Parameters Estimation And Application Of Two-Dimensional Harmonics Based On Quaternion

The paper is part of the research item "Two-dimensional harmonic parametersestimation and nonlinear coupling analysis in complex noise" and "Research andapplication of quaternion in parameters estimation of two dimensional harmonics",which are sponsored by the national natural science foundation.Two-dimensional harmonic parameter estimation and nonlinear couplinganalysis are a common problem in the signal processing field, while it is also a veryimportant part in the research of signal processing theory and methods.Two-dimensional harmonic parameter estimation is widely used especially in thefield of radar, sonar, vibration measurement, oceanography, seismology, EEGanalysis, etc. As the great accomplishment has been achieved in theone-dimensional harmonic parameter estimation in recent years, the researchinterest has been naturally set in the research of two-dimensional parameterestimation. It is not only a necessity to continue the research, but also the needs ofpractice. Two-dimensional harmonic can describe the process and the essence moresufficiently, and so it is more valuable in practice. In addition, two-dimensionalharmonic parameter estimation is the basic and representative problem in themulti-dimensional signal processing field. The similar model exists in the problemof T&A (delay of the time and direction of arrival) problem and the 2-D DOAproblem. So the two-dimensional harmonic parameter estimation research canbenefit the study of relevant field a lot. However, the research of thetwo-dimensional harmonic signal is based on the complex model, many methods ofdisposing this problem are disparting dimensional and pairing steps.Quaternion is introduced in the mathematics field by Hamilton in 1843. Becausethe multiplication of quaternion is not match the exchange rule, the research ismore complex than the research of the real and complex number, which is thereason of the delayed development of quaternion and quaternion matrix. But recenttwenty years, it became the hotspot in the research of quaternion and quaternionmatrix. Application of the quaternion is becaming better and better with thedevelopment of robot technique, floatplane control and computer. Quaternion iswidely used especially in the field of gyroscope, robot technique, artificial satellitecarriage control, computer cartoon, image etc. Research goal of the quaternion is tofind the way through which quaternion in the interspace geometry is the same tocomplex number in plane geometry. Two-dimensional harmonic parameterestimation can be seen as problem of interspace geometry.The purpose that we present quaternion model theory & parameter estimationabout two-dimensional harmonic signal is to find a novel method, based onquaternion theory, that could replace traditional complex searching algorithm usedin two-dimensional parameter estimation in order to couple two-dimensionalparameter, and meanwhile improve the imbalance of two-dimensional parameterestimation precision in low SNR which result from the two-dimensionalparameters placed different status when we make use of parameter automaticmatch.The main works in this paper are summed up as follows.After carefully reading a lot of papers on the two-dimensional parameterestimation and quaternion research, we give a comprehensive sum-up of existingmethods, and find that there are three main problems in this field:(1) The method of two dimensional harmonic signal quaternion model is faulty.(2) The great calculation make it to be difficult to extend the estimationmethods.(3) The simplification of the noise background results in the poor adaptabilityof the estimation methods.Based on the former research, the main innovation of this paper represents in thefollowing three aspects.(1) This paper presents a new quaternion model of two dimensional harmonicsignal, and a special two dimensional harmonic estimation method isintroduced in additive white noise, MA noise, and Gaussian colored noisebased on the quaternion model. We illustrate in detail about therelationship between two-dimensional real harmonics model andtwo-dimensional harmonics model based on quaternion. Then, in order tocalculate the high-order statistics of quaternion, we present conjugateexpression of quaternion;In order to calculate the of determinant ofquaternion matrix, we introduce complex expression of quaternion matrix.of two-dimensional harmonics without any shared frequencies.(2) We advance a new method in nonlinear frequency-coupling analysis basedon quaternion model. We show that algorithm of using quaternion intwo-dimensional harmonics can be used in nonlinear frequency-couplinganalysis. We define the corresponding three-order cumulant to restrain theaddition Gauss noise and extract the coupling and coupled frequency.(3) The problem of two dimensional array signal processing is studiedthrough the quaternion model. The method is presented which can extractthe two-dimensional direction-of-arrival of coherent signals, and jointestimation of frequency and direction-of-arrival, and joint estimation ofDoppler frequency and direction-of-arrival based on quaternion model.This method avoids constructing complicated extended matrix, and solvesthe problem of the error parameter pairs and the imbalance of the twodimensional parameter.At the same time, this paper has some groping effects to the following fourproblems.(1) Corresponding connection of quaternion and two dimensional harmonicsignals with different operation rules.(2) Corresponding connection of eigenvalue, eigenvector and singular value ofthe quaternion matrix and two dimensional harmonic signal.(3) Analysis of two dimensional harmonic signal quaternion model innon-Gaussian noise.