BIBO Stability of D-Dimensional Filters

Abstract

Consider the class of d-dimensional causal filters characterized by a d-variate rational function $H(z) = \frac{{P(z)}}{{Q(z)}}$ analytic on the polydisk $\mathbb{D}^d = \{ z = (z_1 , \ldots ,z_d ):|z_i | < 1\}$. The BIBO stability of such filters has been the subject of much research over the past two decades. In this paper we analyze the BIBO stability of such functions and prove necessary and sufficient conditions for BIBO stability of a d-dimensional filter. In particular, we prove if a d-variate filter H(z) analytic on $\mathbb{D}^d$ has a Fourier expansion that converges uniformly on the closure of $\mathbb{D}^d$, then H(z) is BIBO stable. This result proves a long standing conjecture set forth by Bose in [3].