1582 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 58, NO. 4, AUGUST 2011 An Adaptive Algorithm for Real-Time Multi-Tone Estimation and Frequency Tracking of Non-Stationary Signals D. Alves and R. Coelho Abstract—Real-time harmonic estimation and frequency tracking are well known pivotal problems in many engineering domains. Incidentally, similar challenges arise in tokamak plasma diagnostics’ data processing where the spectral complexity of characteristic signals poses additional obstacles. Building up on previous Kalman filter based developments, this paper describes an adaptive real-time algorithm for simultaneous multi-compo- nent frequency tracking and harmonic estimation in noisy signals. Furthermore, the proposed gain adaptation method is shown to improve estimation performance in cases where the amplitude ratio of signal components differs by roughly up to three orders of magnitude. A series of selectively devised tests were carried out for challenging the performance and determining the operational limits of the algorithm 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, results and performance issues are discussed. Index Terms—Adaptive signal processing, diagnostics, predic- tion methods, real-time systems. I. INTRODUCTION T HE problem of harmonic estimation has been a recur- rent issue in several fields of knowledge over the past few decades. In particular, power systems and power delivery are notoriously vulnerable areas where an increased concern and effort has been put into understanding, identifying and miti- gating this effect that can deteriorate the quality, efficiency and safety of electricity consumption. For this purpose, several ap- proaches have been proposed. Neural network-based algorithms have been put forward in [1] and [2], a comb filter configuration together with the Approximate Maximum Likelihood algorithm is presented in [3] as well as several Kalman Filter (KF) based methodologies [4]–[10]. The advantages of using the KF in the harmonic estimation configuration has also motivated interest in other areas such as spectral analysis [11], [12] and fusion re- search diagnostic’s real-time signal processing [13]–[15]. In the particular subject of nuclear fusion research in mag- netically confinement experiments, a few KF based methodolo- gies have already been successfully applied for addressing spe- Manuscript received June 14, 2010; revised March 02, 2011, April 13, 2011, and May 03, 2011; accepted May 14, 2011. Date of publication June 27, 2011; date of current version August 17, 2011. The authors are with the Associação EURATOM/IST, Instituto de Plasmas e Fusão Nuclear-Laboratório Associado, Instituto Superior Técnico, P-1049-001, Lisboa, Portugal (e-mail: dalves@ipfn.ist.utl.pt; rcoelho@ipfn.ist.utl.pt). Digital Object Identifier 10.1109/TNS.2011.2157701 cific issues. When using an implementation compatible with a phase-locked loop, the KF can be used as a synchronous de- tector in noisy environments for measuring the plasma response to external excitation from antennas or as a frequency and am- plitude estimator of non-stationary signals. This is extremely important to provide low latency estimates of radial location and wavenumber of detrimental plasma instabilities thus, as- sisting control algorithms. Also in the field of plasma instabil- ities, the KF can assist feedback control schemes for the sup- pression/stabilization of external kink modes (also called resis- tive wall modes) when a simple linear model for the complex growth rate of the mode is fed into the filter and magnetic coil measurements are considered [16]. KF methods are also used to estimate vessel induced currents in tokamak confined plasmas thereby improving the plasma equilibrium reconstruction [17]. Application to Abel inversion techniques for microwave reflec- tometry estimation of electron density profile was also success- fully evidenced in [18]. Generally speaking the KF, a predictor-corrector real-time native recursive filter, is a general state space system estimator capable of providing its output, at the measurement’s sampling rate, with no prior knowledge other than the state estimation for the previous time instant. Although originally derived for linear systems, soon the need to extend its validity realm led to further developments such as the Extended Kalman Filter (EKF) [19], the Unscented Kalman Filter [20] or the Ensemble Kalman Filter [21]. In this paper, we address the problem of simultaneous real-time estimation of the frequency, phase and amplitude instantaneous quantities of dominant components in noisy signals. Applications can go from real-time synchronous detection to real-time spectral analysis, resonance detection and mode characterisation. In this work we take the Kalman Filter Harmonic Estimator (KFHE) as a starting point along with its generalization to multi-tone estimation. The Extended Kalman Filter Frequency Tracker (EKFFT) is then introduced and a se- ries of tests are performed to address performance issues under various challenging operational conditions. II. THE KALMAN FILTER HARMONIC ESTIMATOR Among all Bayesian-like methods for solving the so-called tracking problem, the KF stands out from the lot by offering analytical expressions for the optimal solution, although under rather restrictive assumptions. In fact, given a time invariant system whose state space evolution is modeled as a Markov process given by (1) (where and represent, respectively, the state variable and the process noise time series) and whose 0018-9499/$26.00 © 2011 IEEE