Quaternion Fourier Transform on Quaternion Fields and Generalizations

TL;DR

This paper explores the quaternion Fourier transform (QFT) for quaternion fields, establishing its properties, relations to real signal Fourier transforms, and extending it to non-commutative multivector generalizations with applications in spacetime algebra.

Contribution

It introduces new generalizations of the quaternion Fourier transform using linear transformations and Clifford algebra, expanding its applicability to multivector fields.

Findings

Different forms of QFT lead to various Plancherel theorems

QFT computation relates to real signal Fourier transforms

New volume-time and spacetime algebra Fourier transforms are proposed

Abstract

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.