Characterization of best Chebychev approximation using the frequency response of IIR digital filters with convex stability

This paper deals with the application of Chebychevʼs approximation theory to IIR digital filter frequency response (FR) approximation. It explores the properties of the frequency response of IIR digital filters as a nonlinear complex approximating function; IIR digital filter frequency response is used to approximate a prescribed magnitude and phase responses. The approximation problem is closely related to optimization. If the set of approximating functions is non-convex, the optimization problem is difficult and may converge to a local minimum. The main results presented in the paper are proposing a convex stability domain by introducing a condition termed “sign condition” and characterization of the best approximation by the Global Kolmogorovʼs Criterion (GKC). The Global Kolmogorovʼs Criteria is shown to be also a necessary condition for the approximation problem. Finally, it is proved that the best approximation is a global minimum. The sign condition can be incorporated as a constraint in an optimization algorithm.