SINGLE CHANNEL SIGNAL SEPARATION USING MAXIMUM LIKELIHOOD SUBSPACE PROJECTION

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1 th Internatinal Sympsium n ICA and BSS (ICA), April -, Nara, Japan 59 SINGLE CHANNEL SIGNAL SEPARATION USING MAXIMUM LIKELIHOOD SUBSPACE PROJECTION Gil-Jin Jang, Te-Wn Lee, and Yung-Hwan Oh Cmputer Science Department, KAIST, Daejn 5-7, Suth Krea Institute fr Neural Cmputatin, UCSD, La Jlla, CA 99-5, USA {jangbal,yhh{@speech.kaist.ac.kr, tewn@ucsd.edu ABSTRACT This paper presents a technique fr etracting multiple surce signals when nly a single channel bservatin is available. The prpsed separatin algrithm is based n a subspace decmpsitin. The bservatin is prjected nt subspaces f interest with different sets f basis functins, and the riginal surces are btained by weighted sums f the prjectins. A fleible mdel fr density estimatin allws an accurate mdeling f the distributins f the surce signals in the subspaces, and we develp a filtering technique using a maimum likelihd (ML) apprach t match the bserved single channel data with the decmpsitin. Our eperimental results shw gd separatin perfrmance n simulated mitures f tw music signals as well as tw vice signals.. INTRODUCTION Etracting multiple surce signals frm a single channel miture is a challenging research field with numerus applicatins. Varius sphisticated methds have been prpsed ver the past few years in research areas such as cmputatinal auditry scene analysis (CASA) [], and independent cmpnent analysis (ICA) []. Separatin techniques f CASA are mstly based n splitting mitures bserved as a single stream int different auditry streams, by building an active scene analysis system fr the acustic events that ccur simultaneusly in the same spectr-tempral regins. The acustic events are distinguished accrding t the rules inspired intuitively r empirically frm the knwn characteristics f the surces. Eample prpsals f CASA are auditry sund segregatin mdels based n harmnic structures f the sunds [, ], autmatic tne mdeling [5], and psych-acustic gruping rules [6]. Recently Rweis [7] presented a refiltering technique t estimate time-varying masking filters that lcalize sund streams in a spectr-tempral regin. In his wrk, surces are suppsedly disjint in the spectrgram and a mask whse value is binary, r, eclusively divides the mied streams cmpletely. These appraches are hwever able t be applied t certain limited envirnments due t the intuitive prir assumptins f the surces such as harmnic mdulatins r tempral cherency f the acustic bjects. Our wrk, while mtivated by this cncept f spectral masking, is free f the assumptin that the spectrgrams are disjint. The main nvelty f the prpsed methd is that the masking filters can have any real value in [, ]. The algrithm recvers the riginal auditry streams by searching fr the maimized lg likelihd f the separated signals, cmputed by the pdfs (prbability density functins) f the prjectins nt the subspaces. Empirical bservatins shw that the prjectin histgram is etremely sparse, and the use f generalized Gaussian distributins [8] yields a gd apprimatin. We use ICA t prvide discriminative, statistically independent subspaces. The theretical basis f this apprach is sparse decmpsitin [9]. Sparsity in this case means that nly a small number f instants in the representatin differ significantly frm zer. ICA maimizes the sparsity f the subspaces and hence reduces the verlap between the surces in the new crdinates. The paper is rganized as fllws. In sec. we define the prblem frmally and describe the prpsed separatin algrithm. We present the eperimental results fr the synthesized eamples in sec.. Finally we cnclude ur methd in sec... SEPARATION ALGORITHM We first define the single channel separatin prblem, and derive the separatin algrithm based n a mai-

2 5 Jang et al.: Single Channel Signal Separatin Using Maimum Likelihd Subspace Prjectin t y A λ k B λ k v u λ v C C ˆ ˆ Figure : Blck diagram f subspace weighting. (A) Input signal y t is prjected nt N subspace. (B) The prjectins v t i are mdulated by weighting signals λ ik (real valued between and ). (C) The separatin prcess finally terminates with summing up the mdulated signals. Figure : Illustratin f desired subspaces. The ellipses represent the distributins f tw different classes. The arrws are the -dimensinal subspaces alng the maimum energy cncentratins f the classes. Prjecting the mitures nt each subspace prvides minimum errr separatin. mum likelihd (ML) estimatin and generalized Gaussian pdf mdeling. Secnd, we eplain hw t btain statistically independent subspaces. Fr simplicity we nly cnsider the case f binary surces and - dimensinal subspaces... Subspace Decmpsitin Let us cnsider a mnaural separatin f a miture f tw signals bserved in a single channel, such that the bservatin is given by y t = t + t, t [, T ], () where t i is the tth bservatin f the i th surce. Nte that superscripts indicate sample indices f time-varying signals and subscripts indicate the surce identificatin. It is cnvenient t assume all the surces t have zer mean and unit variance. The gal is t recver all t i given nly a single sensr input y t. The prblem is t ill-cnditined t be mathematically tractable since the number f unknwns is T given nly T bservatins. The prpsed methd decmpses the surce signals int N disjint subspace prjectins u t ik, each filtered t cntain nly energy frm a small prtin f the whle space: u t ik = P( t i; w k, d k ) = N n= w kn t d k+n i, () where P is a prjectin peratr, N is the number f subspaces, and w kn is the n th cefficient f the k th crdinate vectr w k whse lag is d k. In the same manner, we define the prjectin f the mied signal as v t k = P(y t ; w k, d k ) = u t k + u t k. () Suppse the apprpriate subparts f an audi signal lie n a specific subspace ver shrt times. The separatin is then equivalent t searching fr subspaces that are clse t the individual surce signals. Mre generally, u t ik is apprimated by mdulating the mied prjectins vk t : u t k = λ k v t k, u t k = λ k v t k () λ ik [, ], λ k +λ k =, where latent variables λ ik are weights n the prjectins f subspace k, which is fied ver time. We can adapt the weights t bring prjectins in and ut f the surce as needed. The riginal surces t i are then recnstructed by recmbining u t i, ut i,..., ut in and perfrming the inverse transfrm f the prjectin. Prper chices f the weights λ ik enable the islatin f a single surce frm the input signal and the suppressin f all ther surces and backgrund nises. This apprach, illustrated in fig., frms the basis f many CASA appraches (e.g. [, 6, 7]). The results f such an analysis are ften displayed as a spectrgram shwing energy as a functin f time and frequency. Fr eample, amng musical instruments, it is pssible t distinguish simultaneusly prduced vilin and cell sunds based n the energy distributin in the spectrgram. The energy f a cell sund is usually cncentrated n the lwer bands f the spectrgram, and a vilin sund is distributed in the higher bands. The prjectins, shwn in fig. -A, act like lw- and high-pass filters in this case. Subspace weighting can als be thught f as Wiener filtering. If the riginal surces are knwn, ptimal filters can be cmputed. We might set λ ik t be equal t the rati f energy frm ne surce in subspace k t the sum f energies frm bth surces in the same subspace... ICA Subspaces A set f subspaces that effectively split target surces is essential in the success f the separatin algrithm.

3 th Internatinal Sympsium n ICA and BSS (ICA), April -, Nara, Japan 5 is nly very slightly cncentrated n bth subspaces in this case... Estimating Surce Signals (a) Prjected distributins f Furier basis (b) Prjected distributins f learned ICA basis Figure : Eample plts f subspace prjectins. Prjectins f music signals are dark pints, and thse f speech signals are bright pints. (a) Furier basis: (, y) pair is ne f the utputs f 6 nn-verlapping bandpass filters that divide the range 65-Hz equally. The center frequencies f the and y aes are at the tp f each D plt. (b) (, y) pair is ne f the prjected values f 6 subspaces btained by the ICA learning algrithm. The subspace numbers f the and y aes are displayed at the tp f each D plt. Fig. shws an eample f desired subspaces. Tw ellipses represent tw different surce distributins, whse energy cncentratins are directed by the arrws. If we prject the miture nt the arrws (-dim subspaces), the riginal surces can be recvered with the errr minimized by the principle f rthgnality. T btain an ptimal basis, we adpt ICA, which estimates the inverse-translatin-peratr such that the resulting crdinate can be statistically as independent as pssible []. ICA infers time-dmain basis filters w k generating the mst prbable utputs. The learned basis filters maimize the amunt f infrmatin at the utput, and hence they cnstitute an efficient representatin f the given sund surce. Fig. displays the distributins f subspace prjectins nt Furier and learned ICA bases fr the same data. Althugh the crdinates appear almst uncrrelated (the distributins are spread equally in all directins), there are t much verlaps in Furier subspaces between tw signals fr the weighting in eq. t be applicable. The ICA crdinates are nt nly uncrrelated but als nn-verlapping; the distributins f music signals are bradened in the ais and shrunk in the y ais, and thse f speech signals are cnfigured cnversely. At the furth clumn, the music distributin enclses the speech distributin, meaning that the energy f speech The separatin is equivalent t the estimatin f the weights λ ik, which can be accmplished by simply finding the values that maimize the prbability f the subspace prjectins. The histgrams f natural sunds reveal that p(u t ik ) is highly super-gaussian [, ]. The success f the separatin algrithm fr ur purpse depends highly n hw clsely the ICA density mdel captures the true surce cefficient density. The better the density estimatin, the mre the basis features in turn are respnsible fr describing the statistical structure. The generalized Gaussian distributin mdels a family f density functins that is peaked and symmetric at the mean, with a varying degree f nrmality in the fllwing general frm [8]: p(u) ep ( u q ), (5) where the epnent q regulates the deviatin frm nrmality. The Gaussian, Laplacian, and strng Laplacian natural sund distributins are mdeled by putting q =, q =, and q < respectively. We apprimate the lg prbability density f the prjectins, lg p(u t ik ), accrding t eq. by substituting the unknwn variable u t ik with the prjectins f the single channel bservatin vk t scaled by λ ik: lg p(u t ik) u t q ik = ik λ q ik ik vt k q ik. (6) We cnstrain that λ ik [, ] and λ k +λ k =. The value f λ k then depends slely n λ k, s we need t cnsider λ k nly. We define the bject functin Ψ k f subspace k by the sum f the jint lg prbability density f u t k and ut k, ver the time ais: Ψ k def = t lg p(ut k, ut k ) = t [lg p(ut k ) + lg p(ut k )] = λ q k k C k ( λ k ) q k C k, (7) where C ik = t vt k q ik, whse value is nnnegative and unaffected by λ k. The prblem is equivalent t cnstrained maimizatin in the clsed interval [, ]; we can find a unique slutin either at bundaries ( r ), r at lcal maimum. Fig. illustrates the different lcatins f the slutin accrding t the values f the tw epnents: (A) When bth f the epnents are psitive, Ψ k is cnve, and the slutin is lcal maimum such that Ψ k = λ q k k λ q kc k + ( λ k ) qk q k C k k =, (8)

4 5 Jang et al.: Single Channel Signal Separatin Using Maimum Likelihd Subspace Prjectin Ψ = λ q C ( λ) q C A B C D q > q > q < q < q < q > q > q < dψ/dλ = λ q q C +( λ) q q C A B Surce Training data Surce Training data Surce Testing data ICA Subspaces y Input Miture w k Prjectin weighting GGM GGM Separated Surce Testing data Separated ˆ ˆ : lcal maimum : bundary values Figure : The psitins f ptimal λ k accrding t the epnents q k and q k. (A) When bth are psitive, the bject functin Ψ k is cnve and maimized when the derivative is zer. (B) When bth are negative, the bject functin is cncave and maimized at ne f the bundaries. (C),(D) The bject functin is half cnve and half cncave. The lcal maimum and bundary values shuld be all eamined t find the slutin. which is unique in [, ] and can be fund by Newtn s methd []. (B) When bth are negative, Ψ k is cncave, and the slutin is lcated at ne f the bundaries, either r. (C)-(D) When the signs f the epnents are different, the slutin is lcated at either lcal maimum r bundaries. The cmputatin f the weighting filter fr each cmpnent is dne separately since the cmpnent pdfs are regarded independent in the ICA crdinates.. EVALUATION We nw present eperiments and results that demnstrate the perfrmance f the prpsed subspace filtering technique. We cmpare ur results with standard Furier basis results... Eperiment Setup We have tested the perfrmance f the prpsed methd n single channel mitures f fur different sund types; mnaural signals f rck and jazz music, and male and female speech. We used different sets f speech signals fr training the generalized Gaussian mdel parameters and fr generating the mitures. Fr the miture generatin, ne sentence frm each f the target speak-.5 Figure 5: The flws f training data and testing data. Training data are used t learn the ICA subspaces, and t estimate the generalized Gaussian pdf parameters fr each surce. Testing data are used t generate the mitures which are used in the separatin eperiments. ers mcpm and fdaw was selected frm the TIMIT speech database. The training sets were designed t have sentences fr each gender, each frm 7 randmly chsen males and 7 randmly chsen females. Half f the music sund was used fr training, half fr generating mitures. All signals were dwnsampled t 8kHz, frm riginal.khz (music) and 6kHz (speech). Audi files fr all the eperiments are accessible at the website... Implementatin As stated abve, the data are divided int training and testing data. The training data are used fr learning ICA subspaces, and estimating generalized Gaussian parameters (q ik, variances, etc.) fr mdeling subspace cmpnent pdfs. While learning the subspaces, all the training data f surce and surce are used t reflect the statistical prperties f bth sund surces upn the resultant subspaces. The pdf parameters are estimated separately fr each f surce and surce. The crdinate vectr w k and the generalized Gaussian epnents are used in separating the mitures the testing data. The data flws in ur eperiments are illustrated in fig. 5. The weighting filters are cmputed blck-wise; that is, we chp the input signals int blcks f fied length and assign different weighting filters fr the individual blcks. The cmputatin f the weighting filter at each blck is dne independently f the ther blcks; hence

5 th Internatinal Sympsium n ICA and BSS (ICA), April -, Nara, Japan 5 the weighting becmes mre accurate as the blck length shrinks. Hwever if the blck length is t shrt, the cmputatin becmes unreliable. We perfrmed separatin eperiments with varying blck length t find the ptimal length... Eperimental Results We generated a synthesized miture by selecting tw surces ut f the fur and simply adding them. The prpsed separatin algrithm was applied t recver the riginal surces. The similarity between the input and utput signals is measured by signal-t-nise rati (SNR), which is defined by snr(s, ŝ) [db] = lg t t s (s ŝ), where s is the riginal surce signal and ŝ its estimate. T qualify a separatin result we define a perfrmance measurement functin π as the sum f the increases in the SNR values f the tw recvered surce signals: π(ˆ, ˆ ;, ) = snr(, ˆ ) + snr(, ˆ ), Table : Calculated π values f the separatin results, with standard Furier basis. mi clumn lists the symbls f the surces that are mied t the input. (R, J, M, F) stand fr rck, jazz music, male, and female speech. The ther clumns are the evaluated π values gruped by blck lengths (in millisecnds). The filters are cmputed at every blck. The last rw is the average π. Audi files fr all the results are accessible at the prvided website. mi 5ms ms 5ms 5ms ms R + J R + M..... R + F J + M J + F M + F Average Table : Calculated π values f the separatin results, with learned ICA bases used. mi 5ms ms 5ms 5ms ms R + J R + M R + F J + M J + F M + F Average e e Figure 6: Separatin results f jazz music and male speech with blck length 5ms, ICA basis. The graphs plt the spectrgrams in lgarithmic scale, time as ais and frequency as y ais. In vertical rder the graphs in (a) and (b) are fr: riginal surces ( and ), mied signal ( + ), and the recvered signals. Only the largest % f the spectral cmpnents (in terms f magnitude) are pltted. where ˆ i and i are the recvered surces and the riginal surce signals. Table reprts the separatin results when Furier basis is used, and Table reprts the separatin results f the same data, with the learned ICA bases. The prpsed methd deals with binary mitures nly. The pssible pairs are {(R,J), (R,M), (R,F), (J,M), (J,F), (M,F)}, where the symbls R, J, M, F stand fr rck and jazz music, male and female speech. The perfrmances are measured by π values gruped by blck length, and the ptimal blck length was 5ms in bth cases. With learned ICA subspaces, the perfrmances were imprved mre than db n the average in all blck lengths, cmpared t Furier basis. In terms f the sund surce types, generally mitures cntaining music were recvered mre cleanly than the male-female miture. Fig. 6 plts the spectrgrams f the riginal surces and the recvered results fr the miture f jazz music and male speech. The recvered signals lk very similar t the riginal surces.. CONCLUSIONS We have presented a nvel single channel signal separatin algrithm based n subspace decmpsitin and maimum likelihd filtering. The riginal surce sig-

6 5 Jang et al.: Single Channel Signal Separatin Using Maimum Likelihd Subspace Prjectin 5. REFERENCES e e Figure 7: Separatin results f male and female speech with blck length 5ms, ICA basis. The graphs plt the spectrgrams in lgarithmic scale, time as ais and frequency as y ais. In vertical rder the graphs in (a) and (b) are fr: riginal surces ( and ), mied signal ( + ), and the recvered signals. Only the largest % f the spectral cmpnents (in terms f magnitude) are pltted. nals are recvered by prjecting the input miture nt the given subspaces, mdulating the prjectins, and re-cmbining the prjected signals. The mdulatin filters are btained by the ML estimatin derived by the generalized Gaussian epansin f the prjectin pdf. The subspaces learned by the ICA algrithm achieve gd separatin perfrmance. Eperimental results shwed successful separatins f the simulated mitures f rck and jazz music, and male and female speech signals. The prpsed methd has additinal ptential applicatins including suppressin f envirnmental nise fr cmmunicatin systems and hearing aids, enhancing the quality f crrupted recrdings, and preprcessing fr speech recgnitin systems. On the theretic end, We are currently wrking n alternative appraches t pdf mdeling and bject functin, enhancing discriminacy f the subspaces, and ptimizing the ML estimatin twards real-time prcessing. On the practical end, we are interested in cmparing ur methds t ther single channel denising methds. Nte that mst denising methds are nt suitable fr cmparisn since they are nt suited fr nn-statinary signals such as speech r music mied int the single channel. We are als investigating the use f this apprach in the AURORA database evaluatin task. [] G. J. Brwn and M. Cke, Cmputatinal auditry scene analysis, Cmputer Speech and Language, vl. 8, n., pp. 97 6, 99. [] A. J. Bell and T. J. Sejnwski, An infrmatinmaimizatin apprach t blind separatin and blind decnvlutin, Neural Cmputatin, vl. 7, n. 6, pp., 995. [] H. G. Okun, T. Nakatani, and T. Kawabata, Listening t tw simultaneus speeches, Speech Cmmunicatins, vl. 7, pp. 99, 999. [] D. L. Wang and G. J. Brwn, Separatin f speech frm interfering sunds based n scillatry crrelatin, IEEE Trans. n Neural Netwrks, vl., pp , 999. [5] K. Kashin and T. Tanaka, A sund surce separatin system with the ability f autmatic tne mdeling, in Prc. f Int. Cmputer Music Cnference, 99, pp [6] D. P. W. Ellis, A cmputer implementatin f psychacustic gruping rules, in Prc. th Int. Cnf. n Pattern Recgnitin, 99. [7] S. T. Rweis, One micrphne surce separatin, Advances in Neural Infrmatin Prcessing Systems, vl., pp ,. [8] M. S. Lewicki, A fleible prir fr independent cmpnent analysis, t be published, Neural Cmputatin,. [9] Michael Zibulevsky and Barak A. Pearlmutter, Blind surce separatin by sparse decmpsitin, Neural Cmputatins, vl., n.,. [] A. J. Bell and T. J. Sejnwski, Learning the higher-rder structures f a natural sund, Netwrk: Cmputatin in Neural Systems, vl. 7, n., pp. 6 66, July 996. [] Samer A. Abdallah and Mark D. Plumbley, If the independent cmpnents f natural images are edges, what are the independent cmpnents f natural sunds?, in Prc. f Internatinal Cnference n Independent Cmpnent Analysis and Signal Separatin (ICA), San Dieg, CA, December, pp [] William H. Press, Saul A. Teuklsky, William T. Vetterling, and Brian P. Flannery, Numerical Recipes in C: the art f scientific cmputing, Cambridge University Press, secnd editin, 99.