Digital Signal Processing Reference
In-Depth Information
substitute
p
=
s/
1000 in (4.40) and simplify to get
H(s)
as shown below:
1
.
7783
H(p)
|
p
=
s/
1000
=
s
1000
3
2
s
1000
2
2
s
1000
+
+
+
1
(
1
.
7783
)
10
9
=
(4.41)
s
3
+
(
2
×
10
3
)s
2
+
(
2
×
10
6
)s
+
10
9
=
H(s)
(4.42)
The magnitude of
H(p)
plotted on the normalized frequency scale
shown in
Figure 4.7 is marked as “Example (2).” It is found that the attenuation at the
stopband edge
s
=
5 is about 42 dB, which is more than the specified 30 dB.
It must be remembered that in (4.37)
p
=
1 is the bandwidth of the prototype
filter, and at this frequency,
|
2
has a value of
1
2
or a magnitude of
−
3dB.
Hence formulas (4.31) and (4.34) cannot be used if the maximum attenuation
A
p
in the passband is different from 3 dB. In this case, we modify the function to
the form (4.43), which is the general case:
H(j)
|
1
2
|
H(j)
|
=
(4.43)
2
2
n
1
+
2
)
Now the attenuation at
=
1 is given by 10 log
(
1
+
=
A
p
, from which we
get
2
(
10
0
.
1
A
p
−
1
)
. We may also note that
2
=
=
1 in the previous case when
A
p
=
3. When
A
p
is other than 3 dB, the formulas for calculating
n
and
p
k
are
log
(
10
0
.
1
A
s
1
)
1
)/(
10
0
.
1
A
p
−
−
n
≥
(4.44)
2log
s
0
Example (3)
−
1
−
2
−
3
Example (4)
−
4
Example (2)
−
5
10
°
Frequency in radians/sec-linear scale
Figure 4.7
Magnitude responses of the prototype filters in Examples 4.1-4.3.
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