Digital Signal Processing Reference
In-Depth Information
q
(
t
)
3
t
−2
p
−
p
0
p
2
p
(a)
3
p
1.5
D
n
Q
(
w
)
6
6
3/
p
3/
p
6/5
6/5
3/5
p
3/5
p
n
w
−8
−6
−4
−2
4
−8
−6
−4
−2
4
−1/
p
−2
−1/
p
−2
(b)
(c)
Equation (5.62) provides us with an alternative method for calculating the CTFT
of periodic signals using the exponential CTFS. We illustrate the procedure in
Examples 5.24 and 5.25.
Fig. 5.13. Alternative
representations for the periodic
function considered in Example
5.24. (a) A periodic rectangular
wavefunction
q
(
t
), (b) CTFS
coefficients
D
n
for
q
(
t
), and
(c) the CTFT
Q
(ω)of
q
(
t
).
Example 5.24
Calculate the CTFT representation of the periodic waveform
q
(
t
) shown in
Fig. 5.13(a).
Solution
The waveform
q
(
t
) is a special case of the rectangular wave
x
(
t
) considered in
Example 4.14 with
τ
= π
and
T
=
2
π
. Mathematically,
q
(
t
)
=
3
x
(
t
) with duty cycle
τ/
T
=
1
/
2
.
Using Eq. (4.49), the CTFS coefficients of
s
(
t
) are given by
=
3
n
2
D
n
2
sinc
or
#
3
2
n
=
0
0
n
=
2
k
=
0
=
3
n
2
D
n
2
sinc
=
3
n
π
#
n
=
4
k
+
1
−
3
n
π
n
=
4
k
+
3
.
Substituting
ω
0
=
1 in Eq. (5.62) results in the following expression for the
CTFT:
n
=−∞
D
n
δ
(
ω −
n
)
.
∞
CTFT
←−−→
q
(
t
)
2
π
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