Digital Signal Processing Reference
In-Depth Information
(a)
(b)
h
()
1
0.2
H
(
jf
)
d
0.9
0.8
0.1
0.7
0.6
0
0.5
0.4
-0.1
0.3
0.2
0.1
-0.2
0
0
100
200
300
400
500
n
40
f
d
0
10
20
30
Fig. 5.3
Impulse response a and frequency response b of digital filter from Example 5.2
Thus the substitution for Laplace operator is:
s
¼
B
1
2az
1
þ
z
2
1
z
2
¼
3
:
08
1
1
:
7z
1
þ
z
2
1
z
2
Then, after simple transformations, one obtains the following transfer function
of sought IIR filter:
0
:
0674 1
2z
2
þ
z
ð Þ
1
þ
b
ðÞ
z
1
þ
b
ðÞ
z
2
þ
b
ðÞ
z
3
þ
b
ðÞ
z
4
;
H
ð
z
Þ¼
where
b
ðÞ¼
2
:
672
;
b
ðÞ¼
2
:
99
;
b
ðÞ¼
1
:
674
;
b
ðÞ¼
0
:
413
:
The filter difference equation is then:
y
ðÞ¼
0
:
0674 x
ðÞ
2
xn
2
½
ð
Þ
xn
4
ð
Þ
þ
2
:
672yn
1
ð
Þ
2
:
99yn
2
ð
Þ
þ
1
:
674yn
3
ð
Þ
0
:
413yn
4
ð
Þ:
The impulse and frequency responses of the designed filter are shown in
Fig.
5.3
. One can see that the impulse response lasts for approximately 1.5 cycle of
50 Hz (some 30 samples). The transition band from pass to rejection regions is
quite steep. In case if this slope is not sufficient, one can increase the filter order
(both analog prototype and resulting digital filter), but this usually results also in
longer duration of filter transient response in time domain.
5.2.2 Application of Impulse Response Invariance Method
The impulse response invariance method relies on design of digital filter, which
has impulse response obtained by sampling of impulse response of the analog
origin:
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