REALIZATION OF DIGRAPH FILTERS VIA AUGMENTED GFT

This study proposes a filtering method for directed graph (digraph) signals. In order to realize digraph filtering, a novel graph Fourier transform (GFT), Augmented GFT (AuGFT) , is proposed by defining an Hermitian adjacency matrix. Although there has been the same method to give the adjacency matrix of digraphs, this study defines a novel digraph Laplacian. The existing digraph Laplacian does not give the graph signal variation considering the edge directions, while the novel one does. This paper introduces three important ideas. The first is the definition of a novel degree matrix to give the novel digraph Laplacian. The second is to decompose the symmetric and skew-symmetric components of the novel digraph Laplacian independently into their spectral components. The third is, based on the decomposition, to augment the conventional GFT for digraphs as an invertible real-valued dictionary. The new GFT is shown to provide a practical form of real-valued digraph filtering. The signifiance of the proposed method is verified through simulations of signal filtering on digraphs.