Uncertainty Principles for the Short-time Free Metaplectic Transform

TL;DR

This paper introduces the short time free metaplectic transform (STFMT), a localized time-frequency analysis tool, and explores its properties and uncertainty principles, expanding the theoretical framework for analyzing higher-dimensional signals.

Contribution

The paper proposes the STFMT, defines its properties, and extends various uncertainty principles from quantum mechanics to this new transform, enhancing signal analysis capabilities.

Findings

Defined the STFMT and analyzed its properties.

Extended multiple uncertainty principles to the STFMT domain.

Broadened the theoretical understanding of time-frequency analysis for higher-dimensional signals.

Abstract

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short time free metaplectic transform (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domain