6. IIR Digital Filters

In this chapter we will describe the design of infinite impulse response (IIR) digital filters. The impulse response of an IIR digital filter has an infinite extent or length or duration, hence the name IIR filters. Design of an IIR filter amounts to the determination of its impulse response sequence {h[n]} in the discrete-time domain or to the determination of its transfer function H(e^jΩ) in the frequency domain. The design can also be accomplished in the Z-domain. In fact, this is the most commonly used domain. The theory of analog filters preceded that of digital filters. Elegant design techniques for analog filters in the frequency domain were developed much earlier than the development of digital filters. As a result, we will adopt some of the techniques used to design analog filters in designing an IIR digital filter. In order to facilitate the design of an IIR digital filter, one must specify certain parameters of the desired filter. These parameters can be in the discrete-time domain or in the frequency domain. Once the parameters or specifications are known, the task is to come up with either the impulse response sequence or the transfer function that approximates the specifications of the desired filter as closely as possible. In the discrete-time domain, one of the design techniques is known as the impulse invariance method. In the frequency domain, the design will yield a Butterworth or Chebyshev or elliptic filter. These three design procedures will result in a closed-form solution. Similarly, the impulse invariance technique will also result in a closed-form solution to the design of IIR digital filters. In addition to these analytical methods, an IIR digital filter can also be designed using iterative techniques. These are called the computer-aided design. Let us first describe the impulse invariance method of designing an IIR digital filter. We will then deal with the design in the frequency domain and the computer-aided design.