Subband-based Single-channel Source Separation of Instantaneous Audio Mixtures

Transcription

1 World Alied Sciences Journal 6 (6: , 009 ISSN IDOSI ublicaions, 009 Subband-based Single-channel Source Searaion of Insananeous Audio Mixures Jalil Taghia and Mohammad Ali Doosari Dearmen of Elecrical Engineering, School of Technical and Engineering, Shahed Universiy, Khalie ars Highway, oscode: , Tehran, Iran Absrac: In his aer, a new algorihm is develoed o searae he audio sources from a single insananeous mixure. The algorihm is based on subband decomosiion and uses a hybrid sysem of Emirical Mode Decomosiion (EMD and rincile Comonen Analysis (CA o consruc arificial observaions from he single mixure. In he searaion sage of algorihm, we use Indeenden Comonen Analysis (ICA o find indeenden comonens. A firs he observed mixure is divided ino a finie number of subbands hrough filering wih a arallel bank of IR band-ass filers. Then EMD is emloyed o exrac Inrinsic Mode uncions (IMs in each subband. By alying CA o he exraced comonens, we find uncorrelaed comonens which are he arificial observaions. Then we obain indeenden comonens by alying Indeenden Comonen Analysis (ICA o he uncorrelaed comonens. inally, we carry ou subband synhesis rocess o reconsruc fullband searaed signals. The exerimenal resuls subsaniae ha he roosed mehod ruly erforms he ask of source searaion from a single insananeous mixure. Key words: Blind source searaion indeenden comonen analysis emirical mode decomosiion rincile comonen analysis single-channel audio source searaion subband decomosiion INTRODUCTION The roblem of Blind Source Searaion (BSS includes finding indeenden source signals from heir observed mixures wihou rior knowledge abou he acual mixing channels. The source searaion roblem is of ineres in various alicaions [, ]. In his aer, we consider he searaion of mixed audio signals. Audio source searaion has many oenial alicaions such as music ranscriion, seaker searaion in video conferencing and robus audio indexing in mulimedia analysis. Recording done wih mulile microhones enables echniques which use he saial locaion of sourcen he searaion rocess [3, 4] so ha i ofen makes he searaion ask easier. However, ofen only a single channel recording is available and mehods based on single-channel audio source searaion are more racical in he real world alicaions. There are some exising algorihms of single-mixure audio source searaion [5, 6]. In [7] an indeenden subsace analysis (ISA mehod has been inroduced o searae he comonen sources from heir single mixure. In [8] he ISA mehod has been emloyed o searae he drum racks from olyhonic music. The basic consideraion of he ISA mehod is o decomose he Time-frequency (T sace of a mixed signal as a sum of indeenden source subsaces. The T reresenaion is obained by alying Shor-ime ourier Transform (STT. In his aer, we consider he BSS of a single insananeous mixure of audio signals. We roose a new BSS algorihm ha emloys subband rocessing named subband BSS. By using his algorihm, he observed mixure is convered ino subband domain wih a filer bank so ha we can choose a moderae number of subbands and mainain a sufficien number of samlen each subband. The roosed subbandbased single-channel algorihm emloys a hybrid sysem of Emirical Mode Decomosiion (EMD [9], rincile Comonen Analysis (CA [0] o consruc arificial observaions from he mixure. In searaion sage, Indeenden Comonen Analysis (ICA is used o find indeenden comonens. or his urose, we use he fixed-oin algorihm (asica algorihm [, ] because i rovides fas convergence and acceable searaion. Generally, ICA algorihms suffer scaling and ermuaion ambiguiies. To eliminae he scaling ambiguiy, we roose a new mehod which is based on solving overdeermined equaions. To solve ermuaion ambiguiy, we use a K-means clusering algorihm which is based on Kullback-Leibler Divergence (KLD. Corresonding Auhor: Dr. Jalil Taghia, Dearmen of Elecrical Engineering, School of Technical and Engineering, Shahed Universiy, Khalie ars Highway, oscode: , Tehran, Iran 784

2 World Al. Sci. J., 6 (6: , 009 We have simulaed he roosed algorihm o searae he sources from a single mixure of wo audio signals. In order o demonsrae he roficiency of our algorihm, he searaion erformance of boh our algorihm and ISA mehod is comared. The organizaion of his aer is as follows. The basics of uilized ICA and EMD algorihms are described in Secion II and Secion III, resecively. Secion IV describes he KLD-based K-means clusering algorihm. Secion V describes he roosed signal searaion algorihm. Secion VI resens he exerimenal resuls. inally Secion VII concludes his aer. INDEENDENT COMONENT ANALYSIS Indeenden Comonen Analysis (ICA is a saisical model where he observed daa is exressed as a linear combinaion of underlying laen variables. The laen variables are assumed o be non-gaussian and muually indeenden. The ask is o find ou boh he laen variables and he mixing rocess. The basic ICA model is x( = A s( where x( = [x (,,x m (] T is he vecor of observed random variables, s( = [s (,,s n (] T is he vecor of saisically indeenden variables called he indeenden comonens and A is an unknown consan mixing marix. m and n are he number of observed mixures and original sources resecively and ( T denoes ransose oeraion. In his aer, we choose he fixed-oin algorihm given by [, ] since i is a comuaionally efficien and robus fixed ye algorihm for ICA BSS. I is also named asica algorihm. The indeenden comonenn he ICA model are found by searching a marix W such ha y( = W x( where W = [w,,w n ] T and w i (i =,,,n are m-dimensional weigh vecors. The fixed oin algorihm searches for he exrema of he cos funcion T J ( w = E{G( w x(} G where G: R + {0} R is a smooh even funcion and T E{( w x( } = w = The algorihm requires a re-whiening of he daa. Whiening can be accomlished by CA. The sabilized fixed-oin algorihm for one uni is []: where β = E {w T x(g(w T x(}, µ is a se size arameer ha may change wih he ieraion coun and g( is he derivaive of G( and g ( is he derivaive of g( The one-uni algorihm can be exended o he esimaion of he whole ICA ransformaion y( = W x( []. EMIRICAL MODE DECOMOSITION The emirical mode decomosiion is a signal rocessing echnique o decomose any non-saionary and nonlinear signal ino oscillaing comonens wih some basic roeries. The key benefi of using EMD is ha i is auomaic and fully daa adaive. EMD decomoses a ime series x( ino a sum of band-limied funcions c b ( by emirically idenifying he hysical ime scalenrinsic o he daa. Each exraced mode c b ( named Inrinsic Mode uncion (IM conains wo basic condiions. irs, in he whole daa se, he number of exrema (maxima and minima and he number of zero crossings mus be he same or differ a mos by one. Second, a any oin, he mean value of he enveloe defined by he local maxima and he enveloe defined by he local minima is zero. The firs condiion is similar o he narrow-band requiremen for a saionary Gaussian rocess and he second condiion is a local requiremen induced from he global one and is necessary o ensure ha he insananeous frequency will no have redundan flucuaions anduced by asymmeric waveforms. There are several aroaches for comuing EMD. The following algorihm is emloyed here o decomose signal x( ino a se of IM comonens. Idenificaion of all maxima and minima of he ime series x( Generae he uer and lower enveloes u( and l( resecively, by connecing he maxima and minima searaely wih cubic sline inerolaion. Deermine he local mean as µ ( = [u(+l(]/ An IM should have zero local mean hus we subrac µ ( from he original signal x( as e ( = x( - µ (. Check wheher e ( is an IM or no by checking he wo basic condiions as described above. Reea ses o 5 and so when an IM e ( is obained. w + T w-µ [E{ x(g( w x(} ßw] = T [E{g( w x(} ß] w = w w new + + ( Once he firs IM is obained, define c ( = e ( which is he smalles emoral scale in x( To find he res of he IMs, generae he residue r ( of he daa by subracing c ( from he signal x( as 785

3 World Al. Sci. J., 6 (6: , 009 r ( = x( - c (. The sifing rocess will be coninued unil he final residue is a consan, a monoonic funcion, or a funcion wih only one maximum and one minimum from which no more IMs can be obained. The subsequen IMs and he residues are comued as: r( c( = r(...r ( c ( = r( ( B B B where r B ( is he final residue. A he end of he decomosiion, he signal x( is reresened as: B b B (3 b= x( = c ( + r( where B is he number of IMs and r B ( is he final residue. The IMs are he fundamenals for reresening he ime series daa. Being daa adaive, he fundamenals usually offer a hysically meaningful reresenaion of he underlying rocesses. There is no need for considering he signal as a sack of harmonics and, herefore, EMD ideal for analyzing non-saionary and nonlinear daa [3]. Moreover, EMD uses only a single mixure o exrac IMs. KLD-BASED K-MEANS CLUSTERING In his secion a K-means clusering algorihm based on Kullback-Leibler divergence inroduced in [4] is used in our aer for he grouing rocess. Symmeric KLD measures he relaive enroy beween wo robabiliy mass funcions (z and q(z over a random variable Z as Here: (z q(z KLD(,q = (zlog + q(zlog z Z q(z z Z (z (4 (a Calculae he disances d of ν from w by d = KLD(v,w, =,,,. (b Idenify he cener i closes o ν so ha i = argmin{d}, =,,,. (a Udae he weigh of he i h cener by w i( + = w( i +η((v w i(. (b Udae he learning rae facor η by: η( η ( + = (c Calculae he change c of weighs by: c = w( + w( w( + Check, if c<ε hen so, else reea. The erm ε reresens error hreshold (of he order 0 6 in his mehod. ROOSED SEARATION MODEL In his secion we roose a new algorihm for searaing insananeous audio mixures which are recorded by one microhone. Our single-channel BSS model is shown in ig.. A firs, he observed mixure x( is divided ino D segmens wih shor lengh, in comarison o he whole lengh of he signal, by windowing. The windowing rocess necessary since i considerably reduces he running ime of he algorihm and imroves he searaion erformance when he daa is non-saionary. Then, he algorihm is alied o he windowed signal x (i ( in he i h (i =,,,D segmen o derive searaed signaln each segmen. KLD is used o measure he informaion heoreic disance beween wo basis vecors during K-means clusering whereas radiional K-means clusering uses he Euclidean disance. By using he KLD-based K-means clusering algorihm, vecors are auomaically groued ino clusers corresonding o he given number of sources according o he enroy conained by individual vecors. The KLD-based K-means clusering algorihm is summarized as: Iniialize cluser cener weighs w,w,,w and reea ses o 4 unil convergence is reached. or ieraion : Selec a normalized basis funcion ν (sequenially from reduced se of basis funcions 786 Subband analysis sage: The observed mixure x (i ( is divided ino a finie number of K subbands hrough filering i wih a arallel bank of IR band-ass filers. To execue BSS on real-valued signals, we use he subband analysis filer bank as he cosine modulaed version of a rooye filer h 0 (n of lengh N and cuoff frequency ω c = π/k [5]: nπ h k(n = h 0(ncos[(k ], k =,,...,K (5 K nπ h 0 (n = sincw(n (6 N As shown in ig., h 0 (n is a runcaed sinc( funcion weighed by a Hamming window wih w(n = cos(nπ/N where n =,,,N. The

4 World Al. Sci. J., 6 (6: , 009 ROOSED ALGORITHM ( s ( ( s ( ( s ( s ( x( WINDOWIN G x x x ( ( i (D ROOSED ALGORITHM x( ( i h h k h K SB SB SB ( ( ( (, (, (, ( ( K, ( K, ( ( K, ROOSED ALGORITHM ROOSED ALGORITHM CLUSTERIN G BLOCK (i s ( (i s ( (i s ( (D s ( (D ( s (D s ( (, ( ( ( K, (, ( K, (, ( K, CLUSTERIN G BLOCK g g k g K g g k g K g g k g K S S S s ( s ( ( ( ( SEARATION BLOCK (SB c i ( k, x i ( k, EMD c i ( k, b c ( i ( k, B CA u i ( k, u i ( k, u i ( k, ICA CLUSTERING BLOCK ig. : The roosed single-channel subband-based BSS algorihm. SB denoes Searaion Block. D, B, and K indicae he number of segmens, IMs, original sources and subbands, resecively, where i =,,,D, b =,,,B, =,,, k =,,.,K. indicaes down-samling rae rooye filer h 0 (n is used o derive he analysis and he corresonding synhesis filer banks deiced in ig.. The effecive bandwidh of he decomosed signal in each subband is reduced by a facor of /K comared o he wider bandwidh of he original fullband signal. Then, we emloy decimaion a he down-samling rae (<K o reduce he aliasing 787 roblem. inally, he observed mixure x (i ( is decomosed o K real-valued subband signals : N (i (i x (k, = h k (nx ( n n= (7 where x (i (k, is down-samled and band-limied signal in he k h subband (k =,,,K and = l. denoes he

5 ig. : Imulse resonse of he low-ass real-valued rooye IR filer h 0 (n wih lengh N = 5 ime index a he reduced samling rae for some ineger l. Searaion sage: In his sage, subband decomosiion combined wih a hybrid sysem of EMD and CA consiues a single-channel searaion model ha emloys ICA in he searaion rocess o find indeenden comonens. The rocess shown in ig... Se : The EMD algorihm is alied o he downsamled and band-limied signal x (i (k, in each subband o exrac IMs c (i b (k, where b =,,,B and B is he number of IMs. The dimensionaliy of c (i b (k, is L where L is he lengh of daa segmen. (i Thus, an observaion marix Y (k, is formed in L B each subband for k =,,,K: (i (i T (i T (i T = (8 L B B Y (k, [(c (k,,(c (k,,,(c (k, ] Se : In order o find he uncorrelaed basis comonens which are our arificial observaions, CA algorihm is used. CA imlemened by (i emloying SVD. The SVD of Y (k, is a L B facorizaion of he form Y (k, = U (k, D (k, V (k, (i (i (i ( i T L B L L L B B B where U (i (k, and V (i (k, are orhogonal marices wih orhogonal columns and D (i (k, is a marix of B singular values σ b = σ b b, where σ σ σ B 0 Marix U (i (k, is referred o as a row basis reresening (i he rincial comonens of Y L B(k,. The singular values reresen he sandard deviaions roorional o he amoun of informaion conained in he World Al. Sci. J., 6 (6: , corresonding rincial comonens. A reduced se of basis vecors are seleced from U (i (k, (i (i (i (i.e. UL (k, = [u L (k,,,u L (k,] by using he firs singular values. We se equal o he number of original sources. Se 3: Afer finding he uncorrelaed basis vecorn each subband, asica algorihm (inroduced in Sec. II is alied o each uncorrelaed comonens (i U T (k, ω, o obain indeenden basis vecorn each subband i.e. s (k, = [s (k,,s (k,,...,s (k,] (i (i (i (i T L L L Subband synhesis sage: Generally, ICA has wo ambiguiies []. irs, he variances of he indeenden comonens can no be deermined (i.e. scaling ambiguiy and second, hey can aear a he ouu of he source-esimaing nework in any order (i.e. ermuaion ambiguiy. To eliminae scaling ambiguiy we mulily he searaed ouu signals s (i (k, T by scaling coefficiens o ensure equaliy beween searaed and original signaln each segmen. In order o obain hese coefficiens, we need o solve an overdeermined linear sysem of equaionn he simle case of an insananeous mixure such as our case Q.c = b In his equaion c is an -by- vecor indicaing he unknown coefficiens, b is a L-by- vecor, where L is he lengh of daa, consising of he windowed signal x (i (k, and Q is a L-by- marix comosed of searaed signals where L> (i.e. an over deermined sysem. The leas square soluion o his marix equaion is vecor c ha minimizes he -norm of he residual b-qc over all vecors c A leas square soluion o Q.c = b can be found easily as c = Q + b where ( + denoes seudo-inverse oeraion and c is he unique soluion of minimal -norm. The seudo-inverse Q + of Q can be wrien exlicily in erms of he SVD. The SVD of Q is given by Q = UΣV T where U is L-by-L uniary marix, V is an -by- uniary marix and Σ a real L-by- diagonal marix wih (i,i inerior σ i. The singular values σ i saisfy σ σ σmin(l, 0. Therefore, if he SVD of Q is given by Q = UΣV T hen Q + = UΣ + V T where Σ + is -by-l diagonal wih (i,i enry /σ i if σ i >0 and oherwise 0 Afer finding c coefficiens and alying hem o he aroriae (i searaed ouu signals s T(k, we can eliminae he scaling ambiguiy from he searaed signaln each segmen. Solving he ermuaion ambiguiy is necessary since due o he ermuaion ambiguiy of he ICA

6 algorihm, he order of searaed signaln each subband is no consan. or his urose and obaining fullband searaed signaln each segmen, we need o aly KLD-based K-means clusering algorihm (Sec. IV o all searaed signals o grou hem ino clusers ( is equal o he number of original sources so ha each cluser conains searaed subband signals which are relaed o heir corresonding original source. To reconsruc he fullband searaed signaln each segmen, we erform he following ses: all of he searaed subband signaln each cluser are firs usamled by he inerolaion facor nex filered by he synhesis filers nπ g k(n = g 0(ncos[(k ] K where he baseband synhesis filer g 0 (n is acually a ime reversed coy of he analysis rooye filer h 0 (n (6 equal o g 0 (n = h 0 (N-n- wih n =,,, N and k =,,,K. The subband synhesis sage indicaed in ig.. inally, each fullband searaed signal in he subband synhesis sage ouu is given by: K N (i (i s ( = g k(ns (k,( n k= n= for =,,, where is equal o he number of original sources and denoes he ime index a he resored samling rae, such ha = l/. To achieve he full-lengh searaed signals, we should aroriaely oin he fullband searaed signals in each segmen o hose in oher segmens. The roblem is ha he order of searaed signaln he segmens no he same. Therefore, we should use he KLD-based K-means clusering again o grou searaed signals aroriaely. A las, o reconsruc he full-lengh searaed signals, we oin groued signaln each cluser. The full-lengh searaed signals T are denoed by S = [s,s,...,s]. EXERIMENTAL RESULTS In his secion we examine he searaion erformance of he roosed algorihm for searaing audio signals from a single insananeous mixure. our mixures are considered in he exerimens. One of he wo signals of mixures denoed by m, m, m3 and m4, is male seech and he ohers are rock music, azz music, flue sound and female seech, resecively. All of he mixures have been recorded a a samling rae of 8 khz and 6-bi amliude resoluion. The lengh of hese mixures 5.85 seconds. World Al. Sci. J., 6 (6: , 009 (9 789 Table : Used arameern he simulaion seu Symbol Definiion Value D Number of segmens 4 N Lengh of rooye filer 5 K Number of subbands 3 Down-samling rae How o measure he disorion beween original source and he esimaed one has no comleely rivial soluion. I deends on he mixing sysem and he searaion rocess as well as he field of alicaion. I is sill hard o evaluae he searaion algorihm because of he lack of aroriae erformance measures even in he very simle case of linear insananeous mixures [6]. In his aer, in order o measure he disorion beween he original sources and he esimaed sources, we use he imrovemen of signal-o-noise raio (ISNR as he quaniaive measure of searaion erformance [6]. The ISNR is he difference beween inu and ouu SNRs. The inu SNR (SNR I is defined as: SNR s ( I = 0log x( s( (0 where x( is he mixed signal and s ( is he original signal of he h source. If u ( is he searaed signal (esimaed source signal corresonding o h source, he ouu SNR (SNR O is defined as: SNR s( O = 0log s ( u( ( Then we emloyed ISNR (db = SNR O - SNR I as he erformance measure. The ISNR reresens he degree of suression of he inerfering signals o imrove he qualiy of he arge one. Recenly, i is widely used o measure he searaion qualiy beween he mixed and demixed signal [6]. The higher value of ISNR indicaes beer searaion erformance. Table indicaes arameers used in he simulaion seu of he roosed algorihm which yield he bes resul in our exerimens. The resuls of alying he algorihm o he mixures (m-m4 are shown in Table in erms of ISNR. In order o demonsrae he comeence of he roosed algorihm, he searaion erformance of our algorihm and ISA mehod is also comared in his Table. igure 3 resens he firs eigh subband signals afer alying subband decomosiion o he firs segmen of he observed mixure m4. igure 4

7 World Al. Sci. J., 6 (6: , 009 Table : Searaion resuln erms of ISNR Mixure Mehod ISNR of Sig (db ISNR of Sig (db m (male seech & rock music The roosed algorihm ISA mehod m (male seech & azz music The roosed algorihm ISA mehod m3 (male seech & flue sound The roosed algorihm ISA mehod m4 (male & female seech The roosed algorihm ISA mehod ig. 3: The firs eigh subband signals afer alying subband decomosiion o he firs segmen of he mixure m4 illusraes exraced IM comonens afer alying EMD algorihm o he decomosed signal a he firs subband. inally, he searaed signals afer alying he roosed algorihm o m4 are shown in ig. 5. CONCLUSION In his aer, we roosed a new subband-based BSS algorihm o searae audio signals from a single insananeous mixure. The use of subband BSS enabled us o mainain a sufficien number of samles o esimae he saisicn each subband. The roosed searaion algorihm included a hybrid sysem of EMD, CA o consruc arificial observaions from he single mixure. We used asica in he searaion rocess o find indeenden comonens. Since he ICA algorihms suffer from scaling ambiguiy, a new mehod for removing he ICA scaling ambiguiy was roosed. Moreover, a K-means clusering algorihm based on Kulback-Leibler divergence was used o grou he indeenden basis vecors, suffering from ermuaion ambiguiy of ICA algorihms, ino he number of comonen sourcenside he mixure. I is noiceable ha he roosed algorihm did no require any raining daa. A quaniaive searaion erformance was resened in he exerimenal resuln erms of he ISNR by which we comared he searaion efficiency of our algorihm o ha of he ISA mehod. Exending he roosed searaion model for searaing convoluive mixures and imroving he 790

8 World Al. Sci. J., 6 (6: , 009 ig. 4: Exraced IM comonens and he residue afer alying EMD algorihm o he decomosed signal a he firs subband robusness of he roosed algorihm are concerns for our fuure work. ACKNOWLEDGMENT This aer is funded and suored by Iran Telecommunicaion Research Cener (ITRC. The auhors would like o hank ITRC for is grea suor. REERENCES ig. 5: Waveforms of original sources, he recorded mixure m4 and searaed signals; original male seech and female seech (uer wo, he mixure m4 (middle one, 3 searaed male seech and female seech afer alying he roosed algorihm (lower wo 79. Nandi, A.K., 999. Blind Esimaion Using Higher- Order Saisics, Boson, Mass, Kluwer Academic.. Cichocki, A. and S. Amari, 003. Adaive Blind Signal and Image rocessing, Chicheser, UK, John Wiley and Sons. 3. Dudgeon, D.E. and R.M. Mersereau, 984. Mulidimensional Digial Signal rocessing, Englewood Cliffs, USA, renice Hall. 4. Torkkola, K., 000. Blind searaion of delayed and convolved sources, John Wiley and Sons. 5. Jang, G.J. and T.W. Lee, 003. A maximum likelihood aroach o single channel source searaion. Journal of Machine Learning Research, 4: Jang, G.J., T.W. Lee and Y.H. Oh, 003. Single channel signal searaion using ime-domain basis funcions. IEEE Signal rocessing Leer, 0: 68-7.

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