LejaorderingLSFsforaccurateestimationofpredictorcoefcientsCFPederse - PDF document

1 LejaorderingLSFsforaccurateestimationofp
LejaorderingLSFsforaccurateestimationofpredictorcoefcientsC.F.PedersenDepartmentofElectronicSystems,AalborgUniversity,Denmarkcfp@es.aau.dkAbstractLinearprediction(LP)isthemostprevalentmethodforspectralmodellingofspeech,andlinespectrumpair(LSP)de-compositionisthestandardmethodtorobustlyrepresentthecoefcientsofLPmodels.Specically,theanglesofLSPpoly-nomialroots,i.e.linespectrumfrequencies(LSFs),encodeex-actlythesameinformationasLPcoefcients.TheconversionofLPcoefcientstoLSFsandback,hasreceivedconsiderableattentionsincemid1970swhenLSFswereintroduced.ThepresentpaperdemonstrateshowLejaorderingLSFsre-duceamplicationofroundingerrorswhenconvertingLSFstoLPcoefcients.ThetheorybehindLejaorderingandtheLSFstoLPcoefcientsconversionispresented.Tosupplementthe-ory,numericalexperimentsillustratetheaccuracygainachievedbyLejaorderingLSFspriortoconversion.Accuracyismea-suredastherootmeansquaredeviationbetweenestimatedco-efcientvectorswithandwithoutpriorLejaordering.IndexTerms:Linespectralfrequencies,linearpredictioncoef-cients,Lejaorder1.IntroductionLinearprediction(LP)isthepremiermethodforspectralmod-ellingofspeech.ThecoefcientsrepresentingtheLPmodelare,however,sensitivetoquantizationerrors,i.e.smallerrorsmayleadtodetrimentaldistortionsinthespectraldomain.Inthemid1970s,amethodforrobustrepresentationofLPcoef-cientswasintroduced,cf.[1].Themethod,nowknownaslinespectrumpair(LSP)decomposition,decomposestheLPmodel'sdenominatorpolynomialintoLSPpolynomialswithusefulinherentproperties;see,e.g.[2,3,4]forelaboratepre-sentationsofLSPpolynomialproperties.AnotablepropertyisthatLSPpolynomials'rootsareallunit-modulus;hence,theycanunambiguouslyberepresentedbytheirarguments(frequen-cies).Unit-modulusrootsappearasverticallinesinthespectraldomain,thusthetermlinespectralfrequencies(LSFs).Com-paredtoLPcoefcients,LSFsquantizewellandencodeexactlythesameinformation.Therefore,LSFsarepredominantwhenparameterizinganalysisandsynthesisltersinlinearpredictivecodingof,e.g.speechandaudio.ByexploitingpropertiesofLSPpolynomials'roots,e.g.theyoccurincomplexconjugatepairsandinterlaceontheunitcircle,LSFestimationisdonein1andveryefcienttech-niquesexist.See,e.g.[5]foraneffectiverootestimatorbasedonaChebyshevseriesformulationoftheLSPpolynomials.Matlabc\rincorporatefunctionalitytoconvertLPcoefcientstoLSFsandback;thefunctionsarepoly2lsfandlsf2polyrespectively.IntrinsicMatlabc\rfunctionsare,inthispaper,setintypewriterfont.Bothofthesemethodsarebasedonworkpresentedin[5].However,eveninwellconditionedcases,signicantper-tubationsmayoccurwhencomputingpolynomialcoefcientsfromroots,e.g.LPcoefcientsfromLSFs.Thismaycomeasasurprise,astheprocedureappearssimpleandstraightfor-ward,butroundingerrorstendtoaccumulate.Thepresentpa-perdemonstrateshowtheaccumulationofroundingerrorscanbesuppressedbyorderingtheLSPpolynomialrootspriortocomputingtheLPcoefcients.TheorderingschemeisknownasLejaorderingduetothePolishmathematicianFranciszekLeja,cf.[6].Lejaordering,whichisintime(n2)[7],isnotconductedaspartoflsf2poly.Inspiredby[8],aMatlabc\rimplementationofLejaorderinghasbeenpublishedin[9];theimplementationisutilizedinthepresentpaper.Thepapers[10,11]alsocontaininterestinginsightsintosuppres-sionofroundingerroraccumulationbyLejaorderingandLejasequences.TheresultsofthecurrentpapershowthatbyintroducingtheLejaordering,therootmeansquaredeviation(RMSD)betweentrueandestimatedLPcoefcientvectorsisintheneighborhoodofmachineepsilon,",uptothetestedmaximumcoefcientvectorlengthof160.Inthispaper,"2:22
1016,i.e.IEEE754doubleprecision.TheseresultsaresignicantlybetterthanwhatisobtainedwhenestimatingthecoefcientswithoutpriorLejaorderingwithlsf2poly.EspeciallywithoutLejaorder-ing

2 ,theRMSDincreasesasafunctionofLPcoefcie
,theRMSDincreasesasafunctionofLPcoefcientvectorlength,andforlengthsbeyond50,theroundingerrorsaccumu-latetosuchanextentastodominateLPcoefcientestimations.Theremainderofthispaperisorganizedasfollows.Sec-tion2introducesthepreliminariesofthestudy,i.e.LSPpoly-nomials,LSPdecomposition,LSF,andLejaordering.Section3presentstheproposedmethod,andinsection4themethodistestedwithregardtoLPcoefcientestimationaccuracy.Sec-tion5presentstheresultsofthetestsconductedinthenumericalexperiment,andintheclosingsection,section6,theresultsarediscussedalongwithfutureperspectives.2.PreliminariesThissectionintroducesthepreliminariesofthecurrentpaper,i.e.LSPpolynomials,LSPdecomposition,LSFandLejaorder-ing.2.1.LinespectrumpairpolynomialsDecomposingthedenominatorpolynomialofaLPmodelintoLSPpolynomialsisoftenreferredtoasLSPdecomposition.Tointroducethedecomposition,thefollowingdenitionisuseful.Denition1Palindromicandantipalindromicpolynomial.Arealpolynomial,(x)=PNn=0nxn,ispalindromiciffnNnandanti-palindromiciffnNn.Notethefollowingpropertiesof(anti-)palindromicpolynomi-als;cf.[2,3,4]forelaboratepresentationsofLSPpolynomialproperties. Property1(Anti-)palindromicpolynomials.1:Everyrealpolynomialthathasallofitsrootsontheunitcircleiseitherpalindromicoranti-palindromic.2:Conversely,noteverypalindromicoranti-palindromicpolynomialhasallitsrootsontheunitcircle.InLSPdecomposition,theideaistodene(anti-)palindromicpolynomialswithallrootsontheunitcircle,cf.property1,2.Denition2LSPdecomposition.Anyrealpolynomial,(x),oforderNcanbestatedasthesumofapalindromicpolynomial,p(x),andananti-palindromicpolynomial,q(x):(x)=1 2(p(x)+q(x))wherep(x)=(x)+xN+1(x1)q(x)=(x)xN+1(x1)TheLSPdecompositionisbijectiveandthepolynomialsp(x)andq(x)arereferredtoasLSPpolynomials.Notablepropertiesof(x),p(x),andq(x),provedin[3],are:Property2LSPpolynomials.1:Ifalltherootsof(x)areinsidetheunitcircle,thenalltherootsofp(x)andq(x)areinterlacedontheunitcircle.2:Conversely,iftherootsoftworealpolynomialsofthesameorder,onepalindromicandoneanti-palindromic,e.g.p(x)andq(x),areinterlacedontheunitcircle,thentheirsum,e.g.(x),alwayshasallitsrootsinsidetheunitcircle.Polynomialp(x)hasarealrootat-1,andq(x)hasarealrootat1;allotherrootsoccurincomplexconjugatepairs.Arootvector,e.g.of(x),isdenotedby=[1;2;:::;N]T2N2.2.LinespectrumfrequencyAsLSPpolynomials'rootslieontheunitcircletheycanun-ambiguouslybeexpressedbytheirarguments,i.e.frequencies.Thisleadstothefollowingdenition:Denition3Linespectrumfrequencies.LSFsarethearguments(frequencies)ofLSPpolynomials'roots.SinceLSPpolynomials'rootsoccurincomplexconjugatepairs,exceptforthetworealrootsat1,itsufcestodeterminetheLSFsontheupperhalfunitcircle,i.e.intheinterval]0;[.See[5]foraneffectiverootestimatorbasedonaChebyshevseriesformulationoftheLSPpolynomials.ALSFvector,e.g.fortheN'thorderpolynomial(x),thatleavesroomforthetworealroots'argumentscanbedenotedby=[1;2;:::;N+2]T2N+2Therootsofapolynomialdenethepolynomial'scoefcientsuptoscaling.Hence,theLSFsdenetheLSPpolynomials,p(x)andq(x),whichinturndenetheLPmodel'sdenomina-torpolynomial,(x),cf.denition2.Further,stabilityoftheestimatedLPmodelisensuredas(x)willhaveallrootsin-sidetheunitcircle,cf.property2.ThisillustratesthepathofconversionfromLSFstoLPcoefcients.Rootsontheunitcirclecanunambiguouslyberepresentedbytheirarguments(frequencies)andappearinthespectraldo-mainasverticallines;hence,thenamelinespectralfrequencies.2.3.LejaorderingLejaorderingprovesusefulwhenthecoefcientsoftheLPmodel'sdenominatorpolynomial,(x),aretobedeterminedaccuratelyfromtheLSFs.Intheory,themappingbetweenLPcoefcientsandLSFsisbijectiveuptoorderandscaling,butinnumericalcomputations,accumulationofroundingerrorsc

3 anbecomedetrimental.Inthepresentpaper,Le
anbecomedetrimental.Inthepresentpaper,Lejaorderingiscon-sideredasaremedytoalleviateroundingerroraccumulationintheLSFstoLPcoefcientsconversion.Denition4WeightedLejaordering[8]jnjn1Yi=1jnijmaxnlNjljn1Yi=1jlijforn=1;2;:::;N.Fortherstroot,n=1,theequationreducestoj1jmax1lNjljForthesecondroot,n=2,theequationisj2j
j21jmax2lNjlj
jl1jExample1illustratesthatLejaorderingisnotuniqueasthemaximizationmayyieldmorecandidates.Example1Lejaorderingin1in=[1;2;3;4;5]1=5,2=2 _3,3=4 _1,4=1 _4,5=3 _2 out=[5;2;4;1;3]out=[5;3;1;4;2]Already,implementationsoftheLejaorderingschemeexist,e.g.aMatlabc\rimplementation-inspiredby[8]-ispublishedin[9].Theorderingisintime(n2)[7].3.ProposedmethodTable1outlinestheproposedmethodandalgorithmforestimat-ingLPcoefcientsfromLejaorderedLSFs.Therealcoefcientvectorsfor(x),p(x)andq(x)aredenotedbya,pandqre-spectively.RootvectorsaredenotedbyandLSFvectorsby.TheLPpolynomial,(x),isoforderN. 1Formunit-modulusroots22Nfrom2N.Complexconjugatesincluded.2De-interlaceandformp;q2N,cf.prop.2.3Lejaorderpandq,cf.def.4.4Computecoefcientsap;aq2N+1byexpandingNYn=1(xp;n)andNYn=1(xq;n)5Convolverealroots1intoaqandaprespectively.6Computea1 2(apaq)2N+1,cf.def.2. Table1:Outlineoftheproposedmethodandalgorithm. Thesignicantdifferencebetweenlsf2polyandthepro-posedmethodistheLejaordering,i.e.step3intable1.Theorderingisintime(n2)[7].4.Numericalexperiment4.1.ExperimentsetupInthisnumericalexperiment,theproposedmethod,cf.table1,iscomparedtolsf2poly.AsLejaorderingisappliedtoreduceroundingerroraccumulation,theobjectiveoftheexper-imentistomeasurepotentialimprovementsinaccuracy.Thisisdonebyevaluatingtherootmeansquaredeviation(RMSD)betweentrueLPcoefcientvectorsandvectorsestimatedwithlsf2polyandtheproposedmethod.4.2.DatamaterialThedatamaterialisgeneratedbyconvertingLPcoefcientvec-torsintoLSFvectors.Thecoefcientvectorsarerandomizedandrangeinlength.Theorem1isemployedincoefcientvec-torgenerationtoensurethatallrootsoftheLPmodel'sdenom-inatorpolynomial,(x),lieinsidetheunitcircle.Now,thees-timationscanbecomparedwiththetrueLPcoefcientvectors.Theorem1Enestr¨om-Kakeya[12]If(x)=NXn=0nxnwith01:::N0;thenalltherootsof(x)lieoutsidetheopenunitdisk.Con-versely,ifNN1:::00,thenalltherootsof(x)lieintheclosedunitdisc.Minimumphasesequencesaregenerated,i.e.NN1:::00allin,byreversingthecoefcientorderingofN=1,N1iNir,i2[0;N1],andmakingthepolynomialmonic.TheuniformdistributionisdenotedbyUandrU[0;1].Figure1exempliesacoefcientvectorandthepertainingLSFs.Intheupperpanel,thecoefcientsareorderedindescendingpowers,i.e.howMatlabc\rorderspolynomialcoefcients. 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 Coefficient index [1] a [1] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 RealImaginary Figure1:Upper:Coefcientvector,a,fromthedataset,N80.Lower:LSFscomputedfromtheexamplevectorabove.5.ResultsIngure2,atypicalexampleoferrorbetweenthetrueandtheestimatedcoefcientvectorsisillustrated.Thedatasetisasin-gleminimumphasesequenceoflength80. 0 10 20 30 40 50 60 70 80 -1000 -500 0 500 1000 Coefficient index [1] a- u [1] 0 10 20 30 40 50 60 70 80 -4 -2 0 2x 10-14 Coefficient index [1] a- o [1] Figure2:Typicalinstanceoferrorbetweentrue,a,andesti-matedcoefcientvectors.Upper:WithoutLejaordering,u.Lower:WithLejaordering,o.Figure3illustratestheRMSDbetweentrueandestimatedco-efcientvectors.Thedatasetconsistsof31minimumphasesequencesthatrangeinlength50-80.Therangehasbeencho-sentoillustratetheabruptaccumulationofinaccuracieswhenLSPpolynomialrootsarenotLejaordered. 50 55 60 65 70 75 80 0 50 100 150 200 Sequence length [1]RMSD [1] 50 55 60 65 70 75 80 0.5 1 1.5 2x 10-14 Sequence length [1]RMSD [1]

4 Figure3:RMSDbetweentrueandestimatedcoef
Figure3:RMSDbetweentrueandestimatedcoefcientvec-tors.Upper:WithoutLejaordering.Lower:WithLejaorder-ing.Table2liststhemeanandstandarddeviationof14RMSDpopulations.Foreachcoefcientsequencelength,N40;60;:::;160,theRMSDsbetween50trueandestimatedco-efcientvectorsareobtained.Again,estimationsaredonewithandwithoutLejaordering. WithoutLejaorderingWithLejaorderingFormat:;Format:; N=402:27
1081:00
1084:90
10151:71
1015N=602:08
1039:46
1049:62
10153:12
1015N=801:73
1027:04
1011:63
10145:03
1015N=1001:93
1076:99
1062:10
10146:71
1015N=1201:70
10127:65
10112:56
10146:92
1015N=1401:82
10177:97
10163:20
10141:02
1014N=1602:03
10227:73
10214:78
10141:67
1014 Table2:PopulationmeanandstandarddeviationofRMSD.Thepopulationsizeis50foreachN.6.DiscussionTheresultsexpressdifferencesinroundingerroraccumulationwithandwithoutLejaorderingtheLSPpolynomialroots.Itisevident,cf.gure3andtable2,thattheerrorsaccumulatetosuchanextentastodominatetheLPcoefcientestimationswhenLejaorderingisnotapplied.Quantizationorroundinger-rorsareunavoidablewheneveracontinuousspaceisdiscretizedtoallowfornumericalevaluation.Inthepresentpaper,thecom-putationshavebeendoneinMatlabc\rusingIEEE754doubleprecisionoatingpointnumbers,i.e.themachineepsilonis"=2522:22
1016.Thatis,between2nand2n+1thenumbersareequispacedwithincrementsof2n52;asnin-creases,thespacingincreases.Thespacingbetween1and2,i.e.n=0,yields".Theroundingprocedureisround-to-nearestandround-half-up.Hence,themaximumrelativeerrorinducedbyroundingtheresultofasinglearithmeticoperationis"=2broadly,theroundinglevelisabout16decimaldigits.Inspeechprocessing,theorderofalinearpredictivemodelistypically10-12.Tot4resonantpeaks,i.e.formants,8polesarerequired;afewextrapolesmayincreasemodellingaccu-racy.However,thedecreaseinpredictionerrorasfunctionofmodelorderisnotpronouncedbeyondorder10-12.Fromthispracticalviewpoint,theresultsinthepresentpaperaremostlyoftheoreticalinterest.7.References[1]F.Itakura,“Linespectrumrepresentationoflinearpredictorco-efcientsofspeechsignals,”JournaloftheAcousticalSocietyofAmerica,vol.57,pp.S35(A),April1975.[2]HansW.Sch¨ussler,“Astabilitytheoremfordiscretesystems,”IEEETransactionsonAcoustics,SpeechandSignalProcessing,vol.24,pp.87–89,Feb.1976.[3]FrankK.SoongandBiing-HwangJuang,“Linespectrumpair(LSP)andspeechdatacompression,”inAcoustics,Speech,andSignalProcessing.IEEE,1984,vol.9.[4]T.B¨ackstr¨omandC.Magi,“Propertiesoflinespectrumpairpoly-nomials-Areview,”SignalProcessing,vol.86,no.11,pp.3286–3298,February2006.[5]PeterKabalandRaviPrakashRamachandran,“ThecomputationoflinespectralfrequenciesusingChebyshevpolynomials,”IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.34,no.6,pp.1419–1426,Dec.1986.[6]F.Leja,“Surcertainessuitesli´eesauxensemblesplanetleurapplicationalarepr´esentationconforme,”Ann.Polon.Math.,vol.4:8-13,1957.[7]StefanoDeMarchi,“OnLejasequences:Someresultsandap-plications,”Elsevier,AppliedMathematicsandComputation,vol.152,pp.621–647,2004.[8]No¨elM.Nachtigal,LotharReichel,andLloydN.Trefethen,“AhybridGMRESalgorithmfornonsymmetriclinearsystems,”SIAM,JournalofMatrixAnalysisandApplications,vol.13,no.3,pp.796–825,July1992.[9]MarkusLangandBernhard-ChristianFrenzel,“Anewandef-cientprogramforndingallpolynomialroots,”Tech.Rep.TR93-08,RiceUniversityECE,Houston,TX,USA,1993.[10]LotharReichel,“TheapplicationofLejapointstoRichardsonit-erationandpolynomialpreconditioning,”Elsevier,Linearalgebraanditsapplications,vol.154-156,pp.389–414,Aug-Oct1991.[11]J.Baglama,D.Calvetti,andL.Reichel,“FastLejapoints,”Elec-tronicTransactionsonNumericalAnalysis,vol.7,pp.124–140,1998.[12]P.BorweinandT.Erd´elyi,PolynomialsandPolynomialInequal-ities,Springer,1995.