Digital Signal Processing Reference
In-Depth Information
applying the time-reversal property to the answer in Example 4.18. Using the
latter approach, the CTFS coefficients
P
n
for
p
(
t
) are given by
=
1
−
n
4
=
1
n
4
−
j(
−
n
)
π/
4
e
j
n
π/
4
.
P
n
=
S
−
n
4
sinc
e
4
sinc
(4.60)
Equation (4.60) can also be obtained directly by applying the time-shifting
property (
t
0
=−π/
2) to the waveform in Fig. 4.14(a) in Example 4.14.
The function
p
(
t
) can now be represented as an exponential CTFS as follows:
n
=−∞
P
n
e
j
n
ω
0
t
n
=−∞
sinc
∞
∞
=
1
4
n
4
e
j
n
π/
4
e
j
nt
p
(
t
)
=
n
=−∞
sinc
∞
=
1
4
n
4
e
j
n
(
t
+π/
4)
,
where the fundamental frequency
ω
0
is set to 1.
Time scaling
If a periodic signal
x
(
t
) with period
T
0
is time-scaled, the CTFS
spectra are inversely time-scaled. Mathematically,
t
a
CTFS
←−−→
CTFS
←−−→
if
x
(
t
)
D
n
then
x
D
an
,
(4.61)
where the time period of the time-scaled signal
x
(
t
/
a
) is given by (
T
0
/
a
).
Example 4.20
Calculate the exponential CTFS coefficients of the periodic function
r
(
t
) shown
in Fig. 4.19. Represent the function as a CTFS.
Solution
From Fig. 4.19, it is observed that
r
(
t
) (with
T
0
=
π
) is a time-scaled version
of
x
(
t
) (with
T
0
=
2
π
) plotted in Fig. 4.14. The relationship between
r
(
t
) and
x
(
t
)isgivenby
r
(
t
)
=
2
x
(2
t
)
.
Using the time-scaling and linearity properties,
CTFS
←−−→
CTFS
←−−→
(4.62)
2
D
n
/
2
.
if
x
(
t
)
D
n
then
2
x
(2
t
)
r
(
t
)
2
Fig. 4.19. Periodic signal
r
(
t
) for
Example 4.20 obtained by
time-scaling Fig. 4.14.
t
p
p
p
p
−
p
−
−
0
p
2
2
8
8
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