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Convolution

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
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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,
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and let tex2html_wrap_inline3170 be as shown in Fig. 13, defined by
equation330

 

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:
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tex2html_wrap_inline2984 David Marshall 1994-1997