Signal Variation Metrics and Graph Fourier Transforms for Directed Graphs

In this paper we consider the problem of constructing graph Fourier transforms (GFTs) for directed graphs (digraphs), with a focus on developing multiple GFT designs that can capture different types of variation over the digraph node-domain. Specifically, for any given digraph we propose three GFT designs based on the polar decomposition. Our method is closely related to existing polar decomposition based GFT designs, but with added interpretability in the digraph node-domain. Each of our proposed digraph GFTs has a clear node domain variation interpretation, so that one or more of the GFTs can be used to extract different insights from available graph signals. We demonstrate the benefits of our approach experimentally using M-block cyclic graphs, showing that the diffusion of signals on the graph leads to changes in the spectrum obtained from our proposed GFTs, but cannot be observed with the conventional GFT definition.