Several important optical effects can be described in terms of convolutions.

Let us examine the concepts using 1D continuous functions.

The convolution of two functions *f*(*x*) and *g*(*x*), written *f*(*x*)**g*(*x*), is defined
by the integral

For example, let us take two top hat functions of the type
described earlier. Let be the top hat function shown in
Fig. 11,

and let be as shown in Fig. 13,
defined by

**Fig. 13 Another top hat: **

- is the reflection of this function in the vertical
axis,
- is the latter shifted to the right by a distance
*x*. - Thus for a given value of
*x*, integrated over all is the area of overlap of these two top hats, as has unit height. - An example is shown for
*x*in the range in Fig. 14.

**Fig. 14 Convolving two top hats**

If we now consider *x* moving from to , we can see
that

- for or , there is no overlap;
- as
*x*goes from -1 to 0 the area of overlap steadily increases from 0 to 1/2; - as
*x*increases from 0 to 1, the overlap area remains at 1/2; - and finally as
*x*increases from 1 to 2, the overlap area steadily decreases again from 1/2 to 0. - Thus the convolution of
*f*(*x*) and*g*(*x*),*f*(*x*)**g*(*x*), in this case has the form shown in Fig. 15,

**Fig. 15 Convolution of two top hats**

Mathematically the convolution is expressed by:

David Marshall 1994-1997