Abstract
The approximation problem of a filter function of even and odd order is solved mathematically in this paper most directly applying the proposed Christoffel-Darboux formula for two continual orthogonal polynomials on the equal finite segment. As a result, a linear phase digital finite impulse response (FIR) filter function is generated in compact explicit form. In addition, a new difference equation and structure of linear phase digital FIR filter are proposed. Two examples of the extremely economic FIR filter (with four adders and without multipliers) designed by the proposed technique are presented. The proposed solutions are extremely efficiency in regard to energy consumption. DOI: http://dx.doi.org/10.5755/j01.eee.18.8.2639
Highlights
Theory of filtering has wide applications in various frequency ranges and technologies for analog and digital signals [1]
The global Christoffel-Darboux formula for four orthonormal polynomials on two equal finite segments for generating the linear phase two-dimensional finite impulse response (FIR) digital filter functions has been proposed in a compact explicit form [1]
An original approach to a linear phase selective low-pass digital FIR filter design is presented in this paper
Summary
Theory of filtering has wide applications in various frequency ranges and technologies for analog and digital signals [1]. The classical Christoffel-Darboux formula [2], [3] for continual orthogonal polynomials is shown to be an important identity with extremal properties for that purpose This formula for classical Jacobi orthogonal polynomials and for all their particular solutions (Gegenbauer, Legendre and Chebyshev polynomials of the first and second kind) has been applied for generating new class filter functions [4], [5]. The global Christoffel-Darboux formula for four orthonormal polynomials on two equal finite segments for generating the linear phase two-dimensional FIR digital filter functions has been proposed in a compact explicit form [1]. The Christoffel-Darboux formula for two orthogonal polynomials on the equal finite segment is proposed in a compact explicit form This formula can be most directly applied in generating linear phase selective low-pass digital FIR filter functions as demonstrated here. Relevant examples of the linear phase selective low-pass digital FIR filters designed by the proposed technique are presented
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