On the Upper Bound of Filter Length in the Design of Polynomial Graph Filter

Abstract

In this paper, the upper bound of filter length in the design of polynomial graph Alter is studied. First, the characteristic polynomial of graph Laplacian matrix is used to show that the upper bound of filter length is the number of vertex of graph. Then, two methods are presented to reduce the transfer matrix of graph filter if its length is greater than upper bound. One is degree reduction method, the other is long division method. Next, the numerical examples of cycle graph and path graph are illustrated to show the effectiveness of the proposed reduction methods. Finally, the lowpass graph filter design in the temperature data sensor network is applied to demonstrate the usefulness of proposed filter length reduction methods.