Digital Signal Processing Reference
In-Depth Information
where
ω
c1
and
ω
c2
are, respectively, referred to as the lower cut-off and higher
cut-off frequencies of the ideal bandstop filter. A bandstop filter can be imple-
mented from a bandpass filter using the following relationship:
H
bs
(
ω
)
=
A
−
H
bp
(
ω
)
.
(7.6)
The ideal bandstop filter is the converse of the ideal bandpass filter as it elimi-
nates a certain range of frequencies (
ω
c1
≤ ω ≤ ω
c2
) from the input signal.
In the above discussion, we used the transfer function to categorize different
types of frequency selective filters. Example 7.1 derives the impulse response
for ideal lowpass and highpass filters.
Example 7.1
Determine the impulse response of an ideal lowpass filter and an ideal highpass
filter. In each case, assume a gain of
A
within the pass band and a cut-off
frequency of
ω
c
.
Solution
Taking the inverse CTFT of Eq. (7.1), we obtain
ω
c
ω
c
ω
c
A
e
j
ω
t
d
ω =
A
e
j
ω
t
j2
π
t
1
2
π
h
lp
(
t
)
=ℑ
−
1
H
(
ω
)
=
−ω
c
A
j2
π
t
[e
j
ω
c
t
−
j
ω
c
t
]
,
=
−
e
which reduces to
h
lp
(
t
)
=
2j
A
sin(
ω
c
t
)
j2
π
t
=
ω
c
A
π
ω
c
t
π
sinc
.
(7.7)
To derive the impulse response
h
hp
(
t
) of the ideal highpass filter, we take the
inverse CTFT of Eq. (7.3). The resulting relationship is given by
Fig. 7.2. Impulse responses
h
(
t
)
of: (a) ideal lowpass filter and
(b) ideal highpass filter derived
in Example 7.1.
A
δ
(
t
)
−
ω
c
A
π
ω
c
t
π
h
hp
(
t
)
=
A
δ
(
t
)
−
h
lp
(
t
)
=
sinc
.
(7.8)
w
c
t
p
w
c
t
p
A
w
c
A
w
c
(
)
(
)
A
w
c
A
sinc
h
lp
(
t
) =sinc
h
hp
(
t
) =
A
−
p
p
p
t
t
−
4
p
−
3
p
−
2
p
−
p
p
2
p
3
p
4
p
−
4
p
−
3
p
−
2
p
p
p
2
p
3
p
4
p
−
0
0
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
w
c
A
w
c
−
p
(a)
(b)
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