Non-Gaussian Statistical Communication Theory

652 Pages·6.264 MB·English

by  David Middleton(auth.)

NON-GAUSSIAN
STATISTICAL
COMMUNICATION
THEORY
IEEEPress
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J. Anderson B-M. Haemmerli S. Nahavandi
G. W. Arnold M.Lanzerotti T. Samad
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IEEEInformationTheorySociety,Sponsor
NON-GAUSSIAN
STATISTICAL
COMMUNICATION
THEORY
DAVID MIDDLETON
Copyright(cid:1)2012bytheInstituteofElectricalandElectronicsEngineers,Inc.
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LibraryofCongressCataloging-in-PublicationData:
PrintISBN:9780470948477
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
To:
William A. Von Winkle (1930–2007),
Inspiring Director ofResearch (NUSC), andloyal friend;
and tomy Wife,
JoanBartlett Middleton,
who provided theessential supportand encouragement
needed increatingthis book and infulfillingmy life.
CONTENTS
Foreword xv
VisualizingtheInvisible xvii
Acknowledgments xxi
AbouttheAuthor xxiii
Editor’sNote xxv
Introduction 1
1 ReceptionasaStatisticalDecisionProblem 15
1.1 SignalDetectionandEstimation, 15
1.2 SignalDetectionandEstimation, 17
1.2.1 Detection, 17
1.2.2 TypesofExtraction, 20
1.2.3 OtherReceptionProblems, 21
1.3 TheReceptionSituationinGeneralTerms, 22
1.3.1 Assumptions:Space–TimeSampling, 22
1.3.2 TheDecisionRule, 25
1.3.3 TheDecisionProblem, 26
1.3.4 TheGenericSimilarityofDetectionandExtraction, 27
1.4 SystemEvaluation, 27
1.4.1 EvaluationFunctions, 27
1.4.2 SystemComparisonsandErrorProbabilities, 30
1.4.3 Optimization:BayesSystems, 31
1.4.4 Optimization:MinimaxSystems, 32
1.5 ASummaryofBasicDefinitionsandPrincipalTheorems, 35
1.5.1 SomeGeneralPropertiesofOptimumDecisionRules, 35
1.5.2 Definitions, 36
1.5.3 PrincipalTheorems, 37
viii CONTENTS
1.5.4 Remarks:PriorProbabilities,CostAssignments,andSystem
Invariants, 38
1.6 Preliminaries:BinaryBayesDetection, 40
1.6.1 FormulationI:BinaryOn–OffSignalDetection, 42
1.6.2 TheAverageRisk, 43
1.6.3 CostAssignments, 43
1.6.4 ErrorProbabilities, 45
1.7 OptimumDetection:On–OffOptimumProcessingAlgorithms, 46
1.7.1 TheLogarithmicGLRT, 48
1.7.2 RemarksontheBayesOptimalityoftheGLR, 48
1.8 SpecialOn–OffOptimumBinarySystems, 50
1.8.1 Neyman–PearsonDetectionTheory, 50
1.8.2 TheIdealObserverDetectionSystem, 51
1.8.3 MinimaxDetectors, 52
1.8.4 MaximumAposteriori(MAP)DetectorsfromaBayesian
Viewpoint, 53
1.8.5 BayesianSequentialDetectors, 57
1.9 OptimumDetection:On–OffPerformanceMeasuresandSystem
Comparisons, 57
1.9.1 ErrorProbabilities:OptimumSystems, 58
1.9.2 ErrorProbabilities:SuboptimumSystems, 65
1.9.3 DecisionCurvesandSystemComparisons, 66
1.10 BinaryTwo-SignalDetection:DisjointandOverlappingHypothesis
Classes, 69
1.10.1 DisjointSignalClasses, 69
1.10.2 OverlappingHypothesisClasses, 70
1.11 ConcludingRemarks, 73
References, 74
2 Space–TimeCovariancesandWaveNumberFrequencySpectra:
I.NoiseandSignalswithContinuousandDiscreteSampling 77
2.1 InhomogeneousandNonstationarySignalandNoiseFieldsI:
Waveforms,BeamTheory,Covariances,andIntensitySpectra, 78
2.1.1 SignalNormalization, 79
2.1.2 InhomogeneousNonstationary(Non-WS-HS)Noise
Covariances, 80
2.1.3 NarrowbandFields, 83
2.1.4 NoiseandSignalFieldCovariances:NarrowbandCases, 88
2.2 ContinuousSpace–TimeWiener–KhintchineRelations, 91
2.2.1 Directly Sampled Approximation of the W–Kh Relations
(Hom-Stat Examples), 93
2.2.2 ExtendedWiener–KhintchineTheorems:Continuous
InhomogeneousandNonstationaryRandom(Scalar)Fields, 95
2.2.3 TheImportantSpecialCaseofHomogeneous—Stationary
Fields—FiniteandInfiniteSamples, 100
2.3 TheW–KhRelationsforDiscreteSamplesintheNon-Hom-Stat
Situation, 102
2.3.1 TheAmplitudeSpectrumforDiscreteSamples, 102
2.3.2 PeriodicSampling, 107
CONTENTS ix
2.4 TheWiener–KhintchineRelationsforDiscretelySampledRandom
Fields, 108
2.4.1 DiscreteHom-StatWiener–KhintchineTheorem:
PeriodicSamplingandFiniteandInfiniteSamples, 110
2.4.2 Comments, 112
2.5 ApertureandArrays—I:AnIntroduction, 115
2.5.1 Transmission:AperturesandTheirFourierEquivalents, 116
2.5.2 Transmission:ThePropagatingFieldandItsSource
Function, 120
2.5.3 PointArrays:DiscreteSpatialSampling, 126
2.5.4 Reception, 129
2.5.5 NarrowbandSignalsandFields, 134
2.5.6 SomeGeneralObservations, 137
2.6 ConcludingRemarks, 138
References, 139
3 OptimumDetection,Space–TimeMatchedFilters,andBeam
ForminginGaussianNoiseFields 141
3.1 OptimumDetectionI:SelectedGaussianPrototypes—Coherent
Reception, 142
3.1.1 OptimumCoherentDetection.CompletelyKnown
DeterministicSignalsinGaussNoise, 142
3.1.2 Performance, 146
3.1.3 ArrayProcessingII:BeamFormingwithLinearArrays, 150
3.2 OptimumDetectionII:SelectedGaussianPrototypes—Incoherent
Reception, 154
3.2.1 IncoherentDetection:I.NarrowbandDeterministic
Signals, 154
3.2.2 IncoherentDetectionII.DeterministicNarrowbandSignals
withSlowRayleighFading, 169
3.2.3 IncoherentDetectionIII:NarrowbandEquivalentEnvelope
Inputs—Representations, 172
3.3 OptimalDetectionIII:SlowlyFluctuatingNoise
Backgrounds, 176
3.3.1 CoherentDetection, 176
3.3.2 NarrowbandIncoherentDetectionAlgorithms, 180
3.3.3 IncoherentDetectionofBroadbandSignalsinNormal
Noise, 183
3.4 BayesMatchedFiltersandTheirAssociatedBilinearandQuadratic
Forms,I, 188
3.4.1 CoherentReception:CausalMatchedFilters(Type1), 190
3.4.2 IncoherentReception:CausalMatchedFilters(Type1), 192
3.4.3 IncoherentReception-RealizableMatchedFilters;Type2, 195
3.4.4 Wiener–KolmogoroffFilters, 198
3.4.5 Extensions:Clutter,Reverberation,andAmbientNoise, 200
3.4.6 MatchedFiltersandTheirSeparationinSpace
andTimeI, 202
3.4.7 SolutionsoftheDiscreteIntegralEquations, 207
3.4.8 SummaryRemarks, 214
x CONTENTS
3.4.9 Signal-to-NoiseRatios,ProcessingGains,andMinimum
DetectableSignals.I, 214
3.5 BayesMatchedFiltersintheWaveNumber–Frequency
Domain, 219
3.5.1 FourierTransformsofDiscreteSeries, 219
3.5.2 IndependentBeamFormingandTemporal
Processing, 230
3.6 ConcludingRemarks, 235
References, 235
4 MultipleAlternativeDetection 239
4.1 Multiple-AlternativeDetection:TheDisjointCases, 239
4.1.1 Detection, 240
4.1.2 MinimizationoftheAverageRisk, 242
4.1.3 GeometricInterpretation, 244
4.1.4 Examples, 245
4.1.5 ErrorProbabilities,AverageRisk,andSystem
Evaluation, 250
4.1.6 AnExample, 253
4.2 OverlappingHypothesisClasses, 254
4.2.1 Reformulation, 255
4.2.2 MinimizationoftheAverageRiskforOverlappingHypothesis
Classes, 257
4.2.3 Simple(Kþ1)-aryDetection, 259
4.2.4 ErrorProbabilities,AverageandBayesRisk,andSystem
Evaluations, 260
4.3 DetectionwithDecisionsRejection:NonoverlappingSignal
Classes, 262
4.3.1 Optimum(Kþ1)-aryDecisionswithRejection, 264
4.3.2 Optimum(Kþ1)-aryDecisionwithRejection, 265
4.3.3 ASimpleCostAssignment, 266
4.3.4 Remarks, 267
References, 270
5 BayesExtractionSystems:SignalEstimationandAnalysis,
p(H )=1 271
1
5.1 DecisionTheoryFormulation, 272
5.1.1 NonrandomizedDecisionRulesandAverage
Risk, 272
5.1.2 BayesExtractionWithaSimpleCostFunction, 274
5.1.3 BayesExtractionWithaQuadraticCostFunction, 278
5.1.4 FurtherProperties, 281
5.1.5 OtherCostFunctions, 283
5.2 CoherentEstimationofAmplitude(DeterministicSignalsandNormal
Noise,p(H )¼1), 287
1
5.2.1 CoherentEstimationofSignalAmplitudeQuadraticCost
Function, 287
5.2.2 CoherentEstimationofSignalAmplitude(SimpleCost
Functions), 290
CONTENTS xi
5.2.3 Estimationsby(Real)uFilters, 291
5.2.4 BiasedandUnbiasedEstimates, 293
5.3 IncoherentEstimationofSignalAmplitude(DeterministicSignals
andNormalNoise,p(H )¼1), 294
1
5.3.1 QuadraticCostFunction, 294
5.3.2 “Simple”CostFunctionsSCF (IncoherentEstimation), 298
1
5.4 WaveformEstimation(RandomFields), 300
5.4.1 NormalNoiseSignalsinNormalNoiseFields
(QuadraticCostFunction), 300
5.4.2 NormalNoiseSignalsinNormalNoiseFields
(“Simple”CostFunctions), 301
5.5 SummaryRemarks, 304
References, 305
6 JointDetectionandEstimation,p(H )(cid:1)1:I.Foundations 307
1
6.1 JointDetectionandEstimationunderPriorUncertainty½pðH Þ(cid:1)1(cid:2):
1
Formulation, 309
6.1.1 Case1:NoCoupling, 312
6.1.2 Case2:Coupling, 314
6.2 OptimalEstimation[p(H )(cid:1)1]:NoCoupling, 315
1
6.2.1 QuadraticCostFunction:MMSEandBayesRisk, 316
6.2.2 SimpleCostFunctions:UMLEandBayesRisk, 319
6.3 SimultaneousJointDetectionandEstimation:
GeneralTheory, 326
6.3.1 TheGeneralCase:StrongCoupling, 326
6.3.2 SpecialCasesI:BayesDetectionandEstimationWithWeak
Coupling, 331
6.3.3 SpecialCasesII:FurtherDiscussionofg* forWeakor
p<1jQCF
NoCoupling, 333
6.3.4 EstimatorBiasðp(cid:1)1Þ, 336
6.3.5 RemarksonIntervalEstimation,pðH Þ(cid:1)1, 338
1
6.3.6 DetectionProbabilities, 339
6.3.7 WaveformEstimationðp(cid:1)1Þ:CoupledandUncoupledD
andE, 341
6.3.8 ExtensionsandModifications, 342
6.3.9 SummaryRemarks, 345
6.4 JointDandE:Examples–EstimationofSignalAmplitudes
[p(H )(cid:1)1], 350
1
6.4.1 AmplitudeEstimation,p(H )¼1, 352
1
6.4.2 BayesEstimatorsandBayesError,pðH Þ(cid:1)1, 355
1
6.4.3 PerformanceDegradation,p<1, 358
6.4.4 AcceptanceorRejectionoftheEstimator:Detection
Probabilities, 367
6.4.5 RemarksontheEstimationofSignal
IntensityI (cid:3)a2, 371
o o
6.5 SummaryRemarks,p(H) (cid:1)1:I—Foundations, 378
1
References, 379