Digital Signal Processing Reference
In-Depth Information
ω
|
H
(
) |
G
m
2
)
G
m
/
√
(1+
ε
n
= 2
n
= 3
f
, Hz
f = f
c
0
Fig. 1.33
The magnitude frequency response of a Chebychev-I LPF
where G
m
is the maximum gain, n is the filter order, x
c
is the cutoff frequency, e is
a constant that determines the ripple in the passband of the magnitude response,
and C
n
is the nth order Chebychev Polynomial defined by the iterations:
C
o
ð
x
Þ¼
1
;
C
1
ð
x
Þ¼
x
;
C
n
þ
1
ð
x
Þ¼
2xC
n
ð
x
Þ
C
n
1
ð
x
Þ
For example, C
2
(x) = 2x
2
- 1, C
3
(x) = 4x
3
- 3x, C
4
(x) = 8x
4
- 8x
2
? 1.
At x = 0 (DC), the gain G
o
is:
G
m
;
n odd
G
m
=
p
G
o
¼
ð
1
:
46
Þ
1
þ
e
2
;
n even
while at x = x
c
(cutoff), the gain is:
1
þ
e
p
8
n
: ð
1
:
47
Þ
For 0 \ x \ x
c
, the gain fluctuates between G
m
and G
c
¼
G
m
=
G
c
¼
G
m
=
p
1
þ
e
2
(see
Fig.
1.33
). The maximum power gain variation, called ripple, is given by:
:
r (dB)
¼
10 log
10
1
þ
e
2
Like the B-LPF, the nth-order C-LPF has n poles and no zeros. However, for
the same order n, the C-LPF has a sharper cutoff than the B-LPF. Hence, if the
ripple is not disadvantageous in a specific application, Chebychev filters are
preferred over Butterworth filters. This is so because with Chebychev filters one
typically needs a lower order and hence, less hardware to achieve the same cutoff
sharpness.
1.5.4 Design of Butterworth and Chebychev-I LPF's
This subsection addresses the design of both the filter transfer function H(s) and
physical hardware components needed for building Butterworth and Chebychev
filters. The physical components are an appropriate combination of resistors (R),
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