Optimum recursive digital filters with zeros on the unit circle

In this paper we present an efficient algorithm for designing recursive digital filters with optimum magnitudes in the Chebychev sense, all zeros on the unit circle, and different order numerators and denominators. This algorithm takes advantage of the well-known relations between the poles and zeros of analog filters having an equiripple amplitude response either in the passband or stopband. The algorithm requires thus only one approximation interval. This makes it more efficient than the algorithm of Martinez and Parks [1],which works separately with the numerator and denominator. The number of multiplications in the resulting filters is discussed and the optimal orders for numerator and denominator polynomials are considered. A simple explanation for the effect of an extra ripple [1] and for the minimum attainable passband ripple is given.