The generalized eigendecomposition approach to the blind source separation problem

This paper proposes a novel formulation of the generalized eigendecomposition (GED) approach to blind source separation (BSS) problems. The generalized eigendecomposition algorithms consider the estimation of a pair of correlation matrices (a matrix pencil) using observed sensor signals. Each of various algorithms proposed in the literature uses a different approach to form the pencil. This study proposes a linear algebra formulation which exploits the definition of congruent matrix pencils and shows that the solution and its constraints are independent of the way the matrix pencil is computed. Also an iterative eigendecomposition algorithm, that updates separation parameters on a sample-by-sample basis, is developed. It comprises of: (1) performing standard eigendecompositions based on power and deflation techniques; (2) computing a transformation matrix using spectral factorization. Another issue discussed in this work is the influence of the length of the data segment used to estimate the pencil. The algorithm is applied to artificially mixed audio data and it is shown that the separation performance depends on the eigenvalue spread. The latter varies with the number of samples used to estimate the eigenvalues.

[1]  Dominique Martinez,et al.  Nonlinear blind source separation using kernels , 2003, IEEE Trans. Neural Networks.

[2]  A.M. Tome.  Blind source separation using a matrix pencil , 2000, Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks. IJCNN 2000. Neural Computing: New Challenges and Perspectives for the New Millennium.

[3]  Andrzej Cichocki,et al.  Adaptive blind signal and image processing , 2002 .

[4]  Zhi Ding,et al.  A matrix-pencil approach to blind separation of non-white sources in white noise , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[5]  Lang Tong,et al.  Indeterminacy and identifiability of blind identification , 1991 .

[6]  Fabian J. Theis,et al.  Blind Source Separation of Water Artefacts in NMR Spectra using a Matrix Pencil , 2003 .

[7]  Lucas C. Parra,et al.  Blind Source Separation via Generalized Eigenvalue Decomposition , 2003, J. Mach. Learn. Res..

[8]  Michael J. Kirby,et al.  Blind source separation using the maximum signal fraction approach , 2002, Signal Process..

[9]  Aapo Hyvärinen,et al.  A Fast Fixed-Point Algorithm for Independent Component Analysis , 1997, Neural Computation.

[10]  Eric Moulines,et al.  A blind source separation technique using second-order statistics , 1997, IEEE Trans. Signal Process..

[11]  Jean-François Cardoso,et al.  Equivariant adaptive source separation , 1996, IEEE Trans. Signal Process..

[12]  Zhi Ding,et al.  A two-stage algorithm for MIMO blind deconvolution of nonstationary colored signals , 2000, IEEE Trans. Signal Process..

[13]  Andrzej Cichocki,et al.  Adaptive Blind Signal and Image Processing - Learning Algorithms and Applications , 2002 .

[14]  Antoine Souloumiac,et al.  Blind source detection and separation using second order non-stationarity , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[15]  Juha Karhunen,et al.  Principal component neural networks — Theory and applications , 1998, Pattern Analysis and Applications.

[16]  Schuster,et al.  Separation of a mixture of independent signals using time delayed correlations. , 1994, Physical review letters.

[17]  James V. Stone.  Blind Source Separation Using Temporal Predictability , 2001, Neural Computation.

[18]  Michael Kirby,et al.  A SOLUTION PROCEDURE FOR BLIND SIGNAL SEPARATION USING THE MAXIMUM NOISE FRACTION APPROACH: ALGORITHMS AND EXAMPLES , 2001 .

[19]  Ana Maria Tomé.  AN ITERATIVE EIGENDECOMPOSITION APPROACH TO BLIND SOURCE SEPARATION , 2001 .

[20]  Zhi Ding,et al.  A matrix-pencil approach to blind separation of colored nonstationary signals , 2000, IEEE Trans. Signal Process..

[21]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .