Optimal Fractional Fourier Filtering for Graph Signals

Graph signal processing has recently received considerable attention. Several concepts, tools, and applications in signal processing such as filtering, transforming, and sampling have been extended to graph signal processing. One such extension is the optimal filtering problem. The minimum mean-squared error estimate of an original graph signal can be obtained from its distorted and noisy version. However, the best separation of signal and noise, and thus the least error, is not always achieved in the ordinary Fourier domain, but rather a fractional Fourier domain. In this work, the optimal filtering problem for graph signals is extended to fractional Fourier domains, and theoretical analysis and solution of the proposed problem are provided along with computational cost considerations. Numerical results are presented to illustrate the benefits of filtering in fractional Fourier domains.

I. Introduction

Discrete signal processing on graphs, or graph signal processing (GSP) is concerned with analyzing data which reside on irregular and/or complex structures [1]–[14]. In applications including social, neural, and sensor networks, data can be modeled on the vertices of a weighted graph [2], [4]. While conventional signal processing formulations may be inadequate to model and analyze such data, GSP may make possible the extraction of intrinsic complex relationships in such irregular and complex data [2], [10].