
Convolution 1


Cuthbert Nyack
The mathematical expression for the convolution of 2 functions
f1(t) and f2(t) is given by the equation below:
The proceedure for calculating the convolution is illustrated
diagrammatically below. First express f1(t) as f1(tau), then f2(t)
as f2(tau) and as f2(t  tau). For the case illustrated
below f2(t) = 1  2t and f2(tau) = 1  2tau for tau between
0 and 0.5. Therefore f2(tau) = 1 + 2tau for tau between 0
and 0.5 and f2(t  tau) = 1  2(t  tau) for tau between
t and t  0.5. The green curve below shows t = 0.7 and therefore
f2(t  tau) = 0.4 + 2tau for tau between 0.2 and 0.7.
The product f1(tau)f2(t  tau)
function gives the convolution of the 2 functions, when integrated
over all tau for which the product is nonzero, at this
particular value of t. Depending on the analytic expressions for
f1 and f2, one integration may be sufficient to obtain an
analytic expression for the convolution. In the case below
the integration must be carried out 3 times to obtain 3
expressions for the convolution covering the 3 ranges of t
(0 to 0.5), (0.5 to 1.0) and (1.0 to 1.5). In cases where
f1 and f2 are available only in numerical form(eg. sampled data)
then the integral must be evaluated for all values of t for
which the convolution is desired.
For the value of t = 0.7, f2(t  tau) is shown in green and the
product f1(tau)f2(t  tau) is just equal to f2(t  tau) because
f1(tau) = 1 over the range of tau for which f2(t  tau) is
nonzero. The convolution therefore is the integral of f2(t  tau)
and is the area under the green triangle(shaded as striped purple.
Since the height of
the triangle is 1 and its width is 1/2, then its area is 1/4
which is the convolution for that value of t. In fact this is
also the case for t between 0.5 and 1. For t between 0 and 0.5
f2(t  tau) is shown in cyan for t = 0.15 and the product is
shown as the area in yellow. This area is (t  t*t) and is the
value of the convolution for t between 0 and 0.5.
For t between 1 and 1.5 the product function is shown as the
area shaded in red for t = 1.3. This area is (t*t  3*t + 9/4) and is the
convolution for t between 1.0 and 1.5
For t less than 0 or greater than 1.5, the convolution is
zero.
The applet below shows the convolution as t is changed.
The purple line is f1(tau), the orange line is f2(t  tau)
and moves as t is changed by the scrollbar. f2(t  tau) is
shifted upwards slightly to distinguish it from f1(tau). The red line at
the bottom shows the convolution of the 2 functions. As t
is changed, the filled green area shows the area under the
product function f1(tau)f2(t  tau) and the green line at the
bottom shows the value(area of green shading) of the convolution
at the current value of t. The gain scrollbar can be used to
expand the plot of the convolution.
The Applet below shows the convolution of the same 2 functions
as above. In this case f2 is fixed and f1 is moved. Although
the shaded green area may appear reversed or shifted, its
magnitude is the same as in the above case for any given
value of t. As above the shifted applet is displaced upwards
slightly to avoid overlap with the stationary one. The expressions
for the convolution in the 3 ranges of interest can also be
evaluated with f2 fixed and f1 movable. The result will be the
same as the expressions above.
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COPYRIGHT © 1997 Cuthnbert A. Nyack.