Digital Signal Processing Reference
In-Depth Information
Im
Im
Im
Im
p
l
g
r
,
y
i
d
.
,
©
,
L
s
1
1 Re
Re
1
1 Re
Re
λ
p
∗
(a) (b)
Figure 7.19
Pole-zero plots for the leaky integrator and the simple resonator.
want to extract. We will then choose a numerator so that the magnitude
is unity at the frequency of interest. The one extra detail is that, since we
want a real-valued filter, we must place a complex conjugate pole as well.
The resulting filter is called a resonator and a typical pole-zero plot is shown
in Figure 7.19. The
z
-transform of a resonator at frequency
ω
0
is therefore
e
j
ω
0
and by its conjugate:
determined by the pole
p
=
λ
G
0
G
0
H
(
z
)=
)
=
(7.24)
pz
−
1
p
∗
z
−
1
z
−
1
2
z
−
2
(
1
−
)(
1
−
1
−
(
2
λ
cos
ω
)
+
λ
0
The numerator value
G
0
is computed so that the filter's gain at
±
ω
0
is one;
since in this case
H
)
=
H
)
,wehave
e
j
ω
0
e
−j
ω
0
(
(
1
+
λ
2
G
0
=(
1
−
λ
)
−
2
λ
cos 2
ω
0
The magnitude and phase of a resonator with
λ
=
0.9 and
ω
=
π/
3are
0
shown in Figure 7.20.
A simple variant on the basic resonator can be obtained by considering
the fact that the resonator is just a bandpass filter with a very narrow pass-
band. As for all bandpass filters, we can therefore place a zero at
z
1and
sharpen its midband frequency response. The corresponding
z
-transform
is now
=
±
1
−
z
−
2
H
(
z
)=
G
1
1
−
(
2
λ
cos
ω
)
z
−
1
+
λ
2
z
−
2
0
with
G
0
G
1
=
2
(
1
−
cos 2
ω
)
0
ThecorrespondingmagnituderesponseisshowninFigure7.21.
