Digital Signal Processing Reference
In-Depth Information
n
(even)
Hv
P
()
n
(odd)
1
1
Hv
()
P
1
22
Cv
()
1
n
2
1
1
cos
n
cos
( )
v
v
1
C
n
v
()
1
cosh
n
cosh
( )
v
v
1
A
s
v
v
p
1
0
v
s
FIGURE 8.14
Normalized Chebyshev magnitude response function.
cosh
1
v
s
¼
ln
q
v
v
s
þ
2
s
1
(8.32)
As shown in
Figure 8.14
, the magnitude response for the Chebyshev lowpass prototype with an odd-
numbered order begins with a filter DC gain of 1. In the case of a Chebyshev lowpass prototype with an
even-numbered order, the magnitude starts at a filter DC gain of 1
=
p
2
1
þ
ε
. For both cases, the filter
p
1
þ
ε
2
gain at the normalized cutoff frequency
v
p
¼
1is1
=
.
Similarly, Equations
(8.33) and (8.34)
must be satisfied:
1
1
þ
ε
p
A
P
dB ¼
20
$
log
10
(8.33)
2
0
1
1
1
þ
ε
@
A
A
s
dB ¼
20
$
log
10
q
(8.34)
v
s
2
C
2
n
The lowpass prototype order can be solved in Equation
(8.35)
:
2
¼
10
0
:
1
A
p
1
(8.35a)
ε
cosh
1
10
0
:
1
A
s
1
ε
0
:
5
2
n
(8.35b)
cosh
1
ðv
s
Þ
p
x
where cosh
1
Þ
,
ε
is the absolute ripple parameter.
The normalized stopband frequency
v
s
can be determined from the frequency specifications of
the analog filter in
Table 8.6
. Then the order of the lowpass prototype can be determined by
Equation (8.29)
for the Butterworth function and Equation
(8.35b)
for the Chebyshev function.
Figure 8.15
gives frequency edge notations for analog lowpass and bandpass filters. The notations for
analog highpass and bandstop filters can be defined correspondingly.
ðxÞ¼
ln
ðx þ
2
1
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