Wavelet Tutorial_Classic

WELCOMEWelcometoTheWaveletTutorial!ItwasendofOctober1994.Ihadrecentlystudiedfundamentalsofwavelettransformasmygraduationprojectatmyundergraduateinstitution,andIwasplanningtousethistechniqueinanalyzingsignalsofbiologicaloriginformyMaster'sdegreethesis.MymajorprofessorsuggestedthatIshouldworkonEEGsignals,sincetheyarelessstudiedcomparedtomanyotherbiologicalsignals.This,ofcourse,wouldrequireadatabaotherbiologicalsignals.This,ofcourse,wouldrequireadatabaseofEEGsignals.IwasindesperateneedoffindingadatabaseofEEGsignalswhenIdecidedtousethisprojectformyMaster'sdegreethesis,MultiresolutionWaveletAnalysisofEventRelatedPotentialsfortheDetectionofAlzheimer'sDisease.Thehospitalswererefusingtocooperateinsharingtheirfiles,statingthatallpatientfileswereconfidential.IhavethendecidedtosearchtheInternethopingtofindpeoplewhomayhaveadatabasethatmightbeofanyusetome.ItoldthemthatIwouldusethewavelettransformtoanalyzeEEGsignals,andaskedthemiftheyhadsuchdatatosharewithme.ThemajorityofthemailsIhavereceivedfromthemwereofthetype:Sorry!WedonothaveanyEEGdata,butwhatisthiswaveletbusinessanyway?Ifyoucanprovidesomeinformation,wemaybeabletodirectyou...SoIrepliedandtriedtoexplainwhatIwasafter.Itdidn'ttakemetoolongtorealizethatIwaswritinga4-6pagesofinformationonwavelettransformalloveragaineverytimesomeoneaskedformoreinformation.Furthermore,sincemostofthosepeoplewerefromthemedicalcommunityandhadlittleornobackgroundinsignalprocessing,Ihadtostartfromthedefinitionofatransform.Tryingtoexplainarelativelynewsignalprocessingtechniquebackedwithahighlycomplexmathematicaltheory,startingfromthedefinitionofthetransformwasnoeasytask.OnethingIsufferedwhileIwaslearningthebasicsofthewavelettransformisthefactthatthemajorityofthearticlesandbooks(ifnotallofthem)arewrittenbymathpeople,forthemathpeople,inalanguagewhichevenmostofthemathpeoplethemselvescannotunderstandwhatisgoingon.IrememberthatIgotfrustratedwithallthoseequations,tryingtofigureouthowandwheretousethem.IwassofrustratedatthattimethatIdecidedtowritemyownbookinsomeday.WhenIreceivedsomanymailsaboutthewavelettransform,Ithoughtthatwritingatutorialcouldbeastartingpointformyfuturedreamofwritingmyownbookofwavelettrasforms.IknewthatIhadtoputitinsimplewordstomakeitunderstandabletothosepeople.Thisishowthistutorialwasfirstcreated.Inthefirstversionofthetutorial,therewereabsolutelynoequations,anditsimplyconsistedofbasicconceptswhatwavelettransformisallabout.Ireceivedanunexpectednumberofrepliesfrommanypeoplearoundalltheworldwhowerepleasantlysurprisedinhowsimplewordswavelettransformcanbeexplained.Theyaskedmetogivemoreinformation,goingintoalittlemoredetail.IhavethendecidedtowriteacompletetutorialcoveringeverythingfromFouriertransformstoshorttimeFouriertransformandwavelettransforms.PartIofthistutorialpresentsanoverviewofthebasicconceptsthatareofimportanceinunderstandingthewavelettheory.Thispartisstrictlyforthosewhohavenobackgroundinsignalprocessing,somehowheardthatsomewaveletthingorotheristhewaytogo.Thispartsummarizestheconceptoftransforming,andtalksaboutwhenandwhyFouriertransform,byfarthemostoftenusedtransforminsignalprocessing,mightnotbeasuitabletechniquetouse.PartIIintroducestheShortTermFourierTransform(STFT),whichhasbeenusedtoobtaintime-frequencyrepresentationsofnon-stationarysignals.IthinkitisimportanttofullyunderstandSTFT,sincewavelettransformwasdevelopedasanalternativetotheSTFT,toovercomesomeproblemsthatareinherenttoit.Bytheendofthispart,thereadershouldbecomfortablewhyandwhenwavelettransformneedstobeused.PartIIIintroducesthecontinuouswavelettransform(CWT),explaininghowtheproblemsinherenttotheSTFTaresolved.Thispartgivesanintroductiontothemathematicalbackboneofthewavelettransform.AlsogiveninthispartareacoupleexamplesthatactuallyshowhowWTofasignallooklike,somethingIcouldnotfindinanyofthearticlesorbooksIhavereadonWT.PartIVtalksaboutthediscretewavelettransform,averyeffectiveandfasttechniquetocomputetheWTofasignal.Finally,abibliographyisincludedforthosewhoneedmorethanwhatisgiveninthistutorial.IwouldliketonotethatIamnotanexpertonwavelettransform,butjustauserofthismethod.Itistherefore,possiblethatImighthavemissedsomeimportantpoints,orevenmighthavegivenfalseinformation.Shouldyoufindanyincomplete,inconsistent,orincorrectinformationpleasefeelfreetoinformme.Iwillappreciateanycommentsonthistutorial.Thisisabsolutelynecessarytomakethistutorialcompleteandaccurate.Iwillbemostgratefultothosesendingtheiropinionsandcomments.Iwillbethroughlyhappy,ifIcanbeofanyservicetoanyonewhowouldliketolearnwavelettransformwiththistutorial.RobiPOLIKAR06/06/1995,329DurhamComputationCenter,IowaStateUniversityAmes,IOWA,50011THEWAVELETTUTORIALPARTIbyROBIPOLIKARFUNDAMENTALCONCEPTS&ANOVERVIEWOFTHEWAVELETTHEORYSecondEditionWelcometothisintroductorytutorialonwavelettransforms.Thewavelettransformisarelativelynewconcept(about10yearsold),butyettherearequiteafewarticlesandbookswrittenonthem.However,mostofthesebooksandarticlesarewrittenbymathpeople,fortheothermathpeople;stillmostofthemathpeopledon'tknowwhattheothermathpeoplearetalkingabout(amathprofessorofminemad