Parameters Estimation Of Two-dimensional Harmonics Signal Based On Quaternion Model

The dissertation is a part of"Research and application of quaternion in parameters estimation of two dimensional harmonics", which is sponsored by the national nature science foundation (60572069).Two dimensional harmonic parameters estimation is often met in the signal processing, especially in the radar, sonar, vibration measurement and earthquake signal processing etc. However, because of the limitation of research level and method, many researchers have to describe two dimensional signals as one dimensional, resulting in the valuable information lost. However, the researches of the two dimensional harmonic signal always bases on the complex model, and it is difficulty to solve the relative problems with methods based on the complex model.From 1990s, the application of quaternion to digital signal processing is more and more extensive, especially in the field of gyroscope, robot technique, artificial satellite carriage control, computer cartoon, image etc. Many researchers devote themselves to the quaternion to find the way in which we can describe interspace geometry with quaternion as plane geometry with complex in. Two dimensional harmonic parameters estimation can be seen as problem of interspace geometry. According to above reasons, the dissertation that parameters estimation of two-dimensional harmonic based on quaternion is presented.The dissertation is divided into six chapters.In chapter one, the existing research state and main problems of 2-D harmonic parameters estimation are summarized, the current research state and significance of the nonlinear phase coupling is generalized; The relationship between 2-D harmonic parameters estimation and other researches is introduced, and the development of quaternion and quaternion matrix in signal processing is presented. Finally, we determine my study in this dissertation.In chapter two, some basic knowledge which would be used is introduced,such as high-order cumulant, high-order moment, high-order spectra, quaternion and quaternion matrix.In chapter three, basic methods for parameter estimation of 2-D harmonic are introduced and classified according to the coupling mode, at the same time, some representative methods are introduced in detail.In chapter four, two dimensional cubic nonlinear phase coupling frequency estimation based on quaternion in Gaussian noise is presented. Two kinds of four-order cumulant of two dimensional harmonic signal based on quaternion are defined and the signal of coupling and coupled frequency without noise is obtained separately, then the two dimensional harmonic coupling and coupled frequencies in Gaussian noise are estimated by means of the slice spectra approach.In chapter five, the inner correlation of two dimensional harmonic signal auto-correlation matrix singular value decomposition of quaternion and two dimensional harmonic signal frequency is presented, and the two dimensional harmonic frequency in white noise is estimated; meanwhile, the inner correlation of two dimensional harmonic signal high-order cumulant matrix singular value decomposition of quaternion and two dimensional harmonic signal frequency is presented, and two dimensional harmonic frequency in Gaussian noise is estimated. The singular value decomposition of quaternion matrix presented in the paper not only can obtain the number of signal, but also can estimate the harmonic frequency effectively, and reduce the calculation quantity. So we say, the method presented in this dissertation can offer us a new approach to research two dimensional harmonic parameter estimation and the relative areas.In chapter six, a brief summary of this dissertation is given, and some current problems and further research aspects are put forward.Based on the former researches, the main creative work of this dissertation includes four aspects.(1) Two dimensional cubic nonlinear phase coupling frequency estimation based on quaternion in Gaussian noise is presented. In this dissertation, according to two dimensional harmonic signal based on quaternion, two kinds of special fourorder cumulant are defined, and two Vandermonde matrices about two kinds of slice spectra are constructed; Then the two-dimensional harmonic coupling and coupled frequencies are estimated by means of spectra peak value searching.(2) The method that estimating two dimensional harmonic frequencies in white noise with singular value decomposition of quaternion matrix is presented. The inner correlation of two dimensional harmonic signal auto-correlation matrix singular value decomposition of quaternion and two dimensional harmonic signal frequency is presented, then the two dimensional harmonic frequency in white noise is estimated with the orthogonal of signal subspace and noise subspace obtained from its SVDQ.(3) The method that estimating two dimensional harmonic frequencies in Gaussian noise with singular value decomposition of quaternion matrix is presented. The inner correlation of two dimensional harmonic high-order cumulant matrix singular value decomposition of quaternion and two dimensional harmonic signal frequency is presented, then the two dimensional harmonic frequency in Gaussian noise is estimated with the orthogonal of signal subspace and noise subspace obtained from its SVDQ.At the same time, this dissertation has certain directive effects on the following four problems.(1) Robustness of 2-D harmonic parameter estimation;(2) The further research on 2-D polynomial;(3) Two dimensional harmonic parameter estimation research based on quaternion model in complex noise;(4) The application of singular value decomposition of quaternion to digital signal processing.