A Matrix Pseudoinversion 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 <i>A</i> added to a block of dyads (represented as <i>BB</i><sup>H</sup>) as follows: (<i>A</i>+<i>BB</i><sup>H</sup>)<sup>-1</sup>= <i>A</i><sup>-1</sup>- <i>A</i><sup>-1</sup><i>B</i>(<i>I</i> + <i>B</i><sup>H</sup><i>A</i><sup>-1</sup><i>B</i>)<sup>-1</sup><i>B</i><sup>H</sup><i>A</i><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 <i>A</i> is singular and present a matrix pseudoinversion lemma along with some illustrative examples. Based on this result, we propose a block-based adaptive multichannel superexponential algorithm. We present simulation results for the performance of the block-based algorithm in order to show the usefulness of the matrix pseudoinversion lemma.

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