| The problem of two-dimensional (2-D) harmonic parameter estimation in complex noise, usually called two-dimensional harmonic retrieval, is one of the most frequently problems in the field of multidimensional signal processing, and it is also an important problem in statistical signal processing. It is because in many applications such as source localization, radar imaging, vibrational analysis of circularly shaped objects, nuclear magnetic resonance spectroscopy, wireless communication channel estimation, geophysics, biomedical science, array signal processing, acoustic, the corresponding signals can be well described by the2-D harmonic model. Most of the works so far focus on the additive noise case of2-D, but the harmonic signals in complex noise (usually called multiplicative and additive noise) often occur in practical applications. For example, in underwater acoustic application, the multiplicative noise describes the effects acoustic waves due to the fluctuations caused by the medium, changing orientation and interference from scatters of the target. The signal model in multiplicative noise can also be applied to the fading channel, earthquake signal processing, image processing and so on. However, the observed data is influenced not only by the complex noise but also by the outliers, so that the observed values deviate from the true value far away. Many natural, as well as man-made, sources of outliers exist, including atmospheric disturbance to cellular telephone in wireless communication. In underwater signals, outliers may arise from ice cracking in arctic region. When the texture contains outliers the texture may be completely perturbed. If only considering the noise removed, the value texture cannot be recovered. In some regions of seismic image which include outliers can cause an erroneous interpretation. Hence it is very important to study the2-D harmonics in noise and outliers, which is beneficial to the further extension of the harmonic retrieval in practical applications.The so-called subspace method, namely the sample covariance matrix divides observation data space into signal subspace and its orthogonal complement (i.e. noise subspace) by singular value decomposition. Signal subspace is spanned by eigenvectors corresponding to large eigenvalues (usually called signal eigenvalues) of covariance matrix while noise subspace is spanned by eigenvectors corresponding to small eigenvalues (usually called noise eigenvalues) of covariance matrix. The subspace method has played a very important role in harmonic parameter estimation and significant researches consist of three stages. As early as1973and1979, Pisarenko, Schmidt and Bienvenu proposed pisarenko harmonic decomposition method and multiple signal classification method, respectively. The pisarenko harmonic decomposition method can be considered as the simplest subspace method, and also can be considered as the simplest form of multiple signal classification method. The second stage is that in1986and1989, Puaiarj, Roy, Kallhat and Vibegr proposed estimation of signal parameters by rotational invariance techniques. The third stage is that in1991, Vibegr and others proposed subspace fitting algorithm. Because the subspace method could achieve higher accuracy and has good statistical properties it became a classic algorithm for estimation of harmonic parameters. The related researches include subspace methods with singular value decomposition and subspace methods without singular value decomposition. However, to date, most researches for harmonic parameter estimation based on subspace method concentrated on the additive noise, and just a few researches discuss the case of complex noise. Therefore, it is meaningful to study the theories and the algorithms of subspace method for harmonic parameter estimation in complex noise. The frequency parameter estimation is the most important problem in the premise of harmonic components known. This is because frequency parameters are nonlinear parameters. Once the frequency estimation is obtained the phase and amplitude estimation which become linear problems are easily obtained. Besides, the estimation of frequencies influences directly the estimation accuracy of the phase and amplitude. The purpose of this dissertation is to systematically study the subspace method based on singular value decomposition and its statistical analysis of2-D harmonic parameter estimation in complex noise in order to provide the theoretical foundation, fast and efficient algorithms for further researches on the related methods.The major work of this dissertation can be summarized as the following three aspects.1、The signal subspace method is proposed for the frequency estimation of2-D harmonics in complex noise.(1) For2-D harmonics in complex noise, by using the singular value decomposition of data (or squared data) matrix and the rotational invariance of signal subspace and the Least Squares Estimation (LSE) technology, the inherent relation between the frequencies of harmonics in complex noise and the data matrix is derived, which can used to estimate the frequencies of2-D harmonics in complex noise. Meanwhile, the estimated frequencies are automatically paired. The computer simulations showed that for2-D Harmonics in nonzero-mean multiplicative noise, the estimators are effective in case of short data length and low signal-to-noise ratio (SNR); for2-D harmonics in zero-mean multiplicative noise, the data length and SNR do influence the estimators.(2) For2-D harmonics in complex noise, by using the singular value decomposition of data (or squared data) matrix, the linear prediction of dominant singular vectors and the Weighted Least Squares Estimation (WLSE) technology, the inherent relation between the frequencies of harmonics in complex noise and the data matrix is derived, which can used to estimate the frequencies of2-D harmonics in complex noise. Meanwhile, the paring algorithm is proposed to identify the frequency pairs. The computational efficiency and estimation accuracy are demonstrated via computer simulations.2、The noise subspace method is proposed for the frequency estimation of2-D harmonics in complex noise.(1) For2-D harmonics in complex noise, by observing the characteristics of data (or squared data) matrix, the inherent relation between the frequencies of harmonics in complex noise and the characteristic matrix is derived. Based on this relationship the special data matrix is constructed. Using the singular value decomposition of data (or squared data) matrix, the noise subspace and the characteristic matrix, the frequencies of harmonics in complex noise are estimated.(2) For2-D harmonics in complex noise, the statistical analysis of the proposed estimators based noise subspace method has been proposed. By statistical analysis, it can be theoretically proved that the proposed estimators for harmonic frequencies in complex noise are strong consistency.3、The signal subspace method and M-estimator are proposed for the frequency estimation of2-D harmonics in noise and outliers.For2-D harmonics in noise and outliers, by using efficient signal subspace method and robust M-estimation, a new robust iterative algorithm is proposed. It is observed that the proposed method performs better than traditional parameter methods in terms of estimation accuracy and computational complexity.The innovative pursuits in this dissertation can be summarized as the following three aspects.1、The signal subspace and noise subspace method for harmonica in additive noise have been successfully generalized to the case of2-D harmonics in nonzero-mean multiplicative noise respectively, and then further generalized to the case of2-D harmonics in zero-mean multiplicative noise respectively.2、The strong consistency of estimators based on the noise subspace has been theoretically derived and experimentally validated for all the considered harmonics, which provided the theoretical foundation for the further researches on algorithm based noise subspace method.3、The proposed signal subspace method has also been successfully generalized to the case of2-D harmonics with noise and outliers, which provided a parameter method with a fast and accurate manner for2-D harmonics with noise and outliers.The proposed estimation technique uses efficient signal subspace and robust M-estimation, in a sequential, for estimation of the harmonic frequencies. As the proposed approach was based on M-estimation technique, the estimators obtained were robust to presence of outliers in the data.The dissertation consists of the following seven chapters.Chapter One first introduces the background and significance of the problem of2-D harmonic parameter estimation in complex noise, next summarizes the developmental history and current situation on this topic, then presents some main problems of current researches on this issues, finally determines the research contents and significance.Chapter Two introduces the fundamental knowledge in matrix analysis and M-estimatiors related to this dissertation, which provides the appropriate theoretical tool for the researches in this dissertation.Chapter Three studies the signal subspace method for frequency estimation of2-D harmonics in complex noise. For the case of nonzero-mean multiplicative noise, the signal subspace method is proposed to the frequencies by using the rotational invariance of signal subspace and the LSE technology. For the case of zero-mean multiplicative noise, the signal subspace method is proposed to the frequencies by using the rotational invariance of subspace and the LSE technology. Finally, the numerical results validate the feasibility of the proposed signal subspace method.Chapter Four studies the signal subspace method for frequency estimation of2-D harmonics in complex noise. For the case of nonzero-mean multiplicative noise, the signal subspace method is proposed to the frequencies by using the linear prediction of dominant singular vectors and the WLSE technology. For the case of zero-mean multiplicative noise, the signal subspace method is proposed to the frequencies by using the linear prediction of dominant singular vectors and the WLSE technology. Finally, the numerical results validate the feasibility of the proposed signal subspace method.Chapter Five studies the noise subspace method for frequency estimation of2-D harmonics in complex noise. For the case of nonzero-mean multiplicative noise, the noise subspace method is proposed to the frequencies. For the case of zero-mean multiplicative noise, the noise subspace method is proposed to the frequencies. By statistical analysis, the strong consistency of estimators based on the noise subspace is theoretically derived. Moreover, the numerical results validate the feasibility of the proposed noise subspace method. Finally, for comparison, the proposed method of Chapter Three and Chapter Four as well as Chapter Five are included for the considered model.Chapter Six studies the signal subspace method for frequency estimation of2-D harmonics in noise and outliers. The proposed method is based on efficient signal subspace method and robust M-estimation techniques. It is observed that the proposed method performs well in terms of accuracy and computational complexity.Chapter Seven gives a brief summary of this dissertation, and also put forward some suggestions for further researches related to the work on this dissertation. |
PDF Full Text Request