Abstract-The framework of graph signal processing was conceived in the last decade with the ambition of generalizing the tools from classical digital signal processing to the case in which the signal is defined over an irregular structure modelled by a graph. Instead of discrete time -what one would call a regular 1-D domain, in which a signal sample is adjacent to only two neighbors and for any pair of contiguous samples the distance is the same -the signals here are defined over graphs and, therefore, the distance and relations between adjacent samples vary along the signal. For instance, one may consider the temperature signal defined from the data of a sensor mesh network. When creating the tools in such a scenario, many challenges arise even with basic concepts of the classical theory. In this paper, the core ideas of graph signal processing are presented, focusing on the two main frameworks developed along the years, and a couple of examples and applications are shown. We conclude drawing attention to a few of the many open opportunities for further studies in the field.

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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 we derive 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, we can obtain graph filters that approximate an ideal filter better than those designed using the ordinary adjacency matrix. An additional example dealing with noise removal from graph signals illustrates the gain provided by the mentioned filter design strategy.