Digital Signal Processing Reference
In-Depth Information
2
y
1
[
k
]
x
1
[
k
]
0.91
k
k
−1
−2
0
0123456789 0
−1
−2
0
0123456789 0
(a)
(b)
2
y
2
[
k
]
x
2
[
k
]
0.91
0.84
k
k
0
0 123456789 0
−1
−2
0
0123456789 0
−1
−2
(c)
(d)
4
2
x
3
[
k
]
y
3
[
k
]
0.91
0.91
0.91
4
k
k
−1
−2
0
0123456789 0
−1
−2
0
0123
56789 0
0.76
(e)
(f)
Fig. 2.12. Input-output pairs of
the linear DT system specified in
Example 2.2(b). Parts (a)-(f) are
discussed in the text.
Solution
From Eq. (2.39), it follows that:
m
1
(
t
)
→
[1
+
0
.
2
m
1
(
t
)] cos(2
π
10
8
t
)
=
s
1
(
t
)
and
m
2
(
t
)
→
[1
+
0
.
2
m
2
(
t
)] cos(2
π
10
8
t
)
=
s
2
(
t
)
,
giving
α
m
1
(
t
)
+ β
m
2
(
t
)
→
[1
+
0
.
2
α
m
1
(
t
)
+ β
m
2
(
t
)
] cos(2
π
10
8
t
)
= α
s
1
(
t
)
+ β
s
2
(
t
)
.
Therefore, the AM system is not linear.
2.2.2 Time-varying and time-invariant systems
A system is said to be time-invariant (TI) if a time delay or time advance of the
input signal leads to an identical time-shift in the output signal. In other words,
except for a time-shift in the output, a TI system responds exactly the same
way no matter when the input signal is applied. We now define a TI system
formally.
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