小波功率谱相关译文.docx

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小波功率谱相关译文

译文

THECONTINUOUSWAVELETTRANSFORM

ThecontinuouswavelettransformwasdevelopedasanalternativeapproachtotheshorttimeFouriertransformtoovercometheresolutionproblem.ThewaveletanalysisisdoneinasimilarwaytotheSTFTanalysis,inthesensethatthesignalismultipliedwithafunction,{\itthewavelet},similartothewindowfunctionintheSTFT,andthetransformiscomputedseparatelyfordifferentsegmentsofthetime-domainsignal.However,therearetwomaindifferencesbetweentheSTFTandtheCWT:

1.TheFouriertransformsofthewindowedsignalsarenottaken,andthereforesinglepeakwillbeseencorrespondingtoasinusoid,i.e.,negativefrequenciesarenotcomputed.

2.Thewidthofthewindowischangedasthetransformiscomputedforeverysinglespectralcomponent,whichisprobablythemostsignificantcharacteristicofthewavelettransform.

Thecontinuouswavelettransformisdefinedasfollows

Equation3.1

Asseenintheaboveequation,thetransformedsignalisafunctionoftwovariables,tauands,thetranslationandscaleparameters,respectively.psi（t）isthetransformingfunction,anditiscalledthemotherwavelet.Thetermmotherwaveletgetsitsnameduetotwoimportantpropertiesofthewaveletanalysisasexplainedbelow:

Thetermwaveletmeansasmallwave.Thesmallnessreferstotheconditionthatthis（window）functionisoffinitelength（compactlysupported）.Thewavereferstotheconditionthatthisfunctionisoscillatory.Thetermmotherimpliesthatthefunctionswithdifferentregionofsupportthatareusedinthetransformationprocessarederivedfromonemainfunction,orthemotherwavelet.Inotherwords,themotherwaveletisaprototypeforgeneratingtheotherwindowfunctions.

ThetermtranslationisusedinthesamesenseasitwasusedintheSTFT;itisrelatedtothelocationofthewindow,asthewindowisshiftedthroughthesignal.Thisterm,obviously,correspondstotimeinformationinthetransformdomain.However,wedonothaveafrequencyparameter,aswehadbeforefortheSTFT.Instead,wehavescaleparameterwhichisdefinedas$1/frequency$.ThetermfrequencyisreservedfortheSTFT.Scaleisdescribedinmoredetailinthenextsection.

TheScale

Theparameterscaleinthewaveletanalysisissimilartothescaleusedinmaps.Asinthecaseofmaps,highscalescorrespondtoanon-detailedglobalview（ofthesignal）,andlowscalescorrespondtoadetailedview.Similarly,intermsoffrequency,lowfrequencies（highscales）correspondtoaglobalinformationofasignal（thatusuallyspanstheentiresignal）,whereashighfrequencies（lowscales）correspondtoadetailedinformationofahiddenpatterninthesignal（thatusuallylastsarelativelyshorttime）.Cosinesignalscorrespondingtovariousscalesaregivenasexamplesinthefollowingfigure.

Figure3.2

Fortunatelyinpracticalapplications,lowscales（highfrequencies）donotlastfortheentiredurationofthesignal,unlikethoseshowninthefigure,buttheyusuallyappearfromtimetotimeasshortbursts,orspikes.Highscales（lowfrequencies）usuallylastfortheentiredurationofthesignal.

Scaling,asamathematicaloperation,eitherdilatesorcompressesasignal.Largerscalescorrespondtodilated（orstretchedout）signalsandsmallscalescorrespondtocompressedsignals.Allofthesignalsgiveninthefigurearederivedfromthesamecosinesignal,i.e.,theyaredilatedorcompressedversionsofthesamefunction.Intheabovefigure,s=0.05isthesmallestscale,ands=1isthelargestscale.

Intermsofmathematicalfunctions,iff（t）isagivenfunctionf（st）correspondstoacontracted（compressed）versionoff（t）ifs>1andtoanexpanded（dilated）versionoff（t）ifs<1.

However,inthedefinitionofthewavelettransform,thescalingtermisusedinthedenominator,andtherefore,theoppositeoftheabovestatementsholds,i.e.,scaless>1dilatesthesignalswhereasscaless<1,compressesthesignal.Thisinterpretationofscalewillbeusedthroughoutthistext.

COMPUTATIONOFTHECWT

Interpretationoftheaboveequationwillbeexplainedinthissection.Letx（t）isthesignaltobeanalyzed.Themotherwaveletischosentoserveasaprototypeforallwindowsintheprocess.Allthewindowsthatareusedarethedilated（orcompressed）andshiftedversionsofthemotherwavelet.Thereareanumberoffunctionsthatareusedforthispurpose.TheMorletwaveletandtheMexicanhatfunctionaretwocandidates,andtheyareusedforthewaveletanalysisoftheexampleswhicharepresentedlaterinthischapter.

Oncethemotherwaveletischosenthecomputationstartswiths=1andthecontinuouswavelettransformiscomputedforallvaluesofs,smallerandlargerthan``1''.However,dependingonthesignal,acompletetransformisusuallynotnecessary.Forallpracticalpurposes,thesignalsarebandlimited,andtherefore,computationofthetransformforalimitedintervalofscalesisusuallyadequate.Inthisstudy,somefiniteintervalofvaluesforswereused,aswillbedescribedlaterinthischapter.

Forconvenience,theprocedurewillbestartedfromscales=1andwillcontinuefortheincreasingvaluesofs,i.e.,theanalysiswillstartfromhighfrequenciesandproceedtowardslowfrequencies.Thisfirstvalueofswillcorrespondtothemostcompressedwavelet.Asthevalueofsisincreased,thewaveletwilldilate.

Thewaveletisplacedatthebeginningofthesignalatthepointwhichcorrespondstotime=0.Thewaveletfunctionatscale``1''ismultipliedbythesignalandthenintegratedoveralltimes.Theresultoftheintegrationisthenmultipliedbytheconstantnumber1/sqrt{s}.Thismultiplicationisforenergynormalizationpurposessothatthetransformedsignalwillhavethesameenergyateveryscale.Thefinalresultisthevalueofthetransformation,i.e.,thevalueofthecontinuouswavelettransformattimezeroandscales=1.Inotherwords,itisthevaluethatcorrespondstothepointtau=0,s=1inthetime-scaleplane.

Thewaveletatscales=1isthenshiftedtowardstherightbytauamounttothelocationt=tau,andtheaboveequationiscomputedtogetthetransformvalueatt=tau,s=1inthetime-frequencyplane.

Thisprocedureisrepeateduntilthewaveletreachestheendofthesignal.Onerowofpointsonthetime-scaleplaneforthescales=1isnowcompleted.

Then,sisincreasedbyasmallvalue.Notethat,thisisacontinuoustransform,andtherefore,bothtauandsmustbeincrementedcontinuously.However,ifthistransformneedstobecomputedbyacomputer,thenbothparametersareincreasedbyasufficientlysmallstepsize.Thiscorrespondstosamplingthetime-scaleplane.

Theaboveprocedureisrepeatedforeveryvalueofs.Everycomputationforagivenvalueofsfillsthecorrespondingsinglerowofthetime-scaleplane.Whentheprocessiscompletedforalldesiredvaluesofs,theCWTofthesignalhasbeencalculated.

Thefiguresbelowillustratetheentireprocessstepbystep.

Figure3.3

InFigure3.3,thesignalandthewaveletfunctionareshownforfourdifferentvaluesoftau.ThesignalisatruncatedversionofthesignalshowninFigure3.1.Thescalevalueis1,correspondingtothelowestscale,orhighestfrequency.Notehowcompactitis（thebluewindow）.Itshouldbeasnarrowasthehighestfrequencycomponentthatexistsinthesignal.Fourdistinctlocationsofthewaveletfunctionareshowninthefigureatto=2,to=40,to=90,andto=140.Ateverylocation,itismultipliedbythesignal.Obviously,theproductisnonzeroonlywherethesignalfallsintheregionofsupportofthewavelet,anditiszeroelsewhere.Byshiftingthewaveletintime,thesignalislocalizedintime,andbychangingthevalueofs,thesignalislocalizedinscale（frequency）.

Ifthesignalhasaspectralcomponentthatcorrespondstothecurrentvalueofs（whichis1inthiscase）,theproductofthewaveletwiththesignalatthelocationwherethisspectralcomponentexistsgivesarelativelylargevalue.Ifthespectralcomponentthatcorrespondstothecurrentvalueofsisnotpresentinthesignal,theproductvaluewillberelativelysmall,orzero.ThesignalinFigure3.3hasspectralcomponentscomparabletothewindow'swidthats=1aroundt=100ms.

ThecontinuouswavelettransformofthesignalinFigure3.3willyieldlargevaluesforlowscalesaroundtime100ms,andsmallvalueselsewhere.Forhighscales,ontheotherhand,thecontinuouswavelettransformwillgivelargevaluesforalmosttheentiredurationofthesignal,sincelowfrequenciesexistatalltimes.

Figure3.4

Figure3.5

Figures3.4and3.5illustratethesameprocessforthescaless=5ands=20,respectively.Notehowthewindowwidthchangeswithincreasingscale（decreasingfrequency）.Asthewindowwidthincreases,thetransformstartspickingupthelowerfrequencycomponents.

Asaresult,foreveryscaleandforeverytime（interval）,onepointofthetime-scaleplaneiscomputed.Thecomputationsatonescaleconstructtherowsofthetime-scaleplane,andthecomputationsatdifferentscalesconstructthecolumnsofthetime-scaleplane.

Now,let'stakealookatanexample,andseehowthewavelettransformreallylookslike.Considerthenon-stationarysignalinFigure3.6.ThisissimilartotheexamplegivenfortheSTFT,exceptatdifferentfrequencies.Asstatedonthefigure,thesignaliscomposedoffourfrequencycomponentsat30Hz,20Hz,10Hzand5Hz.

Figure3.6

Figure3.7isthecontinuouswavelettransform（CWT）ofthissignal.Notethattheaxesaretranslationandscale,nottimeandfrequency.However,translationisstrictlyrelatedtotime,sin