【作者】 边家文；

【导师】 李宏伟；

【作者基本信息】 中国地质大学 ， 地球探测与信息技术， 2010， 博士

【摘要】 谐波恢复问题是信号处理领域的一个典型问题,同时也是统计信号处理研究的一个重要内容。它在声纳、雷达、地球物理、无线通信、射电天文学、核磁共振、声学等众多领域有广泛的应用,而且最近的研究结果表明,它也是机器人设计以及柔性空间结构控制的基础。这个问题依据背景噪声的复杂程度可以分成两大类,一类是加性噪声中的谐波恢复(又称为常数振幅谐波参数估计),一类是乘性和加性噪声中的谐波恢复(又称为随机振幅谐波参数估计),而加性噪声中根据功率谱的不同又可分为：白噪声中的谐波恢复和有色噪声中的谐波恢复。依据信号的维数又可分为一维谐波恢复和二维谐波恢复。尽管迄今为止的大部分研究工作集中在加性噪声背景下的谐波恢复问题的讨论方面,但是在实际应用中,经常出现复杂噪声背景中的谐波谐复问题。例如：在水声信号处理中,乘性噪声可以描述由媒质、流向变化和目标散射引起的随机波动对声波的影响。因此将这一类型的观测数据建模为乘性和加性噪声中的谐波信号具有更实际的意义,由此进行的信号模型的分析与求解能更充分地提取数据所含的信息。现代数字信号处理的趋势之一为在复杂噪声背景下快速准确的估计信号参数,提供能用于在线实施的在线算法,然而作为在线算法要求算法不仅计算量小,稳定,估计精度高,还需要估计量有很高的关于样本的收敛速度,对噪声有很强的适应能力。现有的一维以及二维谐波恢复参数化方法(非迭代方法)和非参数化方法(迭代方法)在在线估计方面主要有以下不足：已有的非参数化方法如Gauss-Newton方法[1]往往存在计算复杂,收敛速度比较慢,收敛不够稳定,估计严重依赖于初值的问题。参数化方法算法虽然估计性能较为稳定,但是计算比较复杂,关于样本的收敛速度不够快,对噪声的适应能力不够强。最小二乘估计(LSE)[2]和最大似然估计(MLE)[[3]虽然具有很高的估计精度以及最快的关于样本的收敛速度,其中LSE对于一维频率{ωj}的估计的收敛速度达到Op(M-3/2)(其中M为样本容量),对于二维频率对{(ω1k，ω2k)}的关于样本的收敛速度分别达到Op(M-3/2N-1/2),Op(M-1/2N-3/2)(这里M和N为二维样本容量,Op(N-δ)(δ>0)表示Op(N-δ)Nδ以概率有界),然而这些算法往往要基于目标函数在二维频率空间进行多维搜索,计算量比较大而难以实施。若能找到LSE的一种等效算法同时使得该算法计算量很小,计算简便,对噪声适应能力很强,将可以用于在线估计而非常具有实际意义,然而单一的一种算法却很难以具备以上所有的优点。最近印度学者S. Nandi和D. Kundu提出一了一种加性噪声情形下一维谐波频率估计的高精度迭代(HA1)算法。该方法分为两阶段来估计频率,第一阶段得到频率的一个初估计,第二阶段通过三步迭代来逐步提高估计的精度和估计的收敛速度,可以证明经过三步迭代之后的估计量可以达到LSE的估计精度及收敛速度,由于只用了三步迭代,所以计算量很小。除此之外,HAI算法能得到估计量的渐近分布,其计算量不会随着维数增大而变大,可推广到更高维。由以上可以看到,通过联合两种算法分阶段来估计一维以及二维谐波频率,有可能使得算法能同时兼有参数化方法以及非参数化方法的优点即高的估计精度,很快的关于样本的收敛速度,对噪声适应能力强,计算量小以及计算简便易于实施的优点,故可作为一维以及二维谐波频率参数估计的在线估计算法。受[2]的启发,本文将此种具有优良统计性能加性噪声一维情形下的HAI算法拓广到用于非零均值以及零均值乘性噪声情形,然后进一步将其拓广用于二维加性噪声以及非零均值乘性噪声,零均值乘性噪声情形。其中对于加性噪声以及乘性噪声分别为白噪声以及平稳有色噪声两种情形作了对比研究,对于加性噪声分别为实噪声和复噪声作了对比研究,对于二维情形下非零均值乘性噪声以及零均值乘性噪声情形下的HAI算法分别应用到估计纹理图案的第一以及第二成份频率参数。最后提出了二维样本差异较大情形下的基于抽取技巧的HAI算法。HAI算法的拓广过程中第二阶段统计量的构造是本文的难点,要根据待估计参数的特征来构造合适的统计量,特别是对于零均值乘性噪声情形下的估计,若采用非零均值乘性噪声情形下的HAI算法估计会失败,需要通过将信号平方后利用乘性噪声方差来提取并估计频率参数。HAI算法的特色在于：把估计分为两个阶段,可以将两种算法的优点融合在一起,第二阶段通过构造统计量建立迭代式逐步提高估计的精度,可以证明只需要三步即可收敛,故使得算法稳定,计算量很小。三步迭代过程相当于将观测信号统计特征连续利用了三次,所以在样本较小时就有相当高的估计精度,而且可以证明三步迭代之后的估计量达到和LSE相同的收敛速度和估计精度。另外通过理论证明以及仿真实验还可以看到：(1)本文所考虑的多种复杂噪声背景下的一维以及二维谐波频率参数的HAI估计均为无偏估计以及一致估计,估计的渐近分布均为正态分布；(2)本文所考虑的各种复杂噪声背景下的HAI算法对白噪声以及平稳有色噪声均有很强的适应能力；白噪声情形下的估计性能要好于平稳有色噪声情形下的估计性能；对于非零均值乘性噪声情形,复加性噪声与实加性噪声情形下的估计性能相当；对于零均值乘性噪声情形,实噪声情形下的估计会在原点产生伪频率。本文的创新点主要体现在以下几个方面：(1)提出了一维以及二维各种复杂噪声背景下的谐波频率参数的高精度迭代算法,该算法兼有参数化方法以及非参数化算法的优点,计算量小,稳定,估计的收敛速度快,对噪声的适应能力强,适合作为一维以及二维谐波频率参数估计的在线算法；(2)系统分析并证明了HAI估计量为频率参数的无偏估计以及一致估计,并得到了估计量的渐近分布；(3)运用所提出的二维非零均值以及零均值情形下的HAI算法估计纹理图案的第一以及第二成份频率参数进而恢复纹理图案；(4)提出了基于样本抽取技巧的HAI算法,改善了二维情形下二维样本差异较大时的估计的分辨率不足以及估计方差严重偏大的问题。更多还原

【Abstract】 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.更多还原