The High Accuracy Estimator For Estimating The Frequencies Of Harmonics In Complex Noise Basing On Iterative Algorithm

Harmonic retrieval (HR) problem in complex noise is one of the most encountered problems in the area of signal processing, and constitutes a significant part of statistical signal processing research. The theory of HR can be applied to many fields, such as sonar, radar imaging, geophysics, radio communication, radio astronomy, nuclear magnetic resonance, acoustic and so on. The recently research shows that it is also the basis of the design of robots and the control of structure of flexible space. The problem of HR can be classified into two classes according to the complexity degrees of the noise:one is HR in additive noise (constant amplitude condition) and another is HR in multiplicative and additive noise (random amplitude condition), and HR in additive noise can be subdivided into HR in whit noise and HR in color noise according to the power spectra density(PSD). It can also be subdivided into one-dimensional (1-D) HR and two-dimensional (2-D) HR. Most of the works so far focus on HR in additive noise, but one may encounters the harmonics signals in complex noise. For example, in underwater acoustic applications, the multiplicative noise can describe the effects on acoustic waves due to fluctuations caused by the media, change orientation and interference from scatters of the targets. So it will make better sense to model this kind of data as a signal corrupted by multiplicative and additive noise in practice and it will be better to extract the useful information sufficiently in the process of analyzing and solving according to this assume.One of the trends of modern digital signal processing is estimating the parameters of signals fast and accurately and providing the online algorithm. But as an online algorithm, it should be computationally efficient, robust and highly accurate, at the same time the estimators should have high convergence rate about the sample and strong adaptability to the noise. The insufficiencies of the existing parametric (non-iterative) methods and non-parametric(iterative) methods for online implementation lie as follows:the existing non-parametric method such as Gauss-Newton method has the drawbacks of instability of convergence and severely dependence on starting value. The parametric methods own the merit of stability, but they also have the drawbacks in the aspects of complexity of computation, slow convergence rate about the sample, weak adaptability to different noises. The Least Squares Estimators (LSE) and Maximum Likelihood Estimators (MLE) can both give very accurate estimators for the frequencies and have the best convergence rate, the convergence rate for the LSE of the 1-D frequencies{ωj} is Op (M-3/2) (here M is the sample size) and Op(M-3/2N-1/2), Op(M-1/2N-3/2) for the 2-D frequencies pair {(ω1k,ω2k)} (here M and N are the 2-D sample size, Op(N-δ) (δ>0) means that Op(N-δ)Nδis bounded in probability). The two methods both need multidimensional search among the parametric space, so it is exhausted and hard for implementation in practice. It will make good sense to find an equivalent algorithm for the LSE which at the same time has the merits of high accuracy and best convergence rate, very low computation load and easy for computing, strong adaptability to complex noise. But it is difficult to find a single algorithm owing so many merits. Recently, a high accuracy iterative algorithm (HAI) is proposed by S. Nandi and D. Kundu, the scholars of India to estimate the frequencies of 1-D harmonics in additive noise. The estimation is based on the statistics of the observed signals and the procedure is divided into two stages. The first stage is to obtain the initial estimators of the frequencies which are refined by a three step iterative process at the second stage. It can be proved that the HAI estimator attains the convergence rate of the LSE and it needs little time to work since only three iterative steps are needed. Moreover, we can obtain the asymptotic distribution of the HAI estimators for frequencies and the computation of the HAI algorithm will not increase much with the increase of the dimension.It can be seen above that a two-stage joint algorithm can possibly own both the merits of parametric and non-parametric methods which are the high precision, best convergence rate about the sample size, strong adaptability to complex noise, little computation and easy for implementation, so it is suitable to serve as online algorithm for the frequencies of 1-D and 2-D harmonics. Stimulated by [2], we generalize the statistical excellent 1-D HAI algorithm from the additive noise to non-zero mean multiplicative noise and zero mean multiplicative noise, then further generalize it to 2-D harmonics model including:additive noise condition, non-zero mean multiplicative noise and zero mean multiplicative noise condition. The contrast between white noise and stationary color noise is also researched, as well as for the real and complex additive noise condition. On the basis of the 2-D HAI algorithm of non-zero mean multiplicative and zero mean multiplicative noise, the estimation for the frequencies of first component and second component of texture is studied. Finally, the sample based extractive HAI algorithm is put forward for the condition when the difference between the two dimension samples is relatively too large. The construction of the statistics is the difficulty of this thesis. It should be constructed according to the characters of the parameter to be estimated. Especially for the zero-mean multiplicative noise condition, the frequencies can't be estimated through the HAI algorithm in the non-zero mean multiplicative noise condition, the square of the observed signal is needed so that the variance of the multiplicative can be used to draw and estimate the frequencies. The significance of the HAI algorithm lies as follows:the merits of two algorithms can be integrated into one combined algorithm." The iterative item is constructed basing on the statistics at the second stage of the HAI algorithm so that the precision is improved step by step. It is proved that the HAI algorithm converges in three steps so that the algorithm is robust and computationally efficient. The statistical character is utilized three times for the observed signals in the three iterative process, so the accuracy is very high for the estimation of the frequencies when the sample size is very small. It is proved that the HAI estimators have the same convergence rate and precision with the LSE after three iterative steps. Moreover, it is observed through the theory and simulation experiments that:(1) The estimators for all the noise conditions considered are unbiased and consistent and the asymptotic distribution for the HAI estimators considered are all normal.(2) The HAI algorithms for all the conditions considered have strong adaptability for white and stationary color noise, while the performance for the white noise is better than the stationary color noise. The performance of the real additive noise is similar to the complex additive noise for the non-zero mean multiplicative noise condition, while the false frequency at the zero point will appear when the additive noise is real for the zero mean multiplicative noise condition.The innovative pursuits in the dissertation can be summarized as the following four aspects:(1) The HAI algorithms for estimating the frequencies of 1-D harmonics and 2-D harmonics are put forward in multiple complex noise condition. The HAI algorithm has both the merits of the parametric method and non-parametric method which are computationally efficient, robust, fast convergence rate and strong adaptability to noise. So it is suitable to be served as online algorithm.(2) The unbiasedness and consistency are analyzed and proved systematically for the estimation of frequencies, the asymptotic distributions are also obtained.(3) The 2-D HAI algorithm for non-zero mean multiplicative and zero mean multiplicative noise are applied to estimate the frequencies of first and second component of texture respectively and further restore the texture.(4) The sample based extractive HAI algorithm is put forward so as to raise the discrimination for the 2-D frequencies estimation and decrease the variance for the estimators.