Design of IIR digital filters based on eigenvalue problem

This paper presents a new method for designing IIR digital filters with optimum magnitude response in the Chebyshev sense and different order numerator and denominator. The proposed procedure is based on the formulation of a generalized eigenvalue problem by using Remez exchange algorithm. Since there exist more than one eigenvalue in the general eigenvalue problem, we introduce a very simple selection rule for the eigenvalue to be sought for where the rational interpolation is performed if and only if the positive minimum eigenvalue is chosen. Therefore, the solution of the rational interpolation problem can be obtained by computing only one eigenvector corresponding to the positive minimum eigenvalue, and the optimal filter coefficients are easily obtained through a few iterations. The design algorithm proposed in this paper not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step because it has been reduced to the computation of the positive minimum eigenvalue. Some properties of the filters such as lowpass filters, bandpass filters, and so on are discussed, and several design examples are presented to demonstrate the effectiveness of this method.