Graph Fourier Transform based on Directed Laplacian

TL;DR

This paper introduces a new Graph Fourier Transform for directed graphs using a directed Laplacian and Jordan eigenvectors, enabling natural frequency interpretation and polynomial filtering.

Contribution

It redefines GFT for directed graphs within the DSP_G framework using a novel shift operator based on the directed Laplacian.

Findings

Achieves natural frequency ordering and interpretation.

Enables polynomial LSI filters in the directed Laplacian domain.

Provides a new framework for analyzing directed graph signals.

Abstract

In this paper, we redefine the Graph Fourier Transform (GFT) under the DSP$_\mathrm{G}$ framework. We consider the Jordan eigenvectors of the directed Laplacian as graph harmonics and the corresponding eigenvalues as the graph frequencies. For this purpose, we propose a shift operator based on the directed Laplacian of a graph. Based on our shift operator, we then define total variation of graph signals, which is used in frequency ordering. We achieve natural frequency ordering and interpretation via the proposed definition of GFT. Moreover, we show that our proposed shift operator makes the LSI filters under DSP$_\mathrm{G}$ to become polynomial in the directed Laplacian.