Radar-Sonar Signal Processing and Gaussian Signals in Noise. Detection, Estimation, and Modulation Theory

Detection, Estimation, and Modulation Theory

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Detection, Estimation, and Modulation Theory Radar-Sonar Processing and Gaussian Signals in Noise

HARRY L. VAN TREES George Mason University

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Weinheim • Brisbane- • Singapore • Toronto

This text is printed on acid-free paper. © Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act. without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive. Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging in Publication Data is available. ISBN 0-471-10793-X Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To Diane and Stephen, Mark, Kathleen, Patricia, Eileen, Harry, and Julia and the next generation— Brittany, Erin, Thomas, Elizabeth, Emily, Dillon, Bryan, Julia, Robert, Margaret, Peter, Emma, Sarah, Harry, Rebecca, and Molly

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Preface for Paperback Edition

In 1968, Part I of Detection, Estimation, and Modulation Theory [VT68] was published. It turned out to be a reasonably successful book that has been widely used by several generations of engineers. There were thirty printings, but the last printing was in 1996. Volumes II and III ([VT71a], [VT71b]) were published in 1971 and focused on specific application areas such as analog modulation, Gaussian signals and noise, and the radar-sonar problem. Volume II had a short life span due to the shift from analog modulation to digital modulation. Volume III is still widely used as a reference and as a supplementary text. In a moment of youthful optimism, I indicated in the the Preface to Volume III and in Chapter III-14 that a short monograph on optimum array processing would be published in 1971. The bibliography lists it as a reference, Optimum Array Processing, Wiley, 1971, which has been subsequently cited by several authors. After a 30-year delay, Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory will be published this year. A few comments on my career may help explain the long delay. In 1972, MIT loaned me to the Defense Communication Agency in Washington, D.C. where I spent three years as the Chief Scientist and the Associate Director of Technology. At the end of the tour, I decided, for personal reasons, to stay in the Washington, D.C. area. I spent three years as an Assistant Vice-President at COMSAT where my group did the advanced planning for the INTELSAT satellites. In 1978, I became the Chief Scientist of the United States Air Force. In 1979, Dr. Gerald Dinneen, the former Director of Lincoln Laboratories, was serving as Assistant Secretary of Defense for C3I. He asked me to become his Principal Deputy and I spent two years in that position. In 1981,1 joined M/A-COM Linkabit. Linkabit is the company that Irwin Jacobs and Andrew Viterbi had started in 1969 and sold to M/A-COM in 1979. I started an Eastern operation which grew to about 200 people in three years. After Irwin and Andy left M/A-COM and started Qualcomm, I was responsible for the government operations in San Diego as well as Washington, D.C. In 1988, M/ACOM sold the division. At that point I decided to return to the academic world. I joined George Mason University in September of 1988. One of my priorities was to finish the book on optimum array processing. However, I found that I needed to build up a research center in order to attract young research-oriented faculty and

vn

viii

Preface for Paperback Edition

doctoral students. The process took about six years. The Center for Excellence in Command, Control, Communications, and Intelligence has been very successful and has generated over $300 million in research funding during its existence. During this growth period, I spent some time on array processing but a concentrated effort was not possible. In 1995, I started a serious effort to write the Array Processing book. Throughout the Optimum Array Processing text there are references to Parts I and III of Detection, Estimation, and Modulation Theory. The referenced material is available in several other books, but I am most familiar with my own work. Wiley agreed to publish Part I and III in paperback so the material will be readily available. In addition to providing background for Part IV, Part I is still useful as a text for a graduate course in Detection and Estimation Theory. Part III is suitable for a second level graduate course dealing with more specialized topics. In the 30-year period, there has been a dramatic change in the signal processing area. Advances in computational capability have allowed the implementation of complex algorithms that were only of theoretical interest in the past. In many applications, algorithms can be implemented that reach the theoretical bounds. The advances in computational capability have also changed how the material is taught. In Parts I and III, there is an emphasis on compact analytical solutions to problems. In Part IV, there is a much greater emphasis on efficient iterative solutions and simulations. All of the material in parts I and III is still relevant. The books use continuous time processes but the transition to discrete time processes is straightforward. Integrals that were difficult to do analytically can be done easily in Matlab*. The various detection and estimation algorithms can be simulated and their performance compared to the theoretical bounds. We still use most of the problems in the text but supplement them with problems that require Matlab® solutions. We hope that a new generation of students and readers find these reprinted editions to be useful. HARRY L. VAN TREES Fairfax, Virginia June 2001

Preface

In this book I continue the study of detection, estimation, and modulation theory begun in Part I [1]. I assume that the reader is familiar with the background of the overall project that was discussed in the preface of Part I. In the preface to Part II [2] I outlined the revised organization of the material. As I pointed out there, Part III can be read directly after Part I. Thus, some persons will be reading this volume without having seen Part II. Many of the comments in the preface to Part II are also appropriate here, so I shall repeat the pertinent ones. At the time Part I was published, in January 1968,1 had completed the "final" draft for Part II. During the spring term of 1968, I used this draft as a text for an advanced graduate course at M.I.T. and in the summer of 1968, 1 started to revise the manuscript to incorporate student comments and include some new research results. In September 1968, I became involved in a television project in the Center for Advanced Engineering Study at M.I.T. During this project, I made fifty hours of videotaped lectures on applied probability and random processes for distribution to industry and universities as part of a self-study package. The net result of this involvement was that the revision of the manuscript was not resumed until April 1969. In the intervening period, my students and I had obtained more research results that I felt should be included. As I began the final revision, two observations were apparent. The first observation was that the manuscript has become so large that it was economically impractical to publish it as a single volume. The second observation was that since I was treating four major topics in detail, it was unlikely that many readers would actually use all of the book. Because several of the topics can be studied independently, with only Part I as background, I decided to divide the material into three sections: Part II, Part III, and a short monograph on Optimum Array Processing [3]. This division involved some further editing, but I felt it was warranted in view of increased flexibility it gives both readers and instructors.

x

Preface

In Part II, I treated nonlinear modulation theory. In this part, I treat the random signal problem and radar/sonar. Finally, in the monograph, I discuss optimum array processing. The interdependence of the various parts is shown graphically in the following table. It can be seen that Part II is completely separate from Part III and Optimum Array Processing. The first half of Optimum Array Processing can be studied directly after Part I, but the second half requires some background from Part III. Although the division of the material has several advantages, it has one major disadvantage. One of my primary objectives is to present a unified treatment that enables the reader to solve problems from widely diverse physical situations. Unless the reader sees the widespread applicability of the basic ideas he may fail to appreciate their importance. Thus, I strongly encourage all serious students to read at least the more basic results in all three parts. Prerequisites Part II

Chaps. 1-5, 1-6

Part III Chaps. III-l to III-5 Chaps. III-6 to III-7 Chaps. III-8-end

Chaps. 1-4, 1-6 Chaps. 1-4 Chaps. 1-4, 1-6, III-l to III-7

Array Processing Chaps. IV-1, IV-2 Chaps. IV-3-end

Chaps. 1-4 Chaps. III-l to III-5, AP-1 to AP-2

The character of this book is appreciably different that that of Part I. It can perhaps be best described as a mixture of a research monograph and a graduate level text. It has the characteristics of a research monograph in that it studies particular questions in detail and develops a number of new research results in the course of this study. In many cases it explores topics which are still subjects of active research and is forced to leave some questions unanswered. It has the characteristics of a graduate level text in that it presents the material in an orderly fashion and develops almost all of the necessary results internally. The book should appeal to three classes of readers. The first class consists of graduate students. The random signal problem, discussed in Chapters 2 to 7, is a logical extension of our earlier work with deterministic signals and completes the hierarchy of problems we set out to solve. The

Preface

xi

last half of the book studies the radar/sonar problem and some facets of the digital communication problem in detail. It is a thorough study of how one applies statistical theory to an important problem area. I feel that it provides a useful educational experience, even for students who have no ultimate interest in radar, sonar, or communications, because it demonstrates system design techniques which will be useful in other fields. The second class consists of researchers in this field. Within the areas studied, the results are close to the current research frontiers. In many places, specific research problems are suggested that are suitable for thesis or industrial research. The third class consists of practicing engineers. In the course of the development, a number of problems of system design and analysis are carried out. The techniques used and results obtained are directly applicable to many current problems. The material is in a form that is suitable for presentation in a short course or industrial course for practicing engineers. I have used preliminary versions in such courses for several years. The problems deserve some mention. As in Part I, there are a large number of problems because I feel that problem solving is an essential part of the learning process. The problems cover a wide range of difficulty and are designed to both augment and extend the discussion in the text. Some of the problems require outside reading, or require the use of engineering judgement to make approximations or ask for discussion of some issues. These problems are sometimes frustrating to the student but I feel that they serve a useful purpose. In a few of the problems I had to use numerical calculations to get the answer. I strongly urge instructors to work a particular problem before assigning it. Solutions to the problems will be available in the near future. As in Part I, I have tried to make the notation mnemonic. All of the notation is summarized in the glossary at the end of the book. I have tried to make my list of references as complete as possible and acknowledge any ideas due to other people. Several people have contributed to the development of this book. Professors Arthur Baggeroer, Estil Hoversten, and Donald Snyder of the M.I.T. faculty, and Lewis Collins of Lincoln Laboratory, carefully read and criticized the entire book. Their suggestions were invaluable. R. R. Kurth read several chapters and offered useful suggestions. A number of graduate students offered comments which improved the text. My secretary, Miss Camille Tortorici, typed the entire manuscript several times. My research at M.I.T. was partly supported by the Joint Services and by the National Aeronautics and Space Administration under the auspices of the Research Laboratory of Electronics. I did the final editing

xii

Preface

while on Sabbatical Leave at Trinity College, Dublin. Professor Brendan Scaife of the Engineering School provided me office facilities during this period, and M.I.T. provided financial assistance. I am thankful for all of the above support. Harry L. Van Trees Dublin, Ireland,

REFERENCES [1] Harry L. Van Trees, Detection, Estimation, and Modulation Theory, Pt. I, Wiley, New York, 1968. [2] Harry L. Van Trees, Detection, Estimation, and Modulation Theory, Pt. II, Wiley, New York, 1971. [3] Harry L. Van Trees, Optimum Array Processing, Wiley, New York, 1971.

Contents

1 Introduction

1

1.1 Review of Parts I and II 1.2 Random Signals in Noise 1.3 Signal Processing in Radar-Sonar Systems

1 3 6

References

1

2 Detection of Gaussian Signals in White Gaussian Noise

8

2.1 Optimum Receivers 2.1.1 Canonical Realization No. 1: Estimator-Correlator 2.1.2 Canonical Realization No. 2: Filter-Correlator Receiver 2.1.3 Canonical Realization No. 3: Filter-Squarer-Integrator (FSI) Receiver 2.1.4 Canonical Realization No. 4: Optimum Realizable Filter Receiver 2.1.5 Canonical Realization No. AS: State-variable Realization 2.1.6 Summary: Receiver Structures 2.2 Performance 2.2.1 Closed-form Expression for //(s) 2.2.2 Approximate Error Expressions 2.2.3 An Alternative Expression for JUR(S) 2.2.4 Performance for a Typical System 2.3 Summary: Simple Binary Detection 2.4 Problems

9 15

23 31 32 35 38 42 44 46 48

References

54

16 17 19

xiv

Contents

3 General Binary Detection: Gaussian Processes 3.1 Model and Problem Classification 3.2 Receiver Structures 3.2.1 Whitening Approach 3.2.2 Various Implementations of the Likelihood Ratio Test 3.2.3 Summary: Receiver Structures 3.3 Performance 3.4 Four Special Situations 3.4.1 Binary Symmetric Case 3.4.2 Non-zero Means 3.4.3 Stationary "Carrier-symmetric" Bandpass Problems 3.4.4 Error Probability for the Binary Symmetric Bandpass Problem 3.5 General Binary Case: White Noise Not Necessarily Present : Singular Tests 3.5.1 Receiver Derivation 3.5.2 Performance: General Binary Case 3.5.3 Singularity 3.6 Summary: General Binary Problem 3.7 Problems References

4 Special Categories of Detection Problems 4.1 Stationary Processes: Long Observation Time 4.1.1 Simple Binary Problem 4.1.2 General Binary Problem 4.1.3 Summary: SPLOT Problem 4.2 Separable Kernels 4.2.1 Separable Kernel Model 4.2.2 Time Diversity 4.2.3 Frequency Diversity 4.2.4 Summary: Separable Kernels 4.3 Low-Energy-Coherence (LEC) Case 4.4 Summary 4.5 Problems References

56 56 59 59 61 65 66 68 69 72 74 77 79 80 82 83 88 90 97

99 99 100 110 119 119 120 122 126 130 131 137 137 145

Contents

xv

Discussion: Detection of Gaussian Signals

147

5.1 Related Topics 5.1.1 M-ary Detection: Gaussian Signals in Noise 5.1.2 Suboptimum Receivers 5.1.3 Adaptive Receivers 5.1.4 Non-Gaussian Processes 5.1.5 Vector Gaussian Processes 5.2 Summary of Detection Theory 5.3 Problems References

147 147 151 155 156 157 157 159 164

Estimation of the Parameters of a Random Process

167

6.1 Parameter Estimation Model 6.2 Estimator Structure 6.2.1 Derivation of the Likelihood Function 6.2.2 Maximum Likelihood and Maximum A-Posteriori Probability Equations 6.3 Performance Analysis 6.3.1 A Lower Bound on the Variance 6.3.2 Calculation of J™(A) 6.3.3 Lower Bound on the Mean-Square Error 6.3.4 Improved Performance Bounds 6.4 Summary 6.5 Problems References

168 170 170

Special Categories of Estimation Problems

188

7.1 Stationary Processes: Long Observation Time 7.1.1 General Results 7.1.2 Performance of Truncated Estimates 7.1.3 Suboptimum Receivers 7.1.4 Summary 7.2 Finite-State Processes 7.3 Separable Kernels 7.4 Low-Energy-Coherence Case

188 189 194 205 208 209 211 213

175 177 177 179 183 183 184 185 186

xvi Contents

7.5 Related Topics 7.5.1 Multiple-Parameter Estimation 7.5.2 Composite-Hypothesis Tests 7.6 Summary of Estimation Theory 7.7 Problems References

8 The Radar-sonar Problem References

Detection of Slowly Fluctuating Point Targets

217 217 219 220 221 232

234 237

238

9.1 9.2 9.3 9.4

Model of a Slowly Fluctuating Point Target 238 White Bandpass Noise 244 Colored Bandpass Noise 247 Colored Noise with a Finite State Representation 251 9.4.1 Differential-equation Representation of the Optimum Receiver and Its Performance: I 252 9.4.2 Differential-equation Representation of the Optimum Receiver and Its Performance: II 253 9.5 Optimal Signal Design 258 9.6 Summary and Related Issues 260 9.7 Problems 263 References 273

10 Parameter Estimation: Slowly Fluctuating Point Targets 275 10.1 Receiver Derivation and Signal Design 275 10.2 Performance of the Optimum Estimator 294 10.2.1 Local Accuracy 294 10.2.2 Global Accuracy (or Ambiguity) 302 10.2.3 Summary 307 10.3 Properties of Time-Frequency Autocorrelation Functions and Ambiguity Functions 308

Contents xvii

10.4 Coded Pulse Sequences 10.4.1 On-off Sequences 10.4.2 Constant Power, Amplitude-modulated Waveforms 10.4.3 Other Coded Sequences 10.5 Resolution 10.5.1 Resolution in a Discrete Environment: Model 10.5.2 Conventional Receivers 10.5.3 Optimum Receiver: Discrete Resolution Problem 10.5.4 Summary of Resolution Results 10.6 Summary and Related Topics 10.6.1 Summary 10.6.2 Related Topics 10.7 Problems References

11 Doppler-Spread Targets and Channels 11.1 Model for Doppler-Spread Target (or Channel) 11.2 Detection of Doppler-Spread Targets 11.2.1 Likelihood Ratio Test 11.2.2 Canonical Receiver Realizations 11.2.3 Performance of the Optimum Receiver 11.2.4 Classes of Processes 11.2.5 Summary 11.3 Communication Over Doppler-Spread Channels 11.3.1 Binary Communications Systems: Optimum Receiver and Performance 11.3.2 Performance Bounds for Optimized Binary Systems 11.3.3 Suboptimum Receivers 11.3.4 M-ary Systems 11.3.5 Summary: Communication over Dopplerspread Channels 11.4 Parameter Estimation: Doppler-Spread Targets 11.5 Summary: Doppler-Spread Targets and Channels 11.6 Problems References

313 313 314 323 323 324 326 329 335 336 336 337 340 352

357 360 365 366 367 370 372 375 375 376 378 385 396 397 398 401 402 411

xviii

Contents

12 Range-Spread Targets and Channels 12.1 Model and Intuitive Discussion 12.2 Detection of Range-Spread Targets 12.3 Time-Frequency Duality 12.3.1 Basic Duality Concepts 12.3.2 Dual Targets and Channels 12.3.3 Applications 12.4 Summary: Range-Spread Targets 12.5 Problems References

13 Doubly-Spread Targets and Channels 13.1 Model for a Doubly-Spread Target 13.1.1 Basic Model 13.1.2 Differential-Equation Model for a DoublySpread Target (or Channel) 13.1.3 Model Summary 13.2 Detection in the Presence of Reverberation or Clutter (Resolution in a Dense Environment) 13.2.1 Conventional Receiver 13.2.2 Optimum Receivers 13.2.3 Summary of the Reverberation Problem 13.3 Detection of Doubly-Spread Targets and Communication over Doubly-Spread Channels 13.3.1 Problem Formulation 13.3.2 Approximate Models for Doubly-Spread Targets and Doubly-Spread Channels 13.3.3 Binary Communication over Doubly-Spread Channels 13.3.4 Detection under LEC Conditions 13.3.5 Related Topics 13.3.6 Summary of Detection of Doubly-Spread Signals 13.4 Parameter Estimation for Doubly-Spread Targets 13.4.1 Estimation under LEC Conditions 13.4.2 Amplitude Estimation 13.4.3 Estimation of Mean Range and Doppler 13.4.4 Summary

413 415 419 421 422 424 427 437 438 443

444 446 446 454 459 459 461 472 480 482 482 487 502 516 521 525 525 527 530 533 536

Contents

13.5 Summary of Doubly-Spread Targets and Channels 13.6 Problems References

14 Discussion

xix

536 538 553

558

14.1 Summary: Signal Processing in Radar and Sonar Systems 558 14.2 Optimum Array Processing 563 14.3 Epilogue 564 References 564

Appendix: Complex Representation of Bandpass Signals, Systems, and Processes 565 A.I Deterministic Signals A.2 Bandpass Linear Systems A.2.1 Time-Invariant Systems A.2.2 Time-Varying Systems A.2.3 State-Variable Systems A. 3 Bandpass Random Processes A.3.1 Stationary Processes A.3.2 Nonstationary Processes A.3.3 Complex Finite-State Processes A.4 Summary A.5 Problems References

566 572 572 574 574 576 576 584 589 598 598 603

Glossary

605

Author Index

619

Subject Index

623

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1 Introduction

This book is the third in a set of four volumes. The purpose of these four volumes is to present a unified approach to the solution of detection, estimation, and modulation theory problems. In this volume we study two major problem areas. The first area is the detection of random signals in noise and the estimation of random process parameters. The second area is signal processing in radar and sonar systems. As we pointed out in the Preface, Part III does not use the material in Part II and can be read directly after Part I. In this chapter we discuss three topics briefly. In Section 1.1, we review Parts I and II so that we can see where the material in Part III fits into the over-all development. In Section 1.2, we introduce the first problem area and outline the organization of Chapters 2 through 7. In Section 1.3, we introduce the radar-sonar problem and outline the organization of Chapters 8 through 14.

1.1 REVIEW OF PARTS I AND II

In the introduction to Part I [1], we outlined a hierarchy of problems in the areas of detection, estimation, and modulation theory and discussed a number of physical situations in which these problems are encountered. We began our technical discussion in Part I with a detailed study of classical detection and estimation theory. In the classical problem the observation space is finite-dimensional, whereas in most problems of interest to us the observation is a waveform and must be represented in an infinite-dimensional space. All of the basic ideas of detection and parameter estimation were developed in the classical context. In Chapter 1-3, we discussed the representation of waveforms in terms of series expansions. This representation enabled us to bridge the gap

2 1.1 Review of Parts I and II

between the classical problem and the waveform problem in a straightforward manner. With these two chapters as background, we began our study of the hierarchy of problems that we had outlined in Chapter 1-1. In the first part of Chapter 1-4, we studied the detection of known signals in Gaussian noise. A typical problem was the binary detection problem in which the received waveforms on the two hypotheses were r(f) = *(/) + »(0,

Tt£t£Tf:Hlt

(1)

r(t) = s0(t) + n(t),

T,
(7)

10 2.1 Optimum Receivers

4. We denote the limit of A(rK(t)) as A(r(/)). The test consists of comparing the likelihood ratio with a threshold rj, A[r(t)] ^77.

(8)

As before, the threshold rj is determined by the costs and a-priori probabilities in a Bayes test and the desired PF in a Neyman-Pearson test. We now carry out these steps in detail and then investigate the properties of the resulting tests. The orthonormal functions for the series expansion are the eigenfunctions of the integral equation! A WO = f TfKs(t, uty,(u) du,

T,

If we are using a Bayes test, we must evaluate the infinite sum on the right side in order to set the threshold. On page 22 we develop a convenient closed-form expression for this sum. For the Neyman-Pearson test we adjust y* directly to obtain the desired PF so that the exact value of the sum is not needed as long as we know the sum converges. The convergence follows easily. oo

/

91 s\

oo 9; s

^

fTf

2 In \ 1 +NO/ ^ < Jn Y*Ho

The second term on the left side of (35) is generated physically by either a cross-correlation or a matched filter operation, as shown in Fig. 2.1. The impulse response of the matched filter in Fig. 2.16 is

,.

h(j) =

(0,

(36)

elsewhere.

We previously encountered these operations in the colored noise detection problem discussed in Section 1-4.3. Thus, the only new component in the optimum receiver is a device to generate 1R. In the next several paragraphs we develop a number of methods of generating 1R.

r* JT. (a)

r(t)

h(r) (b)

Fig. 2.1 Generation of 1D.

•ID

Canonical Realization No. I: Estimator-correlator 2.1.1

15

Canonical Realization No. 1: Estimator-Correlator

We want to generate 1R, where Tt

OK")

"oJ T( d(

Filter-squarer receiver (realizable).

The difficulty with all of the configurations that we have derived up to this point is that to actually implement them we must solve (23). From our experience in Chapter 1-4 we know that we can do this for certain classes of kernels and certain conditions on Ti and Tf. We explore problems of this type in Chapter 4. On the other hand, in Section 1-6.3 we saw that whenever the processes could be generated by exciting a linear finitedimensional dynamic system with white noise, we had an effective procedure for solving (44). Fortunately, many of the processes (both nonstationary and stationary) that we encounter in practice have a finite-dimensional state representation. In order to exploit the effective computation procedures that we have developed, we now modify our results to obtain an expression for 1R in which the optimum realizable linear filter specified by (44) is the only filter that we must find. 2.1.4

Canonical Realization No. 4: Optimum Realizable Filter Receiver

The basic concept involved in this realization is that of generating the likelihood ratio in real time as the output of a nonlinear dynamic system.f The derivation is of interest because the basic technique is applicable to many problems. For notational simplicity, we let Tt = 0 and Tf = T in this section. Initially we shall assume that m(t) = 0 and consider only 1R. Clearly, 1R is a function of the length of the observation interval T. To emphasize this, we can write 1R(T | r(«), 0 < u < T) A 1R(T).

(52)

More generally, we could define a likelihood function for any value of time /. /*(' r(u), 0 < u < 0 A 7B(0, (53) where 7^(0) = 0. We can write 1R(T) as

at

Mdt. (54)

t The original derivation of (66) was done by Schweppe [5]. The technique is a modification of the linear filter derivation in [6].

20 2.1

Optimum Receivers

Now we want to find an easy method for generating lR(t). Replacing Tby t in (31), we have ln(t) = ^ [dr KT) f *du h,(r, it: t)r(u), A/o Jo

(55)

Jo

where /^(T, u:t) satisfies the integral equation

%* hfr, ii: f) + fW, *: OK/*, ") «k = K.O-, M), 2 Jo

0 < T, w < r. (56)

[Observe that the solution to (56) depends on t. We emphasize this with the notation h^-, •:/)•] Differentiating (55), we obtain

-

(57)

We see that the first two terms in (57) depend on h±(t, u:t). For this case, (56) reduces to — ° ht(t, ii : r) + f W', « : 0/C/z, «) dz = X. (t, «), Jo

2

0 < u < t.

(58)f

We know from our previous work in Chapter 1-6 that

or

4(0 = fW',":OK»)rf» Jo

(59)

Sr(t) = f hj(u, f : OK") ^ w -

(60)

Jo

[The subscript r means that the operation in (59) can be implemented with a realizable filter.] The result in (60) follows from the symmetry of the solution to (56). Using (59) and (60) in (57) gives iM = T^KO-UO + [dr [du r(r) ^ u : t > r(u)l . NQ\_ Jo Jo dt J

(61)

In Problem 1-4.3.3, we proved that a/>l(

^" :r) = - ^(r, f : 0/»i(r, « : 0, at

0 < r, u < f .

Because the result is the key step, we include the proof (from [7]). t Notice that h^t, u:t) = hor(t,u) [compare (44) and (58)].

(62)

Canonical Realization No. 4: Optimum Realizable Filter Receiver 21 Proof of (62). Differentiating (56) gives ^o 8 ifr."-0 + r ^i(r,z.t) 2 a/ Jo dt

w)

^+

t_

.)/j: (/>|() = 0>

o < r, « < /.

(63) Now replace Ks(t, u) with the left side of (58) and rearrange terms. This gives

x K,(z, u) dz,

0 .s(0 dt.

(72)

From (33),

o 3 s\

1 rT

^-) = --7 N0 / N0 Jo

(73)

We see that whenever we use Canonical Realization No. 4, we obtain the first bias term needed for the Bayes test as a by-product. The second bias term [see (34)] is due to the mean, and its computation will be discussed shortly. A block diagram of Realization No. 4 for generating 1R and /^1] is shown in Fig. 2.7. Before leaving our discussion of the bias term, some additional comments are in order. The infinite sum of the left side of (72) will appear in several different contexts, so that an efficient procedure for evaluating it is important. It can also be written as the logarithm of the Fredholm

Canonical Realization No. 4S: State-variable Realization !

+ HH N •£- I 0

Optimum realizable filter Fig. 2.7

23

H fI dt dt I—»• ^ + /a JQ

?

Optimum realizable filter realization (Canonical Realization No. 4).

determinant [8], oo

2 In

/

*-=i \

91 s\

r

/ oo

\n

1 + ^) = In lid + *V)

NO'

L l«=i

JJ*=2AVo

/ 9 \

±1" M^ •

\N 0 /

(74)

Now, unless we can find D^(2lN0) effectively, we have not made any progress. One procedure is to evaluate !Ps(0 and use the integral expression on the right side of (73). A second procedure for evaluating D&(-} is a by-product of our solution procedure for Fredholm equations for certain signal processes (see the Appendix in Part II). A third procedure is to use the relation »

/

?A S \

»=1

\

N0/

f 2/A'°'

2 ln ! + — =

Tf

(75)

JO

where h^t, t \ z) is the solution to (23) when N0/2 equals z. Notice that this is the optimum unrealizable filter. This result is derived in Problem 2.1.2. The choice of which procedure to use depends on the specific problem. Up to this point in our discussion we have not made any detailed assumptions about the signal process. We now look at Realization No. 4 for signal processes that can be generated by exciting a finite-dimensional linear system with white noise. We refer to the corresponding receiver as Realization No. 4S ("S" denotes "state")2.1.5

Canonical Realization No. 4S: State-variable Realization

The class of signal processes of interest was described in detail in Section 1-6.3 (see pages I-516-I-538). The process is described by a state equation, 0 + G(0u(0,

(76)

24 2.1

Optimum Receivers

where F(0 and G(0 are possibly time-varying matrices, and by an observation equation, s(t) = C(0x(0, (77) where C(f) is the modulation matrix. The input, u(0, is a sample function from a zero-mean vector white noise process, £[u(0ur(r)] = Q d(t - r\ and the initial conditions are £[x(0)] = 0, r

£[x(0)x (0)] A P0.

(78) (79) (80)

From Section 1-6.3.2 we know that the MMSE realizable estimate of s(t) is given by the equations Sr(t) = C(0*(0,

(81)

x(0 = F(0*(0 + 5XOCT(0 j WO - C(0*(01-

(82)

The matrix £P(0 is the error covariance matrix of x(t) — x(/). 5X0 - E[(\(t) - x(0)(xr(0 - xT(0)]. It satisfies the nonlinear matrix differential equations,

(83)

5X0 = F(05XO + 5XOFT(0 - 5XOCr(01- C(05XO + G(OQGr(0. (84)

The mean-square error in estimating s(t) is

fp.(0 = C(05XOCT(0-

(85)

Notice that £p(0 is the error covariance matrix for the state vector and £Ps(f) is the scalar mean-square error in estimating s(t). Both (84) and (85) can be computed either before r(t} is received or simultaneously with the computation of x(f). The system needed to generate 1R and /g] follows easily and is shown in Fig. 2.8. The state equation describing 1R is obtained from (65), lR(t) = — [2r(t)sr(i) — sr\t)],

(86)

where sr(t) is defined by (81)-(84) and 1R A 1R(T).

(87)

Realization of

Matrix

CCt)tp(t)CTW= t,(t)

26 2.1 Optimum Receivers

The important feature of this realization is that there are no integral equations to solve. The likelihood ratio is generated as the output of a dynamic system. We now consider a simple example to illustrate the application of these ideas. Example. In Fig. 2.9 we show a hypothetical communication system that illustrates many of the important features encountered in actual systems operating over fading channels. In Chapter 10, we shall develop models for fading channels and find that the models are generalizations of the system in this example. When Hl is true, we transmit a deterministic signal /(/). When H0 is true, we transmit nothing. The channel affects the received signal in two ways. The transmitted signal is multiplied by a sample function of a Gaussian random process b(t). In many cases, this channel process will be stationary over the time intervals of interest. The output of the multiplicative part of the channel is corrupted by additive white Gaussian noise w(t), which is statistically independent of b(t). Thus the received waveforms on the two hypotheses are

r(t) =f(t)b(t) + w(t\

o < t < T-.HI,

r(t) = w(t),

0

(99)

Then (82) and (84) reduce to

|p(/) = -2*6fP(0 -

^o

and (100)

The resulting receiver structure is shown in Fig. 2.10.

We shall encounter other examples of Canonical Realization No. 4S as we proceed. Before leaving this realization, it is worthwhile commenting on the generation of 1D, the component in the likelihood ratio that arises because of the mean value in the signal process. If the process has a finite state representation, it is usually easier to generate 1D using the optimum realizable filter. The derivation is identical with that in (54)-(66). From (22) and (26)-(28) we have

2 / 1m T 1 I WO = — ( ) — "^r, u:T)m(u) du r(r) dr. N0 Jo L Jo J As before, r>Tdl (t) D dt, Jo Jo dt and dt

„. ... ,, Nn{_ \

. , Jo

(101)

(102)

,-m(t)\ hfr, t'.i)r(r)dr / J o ^I(T, f : Ofri(*> ": Ow(u)r(r)dr drdu du\.. (103)

The resulting block diagram is shown in Fig. 2. 1 1 . The output of the bottom path is just a deterministic function, which we denote by K(t), K(r) A. m(j) _- I hj(t hj(t,tu:f)m(u) u: Jo

du,

0 < t < T.

(104)

Because K(t) does not depend on r(t), we can generate it before any data are received. This suggests the two equivalent realizations in Fig. 2.12. Notice that (101) (and therefore Figs. 2.11 and 2.12) does not require that the processes be state-representable. If the processes have a finite state, the optimum realizable linear filter can be derived easily using state-variable techniques. Using the state representation in (76)-(80) gives

:

v

Optimum realizable filter Fig. 2.11

Generation of/^ using optimum realizable filters.

Sample at t=T

r(t) ^

2

*C\ _ I

VfT

N0K(T

-r^ T)

Matched filter

^

L- /*

Tr .*

\

Fig. 2.12

Generation of 1D.

29

30 2.1 Optimum Receivers

the realization in Fig. 2.13a.f Notice that the state vector in Fig. 2.13a is not x(r), because r(t) has a nonzero mean. We denote it by x(f). The block diagram in Fig. 2.13cr can be simplified as shown in Fig. 2.13b. We can also write lD(t) in a canonical state-variable form: 0

KO, *(0

0

x(0

F(0 ~ 77

NO

(105) where K(t) is defined in Fig. 2.11 and ^p(0 satisfies (84).

(a)

Fig. 2.13

State-variable realizations to generate 1D.

t As we would expect, the system in Fig. 3.12 is identical with that obtained using a whitening approach (e.g., Collins [9] or Problem 2.1.3).

Summary: Receiver Structures 31

Looking at (32) and (34), we see that their structure is identical. Thus, we can generate /^] by driving the dynamic system in (105) with (— w(0/2) instead of r(t). It is important to emphasize that the presence of m(t) does not affect the generation of lR(t) in (86). The only difference is that sr(t) and x(0 are no longer MMSE estimates, and so we denote them by sr(t) and \(t), respectively. The complete set of equations for the non-zero-mean case may be summarized as follows:

/>) = N77 MAO + 2KOUOL Q

(107) sr(0 = C(0x(f),

i(r) = F(0x(r) + -£- |p(OCr(0[KO - C(r)x(0], NO

(108)

(109)

with initial conditions x(0) = 0

(110)

and

/z>(0) = 0.

(Ill)

The matrix £P(0 is specified by (84). The biases are described in (73) and a modified version of (105). This completes our discussion of state-variable realizations of the optimum receiver for the Gaussian signal problem. We have emphasized structures based on realizable estimators. An alternative approach based on unrealizable estimator structures can also be developed (see Problem 1-6.6.4 and Problem 2.1.4). Before discussing the performance of the optimum receiver, we briefly summarize our results concerning receiver structures.

2.1.6 Summary: Receiver Structures In this section we derived the likelihood ratio test for the simple binary detection problem in which the received waveforms on the two hypotheses were r(f) = w(0, Tt < t < Tf:H9, r(0 = *(0 + w(0,

T^t^Tt'.Hi.

(112)

32 2.2 Performance

The result was the test /* +/JD fllni? -/g ] -/I?1,

(H3)

Ho

where the various terms were defined by (31)-(34). We then looked at four receivers that could be used to implement the likelihood ratio test. The first three configurations were based on the optimum unrealizable filter and required the solution of a Fredholm integral equation (23). In Chapter 4 we shall consider problems where this equation can be easily solved. The fourth configuration was based on an optimum realizable filter. For this realization we had to solve (44). For a large class of processes, specifically those with a finite state representation, we have already developed an efficient technique for solving this problem (the Kalman-Bucy technique). It is important to re-emphasize that all of the receivers implement the likelihood ratio test and therefore must have identical error probabilities. By having alternative configurations available, we may choose the one easiest to implement.! In the next section we investigate the performance of the likelihood ratio test.

2.2

PERFORMANCE

In this section we analyze the performance of the optimum receivers that we developed in Section 2.1. All of these receivers perform the test indicated in (35) as where Tr

•(t)hi(t, ii)r(u) dt dit,

(115)

Ti

ID = f \(«)KtO du,

and

(116)

JTi

f The reader may view the availability of alternative configurations as a mixed blessing, because it requires some mental bookkeeping to maintain the divisions between realizable and unrealizable filters, the zero-mean and non-zero-mean cases, and similar separations. The problems at the end of Chapter 4 will help in remembering the various divisions.

Performance

33

From (33) and (34), we recall that 7V/

(118) (119)

To compute PD and PF, we must find the probability that / will exceed y on Hl and H0, respectively. These probabilities are

and = f

/ B (L|// 0 ) oo, we denote it by 1RK, —

-—

(122)

We see that 1RK is a weighted sum of squared Gaussian random variables. The expression in (122) is familiar from our work on the general Gaussian problem, Section 1-2.6. In fact, if the Rt were zero-mean, (122) would be identical with (1-2.420). At that point we observed that if the A/ were all equal, 1RK had a chi-square density with K degrees of freedom (e.g., 1-2.406). On the other hand, for unequal A/, we could write an expression for the probability density but it was intractable for large K. Because of the independence of the R^ the characteristic function and momentgenerating function of 1RK followed easily (e.g., Problem 1-4.4.2). Given the characteristic function, we could, in principle at least, find the probability density by computing the Fourier transform numerically. In practice, we are usually interested in small error probabilities, and so we must know the tails of p^^L \ H^ accurately. This requirement causes the amount of computation required for accurate numerical inversion to be prohibitive. This motivated our discussion of performance bounds and approximations in Section 1-2.7. In this section we carry out an analogous discussion for the case in which K^- oo.

34 2.2

Performance

We recallf that the function pK(s) played the central role in our discussion. From (1-2.444), ^(5) A In tfl(R)|Ho(s)], (123) [The subscript K is added to emphasize that we are dealing with /T-term approximation to r(t}.} Where /(R) is the logarithm of the likelihood ratio

/(R) = In A(R) = in /WR I «J\ \P,IH,(R I «„)/

(124)

and KR)\Ho(s) i& its moment-generating function, 6,»,l//0(s) = E[«' I < E ) |//o], for real 5. Using the definition of/(R) in (124),

(125)

^(s) = In f°° • • • f°° [A|Hl(R | "itf [P,|ff.(R | "o)]1'* (*) - - 2 1

2

m, JV0/2(1 - s) + A,s '

(135)

We now find a closed-form expression for the sums in (134) and (135). First we consider JU>R(S). Both of the sums in (134) are related to realizable linear filtering errors. To illustrate this, we consider the linear filtering problem in which r(«) = j(«) + H-(«), T, < u < t, (136) where s(u) is a zero-mean message process with covariance function Ks(t, u) and the white noise has spectral height N0f2. Using our results in Chapter 1-6, we can find the linear filter whose output is the MMSE point estimate of s(-) and evaluate the resulting mean-square error. We denote this error as £P(t \ s(-), -Af0/2). (The reason for the seemingly awkward notation will be apparent in a moment.) Using (72), we can write the mean-square error in terms of a sum of eigenvalues. 2 in

i=l

\

l+

N0 /

-

N0 jTi

f,.|»(-).A. \

2

(137)

Comparing (134) and (137) leads to the desired result.

(138) Thus, to find f*R(s), we must find the mean-square error for two realizable linear filtering problems. In the first, the signal is s(-) and the noise is white with spectral height N0/2. In the second, the signal is s(-) and the noise is white with spectral height N0/2(l — s). An alternative expression for [4R(s) also follows easily.

(139)

Closed-form Expression for fi(s)

37

Here the noise level is the same in both calculations, but the amplitude of the signal process is changed. These equations are the first key results in our performance analysis. Whenever we have a signal process such that we can calculate the realizable mean-square filtering error for the problem of estimating s(t) in the presence of white noise, then we can find ftR(s}. The next step is to find a convenient expression for fiD(s). To evaluate the sum in (135), we recall the problem of detecting a known signal in colored noise, which we discussed in detail in Section 1-4.3. The received waveforms on the two hypotheses are r(0 = m(f) + nc(t) + w(t), r(t) = nc(t) + w(0,

Ti < t < Tf:H^ T; < / < 7> : H0.

(140)

By choosing the covariance function of nc(t) and w(t) appropriately, we can obtain the desired interpretation. Specifically, we let E(nc(t)nc(u)} = Ks(t, u),

T, 2), we may use the approximation to erfc^ (X) given in Fig. 2.10 of Part I. erfc* (X) ~

e-^2/2,

_L

X > 2.

(165)

VlTrX

Then (164) reduces to 1

(166)

This, of course, is the same answer we obtained when the central limit theorem was valid. The second term in the approximation is obtained by using /3(a) from (163). [2] _

F

Y*

„ t (13) implies that Tf

hwo(t, *)hwo(u, flKHt(*, j8) rfa # = d(t - ii).

(15)

The covariance function of r+(t) on //j is ?V £[r«,(0r.(«) | Hj] = JJfc^f,a)^o(u, j8)Kffl(a, 0)rfa# £ K*(f, u).

(16)

Ti

We now expand rj|t(f) using the eigenfunctions of K*(t, u), which are specified by the equation rzv h*i ^

'J?o < IUn'/• Ho

nw

\JV)

We see that the receiver can be realized as two simple binary receivers in parallel, with their outputs subtracted. Thus, any of the four canonic realizations developed in Section 2.1 (Figs. 2.2-2.7) can be used in each path. A typical structure using Realization No. 1 is shown in Fig. 3.3. This parallel processing structure is frequently used in practice. A second equivalent form of (22) is also useful. We define a function M>» «) A QHo(t, u) - QHi(t, u).

(31)

Then, Tt

*=*

r(t)h^t,u)r(u)dtdu.

(32)

Ti

To eliminate the inverse kernels in (31), we multiply by the two covariance

Various Implementations of the Likelihood Ratio Test

63

It

i

1 *0

f* JT.

'-.

Fig. 3.3

Parallel processing realization of general binary receiver (class Aw).

functions and integrate. The result is an integral equation specifying hA(t, u). Tf

f I dt duKHo(x, O/iA (f, u)KHl(u, z) = KHl(x, z) - KHo(x, z), Ti<x,z< Tf. This form of the receiver is of interest because the white noise level does not appear explicitly. Later we shall see that (32) and (33) specify the receiver for class GB problems. The receiver is shown in Fig. 3.4. Two other forms of the receiver are useful for class Bw problems. In this case, the received waveform contains the same noise process on both hypotheses and an additional signal process on H^ Thus, r(t) = 5(0 + nc(t} + w(t), r(t) = nc(t) + W(0,

T, oo, this function of the eigenvalues of the matrix, KSi, will approach the same function of the eigenvalues of the kernel, KSi(t, «).f We denote the eigenvalues of KSi(t, u) by A*'. Thus,

~ AS1., = N0

/

In l + i=l

\

W. N0

(56)

/

t We have not proved that this statement is true. It is shown in various integral equation texts (e.g., Lovitt [1, Chapter III]).

68 3.4

Four Special Situations

The sum on the right side is familiar from (2.73) as

Thus, the first term in /u(s) can be expressed in terms of the realizable mean-square error for the problem of filtering s^t) in the presence of additive white noise. A similar interpretation follows for the second term. To interpret the third term we define a new composite signal process,

s) Sl(t).

(58)

This is a fictitious process constructed by generating two sample functions s0(t) and s^t) from statistically independent random processes with covariances K0(t, u) and A^(/, «) and then forming a weighted sum. The resulting composite process has a covariance function AcomC, w:j ) = *K£t> «) + 0 - *)*iC, u), 7- < /, u < Tf. (59) We denote the realizable mean-square filtering error in the presence of white noise as $P(t I JcomOX A^/2). The resulting expression for /u(s) is

- £/>[