Advanced Theory of Signal Detection: Weak Signal Detection in Generalized Observations

402 Pages·2002·12.856 MB·English

by  Professor Iickho Song, Assistant Professor Jinsoo Bae, Sun Yong Kim PhD, SrMIEEE (auth.)

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:[e-mail]
AssistantProfessor[insooBae
SejongUniversity
98GunjaDongGwangjinGu
Seoul,143-747
Korea
e-mail:[e-mail]
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)
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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