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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 tex2html_wrap_inline3166 be the top hat function shown in Fig. 11,
and let tex2html_wrap_inline3170 be as shown in Fig. 13, defined by


Fig. 13 Another top hat: tex2html_wrap_inline3170


Fig. 14 Convolving two top hats

If we now consider x moving from tex2html_wrap_inline3196 to tex2html_wrap_inline3198, we can see that


Fig. 15 Convolution of two top hats

Mathematically the convolution is expressed by:

tex2html_wrap_inline2984 David Marshall 1994-1997