Generalized Digital Butterworth Filter Design

Ivan W. Selesnick and C. Sidney Burrus

Department of Electrical and Computer Engineering - MS 366
Rice University, Houston, TX 77251-1892, USA
selesi@ece.rice.edu, csb@ece.rice.edu

Abstract

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This paper presents a formula-based method for the design of IIR filters having more zeros than (nontrivial) poles. The filters are designed so that their square magnitude frequency responses are maximally-flat at $ \om = 0 $ and at $ \om = \pi $ and are thereby generalizations of classical digital Butterworth filters. A main result of the paper is that, for a specified half-magnitude 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. Moreover, for a specified number of zeros and a specified half-magnitude frequency, the method directly determines the appropriate way to split the zeros between $ z = -1 $ and the passband. IIR filters having more zeros than poles are of interest, because often, to obtain a good trade-off between performance and the expense of implementation, just a few poles are best.