Uncertainty Inequalities for 3D Octonionic-valued Signals Associated with Octonion Offset Linear Canonical Transform

TL;DR

This paper introduces the octonion offset linear canonical transform, explores its fundamental properties, and generalizes key uncertainty inequalities for 3D octonionic signals within this framework.

Contribution

It defines the octonion offset linear canonical transform and extends several classical uncertainty inequalities to this new transform.

Findings

Derived the closed-form representation of the transform.

Established properties like inversion, norm split, and energy conservation.

Generalized uncertainty inequalities such as Pitts, logarithmic, and Hausdorff-Young inequalities.

Abstract

he octonion offset linear canonical transform can be defined as a time shifted and frequency modulated version of the octonion linear canonical transform, a more general framework of most existing signal processing tools. In this paper, we first define the and provide its closed-form representation. Based on this fact, we study some fundamental properties of proposed transform including inversion formula, norm split and energy conservation. The crux of the paper lies in the generalization of several well known uncertainty relations for the that include Pitts inequality, logarithmic uncertainty inequality, Hausdorff Young inequality and local uncertainty inequalities.