Digital Signal Processing Reference
In-Depth Information
x
(
t
)
x
(
t
− 3)
2
2
t
t
t
t
−8
−6
−4
−2
0
246
−8
−6
−4
−2
0
246
(a)
(b)
x
(
t
+ 3)
2
Fig. 1.21. Time shifting of a CT
signal. (a) Original CT signal
x
(
t
). (b) Time-delayed version
x
(
t
− 3) of the CT signal
x
(
t
).
and (c) Time-advanced version
x
(
t
t
t
−8
−6
−4
−2
0
246
(c)
+ 3) of the CT signal
x
(
t
).
In other words, a signal time-shifted by
T
is obtained by substituting
t
in
x
(
t
)by
(
t
+
T
). If
T
<
0, then the signal
x
(
t
) is delayed in the time domain. Graphically
this is equivalent to shifting the origin of the signal
x
(
t
) towards the right-hand
side by duration
T
along the
t
-axis. On the other hand, if
T
>
0, then the
signal
x
(
t
) is advanced forward in time. The plot of the time-advanced signal
is obtained by shifting
x
(
t
) towards the left-hand side by duration
T
along the
t
-axis.
Figure 1.21(a) shows a CT signal
x
(
t
) and the corresponding two time-shifted
signals
x
(
t
−
3) and
x
(
t
+
3). Since
x
(
t
−
3) is a delayed version of
x
(
t
), the
waveform of
x
(
t
−
3) is identical to that of
x
(
t
), except for a shift of three time
units towards the right-hand side. Similarly,
x
(
t
+
3) is a time-advanced version
of
x
(
t
). The waveform of
x
(
t
+
3) is identical to that of
x
(
t
) except for a shift
of three time units towards the left-hand side.
The theory of the CT time-shifting operation can also be extended to DT
sequences. When a DT signal
x
[
k
] is shifted by
m
time units, the delayed signal
φ
[
k
] is expressed as follows:
φ
[
k
]
=
x
[
k
+
m
]
.
If
m
<
0, the signal is said to be delayed in time. To obtain the time-delayed
signal
φ
[
k
], the origin of the signal
x
[
k
] is shifted towards the right-hand side
along the
k
-axis by
m
time units. On the other hand, if
m
>
0, the signal
is advanced forward in time. The time-advanced signal
φ
[
k
] is obtained by
shifting
x
[
k
] towards the left-hand side along the
k
-axis by
m
time units.
Figure 1.22 shows a DT signal
x
[
k
] and the corresponding two time-shifted
signals
x
[
k
−
4] and
x
[
k
+
4]. The waveforms of
x
[
k
−
4] and
x
[
k
+
4] are
identical to that of
x
[
k
]. The time-delayed signal
x
[
k
−
4] is obtained by shifting
x
[
k
] towards the right-hand side by four time units. The time-advanced signal
x
[
k
+
4] is obtained by shifting
x
[
k
] towards the left-hand side by four time
units.
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