现代数字信号处理Advanced Digital Signal Processing-ch5 wavel

AdvancedDigitalSignalProcessing

(ModernDigitalSignalProcessing)

Chapter5Time-FrequencyAnalysisandWaveletTransform

编辑pptGeneralexpression5.1LinearTransformInnerproduct

编辑pptWave&WaveletTransformWavesWavesarenon-compact(infinite)supportfunctionsNon-compactsupportfunction:Thefunctionsextendtoinfinityinbothdirections.编辑pptWaveletsWaveletsarecompact(finite)supportfunctions.TheyvarywithfrequencyaswellaspositionCompactsupportfunction:Thefunctionsareinalimitduration.编辑pptWave&wavelettransformwaveswaveletsWavetransform(wide-sense)wavelettransform编辑pptOrthogonaltransformOrthogonalbasisfunctionOrthogonal(orthonormal)transformIfc=1,theng(ω,t)isorthonormalbasisfunction.编辑pptUnderstandingofthe(orthogonal)transformIntuitiveinterpretationoforthogonaltransformForagivenωi,Iff(t)isorthogonalwithg(ωi,t),thenF(ωi)=0,i.e.thereisnocomponentcorrespondingtothebasisg(ωi,t)inf(t).Otherwise,thecomponentsoff(t)correspondingtothebasisg(ωi,t)willcomposeF(ωi)inωspace.Ontheotherhand,thecomponentsoff(t)correspondingtothebasisg(ωi,t)isorthogonalwithanybasisg(ωj,t),i≠jandwillcontributenothingtoF(ωj).Decompositioneffect编辑pptGeometricinterpretationoforthogonaltransformNon-orthogonaltransform

Theorthogonaltransformoff(t)isaprojectionoff(t)intoaorthogonalbasisspaceformedby{g(ωi,t)},i=1,2,…OrthogonalProjectionNon-OrthogonalProjectionThesamecomponentoff(t)mayprojectintodifferentbases.Redundancywillprobablyexistinthetransformresults.编辑pptFouriertransform(FT)i.e.theFTisanorthonormalwavetransform.编辑pptNon-Stationary(Time-Variant)SignalStationary(time-invariant)signal编辑pptNon-stationary(time-variant)signal

x1(t)

x2(t)x3(t)x4(t)编辑pptFTofnon-stationary(time-variant)signalSignalsaredifferent,butspectrumsaresimilar编辑pptDeficiencyofwavetransform(e.g.FT)Wavetransformsarenotsuitablefortime-variantsignalsincetheydon’tincludeposition(time)informationinthetransformresults(e.g.FTanalyzestheglobalfrequencydistributionofasignal,butitcannotcharacterizethelocalbehaviorofthesignal).编辑pptBasicIdea5.2Time-FrequencyAnalysisInFT,thelocalbehaviorofasignalisnotrepresentedinthesignal’sfrequencyspectrumTheFTisnotthemostproperrepresentationforthetime-variantsignalsorthesignalscontainingtransientorlocalizationcomponentsTime-frequencyanalysis:characterizingthetimeandfrequencyinformationofasignalsimultaneouslyinitsspectrum编辑pptExamplesofTime-FrequencyAnalysis编辑ppt编辑pptMainToolsofTime-FrequencyAnalysisShorttimeFouriertransform(STFT)WavelettransformWignerdistribution(WD)Quadrictransform(non-lineartransform)Time-frequencydistributionWiger-Villedistribution(non-stationaryrandomsignal)编辑pptDefinitionSTFTofcontinuoustimesignalx(t)STFTofdiscretetimesignalx(n)5.3ShortTimeFourierTransformwherew(t)isarealfinite-widthwindowfunctionwhichslidesalongx(t)

wherew(n)isarealfinite-lengthwindowsequencewhichslidesalongx(n)

编辑pptTheresultofSTFTisa2-Dfunctionwhichreflectsthesignalspectrumvariedwithtime.FTofWindowedx(t)编辑pptisacompactsupportfunction(wavelet),andTheSTFTisWide-sensewavelettransformThesupportwidthofthewavelet(i.e.thewidthofthewindow)isconstantforallfrequencycomponents.编辑pptTheConflictingRequirementsbetweentheFrequencyResolution&theTimeResolutioninSTFTFrequencyresolutionrequirementThewindowwidthTshouldbewideenoughtogivethedesiredfrequencyresolution.TimeresolutionrequirementThewindowwidthTshouldbenarrowenoughsoasnottoblurthetimedependentevents,i.e.thesignalsegmentincludedinthewindowcanbetreatedasstationaryapproximately.编辑ppt编辑pptPartitionoftime-frequencyplaneinSTFTt编辑pptProblemsofSTFTHeisenberguncertaintyprincipleSTFTisredundantrepresentation

NotgoodforcompressionThesameandtthroughttheentireplane!

Wecannotperfectlylocalizeeventsintimeandfrequencysimultaneously!编辑pptMulti-ResolutionAnalysis(MRA)BasicideaThehighfrequencycomponentsvaryrapidlyintime.Arelativelyshortsignalsegmentcancharacterizethemproperly,hencearelativelynarrowtimewindowcanbeused(hightimeresolutionandlowfrequencyresolution).Oncontrary,thelowfrequencycomponentsvaryslowlyintimeandarelativelywidetimewindowshouldbeused(highfrequencyresolutionandlowtimeresolution).Higherfrequency

Morenarrowtimewindow

Lowerfrequency

Widertimewindow

Highertimeresolution

Higherfrequencyresolution

编辑ppt编辑pptPartitionoftime-frequencyplaneMRA

Differenttimeandfrequencyresolutionsareadoptedtothedifferentfrequency(scale)componentsofsignalatsametime编辑ppt5.4ContinuousWaveletTransform(CWT)DefinitionCWT

whereismother(basis)waveletwhichsatisfies编辑pptScaling&translationofmotherwavelet

whereaisscaling(dilation)parameterandbistranslation(shifting)parameter.isthebasisfunctionofCWT.Itiscalledtheanalysiswavelet.编辑pptScaling(dilation)编辑pptTranslation(shifting)编辑pptScalingandtranslation编辑pptRepresentingCWTinFrequencyDomain编辑pptIfthecentralfrequencyoftheFTofisω0,anditsbandwidthisB,

thenthecentralfrequencyandbandwidthoftheFTofareω0/aandB/arespectively,i.e.PropertiesofwaveletFrequencyspectrumanalysisabilityIfthewaveletisaband-passfilterwithrelativelynarrowpassband,thenthewaveletwithdifferentacancharacterizethedifferentfrequencycomponentsofasignal.Constantqualityfactor编辑pptInverseCWT

(ICWT)AdmissibleconditionwhereistheFTofThesatisfiestheadmissibleconditionisaadmissiblewavelet.Abasicrestrictionforconstructingamotherwavelet

编辑pptICWT编辑pptExamplesofMotherWavelets编辑pptPropertiesofCWTLinearityTimeshifting

编辑pptScalingMoyaltheorem(innerproducttheorem)

EnergyofWT

编辑pptReproducingkernelequationICWTReproducingkernel:thedependencebetween编辑pptRedundancyofCWTReproducingkernelequation编辑ppt5.5DiscreteWaveletTransform(DWT)DefinitionDiscretizingoftheScaling&TranslationFactor

mother(basis)waveletBasiswithlargerscaleLowersamplingrate编辑pptDWT

:DWTorwaveletseriesUsually,areadopted,then编辑pptWaveletFrameRequirementsforDiscreteWaveletBasisCompleteness

Cancharacterizethex(t)completely?ReversibilityCanx(t)berestoredfromstably?UniversalityWhetheranyx(t)canberepresentedbyalinearcombinationofthewaveletbasis编辑pptCompletenessUniquenesscontinuity编辑pptReversibilityUniquenesscontinuity编辑pptFrameLetbeaclusteroffunctionsinHilbertspaceH,ifforanyfunction,itisheldthatthenisaframe. Moreover,ifA=B,thenisatightframeand编辑pptIfA=B=1,then,henceisasetoforthogonalbasesinHspace.Suchasetofbasesisorthonormal

if编辑pptDualframe

wheresatisfiesRestoringanditiscalledthedualframeof编辑pptForconvenience,whenA≠Bbut,itisusuallyapproximatedaswhereIfA=B,then编辑pptWaveletframeIfforanyfunctionx(t),thewaveletbasisfunctionsatisfiesthenisawaveletframe.Itsdualwaveletframeiswhichsatisfies编辑pptIfA=B,thenorIfA≠B,thenand编辑ppti.e.isanadmissiblewavelet.Ifisawaveletframe,thenitmeetsthethreerequirementsfordiscretewaveletbasisproposedbefore,and编辑pptDesigningOrthonormalWaveletBasiswithMRAOrthogonalwaveletbasis:removingtheinformationredundancyinthedataaftertransformation

thenisorthogonal,andifC=A,thenisorthonormal.IfReproducingkernelequationinDWTwithtightframe:编辑pptMRAdisse