Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 58

narrowband filter (NBF), i.e. NF ¼ 1 B ð z Þ= A ð z Þ . This leads to a simple difference

form

1 þ r 2
2

1 þ r 2
2

y k ¼ rpy k 1 r 2 y k 2 þ

u k rpu k 1 þ u k 2 : ð 3 : 21 Þ

The above notch filter can be written in a different way to understand it better by
defining

' ¼ cos 1 rp =ð 1 þ r 2

¼ 2 f and ½ Þ and rewriting the filter as

ð 1 re j z 1 Þð 1 re j z 1 Þ y k ¼ð 1 e j ' z 1 Þð 1 e j ' z 1

Þ u k : ð 3 : 22 Þ

The physical significance is that is the natural resonant angular frequency, while

at angular frequency ' the frequency response of the system is maximum for the
second-order system. This helps to give a feel for its operation while implementing
the filter.

1

< --- Amplitude spectrrum

0.8

0.6

0.4

< --- Phase is 0 to -360

°

0.2

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 3.17 Notch filter

3.6.1 Overview
We have covered a variety of filters and we have seen that only the denominator
polynomial makes a significant difference in the NBF, APF and NF; and even in the
NCO, taking r ¼ 1 is a point to be noted. A thorough understanding of first- and
second-order systems is the basis of understanding any digital filter.

3.7 Other Autoregressive Filters
Autoregressive filters are very commonly used, and excellent software packages can
obtain the coefficients for given filter specifications. Most of the popular filters, such
as Chebyshev and Butterworth (Figure 3.18), are cascaded second-order filters of