Harmonic Retrieval In Complex Noise

Harmonic retrieval in complex noise is one of the most frequently encountered problems inpractice, and constitutes a significant part of statistical signal processing research. The theory ofharmonic retrieval can be applied to many fields, such as sonar, radar imaging, geophysics, radioastronomy, radio communications and so on. The problem of harmonics retrieval can beclassified into two classes according to the complex degrees of noise: one is harmonics retrievalin additive noise; another is harmonics retrieval in complex noise—multiplicative and additivenoise. Most of the works so far focus on harmonics problems of additive noise. However, inpractice application, one may encounter harmonics signals in complex noise. For example, inunderwater acoustic applications, the multiplicative noise can describe effects on acoustic wavesdue to fluctuations caused by the media, changing orientation and interference from scatters ofthe target. Relatively much less attention has been paid to estimating harmonics in complexnoise.This paper presents the estimation of number of harmonics in complex noise using theenhanced matrix. And then, study the algorithm of the parameter estimation of two-dimensionalharmonics in multiplicative and additive noise and the strong compatibility and convergence rateof the estimators using the theory of cyclic cumulants.This dissertation consists of six chapters.In chapter one, we summarized current research state and the significance of harmonicretrieval in complex noise, showed the development of quaternion in signal processing, andpresented the main problems in harmonic retrieval in complex noise. Finally, we determined ourstudy in this dissertation.In chapter two, we introduced some foundational knowledge which would be used later,such as matrix algebra, high-order cumulant and cyclic-stationary.In chapter three, the estimation of number of harmonics in additive noise is studied. Weconstruct an enhanced matrix only from the data sample. We construct an enhanced matrix fromthe data samples. By analyzing the eigenvalues of the covariance matrix of the enhanced matrix,we get the relation between the eigenvalues and the numbers of harmonics in multiplicative andadditive noise. We present the algorithm for estimating the number of harmonics inmultiplicative and additive noise using enhanced matrix based on the relation, and we analyze the theoretical property of the proposed algorithm. Simulation results are presented to demonstratethe effectiveness of the proposed algorithm.In chapter four, we study the estimation of the number of harmonics in multiplicative andadditive noise. An enhanced matrix only from the data samples is constructed. By analyzing theeigenvalues of the covariance matrix of the enhanced matrix, we get the relation between theeigenvalues and the numbers of harmonics in multiplicative and additive noise. We present thealgorithm for estimating the number of harmonics in multiplicative and additive noise usingenhanced matrix based on the relation. Simulation results are presented to demonstrate theeffectiveness of the proposed algorithm.In chapter five, the harmonic retrieval two-dimensional harmonics is multiplicative andadditive noise. At first, we introduce the model of the problem and some algorithm. According tothe properties that two-dimensional cyclic cumulants, we present the executable algorithms ofharmonic retrieval in multiplicative and additive noise based on first-order, second-order andthird-order cyclic cumulants, respectively. And we prove the strong compatibility andconvergence rate of the estimators.In chapter six, we present the method of parameters estimation of the two-dimensional inmultiplicative and additive noise. At first, we give the definition of the two-dimensional Chirp Ztransform and compare the spectral resolution with the two-dimensional Fourier transform. Wefound the two-dimensional Chirp Z transform can improve the spectral resolution. According tothis, we present the algorithm of parameters estimation of two-dimensional harmonics inmultiplicative and additive noise. The proposed algorithm can improve the frequency resolutionand the accuracy based on two-dimensional Fourier transform. Simulation results are presented todemonstrate the effectiveness of the proposed algorithm.Finally, a brief summary of the dissertation is given. The suggestion for future researchesrelated to the harmonic parameter estimation in noise is put forward.