稳健的超分辨算法研究——基于信号分离理论的算法研究
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摘要
阵列超分辨方向估计方法以其优异的分辨和估计性能使其在众多领域内展
现了广阔的应用前景。然而目前的超分辨处理方法是基于对信号模型的准确已知
的假设下而建立的，当信噪比不高、阵列输出数据长度较短，以及存在系统误差
时，这些已知条件就发生一定的偏离，超分辨方法的性能随之大大下降，这严重
限制了超分辨技术在实际中的应用。因此，稳健的超分辨处理方法是目前受到广
泛关注的研究课题。本文重点的是在提出一类基于信号分离理论的超分辨算法的
基础上，展开对一种可以对环境加性噪声及系统误差的假设可松弛的超分辨算法
的较为深入的讨论。全文的主要工作概括如下：
对所谓的松弛（RELAX）算法从数据域推广到相关域，将原算法的迭代
减法等价为矩阵算子的递归迭代；将原来算法对于信号波形参数与方位参数的递
归迭代拟合变为一个简单的信号波形参数的解析式与方位参数最小二乘迭代拟
合。通过在相关域的讨论，提出了以信号分离理论为基础实现信号分辨的概念。
对于利用递归迭代的矩阵算子投影实现信号分离的有效性作了证明。
在矩阵算子实现信号分离的前提下，建立了关于信号方位参数的多维
优化问题—即所谓的多维（MD-）松弛（RELAX）算法。通过建立这个多维优化问
题的交替分离（AS）解法，归结出了与松弛（RELAX）算法在原相关域内完全一致的
最小二乘拟合问题，使原来的矩阵算子的递归迭代关系变成为相应矩阵算子的线
性方程组。通过与最大似然（ML）算法的交替投影（AP）解法的比较，提出了恰当
（Exact-fitting）拟合与非恰当（Nonexact-fitting）拟合的概念。基于这些讨论，提出了
改进的（Im-）松弛（RELAX）算法，这个算法实质就是松弛（RELAX）算法的交替分离
（AS）解法。
根据松弛（RELAX）算法实现信号分辨其对应矩阵算子应该满足信号分
离的前提，导出了松弛（RELAX）算法实现信号分辨的一些特性函数，讨论了在理
想情况下以及有系统误差时，松弛（RELAX）算法实现信号分辨以及未实现信号分
辨时这些特征函数的性态。
对松弛（RELAX）算法与最大似然（ML）算法，以及松弛（RELAX）算法的
交替分离（AS）解法与最大似然（ML）算法的交替投影（AP）解法进行了多个方面的
比较，发现了松弛（RELAX）算法的交替分离（AS）解法在实现当前信号方位参数的
新的估计时对目前估计的最大依赖性，以及最大似然（ML）算法的交替投影（AP）
解法在实现当前信号方位参数的新的估计时对目前估计的非直接依赖性。这种最
大依赖性使得松弛（RELAX）算法的交替分离（AS）解法收敛速度非常慢-这是松
弛（RELAX）算法的交替分离（AS）解法的严重缺陷，这种最大依赖性使得松弛
（RELAX）算法的交替分离（AS）解法在有系统误差时能够有效地抑制局部极值，这
是它的一个重要性能。我们提出了避免其缺陷而有效地利用其特征的AS-AP联
合处理方法。
对松弛aa人刀算法关于环境加性噪声以及系统误差的假设可松弛的原理
进行了分析讨论．
．＞ 对最大似然算法的相关算法进行了讨论，指出IMP算法、AN’PA算
．法，M算法挪是最大似然算法的一维解法，而IMP算法、ANPAX法本质上
仍是A豆算法．
＋讨讼了时空二羹刚m囚C 得出了在倍号发生时域、特
姚空域荣并时时空二耀亚羹方廷两面挝，以孤臼了时闺羹戳蚣曰的雕
OOM．
冬 讨用了 羹阵的系统俱盏狙舌间回，在惺簿＠小二柔板仓准回的菌
臼下，采用蹦洞元址臼，合问回的馒戳与阵 致．x利用对阵元个
戳的瞩回已问，樱吕对系羹课蠢的羹计槽征田不同回设，羹出了相度的约柬参
件，幢系羹误望的训合闷回虎为召钓原的优化口阻由于阵无个戳是已知的，所
以这个优亿口回肋复来计翼可摄＠克民因而系挽误垄的拟合问回实质成为一些
而单的判 稷．
With excellent resolution and estimation performance, The super-resolution array processing methods have shown their prospect of application in many fields. Nevertheless , because the excellence performances of the methods are based on the accurate knowledge of the data model , the performance of super-resolution methods degraded extremely when the signal-to-noise-ratio is not very high , and the snapshots is not very more or the system errors are existing in the array. This prevent greatly their application in practical system. So , the study of the robust processing methods are main subject in the current .This dissertation is focused on proposing a category of the signals separation, and lay the foundation on the category and drive on the study about the algorithm that can relaxation the h~othcsis of the additive noise and the system errors . The content can be outlined as follows:
?A genemlintion of thc(DD-)RELAX from the data domain to the correlation domain , i.e the CD-RELAX, was provided . The equivalence of the iterative subtraction and the recursion iterative of the definite matrix operators was proved , the tranuibrmation of the recursion iterative fitting between the azimuth and the waveform to the analytical form of the waveform and the iterative fitting of the azimuth was made . With the analysis of the RELAX in the correlation domain, the theory of the signals resolution that based on the signals separation was established, and the effect of the signal separation with the projection of the iterative matrix operators was also proved.
?Under the hypothesis of the given matrix operators satisfying the signals separation, the multi-dimensional optimum about the signal azimuth, and the multidimensional(MD-)RELAX , was established. To establish the Alternating Separation (AS)algorithm of the (MD-)RELAX , that same one dimensional optinluni as the CDRELAX was derived and the recurrent iterative operators relation was replaced by the operators liner equation . By comparing the CD-RELAX with the Alternating Projection of the Maximum Likelihood(ML-AP) , the opinion of the exact fitting and the non-exact fitting was proposed . Based on the discussion , the improved RELAX was proposed , that it is the RELAX-AS.
~. On the characteristic of the RELAX , some functions was defined, the regularity and irregularity , as in the ideal case and the non ideal case, are discussed, respectively.
C?The comparison is made about the RELAX and ML , the RELAX-AS and the ML-AP , The weightiest characteristic of the current estimation of the signal parameter in getting the new estimation in the RELAX-AS was found . The character is relative to the convergent slowness and the restraint of the local extreme ; The
慍
indirect relation of the current estimation of the signal parameter in getting the new estimation in the ML-AP was also point out . The method about rid ofthe shortcoming and usc the characters of the ML-AP and RELAX-AS , the RELAX-AS and ~AP approach, w also ?
?The discussion of the relative algorithm of the Maximum Likelihood ii made , the conchnion abont that IMP. ANPA and Al?are all one dimensional algorithm of the Maximum Likelihood, the intrinsic of the IMP and ANPA are same as that of the AP was derived.
?The asymptotic of the two dimensional MUSIC of the spatial and temporal processing is established, The effect of the two dimensional processing under the spatial or the temporal wmcx is proved, and the optimum length of the temporal filtering is provided.
晘 The statistical fitting of the gain and phase errors is discussed , on guaranteeing the aitariom of the least squares , using the separation and elimination approach , making the dimension of the errors fitting equal to the number of the sensors .Under the different hypothesis of the statistical characteristic of the errors, the different constrains are proposed, which make the errors fitting as a constrains optimum Since the complicate algorithm can be done for definite array before the app cati
现了广阔的应用前景。然而目前的超分辨处理方法是基于对信号模型的准确已知
的假设下而建立的，当信噪比不高、阵列输出数据长度较短，以及存在系统误差
时，这些已知条件就发生一定的偏离，超分辨方法的性能随之大大下降，这严重
限制了超分辨技术在实际中的应用。因此，稳健的超分辨处理方法是目前受到广
泛关注的研究课题。本文重点的是在提出一类基于信号分离理论的超分辨算法的
基础上，展开对一种可以对环境加性噪声及系统误差的假设可松弛的超分辨算法
的较为深入的讨论。全文的主要工作概括如下：
对所谓的松弛（RELAX）算法从数据域推广到相关域，将原算法的迭代
减法等价为矩阵算子的递归迭代；将原来算法对于信号波形参数与方位参数的递
归迭代拟合变为一个简单的信号波形参数的解析式与方位参数最小二乘迭代拟
合。通过在相关域的讨论，提出了以信号分离理论为基础实现信号分辨的概念。
对于利用递归迭代的矩阵算子投影实现信号分离的有效性作了证明。
在矩阵算子实现信号分离的前提下，建立了关于信号方位参数的多维
优化问题—即所谓的多维（MD-）松弛（RELAX）算法。通过建立这个多维优化问
题的交替分离（AS）解法，归结出了与松弛（RELAX）算法在原相关域内完全一致的
最小二乘拟合问题，使原来的矩阵算子的递归迭代关系变成为相应矩阵算子的线
性方程组。通过与最大似然（ML）算法的交替投影（AP）解法的比较，提出了恰当
（Exact-fitting）拟合与非恰当（Nonexact-fitting）拟合的概念。基于这些讨论，提出了
改进的（Im-）松弛（RELAX）算法，这个算法实质就是松弛（RELAX）算法的交替分离
（AS）解法。
根据松弛（RELAX）算法实现信号分辨其对应矩阵算子应该满足信号分
离的前提，导出了松弛（RELAX）算法实现信号分辨的一些特性函数，讨论了在理
想情况下以及有系统误差时，松弛（RELAX）算法实现信号分辨以及未实现信号分
辨时这些特征函数的性态。
对松弛（RELAX）算法与最大似然（ML）算法，以及松弛（RELAX）算法的
交替分离（AS）解法与最大似然（ML）算法的交替投影（AP）解法进行了多个方面的
比较，发现了松弛（RELAX）算法的交替分离（AS）解法在实现当前信号方位参数的
新的估计时对目前估计的最大依赖性，以及最大似然（ML）算法的交替投影（AP）
解法在实现当前信号方位参数的新的估计时对目前估计的非直接依赖性。这种最
大依赖性使得松弛（RELAX）算法的交替分离（AS）解法收敛速度非常慢-这是松
弛（RELAX）算法的交替分离（AS）解法的严重缺陷，这种最大依赖性使得松弛
（RELAX）算法的交替分离（AS）解法在有系统误差时能够有效地抑制局部极值，这
是它的一个重要性能。我们提出了避免其缺陷而有效地利用其特征的AS-AP联
合处理方法。
对松弛aa人刀算法关于环境加性噪声以及系统误差的假设可松弛的原理
进行了分析讨论．
．＞ 对最大似然算法的相关算法进行了讨论，指出IMP算法、AN’PA算
．法，M算法挪是最大似然算法的一维解法，而IMP算法、ANPAX法本质上
仍是A豆算法．
＋讨讼了时空二羹刚m囚C 得出了在倍号发生时域、特
姚空域荣并时时空二耀亚羹方廷两面挝，以孤臼了时闺羹戳蚣曰的雕
OOM．
冬 讨用了 羹阵的系统俱盏狙舌间回，在惺簿＠小二柔板仓准回的菌
臼下，采用蹦洞元址臼，合问回的馒戳与阵 致．x利用对阵元个
戳的瞩回已问，樱吕对系羹课蠢的羹计槽征田不同回设，羹出了相度的约柬参
件，幢系羹误望的训合闷回虎为召钓原的优化口阻由于阵无个戳是已知的，所
以这个优亿口回肋复来计翼可摄＠克民因而系挽误垄的拟合问回实质成为一些
而单的判 稷．
With excellent resolution and estimation performance, The super-resolution array processing methods have shown their prospect of application in many fields. Nevertheless , because the excellence performances of the methods are based on the accurate knowledge of the data model , the performance of super-resolution methods degraded extremely when the signal-to-noise-ratio is not very high , and the snapshots is not very more or the system errors are existing in the array. This prevent greatly their application in practical system. So , the study of the robust processing methods are main subject in the current .This dissertation is focused on proposing a category of the signals separation, and lay the foundation on the category and drive on the study about the algorithm that can relaxation the h~othcsis of the additive noise and the system errors . The content can be outlined as follows:
?A genemlintion of thc(DD-)RELAX from the data domain to the correlation domain , i.e the CD-RELAX, was provided . The equivalence of the iterative subtraction and the recursion iterative of the definite matrix operators was proved , the tranuibrmation of the recursion iterative fitting between the azimuth and the waveform to the analytical form of the waveform and the iterative fitting of the azimuth was made . With the analysis of the RELAX in the correlation domain, the theory of the signals resolution that based on the signals separation was established, and the effect of the signal separation with the projection of the iterative matrix operators was also proved.
?Under the hypothesis of the given matrix operators satisfying the signals separation, the multi-dimensional optimum about the signal azimuth, and the multidimensional(MD-)RELAX , was established. To establish the Alternating Separation (AS)algorithm of the (MD-)RELAX , that same one dimensional optinluni as the CDRELAX was derived and the recurrent iterative operators relation was replaced by the operators liner equation . By comparing the CD-RELAX with the Alternating Projection of the Maximum Likelihood(ML-AP) , the opinion of the exact fitting and the non-exact fitting was proposed . Based on the discussion , the improved RELAX was proposed , that it is the RELAX-AS.
~. On the characteristic of the RELAX , some functions was defined, the regularity and irregularity , as in the ideal case and the non ideal case, are discussed, respectively.
C?The comparison is made about the RELAX and ML , the RELAX-AS and the ML-AP , The weightiest characteristic of the current estimation of the signal parameter in getting the new estimation in the RELAX-AS was found . The character is relative to the convergent slowness and the restraint of the local extreme ; The
慍
indirect relation of the current estimation of the signal parameter in getting the new estimation in the ML-AP was also point out . The method about rid ofthe shortcoming and usc the characters of the ML-AP and RELAX-AS , the RELAX-AS and ~AP approach, w also ?
?The discussion of the relative algorithm of the Maximum Likelihood ii made , the conchnion abont that IMP. ANPA and Al?are all one dimensional algorithm of the Maximum Likelihood, the intrinsic of the IMP and ANPA are same as that of the AP was derived.
?The asymptotic of the two dimensional MUSIC of the spatial and temporal processing is established, The effect of the two dimensional processing under the spatial or the temporal wmcx is proved, and the optimum length of the temporal filtering is provided.
晘 The statistical fitting of the gain and phase errors is discussed , on guaranteeing the aitariom of the least squares , using the separation and elimination approach , making the dimension of the errors fitting equal to the number of the sensors .Under the different hypothesis of the statistical characteristic of the errors, the different constrains are proposed, which make the errors fitting as a constrains optimum Since the complicate algorithm can be done for definite array before the app cati
引文
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