Graph Fourier Transform Enhancement Through Envelope Extensions

Abstract

Many real-world networks are characterized by directionality; however, the absence of an appropriate Fourier basis hinders the effective implementation of graph signal processing techniques. Inspired by discrete signal processing, where embedding a line digraph into a cycle digraph facilitates the powerful Discrete Fourier Transform (DFT) for signal analysis, addressing the structural complexities of general digraphs can help overcome the limitations of the Graph Fourier Transform (GFT) and unlock its potential. The DFT functions as a GFT for both cycle graphs and Cayley digraphs defined over the finite cyclic group $\mathbb{Z}_{N}$. In this work, we present a systematic framework to identify a class of Cayley digraphs capable of encompassing a given directed graph. By embedding the original directed graph into such Cayley digraphs and employing envelope extensions that inherently support the GFT, we enable the use of the GFT associated with the extended structures for signal analysis. Among the candidate envelope extensions, optimal performance is achieved by selecting one that satisfies a set of pre-specified structural properties. The resulting GFT is numerically stable and provides a basis that approximates the eigenstructure of the adjacency matrix associated with the original digraph. It is shown that the envelope extensions possess a convolution product, with their GFT fulfilling the convolution theorem. Additionally, shift-invariant graph filters (systems) are described as the convolution operator, analogous to the classical case. This allows the utilization of systems for signal analysis.