Digital Signal Processing Reference
In-Depth Information
y
(
t
)
x
(
t
)
y
(
t
t
0
)
Delay
t
0
System
(a)
x
(
t
t
0
)
x
(
t
)
y
d
(
t
)
Delay
t
0
System
(b)
Figure 2.44
Test for time invariance.
for all , where indicates the transformation that describes the system's
input-output relationship. In other words, a time-invariant system does not change
with time; if it is used today, it will behave the same way as it would next week or
next year. A time-invariant system is also called a
fixed system
.
A test for time invariance is illustrated in Figure 2.44. The signal
t
0
T[x(t - t
0
)]
y(t - t
0
)
is
obtained by delaying
y
(
t
) by
t
0
seconds. We define
y
d
(t)
as the system output for the
delayed input
x(t - t
0
),
such that
y
d
(t) = T[x(t - t
0
)]
The system in Figure 2.44 is time invariant, provided that
y(t - t
0
) = y
d
(t).
(2.74)
A system that is not time invariant is
time varying
.
As an example of time invariance, consider the system
y(t) = e
x(t)
.
From (2.73) and (2.74),
`
x(t - t
0
),
= e
x(t - t
0
)
`
y
d
(t) = y(t)
= y(t)
t - t
0
,
and the system is time invariant.
Consider next the system
y(t) = e
-t
x(t).
In (2.73) and (2.74),
`
x(t - t
0
)
= e
-t
x(t - t
0
)
y
d
(t) = y(t)
and
`
t - t
0
= e
-(t - t
0
)
x(t - t
0
).
y(t)
The last two expressions are not equal; therefore, (2.74) is not satisfied, and the sys-
tem is time varying.
