Iickho Song· Iinsoo Bae.Sun Yong AdvancedTheoryofSignalDetection ONLINELIBRARY Engineering http://www.springer.de/engine-de/ Springer-Verlag Berlin Heidelberg GmbH Iickho Song· Iinsoo Bae Sun Yong Kim Advanced Theory of Signal Detection Weak Signal Detection in Generalized Observations With 116Figures and 57Tables , Springer ProfessorIickhoSong KoreaAdvanced Institute ofScienceandTechnology Dept.EE KAIST,373-1 GuseongDong YuseongGu, Daejeon,305-701 Korea e-mail:[email protected] AssistantProfessor[insooBae SejongUniversity 98GunjaDongGwangjinGu Seoul,143-747 Korea e-mail:[email protected] SunYongKim,PhD,SrMIEEE DepartmentofElectronicEngineering KonkukUniversity 1Hwayang Dong, Gwangjin Gu Seoul 143-701 Korea e-mail:kimsytskkucc.konkuk.aekr ISBN978-3-642-07708-1 ISBN978-3-662-04859-7(eBook) DOI10.1007/978-3-662-04859-7 LibraryofCongressCataloging-in-Publication Data Song.Iickho: AdvancedTheoryofSignalDetection:WeakSignalDetectioninGeneralizedObservations; With57TablesIIickhoSong;[insooBae;SunYongKim.- Berlin;Heidelberg;NewYork;Barcelona; HongKong;London;Milan;Paris;Tokyo:Springer,2002 (SignalsandCommunicationTechnology) Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproductiononmicrofilmorinotherways,andstorageindatabanks.Duplicationof thispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLaw ofSeptember9,1965,initscurrentversion, andpermissionforusemustalwaysbeobtainedfrom Springer-Verlag.Violations areliableforprosecutionactunderGermanCopyrightLaw. http://www.springer.de © Springer-VerlagBerlinHeidelberg2002 OriginallypublishedbySpringer-VerlagBerlinHeidelbergNew Yorkin2002. Softcoverreprintofthehardcover lstedition2002 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsence ofaspecificstatement,that such names areexempt from the relevant protectivelawsandregulationsandthereforefreeforgeneral use. Typesetting:Camera-readycopyfromauthors Cover-Design: Design&Production. Heidelberg Printedonacid-freepaper SPIN:10860664 6213020/kk 543210 To The Lovely Members of Our Families and Academic Family Preface Wehave some time ago noticed that findinga book dealing withtopics in the ad vanced theoryandapplicationsofsignaldetection isnotquiteaneasy matter.This iscontrasted withthatthere arenumerous booksonthemoregeneral subjectofde tectionandestimation.Frankly,ourexperience andexpertiseisonlyonsomepartial portions of the theory and recent topics of signal detection.This book istherefore meanttoinclude notalltheadvanced andinteresting topicsinthetheoryandappli cationsofsignaldetection, butjustonlysomesubsetsofthem:somesuchimportant and interestingtopics and issues as distributed signal detection and sequential de tectionarenotconsideredonlyduetoourlimitedknowledgeandcapacity. The goalwehaveinmind forthisbookistopresent several advanced topicsin signal detection theory and thereby help readers gain novel ideas and insights.In thisbook, wehavetriedtocompletelypresentinaunifiedwaythethemeoflocally optimum detection ofsignals ingeneralized observations. Amongourhope isthus that the readers would be able to understand the concepts and fundamentals of a generalized observation model as applied to signal detection problems.This book willalso allowthe readers, whether theyare students, academics, practitioners,or researchers, tohaveanexpanded viewonsignal detection. Although the work described in this book for each area is given in separate chapters, the general philosophy of the underlying key concepts, especially local optimality, nonparametricity, and robustness, will permeate the entire book. This book can logically be divided intothree parts, though itis notexplicitly indicated in the table ofcontents. The first part, Chapters 2 through 4 in addition to some sections ofChapter I,contains theapplicationofstatisticalhypothesistestingtothe problem of weak signaldetection inageneralizedobservation model. Asymptotic andfinitesample-sizeperformancecharacteristicsofseveraldetectorsincludingthe locallyoptimumdetectors areconsideredforcomparison.Thesecondpart,Chapters 5and6inaddition tosome sections ofChapter I, dealswithlocally optimum rank detectors. The locally optimum rank detectors are nonparametric signal detectors having bases in thesignand rank statisticsoftheobservations. Asinthefirstpart, asymptotic and finite sample-size performance characteristics of several detectors including the locally optimumrank detectors are considered for comparison. The lastpartdeals withdetectionschemes undertwodistinctandinterestingobservation scenarios. Detection ofsignals in weakly-dependentnoise, a good approximation of high speed sampling communication systems, is analyzed inChapter 7, where VIII the noise processisassumed tobe dependent toacertaindegree. The combination offuzzy set theory and signaldetection theory consideredin Chapter8 is another unique topic inthisbook:interestingresultsonthelocallyoptimumfuzzy detection ofknownandrandomsignals areincluded. Inadditiontoitsdeliberateorientationtoandcomprehensivetreatmentofsignal detectionin a generalized observationmodel, this bookhas a number offeatures thatenhanceitsstatusboth asatextbookonadvancedsignal detectionandalso asa usefulreferencevolume.Each chapterbeginswithabriefdiscussionofitsintentand ends with achaptersummary;results aremotivatedanddevelopedasthoroughlyas possible;and proofs are providedforallimportantfacts and results thatare notob vious,eitherdirectly inthebookorasproblems(some tobetackled bythereaders). Asthisbookdeals withadvancedtopicsinsignal detectiontheory,thisbookismore usefultothose whohavecompletedanintroductorycourse onsignaldetection.Yet, the necessarybackgroundassumedisanexposuretothebasictheory ofprobability andrandomprocessesandintroductorydetectiontheory: therefore,thisbookshould beusefultopracticingengineersandresearchersaswellas academicsandstudents. Readers might also use this bookas ahandbookoflocally optimumdetection.We are quite sure that any person interestedin locally optimum detection will findit pleasureand rewardingtogainnovelideasand insightsfrom thisbook. We would like toacknowledgethecontributionsofmany individualswho over theyears have providedstimulatingdiscussionsofresearchproblems,opportunities to strive for thesolutionsand findapplicationsofthe results, and valuable sugges tions and comments: these all have been crucial and essentialinthe completionof thisbook.Specifically,weexpressourdeepestappreciationtoProfessorsSouguil1. M.AnnandSaleemA.Kassamwithoutwhoseexceptionallyexcellentandthorough guidancelong time ago this attemptwouldhave never been possibleor realizedin anysense. Weexpress ourgratefulappreciationtoallthemembersoftheStatistical SignalProcessingLaboratory,KoreaAdvancedInstituteofScienceandTechnology (KAIST),especiallytoSoRyoungPark and SeokhoYoon,fortheir invaluablehelp and suggestionsinpreparingthemanuscriptandfigures ofthisbook. The research projects leading to this bookhave been financially supportedby many grantsincludingthose from Korea ScienceandEngineeringFoundation,Ko reaResearchFoundation,andMinistryofInformationandCommunication:most of all, the supportfrom the YoungScientistsAward tothe first author in2000should behighlyappreciated. November2001 IickhoSong JinsooBae Sun YongKim KAIST SejongUniversity KonkukUniversity Daejeon Seoul Seoul Korea Korea Korea Contents 1. PRELIMINARIES .......................................... 1 1.1 AnOverview ..................................... 1 1.1.1 DetectionofDiscrete-TimeSignals ..................... 1 1.1.2 Organization oftheBook.............................. 2 1.2 Locally Optimum Detection.................................. 4 1.2.1 BasicConcepts. ..................................... 4 1.2.2 Methods inPerformance Comparison .............. 5 1.3 ObservationModels 8 1.3.1 AdditiveNoiseModel ............... ................. 8 1.3.2 AGeneralized ObservationModel ...................... 8 1.3.3 Assumptions........................................ 10 1.4 ReparametrizationoftheGeneralizedObservationModel ......... 14 1.5 NoiseProbabilityDensity Functions........................... 17 1.5.1 Generalized GaussianDistribution ... .. .. 17 1.5.2 Generalized CauchyDistribution ....................... 19 1.5.3 Student'st-Distribution............................... 21 1.5.4 LogisticDistribution ................................. 23 1.5.5 BivariateGaussian Distribution .. ...................... 24 1.5.6 Bivariatet-Distribution ............................... 26 1.6 RankStatistics andScoreFunctions ........................... 29 1.6.1 Sign,Order,andRankStatistics ........................ 29 1.6.2 ScoreFunctions 31 1.6.3 ApproximationstoandAsymptoticAveragesofScoreFunc- tions 32 1.7 Summary .... ...................................... ... .... 37 Problems....................................................... 38 Appendix1.1 VariousExpressions andPropertiesofScoreFunctions .. 42 Appendix1.2 Sums andWeightedSumsofScoreFunctions .......... 45 X Contents 2. LOCALLYOPTIMUMDETECTIONOF KNOWNSIGNALS. .... 59 2.1 Introduction ............................................... 59 2.2 DetectioninGeneralizedObservations. ........................ 60 212.1 LocallyOptimumTestStatistic. ........................ 60 2.2.2 ObservationsandComments........................... 62 2.2.3 ExamplesofLocallyOptimumDetectors ................ 64 2.3 PerformanceoftheLocallyOptimumDetectors................. 69 2.3.1 AsymptoticPerformance. ............................. 69 2.3.2 FiniteSample-SizePerformance. ....................... 78 2.4 Summary......... ......................... ............... 79 Problems....................................................... 81 3. LOCALLYOPTIMUMDETECTIONOF RANDOM SIGNALS .,. 85 3.1 Introduction............................................... 85 3.2 LocallyOptimumTestStatistic ............................... 87 3.2.1 TestStatisticinMultiplicativeNoise .................... 88 3.2.2 TestStatisticinSignal-DependentNoise................. 91 3.2.3 ExamplesoftheLocallyOptimumDetectors............. 94 3.3 PerformanceoftheLocallyOptimumDetectors................. 97 3.3.1 AsymptoticPerformanceCharacteristics................. 97 3.3.2 AsymptoticRelativeEfficienciesforSpecificNoiseDistri- butions .." 99 3.3.3 Asymptotic Relative Efficienciesfor the Additive Noise Model 104 3.3.4 FiniteSample-SizePerformance 107 3.4 Summary '.'.. 111 Problems 112 Appendix3.1 EfficaciesofRandomSignalDetectors 114 4. LOCALLYOPTIMUMDETECTIONOF COMPOSITESIGNALS 123 4.1 Introduction 123 4.2 CompositeSignalDetectioninAdditiveNoise 124 4.2.1 ObservationModel 124 4.2.2 LocallyOptimumTestStatistic 125 4.2.3 StructuresofLocallyOptimumDetectors 127 4.2.4 ExamplesoftheLocallyOptimumDetectors 129 4.2.5 PerformanceCharacteristics 131 4.3 CompositeSignalDetectioninMultiplicativeNoise 143 4.3.1 ObservationModel 143 4.3.2 LocallyOptimumTestStatistic 143 4.3.3 PerformanceoftheLocallyOptimumDetectors 145 4.4 CompositeSignalDetectioninSignal-DependentNoise 151 4.4.1 ObservationModel 151 4.4.2 DetectorTestStatisticandStructures 151 4.4.3 PerformanceCharacteristics 156 Contents XI 4.5 Summary 164 Problems - 165 Appendix4.1 EfficaciesinAdditiveNoise 167 Appendix4.2 LocallyOptimumTestStatisticforCompositeSignals 172 Appendix4.3 ApplicationsofL'Hospital'sRule 181 5. KNOWNSIGNALDETECTIONWITHSIGNSANDRANKS 185 5.1 Introduction 185 5.2 LocallyOptimumRankDetectionofKnownSignals 187 5.2.1 DetectioninAdditiveNoise 187 5.2.2 DetectioninMultiplicativeNoise 192 5.2.3 DetectioninSignal-DependentNoise 195 5.2.4 ExamplesofScoreFunctions 200 5.3 Median-ShiftSignDetection 204 5.3.1 TestStatisticoftheMedian-ShiftSignDetector 204 5.3.2 OptimumMedian-ShiftValue 206 5.3.3 PerformanceCharacteristics 214 5.4 Summary 222 Problems 223 Appendix5.1 ScoreFunctionsforSomeSpecificDistributions 232 6. RANDOMSIGNALDETECTIONWITHSIGNSANDRANKS 239 6.1 Introduction 239 6.2 RandomSignalDetectioninAdditiveNoise 240 6.2.1 LocallyOptimumRankTestStatistic 240 6.2.2 MultipleInputandTwoSampleDetection 242 6.2.3 PerformanceCharacteristics 245 6.3 RandomSignalDetectioninMultiplicativeandSignal-Dependent Noise 252 6.3.1 DetectioninMultiplicativeNoise 252 6.3.2 DetectioninSignal-DependentNoise 255 6.4 CompositeSignalDetection 259 6.4.1 DetectioninAdditiveNoise 259 6.4.2 DetectioninMultiplicativeNoise 261 6.4.3 DetectioninSignal-DependentNoise 264 6.5 ExamplesofScoreFunctions 267 6.6 Summary 274 Problems 275 Appendix6.1 DerivationoftheTestStatistic 279 Appendix6.2 EfficaciesofDetectors 280