Digital Signal Processing Reference
In-Depth Information
2
1
1.5
0.8
0.6
1
0.4
0.5
0.2
t
w
0
0
−1
−2
p
−1.5
p
−
p
−0.5
p
0
0.5
pp
1.5
p
2
p
(a)
(b)
20
1
0.8
15
0.6
10
0.4
5
0.2
k
W
0
0
0
5
10
15
20
25
30
35
40
45
50
−2
p
−1.5
p
−
p
−0.5
p
0
0.5
pp
1.5
p
2
p
−5
(c)
(d)
Comparing the magnitude spectrum
H
(
ω
)
of the LTIC system with the
magnitude spectrum
H
(
Ω
)
of the LTID system plotted in Figs. 16.2(b) and (d),
respectively, we observe two major differences. First, the magnitude spectrum
H
(
Ω
)
is periodic with a period of 2
π
. Secondly, the magnitude spectrum
H
(
Ω
)
is scaled by a factor of 1
/
T
in comparison with
H
(
ω
)
. In order to
obtain a DT filter with a DC amplitude gain of the same value as that of the CT
filter, we multiply the sampled impulse response
h
[
k
] by a factor of
T
:
Fig. 16.2. Impulse invariance
method used for transforming
analog filters to digital filters in
Example 16.1. (a) Impulse
response
h
(
t
) and
(b) magnitude spectrum
H
(ω)of
the analog filter. (c) Impulse
response
h
[
k
] and
(d) magnitude spectrum
H
(
Ω
)
of the transformed digital filter.
−α
kT
u
(
k
)
.
h
[
k
]
=
Th
(
kT
)
=
T
e
(16.11)
Alternatively, the following transform pair can be used for the impulse invari-
ance transformation:
1
s
+ α
T
1
−
e
−α
T
z
−
1
zT
z
−
e
−α
T
.
impulse invariance
←−
−→
or
(16.12)
Example 16.2 illustrates the application of Eq. (16.12) in transforming a But-
terworth lowpass filter into a digital lowpass filter.
Example 16.2
Consider the following Butterworth filter:
81
.
6475
s
2
+
12
.
7786
s
+
81
.
6475
.
H
(
s
)
=
Use the impulse invariance transformation to derive the transfer function of the
equivalent digital filter.
Search WWH ::
Custom Search