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392 Chapter 8 IIR FILTER DESIGN
A quantitative measure of the width of the peak is the 3dB bandwidth of
the filter, denoted as ∆(ω). For values of r close to unity,
∆ω ≈ 2(1 − r) (8.14)
Figure 8.3 illustrates the magnitude and phase responses of a digital resonator
with ω 0 = π/3, r =0.90. Note that the phase response has its
greatest rate of change near the resonant frequency ω r ≈ ω 0 = π/3.
This resonator has two zeros at z =0.Instead of placing zeros at the
origin, an alternative choice is to locate the zeros at z =1and z = −1.
This choice completely eliminates the response of the filter at the frequencies
ω =0and ω = π, which may be desirable in some applications. The
corresponding resonator has the system function
H(z) =
G(1 − z −1 )(1 + z −1 )
(1 − re jω0 z −1 )(1 − re −jω0 z −1 )
and the frequency response characteristic
1 − z −2
= G
1 − (2r cos ω 0 )z −1 + r 2 z −2 (8.15)
H ( e jω) 1 − e −j2ω
= G
[1 − re j(ω0−ω) ][1 − re −j(ω0+ω) ]
(8.16)
where G is a gain parameter that is selected so that ∣ ( )∣ H e
jω 0 ∣ =1.
The introduction of zeros at z = ±1 alters both the magnitude and
phase response of the resonator. The magnitude response may be expressed
as
(
∣ H e
jω )∣ ∣
N(ω)
= G
(8.17)
D 1 (ω)D 2 (ω)
where N(ω) isdefined as
N(ω) = √ 2(1 − cos 2ω) (8.18)
Due to the presence of the zeros at z = ±1, the resonant frequency of the
resonator is altered from the expression given by (8.13). The bandwidth
of the filter is also altered. Although exact values for these two parameters
are rather tedious to derive, we can easily compute the frequency response
when the zeros are at z = ±1 and z =0,and compare the results.
Figure 8.4 illustrates the magnitude and phase responses for the cases
z = ±1 and z =0,for pole location at ω = π/3 and r =0.90. We observe
that the resonator with z = ±1 has a slightly smaller bandwidth than
the resonator with zeros at z =0.Inaddition, there appears to be a very
small shift in the resonant frequency between the two cases.
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