Graph fractional Fourier transform

Abstract

The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes one definition of FRFT based on specific graph structure with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This work introduces a unified framework for GFRFT that supports multiple definitions with any graph structure and transform order. The proposed approach also allows faster transform matrix computations on large graphs and learnable transform order. Additionally, data sources on each vertex can also continually provide time-series signals such that graph signals are time varying. Joint time-vertex Fourier transform (JFT), with the associated framework of joint time-vertex (JTV) processing, provides spectral analysis tools for such signals. Just as the FRFT generalizes FT, we propose the joint time-vertex fractional Fourier transform (JFRFT) to generalize JFT. JFRFT provides an additional fractional analysis tool for JTV processing by extending temporal and vertex domains to fractional orders. Recently, the classical optimal Wiener filtering problem is introduced to JTV domain. However, the ordinary Fourier domain is not always optimal for separating the signal and noise; one can reach the smaller error in a fractional domain. We generalize the optimal Wiener filtering to the proposed JFRFT domains and provide a theoretical analysis and solution to the problem in the fractional JTV domains. We numerically verify our claims by presenting computational cost analysis and experiments with comprehensive comparisons to state-of-the-art approaches.