扩展的“序贯钻孔”联合同余变换及其在高斯独立向量分析中的应用
The Extended "Sequentially Drilled" Joint Congruence Transformation and its Application in Gaussian Independent Vector Analysis
摘要：
独立向量分析（Independent Vector Analysis，IVA）是近年来独立分量分析（Independent Component Analysis，ICA）在多个混合信号集合中的扩展，其中每个集合中的源信号是独立的，但可能依赖于其他集合中的源信号。在半盲IVA（或ICA）框架中，关于源的概率分布的信息可能是可用的，从而产生最大似然（ML）分离。在最近的工作中，我们已经表明，在多变量高斯模型下，具有任意时间协方差矩阵（平稳或非平稳）的源信号，ML分离需要解决一组矩阵的“顺序钻孔”联合同余（SeDJoCo）变换，这让人想起（但不同于）经典的联合对角化。在本文中，我们将我们的结果扩展到IVA问题，展示了高斯模型（具有任意协方差和互协方差矩阵）的ML解决方案如何采用扩展SeDJoCo问题的形式。我们制定的扩展问题，推导出一个解决方案的存在条件，并提出了两个迭代求解算法。此外，我们推导出诱导的Cramér-Rao下界（iCRLB）的干扰源比（ISR）矩阵，并通过模拟演示如何通过解决扩展的SeDJoCo问题获得ML分离确实达到iCRLB（渐近），而不是其他分离方法，不能利用先验知识的源的分布。
英文摘要：
Independent Vector Analysis (IVA) has emerged in recent years as an extension of Independent Component Analysis (ICA) into multiple sets of mixtures, where the source signals in each set are independent, but may depend on source signals in the other sets. In a semi-blind IVA (or ICA) framework, information regarding the probability distributions of the sources may be available, giving rise to Maximum Likelihood (ML) separation. In recent work we have shown that under the multivariate Gaussian model, with arbitrary temporal covariance matrices (stationary or non-stationary) of the source signals, ML separation requires the solution of a "Sequentially Drilled" Joint Congruence (SeDJoCo) transformation of a set of matrices, which is reminiscent of (but different from) classical joint diagonalization. In this paper we extend our results to the IVA problem, showing how the ML solution for the Gaussian model (with arbitrary covariance and cross-covariance matrices) takes the form of an extended SeDJoCo problem. We formulate the extended problem, derive a condition for the existence of a solution, and propose two iterative solution algorithms. In addition, we derive the induced Cramér-Rao Lower Bound (iCRLB) on the resulting Interference-to-Source Ratios (ISR) matrices, and demonstrate by simulation how the ML separation obtained by solving the extended SeDJoCo problem indeed attains the iCRLB (asymptotically), as opposed to other separation approaches, which cannot exploit prior knowledge regarding the sources' distributions.
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