Widely-Linear MMSE Estimation of Complex-Valued Graph Signals

TL;DR

This paper introduces a graph signal processing-based widely-linear MMSE estimator for complex-valued signals that outperforms linear estimators and approaches the performance of the optimal WLMMSE, with reduced computational complexity.

Contribution

The paper develops the GSP-WLMMSE estimator, a novel approach that combines graph signal processing with widely-linear estimation for complex signals, reducing complexity while maintaining low MSE.

Findings

GSP-WLMMSE always has equal or lower MSE than GSP-LMMSE.

GSP-WLMMSE matches WLMMSE performance under certain conditions.

Simulation results show GSP-WLMMSE outperforms GSP-LMMSE and approaches WLMMSE.

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. 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 diagonal 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 then state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In the simulations, we investigate two synthetic estimation problems (with linear and nonlinear models) and the 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.