Frames and vertex-frequency representations in graph fractional Fourier domain

Vertex-frequency analysis, particularly the windowed graph Fourier transform (WGFT), is a significant challenge in graph signal processing. Tight frame theories is known for its low computational complexity in signal reconstruction, while fractional order methods shine at unveil more detailed structural characteristics of graph signals. In the graph fractional Fourier domain, we introduce multi-windowed graph fractional Fourier frames (MWGFRFF) to facilitate the construction of tight frames. This leads to developing the multi-windowed graph fractional Fourier transform (MWGFRFT), enabling novel vertex-frequency analysis methods. A reconstruction formula is derived, along with results concerning dual and tight frames. To enhance computational efficiency, a fast MWGFRFT (FMWGFRFT) algorithm is proposed. Furthermore, we define shift multi-windowed graph fractional Fourier frames (SMWGFRFF) and their associated transform (SMWGFRFT), exploring their dual and tight frames. Experimental results indicate that FMWGFRFT and SMWGFRFT excel in extracting vertex-frequency features in the graph fractional Fourier domain, with their combined use optimizing analytical performance. Applications in signal anomaly detection demonstrate the advantages of FMWGFRFT.