Digital Signal Processing Reference
In-Depth Information
1.25
1.25
1
1
0.75
0.75
0.5
0.5
0.25
0.25
k
0
k
0
−4
−2
1 0
−4
−2
1 0
(a)
(b)
1.25
1
Fig. 1.24. Time shifting of the
DT sequence in Example 1.15.
(a) Original DT sequence
x
[
k
].
(b) Time-delayed version
x
[
k
− 2] of
x
[
k
].
(c) Time-advanced version
x
[
k
+ 2] of
x
[
k
].
0.75
0.5
0.25
0
k
−4
−2
1 0
(c)
Table 1.1. Values of the signals
p
[
k
] and
q
[
k
]
k
−
2
−
1
p
[
k
]
0 . 2
0 . 4
0 . 6
0 . 8
q
[
k
]
0 . 2
0 . 4
0 . 6
0 . 8
By changing the independent variable from
m
to
k
and simplifying, we
obtain
0
.
2(
k
+
2)
−
2
≤
k
≤
3
q
[
k
]
=
x
[
k
+
2]
=
0
elsewhere.
Values of
q
[
k
], for
−
2
≤
k
≤
7, are shown in Table 1.1, and the stem plot for
q
[
k
] is plotted in Fig. 1.24(c).
As in Example 1.14, we observe that the waveform for
p
[
k
]
=
x
[
k
−
2] can
be obtained directly by shifting the waveform of
x
[
k
] towards the right-hand
side by two time units. Similarly, the waveform for
q
[
k
]
=
x
[
k
+
2] can be
obtained directly by shifting the waveform of
x
[
k
] towards the left-hand side
by two time units.
1.3.2 Time scaling
The
time-scaling
operation compresses or expands the input signal in the time
domain. A CT signal
x
(
t
) scaled by a factor
c
in the time domain is denoted by
x
(
ct
). If
c
>
1, the signal is compressed by a factor of
c
. On the other hand, if
0
<
c
<
1 the signal is expanded. We illustrate the concept of time scaling of
CT signals with the help of a few examples.
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