A Remez-type algorithm for designing digital filters composed of all-pass sections based on phase approximations

An efficient Remez-type algorithm is introduced for determining an allpass section in such a way that its phase response minimizes in the Chebyshev sense a given weighted error function on a closed subset of [0, π]. The effectiveness of the algorithm is based on two facts. First, for the best solution of an allpass filter of order N, if it exists, the weighted error function achieves the peak absolute value with alternating signs at least at N+1 consecutive points in the approximation subset. Second, the phase response of an allpass filter of order N can be forced to take on the given values at N points by using the recurrence formulas of Henk. The algorithm can be applied in a straightforward manner for designing phase equalizers, filters proving an arbitrary noninteger delay, and special approximately linear-phase Hilbert transformers as well as filters with transfer function of the form H(z)=[z^-M+A(z)]/2 with A(z) being an allpass filter. Furthermore, it can be applied for designing wave lattice filters (parallel connection of two stable allpass filters) having several passband and stopband regions and arbitrary weightings in these regions. Several examples are included illustrating the flexibility and efficiency of the proposed design technique