Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 61

Unlike the pole-zero plot of the Chebyshev filter, we see that the focal point of the
Butterworth filter plot is away from the origin. The pole trace follows the equation
x 0 : 1582 ¼ 0 : 1651y 2 , as shown in Figure 3.18. Once again, the zeros are

positioned at 180 on the unit circle.
By now you have probably understood why we have focused on second-order
filters. Almost all the filters can be realised by cascading second-order filters. For
detailed design methods, consult a book on filter design.

3.8 Adaptive Filters
An Adaptive filter is a filter whose characteristics can be modified to achieve an
objective by automatic adaptation or modification of the filter parameters. We
illustrate this using a simple unity-gain BPF filter described in Section 3.4 via
(3.17). We demonstrate this idea of adaptation by taking a specific criterion. In
general, the criteria and the choice of feedback for adjusting parameters make an
adaptive filter robust.

u k x k

Filter ( r , p )

Criteria

Adjust r and/or p

Figure 3.20 Adaptive filter

. The parameter r is
in (0,1). By choosing r ¼ 1 the filter becomes marginally stable and this must be
avoided.

In this particular case, 2 p 2 since we have p ¼ 2 cos

3.8.1 Varying r
The effect of increasing the radial pole position r on the frequency response curve is
that it becomes sharper and the amplitude increases. We also notice that as the value
of r is reduced, the peak frequency changes. We show this by varying the value of r
from 0.65 to 0.95. The value of p is kept constant at p ¼ 1 : 5321 ð ¼ 40 Þ . The plot
is shown in Figure 3.21. The Peak frequency is related to the radial pole position r
and the angular measure of the pole

as

25p ð r þ r 1 cos pk ¼ 0 : Þ; ð 3 : 25 Þ

where pk is peak angular frequency. Notice that when r ! 1 we get pk ! .