Elementary Matrix Decomposition Algorithm for Symmetric Extension of Laurent Polynomial Matrices and Its Application in Construction of Symmetric M-Band Filter Banks

Abstract

In this paper, we develop a novel and effective algorithm for the construction of perfect reconstruction filter banks (PRFBs) with linear phase. In the algorithm, the key step is the symmetric Laurent polynomial matrix extension (SLPME). There are two typical problems in the construction: (1) For a given symmetric finite low-pass filter $\mathbf {a}$ with the polyphase, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is $\mathbf {a}$. (2) For a given dual pair of symmetric finite low-pass filters, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is $\mathbf {a}$, while its low-pass band of the synthesis filter bank is $\mathbf {b}$. In the paper, we first formulate the problems by the SLPME of the Laurent polynomial vector(s) associated to the given filter(s). Then we develop a symmetric elementary matrix decomposition algorithm based on Euclidean division in the ring of Laurent polynomials, which finally induces our SLPME algorithm.