Efficient solution of block linear systems with Toeplitz entries using a channel decomposition technique

Abstract

Abstract This paper is concerned with the development of efficient solvers for block linear systems with Toeplitz entries. Fast block Toeplitz solvers are required in a wide range of applications in the area of multichannel and multidimensional digital signal processing. The presentation and the derivation of all algorithms proposed in this paper are based on the context of multichannel FIR Wiener filtering. A novel channel decomposition technique is applied to obtain fast, order recursive algorithms that require scalar operations, only. The proposed algorithms can accommodate multichannel filter of different filter orders and manage to get free of matrix operations. Two basic algorithm structures are derived, the Levinson type and the Schur type. They are both based on suitable permutations that unravel the recursive nature of the pertinent matrices and enable the development of fast block linear system solvers. Normalized versions are also derived and a simple test for detecting the positive definiteness of a block matrix with Toeplitz entries, is obtained. An efficient lattice structure, that requires scalar operations, is also derived and is subsequently used to obtain Schur-type recursions. The Schur-type algorithm admits full parallelism and, if parallel processing environment is available, it reduces processing time by an order of magnitude. To illustrate the structure of the proposed algorithms, Matlab code is provided for the fast Levinson-type algorithm, as well as for the Schur counterpart.