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 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.