The two-sided Gabor quaternion Fourier transform and some uncertainty principles

TL;DR

This paper introduces the Gabor quaternion Fourier transform (GQFT), extending the classical Fourier analysis to quaternion signals, and establishes its fundamental properties and uncertainty principles.

Contribution

It presents the first definition and analysis of the GQFT, including key properties and uncertainty principles for quaternion-valued signals.

Findings

GQFT generalizes classical Fourier transform to quaternion signals

Plancherel and inversion formulas are established for GQFT

Heisenberg uncertainty principles are proved for GQFT

Abstract

In this paper, we define a new transform called the Gabor quaternionic Fourier transform (GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals, we give several important properties such as the Plancherel formula and inversion formula. Finally, we establish the Heisenberg uncertainty principles for the GQFT.