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

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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. Permission for books published in Europe: [email protected] Permissions for books published in the United States of America: [email protected] 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 .
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