Biomedical Engineering Reference
In-Depth Information
1.3 Linear time invariant systems
In signal processing there is an important class of systems called linear time in-
variant systems—LTI (in case of sampled signals this is sometimes named linear
shift invariant). We can think of the system as a box which modifies the input with a
linear operator
L
to produce the output:
out put
input
L
{
input
}
The basic properties of such a system are:
1. Linearity: superposition of inputs produces superposition of outputs, formally:
if an input
x
1
(
t
)
produces output
y
1
(
t
)
:
L
{
x
1
(
t
)
}
=
y
1
(
t
)
and input
x
2
(
t
)
produces output
y
2
(
t
)
L
{
x
2
(
t
)
}
=
y
2
(
t
)
then the output of the superposition will be:
L
{
a
1
x
1
(
t
)+
a
2
x
2
(
t
)
}
=
a
1
L
{
x
1
(
t
)
}
+
a
2
L
{
x
2
(
t
)
}
=
a
1
y
1
(
t
)+
a
2
y
2
(
t
)
2. Time invariance: for a given input the system produces identical outputs no
matter when we apply the input. More formally:
if
L
)
The important property of LTI systems is that they are completely characterized by
their impulse response function. The impulse response function can be understood
as the output of the system due to the single impulse
2
at the input. It is so, because
we can think of the input signal as consisting of such impulses. In case of a discrete
signal it is very easy to imagine. In case of continuous signal we can imagine it as a
series of infinitely close, infinitely narrow impulses. For each input impulse, the sys-
tem reacts in the same way. It generates a response which is proportional (weighted
by the amplitude of the impulse) to impulse response function. The responses to con-
secutive impulses are summed up with the response due to the former inputs
(Figure
1.4).
Such operation is called
convolution
. For convolution we shall use symbol:
{
x
(
t
)
}
=
y
(
t
)
then
L
{
x
(
t
+
T
)
}
=
y
(
t
+
T
.
The process is illustrated in Figure 1.4. Formally the operation of the LTI system
can be expressed in the following way. Let's denote the impulse response function
as
h
˙
[
n
]
. Next, let us recall the definition of the Kronecker delta:
1if
n
=
0
δ
[
n
]=
0
,
and
n
∈
Z
(1.15)
0if
n
=
2
In case of continuous time system the impulse is the Dirac's delta—an infinitely sharp peak bounding
unit area; in case of discrete systems it is a Kronecker delta—a sample of value 1 at the given moment in
time.
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