Monotonic, critical monotonic, and nearly monotonic low‐pass filters designed by using the parity relation for Jacobi polynomials

Abstract

A new class of continuous-time low-pass filter using a set of Jacobi polynomials, with all transmission zeros at infinity, is described. The Jacobi polynomial has been adapted by using the parity relation for Jacobi polynomials in order to be used as a filter approximating function. The resulting class of polynomials is referred to as a pseudo Jacobi polynomials, because they are not orthogonal. The obtained magnitude response of these filters is more general than the magnitude response of the classical ultraspherical filter, because of one additional degree of freedom available in pseudo Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripples or sharper cutoff slope. Monotonic, critical monotonic, or nearly monotonic passband filter approximating functions can be also generated. It is shown that proposed pseudo Jacobi polynomial filter approximation also includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, and many transitional filter approximations, as its special cases. Several examples are presented, and detailed formulas including the practical suggestions for their efficient implementation are also provided. The proposed nearly monotonic filter is compared with the least-square-monotonic filters, designed as critical monotonic, in details. The advantages of the new filters are discussed.

Chained-Function Filter Synthesis Based on the Modified Jacobi Polynomials

“…If α = −0.5 and β increases, the ripples in the passband decrease smoothly to be unequal and smaller in magnitude. For β > 1.5 the passband response is nearly monotonic [7], but the rejection slope factor is much steeper than a Butterworth filter cutoff slope. On the other hand, for −1 < β < −0.5 the passband ripples are unequal, but in magnitude are larger than 1.…”

“…The characteristic function, used for the proposed analog filter design, can be either n-the degree modified Jacobi polynomial [7] or a chained-function formed as the product of ν lower degree modified Jacobi polynomials, called modified Jacobi seed functions. Thus, a new family of characteristic, or generating, functions of the same degree n, called modified Jacobi chained-function (mJCF), is given by…”

Lowpass filters with almost‐maximally flat passband and Chebyshev stopband attenuation

“…Introduction: In the recently published paper [1], the authors have reported that the modified Jacobi polynomials can be used to construct a useful allpole lowpass filter functions. For the given lowpass filter degree, two parameters of the modified Jacobi polynomial can be used to a trade-off between passband magnitude response having ripples or being nearly monotonic, transition band width, or group delay deviation in the passband [2].…”

The Kharitonov theorem and robust stabilization via orthogonal polynomials

“…Orthogonal polynomials [6], [39] play an important role in a number mathematical areas. On one hand, orthogonal polynomials have been extensively used in applications for solving practical problems, such as in signal processing [32] and in filter design [38], [30]. On the other hand, the zeros of a certain family of orthogonal polynomials can be interpreted as the electrostatic energy for a system of a finite number of charges; see [43].…”