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Some Special Filter Types 393
Magnitude
1
0.8
0.6
0.4
0.2
Magnitude Response
0
–1 –1/3 0 1/3 1
Radians / π
Phase Response
0.5
0
–0.5
–1 –1/3 0 1/3 1
ω in π units
FIGURE 8.4 Magnitude and phase responses of digital resonator with zeros at
z = ±1 (solid lines) and z =0(dotted lines) for r =0.9 and ω 0 = π/3
8.2.2 NOTCH FILTERS
A notch filter is a filter that contains one or more deep notches or, ideally,
perfect nulls in its frequency response. Figure 8.5 illustrates the frequency
response of a notch filter with a null at the frequency ω = ω 0 . Notch filters
are useful in many applications where specific frequency components must
be eliminated. For example, instrumentation systems require that the
power line frequency of 60 Hz and its harmonics be eliminated.
To create a null in the frequency response of a filter at a frequency
ω 0 ,wesimply introduce a pair of complex-conjugate zeros on the unit
circle at the angle ω 0 . Hence, the zeros are selected as
Then, the system function for the notch filter is
z 1,2 = e ±jω0 (8.19)
H(z) =b 0 (1 − e jω0 z −1 )(1 − e −jω0 z −1 )
= b 0 (1 − (2 cos ω 0 )z −1 + z −2 ) (8.20)
where b 0 is a gain factor. Figure 8.6 illustrates the magnitude response of
a notch filter having a null at ω = π/4.
The major problem with this notch filter is that the notch has a relatively
large bandwidth, which means that other frequency components
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