Matrices of finite abelian groups, Finite Fourier Transform and codes

TL;DR

This paper generalizes the concept of Fourier Transforms to finite groups, including non-abelian ones, by exploring their algebraic structures and linear representations, with practical examples for engineering applications.

Contribution

It introduces a comprehensive algebraic framework for finite group Fourier Transforms, extending beyond cyclic groups to all finite groups, and clarifies their underlying structures.

Findings

Unified algebraic perspective on FFT for finite groups

Illustrated applications with engineering examples

Enhanced understanding of group representations in signal processing

Abstract

Finite (or Discrete) Fourier Transforms (FFT) are essential tools in engineering disciplines based on signal transmission, which is the case in most of them. FFT are related with circulant matrices, which can be viewed as group matrices of cyclic groups. In this regard, we introduce a generalization of the previous investigations to the case of finite groups, abelian or not. We make clear the points which were not recognized as underlying algebraic structures. Especially, all that appears in the FFT in engineering has been elucidated from the point of view of linear representations of finite groups. We include many worked-out examples for the readers in engineering disciplines.