Linear canonical space-time transform and convolution theorems

TL;DR

This paper introduces a linear canonical space-time transform for 16-dimensional signals valued in Clifford algebra, establishing its properties and a convolution theorem, extending Fourier analysis concepts to this new transform.

Contribution

The paper defines a novel linear canonical space-time transform for Clifford algebra-valued signals and derives its convolution theorem, expanding the mathematical framework for signal processing.

Findings

Defined the linear canonical space-time transform and its properties

Established the convolution operator and theorem for the transform

Derived the convolution theorem for the two-sided transform

Abstract

Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time $C\ell_{3,1}$-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding convolution theorem are proposed. Third, the convolution theorem associated with the two-sided linear canonical space-time transform is derived.