An adaptive real-time multi-tone estimator and Frequency Tracker for non-stationary signals

Harmonic estimation and frequency tracking in real-time are well known pivotal problems in fields like Power Systems/Delivery/Electronics, Telecommunications, Acoustics, Speech and Signal Processing. Incidentally, these subjects also dwell in other (less common) engineering and scientific areas, such as thermonuclear fusion research and, in particular, tokamak plasma diagnostics data processing, where the spectral complexity of characteristic signals imposes frequent challenges. Numerous techniques have been proposed to address these problems. In the overwhelming majority of cases real-time harmonic estimation and frequency tracking have been addressed separately, either by convenience or necessity. Some proposals have employed Kalman Filters (KFs) and KF derived methodologies more or less sophisticated although never dealing with both topics concurrently. The KF is an essential pillar in control theory and it´s merits are well established in a wide range of applications. Although addressing linear systems in its original concept, natural evolutions of the KF for the modelling of nonlinear systems have emerged since, among which the Extended Kalman Filter, a standard for example in GPS and navigation systems. In this paper, a comprehensive approach to multi-tone estimation and frequency tracking using KF techniques is presented. Both the Kalman Filter Harmonic Estimator (KFHE) and the Extended Kalman Filter (EKF) Frequency Tracker (EKFFT) are introduced, along with their generalisations to multi-tone analysis. A series of selectively devised tests were carried out for challenging the performance and determining the operational limits of the EKFFT when aiming to provide accurate estimates, in real-time, of both instantaneous amplitude and phase plus the instantaneous frequency evolution of dominant tones in noisy signals. Finally, a robust algorithm is proposed for achieving the intended goal and conclusions are drawn.