Approaches to multiharmonic frequency tracking and estimation

Abstract

The problem of tracking the possibly time-varying fundamental frequency of a noisy multiharmonic signal, along with its time-varying and unknown harmonic amplitudes, is one of great practical interest, and stems, for example, from problems in sonar signal processing. This thesis continues the theoretical and simulation study of a novel frequency tracker for the multiharmonic case, the Extended Kalman Filter (EKF), first reported in [1]. In addition, the single tone and multiharmonic Maximum Likelihood estimators (MLE{u2019}s) are studied, in their application to the problem of estimating a constant frequency for a sinusoidal signal and a multiharmonic signal respectively. The EKF is applied to the multiharmonic estimation problem, and its performance compared with the CR bounds. For high SNR (signal-to-noiseratio), the EKF is shown to be efficient, (i.e., to have performance that meets the CR bounds). An important averaging approximation is introduced and applied in the calculation of the EKF performance. The notion of a complex analytic signal is clarified. Prompted by a key theoretical result of [1], the performance of an EKF applied to the frequency tracking problem (where the harmonic amplitudes are assumed constant and known) is analyzed (with the aid of the averaging approximation). In some cases, the performance is determined explicitly without resorting to bounds. A close relationship to the well known problem of FM (frequency modulation) demodulation and the associated notion of PLL's (phase locked loops) is observed, and an important parameter termed the effective SNR arises. In addition, the existence of a threshold effect (a dramatic collapse in the performance of an estimator, evident as the SNR is lowered) is demonstrated for the EKF. A simplified continuous time model of the EKF, termed the CPLL (coupled phase locked loop), is derived. As a multiharmonic generalization of the well known PLL, it is successfully analyzed using techniques adapted from those for the PLL. The performance of the CPLL is analyzed in the presence and absence of measurement noise, and its rough equivalence to the EKF is demonstrated. The importance of not overestimating the number of harmonics in the received signal, prior to tracking of the signal{u2019}s frequency, is demonstrated by recourse to the CPLL analysis. Two distinct philosophies of analyzing the threshold effect are identified and discussed - the so-called black box and internal philosophies. The importance of understanding the causes of the threshold effect and the practical desirability of being able to predict its occurrence is made clear. The application of these two different approaches gives new and powerful ways of predicting the point at which threshold occurs, for both the single tone and multiharmonic MLE's. One approach, the black box, leads to the discovery of threshold "indicator" quantities that are calculable from readily available, prior information, for both the single tone and significantly, the mutiharmonic MLE{u2019}s. The other approach (internal) leads to a way of theoretically calculating performance curves clearly demonstrating the threshold effect for the multiharmonic MLE, which till now required Monte Carlo simulation. In addition, the success of the black box approach applied to the multiharmonic MLE has enabled the formation of a strong conjecture as to the characterization of threshold for the EKF/CPLL, a difficult and unsolved problem. Show more