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

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: