On the Shift Operator and Optimal Filtering in Graph Signal Processing

Defining a sound shift operator for signals existing on a certain graphical structure, similar to the well-defined shift operator in classical signal processing, is a crucial problem in graph signal processing. Since almost all operations, such as filtering, transformation, prediction, etc., are directly related to the graph shift operator. We define a unique shift operator that satisfies all properties the shift operator as in the classical signal processing, especially this shift operator preserves the energy of a graph signal. Our definition of graph shift operator negates the shift operator defined in the literature as the graph adjacency matrix, which generally does not preserve the energy of a graph signal. We show that any graph shift invariant graph filter can be written as a polynomial function of the graph shift operator and that the adjacency matrix of a graph is indeed a linear shift invariant graph filter with respect to the graph shift operator. Furthermore, we introduce the concepts of the finite impulse response (FIR) and infinite impulse response (IIR) filters similar to the classical signal processing counterparts and obtain an explicit form of such filters. Based on the defined shift operator, we obtain the optimal filtering on graphs, i.e., the corresponding Wiener filtering on graph, and elaborate on the structure of such filters for any arbitrary graph structure. We specially treat the directed cyclic graph and show that the optimal linear shift invariant filter is indeed the well-known Wiener filter in classical signal processing. This result show that, optimal linear time invariant filters for time series data is a subset of optimal graph filters. We also elaborate on the best linear predictor graph filters, optimal filters filter for product graphs.