Digital Signal Processing Reference
In-Depth Information
periodic extension
x
T
(
t
) is shown in Fig. 5.1(b). Using Eq. (5.3), the exponential
CTFS coefficients of the periodic extension are given by
T
0
/
2
=
1
T
0
=
1
T
0
D
n
−
j
n
ω
0
t
d
t
−
j
n
ω
0
t
d
t
.
x
(
t
)e
x
(
t
)e
T
0
−
T
0
/
2
Since
x
T
(
t
)
=
x
(
t
) within the range
−
T
0
≤
t
≤
T
0
, the above expression
reduces to
T
0
/
2
∞
=
1
T
0
=
1
T
0
=
1
D
n
−
j
n
ω
0
t
d
t
−
j
n
ω
0
t
d
t
x
(
t
)e
x
(
t
)e
T
0
X
(
ω
)
ω=
n
ω
0
,
(5.63)
−∞
−
T
0
/
2
which is the relationship between the CTFT of the aperiodic signal
x
(
t
) and the
CTFS coefficients of its periodic extension
x
T
(
t
). In other words, we can derive
the exponential CTFS coefficients of a periodic signal with period
T
0
from the
CTFT using the following steps.
(1) Compute the CTFT
X
(
ω
) of the aperiodic signal
x
(
t
) obtained from one
period of
x
T
(
t
)as
x
T
(
t
)
−
T
0
/
2
≤
t
≤
T
0
/
2
x
(
t
)
=
0
elsewhere.
(2) The exponential CTFS coefficients
D
n
of the periodic signal
x
T
(
t
) are given
by
=
1
D
n
T
0
X
(
ω
)
ω=
n
ω
0
,
where
ω
0
denotes the fundamental frequency of the periodic signal
x
T
(
t
)
and is given by
ω
0
=
2
π/
T
0
.
Example 5.26
Calculate the exponential CTFS coefficients of the periodic signal
x
T
(
t
) shown
in Fig. 5.13(a).
Solution
Step 1
The aperiodic signal representing one period of
x
T
(
t
)isgivenby
t
π
x
(
t
)
=
3 rect
.
Using Table 5.2, the CTFT of the rectangular gate function is given by
t
π
ω
2
CTFT
←−−→
3 rect
3
π
sinc
.
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