Biomedical Engineering Reference
In-Depth Information
FIGURE 1.4:
Idea of output production by an LTI system—convolution of input
spikes with the impulse response function. The impulse response of the LTI system is
multiplied by the input impulse. This is the response to the current input. The current
response is added to the time-shifted responses to the previous inputs.
[
]
Using this function any discrete sequence
x
n
can be expressed as:
]=
∑
k
x
[
n
x
[
k
]
δ
[
n
−
k
]
(1.16)
and the output of the LTI system at time
n
due to single impulse at moment
k
as
3
:
h
k
[
n
]=
L
{
δ
[
n
−
k
]
}
=
h
[
n
−
k
]
(1.17)
The output of the LTI system
y
[
n
]=
L
{
x
[
n
]
}
(1.18)
can be computed by substituting (1.16) into (1.18):
{
∑
k
y
[
n
]=
L
x
[
k
]
δ
[
n
−
k
]
}
(1.19)
Due to the linearity of
L
and property (1.17) we have:
]=
∑
k
]
}
=
∑
k
y
[
n
x
[
k
]
L
{
δ
[
n
−
k
x
[
k
]
h
[
n
−
k
]=(
x
h
)[
n
]
(1.20)
˙
3
In this formula we also use the time invariance property.
Search WWH ::
Custom Search