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## Butterworth filters

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 8.17: Replacing a real-valued pole or zero (shown as a solid dot) with an array of four of them (circles) as for a Butterworth filter. In this example we get four new poles or zeros as shown, lying along the circle where .

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.

Figure 8.18: Butterworth low-pass filter with three poles and three zeros: (a) pole-zero plot. The poles are chosen for a cutoff frequency ; (b) frequency responses for four filters with the same pole configuration, with different placements of zeros (but leaving the poles fixed). The low-pass filter results from setting for the zeros; the two shelving filters correspond to and , and finally the high-pass filter is obtained setting . The high-pass filter is normalized for unit gain at the Nyquist frequency, and the others for unit gain at DC.

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