Digital Signal Processing Reference
In-Depth Information
A CT system with
x
(
t
)
→
y
(
t
) is time-invariant iff
x
(
t
−
t
0
)
→
y
(
t
−
t
0
)
(2.40)
for any arbitrary time-shift
t
0
. Likewise, a DT system with
x
[
k
]
→
y
[
k
]is
time-invariant iff
x
[
k
−
k
0
]
→
y
[
k
−
k
0
]
(2.41)
for any arbitrary discrete shift
k
0
.
Example 2.4
Consider two CT systems represented mathematically by the following input-
output relationship:
y
(
t
)
=
sin(
x
(
t
));
(i) system I
(2.42)
(ii) system II
y
(
t
)
=
t
sin(
x
(
t
))
.
(2.43)
Determine if systems (i) and (ii) are time-invariant.
Solution
(i) From Eq. (2.42), it follows that:
x
(
t
)
→
sin(
x
(
t
))
=
y
(
t
)
and
x
(
t
−
t
0
)
→
sin(
x
(
t
−
t
0
))
=
y
(
t
−
t
0
)
.
Since sin[
x
(
t
−
t
0
)]
=
y
(
t
−
t
0
), system I is time-invariant. We demonstrate
the time-invariance property of system I graphically in Fig. 2.13, where a time-
shifted version
x
(
t
−
1) of input
x
(
t
) produces an equal shift of one time unit
in the original output
y
(
t
) obtained from
x
(
t
).
(ii) From Eq. (2.43), it follows that:
x
(
t
)
→
t
sin(
x
(
t
))
=
y
(
t
)
.
If the time-shifted signal
x
(
t
−
t
0
) is applied at the input of Eq. (2.43), the new
output is given by
x
(
t
−
t
0
)
→
t
sin(
x
(
t
−
t
0
))
.
The shifted output
y
(
t
−
t
0
)isgivenby
y
(
t
−
t
0
)
=
(
t
−
t
0
) sin(
x
(
t
−
t
0
))
.
Since
t
sin[
x
(
t
−
t
0
)]
=
y
(
t
−
t
0
), system II is not time-invariant. The time-
invariance property of system II is demonstrated in Fig. 2.14, where we
observe that a right shift of one time unit in input
x
(
t
) alters the shape of the
output
y
(
t
).
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