Discrete Linear Canonical Transform on Graphs: Uncertainty Principle and Sampling

TL;DR

This paper introduces a new uncertainty principle for the graph linear canonical transform (GLCT), develops sampling theory based on this principle, and proposes optimal sampling strategies for graph signals, validated through simulations.

Contribution

It establishes the first uncertainty principle for GLCT and formulates a novel sampling theory for bandlimited graph signals in this domain.

Findings

Uncertainty principle for GLCT demonstrated

Sampling conditions for bandlimited signals derived

Optimal sampling operators proposed and validated

Abstract

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications.