# A Generalization of the Octonion Fourier Transform to 3-D Octonion-Valued Signals -- Properties and Possible Applications to 3-D LTI Partial Differential Systems

## TL;DR

This paper extends the octonion Fourier transform to 3-D octonion-valued signals, establishing its properties, and explores its applications to linear PDE systems, enhancing signal processing and system analysis tools.

## Contribution

It generalizes the OFT to octonion-valued functions, proves key properties analogous to classical Fourier transform, and links it to 3-D LTI PDE systems.

## Key findings

- OFT is well-defined for octonion-valued functions.
- Classical Fourier properties have direct octonion equivalents.
- The generalized OFT aids analysis of 3-D LTI systems.

## Abstract

The paper is devoted to the development of the octonion Fourier transform (OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a continuation and generalization of earlier work by Blaszczyk and Snopek, where they proved few essential properties of the OFT of real-valued functions, e.g. symmetry properties. The results of this article focus on proving that the OFT is well-defined for octonion-valued functions and almost all well-known properties of classical (complex) Fourier transform (e.g. argument scaling, modulation and shift theorems) have their direct equivalents in octonion setup. Those theorems, illustrated with some examples, lead to the generalization of another result presented in earlier work, i.e. Parseval and Plancherel Theorems, important from the signal and system processing point of view. Moreover, results presented in this paper associate the OFT with 3-D LTI systems of linear PDEs with constant coefficients. Properties of the OFT in context of signal-domain operations such as derivation and convolution of $\mathbb{R}$-valued functions will be stated. There are known results for QFT, but they use the notion of other hypercomplex algebra, i.e. double-complex numbers. Considerations presented here require defining other higher-order hypercomplex structure, i.e. quadruple-complex numbers. This hypercomplex generalization of the Fourier transformation provides an excellent tool for the analysis of 3-D LTI systems.