The faces of Convolution: from the Fourier theory to algebraic signal processing

TL;DR

This paper provides an overview of convolution across Fourier and algebraic signal processing theories, discussing their relationships and exploring the possibility of a unified approach.

Contribution

It offers a comprehensive comparison of convolution in classical Fourier and algebraic signal processing theories, proposing insights on unification.

Findings

Convolution concepts vary across theories but share underlying principles.

Potential for a unified convolution framework is discussed.

The article clarifies differences and similarities among convolution approaches.

Abstract

In this expository article, we provide a self-contained overview of the notion of convolution embedded in different theories: from the classical Fourier theory to the theory of algebraic signal processing. We discuss their relations and differences. Toward the end, we provide an opinion on whether there is a consistent approach to convolution that unifies seemingly different approaches by different theories.