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WaveletToolboxComputationVisualizationProgrammingUser’sGuideVersion1MichelMisitiYvesMisitiGeorgesOppenheimJean-MichelPoggiForUsewithMATLAB®HowtoContactTheMathWorks:508-647-7000Phone508-647-7001FaxTheMathWorks,Inc.Mail24PrimeParkWayNatick,MA01760-1500@mathworks.comTechnicalsupportsuggest@mathworks.comProductenhancementsuggestionsbugs@mathworks.comBugreportsdoc@mathworks.comDocumentationerrorreportssubscribe@mathworks.comSubscribinguserregistrationservice@mathworks.comOrderstatus,licenserenewals,passcodesinfo@mathworks.comSales,pricing,andgeneralinformationWavletToolboxUser’sGuide©COPYRIGHT1996-1997byTheMathWorks,Inc.AllRightsReserved.Thesoftwaredescribedinthisdocumentisfurnishedunderalicenseagreement.Thesoftwaremaybeusedorcopiedonlyunderthetermsofthelicenseagreement.Nopartofthismanualmaybephotocopiedorrepro-ducedinanyformwithoutpriorwrittenconsentfromTheMathWorks,Inc.U.S.GOVERNMENT:IfLicenseeisacquiringthesoftwareonbehalfofanyunitoragencyoftheU.S.Government,thefollowingshallapply:(a)forunitsoftheDepartmentofDefense:RESTRICTEDRIGHTSLEGEND:Use,duplication,ordisclosurebytheGovernmentissubjecttorestric-tionsassetforthinsubparagraph(c)(1)(ii)oftheRightsinTechnicalDataandComputerSoftwareClauseatDFARS252.227-7013.(b)foranyotherunitoragency:NOTICE-Notwithstandinganyotherleaseorlicenseagreementthatmaypertainto,oraccompanythedeliveryof,thecomputersoftwareandaccompanyingdocumentation,therightsoftheGovernmentregardingitsuse,reproductionanddisclosureareassetforthinClause52.227-19(c)(2)oftheFAR.Contractor/manufacturerisTheMathWorksInc.,24PrimeParkWay,Natick,MA01760-1500.MATLAB,Simulink,HandleGraphics,andReal-TimeWorkshopareregisteredtrademarksandStateflowandTargetLanguageCompileraretrademarksofTheMathWorks,Inc.Otherproductorbrandnamesaretrademarksorregisteredtrademarksoftheirrespectiveholders.PrintingHistory:March1996Firstprinting%FAX)@ivContentsPrefaceAbouttheAuthors..........................................xvAcknowledgments.........................................xviWhatistheWaveletToolbox?............................xviiHowtoUseThisGuide...............................xviiiForMoreBackground.....................................xixInstallation.................................................xxSystemRecommendations..............................xxPlatform-SpecificDetails...............................xxWindowsFonts.....................................xxOtherPlatformsFonts..............................xxiMouseCompatibility................................xxiTypographicalConventions..............................xxii1Wavelets:ANewToolforSignalAnalysisFourierAnalysis..........................................1-3Short-TimeFourierAnalysis..............................1-4WaveletAnalysis..........................................1-5WhatCanWaveletAnalysisDo?........................1-5vContentsWhatisWaveletAnalysis?.................................1-7NumberofDimensions................................1-7TheContinuousWaveletTransform.......................1-8Scaling.............................................1-9Shifting............................................1-10FiveEasyStepstoaContinuousWaveletTransform.......1-10ScaleandFrequency.................................1-13TheScaleofNature..................................1-13What’sContinuousAbouttheContinuousWaveletTransform?..................................1-15TheDiscreteWaveletTransform.........................1-16One-StageFiltering:ApproximationsandDetails..........1-16Multiple-LevelDecomposition..........................1-19NumberofLevels..................................1-19WaveletReconstruction..................................1-20ReconstructionFilters................................1-21ReconstructingApproximationsandDetails..............1-21RelationshipofFilterstoWaveletShapes................1-23TheScalingFunction...............................1-25MultistepDecompositionandReconstruction.............1-25WaveletPacketAnalysis..................................1-27HistoryofWavelets.......................................1-29AnIntroductiontotheWaveletFamilies.................1-30Haar..............................................1-31Daubechies.........................................1-31Biorthogonal........................................1-32Coiflets............................................1-33Symlets............................................1-33Morlet.............................................1-34MexicanHat........................................1-34Meyer.............................................1-35vi2UsingWaveletsContinuousWaveletAnalysis(One-Dimensional)..........2-3ContinuousAnalysisUsingtheCommandLine............2-3ContinuousAnalysisUsingtheGraphicalInterface.........2-7ImportingandExportingInformationfromtheGraphicalInterface...........................2-11LoadingSignalsintotheContinuousWavelet1-DTool...2-11SavingWaveletCoefficients.........................2-12One-Dimensional