Design of recursive digital filters with optimum magnitude and attenuation poles on the unit circle

A class of infinite impulse response (IIR) digital filters with optimum magnitude in the Chebyshev sense, arbitrary attenuation in the passband and stopband, all zeros on the unit circle, and different order numerator and denominator is discussed. Several properties of low-pass filters of this type are described, such as the effect of an extra ripple in the passband and the minimum attainable passband ripple for a given order. An algorithm for the design of these filters is presented, which given the order of the filter, passband edge, stopband edge and passband ripple minimizes the stopband ripple. Alternatively, the stopband ripple can be fixed and the passband ripple minimized. This is done by working with the numerator and denominator separately. This algorithm is fast compared to other existing design procedures. Several examples are presented and compared with the classical elliptic filters. Filters are described which meet the same tolerance scheme as an elliptic filter with fewer multiplications.