Approximation of multivariable functions by M-D transfer functions with separable denominator
Multidimensional Systems and Signal Processing
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Least-squares approximation of FIR by IIR digital filters
IEEE Transactions on Signal Processing
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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.