A filter with one real pole and one real zero can be configured as a shelving filter, as a high-pass filter (putting the zero at the point ) or as a low-pass filter (putting the zero at ). The frequency responses of these filters are quite blunt; in other words, the transition regions are wide. It is often desirable to get a sharper filter, either shelving, low- or high-pass, whose two bands are flatter and separated by a narrower transition region.

A procedure borrowed from the analog filtering world transforms real,
one-pole, one-zero filters to corresponding
*Butterworth filters*,
which have narrower transition regions. This procedure is described clearly
and elegantly in the last chapter of [Ste96]. The derivation
uses more mathematics background than we have developed here, and we will simply
present the result without deriving it.

To make a Butterworth filter out of a high-pass, low-pass, or shelving
filter, suppose that either the pole or the
zero is given by the expression

where is a parameter ranging from 1 to . If this is the point , and if it's .

Then, for reasons which will remain mysterious, we replace the point (whether
pole or zero) by points given by:

where ranges over the values:

In other words, takes on equally spaced angles between and . The points are arranged in the complex plane as shown in Figure 8.17. They lie on a circle through the original real-valued point, which cuts the unit circle at right angles.

A good estimate for the cutoff or transition frequency defined by
these circular collections of poles or zeros is simply the spot where the
circle intersects the unit circle, corresponding to
. This gives
the point

which, after some algebra, gives an angular frequency equal to

Figure 8.18 (part a) shows a pole-zero diagram and frequency response for a Butterworth low-pass filter with three poles and three zeros. Part (b) shows the frequency response of the low-pass filter and three other filters obtained by choosing different values of (and hence ) for the zeros, while leaving the poles stationary. As the zeros progress from to , the filter, which starts as a low-pass filter, becomes a shelving filter and then a high-pass one.