非正交联合对角化理论及其在盲源分离中的应用研究

Non-orthogonal joint diagonalization is widely used in the fields of direction of arrival estimation, sound source location, blind equalization and so on. It is also a unique solution for blind source separation. However, the result of non-orthogonal joint diagonalization is usually not ideal, and the optimization algorithm is also very easy to obtain ill-conditioned solution, which greatly restricts its application. Therefore, it is very challenging to build a non-orthogonal joint diagonalization performance measurement model which is consistent with the objective evaluation and acquire well-conditioned solutions. The project will intend to explore coupling manners between matrix’s diagonal elements and off-diagonal elements and nesting patterns between the direct form and indirect form of joint diagonalization model, reveal the impact mechanism of these coupling and nesting processing on the performance of non-orthogonal joint diagonalization, establish effective performance measuring model which is easy to optimize; and through elaborately designing the penalty terms or constraint conditions of the optimization problem, intend to explore the way to avoid the ill conditioned solution. Moreover, as far as blind source separation is concerned, we will fully excavate source signals’ characteristics beneath the data model, and reveal the potential relationship between the condition under which the target matrix set can be jointly diagonalized and source signals’ characteristics, explore construction methods of the target matrix set and successfully accomplish the separation of source signals. The implementation of the project will not only enrich and enhance the theoretical results of the research on non-orthogonal joint diagonalization, but also promote and facilitate its wider and more effective applications in many areas such as blind source separation.

非正交联合对角化在波达方向估计、声源定位、盲均衡等领域中应用广泛，也是一种独特的盲源分离解决方法。然而，非正交联合对角化的结果通常不理想，优化算法也极易得到病态解，极大地制约了其应用。因此，构建与客观评价一致的非正交联合对角化性能度量模型并获得良态解富有挑战性，亟待突破。项目拟探索矩阵对角元素与非对角元素之间的耦合方式、联合对角化直接形式与间接形式模型之间的嵌套模式，揭示这些耦合、嵌套处理对非正交联合对角化性能的影响机理，构建易于优化的、有效的性能度量模型；精心设计最优化问题的惩罚项或约束条件，探索避免病态解的方法。针对盲源分离，拟充分挖掘数据模型中蕴含的源信号特性，揭示目标矩阵集可被联合对角化的条件与源信号特性之间的潜在关系，探索目标矩阵集的构造方法，实现源信号的成功分离。项目的实施不仅会丰富和提升非正交联合对角化的理论成果，而且会推动和促进其在盲源分离等诸多领域更加广泛而有效的应用。

联合对角化是一种非常重要的多维信号处理方法，在波达方向估计、雷达信号处理、声源定位、多模态流形学习、图像目标识别等领域得到了广泛应用，也为盲源分离提供了一个独特而新颖的解决方案。对与观测信号有关的一组目标矩阵进行联合对角化处理，优化处理得到的对角化器也就是所需的分离矩阵。利用源信号的各种不同特性，由传感器观测信号可以构造出形形色色的具有上述特点的目标矩阵集，进而通过对的联合对角化得到对角化器（分离矩阵），最终能实现源信号的成功分离。首先，为了客观而有效度量对角化结果优劣的模型，达到对角元素尽可能大而非对角元素足够小的理想化联合对角化目标，我们构造了同时考虑对角元素与非对角元素、融合联合对角化直接形式模型和间接形式模型的度量函数。其次，得到了满足非正交联合对角化对退化解和奇异解免疫要求的有效方法，实现了对角化器各列之间完全解耦，提出了迭代初值的简单有效设置方法。同时我们注意到条件数越小意味着矩阵更良态，将其作为正则项加入最小化问题中，确保了矩阵避免奇异解。充分利用源信号的非平稳性、非白性或非高斯性，提出一个统一的代价函数。通常来说，独立同分布的统计特性是非常苛刻的要求，如何充分利用信号的时间结构将非常有用。自然信号如语音信号、神经生理信号具有丰富的动态时间结构，对其加以利用是有益的。构造了一个相关矩阵，并采用自回归模型对其脉冲响应系数进行预测：利用时频分析特性，得到一个非常简单有效的选取单个源信号时-频点算法。提出了基于非线性频偏的频率分集阵多目标跟踪与定位方法。提出两种微多普勒特征提取方法。最后，在目标矩阵集的构造过程中充分挖掘和有效利用了源信号的各种特性，确保了各个矩阵本身可被近似联合对角化，实现了对感兴趣目标的识别及其参数的有效估计。项目取得的成果丰富了非正交联合对角化的理论成果，推动了盲源分离在目标识别、参数估计、频率分集阵、图像识别、调制识别等领域中的广泛应用。