矩阵特征分解二阶修正算法

Second-Order Correction for Eigendecomposition of Data Covariance Matrices

本文研究了时变协方差矩阵特征分解的自适应更新问题,提出了矩阵特征分解二阶修正算法。首先将矩阵秩-l更新与矩阵一阶扰动问题联系起来;然后利用矩阵扰动理论将当前时刻的协方差矩阵特征值和特征向量展开为无穷级数形式,当扰动项趋于零时它们分别趋于前一时刻对应特征值和特征向量;将扰动级数二阶以后的所有项省略得到二阶修正算法;特别地研究了最小特征值重合时信号子空间的更新问题;在系统平稳和非平稳两种条件下分别进行仿真验证算法的性能,并且和一阶修正算法比较,仿真结果表明本文提出的方法具有更高的估计精度。

In this paper, new algorithm for adaptive eigendecomposition of time-varying data covariance is presented. We rewrite the rank-1 updating as a first-order perturbation problem. According to the first-order perturbation analysis, the eigenvalues and normalized eigenvectors of k-th time can be expended in power series, which converge to the respective eigenvalues and normalized eigenvectors of (k-1)-th time when the modification trend to zero. By neglecting all terms after the second-order term, we obtain the tractable second-order correction algorithm. Especially, when the minimal eigenvalues are equal, the signal subspace is updated adaptively, and the noise subspace is let alone. Under the stationary and nonstationary scenarios, the simulations are carried out. The results show that the proposed second-order correction algorithm possesses the higher accuracy than the first-order correction, as expected.