Digital Signal Processing Reference
In-Depth Information
We can further write the transfer function as
HðsÞ¼
ðs þ 1Þ1
ðs þ 1Þ
ðs þ 1Þ
2
þ2
2
0:5
2
ðs þ 1Þ
2
þ2
2
ðs þ 1Þ
2
þ2
2
¼
From the Laplace transform table (Appendix B), the analog impulse response can easily be found as
h
t
¼ e
t
cos
2t
u
t
0:5e
t
sin
2t
u
t
Sampling the impulse response hðtÞ using a sampling interval T ¼ 0:1 and using the scale factor of T ¼ 0:1,
we have
Th
n
¼ ThðtÞj
t¼nT
¼ 0:1e
0:1n
cos
0:2n
u
n
0:05e
0:1n
sin
0:2n
u
n
Applying the z-transform (Chapter 5) leads to
H
z
¼ Z
0:1e
0:1n
cos
0:2n
u
n
0:05e
0:1n
sin
0:2n
u
n
0:1z
z e
0:1
cos
0:2
z
2
2e
0:1
cos
0:2
z þ e
0:2
0:05e
0:1
sin
0:2
z
z
2
2e
0:1
cos
0:2
z þ e
0:2
¼
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency (Hz)
100
50
0
-50
-100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency (Hz)
FIGURE 8.30
Frequency responses. The line of “x”s represents frequency responses of the analog filter; the solid line
represents frequency responses of the designed digital filter.
Search WWH ::
Custom Search