Design of Variable Polynomial Graph Filter Using Fractional-Order Graph Laplacian Matrix

A polynomial graph filter is one of the widely used systems for processing irregular data collected from various complex networks. To increase the flexibility of the conventional constant graph filter in use which has a fixed spectral response, a more general type of graph filter called variable graph filter (VGF) was used in this study. Three steps used in the proposed VGF design method include the following. (1) The constant prototype graph filter in which a basic building matrix is chosen as the scaled graph Laplacian matrix (SGLM) is designed by the Lp-norm minimization method. (2) The SGLM of a prototype filter is replaced by the fractional-order SGLM such that the cut-off frequency of the filter is adjusted by changing the value of the fractional order. The spectral mapping relation between the original and mapped frequency is derived. (3) An implementation structure of VGF is provided by using direct-form realization. The fractional order in the structure is adjusted without redesigning the graph filter. Finally, the denoising application of the sensor network was used to determine the optimal fractional order value for obtaining the best signal-to-noise ratio (SNR) and demonstrate the effectiveness of the designed VGF,