信号检测是指利用观测数据判别所感兴趣的信号是否存在，信号恢复是指从观测数据中提取具体信号或估计信号的关键参数。稀疏是自然界常见信号的天然特征。联合稀疏是指多个稀疏信号享有相同的非零元素位置。在信号处理中开发联合稀疏特性，有助于提升信号检测、恢复性能。本文围绕联合稀疏信号检测和恢复这一课题，主要研究联合稀疏信号检测方法及其检测性能界限、联合稀疏信号恢复方法及其在雷达成像问题中的应用。论文的主要工作和创新点如下：1） 针对联合稀疏信号检测问题，提出了基于局部最大势检验的信号检测方法。推导得出了该方法在模拟数据、低精度量化数据、高斯噪声和非高斯噪声情形下的理论检测性能。在基于量化数据的检测框架下，推导了最优的量化器形式，分析了量化带来的检测性能损失，并给出了补偿该性能损失的策略。相对于已有的检测方法，所提出的检测方法在未显著损失检测性能的前提下，降低了检测过程的计算负担和数据传输负担。2） 针对联合稀疏信号恢复问题，提出了基于前瞻基信号选择的信号恢复方法。基信号对应于稀疏信号中非零元素的位置。该方法评价了当前迭代基信号的选择对未来迭代信号重建误差的影响。理论证明，该方法提升了基信号选择的稳定性。以多通道雷达成像为例，基于实测雷达数据的实验结果表明，相比于已有方法，该方法有效提升了联合稀疏信号的恢复精度，减少了雷达图像中杂点的数量。3） 针对联合稀疏信号恢复问题，提出了基于双块稀疏性的信号恢复方法。该方法在信号联合稀疏性的基础上进一步开发了信号内部的成簇特性。基于实测雷达数据的实验结果表明，相比于已有方法，该方法提升了雷达图像中非零像素点在目标区域内的聚集程度，抑制了目标区域外的能量泄露，从而提升了雷达成像质量。进一步，为降低雷达成像系统的硬件成本，把该方法拓展到单比特量化数据采集情形。基于实测雷达数据的实验表明，该方法显著提升了基于单比特数据的雷达成像质量。
The task of signal detection is deciding whether signals of interest exist by using their observed data. Furthermore, signals are reconstructed or their key parameters are estimated from the observations in the task of signal recovery. Sparsity is a natural characteristic of most of signals in practice. The fact that multiple sparse signals share the common locations of dominant coefficients is called by joint sparsity. In the context of signal processing, joint sparsity model results in higher performance of signal detection and recovery. This thesis focuses on the task of detecting and reconstructing signals with joint sparsity. The main contents include key methods for detection of joint sparse signals and their corresponding theoretical performance analysis, and methods for joint sparse signal recovery and their application in the context of radar imaging. The main contribution of this thesis is as follows:1) For the problem of detection of joint sparse signals, a method is proposed based on the strategy of locally most powerful test. The theoretical detection performance of this method is provided in the cases of analog observations, coarsely quantized observations, Gaussian noise and non-Gaussian noise, respectively. For the problem of signal detection with quantized observations, the thresholds of optimal quantizer are solved, and the detection performance loss caused by quantization is quantitatively evaluated with the optimal quantizer. The strategy of compensating for the detection performance loss caused by quantization is also provided. Compared with existing detection methods, the proposed method significantly reduces the computational and communication burden without noticeable detection performance loss. 2) For the problem of recovery of joint sparse signals, a method is proposed based on the selection of basis-signals with the look-ahead strategy. Basis-signals correspond to the locations of non-zero values in sparse signals. This method evaluates the effect of the selection of basis-signals on future recovery error in the iterative process. Theoretical analysis indicates that, this method improves the stability in selection of basis-signals. The application of this method in the field of multiple-channel radar imaging is considered. Experiments based on real radar data demonstrate that, this method improves the accuracy of signal recovery with joint sparsity and reduces the number of artifacts in radar images. 3) For the problem of recovery of joint sparse signals, a method is proposed based on the two-level block sparsity, which combines not only the joint sparsity of multiple signals but also the clustering structure in each sparse signal. Experimental results based on real radar data show that, compared with existing methods, the dominant pixels in radar images generated by the proposed method are more concentrated in the target area, and there is less energy leak in the non-target area, i.e., better imaging quality is provided by the proposed method. Furthermore, this method is extended to the 1-bit quantization scenario to reduce the hardware consumption of system for radar imaging. Experiments based on real radar data demonstrate that, the proposed method based on the two-level block sparsity significantly improves the quality of 1-bit radar imaging.
