Digital Signal Processing Reference
In-Depth Information
Fig. 2.38 Implementation
schemes for a sinusoidal dig-
ital oscillator
sin (
b
)
y
(
n
)
δ
(
n
)
z
−1
2 cos(
b
)
z
−1
− 1
H
(
z
)
Im ( z )
b =
Ω
o
Re ( z )
− b
Ω
=
−
Ω
o
0
Ω
o
−
π
π
ω
T
s
Fig. 2.39
Magnitude response and pole-zero diagram of the digital oscillator
2.6.9.4 The Digital Resonator
The magnitude response of the digital oscillator was seen to consist of two very
sharp two spikes. Analogously, one can design a digital resonator, which is a
narrowband BPF centered around a resonant frequency f
o
. Such a resonator could
be useful for extracting a fundamental sinusoid (with frequency f
o
) corrupted by
harmonics or other sinusoids. Note that if one attempts to design a resonator using
classical filter design techniques, one needs a prohibitively large filter order. To
ensure stability, the poles should be inside the unit circle, i.e., p
1
;
2
¼
re
X
o
;
with
r very near to 1. A possible transfer function for the resonator can then be:
1
ð
z
p
1
Þð
z
p
2
Þ
¼
1
z
2
2r cos
ð
X
o
Þ
z
þ
r
2
:
H
1
ð
z
Þ¼
ð
2
:
40
Þ
The magnitude response for this resonator is shown in Fig. (
2.40
) as the dotted
curve. Note that this curve has non-zero values well beyond the frequency of
interest, X
o
.
A transfer function which also satisfies the above criteria with better stopband
attenuation is:
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