Optimal design using min - max criteria for two-dimensional recursive digital filters

This paper concerns design techniques for two-dimensional recursive digital filters in the frequency domain. The filters are designed so as to approximate simultaneously prescribed specifications of magnitude and group delay by treating the coefficients of the transfer function as decision variables. These approximation problems are formulated as minimizing the maximum errors between the filter's response and the desired one with respect to the magnitude and group delay characteristics. The following two types of optimization techniques are proposed: (1) the Min-Max Type Optimization Satisfaction Method and (2) the Min-Max Type Multi-Objective Optimization Method. The former is a method that optimizes (approximates) the magnitude characteristic in a min-max sense under the satisfaction condition requiring that the maximum error of group delay should be kept below the tolerance level. The latter is a method that optimizes the vector-valued objective function of the maximum errors of magnitude and group delay. The optimal solution for this multiobjective problem is obtained by combining the "satisfaction approach" and the "maximum component minimization technique" for vector optimization. Since these problems include maximum-valued functions in the performance indices, as a nondifferentiable optimization technique, we apply Mifflin's algorithm using generalized gradients. Several numerical examples are presented.