Biomedical Engineering Reference
In-Depth Information
f
(- )
0
FIGURE 10.15
:
Second step. The step function is folded about the
y
-axis,
f
(
−
λ
)
Figure 10.18 shows the folding and translation steps. Note that the translation
is to the right. The function
f
(
t
−
τ
)
−
τ
) can be thought of as
f
{−
(
τ
−
t
)
}
, then
f
(
τ
is (
where
t
), which results in movement to the right (positive direction).
In the next example, illustrated in Fig. 10.19, the original does not begin at
t
τ
−
0,
but is delayed by some magnitude
a
. The amount of displacement of the function is
measured from the position of
f
(
=
0. Note that the amount
of delay is carried in the folding about the
y
-axis, which implies that the zero values must
be accounted for in calculation of the convolution integral.
−
τ
) which is
f
(
t
−
τ
)at
t
=
10.1.2 Steps in the Complete Convolution Process
The entire process of convolution requires the five steps listed as follows:
1.
Change of variable
2.
Folding about the vertical
y
-axis
3.
Translation along the horizontal
x
-axis
4.
Multiplication of the two functions,
f
1
(
τ
) and
f
2
(
t
−
τ
)
5.
Integration of areas
The process might be synopsized as determining the area under the curves (prod-
uct) as the folded function is slid along the horizontal axis. Note the direction of trans-
lation; the conventional notation is as follows:
f
(- )
0
FIGURE 10.16
:
Third step. Translation of the function along the
x
-axis some distance,
t
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