The Constant Q Transform Benjamin Blankertz Introduction The constant Q transform as - PDF document

TheConstantQTransform[BenjaminBlankertz]1IntroductionTheconstantQtransformasintroducedin[Brown,1991]isverycloserelatedtotheFouriertransform.LiketheFouriertransformaconstantQtransformisabankoflters,butincontrasttotheformerithasgeometricallyspacedcenterfrequencies ),wheredictatesthenumberofltersperoctave.Tomakethelterdomainsadjectantonechoosesthebandwidthofthe-thlteras
1).Thisyieldsaconstantratiooffrequencytoresolution =(2 .WhatmakestheconstantQtransformsousefulisthatbyanappropriatechoicefor(minimalcenterfrequency)andthecenterfrequenciesdirectlycorrespondtomusicalnotes.Forinstancechoosing=12andasthefrequencyofmidinote0makesthe-thcq-bincorrespondthemidinotenumberAnothernicefeatureoftheconstantQtransformisitsincreasingtimeresolutiontowardshigherfrequencies.Thisresemblesthesituationinourauditorysystem.Itisnotonlythedigitalcomputerthatneedsmoretimetoperceivethefrequencyofalowtonebutalsoourauditorysense.Thisisrelatedtomusicusuallybeinglessagitatedinthelowerregisters.2DerivingFilterParametersfortheConstantQTransformToderivethecalculationoftheconstantQtransformofsomesequencewebeginwithaninspectionofthefamiliarformulaofaFourierlterattn]e�2inz=NEachcomponentoftheconstantQtransform,inthesequelcalledcq-bin,willbecalculatedassuchalter,butsuitablevaluesforandwindowlengthhavetobefoundinordertomatchthepropertiesdiscussedabove.Thebandwidthofthelter(1)is
denotesthesamplingrate)independentlyof.Thusthedesiredbandwidth
canberealizedbychoosingawindowoflength
k=Qfs .Thefrequencytoresolutionratioofthelterin(1)is .Toachieveaconstantvalue forthefrequencytoresolutionratioofeachcq-binonehastoselectThusforintegervaluesthe-thcq-binisthe-thDFT-binwithwindowlength Summerizingwegetthefollowingrecipeforthecalculationofacon-stantQtransform:Firstchooseminimalfrequencyandthenumberofbinsperoctaveaccordingtotherequirementsoftheapplication.Themax-imalfrequencymaxonlyaectsthenumberofcq-binstobecalculatedmax :=(2 andforkKset k]:= n]wNk[n]e�2inToreducespectralleakage(cf.[Harris,1978]),itisadvisabletousethelterinconjunctionwithsomewindowfunction::n]:nNissomeanalysiswindowoflength.WefollowedJudithBrownbyusingHammingwindows.3DirectImplementationForcomparisonrsttheusualFT-algorithmisgiven,notinitscompactft=(x.*win)*exp(-2*pi*i*(0:N-1)'*(0:N-1)/N),butwithaloop.functionft=slowFT(x,N)fork=0:N-1;ft(k+1)=x(1:N)*(hamming(N).*exp(-2*pi*i*k*(0:N-1)'/N))/N;AndhereisthedirectimplementationoftheconstantQtransform.functioncq=slowQ(x,minFreq,maxFreq,bins,fs)Q=1/(2(1/bins)-1);fork=1:ceil(bins*log2(maxFreq/minFreq));N=round(Q*fs/(minFreq*2((k-1)/bins)));cq(k)=x(1:N)*(hamming(N).*exp(-2*pi*i*Q*(0:N-1)'/N))/N; denotestheleastintergergreaterthanorequalto 4AnEcientAlgorithmSincethecalculationoftheconstantQtransformaccordingtotheformula(5)isverytimeconsuming,anecientalgorithmishighlydesirable.UsingmatrixmultiplicationtheconstantQtransformofarowvectoroflengthforallkK)iswhereisthecomplexconjugateofthetemporalkernel;k n]e2in0otherwiseSincethetemporalkernelisindependentoftheinputsignalonecanspeedupsuccessiveconstantQtransformsbyprecalculating.Butthisisverymemoryconsumingandsincetherearemanynonvanishingelementsinthecalculationofthematrixproductstilltakesquiteawhile.LuckilyJudithBrownandMillerPuckettecameupwithaverycleverideaforimprovingthecalculation[BrownandPuckette,1992].Theideaistocarryoutthematrixmultiplicationinthespectraldomain.Sincethewin-dowedcomplexexponentialsofthetemporalkernelhaveaDFTthatvan-ishesalmosteverywhereexceptfortheimmediatevicinityofthecorrespond-ingfrequencythespectralkernel:=DFT()(onedimensionalDFTsappliedcolumnwise)isasparsematrix(aftereliminatingcomponentsbelowsomethresholdvalue).Thisfactcanbeexploitedforthecalculationofowingtotheidentity n]yft[n](9)whereandaresequencesoflengthanddenotetheirunnor-malizeddiscreteFouriertransform.ApplyingthisidentitytotheformulaofconstantQtransform(5)usingdenitions(7)and(8)yields NXx[n]S In(7)wecenterthelterdomainsontheleftfortheeaseofnotation.Right-centeringismoreappropriateforreal-timeapplications.Middle-centeringhastheadvantageofmakingthespectralkernel(8)real. orequivalentlyinmatrixnotation DuetothesparsenessofthecalculationoftheproductinvolvesessentiallylessmultiplicationsthanTheFouriertransformsthatariseintheecientalgorithmshouldofcoursebecalculatedusingFFTs.Tothisendischosenasthelowestpowerof2greaterthanorequalto(whichisthemaximumofall).Thecalculationofthespectralkernelisquiteexpensive,buthavingdonethisonceallsucceedingconstantQtransformsareperformedmuchfaster.5ImplementationoftheEcientAlgorithmfunctionsparKernel=sparseKernel(minFreq,maxFreq,bins,fs,thresh)ifnarginthresh=0.0054;end%forHammingwindowQ=1/(2(1/bins)-1);(3)K=ceil(bins*log2(maxFreq/minFreq));(2)fftLen=2nextpow2(ceil(Q*fs/minFreq));tempKernel=zeros(fftLen,1);sparKernel=[];fork=K:-1:1;len=ceil(Q*fs/(minFreq*2((k-1)/bins)));(4)tempKernel(1:len)=...hamming(len)/len.*exp(2*pi*i*Q*(0:len-1)'/len);(7)specKernel=fft(tempKernel);(8)specKernel(find(abs(specKernel)resh))=0;sparKernel=sparse([specKernelsparKernel]);sparKernel=conj(sparKernel)/fftLen;(10)functioncq=constQ(x,sparKernel)%xmustbearowvectorcq=fft(x,size(sparKernel,1))*sparKernel;(10)AmatlabimplementationoftheecientalgorithmfortheconstantQtransform[BrownandPuckette,1992]codedbyBenjaminBlankertz.Thehasonlytobecalculatedonce.Thismighttakesomeseconds.Afterthat,constantQtransformsofanyrowvectorcanbedonevery ecientlybycalling.Thefunctionisincludedinthedsp-toolbox.ItreturnstheHammingwindowofthegivenlength:hamming(len)=0.46-0.54*cos(2*pi*(0:len-1)'/len)ReferencesJudithC.Brown,CalculationofaconstantQspectraltransform,J.Acoust.Soc.Am.,89(1):425{434,1991.JudithC.BrownandMillerS.Puckette,Anecientalgorithmforthecal-culationofaconstantQtransform,J.Acoust.Soc.Am.,92(5):2698{2701,F.J.Harris,OntheUseofWindowsforHarmonicAnalysiswithDiscreteFourierTransform,in:Proc.IEEE,Bd.66,pp.51{83,1978.