Recently, a class of widely linear (augmented) complex-valued Kalman filters (KFs), that make use of augmented complex statistics, have been proposed for sequential state space estimation of the generality of complex signals. This was achieved in the context of neural network training, and has allowed for a unified treatment of both second-order circular and noncircular signals, that is, both those with rotation invariant and rotation-dependent distributions. In this paper, we revisit the augmented complex KF, augmented complex extended KF, and augmented complex unscented KF in a more general context, and analyze their performances for different degrees of noncircularity of input and the state and measurement noises. For rigor, a theoretical bound for the performance advantage of widely linear KFs over their strictly linear counterparts is provided. The analysis also addresses the duality with bivariate real-valued KFs, together with several issues of implementation. Simulations using both synthetic and real world proper and improper signals support the analysis.

Real-time frequency estimation in three-phase power systems is revisited from the state space point of view, in order to provide a unified framework for frequency tracking in both balanced and unbalanced system conditions. This is achieved by using a novel class of widely linear complex valued Kalman filters, which provide unbiased frequency estimation and are faster converging and more robust to noise and harmonic artifacts than the existing methods. It is shown that the Clarke's transformed threephase voltage is circular for balanced systems and noncircular for unbalanced ones, making the proposed widely linear estimation perfectly suited both to identify the fault and to provide accurate estimation in unbalanced conditions, critical issues where standard models typically fail. The analysis and simulations show that the proposed model outperforms the recently introduced widely linear stochastic gradient-based frequency estimators, based on the augmented complex least mean square. Comprehensive simulations on synthetic and real-world power system data, in both balanced and unbalanced conditions, support the approach.

Abstract-Conventional complex valued signal processing algorithms assume rotation invariant (circular) signal distributions, and are thus suboptimal for real world processes which exhibit rotation dependent distributions (noncircular). In nonlinear sequential state space estimation, noncircularity can arise from the data, state transition model, and state and observation noises. We provide further insight by revisiting the augmented complex unscented Kalman filter (ACUKF) and illuminating its operation in such scenarios. The analysis establishes a relationship between the estimation error and the degree of second order noncircularity (improperness) in the system for the conventional complex unscented Kalman filter (CUKF), and is supported by simulations on both synthetic and real world proper and improper signals.

Motivated by the growing need for robust and accurate frequency estimators at the low-and medium-voltage distribution levels and the emergence of ubiquitous sensors networks for the smart grid, we introduce a distributed Kalman filtering scheme for frequency estimation. This is achieved by using widely linear state space models, which are capable of estimating the frequency under both balanced and unbalanced operating conditions. The proposed distributed augmented extended Kalman filter (D-ACEKF) exploits multiple measurements without imposing any constraints on the operating conditions at different parts of the network, while also accounting for the correlated and noncircular natures of real-world nodal disturbances. Case studies over a range of power system conditions illustrate the theoretical and practical advantages of the proposed methodology.

Bearings only target tracking is concerned with estimating the trajectory of an object from noise-corrupted bearing (phase) measurements. Traditionally this problem has been formulated as real valued for the Cartesian coordinate system or modified polar coordinate system. In this study, the authors introduce the bearings only tracking problem for the complex and quaternion domains to take advantage of the natural representation offered by these domains, for multivariate real signals, as well as the greater insights provided into the dynamics of tracking. Moreover, the authors introduce the augmented complex and quaternion extended Kalman filters for the modelling of second-order non-circular complex and quaternion valued signals, for which a widely linear model is shown to be more suitable than a strictly linear model.