Windowed hypergraph Fourier transform and vertex-frequency representation

In recent decades, many tools have been developed and applications have been made possible thanks to graph signal processing (GSP). In this scenario, relationships between elements always occur in pairs. Recently, this theory has been extended to hypergraphs, where elements can be related in groups of two or more elements, the hypergraph signal processing (HGSP). In this context, we propose a novel hypergraph Fourier transform (HGFT) that deals directly with one-dimensional signals. From this Fourier transform, we define the translation and modulation operators for hypergraphs. With the help of these operators, we introduce a methodology for vertex-frequency analysis on hypergraphs based on a novel windowed hypergraph Fourier transform (WHGFT). Finally, in order to illustrate the effectiveness of the proposed tools, we perform experiments with path, cycle, squid, and random geometric hypergraphs.