数字信号处理：原理、算法与应用

IINTRODUCTION

1.1Signals,Systems,andSignalProcessing2

1.1.1BasicElementsofaDigitalSignalProcessingSystem,4

1.1.2AdvantagesofDigitaloverAnalogSignalProcessing,5

1.2ClassificationofSignals6

1.2.1MultichannelandMultidimensionalSignals,7

1.2.2Continuous-TimeVersusDiscrete-TimeSignals,8

1.2.3Continuous-ValuedVersusDiscrete-ValuedSignals,10

1.2.4DeterministicVersusRandomSignals,11

1.3TheConceptofFrequencyinContinuous-Timeand

Discrete-TimeSignals14

1.3.1Continuous-TimeSinusoidalSignals,14

1.3.2Discrete-TimeSinusoidalSignals,16

1.3.3HarmonicallyRelatedComplexExponentials,19

1.4Analog-to-DigitalandDigital-to-AnalogConversion21

1.4.1SamplingofAnalogSignals,23

1.4.2TheSamplingTheorem,29

1.4.3QuantizationofContinuous-AmplitudeSignals,33

1.4.4QuantizationofSinusoidalSignals,36

1.4.5CodingofQuantizedSamples,38

1.4.6Digital-to-AnalogConversion,38

1.4.7AnalysisofDigitalSignalsandSystemsVersusDiscrete-Time

SignalsandSystems,39

1.5SummaryandReferences39

Problems40

2DISCRETE-TIMESIGNALSANDSYSTEMS

2.1Discrete-TimeSignals43

2.1.1SomeElementaryDiscrete-TimeSignals,45

2.1.2ClassificationofDiscrete-TimeSignals,47

2.1.3SimpleManipulationsofDiscrete-TimeSignals,52

2.2Discrete-TimeSystems56

2.2.1Input-OutputDescriptionofSystems,56

2.2.2BlockDiagramRepresentationofDiscrete-TimeSystems,59

2.2.3ClassificationofDiscrete-TimeSystems,62

2.2.4InterconnectionofDiscrete-TimeSystems,70

2.3AnalysisofDiscrete-TimeLinearTime-InvariantSystems72

2.3.1TechniquesfortheAnalysisofLinearSystems,72

2.3.2ResolutionofaDiscrete-TimeSignalintoImpulses,74

2.3.3ResponseofLTISystemstoArbitraryInputs:TheConvolution

Sum.75

2.3.4PropertiesofConvolutionandtheInterconnectionofLTl

Systems,82

2.3.5CausalLinearTime-InvariantSystems,86

2.3.6StabilityofLinearTime-InvariantSystems,87

2.3.7SystemswithFinite-DurationandInfinite-DurationImpulse

Response.90

2.4Discrete-TimeSystemsDescribedbyDifferenceEquations91

2.4.1RecursiveandNonrecursiveDiscrete-TimeSystems.92

2.4.2LinearTime-InvariantSystemsCharacterizedby

Constant-CoefficientDifferenceEquations,95

2.4.3SolutionofLinearConstant-CoefficientDifferenceEquations.100

2.4.4TheImpulseResponseofaLinearTime-InvariantRecursive

System.108

2.5ImplementationofDiscrete-TimeSystems111

2.5.1StructuresfortheRealizationofLinearTime-Invariant

Systems.111

2.5.2RecursiveandNonrecumveRealizationsofFIRSystems.116

2.6CorrelationofDiscrete-TimeSignals118

2.6.1CrosscorrelationandAutocorrelationSequences,120

2.6.2PropertiesoftheAutocorrelationandCrosscorrelation

Sequences.122

2.6.3CorrelationofPeriodicSequences,124

2.6.4ComputationofCorrelationSequences,130

2.6.5Input-OutputCorrelationSequences,131

2.7SummaryandReferences134

Problems135

3THEZ-TRANSFORMANDITSAPPLICATIONTOTHEANALYSIS

OFLTlSYSTEMS151

3.1Thez-Transform151

3.1.1TheDirectz-Transform.152

3.1.2TheInversez-Transform,160

3.2Propertiesofthez-Transform161

3.3Rationalz-Transforms172

3.3.1PolesandZeros.172

3.3.2PoleLocationandTime-DomainBehaviorforCausalSignals.178

3.3.3TheSystemFunctionofaLinearTime-InvariantSystem,181

3.4Inversionofthez-Transform184

3.4.1TheInversez-TransformbyContourIntegration,184

3.4.2TheInversez-TransformbyPowerSeriesExpansion.186

3.4.3TheInversez-TransformbyPartial-FractionExpansion,[88

3.4.4DecompositionofRationalz-Transforms.195

3.5TheOne-sidedz-Transform197

3.5.1DefinitionandProperties,197

3.5.2SolutionofDifferenceEquations,201

3.6AnalysisofLinearTime-InvariantSystemsinthez-Domain203

3.6.1ResponseofSystemswithRationalSystemFunctions.203

3.6.2ResponseofPole-ZeroSystemswithNonzeroInitial

Conditions.204

3.6.3TransientandSteady-StateResponses.206

3.6.4CausalityandStability.208

3.6.5Pole-ZeroCancellations,210

3.6.6Multiple-OrderPolesandStability,211

3.6.7TheSchur-CohnStabilityTest,213

3.6.8StabilityofSecond-OrderSystems,215

3.7SummaryandReferences219

Problems220

4FREQUENCYANALYSISOFSIGNALSANDSYSTEMS230

4.1FrequencyAnalysisofContinuous-TimeSignals230

4.1.1TheFourierSeriesforContinuous-TimePeriodicSignals.232

4.1.2PowerDensitySpectrumofPeriodicSignals,235

4.1.3TheFourierTransformforContinuous-TimeAperiodic

Signals,240

4.1.4EnergyDensitySpectrumofAperiodicSignals.243

4.2FrequencyAnalysisofDiscrete-TimeSignals247

4.2.1TheFourierSeriesforDiscrete-TimePeriodicSignals,247

4.2.2PowerDensitySpectrumofPeriodicSignals,250

4.2.3TheFourierTransformofDiscrete-TimeAperiodicSignals,253

4.2.4ConvergenceofttxeFourierTransform,256

4.2.5EnergyDensitySpectrumofAperiodicSignals,260

4.2.6RelationshipoftheFourierTransformtothez-Transform,264

4.2.7TheCepstrum,265

4.2.8TheFourierTransformofSignalswithPolesontheUnit

Circle,267

4.2.9TheSamplingTheoremRevisited,269

4.2.10Frequency-DomainClassificationofSignals:TheConceptof

Bandwidth,279

4.2.11TheFrequencyRangesofSomeNaturalSignals,282

4.2.12PhysicalandMathematicalDualities,282

4.3PropertiesoftheFourierTransformforDiscrete-Time

Signals286

4.3.1SymmetryPropertiesoftheFourierTransform,287

4.3.2FourierTransformTheoremsandProperties,294

4.4Frequency-DomainCharacteristicsofLinearTime-Invariant

Systems305

4.4.1ResponsetoComplexExponentialandSinnsoidalSignals:The

FrequencyResponseFunction,306

4.4.2Steady-StateandTransientResponsetoSinusoidalInput

Signals,314

4.4.3Steady-StateResponsetoPeriodicInputSignals,315

4.4.4ResponsetoAperiodicInputSignals,316

4.4.5RelationshipsBetweentheSystemFunctionandtheFrequency

ResponseFunction,319

4.4.6ComputationoftheFrequencyResponseFunction,321

4.4.7Input-OutputCorrelationFunctionsandSpectra,325

4.4.8CorrelationFunctionsandPowerSpectraforRandomInput

Signals,327

4.5LinearTime-InvariantSystemsasFrequency-Selective

Filters330

4.5.1IdealFilterCharacteristics,331

4.5.2Lowpass,Highpass,andBandpassFilters,333

4.5.3DigitalResonators,340

4.5.4NotchFilters,343

4.5.5CombFilters,345

4.5.6All-PassFilters,350

4.5.7DigitalSinusoidalOscillators,352

4.6InverseSystemsandDeconvolution355

4.6.1InvertibilityofLinearTime-InvariantSystems,356

4.6.2Minimum-Phase,Maximum-Phase,andMixed-PhaseSystems,359

4.6.3SystemIdentificationandDeconvolution,363

4.6.4HomomorphicDeconvolution,365

SummaryandReferences367

Problems368

5THEDISCRETEFOURIERTRANSFORM:ITSPROPERTIESAND

APPLICATIONS394

5.1FrequencyDomainSampling:TheDiscreteFourier

Transform394

5.1.1Frequency-DomainSamplingandReconstructionof

Discrete-TimeSignals,394

5.1.2TheDiscreteFourierTransform(DFT),399

5.1.3TheDFTasaLinearTransformation,403

5.1.4RelationshipoftheDFTtoOtherTransforms,407

5.2PropertiesoftheDFT409

5.2.1Periodicity,Linearity,andSymmetryProperties,410

5.2.2MultiplicationofTwoDFTsandCircularConvolution,415

5.2.3AdditionalDFTProperties,421

5.3LinearFilteringMethodsBasedontheDFT425

5.3.1UseoftheDFTinLinearFiltering,426

5.3.2FilteringofLongDataSequences,430

5.4FrequencyAnalysisofSignalsUsingtheDFT433

5.5SummaryandReferences440

Problems440

6EFFICIENTCOMPUTATIONOFTHEOFT:FASTFOURIER

TRANSFORMALGORITHMS448

6.1EfficientComputationoftheDFT:FFTAlgorithms448

6.1.1DirectComputationoftheDFT,449

6.1.2Divide-and-ConquerApproachtoComputationoftheDFT,450

6.1.3Radix-2FFTAlgorithms,456

6.1.4Radix-4FFTAlgorithms,465

6.1.5Split-RadixFFTAlgorithms,470

6.1.6ImplementationofFFTAlgorithms,473

6.2ApplicationsofFFTAlgorithms475

6.2.1EfficientComputationoftheDFTofTwoRealSequences,475

6.2.2EfficientComputationoftheDFTofa2N-PointReal

Sequence,476

6.2.3UseoftheFFTAlgorithminLinearFilteringandCorrelation,477

6.3ALinearFilteringApproachtoComputationoftheDFT479

6.3.1TheGoertzelAlgorithm,480

6.3.2TheChirp-zTransformAlgorithm,482

QuantizationEffectsintheComputationof'theDFT486

6.4.1QuantizationErrorsintheDirectComputationoftheDFT,487

6.4.2QuantizationErrorsinFFTAlgorithms,489

6.5SummaryandReferences493

Problems494

7IMPLEMENTATIONOFDISCRETE-TIMESYSTEMS500

7.1StructuresfortheRealizationofDiscrete-TimeSystems500

7.2StructuresforFIRSystems502

7.2.1Direct-FormStructure,503

7.2.2Cascade-FormStructures,504

7.2.3Frequency-SamplingStructures*,506

7.2.4LatticeStructure,511

7.3StructuresforIIRSystems519

7.3.1Direct-FormStructures,519

7.3.2SignalFlowGraphsandTransposedStructures,521

7.3.3Cascade-FormStructures,526

7.3.4Parallel-FormStructures,529

7.3.5LatticeandLattice-LadderStructuresforIIRSystems,531

7.4State-SpaceSystemAnalysisandStructures539

7.4.1State-SpaceDescriptionsofSystemsCharacterizedbyDifference

Equations,540

7.4.2SolutionoftheState-SpaceEquations,543

7.4.3RelationshipsBetweenInput-OutputandState-Space

Descriptions,545

7.4.4State-SpaceAnalysisinthez-Domain,550

7.4.5AdditionalState-SpaceStructures,554

7.5RepresentationofNumbers556

7.5.1Fixed-PointRepresentatknvofNumbers,557

7.5.2BinaryFloating-PointRepresentationofNumbers,561

7.5.3ErrorsResultingfromRoundingandTruncation,564

7.6QuantizationofFilterCoefficients569

7.6.1AnalysisofSensitivitytoQuantizationofFilterCoefficients,569

7.6.2QuantizationofCoefficientsinFIRFilters,578

7.7Round-OffEffectsinDigitalFilters582

7.7.1Limit-CycleOscillationsinRecursiveSystems,583

7.7.2ScalingtoPreventOverflow,588

7.7.3StatisticalCharacterizationofQuantizationEffectsinFixed-Point

RealizationsofDigitalFilters,590

7.8SummaryandReferences598

Problems600

8DESIGNOFDIGITALFILTERS614

8.1GeneralConsiderations614

8.1.1CausalityandItsImplications,615

8.1.2CharacteristicsofPracticalFrequency-SelectiveFilters,619

8.2DesignofFIRFilters620

8.2.1SymmetricandAntisymmetricFIRFilters,620

8.2.2DesignofLinear-PhaseFIRFiltersUsingWindows,623

8.2.3DesignofLinear-PhaseFIRFiltersbytheFrequency-Sampling

Method,630

8.2.4DesignofOptimumEquirippleLinear-PhaseFIRFilters,637

8.2.5DesignofFIRDifferentiators,652

8.2.6DesignofHilbertTransformers,657

8.2.7ComparisonofDesignMethodsforLinear-PhaseFIRFilters,662

8.3DesignofIIRFiltersFromAnalogFilters666

8.3.1IIRFilterDesignbyApproximationofDerivatives,667

8.3.2IIRFilterDesignbyImpulseInvariance.671

8.3.3IIRFilterDesignbytheBilinearTransformation,676

8.3.4TheMatched-zTransformation,681

8.3.5CharacteristicsofCommonlyUsedAnalogFilters,681

8.3.6SomeExamplesofDigitalFilterDesignsBasedontheBilinear

Transformation,692

8.4FrequencyTransformations692

8.4:1FrequencyTransformationsintheAnalogDomain,693

8.4.2FrequencyTransformationsintheDigitalDomain,698

8.5DesignofDigitalFiltersBasedonLeast-SquaresMethod701

8.5.1Pad~ApproximationMethod,701

8.5.2Least-SquaresDesignMethods,706

8.5.3FIRLeast-SquaresInverse(Wiener)Filters,711

8.5.4DesignofIIRFiltersintheFrequencyDomain,719

8.6SummaryandReferences724

Problems726

9SAMPLINGANDRECONSTRUCTIONOFSIGNALS738

9.1SamplingofBandpassSignals738

9.1.1RepresentationofBandpassSignals,738

9.1.2SamplingofBandpassSignals.742

9.1.3Discrete-TimeProcessingofContinuous-TimeSignals.746

9.2Analog-to-DigitalConverSion748

9.2.1Sample-and-Hold,748

9.2.2QuantizationandCoding,750

9.2.3AnalysisofQuantizationErrors.753

9.2.4OversamplingA/DConverters,756

DESIGNOFDIGITALFILTERS614

8.1GeneralConsiderations614

8.1.1CausalityandItsImplications,615

8.1.2CharacteristicsofPracticalFrequency-SelectiveFilters,619

8.2DesignofFIRFilters620

8.2.1SymmetricandAntisymmetricFIRFilters,620

8.2.2DesignofLinear-PhaseFIRFiltersUsingWindows,623

8.2.3DesignofLinear-PhaseFIRFiltersbytheFrequency-Sampling

Method,630

8.2.4DesignofOptimumEquirippleLinear-PhaseFIRFilters,637

8.2.5DesignofFIRDifferentiators,652

8.2.6DesignofHilbertTransformers,657

8.2.7ComparisonofDesignMethodsforLinear-PhaseFIRFilters,662

8.3DesignofIIRFiltersFromAnalogFilters666

8.3.1IIRFilterDesignbyApproximationofDerivatives,667

8.3.2IIRFilterDesignbyImpulseInvariance,671

8.3.3IIRFilterDesignbytheBilinearTransformation,676

8.3.4TheMatched-zTransformation,681

8.3.5CharacteristicsofCommonlyUsedAnalogFilters,681

8.3.6SomeExamplesofDigitalFilterDesignsBasedontheBilinear

Transformation,692

8.4FrequencyTransformations692

8.4.1FrequencyTransformationsintheAnalogDomain,693

8.4.2FrequencyTransformationsintheDigitalDomain,698

8.5DesignofDigitalFiltersBasedonLeast-SquaresMethod701

8.5.1Pad6ApproximationMethod,701

8.5.2Least-SquaresDesignMethods,706

8.5.3FIRLeast-SquaresInverse(Wiener)Filters.711

8.5.4DesignofIIRFiltersintheFrequencyDomain,719

8.6SummaryandReferences724

Problems726

9SAMPLINGANDRECONSTRUCTIONOFSIGNALS738

9.1SamplingofBandpassSignals738

9.1.1RepresentationofBandpassSignals,738

9.1.2SamplingofBandpassSignals,742

9.1.3Discrete-TimeProcessingofContinuous-TimeSignals.746

9.2Analog-to-DigitalConversion748

9.2.1Sample-and-Hold,748

9.2.2QuantizationandCoding,750

9.2.3AnalysisofQuantizationErrors,753

9.2.4OversamplingA/DConverters,756

9.3Digital-to-AnalogConversion763

9.3.1SampleandHold,765

9.3.2First-OrderHold,768

9.3.3LinearInterpolationwithDelay,771

9.3.4OversamplingD/AConverters,774

9.4SummaryandReferences774

Problems775

10MULTIRATEDIGITALSIGNALPROCESSING782

10.1Introduction783

10.2DecimationbyaFactorD784

10.3InterpolationbyaFactor!787

10.4SamplingRateConversionbyaRationalFactorI/D790

10.5FilterDesignandImplementationforSampling-Rate

Conversion792

10.5.1Direct-FormFIRFilterStructures,793

10.5.2PolyphaseFilterStructures.794

10.5.3Time-VariantFilterStructures.800

10.6MultistageImplementationofSampling-RateConversion806

10.7Sampling-RateConversionofBandpassSignals810

10.7.1DecimationandInterpolationbyFrequencyConversion.812

10.7.2Modulation-FreeMethodforDecimationandInterpolation,814

10.8Sampling-RateConversionbyanArbitraryFactor815

10.8.1First-OrderApproximation.816

10.8.2Second-OrderApproximation(LinearInterpolation),819

10.9ApplicationsofMultirateSignalProcessing821

10.9.1DesignofPhaseShifters.821

10.9.2InterfacingofDigitalSystemswithDifferentSamplingRates,823

10.9.3ImplementationofNarrowbandLowpassFilters,824

10.9.4ImplementationofDigitalFilterBanks,825

10.9.5SubbandCodingofSpeechSignals,831

10.9.6QuadratureMirrorFilters,833

10.9.7Transmultiplexers,841

10.9.80versamplingA/DandD/AConversion,843

10.10SummaryandReferences844

Problems846

11LINEARPREDICTIONANDOPTIMUMLINEARFILTERS852

11.1InnovationsRepresentationofaStationaryRandom

Process852

11.1.1RationalPowerSpectra,854

11.1.2RelationshipsBetweentheFilterParametersandthe

AutocorrelationSequence,855

11.2ForwardandBackwardLinearPrediction857

11.2.1ForwardLinearPrediction,857

11.2.2BackwardLinearPrediction,860

11.2.3TheOptimumReflectionCoefficientsfortheLatticeForwardand

BackwardPredictors,863

11.2.4RelationshipofanARProcesstoLinearPrediction,864

11.3SolutionoftheNormalEquations864

11.3.1TheLevinson-DurbinAlgorithm,865

11.3.2TheSchiirAlgorithm,868

11.4PropertiesoftheLinearPrediction-ErrorFilters873

11.5ARLatticeandARMALattice-LadderFilters876

11.5.1ARLatticeStructure,877

11.5.2ARMAProcessesandLattice-LadderFilters,878

11.6WienerFiltersforFilteringandPrediction880

11.6.1FIRWienerFilter,881

11.6.20rthogonalityPrincipleinLinearMean-SquareEstimation,884

11.6.3IIRWienerFilter,885

11.6.4NoncausalWienerFilter,889

11.7SummaryandReferences890

Problems892

12POWERSPECTRUMESTIMATION898

12.1EstimationofSpectrafromFinite-DurationObservationsof

Signals896

12.1.1ComputationoftheEnergyDensitySpectrum,897

12.1.2EstimationoftheAutocorrelationandPowerSpectrumof

RandomSignals:ThePeriodogram,902

12.1.3TheUseoftheDFTinPowerSpectrumEstimation,906

12.2NonparametricMethodsforPowerSpectrumEstimation908

12.2.1TheBartlettMethod:AveragingPeriodograms,910

12.2.2TheWelchMethod:AveragingModifiedPeriodograms,911

12.2.3TheBlackmanandTukeyMethod:Smoothingthe

Periodogram,913

12.2.4PerformanceCharacteristicsofNonparametricPowerSpectrum

Estimators,916

12.2.5ComputationalRequirementsofNonparametricPowerSpectrum

Estimates,919

12.3ParametricMethodsforPowerSpectrumEstimation920

12.3.1RelationshipsBetweentheAutocorrelationandtheModel

Parameters,923

12.3.2TheYule-WalkerMethodfortheARModelParameters,925

12.3.3TheBurgMethodfortheARModelParameters,925

12.3.4UnconstrainedLeast-SquaresMethodfortheARModel

Parameters,929

12.3.5SequentialEstimationMethodsfortheARModelParameters,930

12.3.6SelectionofARModelOrder,931

12.3.7MAModelforPowerSpectrumEstimation,933

12.3.8ARMAModelforPowerSpectrumEstimation,934

12.3.9SomeExperimentalResults,936

12.4MinimumVarianceSpeCtralEstimation942

12.5EigenanalysisAlgorithmsforSpectrumEstimation946

12.5.1PisarenkoHarmonicDecompositionMethod,948

12.5.2Eigen-decompositionoftheAutocorrelationMatrixforSinusoids

inWhiteNoise,950

12.5.3MUSICAlgorithm,952

12.5.4ESPRITAlgorithm,953

12.5.5OrderSelectionCriteria,955

12.5.6ExperimentalResults,956

12.6SummaryandReferences959

Problems960

ARANDOMSIGNALS,CORRELATIONFUNCTIONS,ANDPOWER

SPECTRAA1

BRANDOMNUMBERGENERATORSB1

CTABLESOFTRANSITIONCOEFFICIENTSFORTHEDESIGNOF

LINEAR-PHASEFIRFILTERSCl

DLISTOFMATLABFUNCTIONSD1

REFERENCESANDBIBLIOGRAPHYR1

INDEX11