Digital Signal Processing, 10th Edition

6Infinite Impulse Response (IIR) Filter Sections

Digital filters with an infinite impulse response (IIR) are discrete linear systems which are governed by a convolution equation based on an infinite number of terms. In principle, they have infinite memory. This memory is achieved by feeding the output back to the input, so they are known as recursive filters. Each element of the set of output numbers is calculated by weighted summation of a certain number of elements of the input set and of the previous output set.

In general, this IIR allows for much more selective filtering functions to be obtained than with finite impulse response (FIR) filters of similar complexity. However, the feedback loop complicates the study of the properties and design of these filters and leads to parasitic phenomena.

When examining IIR filters, it is simpler initially to consider them in terms of first- and second-order sections. Not only are these simple structures useful in introducing the properties of IIR filters. They also represent the most frequently used type of implementation. Indeed, even the most complex IIR filters appearing in practice are generally formed from a set of such sections.

6.1 First-Order Section

Consider a system which, for the set of data x(n), produces the set y(n) such that:

(6.1)

where b is a constant. This is a first-order filter section.

This system’s response ...