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Some Special Filter Types 391
The corresponding system function is
H(z) =
=
b 0
(1 − re jω0 z −1 )(1 − re −jω0 z −1 )
b 0
1 − (2r cos ω 0 )z −1 + r 2 z −2 (8.7)
where b 0 is a gain parameter. The frequency response of the resonator is
H ( e jω) =
b
[ ][ 0
1 − re
−j(ω−ω 0)
1 − re−j(ω+ω0)] (8.8)
Since ∣ ∣H ( e jω)∣ ∣ has its peak at or near ω = ω 0 ,weselect the gain parameter
b 0 so that ∣ ∣ H
(
e
jω )∣ ∣ =1.Hence,
∣ H
(
e
jω 0
)∣ ∣ =
b 0
|(1 − r)(1 − re −j2ω0 )|
Consequently, the desired gain parameter is
=
b 0
(1 − r) √ 1+r 2 − 2r cos 2ω 0
(8.9)
b 0 =(1− r) √ 1+r 2 − 2r cos 2ω 0 (8.10)
The magnitude of the frequency response H(ω) may be expressed as
∣ H
(
e
jω )∣ ∣ =
b 0
D 1 (ω)D 2 (ω)
(8.11)
where D 1 (ω) and D 2 (ω) are given as
D 1 (ω) = √ 1+r 2 − 2r cos(ω − ω 0 )
D 2 (ω) = √ 1+r 2 − 2r cos(ω + ω 0 )
(8.12a)
(8.12b)
Foragiven value of r, D 1 (ω) takes its minimum value (1 − r) atω = ω 0 ,
and the product D 1 (ω)D 2 (ω) attains a minimum at the frequency
( )
1+r
ω r = cos −1 2
cos ω 0
2r
(8.13)
which defines precisely the resonant frequency of the filter. Note that
when r is very close to unity, ω r ≈ ω 0 , which is the angular position of
the pole. Furthermore, as r approaches unity, the resonant peak becomes
sharper (narrower) because D 1 (ω) changes rapidly in the vicinity of ω 0 .
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