Digital Signal Processing Reference
In-Depth Information
By calculating the magnitude and phase of the term exp(j
ω
0
t
)
x
(
t
) on the
left-hand side of the CTFT pair shown in Eq. (5.49), we obtain
h
(
t
)
=
e
j
ω
0
t
x
(
t
)
=
e
j
ω
0
t
x
(
t
)
=
x
(
t
)
;
magnitude
(5.50)
<
h
(
t
)
= <
e
j
ω
0
t
x
(
t
)
= <
e
j
ω
0
t
+ <
x
(
t
)
= ω
0
t
+ <
x
(
t
)
.
phase
(5.51)
In other words, frequency shifting the CTFT of a signal does not change the
amplitude
x
(
t
)
of the signal
x
(
t
) in the time domain. The only change is in the
phase
<
x
(
t
) of the signal
x
(
t
), which is modified by an additive factor of
ω
0
t
.
Example 5.17
In Section 2.1.3, we considered an amplitude modulator used in the AM band of
the radio transmission to transmit an information signal
m
(
t
) to the receiver. In
terms of the information signal
m
(
t
), the amplitude-modulated signal is given
by
s
(
t
)
=
A
[1
+
km
(
t
)] cos(
ω
0
t
)
.
Express the CTFT of the amplitude-modulated signal
s
(
t
) in terms of the CTFT
M
(
ω
) of the information signal
m
(
t
).
Solution
The amplitude-modulated signal is a sum of two terms:
A
cos(
ω
0
t
) and
Akm
(
t
)
cos (
ω
0
t
). In Example 5.13, we calculated the CTFT of the
A
cos(
ω
0
t
)as
CTFT
←−−→
A
cos(
ω
0
t
)
A
π
[
δ
(
ω − ω
0
)
+ δ
(
ω + ω
0
)]
.
By expanding cos(
ω
0
t
), the second term
Akm
(
t
) cos(
ω
0
t
) is expressed as fol-
lows:
Akm
(
t
) cos(
ω
0
t
)
=
1
2
Akm
(
t
)[e
j
ω
0
t
+
e
−
j
ω
0
t
]
.
By using the frequency-shifting property, the CTFT of the terms
m
(
t
) exp(j
ω
0
t
)
and
m
(
t
) exp(
−
j
ω
0
t
) are given by
CTFT
←−−→
CTFT
←−−→
m
(
t
)e
j
ω
0
t
−
j
ω
0
t
M
(
ω − ω
0
)
and
m
(
t
)e
M
(
ω + ω
0
)
.
By using the linearity property, the CTFT of
Akm
(
t
) cos(
ω
0
t
) is then given by
←−−→
1
CTFT
Akm
(
t
) cos(
ω
0
t
)
2
Ak
[
M
(
ω − ω
0
)
+
M
(
ω + ω
0
)]
.
By adding the CTFTs of the two terms, the CTFT of the amplitude-modulated
signal is given by
πδ
(
ω − ω
0
)
+ πδ
(
ω + ω
0
)
+
k
2
M
(
ω − ω
0
)
+
k
CTFT
←−−→
s
(
t
)
A
2
M
(
ω + ω
0
)
.
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