Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems

Abstract

In this paper, we study the invertibility of M-variate Laurent polynomial N times P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given an N times P Laurent polynomial matrix H(z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , ..., z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> ) of degree at most k, we want to find a P times N Laurent polynomial left inverse matrix G(z) of H(z) such that G(z)H(z) = J. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this paper is to prove that H(z) is generically invertible when N - P ges M; whereas when N - P < M, then H(z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us.