Unitary Shift Operators on a Graph

TL;DR

This paper introduces a unitary graph shift operator that preserves energy, derives a graph differential operator, and develops a new graph Fourier transform with orthogonal bases and clear frequency interpretation.

Contribution

It presents a novel unitary shift operator for graph signals, ensuring energy preservation and orthogonal Fourier bases, with applications to graph Hilbert transforms and analytic signals.

Findings

Energy-preserving shift operator demonstrated

Orthogonal graph Fourier bases established

Applications to graph Hilbert transform shown

Abstract

A unitary shift operator (GSO) for signals on a graph is introduced, which exhibits the desired property of energy preservation over both backward and forward graph shifts. For rigour, the graph differential operator is also derived in an analytical form. The commutativity relation of the shift operator with the Fourier transform is next explored in conjunction with the proposed GSO to introduce a graph discrete Fourier transform (GDFT) which, unlike existing approaches, ensures the orthogonality of GDFT bases and admits a natural frequency-domain interpretation. The proposed GDFT is shown to allow for a coherent definition of the graph discrete Hilbert transform (GDHT) and the graph analytic signal. The advantages of the proposed GSO are demonstrated through illustrative examples.