A novel description and mathematical analysis of the Fractional Discrete Fourier Transform

TL;DR

This paper introduces a new mathematical framework for the Fractional Discrete Fourier Transform, providing a closed-form expression and analyzing its properties, which enhances understanding of its rotation-like behavior between time and frequency domains.

Contribution

It presents a novel closed-form expression for the FrDFT matrix and offers a detailed analysis of its properties, advancing the theoretical understanding of the transform.

Findings

FrDFT corresponds to a 2D rotation in time-frequency space

A new closed-form expression for the transformation matrix is derived

Preliminary analysis reveals key properties of the transform

Abstract

I discuss the nature of a Fractional Discrete Fourier Transform (FrDFT) described algorithmically by a combination of chirp transforms and ordinary DFTs. The transform is shown to be consistent with a continuous two-dimensional rotation between the time and frequency domains. I further present a new closed-form expression for the transformation matrix and some preliminary analysis of its properties.