Uncertainty Principle for Measurable Sets and Signal Recovery in Quaternion Domains

TL;DR

This paper extends the uncertainty principle to quaternion domains, enabling improved signal recovery in hypercomplex signal processing by analyzing localization constraints in the Quaternion Fourier transform.

Contribution

It generalizes the classical uncertainty principle to hypercomplex signals using quaternion algebra, facilitating new approaches in signal recovery.

Findings

Uncertainty principle is extended to quaternion domains.

Enhanced signal recovery methods for hypercomplex signals.

Potential applications in advanced signal processing tasks.

Abstract

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the Quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time-limiting data.