Algebraic Signal Processing Theory: Sampling for Infinite and Finite 1-D Space

We derive a signal processing framework, called space signal processing, that parallels time signal processing. As such, it comes in four versions (continuous/discrete, infinite/finite), each with its own notion of convolution and Fourier transform. As in time, these versions are connected by sampling theorems that we derive. In contrast to time, however, space signal processing is based on a different notion of shift, called space shift, which operates symmetrically. Our work rigorously connects known and novel concepts into a coherent framework; most importantly, it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete Fourier transform, and hence can be derived by sampling. The platform for our work is the algebraic signal processing theory, an axiomatic approach and generalization of linear signal processing that we recently introduced.

I. Introduction

The theory of time signal processing in one dimension is the foundation of our discipline. It consists of four closely related variants depending on the nature of the time domain (see Fig. 1): infinite continuous, finite (meaning finite duration) continuous and periodically extended, infinite discrete, and finite discrete and periodically extended. Each case has its own notion of filtering, or, convolution, spectrum, and Fourier transform. For example, infinite discrete-time signal processing has the discrete-time Fourier transform (DTFT) as Fourier transform and the spectrum is periodic, that is, continuous, finite, and periodically extended (see Fig. 1). Note that all visualizations in Fig. 1 are directed, representing the directed flow of time formally captured by the time shift discussed next.