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New class of filter functions generated directly by the modified Christoffel–Darboux formula for classical orthonormal Jacobi polynomials
Author(s) -
Pavlović Vlastimir D.
Publication year - 2013
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.1817
Subject(s) - mathematics , filter (signal processing) , christoffel symbols , composite image filter , jacobi polynomials , filter design , low pass filter , chebyshev filter , m derived filter , network synthesis filters , mathematical analysis , orthogonal polynomials , computer science , image (mathematics) , artificial intelligence , electronic engineering , engineering , computer vision
The new solution of determining the prototype analogue 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. The approximation problem of filter function was solved mathematically, applying the summed Christoffel–Darboux formula for the orthonormal polynomials. The starting point in solving the approximation problem is the direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthonormal polynomials in the finite interval [−1, + 1] in full respect to the weighting function with two free real parameters. Solution for the monotonic and non‐monotonic filter functions is obtained in an explicit form, which is shown to enable generation of the Jacobi filter functions in a simple way by choosing the values of the free real parameters. Moreover, the proposed solution with the same criterion of approximation is used to generate for particular specifications the appropriate best known classical approximation functions of free parameters: the Gegenbauer, Legendre, and Chebyshev filter functions of the first and second kind as well. The example of the proposed all‐pole low‐pass lossless LC filter in the frequency domain for frequency responses of the 11th‐order filter is illustrated. This work is concerned with the approximation of an all‐pole low‐pass filter to design a circuit assuring an optimum realization of the required specifications. The article proposes new class filter functions with an approach to ideal filter magnitude characteristic. Copyright © 2012 John Wiley & Sons, Ltd.