An Adaptive Super-Exponential Deflation Algorithm for Blind Deconvolution of MIMO Systems Using the Matrix Pseudo-Inversion Lemma

Abstract

The multichannel blind deconvolution of finite-impulse response (FIR) or infinite-impulse response (IIR) channels is investigated using the multichannel super-exponential deflation methods. We propose a new adaptive approach to the multichannel super-exponential deflation methods using the matrix pseudo-inversion lemma (which is extended from the matrix inversion lemma in the full-rank case to the rank-degenerate case) and the higher-order cross correlations of the channel and the equalizer outputs. In order to see the effectiveness of the proposed approach, many computer simulations are carried out for time-invariant MIMO channels along with time-variant MIMO channels. It is shown through computer simulations that the proposed approach is effective for even time-variant channels.

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

“…We note here that we do not use Theorem 2, because we are not interested in finding 's but interested in calculating . The results of calculating 's and recovering original sources 's are found in [11] and [14].…”

“…Only when , just the channel changed, and the recursion formula (59) in Lemma 2 was used for calculating the pseudo-inverse of . Therefore, we consider that the value of rank normalized by the length of the deconvolver can be used as an index for detecting the change of the channel and for changing the recursion formulas of the matrix pseudo-inversion lemma from (4)-(6) (see [14] for details).…”

“…Remark 1: In the late 1980s, Ogawa extended the matrix inversion lemma to the singular case and presented an operator pseudo-inversion lemma [15]. Instead of the adding term in (3), he treated a more general adding term , where is an operator and is an positive definite operator, but he gave the operator pseudo-inversion lemma under the condition (11) Therefore, case 1) of Lemma 1 is included in the case he just considered, but cases 2) and 3) are not treated by him. We should note that the above condition does not hold true in a nonstationary environment for blind deconvolution of MIMO systems (see Section IV for details).…”

“…Remark 3: Based on Theorem 2, we have developed a recursive algorithm implementing the recursion formula (62) [14], but we have not yet theoretically analyzed the stability and rate of convergence of the algorithm. See [16] for details of the stability analysis of the standard recursive least-squares (RLS) algorithm and its variants.…”

“…Such a singular case may occur in a situation where a problem dealt with is overdetermined in the sense that it has more equations than unknowns. In particular, we encountered this singular situation when we developed an adaptive version of the super-exponential method (SEM) for the blind deconvolution of a multi-input multi-output (MIMO) system, where the number of its outputs is greater than the number of its inputs [9]- [11].…”