Robust RLS in the Presence of Correlated Noise Using Outlier Sparsity

Relative to batch alternatives, the recursive least-squares (RLS) algorithm has well-appreciated merits of reduced complexity and storage requirements for online processing of stationary signals, and also for tracking slowly-varying nonstationary signals. However, RLS is challenged when in addition to noise, outliers are also present in the data due to, e.g., impulsive disturbances. Existing robust RLS approaches are resilient to outliers, but require the nominal noise to be white-an assumption that may not hold in e.g., sensor networks where neighboring sensors are affected by correlated ambient noise. Prewhitening with the known noise covariance is not a viable option because it spreads the outliers to noncontaminated measurements, which leads to inefficient utilization of the available data and unsatisfactory performance. In this correspondence, a robust RLS algorithm is developed capable of handling outliers and correlated noise simultaneously. In the proposed method, outliers are treated as nuisance variables and estimated jointly with the wanted parameters. Identifiability is ensured by exploiting the sparsity of outliers, which is effected via regularizing the least-squares (LS) criterion with the ℓ
₁
-norm of the outlier vectors. This leads to an optimization problem whose solution yields the robust RLS estimates. For low-complexity real-time operation, a suboptimal online algorithm is proposed, which entails closed-form updates per time step in the spirit of RLS. Simulations demonstrate the effectiveness and improved performance of the proposed method in comparison with the nonrobust RLS, and its state-of-the-art robust renditions.