New class of filter functions generated most directly by Christoffel-Darboux formula for Gegenbauer orthogonal polynomials

Abstract

A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [71, þ1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.

Size analysis surface area of the pass-band and stopband of multiplierless 2D FIR filter

“…Application of extreme properties of Christofel-Darboux sum in the filter theory is described in continuous domain, one-dimensional z domain and in two-dimensional z domain [1][2][3][4][5]. In this paper, we analyze the characteristic of pass-band as well as the characteristics of stop-band of multiplierless linear-phase 2D FIR filter functions published in previous research [6].…”

Design of generalised orthogonal filters: application to the modelling of dynamical systems

“…design of orthogonal signal generators [10], least square approximations [11][12][13], process modelling [14,15], and practical realizations of optimal [11,16] and adaptive systems [17,18]. A c c e p t e d M a n u s c r i p t…”

New class of filter functions generated directly by the modified Christoffel–Darboux formula for classical orthonormal Jacobi polynomials

“…Some of the applications of orthogonal polynomials in theory of electrical filters are considered in articles [1][2][3][4]. Some of the applications of orthogonal polynomials in theory of electrical filters are considered in articles [1][2][3][4].…”

“…Many actual design problems can be solved by choosing a filter function as result of the approximation technique by orthogonal polynomials known as the classical, as they are the Gegenbauer (ultraspherical), Chebyshev (first and second kind), and Legendre (spherical) polynomials. Some of the applications of orthogonal polynomials in theory of electrical filters are considered in articles [1][2][3][4].…”

“…By solving this task, it is found that derived approximation proposed in this work can be used to generate a set of new filter function and a set of all particular solutions for filter functions (Gegenbauer filters, Legendre filters, and Chebyshev of first and second kind filters) [1][2][3][4].…”