Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula

TL;DR

This paper unifies complex and hypercomplex discrete Fourier transforms into a single framework using matrix exponentials, enabling computation with standard matrix operations and simplifying implementation.

Contribution

It introduces a matrix exponential formulation of Euler's formula that unifies various hypercomplex Fourier transforms into a single, computationally accessible framework.

Findings

Transforms can be computed using standard matrix operations.

Examples include quaternion, biquaternion, and Clifford algebra transforms.

The approach simplifies implementation by avoiding hypercomplex libraries.

Abstract

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula $e^{j\theta}=\cos\theta+j\sin\theta$, and a matrix root of -1 isomorphic to the imaginary root $j$. The transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of -1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cl1,1 and Cl2,0. The significance of this result is both in the theoretical unification, and also in the scope it affords for insight into the structure of the various transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier transforms may be computed using standard matrix arithmetic packages without the need for a hypercomplex library, which is of importance in providing a reference implementation for verifying implementations based on hypercomplex code.