Digital Signal Processing Reference
In-Depth Information
j3
p
j1.5
D
n
H
(
w
)
n
w
−8
−6
−4
−2
02468
−8
w
0
−6
w
0
−4
w
0
−2
w
0
2
w
0
4
w
0
6
w
0
8
w
0
0
−j1.5
−j3
p
(a)
(b)
The CTFS coefficients
D
n
and the CTFT
Q
(
ω
) of the periodic rectangular wave
are plotted in Figs. 5.13(b) and (c).
Fig. 5.14. Alternative
representations for the sine
wave considered in Example
5.25. (a) CTFS coefficients
D
n
;
(b) CTFT representation
H
(ω).
Example 5.25
Calculate the CTFT for the periodic sine wave
h
(
t
)
=
3 sin(
ω
0
t
).
Solution
To obtain the CTFS representation of the periodic sine wave, we expand sin(
ω
0
t
)
using Euler's identity. The resulting expression is as follows:
h
(
t
)
=
3 sin(
ω
0
t
)
=
3
2j
[e
j
ω
0
t
−
j
ω
0
t
]
,
−
e
which yields the following values for the exponential CTFS coefficients:
−
j1
.
5
n
=
1
=
D
n
j1
.
5
n
=−
1
0
otherwise.
Based on Eq. (5.62), the CTFT of a periodic sine wave is given by
n
=−∞
D
n
δ
(
ω −
n
ω
0
)
=
j3
π
[
δ
(
ω + ω
0
)
− δ
(
ω − ω
0
)]
.
The CTFS coefficients and the CTFT for a periodic sine wave are plotted in Fig.
5.14. The above result is the same as derived in Example 5.18, with a scaling
factor of 3.
∞
H
(
ω
)
=
2
π
5.8 CTFS coefficients as samples of CTFT
In Section 5.7, we presented a method of calculating the CTFT of a periodic
signal from the CTFS representation. In this section, we solve the converse
problem of calculating the CTFS coefficients from the CTFT.
Consider a time-limited aperiodic function
x
(
t
), whose CTFT
X
(
ω
) is known.
By following the procedure used in Section 5.1, we construct several repetitions
of
x
(
t
) uniformly spaced from each other with a duration of
T
0
. The process is
illustrated in Fig. 5.1, where
x
(
t
) is the aperiodic signal plotted in Fig. 5.1(a). Its
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