Multiplication and convolution of distributions for signal processing theory

In the theory of signal processing, signals are usually classified either by determining whether their time domain is discrete or continuous, or by determining whether they are periodic. However, no comprehensive definitions of multiplication and convolution exist that are consistent with the theories behind all classes, although some important theorems in signal processing involve multiplication and convolution. In order to unite the theories behind these classifications, we will consider tempered distributions. In this paper, we propose an approach to the multiplication and convolution of distributions that is appropriate to signal processing theory, and prove a well-known theorem regarding the impulse response of continuous linear time-invariant systems of tempered distributions in the context of this new approach.