Reversible Joint Hilbert and Linear Canonical Transform Without Distortion

TL;DR

This paper introduces a reversible, undistorted joint transform combining Hilbert and linear canonical transforms, simplifying signal processing by reducing complexity and enabling accurate recovery of signals.

Contribution

It proposes a new set of transforms that jointly perform Hilbert and LCT operations, reducing complexity and ensuring perfect reversibility without distortion.

Findings

Transforms are reversible and undistorted.

Reduces complexity in signal relationships.

Enhances signal processing applications.

Abstract

Generalized analytic signal associated with the linear canonical transform (LCT) was proposed recently by Fu and Li ["Generalized Analytic Signal Associated With Linear Canonical Transform," Opt. Commun., vol. 281, pp. 1468-1472, 2008]. However, most real signals, especially for baseband real signals, cannot be perfectly recovered from their generalized analytic signals. Therefore, in this paper, the conventional Hilbert transform (HT) and analytic signal associated with the LCT are concerned. To transform a real signal into the LCT of its HT, two integral transforms (i.e., the HT and LCT) are required. The goal of this paper is to simplify cascades of multiple integral transforms, which may be the HT, analytic signal, LCT or inverse LCT. The proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals: a real signal, its HT and analytic signal, and the LCT of these three signals. Most importantly, all the proposed transforms are reversible and undistorted. Using the proposed transforms, several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the LCT.