Digital Signal Processing Reference
In-Depth Information
Fig. 5.12. Waveform
f
(
t
) used in
Example 5.16.
f
(
t
)
5
3
t
−3
0
3
7
10
13
Solution
By inspection,
f
(
t
) can be expressed in terms of
g
(
t
)as
f
(
t
)
=
3
t
+
3
3
+
5
t
−
7
3
2
g
2
g
.
We calculate the CTFT of each term in
f
(
t
) separately. By considering the
CTFT pair
g
(
t
)
CTFT
←−−→
G
(
ω
) and applying the time-shifting property with
a
=
3, we obtain
t
3
CTFT
←−−→
3
G
(3
ω
)
.
g
Using the time-shifting property,
t
+
3
3
t
−
7
3
CTFT
←−−→
CTFT
←−−→
3e
j3
ω
G
(3
ω
)
−
j7
ω
G
(3
ω
)
.
g
and
g
3e
Finally, by applying the linearity property, we obtain
3
2
g
t
+
3
3
+
5
t
−
7
3
←−−→
3
2
3e
j3
ω
G
(3
ω
)
+
5
2
CTFT
−
j7
ω
G
(3
ω
)
.
2
g
3e
Expressed in terms of the CTFT of
g
(
t
), the CTFT
F
(
ω
) of the function
f
(
t
)is
therefore given by
F
(
w
)
=
9
2
e
j3
ω
G
(3
ω
)
+
15
−
j7
ω
G
(3
ω
)
.
e
2
5.5.4 Frequency shifting
In the time-shifting property, we observed the change in the CTFT when a signal
x
(
t
) is shifted in the time domain. The frequency-shifting property addresses
the converse problem of how a signal
x
(
t
) is modified in the time domain if its
CTFT is shifted in the frequency domain.
CTFT
←−−→
If x
(
t
)
X
(
ω
)
then
CTFT
←−−→
h
(
t
)
=
e
j
ω
0
t
x
(
t
)
X
(
ω − ω
0
)
,
for
ω
0
∈ℜ,
(5.49)
where
ℜ
denotes the set of real values.
The frequency-shifting property can be proved directly from Eq. (5.10) by
considering the CTFT of the signal exp(j
ω
0
t
)
x
(
t
). The proof is left as an exercise
for the reader (see Problem 5.15).
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