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A Simple Least-Squares Design of M-D IIR Filters with Fixed Separable Denominator Based on Multivariate Division Algorithm

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Abstract

Inthis paper, we propose a simple algorithmic solution to the bestapproximation problem of finding the nearest multivariate rationalfunction, with a fixed separable denominator polynomial, froma given multivariate polynomial, where the numerator polynomialis desired to minimize the integral of the squared error overthe distinguished boundary of the unit polydisc. The proposedalgorithm does not require any numerical integration or numericalroot finding technique because this is realized based on thestandard multivariate division algorithm.

A simple observation of the proposed algorithmleads to an ideal membership problem characterizing the solutionto the problem. A relation of this characterization and a multivariategeneralization of the Walsh's Theorem is also discussed withanother ideal membership problem derived by applying a corollaryof the Hilbert Nullstellensatz to the Walsh's Theorem. Althoughthe discussion to derive the latter ideal membership problemseems to be roundabout, such a characterization would be usefulfor further generalization, for example to some weighted least-squaresapproximation.

Numerical examples demonstratethe practical applicability of the proposed method to designproblems of multidimensional IIR filters.

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Authors and Affiliations

  1. Dept. of Electrical and Electronic Engineering, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Hiroshi Hasegawa, Isao Yamada & Kohichi Sakaniwa

Authors
  1. Hiroshi Hasegawa
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  2. Isao Yamada
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  3. Kohichi Sakaniwa
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Hasegawa, H., Yamada, I. & Sakaniwa, K. A Simple Least-Squares Design of M-D IIR Filters with Fixed Separable Denominator Based on Multivariate Division Algorithm. Multidimensional Systems and Signal Processing 11, 339–358 (2000). https://doi.org/10.1023/A:1008481628833

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