NEAR-COMMON ZEROS IN BLIND IDENTIFICATION OF SIMO ACOUSTIC SYSTEMS

Patrick A. Naylor, Xiang (Shawn) Lin and Andy W. H. Khong

Department of Electrical and Electronic Engineering, Imperial College London, UKE-mail: {p.naylor, shawn.linO4, andy.khong}@imperial.ac.uk

ABSTRACT Section 3.2, we introduce the forced spectral diversity (FSD) con-cept for addressing the detrimental effects of near-common zeros on

The common zeros problem for Blind System Identification aatv S loihs h S ocp mly obnto(BSI) has been well known to degrade the performance of clas- adpiv -Saloihs h S ocp mly obnto

(icBSI) has beenhmswellknothefored ithperformance of clas-quet of spectral shaping filters and effective channel undermodelling tech-sic BSI algorithms and therefore limits performance of subsequent niue whc intr. mrv h efomneo h MFM9 niq~~~~~~~~~~~mues which in turn improve the performance of the NMCFLMSspeech dereverberation. Recently, we have shown that multichannel algorithm. The performance of the proposed approach is illustratedsystems cannot be well identified if near-common zeros are present. in Section 4.In this work, we further study the near-common zeros problem us-ing channel diversity measure. We then investigate the use of forcedspectral diversity (FSD) based on a combination of spectral shap- 2. THE NEAR-COMMON ZEROS PROBLEM IN BLINDing filters and effective channel undermodelling. Simulation results SIMO SYSTEM IDENTIFICATIONshow the effectiveness of the proposed approach.

Index Terms- blind system identification, near-common zeros, 2.1. Review of the NMCFLMS algorithmchannel identifiability condition, forced spectral diversity Consider a typical acoustic environment with one talker and mul-

tiple microphones. This system can be modeled as a linear SIMO1. INTRODUCTION system where the relationship between the source signal s(n) and

mth output xm (n) is given byBlind System Identification (BSI) has attracted significant interestdue to its potential applications in several areas. Since the pioneer- xm (n) = Hm (n)s(n) + bm (n), m 1, 2, .. . M (1)ing work by Tong et. al. [1], many second-order statics (SOS)-basedmethods have been proposed, among which are the subspace [2] and where M is the number of channels, xm (n) [xm (n) xm -the cross-relation [3] algorithms. Recently, these methods have been 1) ... xm(n - L + 1)]T and s(n) [s(n) s(n -

adopted from communications into the domain of acoustic signal 1) ... s(n - 2L + 2)]T. The vector bm(n) [bm(n) bm(n -processing such as for acoustic dereverberation and speech source 1) ... bm(n - L + 1)]T denotes the additive noise and hm (n) =separation. Both closed-form [4] and adaptive [5] algorithms have [hm,o(n) hm,i(n) ... hm,L-1 (n)]T is the vector of channel im-been proposed to identify room impulse responses of length up to pulse response of length L, from which the L x (2L - 1) channelhundreds of taps for acoustic dereverberation. matrix is given as

Most SOS-based BSI methods, such as [2][3][4][5], rely on thechannel identifiability conditions [3] from which one of the condi- [ hm,O((n) ... hm,L-1 (n) ... 0 1tions is that the channels must be co-prime, i.e., no zeros being com- Hm (n) ..mon across all channels. This is because when common zeros exist,the BSI algorithms do not have sufficient information to distinguish 0 hm,o(n) hm,L_1(n)the common zeros of the channels from ones due to the source sig-nal. Unlike algorithms for communications applications where vari- with [*] T being the transpose operator. A system equation can thenous pre-processing techniques, such as linear precoding, can be ap- be derived by concatenating all M outputs of (1) as follows:plied to induce specific statistical properties, algorithms for acousticsignal processing in speech dereverberation do not share the similar x(n) = H(n)s(n) + b(n), (3)flexibility. A common way to mitigate the common zeros problemis therefore to increase the number of microphones, which is com- where x(n) [xT(n) X2 (n) ... xm (n)]T,putationally expensive and practically limited. More recently, the b(n) [bT(n) b T(n) ... bT((n)]T and H(n)problem of near-common zeros (NCZs) has been addressed in [6][7] [HT(n) H T(n) ... HT(n)]T is the ML x (2L - 1) globalwhere results presented showed that the performance of SOS-based channel matrix.BSI algorithms can degrade as zeros of different channels become As we are concentrating on the common zeros problem, hence-close to each other. forth we consider only the noise-free case in this paper. In the ab-

In this paper, we first review blind single-input multiple-output sence of noise, a multichannel system can be identified using the(SIMO) system identification in Section 2, where the problem of cross-relation between the ith and jth channel outputs [5], for i 7&NCZs is discussed and illustrated. We then show experimentally, j, given by x[(rn)h3 (n) =xf(n)hi(n) for i, j =1, 2, ... ., M.in Section 3.1, the relationship between the channel characteristics An error function can then be defined, for i 7&jand the performance of the BSI algorithms such as the normalizedmultichannel fast least-mean-square (NMCFLMS) algorithm [5]. In ei (n) =x[(rn)fi3 (ni- 1) -xT(n)fii(ri- 1), (4)

978-1-4244-2338-5/08/$25.00 ©C2008 IEEE 21 ......HSCMA2008

CCluster 1 Cluster 2

M = 2 [ct=302]< -~~~~~~~~~~~~~~~5 M =4[ct=111]

< -o~~~~~~~~~~~~~~r M=6[c=41]

0~~~~t

n0mM8 [c~=19]z

-10-

Fig. 1. An illustrative example of two NCZ clusters for a three-channelSIMO system. -15 ___________________

0 2 4 6 8

300Time (s)

250 . GMC-ST Fig. 3. Performance variation of NMCFLMS [5] with different number ofo ' channels using WGN input and simulated impulse responses.a,200- M =2 ,

Z'150o set of 100 SIMO systems are generated by placing this source-sensor100 configuration in different positions of the room. Finally, results are

z obtained by averaging across all systems.Two generalized multichannel clustering (GMC) algorithms, the

7 '-M= 8 GMC-DC and GMC-ST proposed in [7] were employed to compute0 1 2 3 4 5 6 the number of NCZs over the generated impulse responses. Fig-

Pairwise tolerance x 10-' ure 2 shows the average number of NCZs clusters, denoted as ct,against the tolerance d with different number of channels M. To

Fig. 2. Number of clusters, ct, found using the GMC-DC and GMC-ST [7] show the corresponding performance variation of the BSI algorithmalgorithms against tolerance d with different number of channels M using againt cZs,owethe empoyednteNMCFLMS orithm [5]owithsiuae* mulersoss agyainst NCZs, we then empDloyed the NMCFLMS algorithm [5] withsimulated impulse responses. '-<

a step-size of 0.5. The signal-to-noise ratio (SNR) is set 60 dB toavoid the misconvergence problem [9]. We use the normalized pro-

where h2i(n) is the estimated impulse response for the ith channel. jection misalignment (NPM) [10] to quantify the BSI estimation er-

the NMCFLMS algorithm [5] is derived by minimizing ror. In Fig. 3, performance variation of the NMCFLMS algorithmUsi (4), with different number of channels is shown, where Ct is found forthe cost function -= 6 x 10-3. As can be seen from Fig. 2 and Fig. 3, the num-

M-1 M ber of clusters decreases when M increases, which results in betterJ(n) =E (n), (5) NPM performance. We also note from Fig. 2 that for all sample

h1(n) 2 i=1 j=i+l systems, there exist no exactly common zeros, i.e., no NCZ clusterswere found for d = 0.

where h(n) = [hT(n) hT(n) ... h7T (n)]T is the channel vectorfor estimated impulse responses and 112 denotes 12-norm. Our ob-jective is to estimate hm (n) by employing only observations xm (n) 3. CHANNEL AND SPECTRAL DIVERSITY WITHfor m = 1,2, ... , M. Note that in this work the channel length L is EFFECTIVE UNDERMODELLINGassumed to be available.

It has been shown in Section 2 that the presence of NCZs results inthe performance degradation of the NMCFLMS algorithm. In addi-

2.2. Effect of near-common zeros on the NMCFLMS algorithm tion, increasing spatial diversity using more microphones is compu-

For an M-channel SIMO system, the channels are said to be co- tationally expensive. In this Section, we show that spectral diversityprime if they do not share the same zeros. However, NCZs are clus- can be utilized to reduce the number of NCZ clusters. We proposeters of zeros that satisfy the following conditions [7]: (i) each cluster to obtain such extra diversity for a SIMO system with NCZs via thecontains only M zeros with each channel contributing to a zero, and use of spectral shaping filters and effective channel undermodelling,(ii) the Euclidean distance between any pair of zeros in a cluster lies to which we refer as the FSD concept.within a pairwise tolerance d where d > 0. We note that given thesetwo conditions, any zero can be included in more than one cluster. 3.1. Channel diversity for SIMO systemsFigure 1 shows an example of two clusters for a three-channel sys-tem where the zeros from each channel are represented by triangles, Consider a M-channel SIMO system defined in Section 2, if thesquares and circles. multiple channels do not share exactly common zeros, the global

To illustrate the number of NCZ clusters in SIMO acoustic sys- channel matrix (3) is of full rank [2], i.e., Rank(H(n)) = 2L - 1.tems, a set of impulse responses was simulated using the method of This indicates that the smallest singular value of H(n), denoted asimages [8] with a linear array of M =8 microphones in a room of v2L-1 (H(n)), is non-zero. If exactly common zeros exist, H(n)dimension 10 x 10 x 3 m. The source is located 1 m in front of the becomes rank deficient and leads to (J2L-1 (H(n)) =0. Accordingmicrophone array with uniform spacings of 8 cm. The sampling rate to [11l], the minimum non-zero singular value of the global channelwas 16 kHz with each channel impulse response having 512 coeffi- matrix, i.e., Amin = 2L-1 (H(n)) can be used as a measure ofcients and White Gaussian Noise (WGN) is used as source signals. A channel diversity of H(n).

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X 108 I() L hx,(n) x Inn)c0 /t 1( h(n) n(n)Ct=302 ct=l 11 BSI

O x 2 n x2(n)(2 3 4 5 6 7 8 h(n(n) P) f2(n) (

Number of channelsOrder L Spectral shaping Order L

Fig. 4. Variation of Amin against the number of channels M for simulated filtersimpulse responses used in Fig. 3.

Fig. 5. Schematic for a two-channel FSD SIMO system.

To show that such measurement can also be used for systems co 0 -

with NCZs, Fig. 4 depicts the variation of Amin against differ- a, - LPFent number of channels over impulse responses used in generating HPFFig. 3. The number of NCZs found using the GMC-ST algorithm [7] o-1001o*|for each data point is also shown for d = 6 x 10-3. As can be seen, 0 0.1 0.2 0.3 0.4 0.5the channel diversity increases with M. This indicates that as chan- Normalised frequency (x27: rad)nel diversity increases, the NPM performance of the NMCFLMSalgorithm can improve correspondingly since the number of NCZs Fig. 6. Frequency responses of the spectral shaping filters.reduces with M. It is also noted from the figure that for simulatedacoustic SIMO systems, the value of Amin is small, which can beexpected to result in a large condition number for H(n) and to fur- However, the channel length employed within the NMCFLMS algo-ther degrade the performance of channel inversion algorithms, such rithm is fixed at L and therefore it is equivalently "blind" to such fil-as the MINT algorithm [12], over acoustic SIMO systems. tering process. This results in an effective channel undermodelling,

i.e., the channel estimates produced by the NMCFLMS algorithm,3.2. Spectral diversity with effective undermodelling denoted as hm (n), actually correspond to an undermodelled SIMO

system of length L.Channel undermodelling was introduced for blind identification of For acoustic SIMO systems, the room impulse responses havemicrowave radio impulse responses with small leading and tailing small tailing taps due to the reverberation tail that still exist aftertaps in [ 1], where it was shown that the m-order least-squares (LS) being filtered by spectral shaping filters. This characteristic is similarmethod [3] or subspace method [2] can estimate impulse responses to the microwave radio impulse responses shown in [11]. Therefore,that are close to the m-order "significant" part of the full impulse we propose to effectively undermodel hm (n) by removing the lastresponses which is obtained by removing small leading and tailing LP- 1 taps of hm (n), i.e.,taps. This can only be achieved when the undermodelled system n Uh n M 1 2 9offers sufficient diversity. The attempt of modelling full-length im- m(n) m ( ) mpulse responses, however, can result in much worse estimation per- where U = [ILXL, OLX(L..-1)I with ILXL and OLX(L -1) beingformance. a L x L identity matrix and a L x (LP- 1) null matrix. As will be

Inspired by results in [11], our proposed FSD concept includes shown in Section 4, the combination of spectral shaping filters andthe use of spectral shaping filters and effective channel undermod- the effective channel undermodelling can result in increased channelelling. For clarity of presentation, FSD in this paper is based on a diversity and reduced the number of NCZ clusters.two-channel system case and a schematic for a two-channel SIMO Finally, we note that the original system can be recovered insystem is shown in Fig. 5. As can be seen, the microphone signal principle since the characteristics of the spectral shaping filters arexm (n) is pre-processed using a pair of spectral shaping filters, e.g., known and could be inverted. In practice, effects of noise ampli-a pair of high- and lowpass filters (LPF and HPF), the resulting filter fication in the inversion process will limit accuracy. However, anoutputs X/m (n) are then given as estimate of s(n) can be obtained without the need to "undo" the ef-

fect of the spectral shaping by inverting hm (n) using, for example,Xm (n) =1§iHm (n)s(n) (6) the MINT algorithm [12]. Similar practical limitations apply but

we have found in our tests that such limitations are more than out-

fotrimx 1 2, wheen m denote th m p a i l weighed by the advantages obtained in the BSI due to the increasedmatrix of dimension L x (L + L~ - 1), channel diversity, resulting in an overall improvement in our ability

[ frn,o fm,Lp-1 o 1 to equalize channels containing NCZs.fm,o fm,L I

j (7)Sm = . . .. . 7 ( ) ~~~~~~4.SIMULATIONSfm,o fm,Lv- I

In this section, we further demonstrate the effect of FSD using the setand fm [fm,0, fm,i,... fm,L__ l'T is the impulse response of of two-channel SIMO systems obtained from the impulse responsethe mth filter of length LP. According to Fig. 5, xm (n) can be library that has been used throughout this paper. A pair of high- andconsidered as the linear convolution between s(n) and an equivalent lowpass shaping filters of length LP 32 was generated with theirSIMO system of length L + LP-1, i.e., magnitude responses shown in Fig. 6.

To demonstrate the effect of FSD on SIMO systems with NCZs,hm (T) =~F hm(n), m 1, 2. (8) we first measure the channel diversity over the set of two-channel

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t O..Original system -Original system-FSD system -21--- FSD system

-4z

-lo_ -6

0) ~~~~~~~~~~~~~~~~~~~~~~~n-8-

-10

-12 \

0 20 40 60 80 100 -14 ___2Index of Impulse responses 0 2 4 6 8

Time (s)

Fig. 7. Variation of channel diversity Amin for the original system hm n Fig. 9. Performance comparison for the NMCFLMS algorithm between theand the FSD system h' (n) for different simulated impulse responses. iginal semfhmm(nc) andsth fSD teN F ( rm).

m original system hm (n) and the FSD system h' (n).

-0OeOriginal system- 250 B- FSD system Inform. Theory, vol. 40, no. 2, pp. 340-349, March 1994.

D 200 ,[2] E. Moulines, P. Duhamel, J. F. Cardoso, and S. Mayrargue, "Subspacea) 1 50c ' methods for the blind identification of multichannel FIR filters," IEEEE 100 Trans. Signal Process., vol. 43, no. 3, pp. 516-525, Feb. 1995.z [3] G. Xu, H. Liu, L. Tong, and T. Kailath, "A least-squares approach

5(t3 3.5 4 4.5 c, to blind channel identification," IEEE Trans. Signal Process., vol. 43,Pairwise Tolerance 6 x 10-3 no. 12, pp. 2982-2993, Dec. 1995.

[4] S. Gannot and M. Moonen, "Subspace methods for multi-microphoneFig. 8. Comparison of ct between the original system hm (n) and the FSD speech dereverberation," EURASIP J. Applied Signal Process., vol.system h' (n) against various tolerance 6. 2003, no. 11, pp. 1074-1090, Oct. 2003.

[5] Y Huang and J. Benesty, "A class of frequency-domain adaptive ap-proaches to blind multichannel identification," IEEE Trans. Signal Pro-

SIMO systems chosen from a set of impulse responses described in cess., vol. 51, no. 1, pp. 11-24, Jan. 2003.Section 2.2, where hm (n) and h' (n) are obtained using (9) in a [6] X. S. Lin, N. D. Gaubitch, and P. A. Naylor, "Two-stage blind iden-similar way as described in [13]. In Fig. 7, the variation of Amin tification of SIMO systems with common zeros," in Proc. Europeanfor the original system hm (n) and the FSD system h' (n) is plot- Signal Process. Conf (EUSIPCO), Sept. 2006.ted against different impulse responses. The FSD system is seen [7] A. W. Khong, X. S. Lin, and P. A. Naylor, "Algorithms for identifyingto provides a consistent improvement in terms of channel diversity clusters of near-common zeros in multichannel blind system identifica-over the original system, which implies that h' (n) can be better tion and equalization," in Proc. IEEE Int. Conf Acoust., Speech, Signalidentified. Using the same set of impulse responses, we then com- Process. (ICASSP), Apr. 2008.pare Fig. 8 the number of NCZ clusters between hm (n) and h' (n) [8] J. B. Allen and D. A. Berkley, "Image method for efficiently simulatingagainst pairwise tolerance d. As can be seen, the FSD system h' (n) small room acoustics," J. Acoust. Soc. Am., vol. 65, no. 4, pp. 943-950,only contains approximately half of the number of NCZs compared April 1979.to the original system hm (n) . [9] R. Ahmad, A. Khong, and P. Naylor, "Proportionate frequency domain

Finally, we show in Fig. 9 that the FSD system is more identi- adaptive algorithms for blind channel identification," in Proc. IEEE Int.fiable than the original system in terms of NPM performance. The Conf Acoust., Speech, Signal Process. (ICASSP), May 2006.parameters used are the same as in Section 2 and previous simula- [10] D. Morgan, J. Benesty, and M. M. Sondhi, "On the evaluation of esti-tions. An approximate 3 dB improvement can be clearly seen which mated impulse responses," IEEE Signal Process. Letters, vol. 5, no. 7,is close to the NPM performance for the case ofM = 6 in Fig. 3. pp. 174-176, July 1998.

[11] A. P. Liavas, P. A. Regalia, and J.-P. Delmas, "Robustness ofleast-squares and subspace methods for blind channel identifi-

5. CONCLUSIONS cation/equalization with respect to effective channel undermodel-ing/overmodeling," IEEE Trans. Signal Process., vol. 47, no. 6, pp.

We have addressed and illustrated the near-common zeros problem 1636-1645, June 1999.for blind system identification. We showed the link between the [12] M. Miyoshi andY. Kaneda, "Inverse filtering of room acoustics," IEEEchannel diversity and the NPM performance of the NMCFLMS al- Trans. Acoust., Speech, Signal Process., vol. 36, no. 2, pp. 145-152,gorithm. The FSD concept for blind identification of acoustic SIMO Feb. 1988.system in the presence of NCZs has been proposed, which combines [13] M. K. Hasan, J. Benesty, P. A. Naylor, and D. B. Ward, "Improvingthe use of spectral shaping filters and the effective channel under- robustness of blind adaptive multichannel identification algorithms us-modelling method. Simulation results based on two-channel case ing constraints," in Proc. European Signal Process. Conf (EUSIPCO),confirmed the effectiveness of the proposed concept. Sep. 2005.

6. REFERENCES

[1] L. Tong, G. Xu, and T. Kailath, "Blind identification and equalizationbased on second order statistics: a time domain approach," IEEE Trans.

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