Generalized digital Butterworth filter design

TLDR

IIR filters having more zeros than poles are of interest because often, to obtain a good tradeoff between performance and implementation complexity, just a few poles are best. This correspondence introduces a new class of infinite impulse response (IIR) digital filters that unifies the classical digital Butterworth filter and the well‑known maximally flat FIR filter. The authors provide new closed‑form expressions and a straightforward design technique for these unified IIR digital filters. The new IIR digital filters have more zeros than poles, with monotonic square‑magnitude responses maximally flat at ω=0 and ω=π, and for a specified cutoff frequency and number of zeros there is only one valid way to split the zeros between z=−1 and the passband, allowing continuous variation of the cutoff frequency.

Abstract

This correspondence introduces a new class of infinite impulse response (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 straightforward 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 /spl omega/=0 and at /spl omega/=/spl pi/. Another result of the correspondence is that for a specified cutoff 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. This technique also permits continuous variation of the cutoff frequency. IIR filters having more zeros than poles are of interest because often, to obtain a good tradeoff between performance and implementation complexity, just a few poles are best.