Using Joint Generalized Eigenvectors of a Set of Covariance Matrix Pencils for Deflationary Blind Source Extraction

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In this paper, we develop a new deflationary blind source extraction (BSE) algorithm that extracts source signals in a sequential fashion via the joint generalized eigenvectors of a set of covariance matrix pencils. The new concept of joint generalized eigenvector is defined. We prove that these vectors can be made unique and identical to the source extraction vectors with properly selected matrix pencils. To resolve the open problem of estimating joint generalized eigenvectors, we develop an approach based on the deflation operation and the proportional property of the joint generalized eigenvectors. Specifically, with the proportional property, we show that the estimation problem can be formulated as an optimization involving a quadratic cost function and a unit-rank matrix constraint. An efficient iterative algorithm is then developed by applying the gradient search, matrix shrinkage, deflation, and symmetry-preserving vectorization techniques. This algorithm estimates the joint generalized eigenvectors and conducts BSE sequentially. Its computational complexity and convergence are analyzed. Simulations demonstrate that this algorithm outperforms many typical BSE or blind source separation algorithms. In particular, the new algorithm is more robust to both heavy noise and ill-conditioned mixing matrices.