快速子空间估计方法研究及其在阵列信号处理中的应用
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摘要
在最优化、微分方程、通信、信号处理和系统科学等领域中，子空间扮演着非常重要的角色，尤其是在阵列信号处理中，子空间的获取是至关重要的。这是因为用于参数估计的大多数高分辨算法均要求准确地估计到信号子空间或噪声子空间。然而，常规的子空间估计方法由于涉及到阵列协方差矩阵的估计及其特征值分解，从而需要较高的计算复杂度。尤其是在阵元数较多的情况下，基于特征值分解的子空间估计方法的运算量是相当可观的，难以满足工程应用中实时处理的要求。另一方面，为了获得阵列协方差矩阵的有效估计，还要求有足够多的样本数据。在小样本支撑或快时变的信号环境中，基于特征值分解的子空间估计方法几乎是失效的。本文的工作是围绕着不需要特征值分解的快速子空间估计(FSE-E)来展开的。本文内容大体可以划分为两部分：子空间的快速估计及其在阵列信号处理中的具体应用。在快速估计到信号子空间或噪声子空间后，我们采用基于信号子空间或噪声子空间的超分辨算法估计信号的波达方向(DOA)参数，从而不但能够有效地降低算法的计算复杂度而且还可以提高参数估计的精度。主要工作概括如下：
1．系统阐述了降维自适应滤波技术的基本算法。为了方便后续章节的讨论，首先给出阵列信号的基本模型。然后，对三种基本的降维自适应滤波算法：主分量方法(PC)、互谱法(CSM)和多级维纳滤波器(MSWF)技术作了介绍，重点讨论了多级维纳滤波器。
2．研究了多级维纳滤波器的若干性质，为第四章提出的快速信源数检测方法提供了理论依据。证明了多级维纳滤波器经过P级分解后，以后各级分解得到的观测数据均是白色的随机过程，而且具有特殊的表达式。这里尸表示信源数。仿真结果与本文多级维纳滤波器的若干性质的理论分析完全吻合。
3．根据多级维纳滤波器的多级分解思想，提出了一种快速子空间估计方法。由Krylov子空间与MSWF的关系，我们建立了信号子空间快速估计的基础，并证明了由该快速方法得到的信号子空间和基于阵列协方差矩阵的特征值分解得到的信号子空间是完全等价的；由基于相关相减结构的多级维纳滤波器(CSA-MSWF)的归一化匹配滤波器的相互正交特性，我们得到噪声子空间
H摘要
...豆......口.....
的估计方法;经MSWF预滤波后，新的观测数据的协方差矩阵是一个三对角
矩阵。利用三对角矩阵的对角元素，我们提出一种信源数检测的有效方法，
并证明了该方法依概率1收敛于常规的AIC和MDL方法。仿真结果也证明了快
速子空间估计方法和新的信源数检测方法的有效性。
4.提出了基于快速子空间估计(FSE一E)算法的DOA超分辨估计方法。针对相
干信源的情况，我们给出了两种平滑的方案。如果阵列协方差矩阵是已知的，
我们采用空间平滑的Lanczos算法快速估计信号子空间和噪声子空间。如果阵
列协方差矩阵是未知的，我们采用空间平滑的CSA一MSWF快速计算信号子空
间和噪声子空间的基矢量。仿真结果表明，基于由快速子空间估计方法得到
的信号子空间的ESPIUT方法可以给出更好的估计精度和更低的计算复杂度。
同样地，采用由快速子空间估计方法得到的噪声子空间的MUsIC算法的性能
和运算量也明显优于常规的MUSIC算法。
5.提出了基于压缩信号子空间的信号子空间拟合方法(CSS一SSF)。我们推导了
一个新的信号子空间拟合的代价函数，并证明了压缩信号子空间可以由
MSWF的前若干个匹配滤波器有效地张成。理论分析和数值结果均表明，
CSS一ssF方法在压缩信号子空间的维数远远小于信源数的情况下仍然有效，
而且其估计性能接近于常规的加权信号子空间拟合方法(WSF)。
6.提出了一种渐近有效的降维加权信号子空间拟合方法(R-WSF)。推导了降维
的加权子空间拟合的代价函数，分析了R-WSF方法的计算复杂度。理论分析
表明，当加权矩阵取不同的值时，R-WSF方法的代价函数可以分别渐近地等
效于确定性最大似然法(DML)和常规WsF方法的代价函数。仿真结果表明，
对独立信号和相干信号DOA的估计，R-WSF方法的估计性能在信噪比较高时
和常规的WSF方法几乎相同，在信噪比较低的情况下优于常规的WSF方法。
此外，R-WSF方法的计算复杂度要明显低于常规的WSF方法。
关键词:维纳滤波器，降维滤波，主分量，互谱，多级维纳滤波器，Lanczos算法，
阵列信号处理，超分辨，参数估计，波达方向，子空间，空间谱，特征值分
解，AIC，MDL。
份越们全号丸班曰农ltJ民翻脸宜
It is interesting to fast estimate signal subspace and noise subspace in the area of optimization, differential equation, communication, signal processing and system science. Especially in the field of array signal processing, it is crucial to correctly estimate the subspaces since a large number of super-resolution algorithms for parameter estimation are based on the signal subspace or the noise subspace. Normally, the conventional method for subspace estimation resorts to estimating the array covariance matrix and performing the eigenvalue decomposition (EVD) to the estimate of the array covariance matrix. However, the EVD technique tends to be computationally intensive, in particular for the case of large array. On the other hand, presuming the formation of a sample covariance matrix indicates that it is necessary to have sufficient sample support. Nevertheless, in many practical applications such as in low sample support or when the signal statistics are rapidly time-varying, we may not have enough sample to form the sample covariance matrix. Therefore, it is very interesting to obtain the subspaces without the estimate of the sample covariance matrix and its EVD. In this thesis, we develop a fast subspace estimation method without eigendecomposition (FSE-E) to acquire the signal subspace and the noise subspace, and exploit the subspace estimations to resolve the narrow-band signals impinging upon a uniform linear array (ULA).· The fundamental algorithms for reduced-rank adaptive filters are systematically addressed. To facilitate the discussion of the coming chapters, the data model of array signals is firstly given. Then, the conventional reduced-rank adaptive filtering algorithms, i. e., the principal component (PC) method, the cross-spectral metric (CSM) and the multi-stage wiener filter (MSWF) is presented. Emphasis is placed on the review of the MS WF.· Some useful properties of the MSWF are studied to provide the theory to develop a fast detection algorithm for the number of signal sources. It is proved that, after P multi-stage decompostition, the array data matrix becomes a temporal white random process. Furthermore, all the data matrices after the Pth stage are also white random processes, and take the special forms. Numerical results are consistent with the
analysis, thereby indicating the discussion of the properties of the MSWF is valid.· Based on the multi-stage decomposition of the MSWF, a fast algorithm for estimating the signal subspace and the noise subspace is proposed. In view of the relationship between the Krylov subspace and the MSWF, a novel basis is derived for fast subspace estimation. The novel signal subspace attained by the fast algorithm is completely equivalent to that yielded by the classical EVD based method. Furthermore, considering the orthogonality of the matched filters of the MSWF based on the correlation subtractive architecture (CSA) (CSA-MSWF), a new method for fast noise subspace estimation is developed. Noting that after pre-filtered, the sample covariance matrix becomes a tridiagonal matrix, we propose two efficient methods for detennining the number of signal sources. Moreover, it is proved that the proposed criterion functions converge to the classical AIC and MDL criterion functions with probability one when the number of snapshots tends to infinity. The effectiveness of the proposed methods is strengthened by numerical results.· Since the signal subspace and the noise subspace are capable of being readily obtained by the fast subspace estimation method without eigendecomposition (FSE-E), the classical subspace based methods, more specifically the MUSIC and ESPRIT algorithms, can be exploited to estimate the directions of arrival (DOAs) of narrow-band signals impinging upon a uniform linear array (ULA). For complete correlated (coherent) signal sources, we proposed two different smoothing schemes to decorrelate the coherent signals. In the case where the sample covariance matrix is given, we perform the spatially smoothed Lanczos algorithm to fast compute
1．系统阐述了降维自适应滤波技术的基本算法。为了方便后续章节的讨论，首先给出阵列信号的基本模型。然后，对三种基本的降维自适应滤波算法：主分量方法(PC)、互谱法(CSM)和多级维纳滤波器(MSWF)技术作了介绍，重点讨论了多级维纳滤波器。
2．研究了多级维纳滤波器的若干性质，为第四章提出的快速信源数检测方法提供了理论依据。证明了多级维纳滤波器经过P级分解后，以后各级分解得到的观测数据均是白色的随机过程，而且具有特殊的表达式。这里尸表示信源数。仿真结果与本文多级维纳滤波器的若干性质的理论分析完全吻合。
3．根据多级维纳滤波器的多级分解思想，提出了一种快速子空间估计方法。由Krylov子空间与MSWF的关系，我们建立了信号子空间快速估计的基础，并证明了由该快速方法得到的信号子空间和基于阵列协方差矩阵的特征值分解得到的信号子空间是完全等价的；由基于相关相减结构的多级维纳滤波器(CSA-MSWF)的归一化匹配滤波器的相互正交特性，我们得到噪声子空间
H摘要
...豆......口.....
的估计方法;经MSWF预滤波后，新的观测数据的协方差矩阵是一个三对角
矩阵。利用三对角矩阵的对角元素，我们提出一种信源数检测的有效方法，
并证明了该方法依概率1收敛于常规的AIC和MDL方法。仿真结果也证明了快
速子空间估计方法和新的信源数检测方法的有效性。
4.提出了基于快速子空间估计(FSE一E)算法的DOA超分辨估计方法。针对相
干信源的情况，我们给出了两种平滑的方案。如果阵列协方差矩阵是已知的，
我们采用空间平滑的Lanczos算法快速估计信号子空间和噪声子空间。如果阵
列协方差矩阵是未知的，我们采用空间平滑的CSA一MSWF快速计算信号子空
间和噪声子空间的基矢量。仿真结果表明，基于由快速子空间估计方法得到
的信号子空间的ESPIUT方法可以给出更好的估计精度和更低的计算复杂度。
同样地，采用由快速子空间估计方法得到的噪声子空间的MUsIC算法的性能
和运算量也明显优于常规的MUSIC算法。
5.提出了基于压缩信号子空间的信号子空间拟合方法(CSS一SSF)。我们推导了
一个新的信号子空间拟合的代价函数，并证明了压缩信号子空间可以由
MSWF的前若干个匹配滤波器有效地张成。理论分析和数值结果均表明，
CSS一ssF方法在压缩信号子空间的维数远远小于信源数的情况下仍然有效，
而且其估计性能接近于常规的加权信号子空间拟合方法(WSF)。
6.提出了一种渐近有效的降维加权信号子空间拟合方法(R-WSF)。推导了降维
的加权子空间拟合的代价函数，分析了R-WSF方法的计算复杂度。理论分析
表明，当加权矩阵取不同的值时，R-WSF方法的代价函数可以分别渐近地等
效于确定性最大似然法(DML)和常规WsF方法的代价函数。仿真结果表明，
对独立信号和相干信号DOA的估计，R-WSF方法的估计性能在信噪比较高时
和常规的WSF方法几乎相同，在信噪比较低的情况下优于常规的WSF方法。
此外，R-WSF方法的计算复杂度要明显低于常规的WSF方法。
关键词:维纳滤波器，降维滤波，主分量，互谱，多级维纳滤波器，Lanczos算法，
阵列信号处理，超分辨，参数估计，波达方向，子空间，空间谱，特征值分
解，AIC，MDL。
份越们全号丸班曰农ltJ民翻脸宜
It is interesting to fast estimate signal subspace and noise subspace in the area of optimization, differential equation, communication, signal processing and system science. Especially in the field of array signal processing, it is crucial to correctly estimate the subspaces since a large number of super-resolution algorithms for parameter estimation are based on the signal subspace or the noise subspace. Normally, the conventional method for subspace estimation resorts to estimating the array covariance matrix and performing the eigenvalue decomposition (EVD) to the estimate of the array covariance matrix. However, the EVD technique tends to be computationally intensive, in particular for the case of large array. On the other hand, presuming the formation of a sample covariance matrix indicates that it is necessary to have sufficient sample support. Nevertheless, in many practical applications such as in low sample support or when the signal statistics are rapidly time-varying, we may not have enough sample to form the sample covariance matrix. Therefore, it is very interesting to obtain the subspaces without the estimate of the sample covariance matrix and its EVD. In this thesis, we develop a fast subspace estimation method without eigendecomposition (FSE-E) to acquire the signal subspace and the noise subspace, and exploit the subspace estimations to resolve the narrow-band signals impinging upon a uniform linear array (ULA).· The fundamental algorithms for reduced-rank adaptive filters are systematically addressed. To facilitate the discussion of the coming chapters, the data model of array signals is firstly given. Then, the conventional reduced-rank adaptive filtering algorithms, i. e., the principal component (PC) method, the cross-spectral metric (CSM) and the multi-stage wiener filter (MSWF) is presented. Emphasis is placed on the review of the MS WF.· Some useful properties of the MSWF are studied to provide the theory to develop a fast detection algorithm for the number of signal sources. It is proved that, after P multi-stage decompostition, the array data matrix becomes a temporal white random process. Furthermore, all the data matrices after the Pth stage are also white random processes, and take the special forms. Numerical results are consistent with the
analysis, thereby indicating the discussion of the properties of the MSWF is valid.· Based on the multi-stage decomposition of the MSWF, a fast algorithm for estimating the signal subspace and the noise subspace is proposed. In view of the relationship between the Krylov subspace and the MSWF, a novel basis is derived for fast subspace estimation. The novel signal subspace attained by the fast algorithm is completely equivalent to that yielded by the classical EVD based method. Furthermore, considering the orthogonality of the matched filters of the MSWF based on the correlation subtractive architecture (CSA) (CSA-MSWF), a new method for fast noise subspace estimation is developed. Noting that after pre-filtered, the sample covariance matrix becomes a tridiagonal matrix, we propose two efficient methods for detennining the number of signal sources. Moreover, it is proved that the proposed criterion functions converge to the classical AIC and MDL criterion functions with probability one when the number of snapshots tends to infinity. The effectiveness of the proposed methods is strengthened by numerical results.· Since the signal subspace and the noise subspace are capable of being readily obtained by the fast subspace estimation method without eigendecomposition (FSE-E), the classical subspace based methods, more specifically the MUSIC and ESPRIT algorithms, can be exploited to estimate the directions of arrival (DOAs) of narrow-band signals impinging upon a uniform linear array (ULA). For complete correlated (coherent) signal sources, we proposed two different smoothing schemes to decorrelate the coherent signals. In the case where the sample covariance matrix is given, we perform the spatially smoothed Lanczos algorithm to fast compute
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