Biomedical Engineering Reference
In-Depth Information
f
(
t
)
0
t
FIGURE 10.13
:
Simple unity step function
Before discussing the complete process of convolution, the reader should know
that calculation of the convolution requires the following steps, which were previously
described.
1.
Change of variable from time,
t
, to delay,
τ
2.
Folding about the vertical axis
3.
Translation
Let us look at an example, using the three steps to visualize and examine the
process with a simple function,
f
(
t
), as shown in Figs. 10.13-10.16.
The first step in the convolution process is the change of independent variable
from a function of time,
f
(
t
), to a function of delay,
f
(
), as in Fig. 10.14.
The second step is to fold the step function about the
y
-axis,
τ
f
(
−
λ
), as in
Fig. 10.15.
The third step is to translate the step function along the
x
-axis some distance,
t
,
as shown in Fig. 10.16. Note that the folded and translated step function is written as
f
(
t
−
τ
).
Let us examine another example; assume the function
f
(
t
)
Ae
−
at
. Figure 10.17
shows the function and the first step, which is the change of variable.
=
f
( )
0
FIGURE 10.14
:
First step. Change the independent variable from a function of time,
f
(
t
), to a
function of delay,
f
(
τ
)
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