Digital Signal Processing Reference
In-Depth Information
x
(
t
)
y
(
t
)
2
2
1
t
t
−4
−3
−2
−1
0
−4
−3
−2
−1
0
(a)
(b)
y
2
(
t
)
x
(
t
− 1)
2
2
1
1
t
t
−4
−3
−2
−1
0
−4
−3
−2
−1
0
(c)
(d)
Example 2.5
Consider two DT systems with the following input-output relationships:
(i) system I
Fig. 2.13. Input-output pairs of
the CT time-invariant system
specified in Example 2.4(i).
(a) Arbitrary signal
x
(
t
).
(b) Output of system for input
signal
x
(
t
). (c) Signal
x
(
t
− 1).
(d) Output of system for input
signal
x
(
t
− 1). Note that except
for a time-shift, the two output
signals are identical.
y
[
k
]
=
3(
x
[
k
]
−
x
[
k
−
2]);
(2.44)
(ii) system II
y
[
k
]
=
kx
[
k
]
.
(2.45)
Determine if the systems are time-invariant.
Solution
(i) From Eq. (2.44), it follows that:
x
[
k
]
→
3(
x
[
k
]
−
x
[
k
−
2])
=
y
[
k
]
x
(
t
)
y
(
t
)
2
2
1
t
t
−4
−3
−2
−1
01234
−4
−3
−2
−1
0
1234
(a)
(b)
x
(
t
−
1)
y
2
(
t
)
2
2
1
1
t
t
−4
−3
−2
−1
0
1234
−4
−3
−2
−1
0
1234
(c)
(d)
Fig. 2.14. Input-output pairs of the time-varying system specified in Example 2.4(ii). (a) Arbitrary signal
x
(
t
). (b) Output of system for input signal
x
(
t
). (c) Signal
x
(
t
− 1). (d) Output of system for input signal
x
(
t
− 1). Note that the output for time-shifted input
x
(
t
− 1) is different from the output
y
(
t
) for the
original input
x
(
t
).
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