压缩感知技术在阵列测向中的应用
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摘要
近几年来，压缩感知理论在信息处理领域得到了广泛的关注。其利用信号的稀疏特性，从少量的信号线性变换的测量值中来恢复具有稀疏特性的目标矢量，大大减少了信号在存储和传输过程中的数据量，具有较大的工程意义。
传统的压缩感知理论只利用了信号的稀疏特性，并没有对信号做进一步的约束。为了扩展压缩感知理论的应用范围，获得更精确的恢复结果，本文考虑了稀疏信号的非零元素有着更多先验信息情况下的恢复算法及压缩采样矩阵存在误差下的鲁棒恢复算法。之后将其应用到阵列测向中的具体领域，获得了良好的效果。本论文所研究的主要内容及其创新点如下：
总结了当前压缩感知的基本理论和重要结论，尤其是稀疏向量中的非零元素具有分块稀疏性质时的模型、算法及恢复条件。重点介绍了分块稀疏信号具有总体稀疏局部密集特性时的恢复算法，并提出了一种基于压缩采样匹配追踪算法的改进分块稀疏信号匹配追踪算法。在最后对支撑集进行剔除，对于非零元素没有充满整个子块时表现出良好的鲁棒性。
考虑到压缩采样矩阵存在工程误差时的影响，提出了抗压缩采样矩阵误差的恢复算法。当压缩采样矩阵受到扰动时，在已知扰动矩阵方差的情况下，推导了其对稀疏恢复中最小二乘逼近的影响。提出了一种自适应的鲁棒算法和一种迭代消减误差的恢复算法，仿真证明这两种算法相对传统算法对压缩采样矩阵受到扰动情况下有着更强的鲁棒性。该研究对压缩感知理论的实用化有着较大的推动作用。
总结了当前主要的用于阵列测向的压缩感知理论算法，包括其发展历程、适用范围、优缺点及恢复条件。重点介绍了基于混合范数约束的压缩感知理论测向算法，详细介绍了其流程和对比了几种算法的恢复性能，在工程中应根据需要适当选取合适的恢复算法。
针对分布源在空间上角度扩展表现出来的空间连续分布特性，本论文将分块稀疏恢复算法应用到分布源测向模型中，建立了分布源在空间过完备基下的信号接收模型，提出了基于分块稀疏模型的压缩感知理论测向算法。之后将整体稀疏局部密集的模型扩展到矩阵形式，给出了全新的稀疏模型概念和稀疏度的定义。基于矩阵的分块稀疏模型，使用凸优化算法和贪婪算法对其进行恢复。前者计算量大但恢复性能更好，后者恢复性能在低信噪比下退化严重但计算量较小。两者相对传统算法性能都有所提升，有着各自的优缺点及适用范围。
压缩采样理论的核心思想之一是压缩采样，其表现为使用信号采样的线性变换值代替原来的采样值。这一思想在波束空间算法中有所体现——即对空间信号进行预波束形成达到减少通道数的目的。本论文详细对比了两种算法的异同，指出了两者各自的优缺点。最后将压缩采样矩阵扰动下的恢复算法进行向量到矩阵的扩展，提出了一种对波束矩阵误差具有鲁棒性的波束域测向算法，使波束域算法更具实用性。
In recent years, compressive sensing theory has received the worldwide attention inthe field of signal processing. Based on this theory, the sparse target vector can berecovered by the few linear transform results of the signals through the application ofthe sparsity of signals. Thus, the data base will be greatly decreased during the storageand transmission of the signals, which makes huge contribution to the electronicengineering domain.
The traditional compressive sensing theory only focused on the sparsity of signalswithout making further research on the constraint of signals. For the widespreadapplication of this theory and more accurate recovery results, this dissertation proposeda new recovery method with the distribution of the non-zero elements. Another newrecovery method also is proposed to recover the sparse signal under the compressivesampling matrix uncertainty. These methods are applied to the array signal processing toobtain the satisfactory results. The main contents and innovative points of thisdissertation are as follows:
The dissertation reviewed the most basic theories and the significant academicfruits in the field of the compressive sensing in recent years. Particularly, the model, thealgorithm and the recovery conditions of the sparse vector with the block sparsenon-zero elements are fully studied in this article. Considering the overall sparse localintensive characteristics of the situation of sparse signal, this dissertation proposed therecovery algorithm which is improved by the block matching pursuit algorithm. Theproposed method has good robustness on the condition that non-zero elements fail tofulfill the entire sub-block, since it removed some of the support set at the end of thealgorithm.
Considering the negative effect caused by the engineering errors in thecompressive sampling matrix, the dissertation proposed a new recovery method tocorrect these engineering errors. When the compressive sampling matrix is disturbed, byknowing the variance of the disturbed matrix, the article made a conclusion on theinfluence of the least square approximation in the sparse recovery. An adaptive robust algorithm and an iterative recovery algorithm will be proposed. The simulation provesthat the two methods, compared with the traditional algorithm, are robust to thecompressive sampling under the disturbance. This research has greatly promoted theapplication of the compressive sampling theory in practice.
The dissertation summarizes the main development for direction of arrivalestimation through the compressive sensing algorithm, including its developing process,the scope of application, the advantages and disadvantages and recovery conditions. Thecompression perception theory based on mixed norm constraint-finding algorithm willbe highlighted in the dissertation. In addition, the detailed processes and the comparisonof the recovery performance by the several algorithms will be thoroughly analyzed inthis article. In engineering practice, it is essential to select the appropriate recoveryalgorithm based on the real conditions and needs.
Considering the space continuous distribution of the angular spread out by thedistributed source, the block sparse recovery algorithm is applied to estimate thedirection of the distributed source. It proposed the direction finding algorithm of thecompressive sensing theory based on the block sparse model. The overall sparse localdense model is extended to a matrix form. A brand new concept of sparse model and thedefinition of sparsity will be introduced. Based on the block sparse matrix model, theconvex optimization and greedy algorithm will be applied to estimate the direction ofarrival. The former method has large computation with better recovery performance,while the latter has the serious degradation performance in low SNR but lesscomputation. The traditional algorithm performance has been improved both by thesetwo methods, which has its own advantages and disadvantages and scope of application.
One of the core concepts of the compressive sampling theory is compressivesampling. The linear transformation value of the signal sampling will replace theoriginal sample values. This concept is reflected in the beam-space algorithm, it meansthat the pre-beam-forming of space signal will achieve the purpose of reducing thenumber of channels. In this dissertation, a detailed comparison of the similarities anddifferences of the two algorithms will be mentioned; both of their advantages anddisadvantages also will be pointed out. Finally, the recovery algorithm under theperturbation of the compressive sampling will be expanded from the vector to thematrix. A beam-space direction finding algorithm with the robustness to the error of beam matrix will be proposed, which makes the beam-space algorithm more practical.
传统的压缩感知理论只利用了信号的稀疏特性，并没有对信号做进一步的约束。为了扩展压缩感知理论的应用范围，获得更精确的恢复结果，本文考虑了稀疏信号的非零元素有着更多先验信息情况下的恢复算法及压缩采样矩阵存在误差下的鲁棒恢复算法。之后将其应用到阵列测向中的具体领域，获得了良好的效果。本论文所研究的主要内容及其创新点如下：
总结了当前压缩感知的基本理论和重要结论，尤其是稀疏向量中的非零元素具有分块稀疏性质时的模型、算法及恢复条件。重点介绍了分块稀疏信号具有总体稀疏局部密集特性时的恢复算法，并提出了一种基于压缩采样匹配追踪算法的改进分块稀疏信号匹配追踪算法。在最后对支撑集进行剔除，对于非零元素没有充满整个子块时表现出良好的鲁棒性。
考虑到压缩采样矩阵存在工程误差时的影响，提出了抗压缩采样矩阵误差的恢复算法。当压缩采样矩阵受到扰动时，在已知扰动矩阵方差的情况下，推导了其对稀疏恢复中最小二乘逼近的影响。提出了一种自适应的鲁棒算法和一种迭代消减误差的恢复算法，仿真证明这两种算法相对传统算法对压缩采样矩阵受到扰动情况下有着更强的鲁棒性。该研究对压缩感知理论的实用化有着较大的推动作用。
总结了当前主要的用于阵列测向的压缩感知理论算法，包括其发展历程、适用范围、优缺点及恢复条件。重点介绍了基于混合范数约束的压缩感知理论测向算法，详细介绍了其流程和对比了几种算法的恢复性能，在工程中应根据需要适当选取合适的恢复算法。
针对分布源在空间上角度扩展表现出来的空间连续分布特性，本论文将分块稀疏恢复算法应用到分布源测向模型中，建立了分布源在空间过完备基下的信号接收模型，提出了基于分块稀疏模型的压缩感知理论测向算法。之后将整体稀疏局部密集的模型扩展到矩阵形式，给出了全新的稀疏模型概念和稀疏度的定义。基于矩阵的分块稀疏模型，使用凸优化算法和贪婪算法对其进行恢复。前者计算量大但恢复性能更好，后者恢复性能在低信噪比下退化严重但计算量较小。两者相对传统算法性能都有所提升，有着各自的优缺点及适用范围。
压缩采样理论的核心思想之一是压缩采样，其表现为使用信号采样的线性变换值代替原来的采样值。这一思想在波束空间算法中有所体现——即对空间信号进行预波束形成达到减少通道数的目的。本论文详细对比了两种算法的异同，指出了两者各自的优缺点。最后将压缩采样矩阵扰动下的恢复算法进行向量到矩阵的扩展，提出了一种对波束矩阵误差具有鲁棒性的波束域测向算法，使波束域算法更具实用性。
In recent years, compressive sensing theory has received the worldwide attention inthe field of signal processing. Based on this theory, the sparse target vector can berecovered by the few linear transform results of the signals through the application ofthe sparsity of signals. Thus, the data base will be greatly decreased during the storageand transmission of the signals, which makes huge contribution to the electronicengineering domain.
The traditional compressive sensing theory only focused on the sparsity of signalswithout making further research on the constraint of signals. For the widespreadapplication of this theory and more accurate recovery results, this dissertation proposeda new recovery method with the distribution of the non-zero elements. Another newrecovery method also is proposed to recover the sparse signal under the compressivesampling matrix uncertainty. These methods are applied to the array signal processing toobtain the satisfactory results. The main contents and innovative points of thisdissertation are as follows:
The dissertation reviewed the most basic theories and the significant academicfruits in the field of the compressive sensing in recent years. Particularly, the model, thealgorithm and the recovery conditions of the sparse vector with the block sparsenon-zero elements are fully studied in this article. Considering the overall sparse localintensive characteristics of the situation of sparse signal, this dissertation proposed therecovery algorithm which is improved by the block matching pursuit algorithm. Theproposed method has good robustness on the condition that non-zero elements fail tofulfill the entire sub-block, since it removed some of the support set at the end of thealgorithm.
Considering the negative effect caused by the engineering errors in thecompressive sampling matrix, the dissertation proposed a new recovery method tocorrect these engineering errors. When the compressive sampling matrix is disturbed, byknowing the variance of the disturbed matrix, the article made a conclusion on theinfluence of the least square approximation in the sparse recovery. An adaptive robust algorithm and an iterative recovery algorithm will be proposed. The simulation provesthat the two methods, compared with the traditional algorithm, are robust to thecompressive sampling under the disturbance. This research has greatly promoted theapplication of the compressive sampling theory in practice.
The dissertation summarizes the main development for direction of arrivalestimation through the compressive sensing algorithm, including its developing process,the scope of application, the advantages and disadvantages and recovery conditions. Thecompression perception theory based on mixed norm constraint-finding algorithm willbe highlighted in the dissertation. In addition, the detailed processes and the comparisonof the recovery performance by the several algorithms will be thoroughly analyzed inthis article. In engineering practice, it is essential to select the appropriate recoveryalgorithm based on the real conditions and needs.
Considering the space continuous distribution of the angular spread out by thedistributed source, the block sparse recovery algorithm is applied to estimate thedirection of the distributed source. It proposed the direction finding algorithm of thecompressive sensing theory based on the block sparse model. The overall sparse localdense model is extended to a matrix form. A brand new concept of sparse model and thedefinition of sparsity will be introduced. Based on the block sparse matrix model, theconvex optimization and greedy algorithm will be applied to estimate the direction ofarrival. The former method has large computation with better recovery performance,while the latter has the serious degradation performance in low SNR but lesscomputation. The traditional algorithm performance has been improved both by thesetwo methods, which has its own advantages and disadvantages and scope of application.
One of the core concepts of the compressive sampling theory is compressivesampling. The linear transformation value of the signal sampling will replace theoriginal sample values. This concept is reflected in the beam-space algorithm, it meansthat the pre-beam-forming of space signal will achieve the purpose of reducing thenumber of channels. In this dissertation, a detailed comparison of the similarities anddifferences of the two algorithms will be mentioned; both of their advantages anddisadvantages also will be pointed out. Finally, the recovery algorithm under theperturbation of the compressive sampling will be expanded from the vector to thematrix. A beam-space direction finding algorithm with the robustness to the error of beam matrix will be proposed, which makes the beam-space algorithm more practical.
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