Convolution revisited

Abstract

Few mathematical operations are more important to the engineer than convolution and transform analysis. In this article, the operation of convolution is explored-starting with discrete rather than continuous convolution because of the relative ease of comprehension involved. With this foundation, the study is extended to continuous convolution. A proof of the convolution theorem will show that convolution and transform analysis are closely related. Of much more interest, however, is an intuitive explanation of why convolution and transform analysis techniques lead to exactly the same solution of a given problem. Perhaps the two most important applications of convolution deal with the analysis of linear systems and the sums of independent random variables-the latter problem being used to introduce discrete convolution.