现代数字信号处理AdvancedDigitalSignalProcessing-ch5wavel

AdvancedDigitalSignalProcessing(ModernDigitalSignalProcessing)Chapter5Time-FrequencyAnalysisandWaveletTransform,Generalexpression,5.1LinearTransform,Innerproduct,Wave&WaveletTransformWaves,Wavesarenon-compact(infinite)supportfunctions,Non-compactsupportfunction:Thefunctionsextendtoinfinityinbothdirections.,Wavelets,Waveletsarecompact(finite)supportfunctions.Theyvarywithfrequencyaswellasposition,Compactsupportfunction:Thefunctionsareinalimitduration.,Wave&wavelettransform,waves,wavelets,Wavetransform,(wide-sense)wavelettransform,OrthogonaltransformOrthogonalbasisfunctionOrthogonal(orthonormal)transform,Ifc=1,theng(,t)isorthonormalbasisfunction.,Understandingofthe(orthogonal)transformIntuitiveinterpretationoforthogonaltransformForagiveni,Iff(t)isorthogonalwithg(i,t),thenF(i)=0,i.e.thereisnocomponentcorrespondingtothebasisg(i,t)inf(t).Otherwise,thecomponentsoff(t)correspondingtothebasisg(i,t)willcomposeF(i)inspace.Ontheotherhand,thecomponentsoff(t)correspondingtothebasisg(i,t)isorthogonalwithanybasisg(j,t),ijandwillcontributenothingtoF(j).,Decompositioneffect,GeometricinterpretationoforthogonaltransformNon-orthogonaltransform,Theorthogonaltransformoff(t)isaprojectionoff(t)intoaorthogonalbasisspaceformedbyg(i,t),i=1,2,OrthogonalProjection,Non-OrthogonalProjection,Thesamecomponentoff(t)mayprojectintodifferentbases.Redundancywillprobablyexistinthetransformresults.,Fouriertransform(FT),i.e.theFTisanorthonormalwavetransform.,Non-Stationary(Time-Variant)SignalStationary(time-invariant)signal,Non-stationary(time-variant)signal,x1(t)x2(t)x3(t)x4(t),FTofnon-stationary(time-variant)signal,Signalsaredifferent,butspectrumsaresimilar,Deficiencyofwavetransform(e.g.FT),Wavetransformsarenotsuitablefortime-variantsignalsincetheydontincludeposition(time)informationinthetransformresults(e.g.FTanalyzestheglobalfrequencydistributionofasignal,butitcannotcharacterizethelocalbehaviorofthesignal).,BasicIdea,5.2Time-FrequencyAnalysis,InFT,thelocalbehaviorofasignalisnotrepresentedinthesignalsfrequencyspectrum,TheFTisnotthemostproperrepresentationforthetime-variantsignalsorthesignalscontainingtransientorlocalizationcomponents,Time-frequencyanalysis:characterizingthetimeandfrequencyinformationofasignalsimultaneouslyinitsspectrum,ExamplesofTime-FrequencyAnalysis,MainToolsofTime-FrequencyAnalysisShorttimeFouriertransform(STFT)WavelettransformWignerdistribution(WD)Quadrictransform(non-lineartransform)Time-frequencydistributionWiger-Villedistribution(non-stationaryrandomsignal),DefinitionSTFTofcontinuoustimesignalx(t)STFTofdiscretetimesignalx(n),5.3ShortTimeFourierTransform,wherew(t)isarealfinite-widthwindowfunctionwhichslidesalongx(t),wherew(n)isarealfinite-lengthwindowsequencewhichslidesalongx(n),TheresultofSTFTisa2-Dfunctionwhichreflectsthesignalspectrumvariedwithtime.,FTofWindowedx(t),isacompactsupportfunction(wavelet),andTheSTFTis,Wide-sensewavelettransform,Thesupportwidthofthewavelet(i.e.thewidthofthewindow)isconstantforallfrequencycomponents.,TheConflictingRequirementsbetweentheFrequencyResolution&theTimeResolutioninSTFTFrequencyresolutionrequirementThewindowwidthTshouldbewideenoughtogivethedesiredfrequencyresolution.TimeresolutionrequirementThewindowwidthTshouldbenarrowenoughsoasnottoblurthetimedependentevents,i.e.thesignalsegmentincludedinthewindowcanbetreatedasstationaryapproximately.,Partitionoftime-frequencyplaneinSTFT,t,ProblemsofSTFTHeisenberguncertaintyprincipleSTFTisredundantrepresentationNotgoodforcompressionThesameandtthroughttheentireplane!,Wecannotperfectlylocalizeeventsintimeandfrequencysimultaneously!,Multi-ResolutionAnalysis(MRA)BasicideaThehighfrequencycomponentsvaryrapidlyintime.Arelativelyshortsignalsegmentcancharacterizethemproperly,hencearelativelynarrowtimewindowcanbeused(hightimeresolutionandlowfrequencyresolution).Oncontrary,thelowfrequencycomponentsvaryslowlyintimeandarelativelywidetimewindowshouldbeused(highfrequencyresolutionandlowtimeresolution).,Higherfrequency,Morenarrowtimewindow,Lowerfrequency,Widertimewindow,Highertimeresolution,Higherfrequencyresolution,Partitionoftime-frequencyplane,MRADifferenttimeandfrequencyresolutionsareadoptedtothedifferentfrequency(scale)componentsofsignalatsametime,5.4ContinuousWaveletTransform(CWT),DefinitionCWT,whereismother(basis)waveletwhichsatisfies,Scaling&translationofmotherwavelet,whereaisscaling(dilation)parameterandbistranslation(shifting)parameter.isthebasisfunctionofCWT.Itiscalledtheanalysiswavelet.,Scaling(dilation),Translation(shifting),Scalingandtranslation,RepresentingCWTinFrequencyDomain,IfthecentralfrequencyoftheFTofis0,anditsbandwidthisB,thenthecentralfrequencyandbandwidthoftheFTofare0/aandB/arespectively,i.e.,PropertiesofwaveletFrequencyspectrumanalysisabilityIfthewaveletisaband-passfilterwithrelativelynarrowpassband,thenthewaveletwithdifferentacancharacterizethedifferentfrequencycomponentsofasignal.Constantqualityfactor,InverseCWT(ICWT)Admissiblecondition,whereistheFTof,Thesatisfiestheadmissibleconditionisaadmissiblewavelet.,Abasicrestrictionforconstructingamotherwavelet,ICWT,ExamplesofMotherWavelets,PropertiesofCWTLinearityTimeshifting,ScalingMoyaltheorem(innerproducttheorem)EnergyofWT,Reproducingkernelequation,ICWT,Reproducingkernel:thedependencebetween,RedundancyofCWT,Reproducingkernelequation,5.5DiscreteWaveletTransform(DWT),DefinitionDiscretizingoftheScaling&TranslationFactor,mother(basis)wavelet,Basiswithlargerscale,Lowersamplingrate,DWT,:DWTorwaveletseries,Usually,areadopted,then,WaveletFrameRequirementsforDiscreteWaveletBasisCompletenessCancharacterizethex(t)completely?ReversibilityCanx(t)berestoredfromstably?UniversalityWhetheranyx(t)canberepresentedbyalinearcombinationofthewaveletbasis,Completeness,Uniqueness,continuity,Reversibility,Uniqueness,continuity,FrameLetbeaclusteroffunctionsinHilbertspaceH,ifforanyfunction,itisheldthatthenisaframe.Moreover,ifA=B,thenisatightframeand,IfA=B=1,then,henceisasetoforthogonalbasesinHspace.Suchasetofbasesisorthonormalif,Dualframe,wheresatisfies,Restoring,anditiscalledthedualframeof,Forconvenience,whenABbut,itisusuallyapproximatedas,where,IfA=B,then,WaveletframeIfforanyfunctionx(t),thewaveletbasisfunctionsatisfiesthenisawaveletframe.Itsdualwaveletframeis,whichsatisfies,IfA=B,then,or,IfAB,then,and,i.e.isanadmissiblewavelet.,Ifisawaveletframe,thenitmeetsthethreerequirementsfordiscretewaveletbasisproposedbefore,and,DesigningOrthonormalWaveletBasiswithMRAOrthogonalwaveletbasis:removingtheinformationredundancyinthedataaftertransformation,thenisorthogonal,andifC=A,thenisorthonormal.,If,Rep