Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)

TL;DR

This paper introduces a generalized two-sided Clifford Fourier transform using two square roots of -1 in Clifford algebras, extending quaternion Fourier transforms and analyzing their properties for signal processing applications.

Contribution

It develops a novel two-sided Clifford Fourier transform framework utilizing two multivector square roots of -1, broadening the scope of Fourier analysis in Clifford algebras.

Findings

The transform's properties include linearity and convolution.

The framework enables splitting multivector signals.

Potential applications in signal processing are discussed.

Abstract

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \textit{multivector square roots} $\in \cl{p,q}$ \textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .