A generalized sampling theorem with application to computer-generated transparencies

A set of generalized Dirac functions (¿ functions) is introduced which resemble the Dirac function (d function) in that their masses equal unity. Different members out of this set are determined by different values of a certain parameter. The sampling theorem—which tells us that a band-limited function can be synthesized by a proper low-pass filtering of a sequence of equidistant d functions having variable masses, these masses being equal to the corresponding sample values of the band-limited function—no longer holds, if the practically unrealizable d functions are replaced by realizable ¿ functions. It must be replaced by a generalized sampling theorem, which tells us what relationship exists between the parameters of the ¿ functions and the sample values of the function to be generated at the ouput of the low-pass filter. Once a set of ¿ functions has been chosen, this relationship can be determined explicitly. An application to the synthesis of coherent optical fields by means of computer-generated transparencies is given.