N-dimensional Heisenberg's uncertainty principle for fractional Fourier transform

TL;DR

This paper derives a sharper N-dimensional uncertainty principle for fractional Fourier transforms, extending classical bounds and demonstrating improved precision with applications in time-frequency and optical systems.

Contribution

It introduces a new lower bound for the N-dimensional fractional Fourier transform uncertainty principle, extending classical results and providing conditions for equality.

Findings

The new uncertainty inequality has a larger lower bound than classical versions.

Simulations confirm the derived bounds are sharper than existing ones.

Applications demonstrated in time-frequency and optical system analysis.

Abstract

A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N-dimensional Heisenberg's uncertainty principle is obtained, and extended from N-dimensional Fourier transform domain to two N-dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals' applications in time-frequency and optical system analysis are also given.