Introduction to Wavelet a Tutorial - Qiao

Workshop118onWaveletApplicationinTransportationEngineering,Sunday,January09,2005FengxiangQiao,Ph.D.TexasSouthernUniversitySSA1D1A2D2A3D3IntroductiontoWaveletATutorialTABLEOFCONTENTOverviewHistoricalDevelopmentTimevsFrequencyDomainAnalysisFourierAnalysisFouriervsWaveletTransformsWaveletAnalysisToolsandSoftwareTypicalApplicationsSummaryReferencesOVERVIEWWaveletAsmallwaveWaveletTransformsConvertasignalintoaseriesofwaveletsProvideawayforanalyzingwaveforms,boundedinbothfrequencyanddurationAllowsignalstobestoredmoreefficientlythanbyFouriertransformBeabletobetterapproximatereal-worldsignalsWell-suitedforapproximatingdatawithsharpdiscontinuities“TheForest&theTrees”Noticegrossfeatureswithalargewindow“Noticesmallfeatureswithasmallwindow”DEVELOPMENTINHISTORYPre-1930JosephFourier(1807)withhistheoriesoffrequencyanalysisThe1930sUsingscale-varyingbasisfunctions;computingtheenergyofafunction1960-1980GuidoWeissandRonaldR.Coifman;GrossmanandMorletPost-1980StephaneMallat;Y.Meyer;IngridDaubechies;waveletapplicationstodayPRE-1930FourierSynthesisMainbranchleadingtowaveletsByJosephFourier(borninFrance,1768-1830)withfrequencyanalysistheories(1807)FromtheNotionofFrequencyAnalysistoScaleAnalysisAnalyzingf(x)bycreatingmathematicalstructuresthatvaryinscaleConstructafunction,shiftitbysomeamount,changeitsscale,applythatstructureinapproximatingasignalRepeattheprocedure.Takethatbasicstructure,shiftit,andscaleitagain.ApplyittothesamesignaltogetanewapproximationHaarWaveletThefirstmentionofwaveletsappearedinanappendixtothethesisofA.Haar(1909)Withcompactsupport,vanishesoutsideofafiniteintervalNotcontinuouslydifferentiable10sincoskkkkxbkxaaxfdxxfa20021dxkxxfakcos120dxkxxfbksin120:functionperiodical2anyForxfTHE1930sFindingbythe1930sPhysicistPaulLevyHaarbasisfunctionissuperiortotheFourierbasisfunctionsforstudyingsmallcomplicateddetailsintheBrownianmotionEnergyofaFunctionbyLittlewood,Paley,andSteinDifferentresultswereproducediftheenergywasconcentratedaroundafewpointsordistributedoveralargerintervaldxxfEnergy220211960-1980CreatedaSimplestElementsofaFunctionSpace,CalledAtomsBythemathematiciansGuidoWeissandRonaldR.CoifmanWiththegoaloffindingtheatomsforacommonfunctionUsingWaveletsforNumericalImageProcessingDavidMarrdevelopedaneffectivealgorithmusingafunctionvaryinginscaleintheearly1980sDefinedWaveletsintheContextofQuantumPhysicsByGrossmanandMorletin1980POST-1980AnAdditionalJump-startByMallatIn1985,StephaneMallatdiscoveredsomerelationshipsbetweenquadraturemirrorfilters,pyramidalgorithms,andorthonormalwaveletbasesY.Meyer’sFirstNon-trivialWaveletsBecontinuouslydifferentiableDonothavecompactsupportIngridDaubechies’OrthonormalBasisFunctionsBasedonMallat'sworkPerhapsthemostelegant,andthecornerstoneofwaveletapplicationstodayMATHEMATICALTRANSFORMATIONWhyToobtainafurtherinformationfromthesignalthatisnotreadilyavailableintherawsignal.RawSignalNormallythetime-domainsignalProcessedSignalAsignalthathasbeentransformedbyanyoftheavailablemathematicaltransformationsFourierTransformationThemostpopulartransformationTIME-DOMAINSIGNALTheIndependentVariableisTimeTheDependentVariableistheAmplitudeMostoftheInformationisHiddenintheFrequencyContent00.51-1-0.500.5100.51-1-0.500.5100.51-1-0.500.5100.51-4-202410Hz2Hz20Hz2Hz+10Hz+20HzTimeTimeTimeTimeMagnitudeMagnitudeMagnitudeMagnitudeFREQUENCYTRANSFORMSWhyFrequencyInformationisNeededBeabletoseeanyinformationthatisnotobviousintime-domainTypesofFrequencyTransformationFourierTransform,HilbertTransform,Short-timeFourierTransform,WignerDistributions,theRadonTransform,theWaveletTransform…FREQUENCYANALYSISFrequencySpectrumBebasicallythefrequencycomponents(spectralcomponents)ofthatsignalShowwhatfrequenciesexistsinthesignalFourierTransform(FT)OnewaytofindthefrequencycontentTellshowmuchofeachfrequencyexistsinasignalknNNnWnxkX1011knNNkWkXNnx10111NjNew2dtetxfXftj2dfefXtxftj2STATIONARITYOFSIGNAL(1)StationarySignalSignalswithfrequencycontentunchangedintimeAllfrequencycomponentsexistatalltimesNon-stationarySignalFrequencychangesintimeOneexample:the“ChirpSignal”STATIONARITYOFSIGNAL(2)00.20.40.60.81-3-2-1012305101520250100200300400500600TimeMagnitudeMagnitudeFrequency(Hz)2Hz+10Hz+20HzStationary00.51-1-0.8-0.6-0.4-0.200.20.40.60.810510152025050100150200250TimeMagnitudeMagnitudeFrequency(Hz)Non-Stationary0.0-0.4:2Hz+0.4-0.7:10Hz+0.7-1.0:20HzOccuratalltimesDonotappearatalltimesCHIRPSIGNALSSameinFrequencyDomain00.51-1-0.8-0.6-0.4-0.200.20.40.60.810510152025050100150TimeMagnitudeMagnitudeFrequency(Hz)00.51-1-0.8-0.6-0.4-0.200.20.40.60.810510152025050100150TimeMagnitudeMagnitudeFrequency(Hz)DifferentinTimeDomainFrequency:2Hzto20HzFrequency:20Hzto2HzAtwhattimethefrequencycomponentsoccur?FTcannottell!NOTHINGMORE,NOTHINGLESSFTOnlyGiveswhatFrequencyComponentsExistintheSignalTheTimeandFrequencyInformationcannotbeSeenattheSameTimeTime-frequencyRepresentationoftheSignalisNeededMostofTransportationSignalsareNon-stationar