Christensen, Mads Græsbøll; Vera-Candeas, Pedro; Somasundaram, Samuel D.; Jakobsson, Andreas

Transcription

1 Dwladed frm vb.aau.dk : April 12, 2019 Aalbrg Uiversitet Rbust Subspace-based Fudametal Frequecy Estimati Christese, Mads Græsbøll; Vera-Cadeas, Pedr; Smasudaram, Samuel D.; Jakbss, Adreas Published i: Prceedigs f the IEEE Iteratial Cferece Acustics, Speech ad Sigal Prcessig DOI (lik t publicati frm Publisher): /ICASSP Publicati date: 2008 Dcumet Versi Accepted authr mauscript, peer reviewed versi Lik t publicati frm Aalbrg Uiversity Citati fr published versi (APA): Christese, M. G., Vera-Cadeas, P., Smasudaram, S. D., & Jakbss, A. (2008). Rbust Subspace-based Fudametal Frequecy Estimati. Prceedigs f the IEEE Iteratial Cferece Acustics, Speech ad Sigal Prcessig, Geeral rights Cpyright ad mral rights fr the publicatis made accessible i the public prtal are retaied by the authrs ad/r ther cpyright wers ad it is a cditi f accessig publicatis that users recgise ad abide by the legal requiremets assciated with these rights.? Users may dwlad ad prit e cpy f ay publicati frm the public prtal fr the purpse f private study r research.? Yu may t further distribute the material r use it fr ay prfit-makig activity r cmmercial gai? Yu may freely distribute the URL idetifyig the publicati i the public prtal? Take dw plicy If yu believe that this dcumet breaches cpyright please ctact us at vb@aub.aau.dk prvidig details, ad we will remve access t the wrk immediately ad ivestigate yur claim.

2 ROBUST SUBSPACE-BASED FUNDAMENTAL FREQUENCY ESTIMATION Mads G. Christese 1, Pedr Vera-Cadeas 2, Samuel D. Smasudaram 3, ad Adreas Jakbss 3 1 Dept. f Electric Systems Aalbrg Uiversity, Demark 2 Telecm. Egieerig Dept. Uiversity f Jaé, Spai 3 Dept. f Electrical Egieerig Karlstad Uiversity, Swede ABSTRACT The prblem f fudametal frequecy estimati is csidered i the ctext f sigals where the frequecies f the harmics are t exact iteger multiples f a fudametal frequecy. This frequetly ccurs i audi sigals prduced by, fr example, stiff-striged musical istrumets, ad is smetimes referred t as iharmicity. We derive a vel rbust methd based the subspace rthgality prperty f MUSIC ad shw hw it may be used fr aalyzig audi sigals. The prpsed methd is bth mre geeral ad less cmplex tha a straight-frward implemetati f a parametric mdel f the iharmicity derived frm a physical istrumet mdel. Additially, it leads t mre accurate estimates f the idividual frequecies tha the methd based the parametric iharmicity mdel ad a reduced bias f the fudametal frequecy cmpared t the perfectly harmic mdel. Idex Terms Acustic sigal aalysis, spectral aalysis, frequecy estimati 1. INTRODUCTION The prblem f estimatig the fudametal frequecy, r pitch perid, f a set f harmically related siusids is e f the classical prblems f sigal prcessig, t least i speech ad audi prcessig where it is imprtat t may applicatis ragig frm aalysis ad cmpressi t separati ad ehacemet. I recet years, it has als fud ew applicatis i music ifrmati retrieval ad it remais a active research tpic. I the ideal case, the frequecies f the harmics are iteger multiples f the fudametal frequecy. Fr may musical istrumets, thugh, the frequecies are t exact iteger multiples f the fudametal. This pheme is kw as iharmicity ad this is the prblem we are ccered with i this paper. Fr particular istrumets, like stiff-striged istrumets such as the pia, the iharmicity is very pruced ad has t be take it accut whe tuig the istrumet [1, 2]. Als fr speech sigals, iharmicity has bee bserved t be f imprtace fr mdelig ad cdig purpses (see, e.g., [3]). There are may reass why this iharmicity shuld be take it accut whe estimatig the fudametal frequecy. Firstly, the assumpti f the frequecies f the harmics beig exact iteger multiples f the fudametal may lead t a bias i the estimated fudametal frequecy. It may als lead t sigificat bias f estimated amplitudes sice the iteger multiples may t capture the peaks f the spectrum. This, i tur, may lead t audible artifacts whe resythesizig the audi. Similarly, biased estimates ca result i a bad fit f the sigal mdel t the data ad lw likelihds, causig icrrect mdel ad rder selectis. Therefre, e wuld expect The wrk f M. G. Christese is supprted by the Parametric Audi Prcessig prject, Daish Research Cucil fr Techlgy ad Prducti Scieces grat a fudametal frequecy estimatr that takes this pheme it accut t be mre rbust tha the estimatrs based the idealized mdel. We will w prceed t defie the prblem at had mathematically. We are here ccered with a set f siusids havig frequecies {ω l } crrupted by a additive white cmplex circularly symmetric Gaussia ise, ǫ(), fr = 0,..., N 1, x() = α l e jωl + ǫ(), (1) where {α l } are the cmplex amplitudes (here csidered uisace parameters); these may easily be fud give the frequecies. I this wrk, it is assumed that the umber f harmics L is kw r fud a priri. Fr the perfectly harmic case, the frequecies f the harmics are exact iteger multiples f a fudametal frequecy ω 0, i.e., ω l = ω 0l with l N, i which case the sigal mdel i (1) is characterized by a sigle liear parameter. As already discussed, this mdel is t always a gd fit. Depedig the istrumet, differet parametric mdels f the iharmicity f the harmics ca be derived frm physical mdels (see, e.g., [4]). A example f such a mdel fr stiff-striged istrumets is ω l = ω 0l 1 + Bl 2 where B 1 is a ukw, psitive stiffess parameter. I this case, the mdel i (1) w ctais tw ukw liear parameters, amely ω 0 ad B, which cmplicates matters. This mdel, which we will refer t as the parametric mdel f the iharmicity, has bee used i Bayesia fudametal frequecy estimati fr audi aalysis i [5] ad siusidal audi cdig i [6]. Recetly, it has bee shw that the subspace rthgality priciple kw frm the MUltiple SIgal Classificati (MUSIC) estimati methd (see, e.g., [7]) ca be used fr fidig the fudametal frequecy ad the mdel rder L jitly fr bth a sigle surce [8] ad multiple surces [9]. I this paper, we csider the prblem f takig iharmicity it accut i a methd based the rthgality prperty. We exted the methd i [8] t the parametric mdel f the iharmicity but this leads t a cmputatially cmplex methd. Istead, we use a alterative mdel that at first sight will appear mre cmplicated ad derive a vel rbust estimatr. Mre specifically, we use a mdel where the frequecies {ω l } are mdeled as ω l = ω 0l + l with { l } beig a set f small ukw perturbatis that are t be estimated alg with the fudametal frequecy ω 0. We refer t this frequecy mdel as the perturbed mdel. The perturbatis { l } shuld be small sice arbitrarily large perturbatis will result i meaigless estimates f ω 0. Such ustructured perturbatis have als bee previusly used i a Bayesia framewrk [10]. Oe wuld expect that the itrducti f additial L liear parameters i the mdel (1) wuld result i a mre cmplicated estimatr, but, as it turs ut, the resultig estimatr is, csiderig the cmplexity f the prblem, rather simple. Aside frm the cmputatial cmplexity, there is ather imprtat reas why e wuld prefer the perturbed mdel ver the

3 parametric e, amely that it is mre geeral, meaig that it ca be expected t hld fr a mre geeral class f sigals. Fr example, the parametric iharmicity mdels are t the same fr strigs with clamped r pied eds (see [4]). While it is here assumed that the mdel rder L i (1) is kw, it ca i fact be fud jitly with the ther liear parameters usig the prpsed methd (c.f. [8]), but this is beyd the scpe f this paper. The rest f this paper is rgaized as fllws. First, i Secti 2, we will review the fudametals f the cvariace matrix mdel ad the subspace rthgality prperty ad shw hw this ca be used fr fidig the fudametal frequecy ad stiffess parameters f the parametric iharmicity mdel. I Secti 3, we the preset the ew rbust estimatr assciated with the perturbed mdel. Sme experimetal results ad examples are give i Secti 4 befre the cclusi i Secti SOME PRELIMINARIES I this secti, we preset the fudametals f the MUSIC algrithm (see, e.g., [7]) ad itrduce sme useful vectr ad matrix defiitis. We will d this fr the sigal mdel i (1) based a set f ucstraied frequecies {ω l } ad the shw hw this ca be applied t the parametric iharmicity mdel. We start ut by defiig x() as a sigal vectr ctaiig M samples f the bserved sigal, i.e., x() = [ x() x( + M 1) ] T with ( ) T detig the traspse. The, assumig that the phases f α l are idepedet ad uifrmly distributed the iterval ( π, π], the cvariace matrix R C M M f the sigal i (1) ca be writte as R = E x() x H () = AVA H + σ 2 I, (2) where E { } ad ( ) H dete the statistical expectati ad the cjugate traspse, respectively. Nte that fr this decmpsiti t hld, the ise eed t be Gaussia. Furthermre, V is a diagal matrix ctaiig the squared amplitudes, i.e., V = diag([ α 1 2 α L 2 ]), ad A C M L a rak L Vadermde matrix, i.e., A = ˆ a(ω 1) a(ω L), (3) where a(ω) = ˆ 1 e jω e jω(m 1) T. Als, σ 2 detes the variace f the additive ise, ǫ(), ad I is the M M idetity matrix. We als te that AVA H has rak L. Fr tatial simplicity, we have mitted the depedecy f A the ukws. Let R = UΛU H be the eigevalue decmpsiti (EVD) f the cvariace matrix. The, U ctais the M rthrmal eigevectrs f R, i.e., U = [ u 1 u M ] ad Λ is a diagal matrix ctaiig the crrespdig eigevalues, λ k, with λ 1 λ 2... λ M. Let G be frmed frm the eigevectrs crrespdig t the M L least sigificat eigevalues, i.e., G = [ u L+1 u M ]. The ise subspace spaed by G will the be rthgal t the Vadermde matrix A, i.e., A H G = 0 which is what we refer t as the subspace rthgality prperty. I practice, ly a estimate f the cvariace matrix is available frm which we ca btai a estimate f the ise subspace G. Frm this matrix, a estimate f the these parameters ca be btaied by miimizig the Frbeius rm as J({ω l }) = Tr A H GG H A, (4) with Tr{ } detig the trace. Based this cst fucti, the fudametal frequecy ω 0 ad the stiffess parameter B f the parametric iharmicity mdel ca be estimated i a straight-frward maer. Specifically, the A matrix is cstructed frm the tw parameters as A = ˆ a(ω B) a(ω0l 1 + BL 2 ). (5) Based (4), we the suggest t btai a estimate f the fudametal frequecy ad stiffess parameter as (ˆω 0, ˆB) = arg mi J(ω0, B), (6) ω 0,B which has t be evaluated fr a large rage f cmbiatis f the tw parameters. We te that the estimatr (6) has t previusly appeared i the literature. 3. ROBUST SUBSPACE METHOD The questi is w hw the fudametal frequecy ad the idividual frequecies ca be fud fr the perturbed mdel where ω l = ω 0l + l, i.e., the Vadermde matrix ctaiig the cmplex siusids is w characterized by ω 0 ad { l } as A = ˆ a(ω 0 + 1) a(ω 0L + L). (7) Hwever, direct miimizati f the cst fucti i (4) will t ly be very cmputatially demadig sice we w have L+1 liear parameters, but it will als t lead t ay meaigful estimates f the parameters sice we have ctrl ver the distributi f the perturbatis { l }. Istead, we prpse t estimate the parameters by redefiig the cst fucti i (4) as J(ω 0, { l }) = Tr A H GG H A + P({ l }), (8) where P({ l }) is the pealty fucti which is a -decreasig fucti f a metric with P({0}) = 0. Als, it is desirable that the pealty fucti is additive ver the harmics. Therefre, a atural chice is P({ l }) = P L ν l l p with p 1 which pealizes large perturbatis l. Als, {ν l } is a set f psitive regularizati cstats, the meaig f which will be discussed later. I arrivig at a cmputatially efficiet estimatr, we first te that the Frbeius rm is additive ver the clums f A, i.e., J(ω 0, { l }) = Tr A H GG H A + = ν l l p (9) a H (ω 0l + l )GG H a(ω 0l + l ) + ν l l p. (10) Furthermre, by substitutig ω l by ω 0l + l ad l by ω l ω 0l i (10), we get the fllwig simplified cst fucti J(ω 0, {ω l }) = a H (ω l )GG H a(ω l ) + ν l ω l ω 0l p. (11) It ca be see that the first term lger depeds the fudametal frequecy r the perturbatis but ly the frequecy f the l th harmic ω l. Furthermre, we ca recgize the first term i (11) as the reciprcal f the MUSIC pseud-spectrum, which has w t be calculated ly ce fr each segmet. It ca als be see that the cst fucti is additive ver idepedet terms ad, therefre, the miimizati f the cst fucti ca be perfrmed idepedetly fr each harmic. The fudametal frequecy estimatr

4 ca thus be rewritte as ( L ) X ˆω 0 = arg mi mi a H (ω l )GG H a(ω l ) + ν l ω l ω ω 0l p 0 {ω l } = arg mi ω 0 mi ω l a H (ω l )GG H a(ω l ) + ν l ω l ω 0l p, where the frequecies {ω l } ad thus perturbatis are als fud implicitly. Frm this, it is w als clear why we required the pealty fucti t be additive ver the harmics. We term this estimatr rbust sice it expected t be mre rbust twards mdel mismatch tha the ideal mdel ad the parametric iharmicity mdel. Fr a give ˆω 0, the frequecies ca simply be fud fr l = 1,..., L as ˆω l = arg mi ω l a H (ω l )GG H a(ω l ) + ν l ω l ˆω 0l p, (12) with ly the pealty term chagig ver l. I the ctext f statistical estimati, the augmetati f a lg-likelihd fucti by such a pealty term will result i a maximum a psteriri estimate with the pealty term beig a lg-prir the perturbatis with a implicit uifrm prir the fudametal frequecy. Fr a Gaussia prir, fr example, we wuld have p = 2 ad ν l = 1/(2σ 2 l ) with σ 2 l beig the variace f the l th harmic. The meaig f the regularizati cstats ca als be clarified frm the fllwig: Fr large ν l, the esuig perturbati will be small ad the estimatr is expected t reduce t the perfectly harmic case, whereas fr ν l clse t zer, the estimatr will reduce t fidig ucstraied frequecies frm which meaigful fudametal frequecy estimate ca be fud. The regularizati cstats ν l ca als be iterpreted as Lagrage multipliers. This meas that the estimatr ca be thught f as a cstraied estimatr with a set f implicit cstraits. I this sese, the methd is cceptually related t the rbust Cap beamfrmer f [11] which is based explicit cstraits. It may be wrth mdifyig the pealty such that less emphasis is put perturbatis fr higher harmics r eve use a asymmetrical pealty sice the parametric iharmicity mdel suggest that the harmics will be higher tha the iteger multiple f the fudametal, but fr w we will defer frm further discussi f this. 4. EXPERIMENTAL RESULTS We will w apply the estimatrs t aalysis f audi sigals. Sice we assume that the rder L is kw, we will fcus a sigle te, amely a pia te C Hz whse spectrgram ad time-dmai sigal are shw i Figure 1. I the experimets, the fllwig cditis are used. The sigal has bee dw-sampled t 8820 Hz t reduce the cmputatial cmplexity. The estimates are btaied i the fllwig way. First, segmets csistig f 30 ms have bee used frm which the discrete-time aalytic sigal ad a cvariace matrix are calculated. The EVD f this matrix is cmputed ad partitied it sigal ad ise subspace eigevectrs with L = 7. The, the cst fuctis f the respective estimatrs are calculated fr a wide rage f ω l (ad B fr the parametric iharmicity mdel) frm which carse estimates are btaied. Fially, these are used t iitialize gradiet-based refiemet methds. Fr the prpsed methd, we use p = 1 (crrespdig t a Laplacia prir) sice this is expected t result i mst perturbatis beig clse t zer while allwig fr a few large es. The regularizati cstats with ν l 1/l have bee used such that larger perturbatis are allwed fr higher harmics. The exact perturbati cstats were determied empirically fr a large set f sigals. We Frequecy [Hz] Pseudspectrum Fig. 1. Spectrgram ad time-dmai sigal f a pia te Frequecy [Hz] Fig. 2. MUSIC (lg) pseud-spectrum f a segmet f the pia te alg with the estimated frequecies f the harmics fr the perfectly harmic mdel (circles), the parametric iharmicity mdel (squares) ad usig the perturbed mdel (crsses). te that it is geerally safer t chse ν l t large tha t small but ther tha that, we have bserved the chice f ν l t t be critical. I the first experimet, we will illustrate the ability f the mdels ad estimatrs t capture the frequecies f the idividual harmics f the pia te. Fr this sigal, the parametric iharmicity mdel is expected t perfrm well. I Figure 2, the lgarithm f the MUSIC pseud-spectrum is shw fr a represetative statiary 30 ms segmet f the sigal i Figure 1 alg with the frequecy estimates btaied usig the perfectly harmic mdel, the parametric iharmicity mdel ad the perturbed mdel. It ca be see that the frequecies btaied usig the perfectly harmic mdel are biased with the ω 0 estimate beig Hz. It ca als be see that the parametric iharmicity mdel captures the peaks fairly accurately resultig i a ω 0 estimate f Hz. The perturbed mdel, hwever, appears t capture all the peaks fr a estimated fudametal frequecy f Hz. It ca be see that the prpsed methd is able t reduce the bias f the perfectly harmic mdel while als givig accurate estimates f the idividual harmics. Next, the estimated fudametal frequecies are estimated i steps f 10 ms. The results are shw i Figure 3 fr the three methds, amely fr the

5 perfectly harmic mdel, the parametric iharmicity mdel, ad the perturbed mdel. The geeral cclusis ca be see t be the same as fr Figure 2. The parametric iharmicity mdel appears t be the mst accurate fr this particular sigal with the estimates beig clse t the Hz f the te. It ca als be see that the perfectly harmic mdel here results i a bias thrughut the durati f the sigal, ad that the perturbed mdel is able t reduce this bias. The remaiig bias is mst likely due t the use f a symmetrical pealty fucti that pealizes egative ad psitive perturbatis equally. We te that the fluctuatis f the estimates at abut 0.85 s are due t mdulatis f the 4 th harmic. Fially, we have ivestigated hw well the mdels are able t capture the frequecies f the idividual harmics i the fllwig way: Give the frequecy estimates, the mdel is fitted t the data usig leastsquares ad the sigal-t-ise rati (SNR) is calculated. I Figure 4, the results are shw. It ca be see that the perfectly harmic mdel des t fit the sigal well, ad that the prpsed methd i fact estimates the frequecies mre accurately tha the parametric mdel, whereby a better fit is btaied. 5. CONCLUSION I this paper, the prblem f rbust fudametal frequecy estimati has bee csidered fr the case where the harmics are t exact iteger multiples f a fudametal frequecy, a pheme cmmly kw as iharmicity that frequetly ca be bserved i sigals prduced by musical istrumets. A ew subspace-based methd has bee prpsed fr fidig the fudametal frequecy ad a set f small perturbatis. We have cmpared this methd t methds based a cmmly used parametric mdel f the iharmicity, which is based a physical musical istrumet mdel, ad the perfectly harmic mdel where the frequecies are exact iteger multiples f the fudametal. The prpsed methd is bth cmputatially simple ad mre geeral tha the estimatr based parametric iharmicity mdel ad has bee fud t lead t a reduced bias cmpared t the perfectly harmic mdel ad mre accurate estimates f the idividual frequecies tha bth the ther methds fr a pia sigal. 6. REFERENCES [1] H. Fletcher, Nrmal vibrati frequecies f a stiff pia strig, i J. Acust. Sc. Amer., 1964, vl. 36(1). [2] J. Lattard, Ifluece f iharmicity the tuig f a pia - measuremets ad mathematical simulati, J. Acust. Sc. Am., vl. 94, pp , [3] E. B. Gerge ad M. J. T. Smith, Speech aalysis/sythesis ad mdificati usig a aalysis-by-sythesis/verlap-add siusidal mdel, IEEE Tras. Speech ad Audi Prcessig, vl. 5(5), pp , Sept [4] N. H. Fletcher ad T. D. Rssig, The Physics f Musical Istrumets, Spriger, 2d editi, [5] S. Gdsill ad M. Davy, Bayesia cmputatial mdels fr iharmicity i musical istrumets, i Prc. IEEE Wrkshp Appl. f Sigal Prcess. t Aud. ad Acust., 2005, pp [6] F. Myburg, Desig f a Scalable Parametric Audi Cder, Ph.D. thesis, Techical Uiversity f Eidhve, [7] P. Stica ad R. Mses, Spectral Aalysis f Sigals, Pears Pretice Hall, Fudametal Frequecy [Hz] Fig. 3. Fudametal frequecies estimates btaied usig the perfectly harmic mdel (dash-dtted), the parametric iharmicity mdel (dashed), ad the perturbed mdel (slid). SNR [db] Fig. 4. Recstructi SNR btaied usig the perfectly harmic mdel (dash-dtted), the parametric iharmicity mdel (dashed), ad the perturbed mdel (slid). [8] M. G. Christese, A. Jakbss, ad S. H. Jese, Jit high-resluti fudametal frequecy ad rder estimati, IEEE Tras. Audi, Speech ad Laguage Prcessig, vl. 15(5), pp , July [9] M. G. Christese, A. Jakbss ad S. H. Jese, Multipitch estimati usig harmic music, i Rec. Asilmar Cf. Sigals, Systems, ad Cmputers, 2006, pp [10] S. Gdsill ad M. Davy, Bayesia harmic mdels fr musical pitch estimati ad aalysis, i Prc. IEEE It. Cf. Acust., Speech, Sigal Prcessig, 2002, vl. 2, pp [11] P. Stica, Z. Wag, ad J. Li, Rbust Cap beamfrmig, IEEE Sigal Prcessig Lett., vl. 10(6), pp , Jue 2003.