טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentDoweck Yaron
SubjectFrequency Analysis of Non-Stationary Signals with
Harmonic Components
DepartmentDepartment of Electrical Engineering
Supervisors Dr. Alon Amar
Professor Israel Cohen
Full Thesis textFull thesis text - English Version


Abstract

In this work we consider the case of chirps with harmonic components. That is, the frequency of each component is an integer multiple of the time-varying frequency of a fundamental signal. Harmonic signals appear in various applications. Such signals occur due to propagation through nonlinear media, for example in systems such as radar, sonar and communication systems and also in speech and music signals and vibrational analysis. Harmonics can also be used deliberately, to increase detectability, for example in tissue ultrasound or echolocations calls of bats, whales and dolphins.


The problem of estimating the fundamental frequency of harmonic signals has received much attention in literature. In all estimation methods, the frequency is assumed to be constant during the observation time. This assumption limits the possible observation time to be very short since constant frequency signals are not common. However, by assuming a model of time-varying frequency, the observation time can be increased and consequently the estimation accuracy is improved. In this work we assume that in each observation time, the signals can be modeled as a sum of harmonic linear chirps where the number of harmonic components is unknown. Under this assumption, we propose estimation methods for the parameters of such signals, i.e. the initial frequency and frequency rate of the fundamental chirp. As opposed to parameter estimation of multi-component chirp signals, in this case the problem involves only the two parameters of the fundamental chirp. Estimation methods of these two parameters for such model have not been presented in literature to date.


We consider two types of signal model. In the first part, we assume that the amplitude of each component is constant during the observation time. We start by presenting statistical model order selection criteria, based on the maximum likelihood estimator, which requires a high resolution two-dimensional search. To overcome this problem, we propose a low-complexity estimator, which we term the harmonic separate-estimate method. We show that the proposed method achieves the performance of the maximum likelihood estimator in moderate to high signal-to-noise ratio, and can successfully replace it in the model order selection criteria.


The second signal model is a more general case of time-varying amplitudes. In this case, each amplitude can be either independent random process, or varying according to some unknown dynamic. We show that the model order selection criteria and the maximum likelihood estimator cannot be applied to this model. We develop a computationally intensive iterative estimator, based on the nonlinear least squares estimator for mono-component chirp signals with random amplitudes, and show that the number of harmonic components can be selected using concentration measures of the spectrum of the signal. We then show how the harmonic separate-estimate method can be extended for random amplitude harmonic chirps. Simulation results show that the proposed low complexity method performs well in high signal-to-noise ratio.