Adaptive filtering algorithms for quaternion-valued signals

Advances in sensor technology have made possible the recoding of three and four-dimensional signals which afford a better representation of our actual three-dimensional world than the ``flat view'' one and two-dimensional approaches. Although it is straightforward to model such signals as real-valued vectors, many applications require unambiguous modeling of orientation and rotation, where the division algebra of quaternions provides crucial advantages over real-valued vector approaches. The focus of this thesis is on the use of recent advances in quaternion-valued signal processing, such as the quaternion augmented statistics, widely-linear modeling, and the HR-calculus, in order to develop practical adaptive signal processing algorithms in the quaternion domain which deal with the notion of phase and frequency in a compact and physically meaningful way. To this end, first a real-time tracker of quaternion impropriety is developed, which allows for choosing between strictly linear and widely-linear quaternion-valued signal processing algorithms in real-time, in order to reduce computational complexity where appropriate. This is followed by the strictly linear and widely-linear quaternion least mean phase algorithms that are developed for phase-only estimation in the quaternion domain, which is accompanied by both quantitative performance assessment and physical interpretation of operations. Next, the practical application of state space modeling of three-phase power signals in smart grid management and control systems is considered, and a robust complex-valued state space model for frequency estimation in three-phase systems is presented. Its advantages over other available estimators are demonstrated both in an analytical sense and through simulations. The concept is then expanded to the quaternion setting in order to make possible the simultaneous estimation of the system frequency and its voltage phasors. Furthermore, a distributed quaternion Kalman filtering algorithm is developed for frequency estimation over power distribution networks and collaborative target tracking. Finally, statistics of stable quaternion-valued random variables, that include quaternion-valued Gaussian random variables as a special case, is investigated in order to develop a framework for the modeling and processing of heavy-tailed quaternion-valued signals.Open Acces

Securing the Distributed Kalman Filter Against Curious Agents

Distributed filtering techniques have emerged as the dominant and most prolific class of filters used in modern monitoring and surveillance applications, such as smart grids. As these techniques rely on information sharing among agents, user privacy and information security have become a focus of concern. In this manuscript, a privacy-preserving distributed Kalman filter (PP-DKF) is derived that maintains privacy by decomposing the information into public and private substates, where only a perturbed version of the public substate is shared among neighbors. The derived PP-DKF provides privacy by restricting the amount of information exchanged with state decomposition and conceals private information by injecting a carefully designed perturbation sequence. A thorough analysis is performed to characterize the privacy-accuracy trade-offs involved in the distributed filter, with privacy defined as the mean squared estimation error of the private information at the honest-but-curious agent. The resulting PP-DKF improves the overall filtering performance and privacy of all agents compared to distributed Kalman filters employing contemporary privacy-preserving average consensus techniques. Several simulation examples corroborate the theoretical results