Generic invertibility of multidimensional FIR filter banks and MIMO systems

  • Authors:

  • Ka L. Law;Robert M. Fossum;Minh N. Do

  • Affiliations:

  • Department of Communication Systems, Technische Universität, Darmstadt, Germany and Department of Mathematics and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaig ...;Department of Mathematics and the Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL;Department of Electrical and Computer Engineering, Coordinated Science Laboratory, and the Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL

  • Venue:
  • IEEE Transactions on Signal Processing
  • Year:
  • 2009

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Abstract

In this paper, we study the invertibility of M-variate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given anN × P Laurent polynomial matrix H(Z1,...,ZM) of degree at most k, we want to find a P × N Laurent polynomial left inverse matrix G(Z) of H(Z) such that G(Z) H(Z)=I. 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 ≥ 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.