The-Wavelet-Tutorial

INDEXTOSERIESOFTUTORIALSTOWAVELETTRANSFORMBYROBIPOLIKAR~polikar/wavelets/wttutorial.html[2010-11-2414:08:17]THEENGINEER'SULTIMATEGUIDETOWAVELETANALYSISTHEWAVELETTUTORIALbyROBIPOLIKARAlsovisitRowan’sSignalProcessingandPatternRecognitionLaboratorypagesTwonewtutorials:PatternRecognition&EnsembleBasedSystemsinDecisionMakingPREFACEPARTI:OVERVIEW:WHYWAVELETTRANSFORM?NEW!–ThankstoNoëlK.MAMALET,thistutorialisnowavailableinFrenchPARTII:FUNDAMENTALS:THEFOURIERTRANSFORMANDTHESHORTTERMFOURIERTRANSFORM,RESOLUTIONPROBLEMSPARTIII:MULTIRESOLUTIONANALYSIS:THECONTINUOUSWAVELETTRANSFORMPARTIV:MULTIRESOLUTIONANALYSIS:THEDISCRETEWAVELETTRANSFORMACKNOWLEDGMENTSINDEXTOSERIESOFTUTORIALSTOWAVELETTRANSFORMBYROBIPOLIKAR~polikar/wavelets/wttutorial.html[2010-11-2414:08:17]Pleasenote:Duetolargenumberofe-mailsIreceive,Iamnotabletoreplytoallofthem.Iwillthereforeusethefollowingcriteriainansweringthequestions:1.Theanswertothequestiondoesnotalreadyappearinthetutorial;2.Iactuallyknowtheanswertothequestionasked.Ifyoudonotreceiveareplyfromme,thentheanswerisalreadyinthetutorial,orIsimplydonotknowtheanswer.Myapologiesfortheinconveniencethismaycause.Iappreciateyourunderstanding.Forquestions,commentsorsuggestions,pleasesendane-mailtoROBIPOLIKARMAINPAGEThankyouforvisitingTHEWAVELETTUTORIAL.Includingyourcurrentaccess,thispagehasbeenvisitedtimessinceMarch07,1999.TheWaveletTutorialishostedbyRowanUniversity,CollegeofEngineeringWebServersTheWaveletTutorialwasoriginallydevelopedandhosted(1994-2000)atLastmajorupdate:January12,2001.WAVELETTUTORIALINTRODUCTION-ROBIPOLIKAR~polikar/WAVELETS/WTpreface.html[2010-11-2414:11:34]WELCOMEWelcometoTheWaveletTutorial!ItwasendofOctober1994.Ihadrecentlystudiedfundamentalsofwavelettransformasmygraduationprojectatmyundergraduateinstitution,andIwasplanningtousethistechniqueinanalyzingsignalsofbiologicaloriginformyMaster'sdegreethesis.MymajorprofessorsuggestedthatIshouldworkonEEGsignals,sincetheyarelessstudiedcomparedtomanyotherbiologicalsignals.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.Bytheendofthisp