Application of wavelets and neural networks to diagnostic system development, 1, feature extraction(3)

WhiteleyandDavis(1992)appliedback-propagationneuralnetworks(BPNN)toconvertnumericalsensordataintosymbolicabstractions.Themajorlimitationofthisapproachisthatitrequirestrainingdatatotrainthemodel rst.

ThemostwellknowntechniqueforsignalanalysisisprobablytheFouriertransformanditistherefore

necessarytomentionedithere.Fouriertransformusessineandcosineasitsbuildingblockstodecomposeafunctionintoasumoffrequencycomponents.How-ever,Fouriertransformdoesnotshowhowfrequencyvarieswithtime,thereforeitisnotabletodetectwhenaparticulareventtookplace.Itmeansthatthenon-sta-tionaryfeatureofthesignalisnotcaptured.Theshort-timeFouriertransformisabletoovercomethislimitationbyslidingawindowoverthesignalintime.Howeverintime-frequencyanalysisofanon-stationarysignal,therearetwocon ictingrequirements.Thewin-dowwidthmustbelongenoughtogivethedesiredfrequencyresolutionbutmustalsobeshortenoughtolosetrackoftimedependentevents.Whileitispossibletooptimisethedesignofwindowshapestooptimise,ortrade-offtimeandfrequencyresolution,thereisafun-damentallimitationonwhatcanbeachieved,foragiven xedwindowwidth(Dai,Joseph&Motard,1994).

3.Featureextractionusingwavelettransform

Averybriefintroductionofwavelettransformationforsignalprocessingisnowpresented.Thenthemethodemployedinthisstudyforfeatureextractionusingwaveletsisintroducedandillustratedusingexamples.

3.1.Signaltransformationusingwa6elets

Wavelettransformationisdesignedtoaddresstheproblemofnon-stationarysignals.Itinvolvesrepre-sentingatimefunctionintermsofsimple, xedbuild-ingblocks,termedwavelets.Thesebuildingblocksareactuallyafamilyoffunctionswhicharederivedfromasinglegeneratingfunctioncalledthemotherwaveletbytranslationanddilationoperations.Dilation,alsoknownasscaling,compressesorstretchesthemotherwaveletandtranslationshiftsitalongthetimeaxis.&

Themotherwaveletsatis es

+

(t)dt=0(1)

andthetranslationandscalingoperationson (t)createsafamilyoffunctions,

=

1a,b(t) t ba

(2)Theparameteraisascalingfactorandstretches(or

compresses)themotherwavelet.Theparameterbisatranslationalongthetimeaxisandsimplyshiftsawaveletandsodelaysoradvancesthetimeatwhichitisactivated.Mathematicallydelayingafunctionf(t)bytdisrepresentedbyf(t td).Thefactor1/ aisusedtoensuretheenergyofthescaledandtranslatedversionsarethesameasthemotherwavelet.