On the Fractionalization of the Shift Operator on Graphs

The theory of graph signal processing has been established with the purpose of generalizing tools from classical digital signal processing to the cases where the signal domain can be modeled by an arbitrary graph. In this context, the present paper introduces the notion of fractional shift of signals on graphs, which is related to considering a non-integer power of the graph adjacency matrix. Among the results derived throughout this work, we prove that the referred fractional operator can be implemented as a linear and shift-invariant graph filter for any graph and verify its convergence to the classical fractional delay when a directed ring graph is considered. By means of a real-world example, we show that, using the proposed operator, graph filters that approximate an ideal filter better than those designed using the ordinary adjacency matrix can be obtained. An additional example dealing with noise removal from graph signals illustrates the gain provided by the mentioned filter design strategy.