Digital Signal Processing Reference
In-Depth Information
1+
d
p
1+
d
p
1
1
1−
d
p
1−
d
p
pass band
stop band
pass band
stop band
d
s
d
s
w
w
0
0
w
p
w
s
w
p
w
s
(a)
(b)
1+
d
p
1+
d
p
1
1
1−
d
p
1−
d
p
pass band
stop band
pass band
stop band
d
s
d
s
w
w
0
0
w
p
w
s
w
p
w
s
(c)
(d)
Further substituting
H
(
s
/j) from Eq. (7.15) leads to the following expression:
Fig. 7.5. Frequency
characteristics of standard
implementations of lowpass
filters of order
N
.
(a) Butterworth filter; (b) Type-I
Chebyshev filter; (c) Type-II
Chebyshev filter; (d) elliptic filter.
1
H
(
s
)
H
(
−
s
)
=
2
N
,
(7.18)
s
j
1
+
where the denominator represents the characteristic function for
H
(
s
)
H
(
−
s
).
The poles of
H
(
s
)
H
(
−
s
) occur at
2
N
s
j
=−
1
=
e
j(2
n
−
1)
π
(7.19)
or
j
(2
n
−
1)
π
2
N
j
π
2
+
j
(2
n
−
1)
π
2
N
s
=
jexp
=
exp
(7.20)
for 0
≤
n
≤
2
N
−
1. It is clear that the 2
N
poles for
H
(
s
)
H
(
−
s
), specified in
Eq. (7.20), are evenly distributed along the unit circle in the complex s-plane.
Of these,
N
poles would lie in the left half of the s-plane, while the remaining
N
poles would be in the right half of the s-plane. To ensure a causal and
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