A truncated polynomial interpolation and its application to polynomially WLS design of IIR filters

In this paper, we propose a simple method to find the optimal rational function, with a fixed denominator, which minimizes an integral of polynomially weighted squared error to given analytic function. Firstly, we present a generalization of the Walsh's theorem. By using the knowledge on the zeros of the fixed denominator, this theorem characterizes the optimal rational function with a system of linear equations on the coefficients of its numerator polynomial. Moreover when the analytic function is specially given as a polynomial, we show that the optimal numerator can be derived without using any numerical integration or any root finding technique. Numerical examples demonstrate the practical applicability of the proposed method.