Digital Signal Processing Reference
In-Depth Information
4.
Obtain the desired transfer function
H
(
z
), or
z
z
+
-
1
()
=
(
)
Hz
Hs
w
=
(
)
(
)
Asz
=-
1
z
+
1
21
19
Exercise D.2: First-Order IIR Highpass Filter
1), obtain a corresponding transfer
function
H
(
z
). Let the bandwidth or cutoff frequency be 1
r
/
s
and the sampling fre-
quency be 5 Hz. From the preceding procedure,
H
(
z
) is found to be
Given a highpass transfer function
H
(
s
)
=
s
/(
s
+
(
)
10
z
z
-
1
()
=
Hz
11
-
9
Exercise D.3: Second-Order IIR Bandstop Filter
Given a second-order analog transfer function
H
(
s
) for a bandstop filter, a corre-
sponding discrete-time transfer function
H
(
z
) can be obtained. Let the lower and
upper cutoff frequencies be 950 and 1050 Hz, respectively, with a sampling frequency
F
s
of 5 kHz.
The transfer function selected for a bandstop filter is
s
ssB
+
++
2
w
2
()
=
r
Hs
2
w
2
r
where
B
and
w
are the bandwidth and center frequencies, respectively. The analog
frequencies are
w
T
2
p
¥
950
D
1
w
=
tan
=
tan
=
0 6796
.
A
1
2
2
¥
5000
w
T
2
p
¥
¥
1050
D
2
w
=
tan
=
tan
=
0 7756
.
A
2
2
2
5000
The bandwidth
B
=w
A
2
-w
A
1
=
0.096 and
w
r
=
(
w
A
1
)(
w
A
2
)
=
0.5271. The transfer func-
tion
H
(
s
) becomes
2
0 5271
0 096
s
+
.
()
=
Hs
(D.1)
s
2
+
.
s
+
0 5271
.
and the corresponding transfer function
H
(
z
) can be obtained with
s
=
(
z
-
1)/
(
z
+
1), or
2
{
(
)
(
)
}
+
z
-
1
z
+
1
0 5271
.
()
=
Hz
2
[
(
)
(
)
]
+
(
)
(
)
+
z
-
1
z
+
1
0 096
.
z
-
1
z
+
1
0 5271
.
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