Open AccessLowpass filters approximation based on the Jacobi polynomials
Author(s) -
Nikola Stojanović,
Negovan Stamenković
Publication year - 2017
Publication title -
facta universitatis. series electronics and energetics/facta universitatis. series: electronics and energetics
Language(s) - English
Resource type - Journals
eISSN - 2217-5997
pISSN - 0353-3670
DOI - 10.2298/fuee1703351s
Subject(s) - jacobi polynomials , mathematics , orthogonal polynomials , gegenbauer polynomials , classical orthogonal polynomials , chebyshev polynomials , chebyshev filter , discrete orthogonal polynomials , polynomial , mathematical analysis , legendre polynomials
A case study related to the design the analog lowpass filter using a set of orthogonal Jacobi polynomials, having four parameters to vary, is considered. The Jacobi polynomial has been modified in order to be used as a filter approximating function. The obtained magnitude response is more general than the response of the classical ultra-spherical filter, due to one additional parameter available in orthogonal Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripple, smaller group delay variation or sharper cutoff slope. Two methods for transfer function approximation are investigated: the first method is based on the known shifted Jacobi polynomial, and the second method is based on the proposed modification of Jacobi polynomials. The shifted Jacobi polynomials are suitable only for odd degree transfer function. However, the proposed modified Jacobi polynomial filter function is more general but not orthogonal. It is transformed into orthogonal polynomial when orders are equal and then includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, Gegenbauer (ultraspherical) filter and many other filters, as its special cases.