Design of nearly linear-phase recursive digital filters by using unconstrained least-pth minimax optimization

A method for the design of nearly linear-phase recursive digital filters is proposed. The recursive filter is assumed be a cascade arrangement of second-order biquadratic sections whose transfer functions are expressed in the polar form. An error function is formulated based on the difference between the actual complex frequency response of the filter and the desired frequency response. Then by using a least-pth minimax algorithm, the required design is obtained. The new method achieves filter stability through the use of a parameterization scheme based on the so-called sigmoid function and it incorporates a mechanism by which an arbitrary prescribed stability margin can be achieved. The optimization engine used in the least-pth algorithm is an unconstrained quasi-Newton algorithm based on the Broyden-Fletcher-Goldfarb-Shanno updating formula and it incorporates a nonuniform adaptive variable sampling technique to prevent spikes in the error function. Several filter design examples demonstrate that the proposed method is very efficient in terms of computational effort required since it is an unconstrained method and, furthermore, it can yield designs that are superior relative to some of the known state-of-the-art methods.