Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

Abstract

The properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle.