It is very important to understand how to perform direct convolution, as well as to have a picture in your mind about graphical convolution and how it works. However, there is a vitally important theorem that relates the convolutional of two functions to their Fourier transforms. Consider the system that we’ve put together in Fig. 1. Our picture of linear systems tells us that we can compute the output in one of two ways. Either we can break up the input into a superposition of shifted and weighted delta functions, pass each one through the system to get a superposition of shifted and weighted impulse responses h(x − x0), and then add them up through a convolution integral. Alternately, we can break up our input into a superposition of weighted complex sinusoids via the Fourier transform, pass each one through our system using the transfer function H(ξ), and then add them back up again through the inverse Fourier transform. We might ask ourselves, “what is the relationship between h(x) and H(ξ)? Given the notation we’ve chosen, we might guess that they are related by a Fourier transform.