The generalized eigendecomposition approach to the blind source separation problem

Abstract

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.