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

Abstract

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel-Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval ½À1, þ1 in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions.

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].…”

Evolutionary algorithms based analog filter approximation and implementation

“…In [12], ABC and PSO algorithms are used for high-pass filter approximation and successful results are obtained. In [13], considering orthogonal polynomials, Christoffel-Darboux formula is proposed for generating a new class of filter functions.…”

3D parametric analysis of the RC active Gegenbauer filters

“…C HRISTOFEL-DARBOUX formula, [1][2], for filter function synthesis in continuous domain is described by analytical method in works [3][4][5][6]. In papers [7][8][9], the Christofel-Darboux formula is applied most directly in the design of one-dimensional z domain multiplierless Iinear phase FIR filters and 2D FIR filter is described by analytical method.…”