A simple least-squares design of M-D IIR filters with fixed separable denominator based on multivariate division algorithm

摘要

In this paper, we propose a simple algorithmic solution to the best approximation problem of finding the nearest multivariate rational function, with a fixed separable denominator polynomial, from a given multivariate polynomial, where the numerator polynomial is desired to minimize the integral of the squared error over the distinguished boundary of the unit polydisc. The proposed algorithm does not require any numerical integration or numerical root finding technique because this is realized based on the standard multivariate division algorithm. A simple observation of the proposed algorithm leads to an ideal membership problem characterizing the solution to the problem. A relation of this characterization and a multivariate generalization of the Walsh's Theorem is also discussed with another ideal membership problem derived by applying a corollary of the Hubert Nullstellensatz to the Walsh's Theorem. Although the discussion to derive the latter ideal membership problem seems to be roundabout, such a characterization would be useful for further generalization, for example to some weighted least-squares approximation. Numerical examples demonstrate the practical applicability of the proposed method to design problems of multidimensional IIR filters.