Uncertainty Principles For the continuous Gabor quaternion linear canonical transform

TL;DR

This paper extends the Gabor transform to the quaternion linear canonical transform, establishing properties and uncertainty principles that enhance time-frequency analysis of nonstationary quaternion signals.

Contribution

It introduces a generalized Gabor quaternion linear canonical transform and derives key properties and uncertainty principles for this new framework.

Findings

Derived Plancherel and inversion formulas.

Proved Heisenberg, Lieb, and logarithmic uncertainty principles.

Established analogs of concentration and Benedick's theorems.

Abstract

Gabor transform is one of the performed tools for time-frequency signal analysis. The principal aim of this paper is to generalize the Gabor Fourier transform to the quaternion linear canonical transform. Actually, this transform gives us more flexibility to studied nonstationary and local signals associated with the quaternion linear canonical transform. Some useful properties are derived, such as Plancherel and inversion formulas. And we prove some uncertainty principles: those including Heisenberg's, Lieb's and logarithmic inequalities. We finish by analogs of concentration and Benedick's type theorems.