This post discusses the stability of IIR filters. This should be the last article in this series on frequency filters, unless I find something else interesting enough to write about. The purpose of this series was to show that digital signal processing is not that difficult of a topic. We started with some really simple finite impulse response low pass filters, discussed better filters, tried all pass, high pass, band pass, and band stop filters, checked out various windows, and recently moved to infinite impulse response filters. We have already populated our music wiki with various such DSP related topics, including some relatively complex infinite impulse response filters, such as the Bessel filter, the Chebychev type I filter, and the Chebychev type II filter.

This series of posts will be complete only if we can also put together some examples of high pass, band pass, and band stop infinite impulse response filters. We will do so here by transforming the Butterworth filter from a couple of posts ago.

Butterworth filters can be tedious to derive and the following article is relatively dense with math. Since there is a lot that we can learn from these filters though, it may make sense to describe the steps used in putting together a few Butterworth filters in some reasonable detail.

Low pass Butterworth filters have the following transfer function.

Consider the filter given by

y(k) = 0.22853 x(k – 1) + 1.19249 y(k – 1) – 0.42804 y(k – 2)

This filter is recursive, in the sense that its output y(k) at a particular sample k depends not only on the values of its input x, but also on the previous values of its output y. The transfer function of this filter is as follows.