On the diagonalization of the discrete Fourier transform

TL;DR

This paper introduces a canonical eigenbasis for the DFT when N is an odd prime, defines a new discrete oscillator transform based on this basis, and presents a fast algorithm for its computation.

Contribution

It provides a novel eigenbasis for the DFT at prime N and develops a new transform with an efficient computation method.

Findings

Canonical eigenbasis for DFT at prime N

Definition of the discrete oscillator transform (DOT)

Fast algorithm for computing DOT in specific cases

Abstract

The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.