A Quartic Polynomial Based Approach to the Frequency Estimation of Noisy Real Sinusoids in Noise

Abstract

In this thesis, we address the problem of frequency estimation of noisy real sinusoids in the newlinepresence of Additive White Gaussian Noise (AWGN). The thesis approaches the problem of newlinefrequency estimation in two parts. The first approach implements the Modified Differential newlineEvolution (MDE) algorithm for frequency estimation. The MDE algorithm falls under the newlinelarger class of genetic algorithms and is, therefore, one among the several iterative approaches newlineto frequency estimation. Iterative algorithms are superior to many non-iterative algorithms newlinesince the accuracy offered by iterative algorithms can seldom be matched with its non-iterative newlinecounterparts. However, iterative approaches are computationally very intensive and hence, can newlinebe challenging to implement for real time frequency estimation problems. newlineThe second and most important contribution of this thesis is the quartic polynomial based newlineapproach to frequency estimation. The frequency estimation problem is formulated as the solution newlineof a quartic (fourth order) polynomial. This work also addresses parameter estimation newlinein general, where, the amplitude and phase estimates can also be derived from the estimated newlinefrequency values. The coefficients of the quartic polynomial are derived from the complex newlinemagnitudes of the three Discrete Fourier Transform (DFT) bins, in which the central bin constitutes newlinethe maximum magnitude value of the DFT coefficients. No estimation algorithm is newlineconsidered valid unless it is corroborated with a theoretical/mathematical validation. This newlinemathematical validation is performed by deriving an expression for the bound of the proposed newlinefrequency estimator which is seen to closely follow the Cramer-Rao Lower Bound (CRLB) for newlinethe frequency estimator. The significance of the quartic polynomial based frequency estimator newlineis that it is a simple, non-iterative approach to frequency estimation that can be easily adopted for real-time frequency estimation. The exact solution for the polynomial is known up to a newlinefourth order polynomial...