线谱估计的快速高分辨去网格化压缩感知方法研究

Line spectral estimation is a classical problem in information and signal processing and has wide applications in communications, radar and so on. On the other hand, the compressed sensing theory was founded by Emmanuel Candes and Terrence Tao among others in 2006. Compressed sensing aims at recovering a high-dimensional signal from its low-dimensional linear samples by exploiting the fact that real-world signals can typically be sparsely represented under an appropriate dictionary. It has been applied in various research areas, including information and signal processing, communications, control, medical imaging, etc. During the past decade, by exploiting sparsity in line spectral estimation, compressed sensing has been successfully applied to and greatly advanced the research of line spectral estimation. Recently, the gridless compressed sensing methods proposed by Emmanuel Candes，Gongguo Tang and the applicant completely resolve the grid mismatch problem of the previous grid-based compressed sensing methods; however, their application is limited by the fact that they suffer from low resolution and high computational complexity. This project aims at solving these limitations by proposing novel gridless compressed sensing methods with high resolution and low computational complexity and applying them in direction of arrival estimation and related problems in radar and communications. The new results are expected to overcome the drawbacks of the existing methods and complement the existing compressed sensing framework.

线谱估计是信息与信号处理的一个基本问题，在通信、雷达等领域有着广泛的应用背景。另一方面，压缩感知技术由Emmanuel Candes和陶哲轩等于2006年建立，它利用了现实世界信号具有稀疏性的特点，成功实现了从小样本恢复大数据，已被广泛应用于信息与信号处理、通信、控制、医学成像等诸多领域。在过去十年中，通过发掘线谱估计中频率的稀疏性，压缩感知技术被成功地引入并极大地促进了线谱估计的研究发展。最近，由Emmanuel Candes，Gongguo Tang及申请人等提出的去网格化压缩感知方法彻底解决了传统压缩感知方法的网格不匹配问题。但是，这些方法在分辨率及计算复杂度上仍存在瓶颈。本项目拟进一步研究去网格化压缩感知方法，开发具有高分辨率和低复杂度的线谱估计算法，并将结果应用解决波达方向估计及相关的雷达等问题。新结果有望补充完善压缩感知的理论体系并为去网格化压缩感知方法在实际中的应用扫清障碍。

线谱估计是信号与信息处理领域的基本问题之一，在雷达、无线通信等工程技术领域应用广泛。近年来，压缩感知技术的提出与应用为线谱估计问题的解决提供了全新的研究思路，推动了这一领域的技术变革，但最新的去网格化（无网）压缩感知技术仍有许多关键问题有待研究与解决。本项目主要研究了三部分内容。首先，面向一般化的多通道线谱估计问题研究了无网压缩感知方法的理论性能，通过创新随机矩阵非渐进性分析、半正定矩阵Hadamard乘积等基本数学理论，刻画了每通道采样量随通道数变化的一般规律。其次，系统刻画了频谱分析应用中可能存在的先验信息，通过发展半正定Toeplitz矩阵分解基本数学理论，在无网压缩感知方法框架下为频率区间已知和概率分布函数已知两种情况下的线谱估计问题提供了快速高分辨凸优化算法。最后，面向线谱估计与阵列信号处理撰写了压缩感知研究综述，揭示了过去二十多年间稀疏优化方法的本质区别与联系，阐明了无网压缩感知方法在算法与理论层面的特有优势。以上内容构成了IEEE Trans. Information Theory、IEEE Trans. Signal Processing等领域顶级期刊上的五篇论文的主要成果、Elsevier信号处理系列丛书中的一部受邀章节，以及2017年第25届欧洲信号处理会议三小时专题课程讲座的核心素材；有力构建了面向线谱估计的无网压缩感知算法和理论框架，为阵列信号处理、雷达、无线通信等应用领域提供了高效的操作算法和强有力的理论指导；后续研究已获得国家自然科学基金委优秀青年基金项目资助。