This paper explains clearly, and for the general scientific reader, all of the basic ideas and techniques of digital signal processing. Previously, this information was not available outside of electronic communication specialties. Studies of our universe require a processing of signals, those from atoms and molecules, from the organs of living creatures out to the energies analyzed in radio astronomy. The paper is mathematical but not obscure. It is oriented around numerical methods and computer applications. Mostowski’s exciting work in mathematical logic is given its first physical application in the convolution discussion.