A Note on the Diagonalization of the Discrete Fourier Transform

TL;DR

This paper derives an explicit analytical formula for the canonical basis of the Discrete Fourier Transform when the size is a prime congruent to 1 modulo 4, extending previous approaches.

Contribution

It provides a new explicit formula for the DFT's canonical basis in the case of prime sizes congruent to 1 mod 4, building on prior analytical methods.

Findings

Explicit formula for DFT basis when N=p, p ≡ 1 (mod 4)

Extension of Gurevich and Hadani's approach to new prime cases

Enhances understanding of DFT structure for specific prime sizes

Abstract

Following the approach developed by S. Gurevich and R. Hadani, an analytical formula of the canonical basis of the DFT is given for the case $N=p$ where $p$ is a prime number and $p\equiv 1$ (mod 4).