Uncertainty Relation of Functions for Kernel-based Transforms Obtained in Quantum Mechanics

TL;DR

This paper derives a generic uncertainty relation for kernel-based transforms using a quantum operator approach, providing a concise method applicable to various transforms like FrFT, GFrT, and LCT.

Contribution

It introduces a quantum operator method to derive uncertainty relations for kernel-based transforms, unifying and simplifying previous approaches.

Findings

Derived a general uncertainty relation for kernel-based transforms.

Applied the method to specific transforms like FrFT, GFrT, and LCT.

Provided explicit uncertainty relations for these transforms.

Abstract

In this paper, the generic uncertainty relation (UR) for kernel-based transformations (KT) of signals is derived. Instead of using the statistics approach as shown in the literature before, here we employ quantum mechanical operator approach for directly deriving the UR for KT's. We are able to do this because we have found the quantum operator realization of KT. Our new method is concise and applicable to any kinds of KT's with continuous and discrete parameters and variables. An explicit result of UR for a family of KT's including FrFT, generalized fractional transformation (GFrT) and linear canonical transformation (LCT) is provided as an application of our new method.