Digital Signal Processing Reference
In-Depth Information
3
x
1
[
k
]=
d
[
k
+ 1]
x
[
k
]
2
1
1
k
k
−1
01
−1
01
(a)
(b)
3
x
2
[
k
]= 2
d
[
k
]
x
3
[
k
]= 3
d
[
k
−1]
2
k
k
−1
01
−1
01
(c)
(d)
Fig. 1.20. The DT functions in
Example 1.13: (a)
x
[
k
], (b),
x
[
k
],
(c)
x
2
[
k
], and (d)
x
3
[
k
]. The DT
function in (a) is the sum of the
shifted DT impulse functions
shown in (b), (c), and (d).
Example 1.13
Represent the DT sequence shown in Fig. 1.20(a) as a function of time-shifted
DT unit impulse functions.
Solution
The DT signal
x
[
k
] can be represented as the summation of three functions,
x
1
[
k
],
x
2
[
k
], and
x
3
[
k
], as follows:
x
[
k
]
=
x
1
[
k
]
+
x
2
[
k
]
+
x
3
[
k
]
,
where
x
1
[
k
],
x
2
[
k
], and
x
3
[
k
] are time-shifted impulse functions,
x
1
[
k
]
= δ
[
k
+
1]
,
x
2
[
k
]
=
2
δ
[
k
]
,
and
x
3
[
k
]
=
4
δ
[
k
−
1]
,
and are plotted in Figs. 1.20(b), (c), and (d), respectively. The DT sequence
x
[
k
] can therefore be represented as follows:
x
[
k
]
= δ
[
k
+
1]
+
2
δ
[
k
]
+
4
δ
[
k
−
1]
.
1.3 Signal operations
An important concept in signal and system analysis is the transformation of a
signal. In this section, we consider three elementary transformations that are
performed on a signal in the time domain. The transformations that we consider
are time shifting, time scaling, and time inversion.
1.3.1 Time shifting
The
time-shifting
operation delays or advances forward the input signal in time.
Consider a CT signal
φ
(
t
) obtained by shifting another signal
x
(
t
)by
T
time
units. The time-shifted signal
φ
(
t
) is expressed as follows:
φ
(
t
)
=
x
(
t
+
T
)
.
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