Digital Signal Processing Reference
In-Depth Information
3.7 Impulse respon
se of LTIC systems
In Section 2.2, we considered several properties of CT systems. Since an LTIC
system is completely specified by its impulse response, it is therefore logical to
assume that its properties are completely determined from its impulse response.
In this section, we express some of the basic properties of LTIC systems defined
in Section 2.2 in terms of the impulse response of the LTIC systems. We consider
the memorylessness, causality, stability, and invertibility properties for such
systems.
3.7.1 Memoryless LTIC systems
A CT system is said to be memoryless if its output
y
(
t
) at time
t
=
t
0
depends
only on the value of the applied input signal
x
(
t
) at the same time instant
t
=
t
0
. In other words, a memoryless LTIC system typically has an input-output
relationship of the form
y
(
t
)
=
kx
(
t
)
,
where
k
is a constant. Substituting
x
(
t
)
= δ
(
t
), the impulse response
h
(
t
)ofa
memoryless system can be obtained as follows:
h
(
t
)
=
k
δ
(
t
)
.
(3.43)
An LTIC system will be memoryless if and only if its impulse response
h
(
t
)
=
0 for
t
=
0.
3.7.2 Causal LTIC systems
A CT system is said to be causal if the output at time
t
=
t
0
depends only on
the value of the applied input signal
x
(
t
) at and before the time instant
t
=
t
0
.
The output of an LTIC system at time
t
=
t
0
is given by
∞
y
(
t
0
)
=
x
(
τ
)
h
(
t
0
− τ
)d
τ.
−∞
In a causal system, output
y
(
t
0
) must not depend on
x
(
τ
) for
τ>
t
0
. This
condition is only satisfied if the time-shifted and reflected impulse response
h
(
t
0
− τ
)
=
0 for
τ>
t
0
. Choosing
t
0
=
0, the causality condition reduces to
h
(
−τ
)
=
0 for
τ>
0, which is equivalent to stating that
h
(
τ
)
=
0 for
τ<
0.
Below we state the causality condition explicitly.
An LTIC system will be causal if and only if its impulse response
h
(
t
)
=
0
for
t
<
0.
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