Design of Graph Filter Using Hilbert Matrix and Vandermonde Matrix

Abstract

The graph filter is a useful tool for processing signals collected from various networks. This paper presents the closed-form design of a graph filter. Depending on whether the graph frequency is known or not, there are two types of designs studied here. The first design is the graph-unaware method, where the graph frequency does not need to be known in advance. In this case, the filter coefficients are determined by minimizing the integral squared errors between the actual response and the ideal response. The optimal solution is then obtained by using the inverse of the Hilbert matrix. The second design is the graph-aware method, where the graph frequency needs to be known in advance. In this case, the filter coefficients are obtained by letting the actual response be equal to the ideal response at the prescribed frequency points. The solution is obtained by using the inverse of the Vandermonde matrix. Finally, the temperature data denoising of sensor networks with different sizes are used to demonstrate the effectiveness of the designed graph filters.