Echo Signal Processing the Kluwer International Series in Engineering and Computer Science Echo Signal Processing

ECHO SIGNAL PROCESSING THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE ECHO SIGNAL PROCESSING

Dennis W. Ricker The Pennsylvania State University

KLUWER ACADEMIC PUBLISHERS Boston I Dordrecht I London Library of Congress Cataloging-in-Publication Data

Ricker, Oennis w. Echo Signal Processing / Oennis W. Ricker. p.em. -(The Kll1\\'cr International Series in Engineering and Computer Science; SECS 725) Includes bibliographical references and index. ISBN 978-1-4613-5016-3 ISBN 978-1-4615-0312-5 (eBook) DOI 10.1007/978-1-4615-0312-5 1. Signal Proeessing. 2. Antennas and Propagation. 1. Title. Series.

Copyright ©2003 bySpringer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2003 Softcover reprint of the hardcover 1st edition 2003 AII rights rcserved. No part of this work may be reproduced, stored in a retrieval systelll, OI' transillitted in any form or by any means, electronic, mechanieal, photoeopying, microfilming, recording, or otherwise, without the written permission from the Publisher, with the exception of any material supplied speeifically for the purpose of being entered and executed on a computer systelll, for exclusive use by the purchaser ofthe work.

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Printed 0/1 acid~fi·ee paper. Contents

1 BASIC SIGNAL THEORY 1 1.1 INTRODUCTION ...... 1 1.2 SIGNALS, POWER, AND ENERGY ...... 1 1.3 FOURIER REPRESENTATION OF SIGNALS. 3 1.3.1 Orthonormal Expansions and Fourier 'fransforms. 3 1.3.2 Fourier 'fransform Properties ...... 5 1.4 THE COMPLEX SIGNAL REPRESENTATION 7 1.5 LINEAR SYSTEMS ...... 16 1.5.1 Definition...... 16 1.5.2 The Impulse Response and Convolution Integral 16 1.5.3 The Transfer Function ...... 18 1.6 STOCHASTIC PROCESSES ...... 19 1.6.1 Fourier Transforms and Power Spectra. 23 1.6.2 The Weiner-Khintchine Theorem . . . . 23 1.6.3 Linear System Response to Stochastic Signals . 27 1. 7 SUMMARy...... 28

2 ECHO ENERGY AND TIME BASE 31 2.1 INTRODUCTION ...... 31 2.2 THE ECHO ENERGY CYCLE . . . . 32 2.3 GEOMETRY AND KINEMATICS. 38 2.4 COLLINEAR MOTION ...... 45 2.5 SIGNAL MODELS ...... 51 2.6 RELATIVISTIC TIME MAPPING . 56 2.7 SUMMARy...... 64

3 DETECTION AND ESTIMATION 69 3.1 INTRODUCTION ... 69 3.2 BINARY DETECTION ...... 70 VI CONTENTS

3.3 MULTIPLE HYPOTHESES...... 83 3.4 NON-RANDOM POINT ECHO DETECTION . . . 86 3.5 SLOWLY FLUCTUATING POINT SCATTERING 101 3.5.1 White Gaussian Noise (WGN) ...... 101 3.6 THE RICIAN MODEL ...... 109 3.7 CORRELATED GAUSSIAN INTERFERENCE 112 3.8 WIDE SENSE STATIONARITY . . . . 120 3.9 PARAMETER ESTIMATION ..... 130 3.9.1 Maximum Likelihood Estimation 130 3.9.2 The Cramer-Rao Bound (CRB) . 135 3.9.3 Multiple Parameters ...... 138 3.9.4 The Role of the Ambiguity function 143 3.10 SUMMARY ...... 146

4 AMBIGUITY FUNCTIONS 153 4.1 INTRODUCTION ...... 153 4.2 MODELS AND NOTATION ...... 154 4.2.1 Wide and Narrowband AF Models 155 4.2.2 The AF of Real Waveforms . . . . 158 4.2.3 Alternative Signal Representations 161 The Wavelet Transform (WT) ., 163 The Wigner-Ville Distribution (WVD) . 164 4.3 AF AND UF PROPERTIES ...... 167 4.3.1 General Properties ...... 168 4.3.2 Auto AF Properties at the Origin ... 175 The Wide band Expansion Derivation for T 177 The Deterministic Model ...... 180 The Narrowband Expansion for the Fluctuating Model...... 184 4.4 THE BANDPASS WAVEFORM . . . . . 185 4.4.1 Ambiguity properties at the origin . 185 Window functions ...... 185 Analytic waveform expansion coefficients 187 4.4.2 Estimation Accuracy and the Ambiguity Error El- lipse ...... 189 4.4.3 Wideband Doppler Resolvency 192 4.4.4 Narrowband Model Error . . . 199 4.4.5 Quadratic Distortion (QD) .. 204 4.5 THE STATIONARY PHASE PRINCIPLE 208 4.6 SUMMARy...... 217 CONTENTS VB

5 WAVEFORMS 225 5.1 INTRODUCTION ...... 225 5.2 AMBIGUITY RESOLUTION...... 229 5.2.1 Woodward's Time Resolution Constant 231 5.2.2 The Wigner-Ville Time-Frequency Criterion. 237 5.3 SPARSE SIGNAL SEQUENCES . . . . . 243 5.3.1 Wideband Ambiguity Consistency 246 5.3.2 Global Ambiguity Properties 248 5.4 FREQUENCY HOP CODES 252 5.4.1 Costas Codes ...... 254 5.4.2 Congruence Codes ...... 259 5.4.3 AF Sidelobe Bounds for Hop Codes 265 5.4.4 Hop Code AF Main Lobe Characteristics 268 5.4.5 Logarithmic Frequency Allocation of Hop Codes 274 5.5 PRN WAVEFORMS ...... 276 5.5.1 Shift Register Sequences ...... 277 Runs and Repeated Segments . . . 280 Generating a PRN Code Sequence 282 Autocorrelation and Power Spectrum 282 5.6 DOPPLER TOLERANCE...... 289 5.6.1 HFM/LPM Waveforms ...... 290 5.6.2 Doppler Resolvent Dilation Tolerant Waveforms 298 5.7 SIMULTANEOUS TRANSMISSION. 307 5.8 SUMMARy...... 311

6 SPREAD SCATTERING AND PROPAGATION 319 6.1 INTRODUCTION ...... 319 6.2 THE LINEAR SPREADING MODELS ...... 321 6.2.1 Time-Frequency Correlation and Scattering Func- tions ...... 325 6.2.2 The Wideband Model ...... 328 6.3 RECEIVER RESPONSE ...... 330 6.3.1 Total Energy and the Detection Index 332 6.3.2 Scattering Function Convolution . . . 335 6.4 PROCESSING GAIN ESTIMATION. . . . . 337 6.4.1 Scattering models; the signal component . 338 6.4.2 Scattering models: the reverberation component 339 6.4.3 Receiver Outputs and Gp for CW, LFM and Sparse FSK Waveforms ...... 343 The Performance of "Thumbtack" Waveforms . . . 350 viii CONTENTS

6.5 RECEIVER OPTIMIZATION ...... 351 6.5.1 The Optimal Mismatched Receiver in WGN . 354 6.5.2 The Composite Cross Ambiguity Function . 359 6.6 POST-DETECTION COMBINING ...... 366 6.6.1 The Estimator-Correlator (EC) . . . . . 366 Generalized Signal to Interference Ratio 373 The EC for Uncorrelated Data . . . . . 376 EC Receiver Operating Characteristics . 378 6.6.2 EC Mismatch Error ...... 383 6.6.3 Diversity Combining ...... 388 Frequency diversity code design. 395 6.7 SUMMARy...... 398

7 THE SPATIAL REPRESENTATION 407 7.1 ARRAYS AND BEAM FORMATION 407 7.1.1 The Rectangular Aperture. 416 7.1.2 The Discrete Aperture. . . 420 7.2 ARRAY PROCESSING...... 423 Continuous Passive Signals 429 7.2.1 Active Receiver Implementation and Parameter Es- timation ...... 433 The CR Bound for Spatial Processing 435 Split Processing ...... 438 7.2.2 Spatially Spread Scattering . . . . 441 7.3 MONOPULSE BEARING ESTIMATION 444 7.3.1 Amplitude Comparison 446 Interference Error ...... 451 7.3.2 Phase Comparison...... 454 Spread Scattering Induced Bearing Bias 460 7.4 SUMMARy...... 463 SYMBOLS...... 469 GLOSSARY. . 472 INDEX ...... 480 List of Figures

1.1 Complex envelope spectra 13 1.2 Spectral Representations . 14 1.3 Impulse sequence . . 17 1.4 Iterated integration . . 24

2.1 The echo energy cycle 33 2.2 Volume reverberation element . 36 2.3 Boundary backscatter element . 37 2.4 Stationary multistatic geometry . 40 2.5 Kinematic multistatic geometry . 41 2.6 Ray cone diagram ...... 43 2.7 Collinear constant velocity trajectory diagram . 46 2.8 Collinear time varying trajectory diagram 48 2.9 Down and cross range components 50 2.10 Relativistic reference frames . 57

3.1 Dice game example ...... 75 3.2 Bimodal likelihood example . 79 3.3 Monotonic likelihood example . 79 3.4 Square law ROC curves .... 82 3.5 Receiver structures for correlated interference 124 3.6 Multi-hypothesis receiver bank 125 3.7 CFAR threshold example .... 127 3.8 Interference effects in estimation 134

4.1 Spectral representations ..... 159 4.2 BPSK spectra and ambiguity functions . 162 4.3 Split LFM waveform Wigner-Ville distribution 165 4.4 Split LFM waveform Choi-Williams distn. (C! = 1) 166 4.5 Split LFM waveform Choi-Williams distn. (C! = 20) 167 4.6 Wide band AF volume expansion ...... 176 x LIST OF FIGURES

4.7 CW ambiguity functions . 187 4.8 LFM ambiguity functions 189 4.9 Ambiguity error ellipse .. 191 4.10 Wide and narrowband phase models 196 4.11 FSK modulation function ...... 202 4.12 Quadratic distortion of ambiguity functions 209 4.13 Graphical LFM ambiguity estimation. 211 4.14 Geometric AF approximation ...... 215 4.15 VFM waveform overlap example ...... 216 4.16 VFM ambiguity function graphical estimate 218 4.17 Computed VFM ambiguity function 218

5.1 Three point scatter distribution. 232 5.2 Ambiguity function examples 233 5.3 Delay-Doppler images . . . . . 234 5.4 Thumbtack ambiguity function 236 5.5 Ambiguity slice ...... 238 5.6 Ambiguity coordinant rotation 238 5.7 Wigner-Ville time-frequency plane 241 5.8 Sparse waveform delay-Doppler support 245 5.9 LFM mesa cross ambiguity function " 247 5.10 Three component waveform delay shift. 252 5.11 Three component PRN ambiguity function 253 5.12 Costas code pattern and hit array example 256 5.13 Linear congruence pattern and hit array for GF(ll), a = b = 3 ...... 262 5.14 Quadratic congruence pattern and hit array for GF(ll), a = b = c = 1 ...... 263 5.15 Hyperbolic congruence pattern and hit array for GF(11), a = 3, b = 0 ...... 264 5.16 Costas pulse train correlation ...... 266 5.17 Upper bound for the sidelobes of a Costas code ambiguity function ...... 267 5.18 Welch-Costas spectra...... 269 5.19 Costas AF mainlobe characteristics. 273 5.20 Maximal length PRN shift register 278 5.21 PRN autocorrelation . . . 284 5.22 PRN spectrum ...... 285 5.23 PRN ambiguity function . . . . . 285 5.24 Welch-Costas ambiguity function 286 LIST OF FIGURES xi

5.25 Welch-Costas 101(2) spectrum ...... 287 5.26 [Welch-Costas 101(2) ambiguity function. 287 5.27 PRN Choi-Williams distribution (CWD) . 288 5.28 Welch-Costas CWD .., 288 5.29 Dilated LPM waveform . 292 5.30 LPM ambiguity function. 295 5.31 LFM ambiguity function . 297 5.32 LFM and LPM ambiguity function comparison 297 5.33 Normalized dilation tolerant frequency function . 303 5.34 Dilation sensitivity factor Ds .... . 305 5.35 Dilation tolerant ambiguity functions . . . 307 5.36 Simultone time series and spectrum. . . . 309 5.37 Simultone bed of nails ambiguity function 310

6.1 Sound velocity profile and ray path plot . 320 6.2 Scattering and ambiguity function convolution 334 6.3 Reverberation scattering function ...... 340 6.4 FSK ambiguity function ...... 343 6.5 Ambiguity function approximation models. . . 344 6.6 Scattering and ambiguity function convolution models 345 6.7 CW /LFM reverberation comparison ...... 355 6.8 CW /LFM in-water reverberation ...... 355 6.9 Scattering function phase plane overlap regions 363 6.10 Estimator correlator block diagram...... 368 6.11 Delay spread scatter distribution example . . . 386 6.12 EC and prescient receiver ROC comparison with J.lt = 5 386 6.13 An in-water example of the frequency selective fading . 391 6.14 Swerling II diversity Pd vs SIR as a function of the number of subpulses ...... 394 6.15 Swerling II SIR (Es/No) vs the number of subpulses as a function of Pd for P f = 10-5 ...... 394 6.16 The difference triangle for the generation of bandwidth efficient diversity codes...... 397

7.1 Generalized backscatter geometry. 408 7.2 Spherical-rectangular coordinants . 411 7.3 Square aperture beam patterns . . 418 7.4 2>' x 5>. and tilted beam examples 419 7.5 A 5>' x 5>' Hanning shaded beam 421 7.6 The discrete line array ...... 424 xu LIST OF FIGURES

7.7 Combined scalar and multidimensional receiver . . . 441 7.8 2-element array and spread scattering example .. . 443 7.9 Amplitude mono pulse receiver with squinted beams. 446 7.10 Example of uniform line array squinted beams 450 7.11 Amplitude monopulse additive ratio example 450 7.12 Phase monopulse receiver ...... 454 7.13 Phase density function at

1.1 Fourier transform properties...... 8

5.1 Welch GF(1l)2 code linear and logarithmic frequency al- locations ...... 275 5.2 Maximal Length Shift Register Connection Vectors 281 5.3 Shift register sequence for ai = ai-l + ai-3 . 283 6.1 Matched filter scatterer response Eis . . . 347 6.2 Matched filter reverberation response E Ir 347 6.3 CW processing gain for all scatterers . . . 348 6.4 LFM processing gains for Br ~ W .... 349 6.5 FSK processing gains for sparse codes with Wisk > (N - l)Br ...... 349 Foreword

This book presents basic and advanced topics in the areas of sig• nal theory and processing as applied to acoustic echo-location (sonar). It is written at the advanced undergraduate or graduate level, and as• sumes that the reader is conversant with the concepts and mathematics associated with introductory graduate courses in signal processing such as linear and complex algebra, Fourier analysis, probability, advanced calculus, and linear system theory. The material is presented in a tuto• rial fashion as a logical development starting with basic principles and leading to the development of topics in detection and estimation theory, waveform design, echo modeling, scattering theory, and spatial process• ing. Examples are provided throughout the book to illustrate impor• tant concepts and especially important relationships are boxed. The book addresses the practical aspects of receiver and waveform design, and therefore should be of interest to the practicing engineer as well as the student. Although much of the book is applicable to the general echo-location problem that includes radar, its emphasis is on acoustic echo location especially in regard to time mapping and the wideband or wavelet description of Doppler. Introductory signal theory material is included in the first chapter to provide a foundation for the material covered in the later chapters. A consistent notational convention is ob• served throughout the book so that the various mathematical entities are readily identified. This is described in the glossary and symbol list. Sonar signals in comparison to radar signals are much more sus• ceptible to motion induced distortion because the ratio of attainable to propagation speed is much higher. The kinematic echo formation process is covered in the second chapter where the concepts of time mapping, space-time ray cones, and generalized trajectory diagrams are introduced to describe the distortion of the signal time base. Linear time mapping or Doppler is the dominant effect but higher order time base distortion effects occur especially when propagation is not collinear. It is then necessary to employ the iterative algorithms described in the chapter to duplicate the distortion effects for receiver syntheSis. Doppler frequency shift is a narrowband phenomenon arising from linear time mapping and only approximates the Doppler effect. Receivers for large time-bandwidth waveforms must also account for envelope dilation as well as Doppler shift in order to avoid performance degradation due to receiver mismatch. Design criteria indicating the tradeoffs between per- XVI formance and receiver complexity are discussed. A comparison between time mapping in the acoustic and electromagnetic realms illustrates the difference between signal propagation in free space and in an acoustic medium. Detection and parameter estimation are treated as a maximum likeli• hood problem. This is not a new concept and is well covered in standard signal processing texts but the approach to the derivation of the likeli• hood ratio and its interpretation is specifically tailored to the acoustic echo-location problem. The classic derivation of the likelihood ratio is presented in terms of a binary hypothesis decision policy that minimizes a cost objective such as the Baye's risk. The Karhunen-Loeve expan• sion for stochastic signals is utilized to form the likelihood ratio from the continuous data time series (echo return). This provides a discrete countable representation of a continuous waveform, and allows for the definition, in a limiting sense, of the required probability density func• tions. Furthermore, it is shown that the ambiguity function associated with echo location arises naturally from the maximum likelihood cri• terion, and that its form depends upon the fundamental assumptions regarding the echo formation process and the interfering noise. A discussion and comparison of the properties of wideband (wavelet domain) and narrowband ambiguity functions is a principal contribu• tion of the book. These include the familiar auto ambiguity functions that arise from the maximum likelihood criterion detection of determin• istic and fluctuating point scatterers in white Gaussian noise. They are treated as a special case of the more general cross ambiguity functions that arise both from the maximum likelihood criterion for detection in correlated noise and the maximization of the detection index when in• terrogating spread scatterers. The specification and design of echo location waveforms is introduced by first describing some of the traditional waveforms such as the CW and LFM and the concept of pulse compression. While waveform design has been extensively treated in a radar context there is a need to discuss more recent developments motivated by the requirements of acoustic echo location. These include the design of waveforms from a wideband or time dilation (wavelet) point of view and the associated issues of Doppler (scale) tolerance, dilation sensitivity, and simultaneous delay• Doppler (time scale) resolution. Original material treating the Wigner• Ville distribution as a tool for waveform design is introduced. The book includes an extensive treatment of what are called noise• like (imaging) waveforms. These are PRN sequences and the classes of xvii frequency shift keyed (FSK), Costas, and congruential codes that have high delay-time scale resolution capabilities. The algebraic derivation of the various classes of hop codes, the properties of their nearly ideal 'thumbtack' ambiguity functions, and a comparison with PRN sequences are provided. The detection of doubly spread scattering processes is addressed upon establishing the background for point scatterer detection and estimation, and having introduced and discussed auto and cross ambiguity functions. Spread scattering theory is presented from a signal processing point of view by modeling the random scattering process as a time varying lin• ear filter with an associated scattering function that describes its delay• Doppler/time dilation energy dispersion properties. It is shown that the expected response of a narrow or wide band matched filter receiver to a spread echo or interference process such as reverberation is the convolu• tion of the signal ambiguity and scattering functions. This relationship is used to guide the choice of signal waveforms and as a mechanism for the derivation of useful expressions for the processing gain of matched filters operating in a clutter or reverberation limited environment. The dis• cussion of scattering functions also includes their use as a means for the introduction of prior information for the formulation of optimal detection strategies such as the optimal mismatched receiver and the estimator cor• relator. The relationship of the scattering function to the time-frequency correlation function serves as an introduction to the problem of time and frequency selective fading and the use of diversity processing to improve detection performance under fading conditions. Beam formation, array processing, bearing estimation, and the gen• eralized space-time maximum likelihood receiver are discussed in the last chapter. Spatially spread scattering is described by a generalized scat• tering function that is a function of delay, Doppler and bearing. The maximum likelihood approach as well as monopulse processing are dis• cussed as alternatives for bearing estimation. The material in this book some of which is original is drawn from a wide selection of open literature sources. The book progresses in an orderly fashion from a discussion of basic signal processing principles, to the kinematic formation of echos and the fundamentals of detection and estimation for simple point scattering processes. Ambiguity functions that arise from the maximum likelihood criterion for Gaussian processes are given a chapter of their own followed by a chapter on the various echo location waveforms and their design. The groundwork established in the earlier chapters serves as the basis for the discussion of the detection of XVlll

doubly spread processes and the extension to spatial processing, beam forming, and bearing estimation in the last two chapters.

Acknowledgments

I would like to thank all who contributed to this endeavor includ• ing: Dave Drumheller of ONR who contributed ideas, suggestions, and material for inclusion in the book, my colleagues at the Applied Re• search Laboratory (ARL) of the Pennsylvania State University for pro• viding a stimulating environment and uncountable discussions regarding sonar signal processing, Professor Edward Titlebaum at the University of Rochester for his insight regarding waveform and processor design, Eric Schott and Steve Harp at ARL for their help in fathoming the in• tricacies of LaTeX, Anthony Cutezo at ARL for help with editing and figures, Jack Sharer at ARL for his aid in implementing many of the computations, and Andy Ward for generating a large number of the il• lustrations. I would like to thank my wife Beth for her patience and encouragement and the Applied Research Laboratory and the Office of Naval Research for their support throughout this project.

D.W. Ricker Applied Research Laboratory Pennsylvania State University