Abstract: In many methods based on second order statistics for blind source separation, the mixing matrix is transformed into an unknown unitary matrix after whitening procedure. A novel symmetrical least square cost function with respect to a column of the unknown unitary matrix is proposed based on the orthogonality between each two different columns of a unitary matrix. A new Triply Iterative Algorithm（TIA） following the gradient descent idea is developed to seek the minimum point of the tri-quadratic cost function by alternately estimating one of the three independent variables parameter subsets. After the convergence of the cost function, the column of the unitary matrix corresponding to the source signal with the highest power can be obtained. With each column being got by utilizing the systemic Multi-Stage Algorithm（MSA）, the unitary matrix can be estimated and then the source signals can be retrieved. Simulation results illustrate that, compared with the classic SOBI method which solves the unitary matrix using successive Givens rotations, MSA has better performance, lower computational complexity, and can accurately retrieve the source signals.

Key words: Blind Source Separation（BSS）, Multi-Stage Algorithm（MSA）, symmetric cost function, least-squares approach, Triply Iterative Algorithm（TIA）

摘要： 在利用二阶统计量实现盲源分离问题中，混迭矩阵经过白化以后转变成了酉矩阵。针对酉矩阵各列之间相互正交的特性，提出一种关于酉矩阵某一列的最小二乘对称代价函数。通过基于梯度下降法的三迭代算法，交替估计三二次代价函数中的各组待定参数，搜索代价函数最小点，从而得到对应能量最大信号源的酉矩阵的一列。利用系统化的多步分解算法（MSA），依次估计酉矩阵的一列，最终得到整个酉矩阵的估计。仿真结果表明，与经典的通过连续Givens旋转求酉矩阵的SOBI算法相比，该算法全局拒噪水平至少改善了9 dB，而所需计算时间仅为SOBI的二分之一，更有效地解决了盲源分离问题。

关键词: 盲源分离, 多步分解算法, 对称代价函数, 最小二乘方法, 三迭代算法