Frequency analysis and filter design for directed graphs with polar decomposition

In this study, we challenge the traditional approach of frequency analysis on directed graphs, which typically relies on a single measure of signal variation such as total variation. We argue that the inherent directionality in directed graphs necessitates a multifaceted analytical approach that incorporates multiple signal variations definitions. Our methodology leverages the polar decomposition to define two distinct variations, each associated with different matrices derived from this decomposition. This approach provides a novel interpretation in the node domain and reveals aspects of graph signals that may be overlooked with a singular measure of variation. Additionally, we develop graph filters specifically designed to smooth graph signals in accordance with our proposed variations. These filters allow for bypassing costly filtering operations associated with the original graph through effective cascading. We demonstrate the efficacy of our methodology using an M-block cyclic graph example, validating our claims and showcasing the advantages of our multifaceted approach in analyzing signals on directed graphs.