Algebraic Signal Processing Theory: An Overview

We give an overview of the algebraic signal processing theory, a recently proposed generalization of linear signal processing (SP). Algebraic SP (ASP) is built axiomatically on top of the concept of a signal model, which is a triple (A, M, Phi), where A is a chosen algebra of filters, M an associated A-module of signals, and Phi generalizes the idea of a z-transform. ASP encompasses standard time SP (continuous and discrete, infinite and finite duration), but goes beyond it, for example, by defining meaningful notions of space SP in one and higher dimensions, separable and non-separable. ASP identifies many known transforms as Fourier transforms for a suitably chosen signal model and provides the means to derive and explain existing and novel transform algorithms. As one example, the discrete cosine transform is in ASP the Fourier transform for the finite space model and possesses general radix Cooley-Tukey type algorithms derived by the theory