Widely-Linear MMSE Estimation of Complex-Valued Graph Signals

Abstract

In this paper, we consider the problem of recovering random graph signals with complex values. For general Bayesian estimation of complex-valued vectors, it is known that the widely-linear minimum mean-squared-error (WLMMSE) estimator can achieve a lower mean-squared-error (MSE) than that of the linear minimum MSE (LMMSE) estimator for the estimation of improper complex-valued signals. Inspired by the WLMMSE estimator, in this paper we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators. We discuss the properties of the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE estimator. The GSP-WLMMSE estimator is based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">diagonal</i> covariance matrices in the graph frequency domain, and thus has reduced complexity compared with the WLMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In simulations, we investigate a synthetic linear estimation problem and the nonlinear problem of state estimation in power systems. For these problems, it is shown that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves similar performance to that of the WLMMSE estimator. Moreover, the sample-mean version of the GSP-WLMMSE estimator outperforms the sample-mean WLMMSE estimator for a limited training dataset and is more robust to topology changes.