Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
[
n
]
y
[
n
n
0
]
Delay
n
0
System
(a)
x
[
n
]
x
[
n
n
0
]
y
d
[
n
]
Delay
n
0
System
Figure 9.29
Test for time invariance.
(b)
A test for time invariance is given by
`
`
y[n]
n-n
0
= y[n]
,
(9.71)
x[n-n
0
]
provided that
y
[
n
] is expressed as an explicit function of
x
[
n
]. This test is illustrated
in Figure 9.29. The signal
y[n - n
0
]
is obtained by delaying
y
[
n
] by
n
0
.
Define
y
d
[n]
as the system output for the delayed input
x[n - n
0
],
such that
x[n - n
0
] : y
d
[n].
The system is time invariant, provided that
y[n - n
0
] = y
d
[n].
(9.72)
A system that is not time-invariant is time varying. An example of a time-invariant
system is
y
1
[n] = e
x[n]
,
y
2
[n] = e
n
x[n]
whereas
is time varying.
The property of
linearity
is one of the most important properties that we consider.
Once again, we define the system input signal to be
x
[
n
] and the output signal to be
y
[
n
].
Linear System
A system is
linear
if it meets the following two criteria:
1.
Additivity. If
x
1
[n] : y
1
[n] and x
2
[n] : y
2
[n],
then
x
1
[n] + x
2
[n] : y
1
[n] + y
2
[n].
(9.73)
2.
Homogeneity. If
x[n] : y[n],
then, with
a
constant,
ax[n] : ay[n].
(9.74)
The criteria must be satisfied for all
x
[
n
] and all
a
.
These two criteria can be combined to yield the
principle of superposition
. A
system satisfies the principle of superposition if
a
1
x
1
[n] + a
2
x
2
[n] : a
1
y
1
[n] + a
2
y
2
[n],
(9.75)
