Uncertainty Principles Associated with the Offset Linear Canonical Transform

TL;DR

This paper extends classical uncertainty principles to the offset linear canonical transform (OLCT), a generalization of the linear canonical transform, and introduces a short-time OLCT with associated bounds and principles.

Contribution

It introduces Donoho-Stark and Amrein-Berthier-Benedicks uncertainty principles for the OLCT and generalizes the short-time LCT to the short-time OLCT with Lieb's uncertainty principle.

Findings

Established Donoho-Stark's uncertainty principle for OLCT

Presented Amrein-Berthier-Benedicks's uncertainty principle for OLCT

Derived Lieb's uncertainty principle for the short-time OLCT

Abstract

As a time-shifted and frequency-modulated version of the linear canonical transform (LCT), the offset linear canonical transform (OLCT) provides a more general framework of most existing linear integral transforms in signal processing and optics. To study simultaneous localization of a signal and its OLCT, the classical Heisenberg's uncertainty principle has been recently generalized for the OLCT. In this paper, we complement it by presenting another two uncertainty principles, i.e., Donoho-Stark's uncertainty principle and Amrein-Berthier-Benedicks's uncertainty principle, for the OLCT. Moreover, we generalize the short-time LCT to the short-time OLCT. We likewise present Lieb's uncertainty principle for the short-time OLCT and give a lower bound for its essential support.