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**Detection,** **Estimation,** **and** **Modulation** **Theory** **Detection,** **Estimation,** **and** **Modulation** **Theory** Radar-Sonar **Processing** **and** **Gaussian** **Signals** **in** **Noise** HARRY L VAN TREES George Mason University New York l A Wiley-Interscience Publication JOHN WILEY & SONS, INC Chichester Weinheim Brisbane Singapore l l l l 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 form or by any means except as permitted either the prior may be reproduced, electronic under Section written stored **in** a retrieval mechanical permission photocopying, system or transmitted recording, scanning 107 or 108 of the I976 United States Copyright of the Publisher or authorization through **in** any or otherwise Act without 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 NY 10158-0012, For ordering Department, John Wiley (212) 850-601 I, fax (212) 850-6008 **and** customer service call l -800~CALL-W & Sons, E-Mail: Inc 605 Third PERMREQ ILEY q ISBNO-471-22109-0 This title is also available **in** print as ISBN O-47 1-10793-X Library of Congress Cataloging **in** Publication Data is available ISBN Printed O-47 1-10793-X **in** the United 10987654321 States of America Avenue, New @ WILEYCOM York, To Diane **and** Stephen, Mark, Kathleen, Patricia, Eileen, Harry, **and** Julia **and** the next generationBrittany, Erin, Thomas, Elizabeth, Emily, Dillon, Bryan, Julia, Robert, Margaret, Peter, Emma, Sarah, Harry, Rebecca, **and** Molly Preface for Paperback Edition **In** 1968, Part I of **Detection,** **Estimation,** **and** **Modulation** **Theory** [VT681 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** ([VT7 1a], [VT7 1b]) were published **in** 197 **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** 197 The bibliography lists it as a reference, Optimum Array Processing, Wiley, 197 1, 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, DC 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 C31 He asked me to become his Principal Deputy **and** I spent two years **in** that position **In** 198 1, I joined MIA-COM Linkabit Linkabit is the company that Irwin Jacobs **and** Andrew Viterbi had started **in** 1969 **and** sold to MIA-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** vii Vlll Prqface for Paperback Edition doctoral students.The processtook about six years The Center for Excellence **in** Command, Control, Communications, **and** Intelligence has been very successful **and** has generatedover $300 million **in** researchfunding during its existence During this growth period, I spentsometime on array processingbut a concentratedeffort was not possible.In 1995, I started a seriouseffort to write the Array **Processing** book Throughout the Optimum Arrav Processingtext there are referencesto Parts I **and** **III** of **Detection,** **Estimation,** **and** **Modulation** **Theory** The referencedmaterial 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 secondlevel graduate coursedealing with more specializedtopics **In** the 30-year period, there hasbeen a dramatic changein 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 implementedthat reach the theoretical bounds The advancesin computational capability have also changedhow the material is taught **In** Parts I **and** III, there is an emphasison compact analytical solutions to problems **In** Part IV there is a much greater emphasison efficient iterative solutions **and** simulations.All of the material **in** parts I **and** **III** is still relevant The books use continuous time processesbut the transition to discrete time processesis straightforward Integrals that were difficult to analytically can be done easily **in** Matlab? The various detection **and** estimation algorithms can be simulated **and** their performance comparedto the theoretical bounds.We still usemost of the problemsin the text but supplementthem with problemsthat require Matlab@solutions We hope that a new generation of studentsand readersfind thesereprinted editions to be useful HARRYL VAN TREES Fairfax, Virginia June 2001 Preface **In** this book continue the study of **detection,** **estimation,** **and** **modulation** **theory** begun **in** Part I [I] 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, I 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, I 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 MIT 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 ix 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 separatefrom 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 seesthe widespread applicability of the basic ideashe 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 I-5, I-6 Part **III** Chaps III-1 to III-5 Chaps III-6 to III-7 Chaps.III-$-end Chaps.I-4, I-6 Chaps.I-4 Chaps.I-4, I-6, 111-lto III-7 Array **Processing** Chaps IV-l, IV-2 Chaps.IV-3-end Chaps.I-4 Chaps.III-1 to III-S, 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 somequestionsunanswered It hasthe characteristics of a graduate level text **in** that it presentsthe material **in** an orderly fashion **and** develops almost all of the necessaryresults internally The book should appeal to three classesof readers The first class consists of graduate students The random **signal** problem, discussedin Chapters 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 Prqface 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, becauseit 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 suggestedthat 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 sometimesfrustrating 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 referencesascomplete aspossibleand 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 suggestionswere 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 auspicesof the Research Laboratory of Electronics I did the final editing xii Prg face while on Sabbatical Leave at Trinity College, Dublin Professor Brendan Scaife of the Engineering School provided me office facilities during this peiiod, **and** M.I.T provided financial assistance I am thankful for all of the above support Harry L Van Trees Dublin, Ireland, REFERENCES [l] 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 Glossary h.u(t94 h,,,(t, k&9 i;(t, L(t, H h,(t9 w 10 k l 49 I2 I, I J(A) Jii J-‘(t, u) Jij Jk(tv d J &)&9 u:s) K&9 &.(t7 K$;i21( t, u) K&9 l V 611 whitening filter filter whose output is white on H, filter corresponding to difference between inverse kernels on two hypotheses (3.31) complex envelope of impulse response of bandpass filter complex realizable whitening filter linear matrix transformation optimum linear matrix filter modified Besselfunction of 1st kind **and** order zero integrals involved **in** Edgeworth series expansion (defined by (2.160)) integrals incomplete Gamma function identity matrix function **in** variance bound elements **in** J-l inverse information kernel elements **in** information matrix kth term approximation to information kernel information matrix (Fisher’s) covariance function of composite **signal** (3.59) covariance function of **signal** covariance of r(t) on ith hypothesis functional square root of Kgol’(t, u) covariance function of x(t) correlation function of Doppler process target correlation function two-frequency correlation function covariance matrix covariance function of %(t) linear transformation of x(t) nth order Laguerre polynomial sufficient statistic likelihood function bias term **in** log likelihood ratio term **in** log likehhood ratio due to deterministic input term on log likelihood ratio due to random input actual sufficient statistic (sensitivity problem) sufficient statistics corresponds to cosine **and** sine components correlator output **in** suboptimum test 612 Glossary f wo A A ef Ag Am A 3db A AXWX a a ma2i a hi ch a ais AT zt*‘i **In** **In** A(A) P(s) pBP@) ,u,&> pD@) pLE&) PLPW tLnb> Pm(S) output of correlator **in** “white-optimum” receiver a parameter which frequently corresponds to a signalto-noise ratio **in** message ERB likelihood ratio likelihood ratio likelihood function signal-to-noise for ratio **in** reference bandwidth Butterworth spectra effective signal-to-noise ratio generalized likelihood ratio parameter **in** phase probability density signal-to-noise ratio **in** 3-db bandwidth covariance matrix of vector x covariance matrix of state vector (= Kx(t, t)) Lagrange multiplier maximum eigenvalue eigenvalue of matrix or integral equation ith eigenvalue, given A eigenvalues of channel quad ratic form eigenvalue of **signal** process total eigenvalue eigenvalues of y, (t ) natural logarithm log likelihood function logarithm to the base a characteristic function of random variable x (or x) generating function of I on H, mean Doppler shift ith coefficient **in** expansion of m(t) mean delay mean-value function of process difference between mean-value functions matrix used **in** colored **noise** derivation mean vector ‘ogarit’~m of41(RI, Ho(s) ,u(s) for bandpass problem ,u(s) for binary symmetric problem component of p(s) due to deterministic **signal** ,u(s) for low energy coherence case ,u(s) for low-pass problem component of ,u(s) due to random **signal** p(s) for simple binary problem Glossary WI N n, n CCR lijCt) P pa WWK(~) prPsK(~) P BP PD 613 ,u(s) for separable kernel case asymptotic form of p(s) complex version of p(s) dimension of observation space number of coefficients **in** seriesexpansion **Gaussian** (or Normal) density with mean m **and** standard deviation numerator of spectrum spectral height (joules) **noise** random process colored **noise** (does not contain white noise) ith **noise** component **noise** component at output of whitening filter MMSE realizable estimate of colored **noise** component MMSE unrealizable estimate of colored **noise** component complex envelope of **noise** process **noise** correlation (matrix numbers) **noise** random variable (or vector variable) Cramer-Rao bound elementsin error covariance matrix variance of ML interval estimate expected value of reaZizaO/epoint estimation error minimum mean-square realizable filtering error of s(t) **in** the presenceof white **noise** with spectral height NJ2 variance of error of point estimate of ith **signal** normalized realizable point estimation error expected value of point estimation error, statistical steady state optimum unrealizable error normalized optimum unrealizable error covariance matrix **in** estimating d(t) steady-state error covariance matrix function **in** optimum receiver equations (9.90) distributed error covariance function matrix power probability of error probability of error for binary FSK system probability of error for binary PSK system power **in** bandpassproblem probability of detection (a conditional probability) 614 Glossary w, 7) w - 4))a 444 pr[*I,pr(9 ii2B % rnD 65 Qb,19 QI& Q,k 4 Q' Q,(u,2) R ’ R(t) R&Y u) i?,&, u} St effective power probability of false alarm (a conditional probability) a priori probability of ith hypothesis power **in** low-pass problem probability of a miss (a conditional probability) one-term approximation to PA,I received power transmitted power f, A} transform of &{ probability density of r, given that Hi is true eigenfunction **Gaussian** density, N(0, 1) ith coordinate function, ith eigenfunction moment generating function of I(R), given Ho moment generating function of random variable x phase of **signal** time-frequency correlation function time-frequency cross-correlation function low passphase function spread cross-ambiguity function cross-correlation matrix between input to message generator **and** additive channel **noise** state transition matrix, time-varying system state transition matrix, time-invariant system probability of event **in** brackets or parentheses bandwidth constrain.t carrier frequency (radians/second) Doppler shift mean frequency Marcum’s Q function inverse kernel on ith hypothesis inverse kernel height of scalar white **noise** drive covariance matrix of vector white **noise** drive inverse matrix kernel transmission rate target range correlation function two-frequency correlation function Bayes risk received waveform (denotes both the random process **and** a sample function of the process) Glossary I’(t) Pii 615 combined received **signal** output when inverse kernel filter operates on r(t) K term approximation output of whitening filter output of S,(m) filter (equivalent to cascading two whitening filters) complex envelope of **signal** process normalized correlation si(t) **and** sj(t) (normalized signals) normalized covariance between two random variables target skewness degradation due to interference covariance matrix of vector white **noise** w(t) observation vector Fourier transform of s(t) spectrum of colored **noise** Fourier transform of Q(T) power density spectrum of received **signal** power density spectrum transform of optimum error **signal** Doppler scattering function scattering function uniform Doppler profile spectrum of reverberation return range scattering function **signal** component **in** r(t), no subscript when only one **signal** **signal** depending on A modulated **signal** actual s(t) (sensitivity context) composite **signal** process(3.58) coefficient **in** expansion of s(t) ith **signal** component random component of **signal** received **signal** realizable MMSE estimate of s(t) **signal** transmitted **signal** on HO **signal** on HI **signal** with unwanted parameters random **signal** **signal** component at output of whitening filter 616 Glossary complex covariance matrix (= f&t)) variance variance on H,, HO mean-square Doppler spread mean-square bandwidth mean-square delay spread mean-square durati on vector **signal** pulse duration initial observat ion time (same as Ti) duration of pulse sequence final observation time initial observation time pulse repetition interval mean (arrival) time round-trip delay time unwanted parameter generalized ambiguity function **signal** ambiguity function cross-ambiguity function generalized spread ambiguity function Doppler-spread ambiguity function phase estimate phase of channel response estimate of 8, transition matrix transpose of matrix conjugate transpose of matrix unit step function input to system elementary rectangular **signal** variable **in** piecewise approximation to Vch(t) envelope of channel response target velocity bandwidth parameter (cps) transfer function of whitening filter channel bandwidth (cps) single-sided transform of inverse of ing filter white **noise** process impulse response of whitening filter complex white **noise** process Glossary 617 a matrix operation whose output vector has a diagonal covariance matrix input to modulator random process estimate of random process random vector state vector augmented state vector state vector for desired operation prefil tered state vector state vecto r, message state vector, **noise** distributed complex state variable kernel **in** singularity discussion (3.15 1) output of differential equation transmitted **signal** observation space subspace of observation space output of whitening filter gain matrix **in** state-variable filter (dh,(t, t)) Author Index Aaron, M R., [lo-631,336 Abend, K., f5-321,156 Abramowitz, M., [6-4], 183 Adams, R L., [7-2],199,207,222 Akima, H., [ 6-91,184 Albers, V O., [8-81,234 Allen, J., [lo-SO], 336 Antonov, V P., [ 13-121,460 Arens, R., [A-3], 566 Ares, M., [ 13-271,470 Athans, M., [2-14],44, [g-5], Austin, M E., [5-161,155 Baggeroer, A 260 B.,[2-15],44, [6-181,184, [g-3], 260, [9-4], 260, [lo-811,307, [ 13-551,544, [ 13-561,544, [A-9], 566, 574,589, [A-13], 604 Balakrishnan, A V., [ 13-261,470, [ 13-741, 475 Barankin, E W., [6-141,184 Bar-David, I., [5’-351,156, [13-511,537, [ 13-671,537 Barker, R H., [ 10-171, 317 Barrow, B B., [g-17], 243 Barton, D K., [8-31,234 Bello, P A., [5-40], 157, [541], 157, [ lo-651,352, [ l-21 ,360, [ l-151 ,366, [ll-211,396, [11-221,396, [12-21,421, (13461,537, [13-61],502,547, (13-661, 537, [13-691,537, [All], 585 Bendat, J S., [6-31, 167 Bennett, W R., [g-2], 239, [A-6], 566 Bergland, G., [ 12-51,426 Beringer, R., [7-31,205 Berkowitz, R S., [84], 234, [g-12], 243, [lo-301,323 Bernfeld, M., [8-61,234, [ lo-141,314 Bhattacharyya, A., [ 3-4],72 Birdsall, T G., [lo-261 , 321 Blackman, R B., [6-21,167 Boltyanskii, V., [g-6], 260 Boyd, D., [S-15], 155 Breipohl, A M., [5-181, 155 Bremmer, H., [ 13-81,460 Brookner, E., [ lo-541,336 Brown, J L., [ll-61,407, [13-531,541 Brown, W M., [A-12], 585,603, [A-17], Bucy, R S., [2-6),19 603 Capon, J., [4-lo], 131, [13-581,546 Carleton, H R., [13-l l] ,460 Cauer, W A., [ lo-61,294 Chernov, L A., [ 130181,460 Clay,C S., Jr., [13-g], 460, [13-151,460 Coate, G T., [8-l 1,234 Collins, L D., [2-7],20, [2-g], 30, [2-111, 39, [2-13],42, [3-2],70,78,79, [ll-51, 390, [A-9], 566,574,589 Cook,C E., [8-61,234, [lo-71,294, [lo-141, 314 Cooley, J W., [12-4], 426, [12-71,426 Cooper, G R., [lo-701,340, [lo-731,340 Craig, S E., [ lo-281,321 Cram&, H., [2-12],39 Crane, R B., [A-12], 585,603 Curran, T F., [5-331,156 Darlington, S., [6-81,184, [104], Davenport, W B., [4-81,136, [s-9], [A-151,601 Davies, I L., [ lo-81,279, 308 Davis, R C., [545], 159 294 154, 619 620 Davisson, L D., [5-281,155 DeBuda, R., [ lo-681,339 DeLong,D F Jr., [13-211,470, [13-221, 470,545 Diamond, J., [ lo-641,336 Dicke, R H., [7-31,205, [ lo-31,294 DiFranco, J V., [8-51,234, [g-23], 272 DiToro, M J., [5-271,155, [lo-621, 336 Doob, J L., [2-26],21, [A-16], 601 Drouilhet, P R., [5-lo], 155 Dugundji, J., [A-7] ,566 Duncan, T., [2-28],21 Edrington, T S., [9-131,243 Elspas, B., (10-241, 321, [ 1049 19336 Esposito, R., [5-48], 159, [9-20 13243, [g-21], 243 Falb, P L., [9-51,260 Faure, P., [ 13-101,460 Feldman, J., [ 3-10],88, [ 3-11],88 Feller, W., [2-lo], 39 Fishbein, W., [ 10-281, 321, [lo-291 ,285, 321 Fortmann, T E., [ 13-26b] ,470 Fowle, E N., [lo-191, 318, [lo-561,336, [ 13-281,470 Frank, R L., (10-341, 323 Frost, P A., (2-291, 21 Fujimura, S., [g-22], 243 Gaarder, N T., [H-U], 537 Gabor, D., [A-2], 566,599 Gagliardi, R., (70131,221, (7-141,221 Gallagher, R G., [5-6],150,160, [1348], 537 Gamkrelidze, R:‘, [g-6], 260 Gassner, R L., [lo-701, 340 George, D A., [ 10-571, 336 Gerst,I., [10-64] , 336 Glaser, E M., (5-111, 155 Gold, B., >[12-61,426 Golomb, S W., [lo-201,321, [lo-271,321, [ lo-321,323 Gray, D L., [lo-581, 336 Green, P E., Jr., (4-l ] , 131, [ l-l ] ,360; [12-l], 421, [13-5],448,517, [13-51,538 549, [13-41], 537, [13-651,556 Grenander, U., [6-51, 184 Groginsky, H L., [5-231,155 Author Index GuilIemin, E A., [ 10451,287 Haggerty, R D., [ lo-561,336 Hagfors,T., [1342],537, (13491,537 Hajek, J., [3-12],88 Hancock, J C., [5-261, 155 Harger, R O., [7-81,221 Hauer, C A., [lo-361,338 Heimiller, R C., [lo-331, 323 Hellinger, E., [ 3-3],72 Helstrom,C W., (441,135, [5-34)) 156, [ lo-39],339,348,349, [ 10471,336, [lo-751,339, [ll-191,366, [A-181,599 Ho, Y C., [5-301,155 Hofstetter, E M., [7-71,221, [lo-121,342, [lo-791,307, [13-211,470, [13-221,470, 545 Horton, J W., [8-g], 234 Huffman, D A., [lo-211,321, [lo-311,323 Huttman, E., [ lo-51,294 Hveding, N., [g-19], 243 ?TO, K., [2-27],21 Jacobs,I M., [5-71,161, [6-lo], 184, (~-10 568 Jakowitz, C V., [5-121,155 Jenkins, G M., [6-l], 167 bc, M., [5-81,154 Kadota, T T., [3-14],97, [5-47], 159 Kailath, T., [2-29],21, [5-381,157, [543], 159, [ll-111,366, [ll-121,366, [13-351, 489,502, [1340],537, [13-641,502 Kaiteris, C., [ 10-151, 314 Kakutani, S., [ 3-5],72 Kalman, R E., [2-6],19 Kanda, T., [ lo-351,322 Kelly, E J., [lo-691,340, [lo-77],294,301, 305,352, [13-3],444,459, [13-161,460, [ 13-28)) 470, [A-3], 566, [A-5], 566 Kennedy, R S., [54), 150, [ ll-3],379,396, 397, [ 12-31,422, [ 13-37],449,5 10,522, 523,551, [13-38],521,548,551 Key, E L., [ lo-561,336 Klauder, J., [1046], 343 Kobayashi, H., [ 10-351, 322 Koschmann, A H., [5-181,155 Kotelnikov, V A., [ 6-71, 184 Koval, J F., (10-361, 338 Author Index Kraft, C H., [3-8],83 Krinitz, A., [ 13-451 ,537 Kurth, R: R., [ 114],383,389, 395, [11-51,390, [12-81,435, 454,456,458,498,506,511,513,514, 515 Kushner, H J., [ 13-711,475 Kyhl, R L., [7-31,205 Nilsson, N J., [ 10-521, Nolte, L W., [5-241,155 391,394, [13-71, Lawton, J G., [5-191,155 Lebow, I L., [13-38],521,548,551 Lee, R C., [5-301, 155 Lerner, E C., [ 13-3],444,459 Lerner, R M., [ 10-111, 308 Levin, M J., [7-5],221,223, [7-61,221, [1343], 537, [13-621,553 Lewis, P A W., [12-71,426 Lichtenstein, M G., [lo-591, 336,350 Lions, J L., [ 139741,475 Lovitt, W., [ 2-8],23, [3-l] ,67 Lucky, R W., [5-201,155, [lo-611,336 McAulay, R J., [6-171,184, [ 10-791, 307 Magi& D T., [5-211,155 Manasse, R., [lo-761,294, [ 13-251,470 Marcum, J I., [g-8], 243, [9-g] ,243, [9-lo], 243,272 Mason, S J., [ lo-21,294 Meditch, J S., [ 13-751,475 Miyakawa, H., [ lo-351,322 Middleton, D., [2-17],9, [2-18],9, [2-191, 9, [2-20],9, [4-31,131, [4-5],131,135, [4-71,131, [4-g], 131, [5-31,148, [5-231, 155, [5-251,155, [5-361,156, [5-371, 156, [548], 159, [74], 214, [ll-161,366, [ l-171,366, [ l-181,366, [ l-201,366, [13-131,460, [ 13-171,460 Mishchenko, E., [g-6], 260 Mitter, S K., [ 13-731,475 Mullen, J A., [5-251,155, [7-111,221, [g-20], 243 Nahi, N E., [Y-IS], 221, [7-14,221 Nakagami, M., [g-14], 243, [g-15], 243, [g-22], 243 NeIin,B D., [ll-211,396, [ll-221,396 Nesenberg, M., [g-18], 243 Nightingale, J A., [ 13-291,475, [ 13-301, 475, [13-721,475 336 Ol’shevskii, V V., [ 13-121,460 Palermo, C J., [A-17], 603 Pawula, R F., [7-g], 221 Peterson, W W., [ 10-231, 321 Pettengill, G., [ 13-521 ,537 PhiIIipson, G A., [ 13-731,475 Pierce, J N., [3-6],78,79,95 Piersol, A G., (6-31, 167 Pontryagin, L S., [g-6], 260 Prager, R H., [ 13-201,462 Price, R., [2-l] ,9,16,17, [2-2],9,16, (2-31, 9,16, [24],9,16, [4-l], 131, [4-21,131, [4-61,136, [5-l], 148, [7-l], 214, [lo-121, 308,342, [11-l], 360, [ll-81,366, [11-g], 366, [ ll-101,366, [12-l], 421, [ 13-51, 448,517,549, [13-39],530,537 Proakis, J G., [5-lo], 155 Purdy, R J., [lo-731,340 Radar, C., [ 12-61,426 Rauch, S., [7-151,221 Reed, I S., [ lo-77],294,301,305, [ 13-21, 444,459, [A-4] ,566, [A-5] ,566, [A-8], 566,601 Reiffen, B., [5-311,156, [ 13-501,537 Reintjes, J F., [8-l], 234 Reis, F B., [10-40], 342 Resnick, J B., [ lo-161,314 Rice, S O., [g-16], 243 Ridenour, L N., [ 13-591,546 Rihaczek, A W., [ lo-131,314 Ristenblatt, M P., [lo-251, 321, [lo-261,321 Rittenbach,O E., [lo-281,321, [lo-291,321 Root, W L., [3-9],85,88, [ 3-13],97, [4-81, 136, [5-91,154, [lo-381,339, [1048], 336, [lo-77],294,301,305, [13-681,537, [A-5], 566, [A-15], 601 Ross, M., [5-331,156 Rubin, W L., [8-51,234, [g-23], 272, [lo-151, 314 Rudnick, D., [4-111,131 Rummler, W D., [ 13-24],470,545, [ 13-341, 482,545 Sakamoto, T., [ lo-351,322 Sakrison, D J., [7-121,221 622 Author Index Scholtz, R A., [lo-321 , 323 Schultheiss, P., [7-lo] , 221 Schwartz, M., [A-6], 560 Schwartz, S C., [5-391, 156 Schweppe, F C., [2-5],19, [lo-581,336 Scudder, H J., [5-131, 155, [5-141,155, [5-291,155 Seidman, L D., 16-171, 184, [g-80], 307 Selin, I., [ 10-531, 336 Sheehan, J A., [ 13-281,470 Shepp, L A., [3-15],88 Sherman, H., [5-311,156 Shuey, R L., [5-121,155 Siebert, W M., [lo-g], 287,288, 308, [lo-lo], 308,309 Siegert, A J F., [2-16],44, [5-81, 154 Skolnik, M I., [8-21,234 Slack, M., [9-l], 239 Slepian, D., [5-46], 159 Sosulin, Y G., [2-21],21, [2-22],21, [2-231, 21, [2-24],21 Spafford, L J., [13-32],482,543, [13-331, 482 Speiser, J M., [10-721, 340 SpiIker, J J., [ 13441,537 Stegun, I A., [6-41,183 Steiglitz, K., [5-221, 155 Stein, S., [A-6], 566 Steinberg, B D., [ 13-571,546 Steinberg, H., [7-lo], 221 Stewart, J L., [ 13-191,462, [ 13-201,462 Stratonovich, R L., [2-21],21, [2-22],21, [2-23],21, [2-25],21 Stutt, C A., [lo-371,339, [1043], 342, [10-44], 339, 351, [13-32],482,543 Swerling,P., [6-151,184, [g-7], 243,263, 272, [g-lo], 243,272, [g-11], 243, [lo-511, 336, [lo-781,307 Swick, D A., [ 10-711, 340 Taki, Y., [IO-X], 322 Thau, F E., [ 13-701,475 Thomas, J B., [5-221, 155 Thompson, J S., [13-231,470 Titlebaum, E L., [ lo-741,339, Tolstoy, I., [ 13-91,460 [ 13-231,470 Tufts, D W., [ 10-631, 336 Tukey, J W., [6-21,167, [12-4], 426 Turin,G L., [3-7],79, [5-21,148, [ll-133, 366, [ l-141,366 Turyn, R., [lo-181 , 318 Tzafestas, S G., [ 13-291,475, [ 13-30 ] ,475, [ 13-721,475 Urick, R J., [8-71,234 Urkowitz, H., [ 10-361, 338, [ lo-551,336, [ 13-311,479 Vane, A B., [7-31,205 Van Trees, H L., [l-l], 1, [l-2], 3, [l-3], 7, [542], 157, [13-l] ,444,459, [134], 444, [13-361,492, [14-l], 563, [A-9], 566,574,589 Ville, J., [lo-l], 279, 308 Viterbi, A J., 15-51, 150, 161 Wada, S., [g-22], 243 Wainstein, L A., [6-111, 184, [13-601,546 Watts, D G., [6-l] , 167 Weaver, C S., [5-171, 155 Westerfeld, E C., [ 13-191 ,462, [ 13-201 , 462 White, G M., [5-121, 155 Wiener, N., [ 13-541 ,541, [ l-71 ,407 Wilcox, C H., [1041], 342, [10-42], 342 Wilson, L R., [5-231, 155 Wintz, P A., 15-261, 155 Wishner, R P., [ 10-691, 340 Wogrin, C., [7-lo], 221 Woodward, P M., [6-61,184, [ lo-81,279, 308, [lo-60],279,294, [A-l], 566 Young, T Y., [lo-59],336,350 Zakai, M., [6-121,184, [6-131,184, [lo-821 , 307, [lo-831,307 Zierler, N., [lo-221 , 321 Zimmerman, H J., [ lo-21,294 Ziv, J., [6-121,184, [6-131, 184, [lo-821, 307, [lo-831,307 Zubakov,V D., [6-111,184, [13-601,546 Zweig, F., [7-lo], 221 Subject Index Accuracy, local, 2.94 global, 302 Ambiguity function, cross, 339, 350,481 definition, 279 Doppler-spread, 400,409 examples, 280,283,285,292,308 generalized spread, 40 ideal, 282 properties, 290, 341 reverberation problem, 462 Amplitude **estimation,** doubly-spread targets, 530 spectrum, 191,204,211,214 Array processing, 563 Asymptotic results, 112 Autocorrelation function, time-frequency, 279 Bandlimited spectrum, 108,116 Bandpass detection problems, 74,77 Barker codes, 316 Bayes test, 10 Bernoulli sequences, 19 Bhattacharyya distance, 180 Binary communication, 69,74,79,111, 375, 484 Binary **detection,** simple, 8,100,244, 366, 419,482 general, 56,80,110 Binary FSK, 377 Binary symmetric detection problem, 68 Bound, Barankin, 184 bias, 198 binary error probability, 380 Chernoff, 15 ellipses, 298 error, 160 mean-square error, 18 3,20 performance of suboptimum receivers, 390 variance, 177 Butterworth spectrum, 104 Canonicalreceiver realizations, Dopplerspreadtargets, 367 No 1, Estimator-Correlator, 15 No 2, FilterCorrelator, 16 No 3, Filter-squarer-integrator, 17 No 4, Optimum realizable filter, 19 No 4S, State-variable realization, Characterizations, doubly-spread targets, 45 point targets, 238 Channels, Doppler-spread, 375 dual, 424 equivalent ,5 2 multiplicative, 26 Nakagami, 24 range-spread, 13 Ray leigh, slowly-fluctuating, 38 Rician , 24 tapped-delay line models, 488 Chernoff bounds, 15 Classification, of **Gaussian** detection problems, 57 Coded pulse sequences, 283,313,344 Colored bandpass noise, detection in, 247,267, 329 Comb filters, 264 Communication, binary, 69,74,79,111,375, 484 Doppler-spread channels, 375,406 doubly-spread channels, 482 M-ary, 396,523 623 624 Subject Index range-spread channels, 32 Complex, envelope, 569 finite-state processes, 89 **Gaussian** random variable, 84 **Gaussian** processes, representation, linear systems, 572 random processes, 576,600 signals, 566,598 state-variables, 574 sufficient statistic, 244 white processes, 82 Composite, hypothesis tests, 219 signals, Correlator , 12 1,245 Efficiency factor, 115, 118 Equivalent channels, 22 Error expressions, approximate, 38 Estimates, truncated, 194 **Estimation,** amplitude, 191,204,2 11,214, 530 Doppler, 275 mean Doppler, 33 mean range, 3 parameters of process, 167, 188 range, 275 velocity, 275 Estimator-correlator, Class B,, 65 simple binary, 15,101 Delay-spreadtargets, 412 Fading, frequency-selective,415 **Detection,** binary, 8,100,244,366,419,482 **Gaussian** **signals** **in** white noise, M-ary, 147,159 range-spread targets, 419,438 stationary processes, 99 target with unknown parameters, 339 vector processes, 52,91 Differential equation, 25 2, 25 model for doubly-spread target, 454 Discrete resolution, 323,324, 346 Dispersive targets, 13 Distributed state-variables, 454,473 Diversity, explicit, 381 eigenvalue, 117 frequency, 126 implicit, 38 minimum, 10 optimum, 130,381,510 system, 510 time, 122 Doppler **estimation,** 275 Doppler shift, 241 Doppler-spread channels, 375 Doppler-spread targets, 357 optimum receivers, 367 state-variable model, 365 Doubly-spread targets, 444 Dual channels, 424 Dual targets, 424 Duality, applications, 428 concepts, 422 time-frequency, Edgeworth series,39 time-selective, 35 Feedback shift register, 19, 345 Filter-squarer receiver, Class B,, 65 parameter **estimation,** SPLOT, 189 parameter **estimation,** 174,278 simple binary, 17,246,249,367 suboptimum, 154,390 Filters, mismatched, 338 Finite-state processes, 209,227,25 1,270, 372,389 Fredholm, determinant, 23,71,150,154, 371,391,396 Frequency-selective fading ,4 15 Frequency-spread targets, 357 Functional square root, 18 Gaussian,processes,8,5 6,99,147 pulse, 283,290 Hermite polynomials,343 Information matrix, 295 Inverse kernels, 11 Kernels, inverse,11 separable, 119, 37 Lagrangemultipliers,259 Laguerre polynomials, 34 Large time-bandwidth signals, 39 Likelihood function, 170,399,529 Likelihood ratio test, 10,61,81,245,366, 483 Linear frequency modulation, 290,292,466 Subject 625 Index Local accuracy, 294 Lowenergy-choerence 373,431,516 Processes, complex finite-state, 369,428, 589 complex Gaussian, 360,415,446,583 conditionally Gaussian, 169 finite-state, 209, 227,251,369,428,589 Gaussian, 8,56,99,147 non-Gaussian, 15 vector, 52,91,141,145,157,185,224 Pseudo-random sequences, 321 (LEC), 13 1,2 13, Matched filter, 121,245 Maximum likelihood equations, 175 M-ary systems, 396,5 23 Mean-square bandwidth, 290,571 Mean-square duration, 290,5 MMSE realizable estimate, 24,103,183, 483 Moment-generating functions, 34 Multipath, resoluable, 128,431 Multiple-parameters, 17, 230 369, Non-Gaussianprocesses,156 On-off sequences, 13 Optimal **signal** design, 258 Optimum linear filters, complex processes, 595 Orthogonal series model, 495 Overspread, channels, 504 targets, 45 Parameterestimation, 167,188 Doppler-spread targets, 398,409 doubly-spread targets, 525 finite-state processes, 209,227 generalized sets, 337 low-energy coherence, 213,229,527 range-spread targets, 436 separable kernels, 211,228 SPLOT, 188,221 Performance, approximations, 38,82 bounds, 79,153,380,502 detection of Doppler-spread targets, 370, 380 detection of point target **in** white noise, 246 ’ general binary, 66,82,92 LEC conditions, 136 M-ary, 151 parameter **estimation,** 177, 194,294,400, 436,531 reverberation, 461 simple binary, 32 typical system, 44 white noise, 32 Phasor diagram, 65 Radar, model, 234 **Radar** uncertainty principle, 309 Radiometer, two-filter, 421 Range **estimation,** 275 Range-scattering function, 16 Range-spread targets, 413 Rayleigh, channel, 118 Realization, canonical receiver, 15, 16, 17, 19 parallel processing, Receivers, adaptive, 155 conventional, 326,461 optimum, 9,63,65,101,102,103,109,114, 245,278,367,421,430,431,477,493 optimum for discrete resolution, 329 optimum **in** reverberation, 472 optimum LEC, 135,421,519 suboptimum, 151,162,205,263,385,433, 519 Resolution, dense, 459 discrete, 323, 346 Reverberation, 459,5 39 Reverberation scattering function, 461 Scattering function, Doppler 361,382 doubly-spread, 448,458,462 ’ range, 416,438 range-invariant, 465 Separable kernels, 119,211, 373 Shift-register sequences, 18 **Signal** design, optimal, 258, 270 Singular tests, 79,83 Sonar, model, 235 Special situations, 68 SPLOT case, 99,188, 372 State-variable representation, complex, 25 1, 574,589 distributed, 454,473,495 ordinary, 23,42,209,227 Suboptimum receivers, 151,162,205,263, 385, 433,519 626 Subject Index Summary, detection of point targets, 260 detection theory, 137,157 discrete resolution, 335 doubly-spread targets **and** channels, 36 estimation theory, 184,220 general binary, 88 range-Doppler **estimation,** 326 range-spread targets, 437 receiver structures, simple binary **detection,** 46 Tests, composite-hypothesis, 219 likelihood ratio, 10,61,81,245,366,483 singular, 79 Tilted variables, 38 Time compression, 24 Time-frequency, autocorrelation function, 279 duality, 421,439 Time-selective fading, 35 Two-filter radiometer, 421 Two-frequency correlation function, 17 Tapped-delaymodels,487 Targets, degenerate, 452 Doppler spread, 357 doubly-spread, 444 dual, 424 range-spread, 13 slowly-fluctuating point, 38 overspread, 45 underspread, 45 Underspread,channels,503 targets, 45 Vector processes,157 Volume invariance, 308 White bandpass noise, 244,263 Whitening, approach, 59 filters, 59, 254 ... results in all three parts Prerequisites Part II Chaps I-5, I-6 Part III Chaps III- 1 to III- 5 Chaps III- 6 to III- 7 Chaps .III- $-end Chaps.I-4, I-6 Chaps.I-4 Chaps.I-4, I-6, 111-lto III- 7 Array Processing. . .Detection, Estimation, and Modulation Theory Radar- Sonar Processing and Gaussian Signals in Noise HARRY L VAN TREES George Mason University New York l A Wiley-Interscience Publication... Array Processing, Wiley, New York, 1971 Contents Introduction 1.1 1.2 1.3 Review of Parts I and II Random Signals in Noise Signal Processing in Radar- Sonar Systems Referewes Detection of Gaussian

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