  Graph signal processing (GSP) facilitates the analysis of high-dimensional
data on non-Euclidean domains by utilizing graph signals defined on graph
vertices. In addition to static data, each vertex can provide continuous
time-series signals, transforming graph signals into time-series signals on
each vertex. The joint time-vertex Fourier transform (JFT) framework offers
spectral analysis capabilities to analyze these joint time-vertex signals.
Analogous to the fractional Fourier transform (FRT) extending the ordinary
Fourier transform (FT), we introduce the joint time-vertex fractional Fourier
transform (JFRT) as a generalization of JFT. The JFRT enables fractional
analysis for joint time-vertex processing by extending Fourier analysis to
fractional orders in both temporal and vertex domains. We theoretically
demonstrate that JFRT generalizes JFT and maintains properties such as index
additivity, reversibility, reduction to identity, and unitarity for specific
graph topologies. Additionally, we derive Tikhonov regularization-based
denoising in the JFRT domain, ensuring robust and well-behaved solutions.
Comprehensive numerical experiments on synthetic and real-world datasets
highlight the effectiveness of JFRT in denoising and clustering tasks that
outperform state-of-the-art approaches.
