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Some Special Filter Types 395
Imaginary Part
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
Pole–zero Plot
–1 –0.5 0 0.5 1
Real Part
Decibel
Radians / π
10
0
Magnitude Response
–50
–1 –1/4 0 1/4 1
1
0.5
0
–0.5
Phase Response
–1
–1 –1/4 0 1/4 1
ω in π units
FIGURE 8.7 Magnitude and phase responses of notch filter with poles (solid
lines) and without poles (dotted lines) for ω 0 = π/4 and r =0.85
In particular, suppose that we select the poles at
Hence, the system function becomes
p 1,2 = re ±jω 0 (8.21)
H(z) =b 0
1 − (2 cos ω 0 )z −1 + z 2
1 − (2r cos ω 0 )z −1 + r 2 z −2 (8.22)
The magnitude of the frequency response ∣ ∣ H
(
e
jω )∣ ∣ of this filter is illustrated
in Figure 8.7 for ω 0 = π/4 and r =0.85. Also plotted in this figure
is the frequency response without the poles. We observe that the effect of
the pole is to introduce a resonance in the vicinity of the null and, thus, to
reduce the bandwidth of the notch. In addition to reducing the bandwidth
of the notch, the introduction of a pole in the vicinity of the null may result
in a small ripple in the passband of the filter due to the resonance
created by the pole.
8.2.3 COMB FILTERS
In its simplest form, a comb filter may be viewed as a notch filter in which
the nulls occur periodically across the frequency band, hence the analogy
to an ordinary comb that has periodically spaced teeth. Comb filters are
used in many practical systems, including the rejections of power-line
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