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Some Special Filter Types 397
(a)
FIGURE 8.8
H ( e jω) for L =4
(b)
(
Comb filters with frequency response H ) L e
jω
obtained from
which may be expressed in the compact form as
H(z) =z −N A(z−1 )
A(z)
(8.29)
where
We observe that
A(z) =
N∑
a k z −k , a 0 =1 (8.30)
k=0
∣ H
(
e
jω )∣ ∣ 2 = H(z)H(z −1 )| z=e jω =1 (8.31)
for all frequencies. Hence, the system is all-pass.
From the form of H(z) given by (8.28), we observe that if z 0 is a pole
of H(z), then 1/z 0 is a zero of H(z). That is, the poles and zeros are
reciprocals of one another. Figure 8.9 illustrates the typical pole-zero pattern
for a single-pole, single-zero filter and a 2-pole, 2-zero filter. Graphs
of the magnitude and phase characteristics of these two filters are shown
in Figure 8.10 for a =0.6 and r =0.9, ω 0 = π/4, where A(z) for the two
filters is, respectively, given as
A(z) =1+az −1
A(z) =1− (2r cos ω 0 )z −1 + r 2 z −2
(8.32a)
(8.32b)
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