A Matrix Pseudo-Inversion Lemma and Its Application to Block-Based Adaptive Blind Deconvolution for MIMO Systems

The matrix inversion lemma gives an explicit formula of the inverse of a positive-definite matrix A added to a block of dyads (represented as BBH) as follows: (A + BB<sup>H</sup>)<sup>-1</sup> = A<sup>-1</sup> - A<sup>-1</sup> B(I + B<sup>H</sup> A<sup>-1</sup> B) <sup>-1</sup> B<sup>H</sup> A<sup>-1</sup>. It is well-known in the literature that this formula is very useful to develop a block-based recursive least-squares algorithm for the block-based recursive identification of linear systems or the design of adaptive filters. We extend this result to the case when the matrix A is singular, and present a matrix pseudo-inversion lemma. Based on this result, we propose a block-based adaptive multi-channel super-exponential algorithm (BAMSEA). We present simulation results for the performance of the block-based algorithm in order to show the usefulness of the matrix pseudo-inversion lemma.

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