IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING VOL - PDF document

IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999609RobustnessofGroup-Delay-BasedMethodforExtractionofSignicantInstantsofExcitationfromSpeechSignalsP.SatyanarayanaMurthyandB.Yegnanarayana,SeniorMember,IEEEAbstractÐInthispaper,westudytherobustnessofagroup-delay-basedmethodfordeterminingtheinstantsofsignicantexcitationinspeechsignals.Theseinstantscorrespondtotheinstantsofglottalclosureforvoicedspeech.Themethodusesthepropertiesoftheglobalphasecharacteristicsofminimumphase 610IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999II.DETERMINATIONOFNSTANTSOFXCITATIONInthissection,webrie ypresentthegroup-delay-basedmethodproposedin[9]and[10]fordeterminingtheinstantsofsignicantexcitationfromspeechsignals,andproposesomerenementstothemethod.Themethodisbasedontheglobalphasecharacteristicsofminimumphasesignals.Sincetheaveragegroup-delayofaminimumphasesystemiszero[11],theaverageslopeofthephasespectrumoftheimpulseresponseofthesystemcorrespondstothelocationoftheexcitationimpulsewithintheanalysisframe[9].Inpractice,thecomputedphasespectrumorthegroup-delayfunctiondependsonthewindowfunctionusedforanalysis.Toreducetheeffectsofthewindowfunctionontheestimatedgroup-delayfunction,itispreferabletocomputethegroup-delayfunctionfromtheLPresidualsignal.Theresidualsignalisalsopreferablebecausesomecharacteristicsoftheglottalsourcecanbeseenbetterintheresidualerrorsignalthaninthespeechsignal.Theaverageslopeofthephasespectrumofthespeechsignalisthesamefortheresidualsignalalso,becausetheinverselteroftheLPanalysisisaminimumphasesystem[12].Theresidualsignalisderivedbyinverselteringthespeechsignal,andtheinverselterisobtainedusingLPanalysis.ForLPanalysis,aframesizeofabout25msforevery10msmaybechosen[9],[10].TheinstantsofsignicantexcitationcanbederivedfromtheLPresidualsignalasfollows[10].Aroundeachsamplinginstanta10mssegmentoftheLPresidualsignalisconsideredandthegroup-delayfunctioniscomputedusingtheformula[13] (1)where and aretheFouriertransformsofthewindowedresidual and respectively.Thegroup-delayfunctionissmoothedusingathree-pointmedianltertoremoveanydiscontinuitiesinthegroup-delayfunction.Thenegativeoftheaverageofthesmoothedgroup-delayfunctioniscalledphaseslope.Thephaseslopevalueiscomputedateachsamplinginstanttoobtainthephaseslopefunction.Iftheinstantofsignicantexcitationwithinaframeisatthemidpointoftheframe,thenthephaseslopeiszero.Thereforethepositivezero-crossingsofthephaseslopefunctioncorre-spondtotheinstantsofsignicantexcitation.Short-time(1±3ms)energyoftheLPresidualsignalaroundtheinstantcanbeusedtorepresentthestrengthofexcitationassociatedwiththeinstant[9],[10].Fig.1(a)±(d)showasegmentofspeechsignal,theLPresidualsignal,thephaseslopefunctionandtheextractedinstantswithestimatedstrengths,respectively.Thespeechsignalshowncorrespondstotheutterance where isavoicedpalatalfricativeasin SometimestheLPresidualsignalmaycontainsomespu-riousimpulseswhichmayresultinwrongestimationoftheinstantsofsignicantexcitation,ascanbeseeninFig.1(d),wherethestrengthsarecomputedusingtheshort-timeenergyoftheresidualsignalcenteredaroundtheestimatedinstantsofsignicantexcitation.Theeffectofthesespuriousimpulsescanbereducedbyenhancingtheregionaroundtheinstants (a) (b) (c) (d) Fig.1.(a)Cleanspeechfortheutterance=dzua=:(b)LPresidualsignalderivedfromthesignalin(a).(c)Phaseslopefunction.(d)Signicantinstants,weightedbytheirstrengths,derivedfromthesignalin(a).(e)Signicantinstants,derivedfromthesignalin(a)usingtheproposedalgorithm.ofsignicantexcitationrelativetootherregionsintheLPresidualsignal.ThiscanbeaccomplishedbyderivingaweightfunctionfortheLPresidualsignal.TheweightfunctionisderivedherebysmoothingtheLPresidualsignalwithaHammingwindowofduration0.75ms(eightsamplesat11kHzsamplingrate).Thissmoothingreducesthenoise uctuationsintheresidualsignal.Theshort-timeenergyofthesmoothedresidualsignaliscomputedateverysampleusingaframesizeof1.4ms(15samplesat11kHzsamplingrate).Theshort-timeenergycurvewillhavelargeamplitudesaroundthesignicantinstants.Theshort-timeenergyisnormalizedtoamaximumvalueofoneandisusedasaweightfunctionfortheresidualsignaltoenhancetheregionsintheresidualsignalaroundthesignicantexcitations.Theweightedresidualsignalisusedtoderivetheinstantsofsignicantexcitation.Thephaseslopefunctionissmoothedusingave-pointHammingwindow.Positivezero-crossingsofthesmoothedphaseslopefunctionareusedastheinstantsofsignicantexcitation.Fig.1(e)showstheplotoftheinstantsderivedafterthese MURTHYANDYEGNANARAYANA:EXTRACTIONOFSIGNIFICANTINSTANTSOFEXCITATIONFROMSPEECHSIGNALS611renements.SomeoftheerrorsintheestimationofinstantsinFig.1(d)arecorrectedinFig.1(e).ThedifferentstepsinthealgorithmforthecomputationoftheinstantsofsignicantexcitationaregiveninFig.9.III.MEASUREOFTRENGTHOFXCITATIONReliabilityoftheextractedinstantsdependsonthestrengthofexcitationaroundtheinstants.In[9],[10]theshort-timeenergyoftheLPresidualsignalwasusedtorepresentthestrengthofexcitationateachinstant.Insomecasesitisdifculttousetheshort-timeenergyaroundtheinstantasameasureofthestrength,especiallywhentheresidualsignalisnoisy,asintheregionBCinFig.1(b).Moreover,thederivedresidualsignalenergydependsontheeffectivenessoftheLPanalysisforthesesegments.Weproposeanalternativemeasureforthestrengthofexcitation,whichisbasedontheuseoftheFrobeniusnorm.In[8]theFrobeniusnormofasignalpredictionmatrix,formedbyusingthesamplesinaframeofabout3ms,wasproposedtolocatetheinstantsofglottalclosure.TheFrobeniusnormwascomputedateachsamplinginstant.ThelocationsofthepeaksintheplotoftheFrobeniusnormasafunctionoftimewereconsideredasthedesiredinstants.InthissectionweproposethattheFrobeniusnorm[14]ofthesignalpredictionmatrix[8]formedbyusingthesamplesina3-msframeofdifferencedspeechcenteredaroundtheidentiedinstantofsignicantexcitationcanbeusedtorepresentthestrengthofexcitationatthatinstant.Consideraframeofthedifferencedspeechsignalwith samples, Assumingalinearpredictionorder thefollowingpredictionerrorvectorcanbeformed: (2)where istheToeplitzsignalpredictionmatrixofdimension ............ .. (3)and istheaugmentedvectorofLPC's Assuming arethesamplesofasignalattheoutputofanall-polesystemexcitedbyaperiodicimpulsetrain,thereisalineardependencebetweenthecolumnvectors whentheinstantofexcitationisnotincludedintheanalysisframe[8].Theerrorvectoristhenzero.Butwhentheinstantofexcitationisincluded,thenormoftheerrorvectorgoesup.Theamplitudesofsignalsamplesinthesignalpredictionmatrixalsogoup,becauseoftheexcitation.Thus,theFrobeniusnormofthesignalpredictionmatrix,computedasthesquarerootofthesumofallsquaredelementsofthematrix,alsogoesup.ThesquareoftheEuclideannormof whichisameasureoftheenergy(strength)ofexcitation,isgivenby (4)where istheFrobeniusnormof Theratio isupperboundedby Ignoringthevariationin comparedto wecanuse asameasureofthestrengthofexcitation.ComputingtheEuclideannormof from(2),weget (5) (6)where istheRayleighquotientof [14].ItisshowninAppendixA[see(A.8)]that (7)where arethesingularvaluesof andarealsotheeigenvaluesof ItisalsoknownthatthesquareoftheFrobeniusnormisthesumofsquaredsingularvalues[15].Sowehavetheinequality (8)since isthearithmeticmeanofsquaredsingularvalues.Itisknownthatallthesingularvaluesriseinmagnitudewhenthereisanexcitationwithintheanalysisframeandfallwhenthereisnoexcitation[8].Therefore,both in(7) in(8)trackthesechanges.Therefore canbeusedasameasureofthestrengthofexcitation.Wenotethatthoughthisisameasureofenergyoftheresidualsignal,itiscomputeddirectlyfromthespeechItistobenotedthatsincethesquareoftheFrobeniusnormofthesignalpredictionmatrixisthesumofsquaresofallsamplesinthematrix,itisnothingbuttheshort-timeenergyofthespeechsignalcomputedfromtheweightedsamplesofthespeechsignal.Toillustratetheneedforameasureforthestrengthofexcitation,letusconsiderthedifferentiatedglottalpulses[Fig.2(a)]generatedusingtheLFmodel[16].Alltheparame-tersofthemodelarekeptconstantexceptthetimeconstantofthereturnphaseandtheinstantofpeakpositiveexcitation.Tovarytherateofclosure,thetimeconstantofthereturnphaseisincreasedfrom0.05±1.5msfromlefttoright.Theamplitudesofthepulsesareprogressivelyscaledup(fromlefttoright)sothatallthepulseshaveequalnegativepeakamplitudes.Thesedifferentiatedglottalpulsesareusedtoexciteanall-polemodeltoobtainasyntheticvoicedsoundshowninFig.2(c).Itshouldbenotedthat,intherst40msofthespeechsignal,thesignalcomponentsduetohigherformantscanbeclearlyseen.Thisisduetothesharpclosingphase,whichresultsinamagnitudespectrumoftheexcitationpulsesthatislesssteep.ThisfeatureisnotseeninthelatterportionofthesignalinFig.2(c)duetothegradualclosingphase.Thesecondderivativeoftheglottalpulseandthetwelfth-orderLPresidualsignalareshowninFig.2(b)and(d),respectively.Fromtheseguresitisevident 612IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999 (a) (b) (c) (d) Fig.2.(a)Differentiatedglottalpulses.(b)Secondderivativeofglottalpulses.(c)Syntheticsignal.(d)Residualsignalderivedfromthesignalin(c).(e)First-orderdifferenceofthesignalin(c).thattheamplitudesoftheexcitationimpulsesarehigherfortheglottalpulseswithsharperclosure.Thestrengthofexcitationishigherforsharperclosure,althoughtheamplitudeandenergyofthespeechsignalinFig.2(c)isnearlythesamethroughoutforalltheglottalpulseshapes.ItshouldbenotedinFig.2(a)thattheenergyconcentrationishigherforthepulsesintheinitialportionthaninthelatterportionofthesignal.IfweconsiderthedifferencedsignalofFig.2(c),asshowninFig.2(e),wenoticethatthestrengthofexcitationisalsoevidentinthedifferencedsignal.Itcanalsobeseenbyconsideringadifferenceoperation onthe transformofthesignal, where correspondstothedifferentiatedglottalpulseexcitation,and correspondstothevocaltractsystem.Wehave Thus,thedifferencedsignalcanbeviewedasasignalthatresultsduetotheexcitationofthevocaltractsystemwiththesecondderivativeoftheglottalpulse.ThesecondderivativeoftheglottalpulseinFig.2(b)andthedifferencedsignalinFig.2(e)bothshowthecharacteristicsofthestrengthofexcitation.TheseguressuggestthattheFrobeniusnormofthedifferencedsignalcanbeusedasameasureofthestrengthofexcitationaroundtheinstantofsignicantexcitation.IV.ROBUSTNESSOFTHEELAYInthissectionweshallexaminetherobustnessofthegroup-delay-basedmethodfortwotypesofdegradations,namely,additiverandomnoiseandecho/reverberation.A.RobustnessAgainstAdditiveNoiseLetusconsideranexcitationsignal consistingofanimpulseofamplitude attime andazero-meanadditivewhiteGaussiannoise TheFouriertransformof is (11)where (12) and arerandomvariablescorrespondingtothemagnitudeandphaseof respectively.Withoutlossofgenerality,thephasespectrum canbeassumedtohaveauniformprobabilitydensityfunctionovertherange [17].Let and bethemagnitudeandphaseof respectively. ItisshowninAppendix-B[see(B.4)]that (14)where denotesensembleaverageand istheexcitationSNR,denedasthelogarithmoftheratioofaverageexcitationsignalpowerpersample totheaveragenoisepowerpersample dB (15)For dB,theupperboundontheexpectedvalueofthemagnitudeof isone.IftheFouriertransformin(11)isevaluatedusingan discreteFouriertransform(DFT),themagnitudeoftheDFT canbeshowntobelessthan with99%condence dB[seeApp.B,(B.7)].Expandingthethird MURTHYANDYEGNANARAYANA:EXTRACTIONOFSIGNIFICANTINSTANTSOFEXCITATIONFROMSPEECHSIGNALS613termontherighthandsideof(13)byTaylorseriesexpansion,thephasetermof(13)canbeapproximatedas Thegroup-delayfunction isgivenby Hence,theaveragevalueofthegroup-delayfunctionisgiven Substituting(16)in(18)andnotingthat isanoddfunctionof andthatthesecondtermin(16)vanishesat wehave i.e.,theaveragevalueofthegroup-delayfunctiongivesthelocationoftheimpulse.Inpractice,thegroup-delayfunctioniscomputedatdiscretefrequencies,andhencethecomputedaveragedeviatesfrom(19).Random uctuationsandspikesappearinthegroup-delayfunction[18].Thesespikesmaybiasthemeanvalueofthegroup-delayfunction.Therefore,itispreferabletousemediansmoothingofthecomputedgroup-delayfunctionbeforecomputingtheaverage.Sofarwehaveconsideredanexcitationsignalcorruptedbyadditivenoise.Letusnowconsideranoisyspeechsignal (20)where isthespeechsignaland istheadditivewhitenoise.Toderivetheinstantsofsignicantexcitation,letusconsidertheLPresidualsignal.ThefrequencyresponseoftheinverselterobtainedfromtheLPanalysisisgivenby (21)where and aretheLPcoefcients(LPC's).Theresidualerrorsignalobtainedafterinverselteringisgivenby (22)where isthecomponentattheoutputoftheinverselterduetothespeechsignal and isthecolorednoiseduetolteringofthewhitenoise Notethateventhoughthespeechsignalisassumedtobetheoutputofanall-polesystem,thenoisysignal correspondstoapole-zerosystem[19].Thepowerspectrumofthecolorednoisecomponent givenby (23) (a) (b) (c) Fig.3.(a)SyntheticspeechofFig.2(c)atanaverageSNRof5dB.(b)LPresidualsignalderivedfromthesignalin(a).(c)Thetruelocationsoftheinstantsofsignicantexcitation.(d)Theinstantsofsignicantexcitationderivedfromthenoisysignalin(a).Thesecondmoment dependsonthefrequency Letusconsidertheworstcasesituation,i.e.,themaximumvalueof Let (24)where isthemaximumvalueof givenby Intheexpressionfor in(15),the isreplacedby Assumingthat theeffective fortheresidualsignalisreduced.Theaboveanalysisisvalidevenwhenthespeechiscorruptedbyadditivecoloredrandomnoise,exceptthat nowalsodependsonthemaximumvalueofthepowerspectrumofthecolorednoise.TherobustnessofestimationoftheinstantofexcitationdependsontheexcitationSNR Foraconstantadditive willdecreaseasthestrengthoftheexcitationdecreases.ThisisillustratedinFig.3foranoisycaseofthesyntheticsignalgeneratedbyexcitinganall-polelterwiththedifferentiatedglottalpulsesofFig.2(a).TheoverallSNRofthespeechsignalis5dB.Notethattheperiodicitycannotbeimmediatelyseenfromthenoisecorruptedspeechsignal.Sinceitisasyntheticcase,thestrengthofexcitationcanbeapproximatedtotheamplitudeofthesecondderivative 614IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999oftheglottalpulseshowninFig.2(b).Fig.3(c)showstheactualinstantsofsignicantexcitation.Fig.3(d)showstheinstantsofsignicantexcitationestimatedfromthenoisyspeechsignal.Thegureshowsthattheaccuracyoftheextractedinstantsdependsontheexcitationsignal-to-noiseratio.ReliabilityoftheextractedinstantsdecreaseswithadecreaseintheexcitationSNR,ascanbeseenfromthedeviationoftheinstantsinFig.3(d)relativetotheinstantsinFig.3(c).TheexcitationSNR isdenedastheratioofthesquareoftheamplitudeoftheimpulseandthenoisepower.NotethateventhoughtheaverageSNRofthespeechsignalisnearlyconstant,i.e.,5dB,theexcitationSNRisdecreasingfromlefttorightonthetimescale.B.RobustnessAgainstEchoandReverberationLetusconsiderthefollowingreverberantsignal foranimpulseofstrength anddelayedby samples. (26)where istheattenuationfactor and isthedelayduetoreverberation.TheFouriertransformationof(26)yields (27)where and arethemagnitudeandphaseoftheFouriertransformof respectively.Takingnaturallogarithmonbothsidesof(27),weget[20] NeglectingthehigherordertermsintheTaylorseriesexpan-sionofthelasttermabove,thephasecomponentisgiven Thegroup-delayis Themeanvalueofthegroup-delay is Forasingleecho,theterm in(28)canbereplaced Theexpressionforthephaseissameasin(29)andhencethegroup-delayforthecaseofechoissameasin(30).Itshouldbenotedthattheaboveanalysisisvalidonlyundermildechoandreverberantconditions Wehavealsoassumedthatthesignalcharacteristicsarestationary.Duetononstationarityofspeechsignals,themodelofreverberationin(26)maynotbevalidinrealsituations.C.RobustnessDuetoWeightingoftheLPResidualSignalInthissection,weshowthatsuitableweightingoftheLPresidualsignalimprovestherobustnessofthealgorithmforextractionoftheinstantsofsignicantexcitation.ThisisbecausetheexcitationSNR canbeimprovedbyweighting,asshownbelow.Considertheimpulse-in-noisesequence in(10).Let beapositivewindowfunctionsuchthat Let betheweightedexcitationsignal,suchthattheimpulseat isgiventhemaximalweightof ByfollowingthestepsintheanalysispresentedinSectionIV-A,wehave (32)where (33)and isthephaseoftheFouriertransformoftheweightedexcitationsequence Theapproximationin(32)isjustiedprovidedthat Assumingthat arezero-meanGaussianrandomvari-ableswithvariance wehavefrom(33) (34)where FollowingthestepsintheanalysispresentedinAppendixB,wedenetheweightedexcitationSNRas dB dB(36)Using(15),weget NotethatforthecasewithoutweightingoftheLPresidual Therefore,from(35)and(37), Foranyotherwindowfunctionwithabroadpeakaroundthelocationoftheimpulsei.e., Thus,thereissomegainintheexcitationSNR.Forthelimitingcaseofaweightfunctionwithanarrowpeakat thegainintheexcitationSNRtendsto MURTHYANDYEGNANARAYANA:EXTRACTIONOFSIGNIFICANTINSTANTSOFEXCITATIONFROMSPEECHSIGNALS615 (a) (b) (c) (d) (e) Fig.4.(a)Cleanspeechfortheutterance=dzua=:(b)StrengthsofexcitationbasedontheFrobeniusnorm.(c)Speechdegradedbyambientnoise.(d)Signicantinstantsderivedfromthesignalin(c).(e)Telephonespeech.(f)Signicantinstantsderivedfromthesignalin(e).V.PVALUATIONOFELAYInthissection,weconsidersomeexamplesofspeechdataunderactualconditionsofdegradation,andexaminetheperformanceofthegroup-delay-basedmethodforextractionoftheinstantsofsignicantexcitation.SincewedonothaveamethodforestimatingtheSNRofthestrengthofexcitationforsignalswithnaturaldegradations,theresultscanonlybeinterpretedfromouraprioriknowledgeofthecharacteristicsoftheexcitationfordifferentcategoriesofsounds.Whereverappropriate,theFrobeniusnormofthedifferencedspeechsignalcanbeusedasameasureofthestrengthofexcitation.Fig.4showstheperformanceofthealgorithmfornoiseandtelephonechanneldegradationsforthesegmentofspeechgiveninFig.1(a).ThestrengthsofexcitationattheextractedinstantscomputedusingtheFrobeniusnormareshowninFig.4(b).Forthissignal,thestrengthofexcitationislowerforthesegment intheregionBCcomparedtotheregionCD.ThenoisyspeechsignalinFig.4(c)correspondstothesamespeechasinFig.4(a),butrecordedbyamicrophoneplaced50cmawayfromthespeaker.ThesignalintheregionABisaffectedbytheadditivenoisemorethanthesignalintheregionCDduetolowersignalamplitudes.HencetheinstantsextractedforthesignalinregionABarenotreliable.Mostoftheextractedinstants[Fig.4(d)]forthesignalintheregionBCarecorrect,eventhoughinFig.4(c)thereappearstobenovisibleperiodicityinthesignalintheregionBC.FromFig.4(b)and(d),itcanbeseenthattheinstantsarecorrectlyextractedforthesignalintheregionCD.TheresultsaresimilarforthecaseoftelephonespeechshowninFig.4(e)and(f).InthetelephonespeechshowninFig.4(e),thesignalintheregionABislostanditissignicantlyattenuatedintheregionBC.Thisisbecausethelowrstformantofthevowel isseverelyattenuatedduetothebandpassnatureofthetelephonechannelcharacteristics.TheerrorsintheregionABareduetolowlevelsofthesignalitselfinthatregion.ItisimportanttonotethatalthoughthesignallevelishighintheregionBCforthecleanspeech,thestrengthofexcitationislowfortheinstantsinthatregion.Hence,theextractedinstantsinthisregionaremorepronetoerrorscomparedtotheextractedinstantsintheregionCD.Asystematicinvestigationwascarriedouttostudytheaccuracyoftheextractedinstantsforsyntheticandnaturalvowels.HistogramsofthespreadoftheerrorsareshowninFigs.5and6forvesyntheticandnaturalvowels and respectively,foranoverallSNRof10dB.AllthesyntheticvowelsaregeneratedbythesameLF-model-baseddifferentiatedglottalpulses.Thelengthofeachpulsewaschosentobe80samples.Inthecaseofthenaturalvowels,theglottalcycledurationvariedfrom9msforvowel to7msforvowel InFig.5,thehistogramforeachsyntheticvowelisobtainedbycomputingthehistogramofdeviationsoftheestimatedinstantsofsignicantexcitationfromthetruelocationsfor50realizationsofnoise.Therearetenglottalcyclesinthesignalforeachvowelandhenceweget500suchdeviationsforeachvowel.InFig.6,thedeviationsareobtainedbysubtractingtheestimatedlocationsfromthelocationsextractedfromthecleanspeechsignal.Largerspreadofthehistogramsindicateslargerdeviationoftheextractedinstantsfromthetruelocationsoftheinstants.Theerrorsaretypicallylargerfortheclosevowels and thanfortheopenvowels and ForthesyntheticcaseshowninFig.5,alltheinstantshavethesamestrengthandhencethespreadoferrorsislesscomparedtothecaseofnaturalvowels.ItisimportanttonotethatthevariationinthespreadoftheerrorsfordifferentvowelsisalsoduetotheartifactsoftheLPanalysis.ForthesyntheticcaseshowninFig.5,thespreadislargerfortheclosevowels and despitetheexcitationstrengthbeingthesameforallthevevowels,becauseofthedominanceoftherstformantintheLPanalysisofthenoise-corruptedsignalsfortheseclosevowels.ThisisalsotrueinthecaseofnaturalvowelsshowninFig.6.Thereisasystematicbiasintheestimatedlocationsoftheinstantsofexcitationfor 616IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999 (a) (b) (c) (d) Fig.5.HistogramoferrorsintheestimatedinstantsforvesyntheticvowelsforSNR10dB.(a)=a=,(b)=e=,(c)=i=,(d)=o==u=:thecaseofsyntheticvowels.Thebiasisabouttwosamplesfortheaverageglottalcyclelengthof80samples.Thatis,thebiasisabout3%.ThebiasmayhavebeencausedduetoweightingtheLPresidualsignalbeforecomputingtheinstantsofexcitation.Theweightfunctiondependsonthenatureofthevoicedsound,andtheextentofdegradationcausedbynoise.Thatiswhythebiasispositiveinsomecasesandnegativeinsomeothercases.Errorsintheextractedinstantswerealsostudiedforutter-ancestakenfromthestandardNTIMIT[21],[22]dataformaleandfemalespeech.SincetheTIMIT[23]datawasavailableforreference,thespreadwasestimatedusingthedeviationsoftheextractedinstantsfortheNTIMITdatafromthosefortheTIMITdata.TheTIMITandNTIMITdatatakenforstudywerelowpasslteredanddownsampledto8kHzbeforeprocessing.TheTIMITandNTIMITdatawasrsttime-alignedbeforethedeviationswerecomputed.ThehistogramsoferrorsforonemalevoiceandonefemalevoiceareshowninFigs.7and8.Thedataforthemalevoicecorrespondstothele:intheTIMIT/NTIMIT (a) (b) (c) (d) Fig.6.HistogramoferrorsintheestimatedinstantsforvenaturalvowelsforSNR10dB.(a),(b),(c),(d),(e)database.Thedataforthefemalevoicecorrespondstothele:.Theinstantsofsignicantexcitationwereextractedonlyfromthevoicedregions,whichwereidentiedusingthephonetictranscriptionlesprovidedwiththeTIMITdatabase.FromFigs.7and8,itcanbeseenthattherearemorevaluesofdeviationinthehistogramofdeviationsforfemalespeechthanforthemalespeech.Thisisbecausetheaveragepitchofthefemalespeakerisabout210Hzandthatofthemalespeakerisabout100Hz.Sotherearemoreglottalcyclesintheutteranceofthefemalespeakerthanforthemalespeaker.ThespreadoferrorsislargerfortheseutterancescomparedtotheerrorsforthevowelsinFig.6,becausetheSNRisdifferentfordifferentsegmentsinthiscase,whereasforvowelsitwasconstant.ThespeechSNRvariesoverarangeof20±50dBfortheutterancestakenfromtheTIMITdataandoverarangeof5±40dBfortheutterancestakenfromtheNTIMITdataforbothmaleandfemalevoices.TheSNRfordifferentsegmentswascomputedastheratiooftheenergyofthesignalsamplestotheenergyofthenoisesamplesinthesilenceregions.Thebiasandspread MURTHYANDYEGNANARAYANA:EXTRACTIONOFSIGNIFICANTINSTANTSOFEXCITATIONFROMSPEECHSIGNALS617 Fig.7.HistogramoferrorsfortheutteranceªShehadyourdarksuitingreasywashwaterallyearºutteredbyamalespeaker. Fig.8.HistogramoferrorsfortheutteranceªShehadyourdarksuitingreasywashwaterallyearºutteredbyafemalespeaker.oftheerrorsinFigs.7and8canbeattributednotonlytothevariationsofSNRfordifferentsegments,butalsototheweightfunctionusedontheLPresidualsignalbeforecomputingtheinstantsofexcitation.VI.CInthispaper,wehavedemonstratedthatthegroup-delay-basedmethodproposedin[9]and[10]isindeedrobustagainstdegradationsinspeechduetoadditivenoiseandchanneldistortion.Therobustnessisduetothefactthattheenergyofthesignalisconcentratedaroundtheinstantofsignicantexcitation,whichforvoicedspeechcorrespondstotheinstantaroundglottalclosure.Wehavediscussedtheimportanceofthestrengthofexcitation,whichcannotbedirectlyinferredfromthespeechsignal.WehaveshownthattheerrorsintheextractedinstantsaresmallformanypracticalsignalssuchasintheNTIMITspeechdata.OUNDSONTHEAYLEIGHLetthesingularvaluedecomposition(SVD)[15]of be wherethecolumnsof and aretheleftandrightsingularvectorsof respectively. isthematrixofsingularvalues Therefore (A.2)So aretheeigenvaluesof and thecolumnsof areitseigenvectors.TheRayleighquotientof isdenedas[14] (A.3)where Assumingthattheeigenvaluesof arealldistinct,itseigenvectorsformanorthonormalbasisin Hence, canbeexpressedas (A.4)where arethecomponentsof w.r.t.the Premultiplyingbothsidesof(A.4)by andnotingthat and areitseigenvaluesandeigenvectors,respectively,wehave Premultiplying(A.5)by andnotingthattheeigenvectorsformanorthonormalset,wehave From(A.3),(A.4),and(A.6),wehave From(A.7),itisclearthat i.e.,theRayleighquotientisboundedbytheextremeeigen-valuesof XCITATIONATIOForthezero-meanGaussiandistributedrandomvariables theFouriertransform isacomplexzero-meanGaussianrandomvariable.Thereforewehave Sincethesquareofthemeanisalwayslessthanthesecondmoment,i.e., (B.2) 618IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSING,VOL.7,NO.6,NOVEMBER1999 Fig.9.Algorithmfordeterminationofinstantsofsignicantexcitation.wehave (B.3)Hence (B.4)where istheexcitationSNR: dB(B.5)Letusconsideran -pointdiscreteFouriertransform(DFT)ofthesequencegivenin(10),computedat Itcanbeshown[24]thattherealandimaginarypartsoftheDFTof and are(real)independentidenticallydistributed(i.i.d.)Gaussianrandomvariablesfor Therefore,the and are UndertheseconditionsthemagnitudeoftheDFTof isRayleighdistributed[24].Sincewehavetheknowledgeofboththemeanandvariance weget whichisindeedclosetotheupperbound givenin(B.4)above.FromthecumulativedistributionfunctionofaRayleighdistribution[25],wemaywrite (B.7)where istheprobabilitythat isless From(B.7),wenotethat withmorethan99%condence,when TheauthorswouldliketothankDr.H.A.Murthyforprovidingthedatarequiredforsomeofthestudiesinthis 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P.SatyanarayanaMurthywasborninKakinada,India,in1971.HereceivedtheB.E.degreeinelectronicsandcommunicationengineeringfromChaitanyaBharathiInstituteofTechnology,Osma-niaUniversity,Hyderabad,theM.Tech.andPh.D.degreesinelectricalengineeringfromtheIndianInstituteofTechnology(IIT),Madras,in1994and1999,respectively.FromJanuarytoJuly1994,hewasaSeniorProjectOfcerintheDepartmentofComputerScienceandEngineering,IIT.HeiscurrentlyaManagerwithSpeechandSoftwareTechnologies,Madras.Hisresearchinterestisinspeechsignalprocessing. B.Yegnanarayana(M'78±SM'84)wasborninIndiaonJanuary9,1944.HereceivedtheB.E.,M.E.,andPh.D.degreesinelectricalcommunica-tionengineeringfromtheIndianInstituteofSci-ence,Bangalore,India,in1964,1966,and1974,HewasaLecturerfrom1966to1974andanAssistantProfessorfrom1974to1978intheDe-partmentofElectricalCommunicationEngineering,IndianInstituteofScience.From1966to1971,hewasengagedinthedevelopmentofenvironmentaltestfacilitiesfortheAcousticLaboratory,IndianInstituteofScience.From1977to1980,hewasavisitingAssociateProfessorofcomputerscienceatCarnegieMellonUniversity,Pittsburgh,PA.HewasaVisitingScientistatISROSatelliteCenter,Bangalore,fromJulytoDecember1980.Since1980,hehasbeenaProfessorintheDepartmentofComputerScienceandEngineering,IndianInstituteofTechnology,Madras.HewasaVisitingProfessorattheInstituteforPerceptionResearch,EindhovenTechnicalUniversity,Eindhoven,TheNetherlands,fromJuly1994toJanuary1995.Since1972,hehasbeenworkingonproblemsintheareaofspeechsignalprocessing.Heispresentlyengagedinresearchactivitiesindigitalsignalprocessing,speechrecognition,andneuralnetworks.Dr.YegnanarayanaisamemberoftheComputerSocietyofIndia,aFellowoftheInstitutionofElectronicsandTelecommunicationsEngineersofIndia,aFellowoftheIndianNationalScienceAcademy,andaFellowoftheIndianNationalAcademyofEngineering.