Generalized Digital Butterworth Filters

Ivan W. Selesnick and C. Sidney Burrus


This paper introduces a new class of IIR digital filters that unifies the classical digital Butterworth filter and the well known maximally flat FIR filter. New closed form expressions are provided, and a straight forward design technique is described. The new IIR digital filters have more zeros than poles (away from the origin) and their (monotonic) square magnitude frequency responses are maximally flat at w=0 and at w=pi. Another result of the paper is that, for a specified cut-off frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z=-1 and the passband. The way in which to split the zeros between z=-1 and the passband is given by a simple table. This technique also permits continuous variation of the cut-off frequency. IIR filters having more zeros than poles are of interest, because often, to obtain a good trade-off between performance and implementation complexity, just a few poles are best.