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396 Chapter 8 IIR FILTER DESIGN
harmonics, and the suppression of clutter from fixed objects in movingtarget
indicator (MTI) radars.
We can create a comb filter by taking our FIR filter with system
function
M∑
H(z) = h(k)z −k (8.23)
k=0
and replacing z by z L , where L is a positive integer. Thus, the new FIR
filter has the system function
H L (z) =
M∑
h(k)z −kL (8.24)
k=0
If the frequency response of the original FIR filter is H ( e jω) , the frequency
response of the filter given by (8.24) is
H L
(
e
jω ) =
M∑
h(k)e −jkLω = H ( e jLω) (8.25)
k=0
Consequently, the frequency response characteristic H L
(
e
jω ) is an L-order
repetition of H ( e jω) in the range 0 ≤ ω ≤ 2π. Figure 8.8 illustrates the
relationship between H L
(
e
jω ) and H ( e jω) for L =4. The introduction of
apole at each notch may be used to narrow the bandwidth of each notch,
as just described.
8.2.4 ALLPASS FILTERS
An allpass filter is characterized by a system function that has a constant
magnitude response for all frequencies, i.e.,
∣ H
(
e
jω )∣ ∣ =1, 0 ≤ ω ≤ π (8.26)
A simple example of an allpass system is a system that introduces a pure
delay to an input signal, i.e.,
H(z) =z −k (8.27)
This system passes all frequency components of an input signal without
any frequency dependent attenuation. It simply delays all frequency components
by k samples.
A more general characterization of an allpass filter is one having a
system function of the form
H(z) = a N + a N−1 z −1 + ···+ a 1 z −N+1 + z −N
1+a 1 z −1 + ···+ a N−1 z −N+1 + a N z −N (8.28)
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