IIR Filter Design

Abstract

Digital IIR filters have an infinitely long impulse response and therefore can be associated with analog filters that have the same characteristics. For this reason, the classical method for digital IIR filter design is based on the design of an analog filter, which is later transformed into an equivalent digital filter through a mapping in the complex plane. The advantage of such a technique lies in the fact that analog filter design and mapping methods for analog-to-digital (A/D) transformation are well known and have sound theoretical foundations. The process is based on a lowpass filter, and frequency transformation methods are later applied if a different type of frequency selectivity is desired. These frequency transformations are again well-known mapping procedures in the complex plane. In this chapter, we will first discuss the design of the main types of lowpass analog filters, namely Butterworth, Chebyshev and elliptic lowpass filters, all of which represent different ways of approximating the desired ideal frequency response. Then we will learn how to transform the analog lowpass filter into an equivalent IIR digital lowpass filter via bilinear transformation, and finally how to transform the IIR lowpass filter into a highpass, bandpass or bandstop filter, if needed. The appendix to this chapter provides deeper insight into the mathematical facets of elliptic design, discussing elliptic integrals of the first kind, Jacobi elliptic functions, and the elliptic rational function on which the transfer function of the analog elliptic filter is based. It must be explicitly noted that all classically-designed lowpass IIR filters are inadequate for specifications with cutoff frequency close to zero and narrow transition band. In such cases, it may be advisable to preliminarily downsample the signal and then design a filter with a less extreme cutoff frequency, allowing for a wider transition band.