Discrete and Fast Fourier Transform Made Clear

TL;DR

This paper offers a clear, elementary derivation of the discrete and fast Fourier transforms using linear algebra, emphasizing their mathematical foundations and generalizations to finite abelian groups.

Contribution

It introduces a simple derivation of Fourier transforms via circulant matrices and extends the approach to Fourier transforms on finite abelian groups, including the Boolean cube.

Findings

Provides an elementary derivation of DFT and FFT

Generalizes Fourier transform to finite abelian groups

Highlights applications in theoretical computer science

Abstract

Fast Fourier transform was included in the Top 10 Algorithms of 20th Century by Computing in Science & Engineering. In this paper, we provide a new simple derivation of both the discrete Fourier transform and fast Fourier transform by means of elementary linear algebra. We start the exposition by introducing the convolution product of vectors, represented by a circulant matrix, and derive the discrete Fourier transform as the change of basis matrix that diagonalizes the circulant matrix. We also generalize our approach to derive the Fourier transform on any finite abelian group, where the case of Fourier transform on the Boolean cube is especially important for many applications in theoretical computer science.