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
A
added to a block of dyads (represented as
BB
^H
) as follows: (
A
+
BB
^H
)
^-1
=
A
^-1
-
A
^-1
B
(
I
+
B
^H
A
^-1
B
)
^-1
B
^H
A
^-1
. 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 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.