The Fractional Fourier Transform on Graphs: Modulation and Convolution

The fractional Fourier transform on graph (GFrFT) is an extension of discrete fractional Fourier transform (DFrFT) based on graph signal processing (GSP). In this paper, we discuss the shift, polynomial filters, delta functions, the convolution and modulation of graph signal in fractional domain. First, we review the basic theory of discrete graph signal processing, concepts of graph signals and GFrFT. Then, similar to the definition of the shift of adjacency matrix in vertex domain, the shift matrix of graph spectral domain in the GFrFT is defined. Finally, the new polynomial filters are used to illustrate the convolution and modulation of fractional graph signals in the vertex domain and graph spectral domain, and the relationship between the delta functions and convolution is discussed.