Uncertainty Principles for Quaternion Windowed Offset Linear Canonical Transform of Two Dimensional Signals

TL;DR

This paper introduces a new quaternion-based time-frequency transform called the quaternion windowed offset linear canonical transform, extending analysis tools for 2D signals with applications to uncertainty principles.

Contribution

It generalizes the offset linear canonical transform to quaternion-valued signals and establishes fundamental properties and uncertainty principles for this new transform.

Findings

Derived inner product relation and energy conservation properties.

Established Heisenberg-Weyl, logarithmic, and local uncertainty principles.

Provided an example demonstrating the transform's application.

Abstract

The offset linear canonical transform encompassing the numerous integral transforms, is a promising tool for analyzing non-stationary signals with more degrees of freedom. In this paper, we generalize the windowed offset linear canonical transform for quaternion-valued signals, by introducing a novel time-frequency transform namely the quaternion windowed offset linear canonical transform of 2D quaternion-valued signals. We initiate our investigation by studying some fundamental properties of the proposed transform including inner product relation, energy conservation, and reproducing formula by employing the machinery of quaternion offset linear canonical transforms. Some uncertainty principles such as Heisenberg-Weyl, logarithmic and local uncertainty principle are also derived for quaternion windowed offset linear canonical transform. Finally, we gave an example of quaternion windowed offset linear canonical transform.