Group delay equalisation of discrete Butterworth tan filters in the continuous domain

Abstract

ABSTRACTA new optimisation method for equalising the group delay frequency response of the discrete transfer function of recursive digital filters, so-called tan filters, is presented in this paper. The method is based on an algorithm that is used for a stable all-pass filter approximation. Allpass filter acts as a group delay equaliser, with the purpose of equalising group delay of the discrete tan filter in a maximal flat sense. The algorithm relies on a set of simultaneous nonlinear equations. These set equations are derived directly from the flatness conditions of the equalised group delay response at the origin in the analogue -domain, in order to obtain the unknown coefficient values of the discrete allpass filter. The iterative method most widely used for solving set nonlinear equations and Symbolic Math Toolbox for Matlab features for finding numerical solutions are presented in detail.In the given examples, firstly a discrete Butterworth tan filter was designed by using transfer function poles in closed form. Secondly, the Butterworth tan filter was augmented with cascade connection of non-minimum phase all-pass discrete filter, in order for the group delay response of the whole filter to be equalised in a maximally flat sense.