflecting surfaces; it does not depend upon the relative phases of the echoes from the individual surface elements. The frequency components of bright spot wander noise lie almost entirely in a low-frequency band since it is associated with major aspect changes of the target. Because bright spot wander noise is an uncorrelated component of target-generated angle tracking noise, a complete elimination of angle noise (as defined above) does not reduce angle tracking noise to zero. Examples of the spectral energy distribution of amplitude noise were shown in Fig. 4-23.^ In the spectra illustrated, the analytical method excluded low-frequency results below 30-40 cps. ^Ibid., p. 3. ■'B. L. Lewis, A. J. Stecca, and D. D. Howard, The Effect of an Automatic Gain Control on the Tracking Performance of a Monopulse Radar, NRL Report 4796, 31 July 1956. ^Source: D. D. Howard, from measurements made at the Naval Research Laboratory, Washint^ton, D. C. 8-3] AUTOMATIC FREQUENCY CONTROL 401 The effects of the spectral energy distribution of closed-loop angle noise and the contributions of low-frequency amplitude noise modulation of tracking error caused by flight path input information are discussed in Paragraph 8-17. 8-3 AUTOMATIC FREQUENCY CONTROL Automatic frequency control circuits are employed as a means of over- coming tuning tolerance and stability problems. The operating frequency of the receiver is compared to a reference. An error signal, related to the difference between the operating frequency and the reference, is generated. The error signal is then applied to the system in such a manner as to reduce the difference to an acceptable value. The general problem of automatic frequency control may be visualized as follows (see Fig. 8-4). Fig. 8-4 Automatic Frequency Control in a Pulsed Radar System. In the case of radar employing a pulsed oscillator as the transmitter, it is required that the receiver be tuned to the transmitter frequency. As discussed in Chapter 7, this is done by mixing the incoming signal with a local oscillator signal. The resulting intermediate-frequency (IF) output then is amplified by bandpass amplifiers designed to operate at a fixed intermediate frequency. With such an arrangement, the receiver tuning depends upon the ability of the local oscillator to follow variations in the transmitted frequency and thereby maintain the difference frequency (IF) at the value for which the bandpass amplifiers were designed. The auto- matic system employed to accomplish the desired regulation of the IF is called an automatic frequency control (AFC). 402 REGULATORY CIRCUITS Automatic frequency control is accomplished by applying the generated difference frequency to an error detector whose reference is the desired IF frequency. Such an error detector is a frequency discriminator. The frequency discriminator provides an output whose magnitude is propor- tional to the error and whose polarity indicates whether the IF frequency is above or below the reference. Since a variation in either the transmitter frequency or the local oscillator frequency produces an error in the IF frequency, these variations can be suppressed by suitable control of the local oscillator frequency if they do not exceed the bandwidth limitation imposed on the feedback control loop by the pulsed data. 8-4 VARIATION OF TRANSMITTER FREQUENCY WITH ENVIRONMENTAL CONDITIONS There are two types of frequency instability which result from the environment in which the transmitter must operate. There are relatively long-term frequency changes which occur incident to the effects of tem- perature, vibration, deterioration, and the like; there are also short-term frequency changes which are the result of a time-varying load impedance connected to the transmitter, and frequency modulation from the heater- supply and power-supply noise. Since the reference in an AFC for a conventional radar set is compared with the difference between the transmitter frequency and the local oscil- lator frequency, corresponding variations in the local oscillator frequency occurring at the same time as the transmitter frequency variations are also important. Fig. 8-5 shows typical frequency variation of a magnetron and a klystron with ambient temperature. Some static and slow frequency differences for typical magnetrons and local oscillators are listed in Table 8-1. Table 8-1 FREQUENCY DEVIATIONS OF TYPICAL MAGNETRONS AND KLYSTRON LOCAL OSCILLATORS Maximum Diference Environmental Factor Frequency {Mc) Scatter of magnetron and oscillator frequencies as received from manufacturer =^ 50 Warmup of radar set =^ 1-5 Temperature =^ 15 Pressure (0 to 50,000 h) altitude - 2.5 Pushing ( ='=10% line-voltage variation) 5.0 Aging; ^ 10 MAGNETRON PULLING 403 20 -10 -20 -30 1 1 Temperature-Frequency Characteristic of 4J50 IVIagnetron and \/-270 Klystron ^Magn 3tron Magne Puis PRF tron 1 3 Length 0.5 juse 1200/se snt 25 Amp )n 5| nator Voltage 300 c ~^ <. ^ Klystrc Mod Reso " ^ ^ . V -270 ^ V / -50 -25 25 50 75 100 TEMPERATURE (°C) 125 Fig. 8-5 Frequency Stability of a Magnetron and Klystron vs. Ambient Temper- ature. The response time of the AFC does not have to be very great to correct for the frequency changes listed in Table 8-1. It is only required that; the controlled frequency can be adjusted over the range. To obtain a wide tuning range, control of both the klystron cavity resonator and the reflector potential may be employed. In many cases only reflector control is required if periodic adjustment to accommodate frequency scatter caused by tube replacement and aging is allowed. Table 8-1 indicates that when a radar set is first energized it is usual for the open-loop frequency error to be rather large. A wide pull in range is therefore required. {Pull-in is the process whereby the error in receiver tuning frequency existing at the instant of an off-frequency input signal is reduced by the AFC operation.) 8-5 MAGNETRON PULLING The single mode equivalent circut of a magnetron is shown in Fig. 8-6. The magnetron is considered as a conventional self-excited power oscillator with the L-C tank circuit inductively coupled to the output transmission line. As will be discussed in Paragraph 11-1, loading mismatch can affect both the frequency and power output of the magnetron. Transient variations of the load admittance occur in scanning antenna-radome configurations. In a conical scanning radar, load admittance variations occur with feedhorn 404 REGULATORY CIRCUITS Ideal Transformer Electronic Admittance R S C_L L Load Admittance Output Voltage Fig. 8-6 Equivalent Circuit of a Magnetron. rotation because of imperfect rotary joints. The nature of tliese transient variations governs the time-response requirements for the AFC. The Rieke diagram is a fundamental performance characteristic of the magnetron which describes the dependence of oscillator power output and frequency on the load. A typical Rieke diagram is shown in Fig. 8-7. Fig. 8-7 Possible Operating Condition of a 4J50 Magnetron in a Typical Airborne Radar Set (Rieke Diagram). 8-6] STATIC AND DYNAMIC ACCURACY REQUIREMENTS 405 Although the Rieke diagram specifies the frequency and power output for any load, the pulling figure of the magnetron is defined by the total fre- quency variation resulting from a load which produces a VSWR of 1.5 when it is changed through a phase of 360°. The load corresponding to this condition is shown as a circle in Fig. 8-7. The total frequency variation caused by pulling might therefore be 13 Mc. A typical conical scanning antenna produces relatively small phase variation. The measured phase variation of a typical load is plotted as sectors A, B, and C in Fig. 8-7. The most unfavorable position for the phase is at Sector A, for which the frequency may be pulled a total of approximately 4 Mc. Pulling can also result from a wide-angle scanning antenna looking through discontinuities in the radome as well as the discontinuities in phase of a rotating feedhorn in a conical scanning radar. Sector C is most favorable for elimination of transient frequency pulling caused by phase changes; however, power output variations are relatively large. The transmitted signal will be amplitude-modulated by this effect, and the resultant amplitude mod- ulation on the received signals introduces errors in antenna pointing. Accordingly, Sector B represents the most favorable alternative from the standpoint of low-frequency pulling and minimum amplitude modulation of the transmitter. It will be observed, however, that these advantages are purchased at the price of lower-than-rated power output. In a well-designed antenna-radome combination, rapid phase changes with the position of the antenna are not usually severe. In a conical scanning radar the greatest pulling effect results from the rotation of the feedhorn. The phase may change quite rapidly with feedhorn position, and the frequency of pulling is therefore high. A typical system has been observed to generate two phase rotations in one revolution of the feedhorn with some abrupt changes. The frequency variation is thus predominantly at frequencies greater than twice the lobing frequency of the antenna. Table 8-2 gives some typical pulling characteristics of a conical scanning system. Table 8-2 PULLING CHARACTERISTICS OF A TYPICAL CONICAL SCANNING SYSTEM Frequency of FM Peak Deviation {Mc) f\ (lobina frequency) 0.5 2/i ^ 1.7 3/i 0.25 4/i 0.25 8-6 STATIC AND DYNAMIC ACCURACY REQUIREMENTS Tuning errors in the radar receiver degrade the output signal-to-noise ratio of the radar. Maximum range performance of the radar is thus a 406 REGULATORY CIRCUITS function of the tuning accuracy of the receiver. To obtain the maximum performance from a radar system the IF bandwidth must be matched to both the received pulse and the tuning error of the receiver. Fig. 8-8 shows the loss in signal-to-noise ratio (sensitivity) as a function of a static tuning error in the receiver. The video bandwidth is considered to be very much larger than the IF bandwidth. s 1 o fe 2 UJ Q- ^ 3 CO 5 o =v ^Br=2.0 Br=3.0 \- ~^\ /_^ -Br=4.0 \ \ \"^ \ Br^l.O" \ ^ \ \ \ \ \ \ \ \ V \ 0.5 1.0 1.5 2.0 AfT (TUNING ERROR x PULSE LENGTH) Fig. 8-8 Loss in Sensitivity with Tuning Error. After R. P. Scott, Proc. IRE (Feb. 1948) p. 185. To obtain maximum system performance a compromise between AFC performance and bandwidth must be made to provide maximum SjN. Maximum performance is obtained with Bt ==1.0 when the tuning error is negligible. In a MOPA (master oscillator power amplifier) system it is feasible to realize this condition. In the case of a separate-pulsed trans- mitting oscillator and a continuous-wave klystron local oscillator, some allowance for the static error must be made. The reference is a frequency discriminator and associated envelope rectifiers. The reference must be tuned to the center frequency of the receiver IF and must be stable within environmental conditions encountered. With an IF frequency of 30 Mc the reference can be made to be accurate within ±50 kc of the IF. When a master oscillator, operating continuously, is employed in the radar to obtain the transmitted signal, much greater accuracy can be obtained by the use of crystal control of the IF frequency. The static accuracy that can be realized in the usual radar employing two oscillators is dependent on the accuracy of the reference and the amount of zero frequency gain it is practical to employ in the AFC feedback loop. The dynamic error characteristic must also be considered in choosing the bandwidth. Since the AFC of a pulsed radar set is a sampled-data feedback control device, the error reduction that can be achieved is dependent on the sampling rate. In practice, the effective bandwidth of a continuous propor- tional error AFC is limited to about one-twentieth the PRF. Tuning errors CONTINUOUS-CORRECTION AFC 407 caused by variations in frequency greater than this effective bandwidth are not significantly reduced by the AFC, and their effect on the signal energy at the output of the receiver must be considered. The bandwidth of the receiver IF is then selected so as to maximize the output S /N in the presence of the error resulting from those tuning errors which cannot be removed by the AFC. In some applications, transmitter pulling is often the determining factor in the accuracy of receiver tuning. In these cases it is not essential that the static error be extremely small, and a simplification of the AFC can be realized by the use of limit-activated correction (see Paragraph 8-8). In tracking applications a further requirement exists that tuning error in the receiver must not produce an error in tracking exceeding a specified amount. Two types of errors arise from the tuning error. The frequency error is converted to amplitude modulation by the IF characteristic. The additional amplitude modulation from this source produces errors in the direction signal of a conical scanning radar. Distortion of the pulse shape also occurs and may produce errors in the measurement of range. 8-7 CONTINUOUS-CORRECTION AFC Continuous-correction AFC constitutes a type of closed-loop operation in which the error continuously tends to be minimized. The residual error is a function of the loop gain. A block diagram of a continuous AFC is shown in Fig. 8-9. The input to the AFC is the frequency difference between the transmitting oscillator and Transmitting W(f) W(f)-f2 Oscillator " Balanced Mixer AFC IF Amplifier Frequency Discriminator / Receiver Local Oscillator h C Filter G(S) e Fig. 8-9 Continuous AFC. the receiver local oscillator. This frequency is measured by the frequency discriminator which is the error detector. A voltage e proportional to the difference between the input frequency and the crossover frequency of the discriminator is applied to the local oscillator through a filter G(s). The output of the filter is a control C which adjusts the receiver local oscillator to minimize e. C may be a mechanical or an electrical output or a com- 408 REGULATORY CIRCUITS bination of the two, depending on the nature of the control mechanism of the oscillator. The characteristics of an AFC in stabilizing the receiver tuning when the transmitter frequency or local-oscillator frequency changes can be expressed as £/. £/rf (8-1) 1 + KG(s) where fei is the frequency error in the receiver IF frequency fdi is the frequency error that would result without the AFC K is the d-c or zero frequency gain of the AFC. (K = discrimi- nator sensitivity X d-c gain of the filter X modulation sensi- tivity of the oscillator) G{s) is the normalized transfer function of the filter. Fig. 8-10 shows the control characteristics of a typical klystron oscillator. Referring to Table 8-1, the largest tuning error that might exist in a typical 5.0 ^ 4.0 ^ 3.0 > m 2.0 z LlJ ^ 1.0 / y -N \ Reflector Sensitivity- / \ XT \ —Power Output / A \ \ / \\ / / \ \ ^ .^^ See ext _ -Ih- -30 -20 -10 10 20 30 CHANGE IN FREQUENCY (Mc) Fig. 8-10 Local Oscillator Characteri.stics. system is ±15 Mc incident to temperature environment, provided that an initial adjustment is made on the AFC whenever a tube is changed. Fig. 8-10 shows that the power output of the local oscillator will vary about 1.5 db and the modulation sensitivity will change by a factor of about 2 (or 6 db) for such a frequency variation. Variations in the power output of the oscillator affect both receiver sensitivity and signal-handling capability. However, the effect of a 1.5-db change is negligible. In cases where such a change cannot be tolerated or in which the frequency variation between the uncontrolled oscillators exceeds ±15 Mc, an additional feedback loop is sometimes employed. This auxiliary loop measures the oscillator power output and adjusts the klystron cavity for maximum output. 8-7] CONTINUOUS-CORRECTION AFC 409 Variation in modulation sensitivity limits the bandwidth that can be employed in the AFC loop. The loop must be designed to have adequate stability with both the highest and lowest gain values. To minimize variations in oscillator sensitivity with tuning, the amplitude of the local oscillator signal is usually made smaller than the amplitude of the sample of the transmitter frequency at the AFC mixer. The IF voltage applied to the discriminator is then proportional to the amplitude of the local oscillator output. The discriminator sensitivity therefore is reduced as the oscillator modulation sensitivity increases. Since loop gain involves the product of the discriminator sensitivity and the oscillator modulation sensitivity, the variation in loop gain that would exist if the IF voltage were maintained constant is reduced. This is shown as an effective modulation sensitivity reduction by the broken curve of Fig. 8-10. Pull-in and hold-in performance of the AFC are determined by super- imposing the oscillator control curves on the discriminator characteristic. Fig. 8-11 shows such a curve. Pull-in has been defined previously. Hold-in Oscillator Control Curves Oscillator Control Curves Fig. 8-11 Discriminator Characteristics. is the maximum frequency interval over which AFC control is effective. Referring to Fig. 8-11, an error /^i in receiver tuning will occur for a frequency deviation Ja\ of the transmitter frequency. The frequency deviation /d2 corresponds to the hold-in range. An error /« 2 results from an oscillator deviation/d2. The pull-in range corresponds tofdz- The tuning error can be/^s or/es'; only the point /e3 is stable, however. For all devia- tions less than/d3 the tuning error is a stable condition corresponding to the intersection of the discriminator curve and the oscillator control curve. 410 REGULATORY CIRCUITS The discriminator curve of Fig. 8-1 1 is the characteristic appearing at the output of G(s) in Fig. 8-9. For high-performance systems G(s) is designed with a large zero frequency or d-c gain. Fig. 8-12 shows the resultant discrim- .Oscillator Control Characteristic Typical Characteristic Where D-C Amplifier Follows Discriminator Fig. 8-12 Discriminator Characteristics. inator curve measured at the oscillator. By increasing the d-c gain for a given IF characteristic the pull-in range is increased. In a number of systems, however, a limited d-c gain follows the IF detectors. To realize a large pull-in range a frequency searcli sweep is applied to the oscillator. The presence of an IF signal in the AFCIF is employed to remove the sweep. A relatively narrow-band IF discriminator can exhibit a large pull-in range by this technique. In a typical system the control required on the oscillator may be ±50 volts, but this voltage is usually at some bias level, e.g. —150 volts. To obtain maximum performance from a given loop, a d-c voltage is added to the output of the IF discriminator so that the control voltage is at —150 considerable energy in modulation sidebands at the IF frequency. The AFC mixer of a pulse radar set is operated as a balanced mixer to minimize frequency tuning error caused by discriminator outputs resulting from the modulation spectrum of the transmitted signal. With the usual IF frequencies employed, narrow pulse-length transmitted signals have considerable energy in modulation sidebands at the IF frequency. A typical IF discriminator design can provide an output of 2 to 3 volts per megacycle with a peak-to-peak separation of the discriminator of 4 to 5 Mc. The output from the discriminator is in the form of video pulses. If these pulses are fed directly into the filter the zero frequency gain required from the filter is K/eT (8-2) 8-7] CONTINUOUS-CORRECTION AFC 411 where Eo is the maximum output voltage required T is the interpulse period fe is the static error due to finite gain of the loop r is the pulse length K is the discriminator gain. For a static error of 50 kc, a pulse length of 1.0 Msec, and an interpulse period of 1000 Msec a typical gain required is 250,000. The required gain can be reduced by the use of a pulse lengthener following the frequency discrimi- nator. With a circuit like that described in Paragraph 7-17 the filter gain is reduced to about 250 \i T = Tp and a = 0. A gain of 250 can be provided with an operational amplifier. This d-c gain can be reduced if the discrimi- nator is designed with higher output or if a video amplifier is employed between the pulse lengthener and the discriminator. Although the loop can provide a static accuracy of 50 kc, the tuning accuracy also includes the inherent accuracy of the reference. With thermionic diodes in the discriminator circuit and with stable capacitances and inductors a reference accuracy on the order of 50 kc can be achieved for the assumed discrimi- nator characteristic. The static accuracy is then expected to be 70 kc, provided that means for adjusting the reference initially to the IF frequency of the receiver exist. The dynamic accuracy will depend on hold-in and pull-in requirements. A single lag network is usually employed; i.e., the operational amplifier is made an integrator. An error only appears as a signal sampled at the pulse repetition frequency; therefore the integrator time constant that can be employed depends on the allowable overshoot. At a given time /i the error in frequency might be /i. This error is applied as a voltage to the integrator. The integrator output will change at a rate determined by the RC and the input voltage corresponding to/i. An overshoot of about 50 per cent of the initial error is a reasonable compromise to obtain good dynamic response. The introduction of a step frequency error then results in an output frequency which is 50 per cent of the initial error but of opposite sign. The output of the AFC thus oscillates about the desired frequency with diminishing error. Inputs incident to pulling of the transmitter may occur at the lobing frequency or multiples thereof. The error reduction that can be accomplished by the AFC thus depends on the ratio of the PRF to the pulling frequency. In typical cases there might be 10 to 20 samples during a cycle of the pulling frequency, and error reductions on the order of 10 to 1 are attained when 50 per cent overshoot is allowed. 412 REGULATORY CIRCUITS 8-8 LIMIT-ACTIVATED AFC Limit-activated AFC constitutes a type which has a switching action whenever the error exceeds a predetermined value. The static error in such a system is never zero, but oscillates about the correct value with a constant peak error. The static error is determined by the limit level for the error, the interpulse period, and the integrator characteristic. Essentially the system is the same as that shown in Fig. 8-9 with G(s) = K /s. The difference between this AFC and the continuous-correction AFC is that the input to G(s) does not produce a change in polarity of the oscil- lator control signal until the frequency error exceeds a certain limit L. The output from the frequency discriminator is applied to an integrator whose output controls the oscillator. A frequency error/ei results in a change of the oscillator frequency at a rate Kfei. An error correction signal can only be obtained in a shortest time T after a signal occurs which causes the oscillator frequency to change. The quantity T is the interpulse period. The output frequency therefore oscillates at a frequency which is some fractional integer of the pulse repetition frequency. Fig. 8-13 shows the -1 5.0 2 4.0 2 3.0 in I 20 1.0 -1^ Y ^ i\ \ ^ Boundary Lines Represent \V\ ^>^ Stable Oscillations. -l2 - <i '^ iim SNX;^^''''^ Shaded Areas are ^^ Areas of Instability O C ui Ll \ / 'k A i«r Peak Freq.Error = fe y w ^■^- ^^1^ 1.0 2.0 3.0 4.0 5.0 E.O.F.= Error Oscillation Frequency Fig. 8-13 Characteristics of a Switched (Limit-Activated) Type Control Loop for Constant Input Conditions. relationship between the parameters A', T, and L. It will be observed that multiple modes of oscillation can occur. These result from the fact that the data sample may occur when the frequency error is less than L. The limit level in a practical system is about 50 kc minimum. With the continuous AFC employing a 100 per cent pulse lengthener the static accuracy is independent of the PRF, provided that the response time of the discriminator is the same for all pulse lengths. (Normally a pulse- length change would be associated with a PRF change.) In the limit- activated correction type of AFC, the static accuracy increases as the PRF increases. 8-9] THE INFLUENCES OF LOCAL OSCILLATOR CHARACTERISTICS 413 The dynamic performance of this type of AFC is determined by K and the peak rate of frequency change occurring in the input signal. When the peak rate of the input frequency change is less than K, an error reduction is obtained and the peak errors are of the same order of magnitude as the static error. When the peak rate of the input exceeds K, the error is not reduced. When the input to the AFC is predominantly a single frequency, the required value of K may be determined by K = o^f ■ (8-3) where co is the angular frequency at which the signal frequency is changing, and / is the peak deviation of the signal frequency. 8-9 THE INFLUENCES OF LOCAL OSCILLATOR CHARACTERISTICS A typical electronic tuning characteristic of the local oscillator is shown in Fig. 8-10. Note that both the power output and the frequency-modulation sensitivity vary with the tuning. A variation in modulation sensitivity means that there will be a variation in the loop gain of the AFC. In the case of a continuous AFC the change in power can be employed to compensate (to some extent) for the variation in modulation sensitivity. However, perfect compensation is not attained. As a result, if a continuous AFC is designed with only electronic tuning capability and it must accommodate dynamic inputs, there is a degradation of the static and dynamic error characteristics at the extremes of the tuning range if the overshoot is selected to be 50 per cent at the middle of the tuning range. As shown in Fig. 8-10, making the local oscillator signal smaller than the transmitter signal reduces the variation in modulation sensitivity, but the variation is still more than 2 to 1 over the tuning range. The static error therefore oscillates when the receiver is tuned at the extremes of its tuning range. A smaller overshoot and somewhat poorer dynamic performance must be accepted if the tuning range is required to exceed 30 Mc >vith a fixed static tuning accuracy. In the limit-activated correction AFC there is no benefit from making the local oscillator signal smaller than the transmitter signal since the rate of correction is independent of the magnitude of the error. (A constant rate of control occurs whenever the error is greater than L.) The static error oscillates at all times and will vary over the tuning range of the receiver in accordance with Fig. 8-13 when the change in modulation sensitivity is inserted into the value for K. When the system is subjected to severe dynamic error requirements, a double loop is sometimes employed using a low-frequency feedback to the resonator of the klystron and a high-frequency feedback to the reflector. 414 REGULATORY CIRCUITS A smaller variation in electronic modulation sensitivity is thereby obtained, allowing a design of 50 per cent or greater overshoot in the step response of the electronic tuning loop and also providing a much greater receiver tuning range than can be obtained with electronic tuning alone. 8-10 RELATION TO RECEIVER IF CHARACTERISTICS The dynamic error in the receiver tuning and the received signal IF characteristic are related. The signal at the output of the IF amplifier will contain pulse amplitude modulation arising from the tuning error of the receiver. When the IF response is perfectly symmetrical about the center frequency and the static error is negligibly small, the modulation at the output will be double the frequency of the frequency modulation of the tuning error. In conical scanning radars the dominant output incident to such effects is thus at two or four times the lobing frequency. The response of the angle demodulators to these frequencies is greatly attenuated by the use of balanced ring demodulators, and additional noise on the direction signal caused by AFC characteristics is then negligible provided that saturation is not present. If the modulation is large there is of course a loss in signal- to-noise ratio which can be determined from the rms error and Fig. 8-8. To minimize the conversion of the tuning error to amplitude modulation the nose of the receiver selectivity is made as flat as possible consistent with the considerations discussed in Chapter 7. 8-11 DISCRIMINATOR DESIGN There are two types of discriminators employed in pulse AFC — the phase discriminator and the stagger-tuned discriminator. The choice between the two depends on the details of the control circuitry. A slightly higher effective transfer impedance can be realized with the stagger-tuned circuit, but symmetry is difficult to maintain. If a video amplifier is employed after the discriminator but prior to the integration, then the phase discriminator is the more attractive choice. Fig. 8-14 shows a typical phase discriminator circuit, and the form of the transfer impedance. In designing the discriminator the network elements can be selected so that j-j = ^5, J-2 = -^7, s ^ = jg, and ^-4 = ^7. The discriminator response is then of the same form as the difference in the envelope response of two stagger-tuned one-pole networks. To obtain the maximum sensitivity from the discriminator the poles are located so that the two equivalent response curves cross at their point of inflection. H is inversely proportional to Ci and C2 and these quaatities are minimized to obtain maximum Z,. ■12] INSTANTANEOUS AFC iC 415 z,= l^o(.)|-|gb(s)| i(s) S(S-S^)(S-Sl)(S-S2) \(S-S^)(S-S;)(S-Se)(S-sl)(S-Sj)\ s(s-s)(s-s;)(s-s, (S-S^)(S-S;)(S-Se)(S-S;)(S-Sy) Fig. 8-14 Frequency Discriminator (Phase Type). 8-12 INSTANTANEOUS AFC When the dynamic inputs are so severe that continuous AFC cannot satisfactorily reduce the tuning error, the possibility exists of using an Instantaneous AFC (I AFC). I AFC is a type in which the error correction is completed before the pulse has ended. Extremely wide bandwidths are required in the AFC-IF amplifier and discriminator in order that negligible time delay may be obtained in these elements. The \FC must include a bidirectional pulse lengthener which is required to have a negligible decrement; the output of the lengthener is the controlled value during the pulse. This type of AFC can potentially provide the best dynamic perform- ance in a sampling-type AFC; however, there are some practical limitations. The discriminator measures the instantaneous frequency and there is negligible lag in the loop, so that if the instantaneous frequency is constant during the pulse the tuning error can be reduced to a value dependent on the gain of the loop. In most cases there are, however, intrapulse frequency changes. In the continuous AFC the average frequency is controlled; in the lAFC the controlled frequency depends on the characteristics of the discriminator and pulse lengthener. The controlled frequency is different from the average when intrapulse frequency variations are large. As a result the static error of an lAFC can be larger than that of a continuous AFC. The wide bandwidth required in the discriminator limits its output 416 REGULATORY CIRCUITS SO that the reference accuracy is poorer than in a continuous AFC. Another lAFC problem is associated with holding the voltage precisely during the interpulse period. The lAFC circuit is not commonly employed because of its limitations and because the dynamic inputs can usually be reduced by proper design of the transmitter and associated circuits so that the continuous AFC is adequate. 8-13 PROBLEMS OF FREQUENCY SEARCH AND ACQUISITION Table 8-1 indicates that in a typical case the transmitter frequency can vary over a greater range than can the electronic tuning of a klystron as shown by Fig. 8-10. It has been noted that it is also not always feasible to utilize a discriminator-IF characteristic which will provide such a wide pull-in range. To cope with this situation it is necessary that a mode of operation be provided which will allow the AFC to search for the trans- mitter frequency, acquire it, and track it. With magnetrons which are tunable it is necessary that two loops be provided for the AFC. A slow- response loop which controls the cavity resonator by thermal or mechanical means is sometimes employed. With fixed-frequency magnetrons, however, the electronic tuning range is usually adequate. In these cases, periodic adjustments can be made to the klystron resonator to accommodate aging or replacement of magnetrons. The slow frequency variations that are then encountered are usually well within the electronic tuning capability of the local oscillator. It is sometimes more economical to provide frequency capture within the electronic tuning range by means of a search sweep of the local oscillator than to provide an IF discriminator which has a pull-in range equal to the maximum frequency difference between the transmitter and local oscillator at the time that the radar set may be energized. A typical frequency range over which the local oscillator must search for the transmitter frequency is 40 Mc. The speed at which this search can occur depends on the bandwidth of the IF discriminator, the interpulse period, and the total search range. A typical discriminator might have a pull-in range of 10 Mc and 10 pulses required for acquisition. The maximum search speed is then equal to 1 jT Mc, where T is the interpulse period. Circuits are provided so that the search sweep signal is automatically removed when the transmitter frequency has been captured. 8-14 AUTOMATIC GAIN CONTROL Radar targets act to modulate the amplitude of the reflected signals in several ways. First of all, range variations can produce variations in the received power of more than 100 db. Secondly, amplitude fluctuations M4] AUTOMATIC GAIN CONTROL 417 caused by target motion which were discussed in Paragraph 8-2 can also be large. These variations in signal strength can seriously interfere with tracking of the target unless steps are taken to protect the receiving system from their effects. This is particularly important in angle tracking where the signal amplitudes in two offset antenna lobes are compared (either sequentially or simultaneously) to generate an angle error. The angle error will normally be directly proportional to differences in the received am- plitudes in the two lobes, so that signal strength variations common to the two lobes must be removed if a usable error signal is to be obtained. Regulation of the received signal level is normally accomplished by an automatic gain control (AGC) circuit. This is a feedback loop which adjusts the receiver gain to maintain the average receiver output at a constant reference value. A simplified block diagram of an AGC loop is shown in Fig. 8-15. In operation, the gain of the IF amplifier presented to its input IF Input First Detector e. F -^ Second Detector — Video Amplifier Amplifier r / AGC e Bias 1 AGC Filter ■^+, L nola\ Regulated Video Out Delay Voltage (Reference) Fig. 8-15 Radar Receiver with AGC Loop. voltage Ci is automatically adjusted by an AGC bias Cg. This bias is developed as the difference of the video output voltage Co and a reference voltage Cd referred to as the AGC delay. The system basically acts to maintain the output equal to the delay. The degree to which this is done when the input fluctuates is determined by the AGC filter in the feedback path. An AGC loop is a nonlinear servo in that the feedback signal Cg is not linearly combined with the input, but acts to modify the gain with which the input is amplified. In a sense, an AGC loop in combination with an angle tracking loop is an example of an adaptive servo system in that the system gain is automatically adjusted to compensate for externally generated variations in the received signal strength. Although basically nonlinear, an approximate linear analysis of the system operation for small deviations of the input from an average operating level is very useful and forms the basis for system design. 418 REGULATORY CIRCUITS In designing an AGC loop, particular attention should be paid to the following four areas of performance which are of primary importance: 1. The steady-state or d-c regulation of the video voltage 2. Attenuation of amplitude fluctuations 3. Fidelity with which intelligence is transmitted 4. AGC loop stability The steady-state regulation determines the degree to which the AGC loop compensates for slow variations in the average signal level caused by changes in target aspect and range. As noted above, such variations can be as great as 100 db. The AGC loop is often required to reduce slow variations of the output level to only a few decibels. For instance, in an angle tracking loop, the loop gain is proportional to the output signal level so that varia- tions in this output produce corresponding variations in loop gain. When the average output varies by 2 to 1, or 6 db, the angle track loop gain will also vary by the same factor, and this may have a serious effect on the overall angle track loop stability and performance. Fluctuations in the strength of radar echoes reflected from aircraft targets have been discussed in Paragraph 4-8, and typical spectra of this amplitude noise for two types of aircraft are shown in Figs. 4-23 and 4-24. In Fig. 4-23 the amplitude noise spectrum for a propeller-driven aircraft illuminated by X-band radiation is shown. Very predominant propeller modulation peaks at harmonics of about 60 cps persist to over 300 cps. In Fig. 4-24, the amplitude noise spectra generated by a B-45 jet bomber illuminated by several wavelengths are shown. With no propeller modu- lation, the spectra all fall off within a few cps. In general, it is desirable for the AGC loop to remove amplitude noise whose frequency components fall within the pass band of the angle tracking loop. Otherwise, modulation of an angular lag error by amplitude fluctuations in the receiver output can produce excessive angle noise. Besides removing noise modulation from the receiver signal, the AGC loop must also transmit intelligence modulatibn without appreciable distortion. This is a critical problem in systems which employ sequential or conical scan lobing to generate angle error signals. For instance, in a conically scanned system, the angle error is contained in the amplitude and phase of a sinusoidal error signal at the scan frequency which may vary because of poor scan rate generator regulation. Generally, the AGC loop must be able to transmit this signal with negligible phase shift or change in amplitude. Since the AGC circuit is a feedback loop, stability questions are im- portant and servomechanism design techniques are applicable. These techniques are applied to a linear small-signal approximation to the nonlinear loop which will be derived in the following paragraph. Adequate 8-15] LINEAR ANALYSIS OF AGC LOOPS 419 stability and dynamic response of the AGC loop is often difficult to achieve in combination with other requirements on d-c regulation (proportional to AGC loop gain) and fidelity of intelligence modulation. For pulsed radars where fast AGC action is desired (common in monopulse systems), methods for analyzmg sampled-data servos must be used and the AGC loop band- width is limited to about half the repetition rate by stability considerations. 8-15 LINEAR ANALYSIS OF AGC LOOPS Design of AGC loops is based upon a first order or linear approximation to the nonlinear action of the IF amplifier for small deviations from average operating points.^ This approximation is illustrated in Fig. 8-1 6a. The Output Delay Input Gif egj Video Amplifier Ih Amplifier Bias eg G2(s) AGC Filter ^ r !. + " L ■^' (a) . ^ AGC Filter Ki=lncremental IF Gain Constant K2=lncremental AGC Loop Gain , e, = Constant (b) Fig. 8-16 Linear Approximation to AGC Loop. upper block diagram in this figure shows the essential components of an AGC loop. The nonlinear relation of the IF amplifier gain to the AGC bias is indicated by Gi(eg). The system equations have the following forms: eo = e,KsG,{e„) (8-4) e,= {ea- eo)G,{s). (8-5) ^B. M. Oliver, "Automatic Volume Control as a Feedback Problem," Proc. IRE, 36, 466-473 (1948). 420 REGULATORY CIRCUITS Small deviations in the output can be related to small deviations in d and eg in the following manner: oei deo (8-6) = KiAei + K2Aeg. The gain factors Ki and K2 represent the incremental gain of the IF amplifier to the input and the incremental AGC loop gain, respectively. Deviations of the bias will be simply related to output deviations through the AGC filter: Ae^ = -G2{s)Aeo. (8-7) The approximate linear feedback loop corresponding to Equations 8-6 and 8-7 is pictured in Fig. 8-1 6b. The output-input ratio for this linear regulating loop will have the following form : Aen 1 KiAei 1 + K2G2(sy (8-8) 8-16 STATIC REGULATION REQUIREMENTS OF AGC LOOPS The transfer function represented by Equation 8-8 gives the small-signal modulation transmission characteristics of the loop. If the zero frequency gain of the AGC filter is assumed unity [G2(0)- = 1], the static gain around the loop is K2 as is indicated in Fig. 8-1 6b. The static regulation performance is directly related to the loop gain K2. In order to display this relation, though, the gain control characteristic of the IF amplifier indi- cated in Equation 8-4 must be ex- amined in detail. Typically, the logarithm of the IF amplifier gain is approximately a linear function of the AGC bias voltage. That is, the slope of the gain-bias curve in decibels per volt is a constant. Fig. 8-17 shows a typical IF amplifier gain control Gain characteristic. Such a linear relation can be generally expressed in the following form: AGC BIAS VOLTAGE Fig. 8-17 Typical IF Amplifi Control Characteristics, 20 logio Gi = ^ + Beg (8-9) 8-16] STATIC REGULATION REQUIREMENTS OF AGC LOOPS 421 The constant B in this equation gives the slope of the gain control charac- teristic in decibels per AGC bias volt. Differentiating with respect to eg-. (201ogio.)i^' = B Gi a eg (8-10) i^' =0.1155. Gi deg Multiplying and dividing the LHS by eiK:i allows us to express it simply as the ratio of the loop gain K2 and the video output <?« by utilizing Equation 8-4. 1 ;^n 0.1155 (8-11) 0.1155. For the d-c or static case with G^iO) equal to unity, changes in the bias are directly proportional to changes in the output. Thus the slope B can be expressed as the ratio of the total change in gain to the change in output voltage : „ gain- change (db) gain change (db) 1 X CiK^G, Co ^O.max ^fl.min ^o.max ^o.min (8-12) Substituting Equation 8-12 into Equation 8-11 yields the following expres- sion for the AGC loop gain: K^ = — fMHf^ X [gain change (db)] (8-13) ^o.max ^o,min It is apparent from this expression that with the linear gain control charac- teristic shown in Fig. 8-17, the loop gain will vary somewhat with the video output eg. Normally, the video output will be well enough controlled that its variation can be neglected and an average value used in Equation 8-13. It is possible to compensate for this variation in the loop gain K2 by introducing a slight curvature in the gain control characteristic. Generally, though, uncontrolled departures from linearity with accompanying uncon- trolled variations in the loop gain are a much more important design factor to consider. To illustrate the use of Equation 8-13, suppose that static input varia- tions of 100 db must be regulated by the AGC loop to output variations of only ±1 db or between 0.89^<, and 1.122 eo. Substituting these numbers into Equation 8-13 yields Mmm= 49.5 = 33.4 db. (8-14) '^' 1.122 - 0.89 422 REGULATORY CIRCUITS Thus, with an AGC loop gain of about 50 or 34 db, input variations of 100 db can be reduced to output variations of only ±1 db. 8-17 DYNAMIC REGULATION REQUIREMENTS OF AGC LOOPS It was previously noted that amplitude noise fluctuations in the receiver output will modulate steady-state lag errors in the angle tracking loop output and can thus produce excessive angle tracking noise. For this reason and also to minimize the possibility of saturation, the AGC loop should be designed to remove most of the input amplitude fluctuations, particularly those within the pass band of the angle tracking loop. Actually, if there were no systematic errors, some slight improvement in the glint noise or deviations in angle of arrival could be achieved with no AGC. The reason for this is that there is a correlation between large deviations of the apparent center of reflection of an aircraft target and deep amplitude fades, since both effects are produced by destructive interference of the reflected signals. An effective AGC will increase the receiver gain to compensate for fades and thus increase the magnitude of the glint deviations. In a practical case, this effect is more than balanced by the benefits of removing spurious modulation from the error signal. Typical results from a simulator study of this problem are shown in Fig. 8-18.^" In this case, the target noise spectrum (amplitude and angle) had a width of 1 cps while the tracking servo had a similar bandwidth. The §;'■ 1.6 1.2 0.8 0.4 Fig. 3L 2L L L 2L LAG ERROR (Units of Target Span,/.) ■18 Effect of AGC on Angle Tracking Noise as a Function of Servo Lag Error. *"J. H. Dunn and D. D. Howard, "The Effects of Automatic Gain Control Performance on the Tracking Accuracy of Monopulse Radar Systems," Proc. IRE 47, 430-435 (1959). 8-18] AGC TRANSFER CHARACTERISTIC DESIGN CONSIDERATIONS 423 mean square tracking error is plotted versus the lag error for no AGC, a 1-cps AGC, and a relatively fast AGC of 12 cps. With small lag errors, less noise is produced with no AGC because of the correlation between amplitude and glint noise noted above. For typical tracking conditions, though, in which appreciable lag errors exist, the tracking noise and no or slow AGC greatly exceeds that associated with fast-AGC designs. Other factors to be considered in the design of the low-frequency response of the AGC loop are the transient recovery time of the receiver from deep fades (as great as 60 db) which should be such that the angle error is not blanked for longer than the angle tracking loop response time. This can be achieved by providing a high enough velocity constant for the AGC loop and allowing sufficient dynamic range in the output. The amplitude noise spectrum from most aircraft targets falls off with frequency approximately as if it were filtered by a single section, low pass, RC filter (see Paragraph 4-8). In order that no particular noise frequencies be emphasized in the output, it is desirable, although not absolutely necessary, that the AGC filter match this spectrum; that is, it should fall off with frequency with a — 1 slope in the frequency region covering the angle tracking pass band. The significant factor in determining the quantitative effects of the AGC on received modulations is the transfer characteristic AGC transfer characteristic = :; — ; — j^ ^ , . - (8-15) 1 -(- KiKj2\S) The required behavior of this function and the open-loop characteristic KiGii^s) will be examined in more detail. 8-18 AGC TRANSFER CHARACTERISTIC DESIGN CONSIDERATIONS From the discussion in Paragraphs 8-14 through 8-17 of factors signifi- cant to the design of a radar receiver automatic gain control, several basic specifications emerge as AGC transfer function desiderata in sequential lobe comparison radars, namely: 1. High gain at low frequencies to provide adequate static regulation. Some system specifications contemplate allowing only ±1 db variation in the modulated envelope output for a range of input signal levels of approximately 100 db. 2. An initial transfer function slope of zero from dc to as high a frequency (approaching the angle tracking bandwidth) as possible. 3. Gain drop-off with a —1 slope on a db-versus-log frequency plot to ensure, in view of the established nature of radar noise, that the receiver output shows no noise emphasis at any particular frequency. 424 REGULATORY CIRCUITS 4. Adequate AGC bandwidth (fast AGC) to ensure isolation from amplitude fluctuation in the received echo and minimization of closed-loop angle tracking noise in the practical employment of the radar system (particularly including recognition of situations where angle tracking errors will exist). Estimates of the half-power frequency of amplitude noise for some target tracking problems are as high as co = 10 (a gain of +20 db or 1-^2^21 = 10 at CO = 20 may be taken as a practical design objective). 5. Adequate AGC bandwidth to ensure suitable transient response. This requirement is another aspect of system demands consistent with item (4) above. 6. The phase of the quantity [1 /(I + K2G2(s))] should not vary exces- sively over the range of angle tracking modulation frequencies surrounding the scan frequency to limit crosstalk effects between the azimuth and elevation angle tracking axes. Phase shifts of up to 5° or 10° can be allowed in most systems although some applications may require this phase shift to be maintained less than a few degrees. As will be shown in Paragraph 9-9, with phase shifts of more than 10°, the antenna has a tendency to spiral or circle, and with even larger shifts, it will become unstable. This phase shift must often be maintained in the face of uncontrolled variations in the incremental loop gain K2 of as much as 10 db and uncontrolled variations in scanning frequency of up to ±5 per cent. 7. In order to ensure adequate loop stability and transient response, a minimum gain margin of 6 db should be maintained for all possible varia- tions of the incremental loop gain. Similarly, a phase margin of from 40° to 50° should be maintained. 8. In pulsed systems, it is necessary to provide a minimum gain margin of at least 6 db at one-half the repetition rate to ensure stable operation. This is particularly important in monopulse systems where rapid AGC action is desired. 8-19 THE MODULATION TRANSMISSION REQUIREMENT When large fluctuations in the AGC loop gain are possible or very small scanning frequency phase shifts are required, special design techniques must be used to maintain the phase of the intelligence signal being transmitted through the system. Two such techniques are available. In the first, additional high-frequency lag and lead terms are incorporated into the AGC filter to provide an open-loop phase shift of 180° at and near the lobing frequency. The phase shift of the closed loop is then zero at the lobing frequency and insensitive to variations in loop gain and lobing rate. A second approach is to attenuate K2G2(s) with a null over the required frequency band so that the maximum closed-loop phase </>« will be limited to a small value regardless of the phase of KoGiis). -20] DESIGN OF AN AGC TRANSFER FUNCTION 425 Using the latter approach and referring to Fig. 8-19, tan (prr, KiG<i 1 + K,G, K,G,. (8-16) Fig. 8-19 The Vector [I + K^Giijo:)]. If, for example, it is required that 0^ be maintained less than 1.5°, then the gain at the lobing frequency- must comply with K2G2 < tan 1.5° = 0.0262 = -31.9 db. (8-17) If it is assumed that the lobing frequency is 50 cps, that its regulation is ±5 per cent or ±2.5 cps, and that the angle tracking loop bandwith is 1 cps, the open-loop attenuation should be at least 31.9 db between 46.5 cps = 292 rad/sec and 53.5 cps = 336 rad/sec in order to maintain the phase shift less than 1.5° over the anticipated range of operating conditions. 8-20 DESIGN OF AN AGC TRANSFER FUNCTION As a trial design, a single time constant RC filter is selected for the AGC filter. This will provide a — 1 slope, which was previously noted as desir- able. The maximum gain is chosen on the basis of the static regulation requirement. In the example of Paragraph 8-16, a static gain of 34 db was required to regulate input variations of 100 db to ±1 db in the output. This requirement is adopted as the loop gain in this example. In Paragraph 8-18, it was mentioned that a practical AGC loop design for a system tracking aircraft targets should have a loop gain of 20 db 5000 10 20 50 100 200 500 1000 2000 ANGULAR FREQUENCY (rad/sec) Fig. 8-20 Trial Design of AGC Open-Loop Transfer Function. 426 REGULATORY CIRCUITS at o) = 20 in order to provide sufficient attenuation of amplitude noise fluctuations. This requirement, in connection with the static gain require- ment, fixes the location of the low-frequency corner of the AGC filter at 4 rad/sec. The AGC loop transfer characteristic thus developed is shown in Fig. 8-20. In order to maintain the fidelity of transmitted modulation, a network will be introduced into the transfer function to provide a null at the lobing frequency as described in Paragraph 8-19. It will be supposed that the phase shift must be kept less than 1.5° and the lobing frequency, its regulation, and the angle tracking bandwidth have the values assumed in the example in Paragraph 8-18. Fig. 8-21 shows a parallel-T net- work which can be used to provide the required null.^' The transfer function of this network is mR — W\/ 1 c 1/ r C/m If m 11 ] < -R < < I J u ~ m + 1 ^ m Fig. 8-21 Parallel-T Null Network with Symmetry Pattern m. Voltage transfer function of network u'^ 1 H- + // (-^) + 1 (8-18) where u = jcoRC = jco /coc CO = angular frequency, rad/sec Wc = null location, rad/sec m = symmetry parameter determining null sharpness. The effect of this network with a null of 50 cps = 314 rad /sec and a value of m = 0.54 is shown in Fig. 8-20 in combination with the single time constant RC filter. The value of m (Fig. 8-21) was selected to provide a total attenuation of 31.9 db (as required in Paragraph 8-19) at 292 rad/sec, the lowest possible modulation frequency. The high-frequency gain margin should be checked, particularly at one-half the PRF. An inspection of Fig, 8-20 shows that a minimum gain margin of 20 db, in comparison with the 6-db requirement, is maintained at all high frequencies. The closed-loop response of the AGC loop indicates most directly the dynamic AGC action in attenuating low-frequency amplitude noise and transmitting modulation frequencies. Fig. 8-22 shows the closed-loop response corresponding to the trial design illustrated in Fig. 8-20. "C. F. White, Tmns/er Characteristics of a Bridged Paratlel-T Network, NRL Report R-3I67, 27 August 1947. -21] THE IF AMPLIFIER CONTROL CHARACTERISTIC 427 10 -10 -20 -30 -40 ^, / \ / / A i / \ y ^^Amplitude y-Phase :r^ j^ \ 5 10 20 50 100 200 ANGULAR FREQUENCY (rad/sec) 120 100 _ <o 80 H Q < 20 500 1000 Fig. 8-22 Closed-Loop Response of AGC, Trial Design. 8-21 THE IF AMPLIFIER CONTROL CHARACTERISTIC The AGC loop must maintain the receiver output constant so that the loop gain of the various tracking loops is negligibly affected by the large variations in input signal power that are encountered. The analysis of Paragraph 8-18 is based on maintaining the average value of the receiver output constant. In receivers which must recover modulation from a PAM signal to obtain an error signal, additional consideration must be given to the distortion of the IF signal as it passes through the amplifier. Ideally it is desirable that gain control be applied to IF tubes which exhibit square- law transfer characteristics. Under these conditions no distortion of the IF signal will be apparent at the demodulated output. Usually the radar receiver must incorporate tubes having good gain bandwidth products. Such tubes are invariably of the sharp cutoff variety and are likely to produce distortion of the IF signal with accompanying excessive variations in the AGC loop gain if proper precautions are not taken. To determine the actual distortion through an IF stage, an accurate description of the transfer characteristic is required. In general any characteristic may be expressed as a power series in ^g, the grid-cathode voltage. If the signal input to the IF stage is a modulated signal ei = A sin (xictiX + m cos Wmf) and if an AGC voltage E\ is applied to the number one grid of the tube, then a convenient measure of the distortion is the change in the effective modulation of the signal at the output of the stage. Thus 1 + 1^3^2(1 W) a, + £i(2^2 + ^azEi) + la^A^l + ^m'^) (8-19) where m' is the fractional modulation at the output and <2i, ^2, and a^ are the first three coefficients of the power series expansion for the transfer characteristic. The distortion of the signal is seen to be a function of the 428 REGULATORY CIRCUITS third derivative of the transconductance. In designing an amplifier it is not convenient to obtain an infinite series expressing the transfer characteristic. The published transconductance curves can, however, be employed to obtain a reasonable estimate of the distortion. The transconductance corresponding to an infinitesimal signal applied to the tube at a grid voltage El is determined, and then the transconductance at the positive and negative peak values of the IF signal is determined. From these three values the distortion is computed from m' _ 1 tn 2 1+^— + ^" 2g. (8-20) When yf and Ei are so large that the negative excursion of the signal extends beyond cutoff, Equation 8-20 is not sufficiently accurate. However, it serves as an estimate of the distortion provided that the tube is cut oflF. Sharp cutoff tubes do not always cut off at the voltages indicated by the tube characteristics. The tubes are only required to exhibit less than a specified maximum value of plate current at cutoff bias. As a result of inadequately controlled cutoff characteristics ^m.mrn does not go to zero for large Ei and J, and the distortion in an actual amplifier is sometimes observed to be much greater than estimated by Equation 8-20. In the design of a gain-controlled amplifier employing sharp cutoff tubes the AGC voltage applied to the stages is therefore restricted so that the peak negative voltage E -\- A does not exceed cutoff. The number of stages in the IF amplifier must then greatly exceed the minimum number determined by gain, bandwidth, and stability requirements. A more suitable arrangement in the radar receiver involves the use of two or three remote cutoff pentodes in the early stages of the amplifier. Gain reductions of 35 per stage with negligible distortion can be obtained with some of the available semiremote cutoff tubes having reasonably good gain bandwidth products. Very little or no AGC is then applied to the remaining stages of the IF amplifier. It is desirable to limit AGC loop gain variations with the input signal level in order to maintain loop stability and the required dynamic perform- ance. AGC loop gain variations of 2 : 1 or 6 db represent a practical goal employed in the design of the IF amplifier. As was noted in Paragraph 8-16, the AGC loop gain is proportional to the derivative of the logarithm of the IF amplifier gain with respect to the AGC bias. The contribution of an individual stage is thus proportional to the derivative of the logarithm of the transconductance curve. Unfor- tunately, this quantity, like the third derivative of the transconductance whose importance was noted above, is not normally specified or controlled in tube manufacture, and large variations can occur in the cutoff region. The use of a greater number of remote or semiremote cutoff tubes, limiting 8-22] THE ANGLE MEASUREMENT STABILIZATION PROBLEM 429 the controlled gain variation to the order of 20 db per stage, will minimize these gain variations. When a gain control range in excess of 50 or 60 db must be provided, it is frequently necessary to control the first stages of the radar receiver. When this is done, control voltage must be provided in a manner that causes the least deterioration of signal-to-noise ratio. In addition to grid-1 control of the amplifier stages, grid-3 or plate and screen control is sometimes employed. Grid-3 control provides minimum third-order distortion, but the screen dissipation is generally excessive when the tubes are operated with reasonable gain bandwidth factor. As a result the best compromise is grid-1 control of remote or semiremote cutoft tubes. 8-22 THE ANGLE MEASUREMENT STABILIZATION PROBLEM In airborne radars the measurement of target angular position is compli- cated by the angular motion of the airborne platform. This paragraph will discuss the general features of this problem and the approaches that are employed to solve it. Subsequent paragraphs will show how a specific stabilization prpblem — the AI radar search and track stabilization problem — might be approached. The techniques and lines-of-reasoning used in this example are typical of those which must be employed for the solution of any airborne radar stabilization problem. The essential features of the problem are illustrated by the simple one-dimensonal representation of Fig. 8-23. The space pointing direction of the antenna Atl is made up of two components: (1) the angle At of the antenna with respect to the aircraft and (2) the space orientation angle of the aircraft A a. Thus changes in the orientation of the aircraft — due either to maneuvering or disturbances from wind gusts, etc. — will cause corresponding changes in the space pointing direction of the antenna. From a tactical standpoint, this situation is undesirable. The line of sight from the radar to the target is relatively independent of radar aircraft orientation (neglecting long-term kinematic effects, it is completely independent). Thus, the effect of aircraft platform motion is to degrade the radar's ability to measure the target's position in space. The term angle stabilization refers to the family of techniques employed to isolate the radar measurements from the degrading influences of aircraft motions. These techniques fall into two general classes: (1) data stabiliza- tion and (2) antenna stabilization. Data Stabilization. With this technique, no changes are made to the basic control loops illustrated in Fig. 8-23. The effects of aircraft motion are compensated in the data-processing system by correcting the antenna angle measurements by appropriate functions of the measured platform motion. 430 REGULATORY CIRCUITS Antenna Command Antenna Antenna Drive Angle A, V A rcraft Disturbance Inputs Aircraft '• ^ Orientation Aircraft Aircraft '+ Angle Maneuver ^A Commands Space Pointing Direction At, Fig. 8-23 Basic Relationships in the Airborne Antenna Drive System. This technique is generally applicable to fan-beam AEW systems and other similar applications where platform motions cause measurement errors but do not cause loss of the target. Data stabilization finds particular favor where the antenna structure is so bulky as to preclude any other approach. Antenna Stabilization. For the vast majority of airborne radar applications — missile seekers, AI radars, side-looking radars, infrared systems — stabilization of the antenna itself usually is required. The basic objectives of such a stabilization system may be derived as follows. From Fig. 8-23, the space pointing direction of the antenna may be expressed : ^TL = Ga X (antenna command) -\- Gd X (aircraft disturbance inputs) + Gm X (maneuver commands) (8-21) where Ga, Gd, and Gm are the transfer functions of the antenna drive and the aircraft. From a tactical standpoint, the desired relationship is ^TL = Ga X (antenna command) = j^tl desired. (8-22) Thus, the stabilization system must have two primary objectives: 1. It must provide control loops which reduce the effective couplings (Gd and Gm) between aircraft and antenna motion. (The required amount of reduction is a function of the expected tactical use requirements). 2. It must provide control means for driving the antenna to the desired space pointing direction. Antenna Stabilization During Search. During the search phase of radar operation, the problem is to maintain surveillance of a predetermined volume of space despite the perturbations caused by platform motion. During this phase, target data are not used for control of the antenna; rather the antenna is driven by open-loop command data to sweep out the -22] THE ANGLE MEASUREMENT STABILIZATION PROBLEM 431 desired space volume. The antenna is commanded to move in a direction opposite to that of the aircraft. The general means for solving the stabiliza- tion problem in this phase are shown in Fig. 8-24. The antenna is driven by A ^ (Stabilization Feedback) Air :raft Disturbar Inputs ce \ Antenna Drive Ga i"n* Aircraft Aircraft J a/9"a ) Maneuver 'An / Ant. Pointing \ Direction Commands An.c Antenna Command TIME-^ SPACE REF Fig. 8-24 Basic Search Stabilization System: Single Axis. a generated command function y^TL.c as shown. A feedback signal Aa provides stabilization by subtracting the aircraft orientation angle from the command angle. Thus, we may write ^TL = Ga{/iTL,c — ^ a) + ^'i A (8-23) AtL = GaATL,c + (1 - Ga)/lA- If Ga is essentially unity over the frequency range o^ Atl,c and A a Aa = Atl,c (8-24) which is the desired result. The critical elements of such a system are seen to be: 1. The accuracy of the angular reference which provides the feedback signal. 2. The closed-loop gain and frequency response of the antenna drive which must be sufficient to follow the input commands Atl,c and the stabilization feedback signals. Generally, the dynamic response require- ments imposed by the command function are the most severe. Additional complications are introduced by the more practical problem of stabilization in two or three axes. While the basic principles remain the same, the problem geometry will involve somewhat complex angular 432 REGULATORY CIRCUITS transformations. These will be discussed later in the example of a detailed search stabilization design for an AI radar. Tracking Stabilization. During tracking, information from the target can be employed to position the antenna. As described in Chapter 6, various techniques such as conical scan or monopulse can be used to create an error signal which indicates the amount of error between the line-or-sight and the antenna pointing direction. Generally, however, the stabilization provided by the radar angle tracking control is not sufficient: a faster, tighter inner stabilization-control loop must be used to provide the neces- sary isolation from aircraft motions. A basic tracking radar stabilization system is shown in Fig. 8-25. ^^ Xhe outer control loop represents the generation of the automatic angle tracking Aircraft Disturbance Inputs Antenna Drive G2 An G,h- Aircraft Maneuvering Inputs ^''-[^g,g,\g,g^P'\i + Gi Go+ Go G Fig. 8-25 Basic Tracking Radar Stabilization System. Jls = sight target line; Et = angular error signal; Jtl = stabilization feedback; G3 = rate-measuring device (gyro). error signal. The design considerations for such control loops are covered in Chapter 9. Stabilization is provided by measuring the space angular rate, Atl, of the antenna and feeding this signal into the antenna drive as shown in Fig. 8-25. This arrangement yields the interrelations among antenna pointing direction line-or-sight inputs and aircraft motions, shown. If the gain of the control loops, the products of G1G2 and G2G3, are much greater than unity for frequencies greater than those inherent in the line- of-sight angle Jls and the aircraft space angle Ja, the equation in Fig. 8-25, Atl becomes r GjG + G2G, ^h.s + 1 + G1G2 + G2G; •^TL (gttg;) '^''' l^Variations of this system are discussed GolG, + G3) Paragraph 8-31. .^.., ^..1 (8-25) (8-26) 8-24] AIRCRAFT MOTIONS 433 Over the low-frequency range of^LS, iGs] « |Gi|, and the equation becomes Jtl = Jls + ~ (8-27) Since G1G2 » 1 in the frequency region of interest, Jtl ~ Als as desired. Of course, these relationships hold only if the loop gains are high and the control loops are stable. These are conflicting requirements which are discussed in the following paragraphs where the design of the stabilization loops is covered in detail. 8-23 AI RADAR ANGLE STABILIZATION The primary function of the AI radar control system is to detect a target and to provide tracking information to the interceptor fire-control system about the target's relative position and motion. The degree of space stabilization that must be provided depends on {a) the magnitude of the interceptor space motions during an attack and {b) the accuracy with which the target position and rates must be known. These topics are considered in more detail in subsequent sections. 8-24 AIRCRAFT MOTIONS The first step in the design of the stabilization system is to obtain a description of the aircraft angular^^ motions that will occur in the detection and tracking phases of interceptor operation. The basic angular motions are the roll, pitch, and yaw of the aircraft. The origins of these motions may be outlined as follows. First of all, the aircraft must maneuver in accordance with the vectoring commands or the fire-control system commands. Superimposed on these desired maneuvering motions are the oscillatory motions resulting from lightly damped aircraft motions which are excited by the control actions and the tendency of the human pilot (or autopilot) to overcorrect an error. Finally angular motions will be excited by disturbances such as wind gusts and release of armament or other stores {interference motions). For purposes of preliminary design of the stabilization loops, these motions may be described in several ways: 1. By the maximum expected roll, pitch, and yaw angles and angular rates and derivatives. These data can be estimated from attack-course studies and knowledge of aircraft operation. 2. By the time-response characteristics of the aircraft in yaw, pitch, and roll incident to impulse inputs. This information can be derived from equations which describe aircraft dynamics. i^Linear motion is not considered here since stabilization control loops are pricipally con- cerned with angular motion. Linear motion is considered in Paragraph 9-18 in systematic errors. 434 REGULATORY CIRCUITS 3. By the frequency-response characteristics of the aircraft in yaw, pitch, and roll. This information can also be derived from the equations which describe aircraft dynamics. 4. By an actual time response made of angular aircraft motions as an ideal attack course is flown. This assumes that an actual aircraft is available or that it can be simulated on an analogue computer and "flown" realistically with an autopilot or human pilot. An ideal, simplified radar and antenna tracking system may be assumed, but noise and approximate error filtering should be included in the simulation. The time responses may be used as follows: {a) A Fourier analysis of the time responses may be made. This may be made by conventional methods, but it usually is not as useful in design synthesis as the other techniques are. {b) Segments of the time response may be represented by sinusoids or parts of sinusoids of various amplitudes and frequencies. These data are particularly useful in the synthesis of antenna stabilization control loop frequency responses. 5. By a statistical description of the aircraft motion. This is usually not available, and the effort required to obtain the power density spectrum is considerable. However, motions due to gust disturbances are better described statistically, as discussed in following sections. The most useful descriptions of aircraft motion for preliminary stabiliza- tions considerations are given by methods (1), (3), and (4). Typical numerical values of modern interceptor aircraft motion described by these methods are given in the following pages. Maximum Disturbances Incident to Aircraft Maneuvers. The controlled interceptor motions, as limited by the aerodynamic charac- teristics of the interceptor, which affect the angle tracking system design, are roll and roll rate, yaw and yaw rate, and pitch and pitch rate. Estimates of a typical interceptor's capabilities are: (a) ROLL Roll angle: +180°, -180° Roll rate: 80° /sec to 90° /sec (b) YAW Oscillation frequency: 3 to 6 rad/sec Sideslip angle: 1-2° (c) PITCH Pitch angle: +180°, -30° Pitch rate : 20° /sec up to 40° /sec Pitch frequency: 2-13 rad/sec Pitch oscillation angle: Pitch rate /pitch frequency = 1.5° to 10° -24] AIRCRAFT MOTIONS 435 The values are typical of maxima that may be encountered. Actually, the kinematics of most attack courses do not require maneuvers of this magnitude; for example, the lead collision type of attack described in Chapter 2 theoretically requires no maneuvering at all once the initial error has been corrected. Despite this fact, however, the aircraft will experience relatively large angular rates during an attack because of lightly damped oscillatory modes in the aircraft response and the marginal stability which characterizes the pilot-aircraft steering loop. The data of Fig. 8-26, which were obtained from a typical simulation pro- gram, illustrate the principle. In this simulation, a human pilot attempted Wfi'irnmn 1(111111111111 M M\\\U\\\\\\\\\\\\\\\ ^ IIIIIIIIIIIIIIW ^m Roll Angle m MilA \\\\\\\M\\\\\\\\W^ 20 TIME - Fig. 8-26 Typical Simulation Results, Showing Aircraft Motions During an Attack. to fly an attack course using information from a display which presented steering error and aircraft roll and pitch angles. The steering error signal was contaminated by radar tracking noise. Both the steering error and the noise were passed through a 0.5-sec filter prior to display. 436 REGULATORY CIRCUITS Under these conditions a major portion of the aircraft motions took place at several relatively well defined frequencies. Rolling motions predomi- nated; these took place at frequencies between 0.5 and 3.0 rad/sec, with maximum rolling rate amplitudes in the range of to 20 deg/sec. Also evident is a yawing oscillation at a frequency of 0.5 rad /sec, and a maximum yawing rate amplitude of 1.2 °/sec, and a small pitching oscillation at a frequency of 6 rad/sec and a maximum pitching amplitude of 2° /sec. Sinusoidal Representation of Disturbances. This method is more useful in determining the stabilization control loop specifications. Specifi- cally, this information may be obtained from actual time responses of a simulated aircraft on an analog computer as was shown in the preceding discussion. Portions of the time responses may be approximated by sine waves, and the amplitudes and frequencies of the sine waves can be recorded for various aircraft motions from several different courses. To study the effect of aircraft motion on tracking-line stabilization, the aircraft motions are converted into motion with respect to the axes of the antenna gimbals. Usually, the antenna has two gimbals. ^^ The azimuth gimbal allows the antenna to rotate about an axis parallel to the aircraft's vertical axis; the elevation gimbal permits the antenna to nod up or down. The basic angle and angular rate relationships for such a two-gimbal system are shown in Fig. 8-27. Aircraft Fore -and -Aft Dire action Aircraft Roll, Pitch, Yaw Rates l^= Azimuth Gimbal Angle dp = Elevation Gimbal Angle Transformations Antenna Rates Due to Aircraft Angular Rates: Azimuth coa = oJ;< cos 6^ sin 8,+ co^ sin d^ sin d, + co, cos e, Elevation w^ = - co^ sin 0^ + o>y cos Q^ Fig. 8-27 Angle and Angular Rate Relationships for a Two-Gimbal Radar Antenna. When the antenna tracking lead angle is large, the aircraft rolling motions appear as azimuth and elevation disturbances as is demonstrated by the cox terms in the transformation relationships in Fig. 8-27. This fact "In some missile applications it is necessary to provide a third gimbal to space-stabilize the antenna in roll. This is not considered here. 8-24] AIRCRAFT MOTIONS 437 is important because the rolling rates can be quite large relative to the yawing and pitching rates (see Fig. 8-26). When the sinusoids representing aircraft motions are transformed into equivalent motions in antenna coordinates, results like those shown in Table 8-3 are obtained.^^ These results are typical of what might be Table 8-3 ANTENNA DISTURBANCE FROM AIRCRAFT MOTION OF FREQUENCY c^d A A (Vsec) Aa{°) 'eak to Peak Peak to Peak oJz) (rad/sec) ZS 33.6 1.04 30 9.55 3.14 20 6.04 3.31 20 4.77 4.19 22 4.48 4.92 5 0.497 10.5 7.5 0.715 10.5 obtained for lead angles of 45 deg from a large sample of the type of simulation data displayed in Fig. 8-26. It is assumed that the magnitude of this motion could disturb either channel of the tracking control loops at any time without further attenuation. It should be emphasized that in a missile system or in an autopilot- controlled aircraft, these motions can be calculated by considering the equations of aircraft motion in three dimensions as it follows a prescribed course, assuming that the aircraft and autopilot design are known well enough to be described mathematically. The resulting data can also be approximated by calculating the aircraft frequency response from its design equations; this is often done. Although the computations may be simplified through the use of matrix notation and block diagram representation, the cross-coupling between the control loops is complex and nonlinear because of trigonometric functions involved, and it is difficult to interpret except for simple cases which are discussed in the next section. The task becomes more difficult, if not impossible, when a human pilot is involved because the human transfer function is not defined to a usable degree of accuracy or with sufficient reliability necessary for realistic results. Therefore, when available, analogue simulation is the most propitious method of obtaining information about aircraft motion in space with or without autopilot control. i^In practice many more values should be used because the nature of these disturbances is essentially random and a large sample should be made to obtain representative data. Note, for example, that a single frequency may have different amplitudes at different times. 438 REGULATORY CIRCUITS Aircraft Transfer Functions. The aircraft cannot maneuver with large amplitudes at high frequencies. This can be shown from simplified transfer functions of the aircraft relating control surface motion to aircraft motion. The transfer functions can be obtained from transforms of the differential equations which describe aircraft motion. When simplified, to eliminate the short term yawing oscillation term, the transfer function relating aircraft heading to control surface position reduces to the form K/S(l + Ts).^^ K and T depend upon the particular aircraft charac- teristics, and a frequency plot of this function should resemble the plotted data described in the preceding paragraph. The general form of the space isolation required by the radar antenna should be the reciprocal of the aircraft response transfer function. The equivalent gain factor of the isolation transfer function ultimately depends upon the amount of isolation needed, the equivalent gain of the aircraft, pilot or autopilot, course computer, and the error presentation as discussed in Paragraph 8-32. Gust Disturbances. As an aircraft flies an attack course, it is sub- jected to winds and turbulence or velocity fluctuations in the surrounding air. Turbulence disturbs the aircraft in a random manner, and its general effect is referred to as a gust disturbance. Because of their random nature, gust disturbances are best determined by measurement and then described by power density spectra. The data of typical measurements and the associated normalized power density spectra are presented in the following documents: {a) An Investigation oj the Power Spectral Density oj Atmospheric Turbu- lence by G. C. Clementson, Report No. 6445-T-31, Instrumentation Laboratory, M.I.T., May 1951. [b) A Statistical Description of Large-Scale Atmospheric Turbulence by R. A. Summers, Report No. T-55, Instrumentation Laboratory, M.I.T., May 1954. The normalized power density spectra may be applied to a specific aircraft by scaling both abscissa and ordinate. The effect on the tracking loop antenna position and rate may be then found by multiplying the gust power density spectrum by the square of the transfer function magnitude relating the disturbance to the antenna position rate in the channel corre- sponding to the direction that the gust disturbances were measured. The square root of the integral of this product is the rms value of the antenna motion or rate. For most tactical situations the effect of gust disturbances is negligible when compared with other factors and it will not be considered further in this text. For high-speed, low-altitude flights, however, gusts i^Actually, the transfer function varies in roll, pitch', and yaw. The most useful transfer functions are those which transform aircraft motion to antenna motion. 8-25] STABILIZATION REQUIREMENTS 439 can sometimes be quite severe and in such cases their effects should be studied as part of the systems design. 8-25 STABILIZATION REQUIREMENTS The stabilization requirements vary for the search and track modes. In search, a lack of space stabilization would allow the search pattern to move with the interceptor from its preassigned space sector, and the desired target might not be found. Deviations from true space stabilization in search are produced mainly by (a) Aircraft angular motions^^ (i) Dynamic antenna control loop errors (c) Incorrect commands to the control loops (d) Inaccuracies in the vertical reference The total deviation from true space stabilization that can be allowed is related to the loss in target detection probability that it produces. ^^ Usually 0.25° to 1.5° deviation from the ideal space stabilized pattern can be permitted. However, the deviation varies as a function of the search angle with respect to the aircraft. In the tracking mode, antenna space stabilization is needed for several reasons: (a) To prevent the course computer from operating on inaccurate antenna motions due to interceptor space motion rather than target sight- line motion in space. ^^ (^) To prevent the antenna beam from drifting off the target because of aircraft motion during short periods when the radar signal fades. (<:) To prevent system instability caused by coupling of the interceptor motion with its commands through resulting antenna space motions. This is a form of positive feedback, because as the interceptor moves the antenna in space, the antenna motion produces signals used by the computer to direct the interceptor farther in the same direction. To prevent instability in this positive feedback loop, it is necessary to have the loop gain ^^ always less than unity. This is most important in systems using an autopilot. The problems involved in providing the necessary isolation in the search and track modes are discussed in the paragraphs that follow. "Another source of error may be produced, especially in some missile systems, from control loop disturbance torques created by an unbalanced antenna undergoing large rotational space accelerations. However, a detailed discussion of this is beyond the scope of the text. i^Detection probability is discussed in Paragraphs 3-9 to 3-14. i^Another source of error may be produced, especially in some missile systems, from control loop disturbance torques created by an unbalanced antenna undergoing large linear or rota- tional space accelerations. However, a detailed discussion of this is beyond the scope of the text. 2fThis includes antenna motion detectors, filters, computer, pilot or autopilot, aircraft transfer functions, and the isolation factor provided by the closed, space stabilized antenna control loops (See Paragraph 8-32). 440 REGULATORY CIRCUITS 8-26 SEARCH PATTERN STABILIZATION The space reference for the search pattern control loops is derived from a vertical gyro. The accuracy of the vertical gyro in maintaining a true vertical need not be extremely good; available vertical gyros provide sufficient accuracy for stabilizing the antenna search pattern during the relatively short times that a particular target is being sought. The vertical gyro used has two degrees of freedom and is provided with position detec- tors that measure the aircraft pitch and roll angles with respect to the vertical. Except for some missile applications, yaw angles are generally not measured because the aircraft yaw motions are better controlled and the search pattern is usually much wider in a horizontal direction (see Para- graph 2-26). 8-27 SEARCH STABILIZATION EQUATIONS Because the antenna motion of a two-gimbal antenna does not include roll correction directly, the aircraft roll motion must be converted into the proper azimuth or elevation commands for the two-antenna control loops. The exact transformation of aircraft motions to antenna commands is rather complex as shown by the following formulae: sin Ea = sin As cos Es sine/) — (cos As cos Es sin 6 — sin Es cos 6) cos 4> (8-28) sin Aa cos Ea = sin As cos Es cos <^ -f (cos As cos Es sin 6 — sin Es cos 0) sin (8-29) where Ea is the elevation antenna angle command to move the antenna with respect to the interceptor Aa is the azimuth antenna angle command to move the antenna with respect to the interceptor Es is the elevation search angle with respect to space As is the azimuth search angle with respect to space </> is the aircraft roll angle in space d is the aircraft pitch angle in space. It is possible to mechanize these equations to within a few ininutes of arc^^ by using several resolvers, as shown in Fig. 8-28; but to simplify the mechanization, the transformation equations are often simplified. This can be done in several different ways. One of the approximations which 2^This inaccuracy is due to components, principally the resolvers. -27] SEARCH STABILIZATION EQUATIONS 441 (cos E5 COS A5 sin B - sin £5 cos d) = Aircraft Roll Angle : Aircraft Pitch Angle £(., Ar = Search Command in Space L 1 Resolver for I each Angle sinE^= sinAj cos £5 sin</)-(cos A5 cos Ej sin 6 - sin £5 cos 6) cos sinA^ cosE^=sinA5 cos £5 cos^ + icos A5 cos £5 sin 9 -sin £3 cos 6) sin <^ Fig. 8-28 Exact Coordinate Conversion Mechanization for a Two-Gimbal An- tenna Search Pattern. creates very little significant error (in the command signals for search angles within 50 deg) is expressed by the following equations: Ea = As sin cf) - (d - Es) cos 4> (8-30) Ja = As cos <p - (d - Es) sin (8-31) These equations are shown mechanized in Fig. 8-29. Elevation Antenna Position Antenna Search Loops 4> = Aircraft Roll Angle \ ^ ^^^^^^^^ e = Aircraft Pitch Angle ) A=As cos</)- (d-Es) sin0 Fig. 8-29 Approximate Coordinate Conversion Mechanization. 442 REGULATORY CIRCUITS 8-28 STATIC AND DYNAMIC CONTROL LOOP ERRORS Perhaps the largest errors in the search-pattern stabilization are engen- dered in the antenna control loops. These errors are reduced to satisfactory limits by proper design of the control-loop gains and bandwidths or by modification of the pattern command signals. Errors to be considered are: (a) Static or steady-state errors due to aircraft motion (i>) Static or steady-state errors due to search-pattern velocity (c) Dynamic ierrors due to changes in search-pattern command signals It may be considered that part of the total allowable search-pattern errors may be allotted to the antenna control loops. For example, a 0.35° steady-state error may be assumed and it may be divided equally between aircraft motion and search-pattern velocity signals; i.e., the allowable error contribution of each source is 0.25°.^^ To provide a means for translating the error requirements into a design specification, a generic form muse be assumed for the search stabilization and drive system transfer function. For the example to follow, the assumed open-loop transfer function will have the form COl < Wo < CO3. lO-->^j s{\ + si^^){\ + .syco,,) The following analysis will demonstrate how the values of Ki,, wi, 0^2, and C03 can be chosen to meet a set of system requirements. Aircraft Motion Errors. To reduce the steady-state errors caused by aircraft motion to 0.25°, the nature of the aircraft motion must be known. For example. Table 8-3 shows the amount of antenna movement that would take place at large lead angles if the antenna were not stabilized. The search stabilization loop generates position command signals which — if computed correctly — are equal and opposite to the disturbance caused by aircraft motion. However, the control loops that drive the antenna with respect to the aircraft have finite gain and bandpass. Thus, the actual position of the antenna will tend to lag the stabilization commands. As shown in Fig. 8-30, the amount of lag depends upon the frequency and magnitude of the input relative .to the open loop gain of the stabilization loop at the input frequency. In order that the lag be kept below 0.25° at all input frequencies, the minimum loop gain must be error specification 0.25- As an example, the input at 1.04 rad/sec is 33.6° peak-to-peak or Xi = 16.8°. Thus, the required open-loop gain of/= 1.04 is 67.2. Similar 22Since aircraft motion is independent of the search pattern, the individual errors may be added by taking the square root of the sum of the squares. 443 Xe ~: 1 Xo — *• Go 1 * Let: Xe= Loop Error Xj- Loop Input Command Xo = Loop Output In Search Loop, X, Contains Signals to Move Antenna and Stabilize it in Space. It is Desired to keep the Loop Error Xe Less Than a Particular Magnitude |Xe| Since I X; I IX, I Go»l Xi X, I IXJ= W — for iGol |X/| As a Function of Frequency is Known and |Go| As a Function of Frequency is Found as Desired Fig. 8-30 Derivation ot Required Loop Transfer Function. calculations can be made for other frequencies, resulting in the circled points shown in Fig.. 8-31. These points define the 7nini7nu?n open-loop Low-Frequency Asymptote For Kv=400 1000 0.1 01 1 I I 1 1 m il I I 1 1 m il I I I m ill i i i ii 1.0 10 100 wi FREQUENCY (rad/sec) Fig. 8-31 Search Loop: Open-Loop Transfer Function. 444 REGULATORY CIRCUITS gain necessary to provide isolation from the expected aircraft angular Search-Pattern Velocity Errors. In addition to providing isolation from aircraft motion, the steady-state error caused by the antenna scanning motion must also be considered. Usually the antenna motion is uniform, and in the azimuth or horizontal space direction it is a constant angular velocity. The steady-state position error is the velocity divided by. the velocity constant. ^^ The constant antenna sweep velocity is determined as discussed in Paragraph 5-7 and is usually between 75 and 150°/sec.2* Therefore, to keep the error below 0.25° for a 100° /sec search velocity, the velocity constant must be ^^qIP"" = 400 sec-i = K^. (8-34) Dynamic Errors Due to Search-Pattern Velocity Changes. Some distortion is likely to occur at the ends of the search pattern because physical control loops cannot follow the rapid changes in command signal which are used to change the antenna velocity and position at the end of a horizontal sweep to another horizontal sweep. ^^ The transient at the end of the sweep requires a longer time to reach a small steady-state value than does the vertical-motion transient because the vertical motion is usually a small position step instead of a large velocity reversal. Since it is desired to resume the steady-state error in the shortest time after the sweep direction is changed, both azimuth and elevation control loops may be designed to realize the desired sweep transient. If there were no aircraft roll, each control channel would have different characteristics. But because a large roll angle transfers much of the horizontal motion to the elevation channel, both channels should be designed to have the characteristics of the azimuth channel in horizontal flight. 2^The specification derived in this manner is somewhat pessimistic because the antenna is not always at the large lead angle used to obtain the data in Table 8-3. 241 f the open loop has the form i(l +s/wi)il + Vcos) K^ is the velocity constant. -^The search pattern may have several forms — a horizontal Palmer scan with several horizontal sweeps spaced at the beam width and a diagonal return, a horizontal sweep and return spaced at the beamwidth, a spiral scan, a circular scan, a vertical scan, or combination of these types. ^^he command changes can be made gradually to eliminate transients, but this usually involves a more elaborate signal generator. Even then, the ideal pattern will not necessarily be obtained, and more time may be required for the antenna sweep to resume the desired constant velocity. -28] STATIC AND DYNAMIC CONTROL LOOP ERRORS 445 Fig. 8-32 shows the nature of the error transient that occurs if the direction of the horizontal sweep changes instantaneously. An important Approx. Decay Transient Approx Bound Error — L*^^ — ^ - 1 \ ^^ Max. Time to Peak ' V— ^ ^^ (1.5<)c<2.0 ) 1 /\^^ a r = Horizontal search sweep velocity 1/^ \ X CO2& 0}^ are defined in (b) below \ f=,0, \ \-2 Slope "' ^ N J 0)^ COj _^ ^A ,r -^ V,. Aircraft '"'^^" ,^n ^M 1 •'- Motion U LO "' "^^ ^ pa]i;;rv^~^-x-J r _£J r^ J (b) Position T t 1 1 1 Pattern Aircraft Generator Motion (9,0) e,0 are assumed constant in (a) (c) Fig. 8-32 Dynamic Transients, (a) Approximate position error transient for linear search sweep, r = horizontal search sweep velocity; 002 and coc are defined in design equations, (b) Open-loop transfer function. Design equations (approxi- mations usually accurate enough for engineering design purposes): (1) Gain: (2) Phase: 180° + (/);„ C02\C0i/ TT /tT CoA /tT 0)2 \ CO^/ A 2/3„ 57.3 (3) Minimum Phase atcj^: t~^ = 2cOm NOTE: Substitute ( — ^ )in Equation 2 when solving for coi or a;2- (c) The search loop (azimuth channel). point is that the peak error of the transient, frequently neglected, can be much larger than the steady-state error, and a relatively large time may be required for it to settle if aj2 in Fig. 8-32 is small. Actually, there are several ways in which the dynamic error can be specified. For example, the time 446 REGULATORY CIRCUITS required for the peak error to decay to within a certain percentage of the final value could be specified, a maximum time for the peak error to occur could be given, or the percentage distortion of the overall pattern dimen- sions which can be tolerated could be specified. Fig. 8-32 indicates approxi- mate but useful relationships between the dynamic transient, the open-loop transfer function, and the steady-state error that help determine the search control-loop design characteristics. For example, if the allowable distortion incident to peak dynamic error is to be within 10 per cent of an overall pattern sweep of 30°, the peak error would be 3°. From Fig. 8-32, the upper bound is about Irl ^c above the negative error, and the approximate peak error above zero is 2f/coc — r/i^„ = 3 deg; and if K, = 400 sec^^ and r is 100° /sec, 100(2K - 1 /400) = 3 and coc = 61.6 rad/sec. This is the search-loop bandwidth. If greater accuracy is required, a more sophisticated pattern command would be necessary with special accelerating and decelerating controls — perhaps nonlinear control for maximum effort. This is not usually necessary, how- ever, to obtain a relatively constant sweep velocity. Other characteristics of the open-loop transfer function are found from stability considerations, and an optimum system can be determined directly from the three following equations relating the corner frequencies, peak phase margin, and the loop gain shown in Fig. 8-32.^^ SEARCH LOOP SYNTHESIS 1. Loop gain equation: ^(^Yco, = K, = 400 (8-35) C0 2\C0l/ CO, ^ /C. ^ 400 6.49 (8-36) coi coc 61. 6 2. Phase equation (frequency response peak = 1.3,^* peak phase margin </)„ = 50.3° at a frequency co^ = ^c cos 0„, = 0.64coc) : (-180° + 50.3°) r-(5-„":,) + (!-S) 57.3 (8-37) _ CO,. ^ 0.69 = ^"^-"'^ + ^' CO 3 CO„j CO 3 where ^m = phase angle of G at co„(. 2''The derivations of these equations are discussed in the paper, "Synthesis of Feedback Control Systems with a Minimum Lead for a Specified Performance," by George S. Axelby in IRE Transactions in Automatic Control, PGAC-1, May 1956. ^The closed-loop frequency response peak A/,, occurs at a frequency co„i = wc cos <^m and sin <^m = ^IM ,, in the optimum, minimum lead system. -28] STATIC AND DYNAMIC CONTROL LOOP ERRORS 3. Minimum phase equation : — — ^ — ' = — (differentiate Equation 8-37) Combining Equations 8-37 and 8-38 2(c02 — CO]) 2(a)2 — OJl) 0.69 and 0.69 Wm 0.64cOc 2co,(a;2/co, - 1) 2a)i(5.49) 447 (8-38) (8-39) (8-40) 0.64a;. (0.64) (61. 6) from which coi = 2.48 and co2 = 16.1 O^r. = 39.4 CO. = 61.6 C0.3 = 114. These are the search loop corner frequencies, bandwidth, and maximum phase frequency in radians per second. A log magnitude phase diagram or Odb 400 (ili-) s{m*') (ii4+' 25.0 39.4 1.6 wc=61.6 Fig. 8-33 Nichols Chart of Search Loop. 448 REGULATORY CIRCUITS Nichols chart is shown in Fig. 8-33.^^ Note that if the corner frequency in Fig. 8-32 had been a double corner, C03 would be doubled; if it had been a triple-corner frequency or equivalent, cos would be tripled, etc. However, Fig. 8-33 would be essentially the same in the crossover frequency region. This example illustrates one method of synthesizing the search control loop directly with a minimum of the usual cut and try effort. The procedure is similar, even if other criteria are used to specify the search loop perform- ance. 8-29 SEARCH LOOP MECHANIZATION Actual circuit details of search loop mechanization cannot be discussed in general terms because the control loop components vary with specifications, with the nature of the power available, and with the antenna size. How- ever, a few general considerations can be discussed. The basic components needed to mechanize each of the two search loops are shown in Fig. 8-34. The coordinate converter needed to correct the Loop Input Signal Corrected 'for Aircraft Space Motion -r Power Amplifier Modulator Actuator I Possible I Tachonneter ■E>- Fig. 8-34 General Block Diagram of Search Loop (One Channel). input signal for aircraft motion is not shown because it was discussed in Paragraph 8-27 and illustrated in Fig. 8-28 or 8-29. The practical problems involved in its construction are those of making proper resolver connections with correct phasing of a-c signals. For the exact transformation, the signals between the six resolvers must pass through isolating amplifiers or phase-shifting devices. The loop-actuating signal, or error signal, is usually an a-c voltage proportional to the difference between the input signal Xa and the controlled antenna position /^a as shown in Fig. 8-34. It is obtained in the exact coordinate transformation from the windings of resolvers, which are mounted on the antenna. In the approximate transformation, the a-c error signal is obtained from the sum of voltages from the vertical gyro roll 29Note that the calculated maximum phase margin is about 2° greater than the desired design value. This discrepancy, conservative but negligible in an actual system, occurs because the locus was calculated exactly from the transfer function which was determined from approximate equations. !-29] SEARCH LOOP MECHANIZATION 449 resolver and a potentiometer on the antenna. The latter is not shown, but it is impHed in the summing symbol in Figs. 8-29 and 8-34. Wire-wound potentiometers may be used in each channel, but induction potentiometers are preferred (especially if the loop gain is high) to prevent oscillations between potentiometer wires. Note: a = l Produces Simplest Form ' — ^^^H!;^ 2 / 1 1 \ 1 ^ ^HRsCz c^rJ'^ C2C1R1/! ( q^C,+ R3C, + R3C, l R, ^ y R^ + Rg / (Ri+R2)CiC2Ri(i Design Formulas Ci= arbitrary '^2~'^iVa~/ K = (W4 + C05-a)2j(-^) "4 , Ifi '^^ (ciRj-'CgRj + C3C1R1R4 e, (s+o:^){s + o^^) (b) VC3R1 + C3R2 ^ C3R4 c^rJ C1C3R1R2R4 Cj = arbitrary Design Formulas C ^1 3 R2(W4CO3-a;jC02) (C04COg-C0iCO2) (oj4 + a)5)(c0j + a;2) SRC >R SRC + 1 (d) Fig. 8-35 RC Compensation Networks. 450 REGULATORY CIRCUITS As in all feedback control systems, the error signal is used to position the antenna in a direction which will decrease the error. Generally, however, the control system characteristics shown in Fig. 8-31 do not exist naturally, and it is necessary to provide some form of compensation. On small, low-power systems this compensation can be provided mechanically by adding extra inertia to the antenna inertia with fluid coupling; but with large antennas where space is limited and the power is relatively high, it is more practical to provide electrical compensation in the form of RC networks as shown in Fig. 8-35. Design formulae are given to illustrate how the network parameters are related to the corner frequencies. Generally, the corner frequencies are chosen so that, in combination with the corner frequencies of other equipment in the loop, the desired overall characteristic shown in Fig. 8-31 is obtained. This is illustrated in Fig. 8-36. It should be noted that networks A and especially B of Fig. 8-35 would be used in series with the low-power circuits in the forward path of the search loop, and that 1.0 \ -1 Gd Desired Loop \ Transfer Function =\ ' Gi Loop Transfer .9 ^ ^-y ^Function Without Vy/ Compensation 1 ,E i- \ \ 5i \ \ Lowest Corner Usually Due ^ o -2\ \ /to Actuator and Load "1^ \ \ / g OJ I s ^ \ 5^ \ Gc Loop Compensating \ \ _2 / Function \ \ 1 -r V \ \ \y |GcHG,HGi| ) ^ \ Xi '\\ -1 \ r \ y Note: Compensation Function Has Slopes Other \ \-3 Than Zero Where the Asymptotic Slopes of G^andX \ Go are Different (Compensation May be \ \ Realized With R-C Networks Shown m Fig. 8-35) ^ \ 1.0 \ \ -3 Fig. 8-36 Method of Determining Control-Loop Compensating Function. 8-29] SEARCH LOOP MECHANIZATION 451 network C, possibly in combination with D, might be used in a tachometer feedback path around the antenna, actuator, and power ampHfier. To use the RC compensating networks, it is necessary to convert the a-c error signal to a d-c signal with a demodulator.^" A peak demodulator is often used in the sear.ch loop because it has (1) less high-frequency noise and (2) a smaller time constant than an averaging demodulator. After the error signal is demodulated and passed through d-c networks and a power amplifier, it is applied to the power actuator which moves the antenna. Generally, the actuator is a two-phase a-c electric motor or a hydraulic actuator controlled with an electrically operated valve. If an a-c motor is used, the d-c actuating signal from the compensating network must be modulated before it is applied to the motor through a power amplifier. This may be done electronically with vacuum tubes or with magnetic amplifiers which provide amplification, modulation, and power in one reliable unit. However, if a hydraulic actuator with a control valve is used, the d-c actuating signal may be used to control the valve. Some signal amplifi- cation may be provided with tubes or with transistors. There are advantages and disadvantages in both types of antenna drives. The electric motor is cheaper and lighter than the hydraulic actuator; it does not need oil lines or rotary joints with oil seals; but it does not run as smoothly at low speeds, it is much less efficient, and it cannot produce as much steady-state torque or velocity as a hydraulic actuator of the same size. In addition, gear trains -with troublesome backlash and friction are needed with electric drives, whereas they are not used with hydraulic actuators.^' Regardless of its type, an actuator must be selected which will provide the required search-pattern velocities and accelerations. The output power of the actuator must be greater than the power required to move the antenna inertia along the search-pattern paths in space in the presence of antenna friction and unbalance as well as aircraft pitch and roll motion which may be directly opposed to the desired antenna space motion. In fact, aircraft motion adds considerably to the required actuator torque, velocity, and power because the actuators move with respect to the airframe to generate a pattern in space rather than with respect to the airframe; thus the expected aircraft motion must be combined with the required search- pattern velocities and accelerations to determine the actuator performance characteristics. Note that the antenna inertia and unbalance in elevation may be less than in azimuth and the actuator may be correspondingly '"Simple RC networks can be directly approximated with "notch" networks or resonant filters for a-c signals; but for airborne systems, this is usually not practical because of the accuracy required and because the carrier frequency varies. 'lit is assumed that the hydraulic actuator discussed here consists of a shaft with a vane enclosed in a housing through which oil may be ported to either side of the vane to produce a shaft rotation. Of course, 360° rotation is not possible with this type of actuator. 452 REGULATORY CIRCUITS smaller. In practice, however, the actuators often have the same size for production economy. Finally, it should be emphasized that the control loop and actuators should not be designed to have a performance much greater than that required, not only because increased size and weight would be involved but also because physical limitations inherent in the gimbals and the antenna structures of a given size place an upper practical limit on linear design, ^^ and as this limit of performance is approached, the cost and complexity of control equipment increases rapidly. Specifically, noise is always present in the search control loops, although not to the extent that it is in other control loops associated with fire-control systems, and its detrimental effects become more of a problem as an attempt is made to increase loop performance. In addition, structural resonant frequencies in the antenna make it nearly impossible to construct a stable loop with a crossover frequency near the resonant frequencies. Thus, there is a practical upper limit for the control loop bandwidth which is governed by the antenna characteristics. Since the performance of the loop is primarily a function of the bandwidth, the performance itself is limited. 8-30 STABILIZATION DURING TRACK As was discussed in Paragraph 8-25, the tracking antenna must be stabilized in space to prevent: {a) system errors caused by the course computer operating on informa- tion from coupling between the interceptor and antenna motions. {b) the antenna radar beam from drifting away from the target during brief periods of radar signal fading (c) instability incident to coupling between the interceptor and antenna motions. A portion of the required space stabilization is provided by the automatic tracking loops discussed in Chapter 9, except during periods when the radar signal fades. However, during normal operation, the typical tracking loop cannot provide effective isolation above frequencies equal to about one-half the track loop bandwidth, or about 3.0 rad/sec for a typical system. On the other hand, the interceptor may have appreciable motion at higher frequencies as indicated in Table 8-3. To provide the necessary space stabilization, a special automatic control loop is designed to move the antenna relative to the aircraft in a direction opposite to that of the aircraft motion in space. To provide space stabilization, the control loop must obtain antenna space motion information. This is obtained and converted into useful electrical signals with gyroscopes mounted on the antenna or on the aircraft, ''^It is possible to devise nonlinear control loops which will provide increased performance in special cases. -31] POSSIBLE SYSTEM CONFIGURATIONS 453 and these signals are used to move the antenna in a direction opposite to undesired space motions. Of course, the correction signals are not exact; the gyros sense space velocity only, not the space position actually desired. However, a control loop with sufficient accuracy and speed of response can reduce antenna space motion to magnitudes much lower than those of the interceptor, and the residual space position errors are further reduced by the track loop which uses the target sight line as a reference. Details of designing these stabilization loops are discussed in the subsequent para- graphs. 8-31 POSSIBLE SYSTEM CONFIGURATIONS Several physical configurations are used to mechanize the stabilization loops. Some of them are shown in Fig. 8-37.^^ Theoretically, all of them Rate Command w + p (Volts) From Radar Rate Command ^ Aircraft Space Angle \ An Aft (Volts) ^ Rate Signals Tracking Line Angle Gj Amplifier, Actuator, Antenna G3 Rate Gyro (a) Gyro Components (Current) From Radar Torque Y- ^'* G4 G5 1 , 1 G2 + t A,i ' '" Torque 1 Rate Signals (b) G4Gyro Gimbal and Detector G 5 Gyro Wheel G 6 Torque Motor Three Channels /Two Channels Ga Aircraft GgAircraft Gyros j^ ^^^ ^i^^ Q Coordinate ^ " Converter GrTachometer — *Gc 1^ Rate Signals •K> (c) Two Channels Fig. 8-37 Three Stabilization Loop Configurations, (a) Rate Gyro, (b) Inte- grating Gyro, (c) Aircraft Gyro. 33Another form is not illustrated. It consists of a gimbaled antenna dish which is rotated at high speeds to become an effective gyroscope. It is used in some missile tracking systems to provide stabilization. 454 REGULATORY CIRCUITS are equivalent in a mathematical sense, but practically the arrangement of the physical equipment is entirely different, and, depending on the appli- cation, the mechanization can create discrepancies between the theoretical and realized loop performance. In fact, in many control loops the mecha- nization characteristics, which are incidental to the primary loop function, may prevent the realization of a workable system. Therefore, it is necessary to know as much as possible about equipment characteristics before a design is finalized. Unfortunately, component characteristics vary widely depending on the application, and it is impossible to discuss them in detail. However, the general principles of operation are outlined below for three different systems. Rate Gyro Stabilization Loop. The rate gyroscope^"* is a self- contained unit which produces an electrical signal, usually a voltage, proportional to a space rate about a particular axis. In the stabilization loop design shown in Fig. 8-37a two rate gyros are used, one to measure space rates about the elevation axis and the other to measure space rates about the azimuth axis of the antenna. Both gyros are mounted on the antenna dish^^ where the antenna space rates are measured directly. As shown in the figure, the space rate {Atl) of the antenna tracking line angle, Atl, is measured by gyro and converted to a voltage proportional to Atl- This voltage is compared with an antenna rate command voltage Vr and the voltage difference, the rate error Er, is used to control the antenna actuators through appropriate amplifiers in a direction which will reduce the rate error. Although only one channel is shown, two control loops are needed — one for each antenna motion — and these loops are interconnected, not only through the common space platform and antenna structure but through the gyroscopes as well, because space accelerations in one channel influence the gyro output voltage in the other channel to a degree which is determined by the gyro characteristics and the gyro orientation. However, unless the stabilization loop is of very large bandwidth, this effect can be made negligible by selecting the proper gyro and by choosing the optimum gyro axis mounting.^^ As shown in the figure, motion of the aircraft angle, Aa^ is also detected by the gyro and converted to a voltage, essentially a rate error, which is used to move the antenna at a rate in space opposite to the aircraft space 34See Locke, Guidance, Chap. 9, pp. 350-353. ^^Note that the effective azimuth gyro gain in volts per unit of velocity is proportional to the cosine of the elevation angle and that it will change during normal operation. For computational purposes this is often desired, but it may affect the stability of the azimuth stabilization loop if the elevation angle becomes large. ^A complete discussion of this problem is given in a paper, "Analysis of Gyro Orientation," by Arthur Mayer in the Transactions of the Professional Group on Automatic Control, PGAC-6, December 1958, p. 93. -31] POSSIBLE SYSTEM CONFIGURATIONS 455 motion. Of course, this principle of operation is common to any stabili- zation loop. Integrating Gyro Stabilization. The integrating rate gyroscope is a self-contained unit which produces an electrical signal, usually a voltage, proportional to the time integral of the torques applied around the gyro gimbal axis. A schematic representation of the integrating gyro and the simplified equations expressing its behavior are shown in Fig. 8-38. Such Command Torque \/p Rate ^^ '^ Command Torque Motor Fig. 8-38 Integrating Rate Gyro Relationships. do = gimbal angular displacement about output axis relative to gyro case Bi — input angular displacement about input axis Ig = moment of inertia of gimbal assembly about the output (gimbal) axis C = viscous damping constant T = Koic = rate command torque H = IrOir = rotor angular momentum Ir = rotor inertia ojr = rotor angular rate DIFFERENTIAL EQUATION or for K = H = Ko (coi S{1 + T, Ko for T« 1 a device may be used as an integral part of the antenna control loop as shown in Fig. 8-37b. The rate command, in the form of variable current, is used to produce a torque on the gimbal holding the gyro wheel, with an electromagnet or torque coil in the gyro. This torque is opposed by the gyroscopic torque of the spinning gyro wheel which is induced primarily by antenna space motion about a particular axis. If these torques are not 456 REGULATORY CIRCUITS equal, the gyro gimbal moves at a rate determined by the amount of viscous damping. The motion is detected with a sensitive transducer, often a microsyn, and transformed into a proportional voltage. This voltage is amplified and applied to the antenna actuator in a direction to reduce the rate error. Aircraft rates are also compensated in the same way. A principal difference between the integrating gyro and the rate gyro control loop is that the steady-state error in the integrating gyro loop is zero" for a constant rate command while the steady-state error in the rate gyro loop has a finite value proportional to the rate command and inversely proportional to the d-c loop gain. However, this steady-state error in the rate loop can be reduced to a negligibly small value without great difficulty. The integrating gyro must have compensating networks to make it stable, whereas the loop with the rate gyro may not need compensating networks, depending on the degree of performance required. Practically, some compensation is usual in both forms of the stabilization loop. Another difference in performance between stabilization loops using the integrating gyro and those using the rate gyro is incident to the saturating characteristics of the gyro. Large steady-state rates do not usually saturate the integrating gyro, because the gyro gimbal is in the forward path of the loop and its motion is proportional to the rate error, which is small. Even if it should saturate, it does not change the loop performance appreciably, because changes in the forward gain of a feedback control loop do not affect the overall loop characteristics or its performance significantly. On the other hand, the rate gyro saturation does occur for large, steady rates and its measuring and performance characteristics effectively change. This is not desirable, nor even allowable in most cases, because the rate gyro must measure antenna space rates accurately for the fire-control computer, it must provide a stabilizing signal proportional to the antenna rate at all times to prevent drift, and it must have the proper transfer function to provide track loop stability. This may be particularly serious for systems using guns because the large random space rates induced in the antenna during periods of gun fire cause rate gyro saturation. Consequently, it is necessary to provide rate gyros with linear rate measuring ranges far in excess of those needed to measure the aircraft space rates for the fire-control computer. Unfortunately, a large, linear measuring range reduces the accuracy of the gyro in the important low rate region. However, with missiles for armament and jet aircraft as an antenna platform, this problem is not serious. In recent years, HIG gyros, '^^ fluid damped and hermetically sealed, have been commonly used in antenna stabilizing loops. They are extremely S'^This neglects the effects of actuator stiction which can make the loop nonlinear and produce a small error. ^Hermetic Integrating Gyros developed at the Massachusetts Institute of Technology. 8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 457 accurate, with low drift rates, but they are expensive and require tempera- ture-compensating circuits to maintain the preset value of damping torque. Generally, this accuracy is not needed to provide antenna stabilization, but is needed for the fire-control computer which uses the gyro signals to provide accurate information about the line-of-sight motion. Aircraft Gyro Stabilization Loops. Another of the many methods of providing antenna stabilization is shown in Fig. 8-37c. Because the aircraft with a fire control system and tracking antenna frequently has autopilot gyroscopes, it is possible to use them to provide the antennas with stabilization signals. This reduces the total number of rate gyros needed in the system, but other components must be added as shown in the simplified figure. ^^ The other components are tachometer and position indicators on the azimuth and elevation antenna actuators and a converter to change the space rate signals from three aircraft coordinates to two antenna coor- dinates. Although the converter is not complex, the conversions must be made accurately, and the three aircraft gyros should be mounted near the antenna base to prevent discrepancies from occurring in the rate measure- ments which are different in various parts of the aircraft owing to structural flexing. From this discussion it is evident that several methods of providing antenna stabilization are available. Although the integrating gyro may have slight advantages over the other systems, the ultimate choice of a system configuration will depend upon the required accuracy, the allowable expense, and the permitted complexity of the application. 8-32 ACCURACY REQUIREMENTS ON THE ANGLE TRACK STABILIZATION LOOP The stabilization accuracy requirements for the angle track stabilization loop are determined from the basic functions of the stabilization loops (Paragraphs 8-25 and 8-30) and from different criteria depending on whether a pilot or autopilot is used, a lead collision course or pursuit course is being flown, or a ballistic or target-seeking missile is being fired. The significance of these factors is discussed in the following paragraphs. Stabilization Accuracy Required to Reduce Aircraft Rate Signal Errors. The magnitude of the error in the antenna rate signal caused by own-ship's motion must be reduced to an acceptable value. This value depends on the nature of the system. If an autopilot is used, it must be ^^Note that there are three aircraft gyros which do not measure the desired antenna rates In azimuth and elevation. Therefore the information in three channels must be converted to two channels using the relative rates and angles of the antenna with respect to aircraft in two dimensions. For simplicity the figure depicts a one-channel conversion. 458 REGULATORY CIRCUITS defined from a specified system error, the allowable miss distance, or hit probability density.^" Actually, for an autopilot system, the criterion involving the magnitude of the rate error due to aircraft motion may be less important than a criterion concerning the amount of attenuation needed from the stabilization loop to prevent system instability through the coupling between the rate commands, autopilot, aircraft, and antenna. In the case of manual control of the aircraft flight path, the specification of the allowable stabilization error is governed by the following basic observations. (1) The stabilization error may be considered as a random error in the measurement of angular rate. Thus the low-frequency component of the stabilization error must be compatible with the specification for angular rate measurement accuracy to avoid inaccuracies in the fire-control compu- tation. (2) The high-frequency components of the stabilization error (greater than 1—2 rad/sec) do not affect the fire-control problem directly because the aircraft heading cannot change this rapidly. However, they do increase the apparent amount of noise on the steering indication, and this does degrade the pilot's ability to fly an accurate course (see Paragraph 12-7).'*^ This degradation is proportional to the rms contribution of stabilization error to the total apparent noise appearing on the pilot's indicator. An example using the basic AI radar problem presented in Chapter 2 is informative in illustrating how these principles might be applied. The applicable specification for the azimuth and elevation channels are (see Paragraph 2-27) : Allowable random error in rate measurement 0.11° /sec rms Allowable magnitude of indicator noise 1° rms Allowable filtering 0.5 second. For purposes of analysis, we will assume that the contributions of the stabilization error should be limited to the following: [Low frequency rate measurement] 7^ ^ ^ ^^n / , , ... . \ Kl< 0.03 /sec rms error due to stabilization errors J [Indicator noise due to high 1 ,. ^ ■, ^ro / c 1 -r • Al S 1.25 /sec rms frequency stabilization errors] ''"For a detailed discussion of this problem see "Control System Optimization to Achieve Maximum Hit or Accuracy Probability Density" by G. S. Axelby, Wescon Record of the IRE, 1957. ''^The fact that the stabilization errors are actually correlated with the pilot-induced motions does not seem to be important (as it is for autopilot applications where the correlation results in degradation of system stability). Thus for analysis of manually flown systems, the rms stabilization error must be combined with the rms noise errors from other sources to produce an equivalent noise error. 8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 459 The problem now is to compute the stabilization loop attenuation Ks as a function of frequency needed to achieve these levels of performance. Table 8-3 will be used to provide aircraft input data. For the low-frequency (less than 2 rad) inputs where 0.7 Atl is the rms value of the sinusoidal inputs from Table 8-3, for example, at co = 1.04, Jtl = 35/2 = 17.5°/sec and Kg < 0.00245'. However, for the high-frequency inputs, the effects of filtering and error sensitivity as a function of angular rate must be taken into account to ascertain the amount of stabilization attenuation needed. The basic expression may be written (J.1Atl\S^) {deHc/oATL)^f where dene Id At l = sensitivity of computed error signal to angular rate inputs Gf = rate noise filter characteristic. From Table 2-3 the value of the angular rate sensitivity factor at the time of firing on a head-on course is 14.2. If a 0.5-second filter is used, all sinusoidal signals above 3 rad /sec passing through it are attenuated by a factor co//cos, where cos is any signal frequency and CO/ is the filter corner frequency equal to 2 rad /sec. Therefore, if Kl is assumed to be equal to 1.25° rms for sinusoidal frequencies and Gf equal to co//co£) for disturbance frequencies greater than 3 rad /sec, L25 ^ 0.125 ^' ^ (0.7)(^tl)(14.2)(co//co^) ~ Wcoz>)^rL' ^^'^^^ However, for sinusoidal motion Atl = Atl^d and ,, . 0.125 0.0625 -'^'S S — -J— = —2 — Wf/lTL ^TL (8-44) Using the values for Atl in Table 8-3, the desired Ks is given in the following table as a function of frequency. A plot of Ks is shown in Fig. 8-39. This is the minimum attenuation that must be provided by the stabilization loop if rate signal errors caused by aircraft motion are to be maintained below acceptable levels on the pilot's indicator. The isolation required for the low-frequency inputs is also indicated. The design of the stabilization loop characteristics to achieve this attenuation is described in Paragraph 8-33. 460 REGULATORY CIRCUITS Table 8-4 Ks AS A FUNCTION OF FREQUENCY ^TL CO/) Ks 9.55/2 3.14 0.013 6.04/2 3.31 0.021 4.77/2 4.19 0.026 4.48/2 4.92 0.028 0.497/2 10.5 0.25 0.715/2 10.5 0.174 1.0 0.1 0.01 / •/ / 7 , , , mITI / '} / -yv. — stabilization Loop Must Have ::: "Attenuation Cliaracteristic / MM ' / 1 •7^ 1 1 1 1 ..,....nJ... 'n ' 'JJIII / b yMe H surement Ac curac Ke q't 10 cOp (rad/sec) 100 Fig. 8-39 Desired Stabilization Loop Attenuation. The magnitude of the error in the antenna rate signal may be specified in other ways. For example, it may be determined as the allowable error in an input to the course computer from a study of overall system error specifi- cation. In this case the total rate error may be specified where the total rate error includes tracking loop inaccuracy as well as unwanted aircraft rates due to aircraft motion. The allowable total error depends on the overall accuracy of the system and it may vary from about 1 mil /sec to 4 mils /sec. In Chapter 9, where tracking loop inaccuracies are considered (Fig. 9-5) it is shown that for a total allowable random error of 0.15° /sec (2.62 mils /sec /channel), the rms error component due to the aircraft motion could be large enough to yield the same stabilization attenuation charac- teristic that is shown in Fig. 8-39. 8-32] ACCURACY REQUIREMENTS ON ANGLE TRACK STABILIZATION 461 Stabilization Accuracy Required to Reduce Position Errors Due to Aircraft Motion. The radar antenna must not deviate from its desired position within a specified angle incident to aircraft motion even if the radar signal should not be present for a short time because of temporary- loss of target return. In discussing the accuracy required to accomplish this, it will be necessary to discuss the position accuracy required and the temporary loss of target return which is often referred to as signal fading. The position accuracy needed again depends on the type of fire control being considered. In some missile systems where a proportional navigation course is flown, only antenna rates of the missile seeker are needed to compute the proper course; angular information is not used. Therefore, the antenna angle need not be maintained very accurately. It is only necessary to keep the antenna positioned toward the target within the antenna beamwidth, which may vary from 1° to 15° depending on the antenna size and radiating frequency. Actually, the stabilization accuracy needed to maintain antenna position is an order of magnitude less than that needed to provide the necessary attenuation of antenna motion in the rate signal; and therefore it is usually not used as a criterion for stabilization loop design for this type of application. In many fire-control systems, however, the antenna position is used to compute the course of the aircraft. From a system analysis, analytical or simulated, the random and bias errors in position may be specified. These values may range between 2 mils and 10 mils as described in Paragraph 2-27 for a typical lead collision course. The total random error should be less than 2.62 mils (0.15°) rms and the bias error should be less that 9.15 rms (0.525°). Tracking loop errors contribute to the total position error, but as will be shown in Chapter 9, where the tracking loop errors are considered, the position error component due to platform motion does not require the attenuation needed to reduce the corresponding rate errors to the specified values. Usual