# Blind Source Separation for Convolutive Mixtures: A Unified Treatment

## Abstract

Blind source separation (BSS) algorithms for time series can exploit three properties of the source signals: nonwhiteness, nonstationarity, and nongaussianity. While methods utilizing the first two properties are usually based on second-order statistics (SOS), higher-order statistics (HOS) must be considered to exploit nongaussianity. In this chapter, we consider all three properties simultaneously to design BSS algorithms for convolutive mixtures within a new generic framework. This concept derives its generality from an appropriate matrix notation combined with the use of multivariate probability densities for considering the time-dependencies of the source signals. Based on a generalized cost function we rigorously derive the corresponding time-domain and frequency-domain broadband algorithms. Due to the broadband approach, time-domain constraints are obtained which provide a more detailed understanding of the internal permutation problem in traditional narrowband frequency-domain BSS. For both, the time-domain and the frequency-domain versions, we discuss links to well-known and also to novel algorithms that follow as special cases of the framework. Moreover, we use models for correlated spherically invariant random processes (SIRPs) which are well suited for a variety of source signals including speech to obtain efficient solutions in the HOS case. The concept provides a basis for off-line, online, and block-on-line algorithms by introducing a general weighting function, thereby allowing for tracking of time-varying real acoustic environments.

## Keywords

Blind Source Separation Convolutive Mixtures Second-Order Statistics Higher-Order Statistics Time Domain Frequency Domain Broadband Approach Spherically Invariant Random Processes## Preview

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## References

- [1]A. Hyvärinen, J. Karhunen, and E. Oja,
*Independent Component Analysis*, Wiley & Sons, Inc., New York, 2001.Google Scholar - [2]M. Zibulevsky and B.A. Pearlmutter, “Blind source separation by sparse decomposition in a signal dictionary,”
*Neural Computation*, vol. 13, pp. 863–882, 2001.CrossRefGoogle Scholar - [3]S. Araki, S. Makino, A. Blin, R. Mukai, and H. Sawada, “Blind Separation of More Speech than Sensors with Less Distortion by Combining Sparseness and ICA,” in
*Proc. Int. Workshop on Acoustic Echo and Noise Control (IWAENC)*, Kyoto, Japan, Sep. 2003, pp. 271–274.Google Scholar - [4]J.-F. Cardoso and A. Souloumiac, “Blind beamforming for non gaussian signals,”
*IEE Proceedings-F*, vol 140, no. 6, pp. 362–370, Dec. 1993.Google Scholar - [5]W. Herbordt and W. Kellermann, “Adaptive beamforming for audio signal acquisition,” in
*Adaptive signal processing: Application to real-world problems*, J. Benesty and Y. Huang, Eds., pp. 155–194, Springer, Berlin, Jan. 2003.Google Scholar - [6]E. Weinstein, M. Feder, and A. Oppenheim, “Multi-channel signal separation by decor-relation,”
*IEEE Trans. on Speech and Audio Processing*, vol 1, no. 4, pp. 405–413, Oct. 1993.Google Scholar - [7]L. Molgedey and H.G. Schuster, “Separation of a mixture of independent signals using time delayed correlations,”
*Physical Review Letters*, vol. 72, pp. 3634–3636, 1994.CrossRefGoogle Scholar - [8]L. Tong, R.-W. Liu, V.C. Soon, and Y.-F. Huang, “Indeterminacy and identifiability of blind identification,”
*IEEE Trans. on Circuits and Systems*, vol. 38, pp. 499–509, 1991.CrossRefGoogle Scholar - [9]S. Van Gerven and D. Van Compernolle,”Signal separation by symmetric adaptive decor-relation: stability, convergence, and uniqueness,”
*IEEE Trans. Signal Processing*, vol. 43, no. 7, pp. 1602–1612, 1995.Google Scholar - [10]K. Matsuoka, M. Ohya, and M. Kawamoto, “A neural net for blind separation of nonstationary signals,”
*Neural Networks*, vol. 8, no. 3, pp. 411–419, 1995.CrossRefGoogle Scholar - [11]M. Kawamoto, K. Matsuoka, and N. Ohnishi, “A method of blind separation for convolved non-stationary signals,”
*Neurocomputing*, vol. 22, pp. 157–171, 1998.CrossRefGoogle Scholar - [12]J.-F. Cardoso and A. Souloumiac, “Jacobi angles for simultaneous diagonalization,”
*SIAM J. Mat. Anal. Appl.*, vol. 17, no. 1, pp. 161–164, Jan. 1996.MathSciNetGoogle Scholar - [13]S. Ikeda and N. Murata, “An approach to blind source separation of speech signals,”
*Proc. Int. Symposium on Nonlinear Theory and its Applications*, Crans-Montana, Switzerland, 1998.Google Scholar - [14]L. Parra and C. Spence, “Convolutive blind source separation of non-stationary sources,”
*IEEE Trans. Speech and Audio Processing*, pp. 320–327, May 2000.Google Scholar - [15]D.W.E. Schobben and P.C.W. Sommen, “A frequency-domain blind signal separation method based on decorrelation,”
*IEEE Trans on Signal Processing*, vol. 50, no. 8, pp. 1855–1865, Aug. 2002.CrossRefGoogle Scholar - [16]H.-C. Wu and J.C. Principe,”Simultaneous diagonalization in the frequency domain (SDIF) for source separation,” in
*Proc. IEEE Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA)*, 1999, pp. 245–250.Google Scholar - [17]C.L. Fancourt and L. Parra, “The coherence function in blind source separation of convolutive mixtures of non-stationary signals,” in Proc.
*Int. Workshop on Neural Networks for Signal Processing (NNSP)*, 2001.Google Scholar - [18]P. Comon, “Independent component analysis, a new concept?”
*Signal Processing*, vol. 36, no. 3, pp. 287–314, Apr. 1994.CrossRefzbMATHGoogle Scholar - [19]A. Cichocki and S. Amari,
*Adaptive Blind Signal and Image Processing*, Wiley & Sons, Ltd., Chichester, UK, 2002.Google Scholar - [20]S. Amari, A. Cichocki, and H.H. Yang, “A new learning algorithm for blind signal separation,” in
*Advances in neural information processing systems*, 8, Cambridge, MA, MIT Press, 1996, pp. 757–763.Google Scholar - [21]J.-F. Cardoso, “Blind signal separation: Statistical principles,”
*Proc. IEEE*, vol. 86, pp. 2009–2025, Oct. 1998.Google Scholar - [22]P. Smaragdis, “Blind separation of convolved mixtures in the frequency domain,”
*Neurocomputing*, vol. 22, pp. 21–34, July 1998.Google Scholar - [23]A.J. Bell and T.J. Sejnowski, “Aninformation-maximisation approach to blind separation and blind deconvolution,”
*Neural Computation*, vol. 7, pp. 1129–1159, 1995.Google Scholar - [24]T. Nishikawa, H. Saruwatari, and K. Shikano, “Comparison of time-domain ICA, frequency-domain ICA and multistage ICA for blind source separation,” in
*Proc. European Signal Processing Conference (EUSIPCO)*, Sep. 2002, vol. 2, pp. 15–18.Google Scholar - [25]R. Aichner, S. Araki, S. Makino, T. Nishikawa, and H. Saruwatari, “Time-domain blind source separation of non-stationary convolved signals with utilization of geometric beam forming,” in Proc.
*Int. Workshop on Neural Networks for Signal Processing (NNSP)*, Martigny, Switzerland, 2002, pp. 445–454.Google Scholar - [26]H. Buchner, R. Aichner, and W. Kellermann, “A generalization of a class of blind source separation algorithms for convolutive mixtures,”
*Proc. IEEE Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA)*, Nara, Japan, Apr. 2003, pp. 945–950.Google Scholar - [27]H. Buchner, R. Aichner, and W. Kellermann, “Blind Source Separation for Convolutive Mixtures Exploiting Nongaussianity, Nonwhiteness, and Nonstationarity,”
*Proc. Int. Workshop on Acoustic Echo and Noise Control (IWAENC)*, Kyoto, Japan, September 2003.Google Scholar - [28]R. Aichner, H. Buchner, S. Araki, and S. Makino, “On-line time-domain blind source separation of nonstationary convolved signals,”
*Proc. IEEE Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA)*, Nara, Japan, Apr. 2003, pp. 987–992.Google Scholar - [29]E. Moulines, O. Ait Amrane, and Y. Grenier, “The generalized multidelay adaptive filter: structure and convergence analysis,”
*IEEE Trans. Signal Processing*, vol. 43, pp. 14–28, Jan. 1995.Google Scholar - [30]H. Brehm and W. Stammler, “Description and generation of spherically invariant speech-model signals,”
*Signal Processing*vol. 12, pp. 119–141, 1987.CrossRefGoogle Scholar - [31]H. Buchner, J. Benesty, and W. Kellermann, “MultichannelFrequency-Domain Adaptive Algorithms with Application to Acoustic Echo Cancellation,” in J. Benesty and Y. Huang (eds.),
*Adaptive signal processing: Application to real-world problems*, Springer-Verlag, Berlin/Heidelberg, Jan. 2003.Google Scholar - [32]H. Gish and D. Cochran, “Generalized Coherence,”
*Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP)*, New York, NY, USA, 1988, pp. 2745–2748.Google Scholar - [33]H.H. Yang and S. Amari, “Adaptive online learning algorithms for blind separation: maximum entropy and minimum mutual information,”
*Neural Computation*, vol. 9, pp. 1457–1482, 1997.Google Scholar - [34]T.M. Cover and J.A. Thomas,
*Elements of Information Theory*, Wiley & Sons, New York, 1991.Google Scholar - [35]S. Haykin,
*Adaptive Filter Theory*, 3rd ed., Prentice Hall, Englewood Cliffs, NJ, 1996.Google Scholar - [36]D.H. Brandwood, “A complex gradient operator and its application in adaptive array theory,”
*Proc. IEE*, vol. 130, Pts. F and H, pp. 11–16, Feb. 1983.Google Scholar - [37]A. Papoulis,
*Probability, Random Variables, and Stochastic Processes*, 3rd ed., McGraw-Hill, New York, 1991.Google Scholar - [38]F.D. Neeser and J.L. Massey, “Proper Complex Random Processes with Applications to Information Theory,”
*IEEE Trans. on Information Theory*, vol. 39, no. 4, pp. 1293–1302, July 1993.CrossRefMathSciNetGoogle Scholar - [39]S. Amari, “Natural gradient works efficiently in learning,”
*Neural Computation*, vol. 10, pp. 251–276, 1998.CrossRefGoogle Scholar - [40]J.D. Markel and A.H. Gray
*Linear Prediction of Speech*, Springer-Verlag, Berlin, 1976.Google Scholar - [41]J.W. Brewer, “Kronecker Products and Matrix Calculus in System Theory,”
*IEEE Trans. Circuits and Systems*, vol. 25, no. 9, pp. 772–781, Sep. 1978.CrossRefzbMATHMathSciNetGoogle Scholar - [42]M.Z. Ikram and D.R. Morgan, “Exploring permutation inconsistency in blind separation of speech signals in a reverberant environment,”
*Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP)*, Istanbul, Turkey, June 2000, vol. 2, pp. 1041–1044.Google Scholar - [43]H. Sawada, R. Mukai, S. Araki, and S. Makino, “Robust and precise method for solving the permutation problem of frequency-domain blind source separation,”
*Proc. IEEE Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA)*, Nara, Japan, Apr. 2003, pp. 505–510.Google Scholar - [44]J.-S. Soo and K.K. Pang, “Multidelay block frequency domain adaptive filter,”
*IEEE Trans. Acoust., Speech, Signal Processing*, vol. ASSP-38, pp. 373–376, Feb. 1990.Google Scholar - [45]P.C.W. Sommen, P.J. Van Gerwen, H.J. Kotmans, and A.J.E.M. Janssen, “Convergence analysis of a frequency-domain adaptive filter with exponential power averaging and generalized window function,”
*IEEE Trans. Circuits and Systems*, vol. 34, no. 7, pp. 788–798, July 1987.CrossRefGoogle Scholar - [46]J. Benesty, A. Gilloire, and Y. Grenier, “A frequency-domain stereophonic acoustic echo canceller exploiting the coherence between the channels,”
*J. Acoust. Soc. Am.*, vol. 106, pp. L30–L35, Sept. 1999.Google Scholar - [47]G. Enzner and P. Vary, “A soft-partitioned frequency-domain adaptive filter for acoustic echo cancellation,” in
*Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP)*, Hong Kong, China, April 2003, vol. 5, pp. 393–396.Google Scholar - [48]R.M.M. Derkx, G.P.M. Egelmeers, and P.C.W. Sommen, “New constraining method for partitioned block frequency-domain adaptive filters,”
*IEEE Trans. Signal Processing*, vol. 50, no. 9, pp. 2177–2186, Sept. 2002.CrossRefGoogle Scholar - [49]F.J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,”
*Proc. IEEE*, vol. 66, pp. 51–83, Jan. 1978.Google Scholar - [50]S. Sawada, R. Mukai, S. de la Kethulle de Ryhove, S. Araki, and S. Makino, “Spectral smoothing for frequency-domain blind source separation,” in
*Proc. Int. Workshop on Acoustic Echo and Noise Control (IWAENC)*, Kyoto, Japan, Sep. 2003, pp. 311–314.Google Scholar - [51]R. Mukai, H. Sawada, S. Araki, and S. Makino, “Real-Time Blind Source Separation for Moving Speakers using Blockwise ICA and Residual Crosstalk Subtraction,”
*Proc. IEEE Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA)*, Nara, Japan, Apr. 2003, pp. 975–980.Google Scholar - [52]J.S. Garofolo et al., “TIMIT acoustic-phonetic continuous speech corpus,” National Institute of Standards and Technology, 1993.Google Scholar