REDUCTION OF CORRELATION COMPUTATION IN THE PERMUTATION OF THE FREQUENCY DOMAIN ICA BY SELECTING DOAS ESTIMATED IN SUBARRAYS

15th European Sgna Processng Conference (EUSIPCO 27), Poznan, Poand, September 3-7, 27, copyrght by EURASIP REDUCTION OF CORRELATION COMPUTATION IN THE PERMUTATION OF THE FREQUENCY DOMAIN ICA BY SELECTING DOAS ESTIMATED IN SUBARRAYS Hao Yuan, Hsash Kawa, and Toshharu Horuch KDDI R&D Laboratores, Inc. 2-1-15 Ohara, Fujmno Cty, Satama 356-852, Japan phone: +(81) 49 278 7588, fax: +(81) 49 278 751 ema: {yuan, Hsash.Kawa, to-horuch}@kddabs.jp web: http://www.kddabs.jp/eng/ndex.htm ABSTRACT Ths paper addresses the permutaton probem n bnd source separaton by frequency doman ndependent component anayss. We propose a method to reduce the correaton cacuaton n agnng permutaton by seectng the DOAs (drecton of arrva) estmated n severa subarrays. When severa subarrays are avaabe, we have more choces to seect a reabe DOA to determne the permutaton. More reabe DOAs, more permutatons can be determned by DOAs nstead of the correaton of the separated sgnas between neghbourng frequency bns, whch s much more compex than DOA estmaton. To obtan accurate DOA n each subarray, we aso propose an mproved DOA estmaton method by averagng the phase dfferences n each subarray. Expermenta resuts show that about 58% of the correaton computaton was reduced when compared wth the case of a snge array, and a more accurate DOA was estmated by the proposed method. 1. INTRODUCTION Frequency doman ndependent component anayss (ICA) [1] s a popuar approach to bnd source separaton due to ts computatona effcency. The approach converts convoutve mxed sgnas nto frequency doman and separates the nstantaneous mxed sgnas wth compex-vaued ICA n each frequency bn. The drawback of the approach s, however, that t suffers from the ambgutes of permutaton and scang between frequency bns. To sove the permutaton ambguty probem a DOA based approach has been proposed [2, 3], whch cassfes the separated sgnas accordng to DOA estmated by searchng the drectvty patterns. The computatona cost can be reduced by cacuatng DOA drecty from the nverse of the separatng matrx [4, 5]. Utzng the correaton between the enveopes of the separated sgnas n neghbourng frequency bns s another approach to agn the permutaton for speech sgnas [6, 7]. Sawada et a. proposed a method of ntegratng the above two, whch fxes permutaton at severa frequency bns by DOA and then permutes the rest accordng to the nter-frequency correaton [8]. Snce the correaton cacuaton costs more tme than that of DOA estmaton, about 7 tmes accordng to [8], the computatona cost w become arge when there are ony a few frequency bns at whch permutaton s fxed by DOA. To reduce the correaton computaton by enargng the number of frequency bns at whch the permutatons are determned by DOA, we propose an approach that seects the DOAs estmated n severa subarrays that are arranged accordng to the ntereement spacng. Constructng arge and sma ntereement spacng subarrays can avod spata aasng at hgh frequency, whe t can obtan a more accurate DOA at ow frequency. If more than one subarray can be constructed we have more choces to seect a DOA whch may be used to determne the permutaton at a frequency bn. Ths enabes us to determne more permutatons by DOA nstead of the correaton of the separated sgnas. Accurate DOA n subarrays provdes more choces when seectng DOA from subarrays. To enhance DOA accuracy, we aso propose a method of DOA estmaton that averages the phase dfferences n each subarray. Snce DOA estmaton performance s dependent on the array arrangement and sensor mspacement may cause estmaton error, ths averagng method reduces the varaton of the estmated DOA. We present the averagng method n Secton 2 and descrbe the reducton of correaton computaton n Secton 3 after revewng the ntegrated method of permutaton by DOA and correaton. Expermenta resuts are descrbed n Secton 4 and we summarze the paper n Secton 5. 2. 3D DOA ESTIMATION METHOD WITH SEPARATING MATRIX OF ICA Snce source sgnas are ocated n 3D space, a 3D DOA estmaton such as [5] s needed rather than usng a near array whch cannot dstngush symmetrca anges wth respect to the array axs as n [4, 8]. In ths secton, after revewng the 3D DOA estmaton method [5] usng the separatng matrx of ICA, we propose an mproved DOA estmaton method by averagng the phase dfferences n subarrays and seectng the frequency bn DOA. 2.1 3D DOA estmaton method wth 3 sensors by ICA We assume that L source sgnas are mxed and receved by an M sensor array, where L M. The observed sgnas n the frequency doman can be expressed as 27 EURASIP 418

15th European Sgna Processng Conference (EUSIPCO 27), Poznan, Poand, September 3-7, 27, copyrght by EURASIP X ( f, t) = H( f) S( f, t) + N( f, t) (1) where X( f, t )=[x 1 ( f, t ),, x M ( f, t )] T s the observed sgna vector, S ( f, t )=[s 1 ( f, t ),, s L ( f, t )] T the source sgna vector and N ( f, t ) the nose vector n frequency f, respectvey. H ( f ) s the M L compex mxng matrx n whch the -th coumn represents the transfer functon from the -th source to the sensors. The superscrpt T denotes transposton. Usng the frequency doman ICA, we can fnd a converged separatng matrx W ( f ) n each frequency bn and the observed sgnas can be separated nto ndvdua sgnas, Y( f, t) = W( f) X( f, t) (2) The separated sgna vector Y ( f, t ) s an estmaton of the source sgna vector S ( f, t ). If the separatng matrx has converged, the pseudo nverse of the separatng matrx can be approxmated by the mxng matrx: + W ( f) = H( f) D( f) P( f) where D ( f ) and P ( f ) s the scang matrx and the permutaton matrx, respectvey. Athough the sgnas are mxed n a reverberant condton, we can approxmate the eements of the mxng matrx n (1) as foows: 2π f T H ( f) = a exp { j r k ( θ, φ ) m m m c } (4) where a m s the gan, rm = ( xm, ym, zm) T the coordnate vector of the m-th sensor, c the veocty of the sgna, and T k ( θ, φ) = ( snθ cos φ,snθsn φ,cosθ) s the ook drecton of the -th sgna. From the reaton between W + ( f ) and H ( f ) n (3), we obtan the phase dfference equatons correspondng to sensor 1 and 2, and sensor 1 and 3, respectvey. T ( r r ) k ( θ, φ ) = A (5) 1 2 12 T ( r r ) k ( θ, φ ) = (6) A 1 3 13 where A arg( W / W )/(2 π f / c), A arg( W / W )/ + + + + = = 12 1 2 13 1 3 (2 π f / c). Wthout oss of generaty, we assume the 3 sensors ocate on the X-Y pane and (5) (6) can be rewrtten as foows: x x y y sn cos 1 2 1 2 θ φ A12 = (7) x x y y snθ sn φ A 1 3 1 3 13 From (7), or from (5) and (6), we can obtan sn θ cosφ and sn θ sn φ by sovng the smutaneous equatons and then, obtan the drecton anges ( θ, φ ) of the sgnas. A two-stage estmaton method was aso proposed n [5]. The frst-stage estmates DOA usng (7) wth a sma ntereement spacng array. The second-stage seects another three-sensor array of arge ntereement spacng accordng to the frst-stage DOA and estmates the DOA at ow frequences. Combnng the estmated DOA n two stages, adoptng the frst-stage DOA at hgh frequences and those of the second-stage at ow frequences, a more accurate DOA can be estmated. (3) 2.2 Improved DOA estmaton method by averagng phase dfferences Snce DOA estmaton performance s affected by the array arrangement and sensor pacement error, we propose the foowng DOA estmaton procedure to dea wth probems of constructng sma and arge ntereement spacng subarrays and averagng the phase dfferences n each subarray. Separate the mxed sgnas wth a of the sensors n frequency bns and obtan the separatng matrx W( f ). Construct severa subarrays from the sensors accordng to whether ther ntereement spacng exceeds the hafwaveength of source sgnas. Average the phase dfferences n a sma ntereement spacng subarray usng (8), and estmate DOA at each frequency bn n the same way as usng (7). Takng the hstogram of frequency bn DOA, or averagng them, 1 1 the DOA of the frst subarray ( θ, φ ) s obtaned. ( x xj) ( y yj) A, j, j, j snθ cosφ, j (8) = ( x ) ( ) sn sn u xv yu y θ φ v Au, v uv, uv, uv, where, j, u and v denote the sensor number n the subarray. Repeat the procedure above for the remanng subarrays. To avod spata aasng, DOA s not cacuated for those hgh frequences at whch the haf waveength s ess than the dstance between the wave fronts at sensors, accordng to ( θ φ ). 1 1, Average the subarray DOA ( θ φ ) to obtan the mean vaue ( ϑ, ϕ ). Seect DOAs of the -th subarray whch satsfes = arg mn { φ ϕ } from a subarrays at each frequency bn and cacuate the fna sgna DOA ( θ, φ ) statstcay wth those seected frequency bn DOAs. 3. REDUCTION OF CORRELATION COMPUTATION BY SUBARRAY DOA In ths secton, havng revewed a combned approach of permutaton by DOA and correaton, we propose a method to reduce the correaton computaton usng the DOA estmated n subarrays. 3.1 Permutaton method by DOA and correaton An approach to permutaton agnment based on the nter-frequency correatons between the enveopes of the separated sgnas was proposed n [6, 7]. The correaton between sgna x(t) and y(t) s defned as: E{ x y} E{ x} E{ y} cor( x, y) = (9) σ σ x y where E{ } means the expectaton and σ s the standard devaton. If the separated sgnas correspond to the same source, the sgna enveopes at neghbourng frequences resembe each other. We cacuate the correaton wth enveopes of separated sgnas whch can be expressed as:, 27 EURASIP 419

15th European Sgna Processng Conference (EUSIPCO 27), Poznan, Poand, September 3-7, 27, copyrght by EURASIP f v () t = Y( f,) t (1) Denote Π as a permutaton and Π f as a permutaton at frequency f correspondng to the nverse of the permutaton matrx n (3). A smpe crteron to determne Π f s to maxmze the sum of the correatons n a range of neghbourng frequences wthn a dstance δ L f g arg max cor ( v (), t v () t f Π Π ( ) Π ) g ( ) g f δ = 1 Π = (11) Ths crteron s based on oca nformaton and has the shortcomng that msagnment n a narrow range of frequences may ead the permutaton to be wrong competey out of that range. To dea wth ths probem, a combned approach to permutaton by DOA and correaton was proposed n [8] to beneft from the advantages of both approaches. The DOA approach s robust snce a msagnment at one frequency does not affect other frequences, whe the correaton approach s precse as ong as sgnas are we separated by ICA snce the measurement s based on the separated sgnas. The combned approach s based on the foowng two steps: Fx the permutatons at some frequences where the reabty of the DOA approach s suffcenty hgh; Decde the permutatons for the remanng frequences based on the neghbourng correaton (11) wthout changng the permutaton fxed by the DOA approach. To fx the permutaton wth DOA approach at certan frequences, 3 crtera are used. Frsty, the number of estmated drectons s the same as that of sources. Secondy, the estmated drectons do not dffer greaty from ther averaged drectons. Thrdy, SIR +, cacuated from the drectvty pattern of the separatng matrx and the estmated drectons, s arger than a gven threshod. 3.2 Method of reducng correaton computaton by DOA estmated from subarrays In ths subsecton, we propose a method to reduce the correaton computaton. The man procedure s the same as [8]: fxng some permutatons by DOA and permutng the remanng frequency bns wth correaton. However our procedure seects the best DOA from severa subarrays and then fxes the correspondng permutaton, whch s presented as: Estmate DOA n each subarray and agn permutaton by DOA. Judge f (12) s satsfed for a gven threshod n the -th subarray. max{ φ φ } γ (12) Judge f (13) s satsfed for a gven threshod at those frequency bns satsfyng (12) for a sgnas. SIR + Th (13) SIR where SIR + s cacuated from the drectvty pattern of the separatng matrx and the estmated drectons. Adopt the permutaton satsfyng both (12) and (13) or seectng the best one f severa permutatons satsfy (12) and (13). Ths permutaton s consdered reabe. Determne the permutaton by correaton for the remanng frequency bns when there are severa contnuous reabe permutatons n the neghbourng frequency bns. 36 cm Y 6 3 4 1 2 5 4cm 12cm X Heght of MIC:75 cm 147 cm Speaker 1 (-36, -64, 69) cm 7 323 cm Coordnates of mcrophones (cm) MIC1 (,,) MIC2 (4,,) MIC3 (,4,) MIC4 (4,4,) MIC5 (12,,) MIC6 (,12,) MIC7 (12,12,) 36 MIC 197 cm 367 cm Speaker 3 (51, 41, 94) cm Speaker 2 (6, -5, 56) cm Chamber heght:25 cm Fgure 1 Layout of soundproof chamber Tabe 1 Parameters for the experment Source sgnas 6 speeches of 5 seconds Drectons of sgnas ( θ,φ ) (measured) (47, -119), (55, -4), (35, 39) Sampng rate 8 khz FFT sze 124 Frame shft 512 Threshod γ n (12) mn{ φ φ } / 4 j j SIR + threshod Th SIR n (13) 1 db Condton to take correaton 3 contnuous reabe bns 4. EXPERIMENTAL RESULTS To show the effectveness of the proposed method, a panar array of 7 omndrectona mcrophones was adopted, whch contans 4 mcrophones wth sma and arge ntereement spacng, respectvey. Wth ths confguraton we can construct sma and arge ntereement spacng subarrays and appy the averagng method to the subarrays. The 7 mcrophone array was paced n a soundproof chamber and 3 oudspeakers were setup around t (shown n Fgure 1). We generated the mxed sgnas by convovng the speech sgnas and the mpuse responses from the oudspeakers to the array n order to cacuate SIR defned as sgna to nterference rato. The speech sgnas were ATR phonetcay-baanced sentences n Japanese [9]. We used the FastICA [1] to separate the mxed sgnas wth a of the mcrophones. The parameters n the experment are sted n Tabe 1. From the 7 mcrophones we constructed the 3 subarrays MIC1234, MIC1567 and MIC2356, whch had sma, arge and mdde ntereement spacng, respectvey. Here MIC2356 was added to ncrease the choces when seectng DOA from the subarrays. Fgures 2 and 3 show the estmated azmuth anges of 3 sgnas n frequency bns by MIC123, as an exampe of a snge array, and the proposed method, respectvey. It s seen that the proposed method acheves a fat and concentratve confguraton of the estmated anges whe MIC123 showed a dspersve resut. Tabe 2 shows the meas- 46 cm 27 EURASIP 42

15th European Sgna Processng Conference (EUSIPCO 27), Poznan, Poand, September 3-7, 27, copyrght by EURASIP Ph(deg.) Ph(deg.) 1 5-5 -1-15 1 5-5 -1-15 Ph 1 Ph 2 Ph 3 5 1 15 2 25 3 35 4 45 5 Frequency bn Fgure 2 Estmated azmuth anges by MIC123 Ph 1 Ph 2 Ph 3 5 1 15 2 25 3 35 4 45 5 Frequency bn Fgure 3 Estmated azmuth anges by proposed method Tabe 2 Measured and estmated DOAs (deg.) θ φ θ φ θ φ 1 1 2 2 3 3 Measured 47-119 55 Averaged error -4 35 39 - MIC123 42-121 43-45 24 4 6 MIC124 4-13 38-36 35 25 9 MIC134 43-127 48-35 32 47 6 MIC234 39-126 43-34 35 37 6 MIC1234 39-126 43-36 32 35 6 Two-stage 48-119 45-45 3 4 4 Proposed 45-12 56-41 32 35 2 ured and the estmated DOA by varous methods. The estmated anges vared when seectng dfferent 3-mcrophone arrays because the DOA estmaton performance s affected by the array confguraton and the error of pacement. The varaton can be decreased by usng (8) to average both the pacement error and the phase dfferences. The averaged absoute dfferences between the measured and estmated DOAs are aso sted n the tabe. Snce the proposed method seects the best DOA from the subarrays to whch the averagng method s apped, t acheved the smaest DOA mean error of 2 degrees n ths case. Fgure 4 shows the SIR cacuated n each frequency bn for MIC123 and the proposed method wth permutaton by DOA and by DOA pus correaton. From ths fgure we can see that some msagnments n MIC123 were corrected by the proposed method wth more reabe DOAs. The ast graph shows the resut that the permutatons were correcty agned by DOA pus correaton. Snce there s amost no speech component beow 2Hz for femaes we dd not take ths range nto account. The number of fxed permutatons by DOA and NRR (Nose Reducton Rate) of the separated sgnas are sted n Tabe 3. The proposed method has more fxed permutatons and hgher NRR than 3-mcrophone arrays when adoptng the DOA approach ony. The NRR by MIC123 (DOA) 4 2-2 -4 5 1 15 2 25 3 35 4 45 5 Frequency bn Proposed (DOA) 4 2-2 -4 5 1 15 2 25 3 35 4 45 5 Frequency bn Proposed (DOA+Correaton) 4 2-2 -4 5 1 15 2 25 3 35 4 45 5 Frequency bn Fgure 4 SIR n frequency bns wth permutaton agned by DOA and DOA pus correaton DOA pus correaton s the same for a cases except for MIC124. Snce we need 3 contnuous reabe permutatons to conduct the correaton cacuaton, the permutaton by correaton was not apped to MIC124 whch has ony 2 permutatons fxed by DOA. To evauate the separaton performance of our method n contrast to the conventona method based on a snge array, we separated 1 combnatons of 3 sources from 3 mae and 3 femae speeches. An overa vew of separaton performance of the 1 speech combnatons s shown n Fgure 5 for the cases of 3-mcrophone arrays, subarray MIC1234, and the proposed method wth permutaton fxed by DOA or by DOA pus correaton, where the SIR + threshod s 1 db. For the 3-mcrophone array we seected the best case of MIC123 and the worst case of MIC124 as exampes. Snce a mcrophone par of arge spacng s not appcabe to hgh frequences, the smaest array, vz. MIC1234, s dscussed here as an exampe of 4-mcrophone snge arrays. We aso show the resuts of the proposed method wthout judgng SIR + when fxng the permutaton by DOA. For the DOA approach, the NRR does not dffer much between MIC123, MIC124 and MIC1234 whe the proposed method gves a superor resut. For the DOA pus correaton approach, MIC124 was worse than other methods because t had ony a few reabe permutatons at some speech combnatons and therefore permutaton by correaton was not apped. The varaton of NRR between dfferent speech combnatons s consdered attrbutabe to the varaton of ndependency between the dfferent speeches, dependng on the contents, the speaker s gender, and so on. For MIC1234 and the proposed method, the NRR resuts are amost the same, but the numbers of permutatons fxed by DOA dffer greaty. The average number for the 1 speech combnatons was 48 for MIC1234 and 17 for the proposed method. Snce we dd not take the ranges beow 2Hz and above 398Hz nto account, we separated the mxed sources at a tota of 484 frequency bns. Therefore the number of frequency bns, at whch permutaton s agned by correaton, s reduced from 436 to 377 (14% reducton). 27 EURASIP 421

15th European Sgna Processng Conference (EUSIPCO 27), Poznan, Poand, September 3-7, 27, copyrght by EURASIP Tabe 3 Fxed bn number by DOA and NRR (db) The number of fxed permutatons by DOA NRR (DOA) NRR (DOA+Cor.) MIC123 5 17.4 19.4 MIC124 2 17.5 17.5 Proposed 89 18.4 19.4 NRR(dB) 2 18 16 14 12 mc123 (D) mc124 (D) Comb. 1 Comb. 2 Comb. 3 Comb. 4 Comb. 5 Comb. 6 Comb. 7 Comb. 8 Comb. 9 Comb. 1 Average over comb. 1~1 mc1234 (D) proposed (D) mc123 (D+C) mc124 (D+C) mc1234 (D+C) proposed (D+C) proposed (no SIR) Fgure 5 Separaton performance wth dfferent permutaton methods for 1 speech combnatons (The DOA approach s denoted by D ; the DOA pus correaton approach by D+C and the approach wthout judgng SIR + by no SIR ) Number of frequecy bns 45 4 35 3 25 2 15 1 5 mc123 Average wthout judgng SIR+ Average wth THSIR=1dB mc124 mc134 mc234 mc1234 proposed Fgure 6 Average number of permutatons fxed by DOA and standard devaton wth and wthout judgng SIR + We aso observed from Fgure 5 that the NRR was amost the same for the proposed method wth and wthout judgng SIR + vaues when determnng the permutaton by DOA. Decreasng the SIR + threshod may provde more reabe permutatons. Fgure 6 shows the average numbers and the standard devatons of fxed permutatons by DOA wth Th SIR =1dB and wthout judgng SIR + for the 1 speech combnatons. Athough the NRR s amost the same between the proposed method and snge arrays, the reabe permutaton numbers are sgnfcanty ncreased wthout judgng SIR +. The numbers of reabe permutatons wthout judgng SIR + were 282, 288 and 42 for MIC123, MIC1234 and the proposed method, respectvey. Subtractng the number of reabe permutatons from the tota frequency bns concerned, the numbers of permutatons determned by correaton were 21, 196 and 82 for MIC123, MIC1234 and the proposed method, respectvey. Ths means that the reductons of the correaton computaton are 59% and 58% when compared wth MIC123 and MIC1234, respectvey. From ths fgure t s aso confrmed that the averagng method n the subarray acheved a better resut than a snge array on obtanng reabe permutatons determned by DOA wthout the assstance of the SIR + crtera. 5. CONCLUSIONS We proposed a reducton method of correaton computaton n determnng a permutaton for the frequency doman ICA, by seectng the DOAs estmated n severa subarrays. Ths method s based on the deas of constructng severa sma and arge ntereement spacng subarrays and seectng the DOAs from a of the subarrays. Ths enabes us to obtan more reabe permutatons fxed by DOA nstead of correaton and therefore reduce the correaton computaton. To estmate a more accurate DOA n each subarray, we aso proposed an mproved DOA estmaton method to reduce the varaton of estmated DOA by averagng the phase dfferences n each subarray. If no more than one subarray can be constructed, the method can not be apped but the averagng method s aso avaabe for DOA estmaton when there are enough mcrophones n the array. Expermenta resuts reveaed, wth a 7-mcrophone array, that the proposed method mproved the DOA estmaton accuracy and reduced the correaton computaton by about 58 % n determnng permutatons compared wth that usng a snge array. REFERENCES [1] P. Smaragds, Bnd separaton of convoved mxtures n the frequency doman, Neurocomputng, vo. 22, pp. 21-34, 1998. [2] S. Kurta, H. Saruwatar, S. Kajta, K. Takeda, and F. Itakura, Evauaton of bnd sgna separaton method usng drectvty pattern under reverberant condtons, n Proc. ICASSP 2, pp. 314-3143, June 2. [3] M. Z. Ikram and D. R. Morgan, A beamformng approach to permutaton agnment for mutchanne frequencydoman bnd speech separaton, n Proc. ICASSP 22, pp. 881-884, May 22. [4] H. Sawada, R. Muka and S. Makno, Drecton of arrva estmaton for mutpe source sgnas usng ndependent component anayss, n Proc. ISSPA 23, vo. 2, 23, pp. 411-414, Juy 23. [5] H. Yuan, M. Yamada and H. Kawa, A DOA estmaton method for 3D mutpe source sgnas usng ndependent component anayss, Proc. EUSIPCO26, Itay, Sept. 26. [6] J. Anemüer and B. Komeer, Amptude moduaton decorreaton for convoutve bnd source separaton, n Proc. ICA, pp. 215-22, June 2. [7] N. Murata, S. Ikeda and A. Zehe, An approach to bnd source separaton based on tempora structure of speech sgnas, Neurocomputng, vo. 41, no. 1-4, pp. 1-24, Oct. 21. [8] H. Sawada, R. Muka S. Arak and S. Makno, A robust and precse method for sovng the permutaton probem of frequency-doman bnd source separaton, IEEE Trans. Speech Audo Processng, vo. 12, pp. 53-538, Sep. 24. [9] M. Abe, Y. Sagsaka, T. Umeda and H. Kuwabara, Speech Database User s Manua, ATR Technca Report, TR-I-166 (n Japanese), 199. [1] E. Bngham and A. Hyvärnen, A fast fxed-pont agorthm for ndependent component anayss of compex-vaued sgnas, Internatona Journa of Neura Systems, vo. 1, no. 1, pp. 1-8, Feb. 2. 27 EURASIP 422