A practical guide to time-frequency analysis in th

APracticalGuidetoTime–FrequencyAnalysisintheStudyofHumanMotorBehavior:TheContributionofWaveletTransformJohannIssartelLudovicMarinUniversityofMontpellier1,Montpellier,FrancePhilippeGaillotUniversityof“Pauetdespaysdel’Adour,”Pau,FranceandCenterforDeepEarthExplorationYokohamaKanagawa,JapanABSTRACT.Theauthorspresentapracticalguideforstudyingnonstationarydataonhumanmotorbehaviorinatime–frequencyrepresentation.Theyexplainthelimitsofclassicalmethodsfound-edexclusivelyonthetimeorfrequencybasisandthenanswerthoselimitswiththewindowedFouriertransformandthewavelettransform(WT)methods,bothofwhicharefoundedontime–frequencybases.TheauthorsstressaninterestintheWTmethodbecauseitpermitsaccesstothewholecomplexityofasignal(intermsoftime,frequency,amplitude,andphase).TheythenshowthattheWTmethodiswellsuitedfortheanalysisoftheinterac-tionbetweentwosignals,particularlyinhumanmovementstudies.Finally,todemonstrateitspracticalapplications,theauthorsapplythemethodtorealdata.Keywords:cross-wavelettransform,motorbehavior,time–frequencybasis,wavelettransformnstudiesofhumanmotorbehavior,experimentersareincreasinglystrivingtowardecologicalsituations.Theanalysisofsuchsituationsissometimesquitedifficultbecauseonemusttakeintoconsiderationtheevolutionofbehaviorovertime.Tothatend,investigatorsneedaspecif-icmethodthatenablesthemtoanalyzesuchchangesovertime.Forinstance,considerthewristmovementsofaten-nisplayeroveranentiresetormatch.Thewristmaymoveindifferentdirections(right,left,forward,orbackward),quicklyorslowly,andwithhighorlowfrequencyandhighorlowamplitude.Thewrist’strajectorybecomesmorecomplexandobviouslynonstationaryasthefrequencycon-tentofthesignalevolveswithtime.Itisquitedifficulttoanalyzethemotorcharacteristicsofsuchanindividualcom-plexsignal,anditisevenmoredifficulttostudytheinter-actionsbetweentwosignals—inthatparticularcase,themovementsoftwotennisplayers’wrists.Inclassicalmeth-odssuchasauto-orintercorrelationandtheFouriertrans-form,oneassumesthestationarityoftheanalyzedsignal.Aboveall,themethodsarefoundedoneitherthetimebasisorthefrequencybasis,butneveronbothsimultaneously.Thosemethodscannotprovideinsightintothetemporalevolutionofthefrequency.Ouraiminthistutorialistopre-sentsomemethodsthatpermitonetoobservefrequencyevolutionwithtimeinanonstationarysignal.Thetwomainmethodsthatallowonetoperformatime–frequencyanaly-sisarethewindowedFouriertransform(WFT)andthewavelettransform(WT).Theyallowonetodepictanon-stationarysignalintermsoftime,frequency,amplitude,and,eventually,phase.Thosemethodsareusedinlotsofexperimentaldomainsbutnotofteninexperimentalpsy-chology.Wewouldliketohelpremedythatdeficiencybyproposingapracticalguidethatletsoneunderstandwhythosemethodsareinterestingandhowtheycanbeappliedinthedomainsinwhichweareinterested.Wehaveorganizedthisarticleinthreedifferentsec-tions.First,weintroducethenotionoftemporalandfre-quencybasesandtheconceptoftime–frequencyanalysisbyproposingtwoapproachestotime–frequencyanalysis,namely,WFTandWT.Inthesecondsectionofthisarti-cle,wepresenttheinteractionbetweentwopeoplebycharacterizingthenatureofcoordinationinahumanbehavioralmotortask.Weillustratetheabilityofthewin-dowedcross-correlationfunction(WCCF),andparticular-lythecross-wavelettransform(CWT),toenableustounderstandtheinteractionsbetweentwononstationarysignals.Inthefirsttwosections,weusesyntheticexam-plestohelpdefinetheadvantagesandlimitationsofeachmethod.Finally,inthethirdsection,weapplytheWTandCorrespondenceaddress:LudovicMarin,UPRESEA2991—MotorEfficiencyandMotorDeficiencyLaboratory,UniversityofMontpellier1,700avenueduPicSaintLoup,34090Montpellier,France.E-mailaddress:[e-mail]139IJournalofMotorBehavior,2006,Vol.38,No.2,139–159Copyright©2006HeldrefPublicationsThomasBardainneUniversityof“Pauetdespaysdel’Adour,”Pau,FranceMarielleCadopiUniversityofMontpellier1,Montpellier,FranceCWTmethodstorealdatatoillustratetheirrelevanceinhumanmovementstudies.Time–FrequencyAnalysesNotionofTemporalandFrequencyAnalyses—LocalandGlobalConceptsTounderstandthetime–frequencyanalysisofasignal,wefirstpresentthelimitsoftemporalandfrequencyanaly-sesofasignal.Toclarifyourdemonstration,wepresentallanalyses(time,frequency,andtime–frequency)withasyn-theticsignal.Thatillustrativesignalservesasatooltohelpusexplainandcomparedifferentmethodsofanalysis.Thesynthetictimeserieshasadurationof204.8s(4,096datapoints,20-Hzsamplingrate).Oneobtainssyntheticsignals1(Figure1D)bysummingthreesinesfromthefollowinggeneralequation:(1)withahigh(0.44Hz,Figure1A),anintermediate(0.16Hz,Figure1B),andalow(0.08Hz,Figure1C)frequency,andanamplitudeA(=150arbitraryunits).Wehaveaddedlocalnonstationarycomponentssuchasamplitudemodulation(Aam)asobtainedfromthefollowingequation,(2)frequencymodulation(Afm),asobtainedfromthefollowingequation,(3)andphaseshifts,commonlyencounteredinmovementstud-ies,tothosethreemainfrequencycomponents.Thehigh-frequencycomponentofs1isamplitude-modulatedinTimeInterval3(102.4–153.6s).Thatamplitudemodulationischaracterizedbyaperiod(Tam)of78.8s(correspondingfre-quency~0.013Hz)andan