Digital Signal Processing Reference
In-Depth Information
Fig. 5.1. Periodic extension of a
time-limited aperiodic signal.
(a) Aperiodic signal and (b) its
periodic extension.
x
T
(
t
)
x
(
t
)
t
t
−
L
0
L
−
T
0
−
L
0
L
T
0
(a)
(b)
repetitions of
x
(
t
) uniformly spaced from each other by duration
T
0
such that
there is no overlap between two adjacent replicas of
x
(
t
). The resulting signal
is denoted by
x
T
(
t
) and is shown in Fig. 5.1(b). Clearly, the new signal
x
T
(
t
)is
periodic with the fundamental period of
T
0
and in the limit
T
0
→∞
x
T
(
t
)
=
x
(
t
)
.
lim
(5.1)
Since
x
T
(
t
) is a periodic signal with a fundamental frequency of
ω
0
=
2
π
/
T
0
radians/s, its exponential CTFS representation is expressed as follows:
∞
D
n
e
j
n
ω
0
t
,
x
T
(
t
)
=
(5.2)
n
=−∞
where the exponential CTFS coefficients are given by
=
1
T
0
D
n
−
j
n
ω
0
t
d
t
.
x
T
(
t
)e
(5.3)
T
0
The spectra of
x
T
(
t
) are the magnitude and phase plots of the CTFS coefficients
D
n
as a function of
n
ω
0
. Because
n
takes on integer values, the magnitude
and phase spectra of
x
T
(
t
) consist of vertical lines separated uniformly by
ω
0
. Applying the limit
T
0
→∞
to
x
T
(
t
) causes the spacing
ω
0
=
2
π/
T
0
in
the spectral lines of the magnitude and phase spectra to decrease to zero. The
resulting spectra represent the Fourier representation of the aperiodic signal
x
(
t
)
and are continuous along the frequency (
ω
) axis. The CTFT for aperiodic signals
is, therefore, a continuous function of frequency
ω
. To derive the mathematical
definition of the CTFT, we apply the limit
T
0
→∞
to Eq. (5.3). The resulting
expression is as follows:
1
T
0
D
n
−
j
n
ω
0
t
d
t
lim
T
0
→∞
=
lim
T
0
→∞
x
(
t
)e
T
0
or
∞
1
T
0
−
j
n
ω
0
t
d
t
D
n
=
lim
T
0
→∞
x
(
t
)e
since lim
T
0
→
0
x
T
(
t
)
=
x
(
t
)
.
(5.4)
−∞
In Eq. (5.4), the term
D
n
denotes the exponential CTFT coefficients of
x
(
t
).
Let us define a continuous function
X
(
ω
) (with the independent variable
ω
)as
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