Vertex-Frequency Analysis on Directed Graphs

Vertex-frequency analysis (VFA) is a useful technique in graph signal processing to extract the correspondence between frequencies and vertices. VFA can be calculated by the windowed graph Fourier transform (WGFT) and the localized graph Fourier transform (LGFT). However, since the inability of the classical graph Fourier transform (GFT) to generate orthogonal bases for directed graphs, the two approaches are confined to undirected graphs. In order to address the absence of VFA on directed graphs, we extend the fundamental concepts of VFA, shifted vertex window function and band-pass transfer filters in spectral domain to directed graphs. We first propose WDGFT based on the shifted window function in vertex domain. The relevant properties, reconstruction formulas, and simulation results of vertex-frequency representation are given. Then, we propose the localized directed graph Fourier transform (LDGFT), where the band-pass transfer filters can capture the localized spectral domain. The transfer functions satisfying the reconstruction condition are discussed. The vertex-frequency representation obtained by LDGFT is provided through numerical examples. Furthermore, we present a polynomial approximation technique to decrease computational cost. The LDGFT can be calculated without any matrix decomposition. Finally, we evaluate the proposed directed VFA framework with two kinds of applications, clustering and malfunction detection. We demonstrate that the proposed VFA framework is a powerful tool for directed graph signal processing.