1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind |
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Authors: | Vlastimir Dragoljub Pavlovi? Neboj?a Stanojko Don?ov Dejan Gradimir ?iri? |
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Affiliation: | 1. Department of electronics, Faculty of Electronic Engineering , University of Ni? , Aleksandra Medvedeva 14, PO Box 73, 18000 , Ni? , Republic of Serbia vlastimir.pavlovic@elfak.ni.ac.rs;3. Department of Telecommunications, Faculty of Electronic Engineering , University of Ni? , Aleksandra Medvedeva 14, PO Box 73, 18000 , Ni? , Republic of Serbia |
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Abstract: | Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented. |
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Keywords: | linear phase 1D FIR filter linear phase 2D FIR filter Chebyshev polynomials of the first kind Christoffel–Darboux formula frequency response selective digital filters |
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