Digital Signal Processing Reference
In-Depth Information
Stability property
. To verify that system (iv) is stable, we solve the following
integral:
∞
∞
−
4
t
)d
t
−
4
t
]
∞
0
h
(
t
)
d
t
=
(1
−
e
=
[
t
−
0
.
25e
=∞,
−∞
0
which shows that system (iv) is not stable.
3.7.4 Invertible LTIC systems
Consider an LTIC system with impulse response
h
(
t
). The output
y
1
(
t
)of
the system for an input signal
x
(
t
) is given by
y
1
(
t
)
=
x
(
t
)
∗
h
(
t
). For the
system to be invertible, we cascade a second system with impulse response
h
i
(
t
) in series with the original system. The output of the second system is
given by
y
2
(
t
)
=
y
1
(
t
)
∗
h
i
(
t
)
.
For the second system to be an inverse of the original system, output
y
2
(
t
) should
be the same as
x
(
t
). Substituting
y
1
(
t
)
=
x
(
t
)
∗
h
(
t
) in the above expression
results in the following condition for invertibility:
x
(
t
)
=
[
x
(
t
)
∗
h
(
t
)]
∗
h
i
(
t
)
=
x
(
t
)
∗
[
h
(
t
)
∗
h
i
(
t
)]
.
The above equation is true if and only if
h
(
t
)
∗
h
i
(
t
)
= δ
(
t
)
.
(3.45)
The existence of
h
i
(
t
) proves that an LTIC system is invertible. At times, it is
difficult to determine the inverse system
h
i
(
t
) in the time domain. In Chapter 5,
when we introduce the Fourier transform, we will revisit the topic and illustrate
how the inverse system can be evaluated with relative ease in the Fourier-
transform domain.
Example 3.11
Determine if systems with the following impulse responses:
(i)
h
(
t
)
= δ
(
t
- 2),
(ii)
h
(
t
)
= δ
(
t
)
− δ
(
t
−
2),
are invertible.
Solution
(i) Since
δ
(
t
−
2)
∗ δ
(
t
+
2)
= δ
(
t
), system (i) is invertible. The impulse
response of the inverse system is given by
h
i
(
t
)
= δ
(
t
+
2)
.
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