稀疏元分析理论及其在欠定盲卷积混叠信号处理中的关键问题研究

High-dimensional data processing arises often in information processing problems, where sparse component analysis (SCA) is one of powerful methods for this problem. We investigate sparse component analysis and its application to underdetermined convolutive blind source separation (CBSS) in this project. The goals of this project are as follows. Theoretically, we focus on three issues: estimation of the number of sparse components, multiple-dominant sparse component analysis and convergence analysis of popular iterative algorithms of sparse representation, which are very hot but challenging research topics both internationally and nationally. By incorporating new mathematical theories and research methods, we would like to find effective ways to solve such theoretical problems, enrich and perfect the theory of SCA. Practically, we investigate the application of SCA to underdetermined-CBSS. At present, underdetermined-CBSS has attached lots of attention around the world but remains unsolved largely. The difficulty of this problem is that the number of unknown channel parameters, which need to be identified, increases sharply, as the number of channel delays increases. In essence, this problem can be cast into a high-dimensional data processing problem and SCA can be applied to it. In this project, we would like to investigate underdetermined-CBSS by SCA and build a solid foundation for this problem

高维信息处理是当今信息处理面临的主流问题，稀疏元分析是高维信息处理的有效方法之一。该项目首先针对稀疏元分析领域中目前国际上关注的三个难点理论问题（即稀疏元数目的估计问题、多活跃稀疏元的分类问题、稀疏表示中两类主流算法的收敛性问题）进行探讨，通过引进新的数学工具和新的研究方法，寻找解决上述难点问题的有效途径，丰富和完善稀疏元分析理论。其次，就稀疏元分析方法在欠定盲卷积混叠信号处理应用中的问题展开探讨。因欠定盲卷积混叠问题是国际上盲信号处理的研究热点和难点，当时延阶数的增加会带来估计参数数量的急速增加，这类问题往往转化为高维数据的处理问题。我们将结合稀疏元分析方法讨论该类问题的盲辨识和盲分离问题，为建立欠定盲卷积混叠信号处理的系统理论奠定基础。

高维信息处理是当今信息处理面临的主流问题，稀疏元分析是高维信息处理的有效方法之一。本项目主要研究内容包括：1）稀疏表示理论及方法；2）基于矩阵或张量分解的盲分离；3）欠定及卷积盲辨识和盲分离算法；4）上述方法在胎儿心电/心音检测、脑电信号特征提取、无线通信、智能电网等方面的应用。通过以上研究，证明了经典稀疏表示算法——“FOCUSS算法”的收敛性，提出了非负张量分解降维两步法通用框架，并分析了用于盲分离的Tucker分解算法的唯一性，设计了基于凸几何分析的多活跃元稀疏盲分离算法，提出了基于充分边界采样的盲可分性条件,还提出了基于密度估计的源数目检测、稀疏子空间表示、基于词典学习的最优稀疏表示等一系列算法。此外，将理论成果应用于复杂心电信号检测，在多导联心电信号检测方面取得了突破，同时研制基于盲分离的北斗自主导航系列芯片，提高了信号处理精度。