On-Line Wavelet Denoising with Application to the

On-LineWaveletDenoisingwithApplicationtotheControlofaReactionWheelSystemFran¸coisChaplais∗EcoledesMinesdeParis35rueSaintHonore77305,Fontainebleau,FrancePanagiotisTsiotras†andDongwonJung‡GeorgiaInstituteofTechnology,Atlanta,GA30332-0150,USAAwavelettransformonthenegativehalfrealaxisisdevelopedusinganaverage-interpolationscheme.Thistransformcanbeusedtoperformcausalwaveletprocessing,suchassignaldenoising,withasmalldelay.Thedelayrequiredtoobtainacceptablede-noisinglevelsisdecreasedbyusingaredundanttransforminsteadofanon-redundantone.Resultsfromtheexperimentalimplementationoftheproposedalgorithmforthedenoisingofafeedbacksignalforcontrollingathree-phasepermanent-magnetsynchronousbrushlessDCmotorthatdrivesareactionwheelsystemarepresented.I.IntroductionWaveletshaverecentlybecomeverypopularinsignalprocessing.Theyallowcompactrepresentationofasignalwithasmallnumberofwaveletcoefficients.Givenasequenceofdataa0={a0[k]}k∈Z,thewavelettransformgeneratesforeachscalej≥1asequenceofwaveletcoefficientsaj={aj[k]}k∈Zanddj={dj[k]}k∈Z.Thedataajcontainthe“salient”(low-resolution)featuresofthesignalanddjaretheresidualfeaturesofthesignalatthisscale.Thecoefficientsajanddj,dj+1,...arethusoftenreferredtoasthecoarseandfinedecompositionofthesignalatscalej.Thesimplestandprobablymostcommonmethodforsignalcompressionviawaveletdecompositionisthresholding.Inthresholdingonekeepsonlythecoefficientsdjwhicharelargerthanaprespecifiedtolerance.1Thresholdingappliedonthecoefficientsofawavelettransformisalsoknowntobeanefficientmethodfordenoisingsignalswithsharptransients.2,3Standardthresholdingistypicallyperformedusingwaveletsonthewholerealline.Thisoftencausessignificantdelaysintheprocessing,becausesomeofthefiltersinvolvedinthecomposition/decompositionphasesofarenotcausal.Whenoperatingonon-linedata(suchasfordenoisingsignalswithinafeedbackcontrolloop)itisimperativetousewaveletsonthenegativehalfrealaxis(thehalfaxisrepresentingpast,knownvaluesofthesignal).Inthiscontext,anydelaysarisingfromtheprocessofdenoisingwillbedetrimentaltotheperformanceorstabilityofthefeedbackloop.Inordertominimizeanydelaysarisingfromstandardwaveletprocessinghereinweproposeawavelettransformmethodthatoperatesonlyonpastdata.Itshouldbementionedatthispointthatalthoughtheword“wavelet”willbeusedthroughout,thebasicpointofviewtakeninthispaperisthatofFIRfilterbanks.4,5Moretothepoint,thevarioussignaldecompositionandreconstructionschemesdescribedhereinwereinspiredbytheaverage-interpolationschemesofDonoho6andSweldens;7seealsoRef.8.AsnotedinRef.7,thisschemecanbemodifiedtoprocesssignalsonthehalf-axis.Weexplicitly(albeitbriefly)showhowthiscanbedone.Moredetailsontheproposedalgorithmscanbefoundelsewhere.9∗Professor,CentreAutomatiqueetSystemes,Email:[e-mail].†AssociateProfessor,SchoolofAerospaceEngineering.Tel:[phone].Fax:[phone].Email:[e-mail].‡Ph.D.candidate,SchoolofAerospaceEngineering,Tel:[phone].Email:[e-mail]-InterpolationSchemeAverysimple–yetclassic–schemeforbuildingadiscretewavelettransformwithdiscretevanishingmo-mentsistouseaverage-interpolation.Thestartingpointofaverage-interpolationistheinterpretationofthedatasamplesasaverages.Specifically,givenasequenceaj,average-interpolationassignstoeachdatasampleaj[k]thedyadicinterval[2j(k−1),2jk]oflength2j.Thediscretewavelettransformusingaverage-interpolationthenproceedsasusualandinvolvestwosteps:thedecompositionphase,whichyieldsthetransformedsignalinwaveletspace,andthereconstructionphase,whichretrievesasignalfromitstrans-form(thatis,itswaveletcoefficients).ThedecompositionandreconstructionofthesignalcanbothbeobtainedusingapairofFIRfilters.Ifthereisnodataprocessingafterthedecompositionphase,thenthisprocessresultsinperfectreconstruction:theoriginaldatasequenceisrecoveredwithouterror.Thewholeprocessforaone-stepdecomposition/reconstructionisshowninFig.1.Thefilters¯hand¯garethelow-andhigh-pass(decomposition)filtersand˜hand˜garethelow-andhigh-pass(reconstruction)filters.ThedetailsoftheclassicalconstructionofthesefilterscanbefoundinRefs.6and7.Nonetheless,herewewilluseanalternativederivationofthetransformstartingfromtheaverage-interpolationframework,togetherwiththeusualFIRfilterpresentation,asillustratedinFig.1.Wefollowthisapproachbecausetheaverage-interpolationpresentationcanbeeasilyusedtoderiveasimilarschemeforprocessingsignalsoverthehalf-axiswhichisthemainobjectiveofthispaper.Inaddition,welaterusethisapproachtoobtainaredundanttransformonthehalf-axis.2isanoversamplingfilterwhichinsertazerobetweeneverysample2isanundersamplingfilterwhichremovesthesampleswithoddindicess222g2ha0~~g_h_a0a1d1Figure1.Perfectreconstructionfilterbanks,usedfortheimplementationofthewavelettransformontherealaxis.A.DecompositionSimilarlytostandardwaveletprocessing,thedataisfirstfilteredbyalow-passfilter¯handahigh-passfilter¯g.Thisseparatesthesignaltolow-frequencyandhigh-frequencycomponents.Theconstructionofthesefiltersisgivennext.1.Low