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 1 Introduction 1
 1.1Signals,NoiseandInformation 1
 1.2SignalProcessingMethods 3
 1.2.1Transform-BasedSignalProcessing 3
 1.2.2Source-FilterModel-BasedSignalProcessing 5
 1.2.3BayesianStatisticalModel-BasedSignalProcessing 5
 1.2.4NeuralNetworks 6
 1.3ApplicationsofDigitalSignalProcessing 6
 1.3.1DigitalWatermarking 6
 1.3.2Bio-medical,MIMO,SignalProcessing 8
 1.3.3EchoCancellation 10
 1.3.4AdaptiveNoiseCancellation 12
 1.3.5AdaptiveNoiseReduction 12
 1.3.6BlindChannelEqualisation 13
 1.3.7SignalClassificationandPatternRecognition 13
 1.3.8LinearPredictionModellingofSpeech 15
 1.3.9DigitalCodingofAudioSignals 16
 1.3.10DetectionofSignalsinNoise 17
 1.3.11DirectionalReceptionofWaves:Beam-forming 18
 1.3.12Space-TimeSignalProcessing 20
 1.3.13DolbyNoiseReduction 20
 1.3.14RadarSignalProcessing
  opplerFrequencyShift 21 1.4AReviewofSamplingandQuantisation 22
 1.4.1AdvantagesofDigitalFormat 24
 1.4.2DigitalSignalsStoredandTransmittedinAnalogueFormat 25
 1.4.3TheEffectofDigitisationonSignalBandwidth 25
 1.4.4SamplingaContinuous-TimeSignal 25
 1.4.5AliasingDistortion 27
 1.4.6NyquistSamplingTheorem 27
 viii Contents
 1.4.7Quantisation 28
 1.4.8Non-LinearQuantisation,Companding 30
 1.5Summary 32
 Bibliography 32
 2 NoiseandDistortion 35
 2.1Introduction 35
 2.1.1DifferentClassesofNoiseSourcesandDistortions 36
 2.1.2DifferentClassesandSpectral/TemporalShapesofNoise 37
 2.2WhiteNoise 37
 2.2.1Band-LimitedWhiteNoise 38
 2.3ColouredNoise
  inkNoiseandBrownNoise 39 2.4ImpulsiveandClickNoise 39
 2.5TransientNoisePulses 41
 2.6ThermalNoise 41
 2.7ShotNoise 42
 2.8Flicker(I/f)Noise 43
 2.9BurstNoise 44
 2.10Electromagnetic(Radio)Noise 45
 2.10.1NaturalSourcesofRadiationofElectromagneticNoise 45
 2.10.2Man-madeSourcesofRadiationofElectromagneticNoise 45
 2.11ChannelDistortions 46
 2.12EchoandMulti-pathReflections 47
 2.13ModellingNoise 47
 2.13.1FrequencyAnalysisandCharacterisationofNoise 47
 2.13.2AdditiveWhiteGaussianNoiseModel(AWGN) 48
 2.13.3HiddenMarkovModelandGaussianMixtureModelsforNoise 49
 Bibliography 50
 3 InformationTheoryandProbabilityModels 51
 3.1Introduction
  robabilityandInformationModels 52 3.2RandomProcesses 53
 3.2.1Information-bearingRandomSignalsvsDeterministicSignals 53
 3.2.2Pseudo-RandomNumberGenerators(PRNG) 55
 3.2.3StochasticandRandomProcesses 56
 3.2.4TheSpaceofVariationsofaRandomProcess 56
 3.3ProbabilityModelsofRandomSignals 57
 3.3.1ProbabilityasaNumericalMappingofBelief 57
 3.3.2TheChoiceofOneandZeroastheLimitsofProbability 57
 3.3.3Discrete,ContinuousandFinite-StateProbabilityModels 58
 3.3.4RandomVariablesandRandomProcesses 58
 3.3.5ProbabilityandRandomVariables–TheSpaceandSubspacesofa
 Variable 58
 3.3.6ProbabilityMassFunction–DiscreteRandomVariables 60
 3.3.7Bayes’Rule 60
 3.3.8ProbabilityDensityFunction–ContinuousRandomVariables 61
 3.3.9ProbabilityDensityFunctionsofContinuousRandomProcesses 62
 3.3.10Histograms–ModelsofProbability 63
 3.4InformationModels 64
 3.4.1Entropy:AMeasureofInformationandUncertainty 65
 3.4.2MutualInformation 68
 Contents ix
 3.4.3EntropyCoding–VariableLengthCodes 69
 3.4.4HuffmanCoding 70
 3.5StationaryandNon-StationaryRandomProcesses 73
 3.5.1Strict-SenseStationaryProcesses 75
 3.5.2Wide-SenseStationaryProcesses 75
 3.5.3Non-StationaryProcesses 76
 3.6Statistics(ExpectedValues)ofaRandomProcess 76
 3.6.1CentralMoments 77
 3.6.1.1Cumulants 77
 3.6.2TheMean(orAverage)Value 77
 3.6.3Correlation,SimilarityandDependency 78
 3.6.4Autocovariance 81
 3.6.5PowerSpectralDensity 81
 3.6.6JointStatisticalAveragesofTwoRandomProcesses 83
 3.6.7Cross-CorrelationandCross-Covariance 83
 3.6.8Cross-PowerSpectralDensityandCoherence 84
 3.6.9ErgodicProcessesandTime-AveragedStatistics 85
 3.6.10Mean-ErgodicProcesses 85
 3.6.11Correlation-ErgodicProcesses 86
 3.7SomeUsefulPracticalClassesofRandomProcesses 87
 3.7.1Gaussian(Normal)Process 87
 3.7.2MultivariateGaussianProcess 88
 3.7.3GaussianMixtureProcess 89
 3.7.4Binary-StateGaussianProcess 90
 3.7.5PoissonProcess–CountingProcess 91
 3.7.6ShotNoise 92
 3.7.7Poisson–GaussianModelforCluttersandImpulsiveNoise 93
 3.7.8MarkovProcesses 94
 3.7.9MarkovChainProcesses 95
 3.7.10HomogeneousandInhomogeneousMarkovChains 96
 3.7.11GammaProbabilityDistribution 96
 3.7.12RayleighProbabilityDistribution 97
 3.7.13ChiDistribution 97
 3.7.14LaplacianProbabilityDistribution 98
 3.8TransformationofaRandomProcess 98
 3.8.1MonotonicTransformationofRandomProcesses 99
 3.8.2Many-to-OneMappingofRandomSignals 101
 3.9SearchEngines:CitationRanking 103
 3.9.1CitationRankinginWebPageRankCalculation 104
 3.10Summary 104
 Bibliography 105
 4 BayesianInference 107
 4.1BayesianEstimationTheory:BasicDefinitions 108
 4.1.1Bayes’Theorem 109
 4.1.2ElementsofBayesianInference 109
 4.1.3DynamicandProbabilityModelsinEstimation 110
 4.1.4ParameterSpaceandSignalSpace 111
 4.1.5ParameterEstimationandSignalRestoration 111
 4.1.6PerformanceMeasuresandDesirablePropertiesofEstimators 112
 4.1.7PriorandPosteriorSpacesandDistributions 114
 x Contents
 4.2BayesianEstimation 117
 4.2.1MaximumAPosterioriEstimation 117
 4.2.2Maximum-Likelihood(ML)Estimation 118
 4.2.3MinimumMeanSquareErrorEstimation 121
 4.2.4MinimumMeanAbsoluteValueofErrorEstimation 122
 4.2.5EquivalenceoftheMAP,ML,MMSEandMAVEEstimatesforGaussian
 ProcesseswithUniformDistributedParameters 123
 4.2.6InfluenceofthePrioronEstimationBiasandVariance 123
 4.2.7RelativeImportanceofthePriorandtheObservation 126
 4.3Expectation-Maximisation(EM)Method 128
 4.3.1CompleteandIncompleteData 128
 4.3.2MaximisationofExpectationoftheLikelihoodFunction 129
 4.3.3DerivationandConvergenceoftheEMAlgorithm 130
 4.4Cramer–RaoBoundontheMinimumEstimatorVariance 131
 4.4.1Cramer–RaoBoundforRandomParameters 133
 4.4.2Cramer–RaoBoundforaVectorParameter 133
 4.5DesignofGaussianMixtureModels(GMMs) 134
 4.5.1EMEstimationofGaussianMixtureModel 134
 4.6BayesianClassification 136
 4.6.1BinaryClassification 137
 4.6.2ClassificationError 139
 4.6.3BayesianClassificationofDiscrete-ValuedParameters 139
 4.6.4MaximumAPosterioriClassification 140
 4.6.5Maximum-LikelihoodClassification 140
 4.6.6MinimumMeanSquareErrorClassification 140
 4.6.7BayesianClassificationofFiniteStateProcesses 141
 4.6.8BayesianEstimationoftheMostLikelyStateSequence 142
 4.7ModellingtheSpaceofaRandomProcess 143
 4.7.1VectorQuantisationofaRandomProcess 143
 4.7.2VectorQuantisationusingGaussianModelsofClusters 143
 4.7.3DesignofaVectorQuantiser:K-MeansClustering 144
 4.8Summary 145
 Bibliography 146
 5 HiddenMarkovModels 147
 5.1StatisticalModelsforNon-StationaryProcesses 147
 5.2HiddenMarkovModels 149
 5.2.1ComparisonofMarkovandHiddenMarkovModels 149
 5.2.1.1Observable-StateMarkovProcess 149
 5.2.1.2Hidden-StateMarkovProcess 149
 5.2.2APhysicalInterpretation:HMMsofSpeech 151
 5.2.3HiddenMarkovModelasaBayesianModel 152
 5.2.4ParametersofaHiddenMarkovModel 152
 5.2.5StateObservationProbabilityModels 153
 5.2.6StateTransitionProbabilities 154
 5.2.7State–TimeTrellisDiagram 154
 5.3TrainingHiddenMarkovModels 155
 5.3.1Forward–BackwardProbabilityComputation 156
 5.3.2Baum–WelchModelRe-estimation 157
 5.3.3TrainingHMMswithDiscreteDensityObservationModels 158
 Contents xi
 5.3.4HMMswithContinuousDensityObservationModels 159
 5.3.5HMMswithGaussianMixturepdfs 160
 5.4DecodingSignalsUsingHiddenMarkovModels 161
 5.4.1ViterbiDecodingAlgorithm 162
 5.4.1.1ViterbiAlgorithm 163
 5.5HMMsinDNAandProteinSequences 164
 5.6HMMsforModellingSpeechandNoise 165
 5.6.1ModellingSpeech 165
 5.6.2HMM-BasedEstimationofSignalsinNoise 166
 5.6.3SignalandNoiseModelCombinationandDecomposition 167
 5.6.4HiddenMarkovModelCombination 168
 5.6.5DecompositionofStateSequencesofSignalandNoise 169
 5.6.6HMM-BasedWienerFilters 169
 5.6.7ModellingNoiseCharacteristics 170
 5.7Summary 171
 Bibliography 171
 6 LeastSquareErrorWiener-KolmogorovFilters 173
 6.1LeastSquareErrorEstimation:Wiener-KolmogorovFilter 173
 6.1.1DerivationofWienerFilterEquation 174
 6.1.2CalculationofAutocorrelationofInputandCross-CorrelationofInputand
 DesiredSignals 177
 6.2Block-DataFormulationoftheWienerFilter 178
 6.2.1QRDecompositionoftheLeastSquareErrorEquation 179
 6.3InterpretationofWienerFilterasProjectioninVectorSpace 179
 6.4AnalysisoftheLeastMeanSquareErrorSignal 181
 6.5FormulationofWienerFiltersintheFrequencyDomain 182
 6.6SomeApplicationsofWienerFilters 183
 6.6.1WienerFilterforAdditiveNoiseReduction 183
 6.6.2WienerFilterandSeparabilityofSignalandNoise 185
 6.6.3TheSquare-RootWienerFilter 186
 6.6.4WienerChannelEqualiser 187
 6.6.5Time-AlignmentofSignalsinMulti-channel/Multi-sensorSystems 187
 6.7ImplementationofWienerFilters 188
 6.7.1ChoiceofWienerFilterOrder 189
 6.7.2ImprovementstoWienerFilters 190
 6.8Summary 191
 Bibliography 191
 7 AdaptiveFilters: Kalman, RLS,LMS 193
 7.1Introduction 194
 7.2State-SpaceKalmanFilters 195
 7.2.1DerivationofKalmanFilterAlgorithm 197
 7.2.2RecursiveBayesianFormulationofKalmanFilter 200
 7.2.3MarkovianPropertyofKalmanFilter 201
 7.2.4ComparisonofKalmanfilterandhiddenMarkovmodel 202
 7.2.5ComparisonofKalmanandWienerFilters 202
 7.3ExtendedKalmanFilter(EFK) 206
 7.4UnscentedKalmanFilter(UFK) 208
 7.5SampleAdaptiveFilters–LMS,RLS 211
 xii Contents
 7.6RecursiveLeastSquare(RLS)AdaptiveFilters 213
 7.6.1MatrixInversionLemma 214
 7.6.2RecursiveTime-updateofFilterCoefficients 215
 7.7TheSteepest-DescentMethod 217
 7.7.1ConvergenceRate 219
 7.7.2Vector-ValuedAdaptationStepSize 220
 7.8LeastMeanSquaredError(LMS)Filter 220
 7.8.1LeakyLMSAlgorithm 220
 7.8.2NormalisedLMSAlgorithm 221
 7.8.2.1DerivationoftheNormalisedLMSAlgorithm 221
 7.8.2.2Steady-StateErrorinLMS 222
 7.9Summary 223
 Bibliography 224
 8 LinearPredictionModels 227
 8.1LinearPredictionCoding 227
 8.1.1Predictability,InformationandBandwidth 228
 8.1.2ApplicationsofLPModelinSpeechProcessing 229
 8.1.3Time-DomainDescriptionofLPModels 229
 8.1.4FrequencyResponseofLPModelanditsPoles 230
 8.1.5CalculationofLinearPredictorCoefficients 232
 8.1.6EffectofEstimationofCorrelationFunctiononLPModelSolution 233
 8.1.7TheInverseFilter:SpectralWhitening,De-correlation 234
 8.1.8ThePredictionErrorSignal 235
 8.2Forward,BackwardandLatticePredictors 236
 8.2.1AugmentedEquationsforForwardandBackwardPredictors 238
 8.2.2Levinson–DurbinRecursiveSolution 238
 8.2.2.1Levinson–DurbinAlgorithm 240
 8.2.3LatticePredictors 240
 8.2.4AlternativeFormulationsofLeastSquareErrorPrediction 241
 8.2.4.1Burg’sMethod 241
 8.2.5SimultaneousMinimisationoftheBackwardandForwardPredictionErrors 242
 8.2.6PredictorModelOrderSelection 242
 8.3Short-TermandLong-TermPredictors 243
 8.4MAPEstimationofPredictorCoefficients 245
 8.4.1ProbabilityDensityFunctionofPredictorOutput 245
 8.4.2UsingthePriorpdfofthePredictorCoefficients 246
 8.5Formant-TrackingLPModels 247
 8.6Sub-BandLinearPredictionModel 248
 8.7SignalRestorationUsingLinearPredictionModels 249
 8.7.1Frequency-DomainSignalRestorationUsingPredictionModels 251
 8.7.2ImplementationofSub-BandLinearPredictionWienerFilters 253
 8.8Summary 254
 Bibliography 254
 9 EigenvalueAnalysisandPrincipalComponentAnalysis 257
 9.1Introduction–LinearSystemsandEigenAnalysis 257
 9.1.1AGeometricInterpretationofEigenvaluesandEigenvectors 258
 9.2EigenVectorsandEigenvalues 261
 9.2.1MatrixSpectralTheorem 263
 9.2.2ComputationofEigenvaluesandEigenVectors 263
 Contents xiii
 9.3PrincipalComponentAnalysis(PCA) 264
 9.3.1ComputationofPCA 265
 9.3.2PCAAnalysisofImages:Eigen-ImageRepresentation 265
 9.3.3PCAAnalysisofSpeechinWhiteNoise 266
 9.4Summary 269
 Bibliography 270
 10PowerSpectrumAnalysis 271
 10.1PowerSpectrumandCorrelation 271
 10.2FourierSeries:RepresentationofPeriodicSignals 272
 10.2.1ThePropertiesofFourier’sSinusoidalBasisFunctions 272
 10.2.2TheBasisFunctionsofFourierSeries 273
 10.2.3FourierSeriesCoefficients 274
 10.3FourierTransform:RepresentationofNon-periodicSignals 274
 10.3.1DiscreteFourierTransform 276
 10.3.2Frequency-TimeResolutions:TheUncertaintyPrinciple 277
 10.3.3Energy-SpectralDensityandPower-SpectralDensity 278
 10.4Non-ParametricPowerSpectrumEstimation 279
 10.4.1TheMeanandVarianceofPeriodograms 279
 10.4.2AveragingPeriodograms(BartlettMethod) 280
 10.4.3WelchMethod:AveragingPeriodogramsfromOverlappedandWindowed
 Segments 280
 10.4.4Blackman–TukeyMethod 282
 10.4.5PowerSpectrumEstimationfromAutocorrelationofOverlapped
 Segments 282
 10.5Model-BasedPowerSpectrumEstimation 283
 10.5.1Maximum–EntropySpectralEstimation 283
 10.5.2AutoregressivePowerSpectrumEstimation 285
 10.5.3Moving-AveragePowerSpectrumEstimation 286
 10.5.4AutoregressiveMoving-AveragePowerSpectrumEstimation 286
 10.6High-ResolutionSpectralEstimationBasedonSubspaceEigen-Analysis 287
 10.6.1PisarenkoHarmonicDecomposition 287
 10.6.2MultipleSignalClassification(MUSIC)SpectralEstimation 289
 10.6.3EstimationofSignalParametersviaRotationalInvariance
 Techniques(ESPRIT) 291
 10.7Summary 293
 Bibliography 293
 11Interpolation–ReplacementofLostSamples 295
 11.1Introduction 295
 11.1.1IdealInterpolationofaSampledSignal 296
 11.1.2DigitalInterpolationbyaFactorofI 297
 11.1.3InterpolationofaSequenceofLostSamples 299
 11.1.4TheFactorsThatAffectInterpolationAccuracy 300
 11.2PolynomialInterpolation 301
 11.2.1LagrangePolynomialInterpolation 302
 11.2.2NewtonPolynomialInterpolation 303
 11.2.3HermitePolynomialInterpolation 304
 11.2.4CubicSplineInterpolation 305
 11.3Model-BasedInterpolation 306
 11.3.1MaximumAPosterioriInterpolation 307
 xiv Contents
 11.3.2LeastSquareErrorAutoregressiveInterpolation 308
 11.3.3InterpolationBasedonaShort-TermPredictionModel 309
 11.3.4InterpolationBasedonLong-TermandShort-termCorrelations 312
 11.3.5LSARInterpolationError 314
 11.3.6InterpolationinFrequency–TimeDomain 316
 11.3.7InterpolationUsingAdaptiveCodeBooks 317
 11.3.8InterpolationThroughSignalSubstitution 318
 11.3.9LP-HNMModelbasedInterpolation 318
 11.4Summary 319
 Bibliography 319
 12SignalEnhancementviaSpectralAmplitudeEstimation 321
 12.1Introduction 321
 12.1.1SpectralRepresentationofNoisySignals 322
 12.1.2VectorRepresentationofSpectrumofNoisySignals 323
 12.2SpectralSubtraction 324
 12.2.1PowerSpectrumSubtraction 325
 12.2.2MagnitudeSpectrumSubtraction 326
 12.2.3SpectralSubtractionFilter:RelationtoWienerFilters 326
 12.2.4ProcessingDistortions 327
 12.2.5EffectofSpectralSubtractiononSignalDistribution 328
 12.2.6ReducingtheNoiseVariance 329
 12.2.7FilteringOuttheProcessingDistortions 329
 12.2.8Non-LinearSpectralSubtraction 330
 12.2.9ImplementationofSpectralSubtraction 332
 12.3BayesianMMSESpectralAmplitudeEstimation 333
 12.4EstimationofSignaltoNoiseRatios 335
 12.5ApplicationtoSpeechRestorationandRecognition 336
 12.6Summary 338
 Bibliography 338
 13ImpulsiveNoise:Modelling,DetectionandRemoval 341
 13.1ImpulsiveNoise 341
 13.1.1DefinitionofaTheoreticalImpulseFunction 341
 13.1.2TheShapeofaRealImpulseinaCommunicationSystem 342
 13.1.3TheResponseofaCommunicationSystemtoanImpulse 343
 13.1.4TheChoiceofTimeorFrequencyDomainforProcessingofSignals
 DegradedbyImpulsiveNoise 343
 13.2AutocorrelationandPowerSpectrumofImpulsiveNoise 344
 13.3ProbabilityModelsofImpulsiveNoise 345
 13.3.1Bernoulli–GaussianModelofImpulsiveNoise 346
 13.3.2Poisson–GaussianModelofImpulsiveNoise 346
 13.3.3ABinary-StateModelofImpulsiveNoise 347
 13.3.4HiddenMarkovModelofImpulsiveandBurstNoise 348
 13.4ImpulsiveNoiseContamination,SignaltoImpulsiveNoiseRatio 349
 13.5MedianFiltersforRemovalofImpulsiveNoise 350
 13.6ImpulsiveNoiseRemovalUsingLinearPredictionModels 351
 13.6.1ImpulsiveNoiseDetection 352
 13.6.2AnalysisofImprovementinNoiseDetectability 353
 13.6.3Two-SidedPredictorforImpulsiveNoiseDetection 355
 13.6.4InterpolationofDiscardedSamples 355
 Contents xv
 13.7RobustParameterEstimation 355
 13.8RestorationofArchivedGramophoneRecords 357
 13.9Summary 358
 Bibliography 358
 14TransientNoisePulses 359
 14.1TransientNoiseWaveforms 359
 14.2TransientNoisePulseModels 361
 14.2.1NoisePulseTemplates 361
 14.2.2AutoregressiveModelofTransientNoisePulses 362
 14.2.3HiddenMarkovModelofaNoisePulseProcess 363
 14.3DetectionofNoisePulses 364
 14.3.1MatchedFilterforNoisePulseDetection 364
 14.3.2NoiseDetectionBasedonInverseFiltering 365
 14.3.3NoiseDetectionBasedonHMM 365
 14.4RemovalofNoisePulseDistortions 366
 14.4.1AdaptiveSubtractionofNoisePulses 366
 14.4.2AR-basedRestorationofSignalsDistortedbyNoisePulses 367
 14.5Summary 369
 Bibliography 369
 15EchoCancellation 371
 15.1Introduction:AcousticandHybridEcho 371
 15.2EchoReturnTime:TheSourcesofDelayinCommunicationNetworks 373
 15.2.1Transmissionlink(electromagneticwavepropagation)delay 374
 15.2.2Speechcoding/decodingdelay 374
 15.2.3Networkprocessingdelay 374
 15.2.4De-Jitterdelay 375
 15.2.5Acousticechodelay 375
 15.3TelephoneLineHybridEcho 375
 15.3.1EchoReturnLoss 376
 15.4Hybrid(TelephoneLine)EchoSuppression 377
 15.5AdaptiveEchoCancellation 377
 15.5.1EchoCancellerAdaptationMethods 379
 15.5.2ConvergenceofLineEchoCanceller 380
 15.5.3EchoCancellationforDigitalDataTransmission 380
 15.6AcousticEcho 381
 15.7Sub-BandAcousticEchoCancellation 384
 15.8EchoCancellationwithLinearPredictionPre-whitening 385
 15.9Multi-InputMulti-OutputEchoCancellation 386
 15.9.1StereophonicEchoCancellationSystems 386
 15.9.2Non-uniquenessProbleminMIMOEchoChannelIdentification 387
 15.9.3MIMOIn-CabinCommunicationSystems 388
 15.10Summary 389
 Bibliography 389
 16ChannelEqualisationandBlindDeconvolution 391
 16.1Introduction 391
 16.1.1TheIdealInverseChannelFilter 392
 16.1.2EqualisationError,ConvolutionalNoise 393
 16.1.3BlindEqualisation 394
 xvi Contents
 16.1.4Minimum-andMaximum-PhaseChannels 396
 16.1.5WienerEqualiser 396
 16.2BlindEqualisationUsingChannelInputPowerSpectrum 398
 16.2.1HomomorphicEqualisation 398
 16.2.2HomomorphicEqualisationUsingaBankofHigh-PassFilters 400
 16.3EqualisationBasedonLinearPredictionModels 400
 16.3.1BlindEqualisationThroughModelFactorisation 401
 16.4BayesianBlindDeconvolutionandEqualisation 402
 16.4.1ConditionalMeanChannelEstimation 403
 16.4.2Maximum-LikelihoodChannelEstimation 403
 16.4.3MaximumAPosterioriChannelEstimation 404
 16.4.4ChannelEqualisationBasedonHiddenMarkovModels 404
 16.4.5MAPChannelEstimateBasedonHMMs 406
 16.4.6ImplementationsofHMM-BasedDeconvolution 407
 16.5BlindEqualisationforDigitalCommunicationChannels 409
 16.5.1LMSBlindEqualisation 410
 16.5.2EqualisationofaBinaryDigitalChannel 413
 16.6EqualisationBasedonHigher-OrderStatistics 414
 16.6.1Higher-OrderMoments,CumulantsandSpectra 414
 16.6.1.1Cumulants 415
 16.6.1.2Higher-OrderSpectra 416
 16.6.2Higher-OrderSpectraofLinearTime-InvariantSystems 416
 16.6.3BlindEqualisationBasedonHigher-OrderCepstra 417
 16.6.3.1Bi-Cepstrum 418
 16.6.3.2Tri-Cepstrum 419
 16.6.3.3CalculationofEqualiserCoefficientsfromtheTri-cepstrum420
 16.7Summary 420
 Bibliography 421
 17SpeechEnhancement:NoiseReduction,BandwidthExtension
 andPacketReplacement 423
 17.1AnOverviewofSpeechEnhancementinNoise 424
 17.2Single-InputSpeechEnhancementMethods 425
 17.2.1ElementsofSingle-InputSpeechEnhancement 425
 17.2.1.1SegmentationandWindowingofSpeechSignals 426
 17.2.1.2SpectralRepresentationofSpeechandNoise 426
 17.2.1.3LinearPredictionModelRepresentationofSpeechandNoise426
 17.2.1.4Inter-FrameandIntra-FrameCorrelations 427
 17.2.1.5SpeechEstimationModule 427
 17.2.1.6ProbabilityModelsofSpeechandNoise 427
 17.2.1.7CostofErrorFunctionsinSpeechEstimation 428
 17.2.2WienerFilterforDe-noisingSpeech 428
 17.2.2.1WienerFilterBasedonLinearPredictionModels 429
 17.2.2.2HMM-BasedWienerFilters 429
 17.2.3SpectralSubtractionofNoise 430
 17.2.3.1SpectralSubtractionUsingLPModelFrequency
 Response 431
 17.2.4BayesianMMSESpeechEnhancement 432
 17.2.5KalmanFilterforSpeechEnhancement 432
 17.2.5.1KalmanState-SpaceEquationsofSignalandNoiseModels433
 Contents xvii
 17.2.6SpeechEnhancementUsingLP-HNMModel 435
 17.2.6.1OverviewofLP-HNMEnhancementSystem 436
 17.2.6.2FormantEstimationfromNoisySpeech 437
 17.2.6.3Initial-CleaningofNoisySpeech 437
 17.2.6.4FormantTracking 437
 17.2.6.5HarmonicPlusNoiseModel(HNM)ofSpeechExcitation 438
 17.2.6.6FundamentalFrequencyEstimation 439
 17.2.6.7EstimationofAmplitudesHarmonicsofHNM 439
 17.2.6.8EstimationofNoiseComponentofHNM 440
 17.2.6.9KalmanSmoothingofTrajectoriesofFormantsandHarmonics440
 17.3SpeechBandwidthExtension–SpectralExtrapolation 442
 17.3.1LP-HNMModelofSpeech 443
 17.3.2ExtrapolationofSpectralEnvelopeofLPModel 444
 17.3.2.1PhaseEstimation 445
 17.3.2.2CodebookMappingoftheGain 445
 17.3.3ExtrapolationofSpectrumofExcitationofLPModel 446
 17.3.3.1SensitivitytoPitch 446
 17.4InterpolationofLostSpeechSegments–PacketLossConcealment 447
 17.4.1PhasePrediction 450
 17.4.2CodebookMapping 452
 17.4.2.1EvaluationofLP-HNMInterpolation 453
 17.5Multi-InputSpeechEnhancementMethods 455
 17.5.1Beam-formingwithMicrophoneArrays 457
 17.5.1.1SpatialConfigurationofArrayandTheDirectionofReception458
 17.5.1.2DirectionalofArrival(DoA)andTimeofArrival(ToA) 459
 17.5.1.3SteeringtheArrayDirection:EqualisationoftheToAs
 attheSensors 459
 17.5.1.4TheFrequencyResponseofaDelay-SumBeamformer 460
 17.6SpeechDistortionMeasurements 462
 17.6.1Signal-to-NoiseRatio–SNR 462
 17.6.2SegmentalSignaltoNoiseRatio– SNR
 462
 seg
 17.6.3Itakura–SaitoDistance–ISD 463
 17.6.4HarmonicityDistance–HD 463
 17.6.5DiagnosticRhymeTest–DRT 463
 17.6.6MeanOpinionScore–MOS 464
 17.6.7PerceptualEvaluationofSpeechQuality–PESQ 464
 Bibliography 464
 18Multiple-InputMultiple-OutputSystems,IndependentComponentAnalysis 467
 18.1Introduction 467
 18.2Anoteoncomparisonofbeam-formingarraysandICA 469
 18.3MIMOSignalPropagationandMixingModels 469
 18.3.1InstantaneousMixingModels 469
 18.3.2Anechoic,DelayandAttenuation,MixingModels 470
 18.3.3ConvolutionalMixingModels 471
 18.4IndependentComponentAnalysis 472
 18.4.1ANoteonOrthogonal,OrthonormalandIndependent 473
 18.4.2StatementofICAProblem 474
 18.4.3BasicAssumptionsinIndependentComponentAnalysis 475
 18.4.4TheLimitationsofIndependentComponentAnalysis 475
 xviii Contents
 18.4.5WhyamixtureoftwoGaussiansignalscannotbeseparated? 476
 18.4.6TheDifferenceBetweenIndependentandUncorrelated 476
 18.4.7IndependenceMeasures;EntropyandMutualInformation 477
 18.4.7.1DifferentialEntropy 477
 18.4.7.2MaximumValueofDifferentialEntropy 477
 18.4.7.3MutualInformation 478
 18.4.7.4TheEffectofaLinearTransformationonMutualInformation479
 18.4.7.5Non-GaussianityasaMeasureofIndependence 480
 18.4.7.6Negentropy:AmeasureofNon-GaussianityandIndependence480
 18.4.7.7FourthOrderMoments–Kurtosis 481
 18.4.7.8Kurtosis-basedContrastFunctions–Approximationsto
 EntropicContrast 481
 18.4.8Super-GaussianandSub-GaussianDistributions 482
 18.4.9Fast-ICAMethods 482
 18.4.9.1Gradientsearchoptimisationmethod 483
 18.4.9.2Newtonoptimisationmethod 483
 18.4.10Fixed-pointFastICA 483
 18.4.11ContrastFunctionsandInfluenceFunctions 484
 18.4.12ICABasedonKurtosisMaximization–ProjectionPursuitGradientAscent 485
 18.4.13JadeAlgorithm–IterativeDiagonalisationofCumulantMatrices 487
 18.5Summary 490
 Bibliography 490
 19SignalProcessinginMobileCommunication 491
 19.1IntroductiontoCellularCommunication 491
 19.1.1ABriefHistoryofRadioCommunication 492
 19.1.2CellularMobilePhoneConcept 493
 19.1.3OutlineofaCellularCommunicationSystem 494
 19.2CommunicationSignalProcessinginMobileSystems 497
 19.3Capacity,Noise,andSpectralEfficiency 498
 19.3.1SpectralEfficiencyinMobileCommunicationSystems 500
 19.4Multi-pathandFadinginMobileCommunication 500
 19.4.1Multi-pathPropagationofElectromagneticSignals 501
 19.4.2RakeReceiversforMulti-pathSignals 502
 19.4.3SignalFadinginMobileCommunicationSystems 502
 19.4.4Large-ScaleSignalFading 504
 19.4.5Small-ScaleFastSignalFading 504
 19.5SmartAntennas–Space–TimeSignalProcessing 505
 19.5.1SwitchedandAdaptiveSmartAntennas 506
 19.5.2Space–TimeSignalProcessing–DiversitySchemes 506
 19.6Summary 508
 Bibliography 508
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