Graph Fractional Fourier Transform: A Unified and Efficient Sampling Theory

Abstract: The graph Fourier transform (GFT) is a fundamental tool in graph signal processing and has recently been extended to the graph fractional Fourier transform (GFRFT). Existing sampling methods in the GFRFT domain are primarily designed to minimize error, whereas a wider range of alternative sampling strategies should be admitted. In this paper, a unified and efficient GFRFT sampling theory is proposed. First, a new definition of graph fractional bandlimited signals is introduced, with the corresponding graph fractional sampling and perfect reconstruction theorem, as well as the associated graph fractional localization operator. Next, several GFRFT sampling strategies are developed based on different criteria, including maximum cutoff frequency, minimum error, and maximum localized basis, along with the corresponding representations of their localization operators. Then, by exploiting a localization operator that jointly considers vertex and spectral localization, a fast sampling set selection method in the GFRFT domain is proposed. Finally, numerical experiments investigate the reconstruction errors and execution time of the proposed sampling methods and evaluate their performance in applications, demonstrating the effectiveness of the unified GFRFT sampling theory and its advantages over GFT methods.