Discrete fractional Fourier transform: Vandermonde approach

TL;DR

This paper introduces a new method for deriving the discrete fractional Fourier transform using Vandermonde matrices, providing a straightforward approach to compute rational and irrational powers of the DFT matrix.

Contribution

It presents a novel Vandermonde matrix-based approach to obtain discrete fractional Fourier transforms from the DFT, inspired by quantum harmonic oscillator concepts.

Findings

New Vandermonde approach simplifies fractional Fourier transform derivation.

Enables computation of rational and irrational powers of DFT matrices.

Provides a clear analogy with quantum harmonic oscillator definitions.

Abstract

Based on the definition of the Fourier transform in terms of the number operator of the quantum harmonic oscillator and in the corresponding definition of the fractional Fourier transform, we have obtained the discrete fractional Fourier transform from the discrete Fourier transform in a completely analogous manner. To achieve this, we have used a very simple method based on the Vandermonde matrices, to obtain rational and irrational powers of the discrete Fourier transform matrices.