Chained-Function Filter Synthesis Based on the Modified Jacobi Polynomials

A new class of filter functions with pass-band ripple which derives its origin from a method of determining the chained function lowpass filters described by Guglielmi and Connor is introduced. The closed form expressions of the characteristic functions of these filters are derived by using orthogonal Jacobi polynomial. Since the Jacobi polynomials can not be used directly as filtering function, these polynomials have been adapted by using the parity relation for Jacobi polynomials in order to be used as a filter approximating function. The obtained magnitude response of these filters is more general than the magnitude response of published Chebyshev and Legendre chained function filter, because two additional parameters of modified Jacobi polynomials as two additional degrees of freedom are available. It is shown that proposed modified Jacobi chained function filters approximation also includes the Chebyshev chained function filters, the Legendre chained function filter, and many other types of filter approximations, as its special cases.