Convolution 2.

Cuthbert Nyack
The mathematical expression for the convolution of 2 functions f1(t) and f2(t) is given by the equation below:-

The 2 functions f1(t) and f2(t) are shown below along with f1(t - t) and f2(t - t).

The analytic expressions for the convolution are shown below with their range of validity.

The applet below shows the convolution as t is changed.

The purple line is f1(t), the orange line is f2(t - t) and moves as t is changed by the scrollbar. f2(t - t) is shifted upwards slightly to distinguish it from f1(t). 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(t)f2(t - t) 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.

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COPYRIGHT © 1997 Cuthbert Nyack.