Graph Fourier transform based on directed Laplacian

Abstract

In this paper, we redefine the graph Fourier transform (GFT) under the DSP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> framework. We consider the Jordan eigenvectors of the directed Laplacian matrix 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 for frequency ordering. We achieve natural frequency ordering as well as interpretation via the proposed definition of GFT. Moreover, we show that our proposed shift operator makes linear shift invariant (LSI) filters under DSP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> to become polynomials in the directed Laplacian.