A Matrix Pseudo-Inversion Lemma for Positive Semidefinite Hermitian Matrices and Its Application to Adaptive Blind Deconvolution of MIMO Systems

Abstract

Abstract-In the simplest case, the matrix inversion lemma gives an explicit formula of the inverse of a positive-definite matrix added to a rank-one matrix as follows:(1 It is well known in the literature that this formula is very useful to develop a recursive leastsquares algorithm for the recursive identification of linear systems or the design of adaptive filters. We extend this result to the case when the matrix is singular and present a matrix pseudo-inversion lemma along with some illustrative examples. Such a singular case may occur in a situation where a given problem is overdetermined in the sense that it has more equations than unknowns. This lemma is important in its own right, but in order to show the usefulness of the lemma, we apply it to develop an adaptive superexponential algorithm for the blind deconvolution of multi-input multi-output systems.Index Terms-Adaptive super-exponential algorithm (SEA), matrix pseudo-inversion lemma, recursive algorithms.

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

“…In particular, we encountered this singular situation when we developed a sample-based adaptive version of the superexponential method for the blind deconvolution of multi-input-multioutput (MIMO) systems, where the number of its outputs is greater than the number of its inputs. It should be noted that our previous work on the matrix pseudoinversion lemma is restricted to the case when the added matrix is a single dyad (i.e., is a column vector) [7]- [9]. Therefore, we can confirm some differences between the present work and the previous one, e.g., see Remark 2 and Tables I and II. After the presentation of the matrix pseudoinversion lemma, we apply this lemma to block-based adaptive blind deconvolution of a MIMO system.…”

A Modified Eigenvector Method for Blind Deconvolution of MIMO Systems Using the Matrix Pseudo-Inversion Lemma

“…Here, the calculation of † R is implemented by using the following algorithm based on the matrix pseudo-inversion lemma proposed in [10]. The reason is that in the case that the pseudo-inverse is calculated using data block, the convergence speed is increased and the computational complexity is reduced, compared with the conventional pseudo-inverse algorithms, for example, the built-in function "pinv" in MATLAB [11]. Therefore, in order to provide a recursive formula based on block data for time-updating of pseudo-inverse, the block index "k" is defined, and then R and † R are described as (k) R and P(k), respectively, where the k-th block of data is defined as…”

A modified eigenvector method for blind deconvolution of MIMO systems using the matrix pseudo-inversion lemma

“…The reason is that in the case that the pseudo-inverse is calculated using data block, the convergence speed is increased and the computational complexity is reduced, compared with the conventional pseudo-inverse algorithms, for example, the built-in function "pinv" in MATLAB [11]. Therefore, in order to provide a recursive formula based on block data for time-updating of pseudo-inverse, the block index "k" is defined, and then R and † R are described as (k) R and P(k), respectively, where the k-th block of data is defined as…”