基于实数多项式的根值最小范数算法

Root minimum norm algorithm based on real polynomial

在根值最小范数算法中需对复数多项式求根,计算量较大。针对此问题,提出了一种基于实数多项式的根值最小范数算法,该算法适用于均匀线性阵列。首先通过保角变换将分布在复平面单位圆的变量映射到实数轴[-1,1],其能够将算法中的复数多项式转换为实数多项式;其次对该实数多项式求根,并从中选出[-1,1]的根值;最后将筛选出的根值代入信号频谱函数中,根据频谱函数的值选择出最优的波达方向估值。理论分析说明该算法比根值最小范数算法的时间复杂度低;仿真实验表明,与根值最小范数相比,在信号和噪声不相关时,该算法的均方根误差略小,在信号和噪声相关时,随着信噪比的增加,该算法的均方根误差逐渐变小。

The root minimum norm algorithm must solve roots of the complex polynomial, but it requires a large amount of calculations. Aiming at solving this problem, this paper proposed a root minimum norm algorithm based on real polynomial, which was suitable for uniform linear array. Firstly, it used conformal transformation to transform variables of distri-buted on the complex plane unit circle into the real line range of [ - 1, 1 ]. Thus, the complex polynomial of root minimum norm algo- rithm became real polynomial. Secondly, from the real polynomial, solved the roots of the algorithm, and those within the range of [ - 1, 1 ] were feasible solutions. Finally, the signal spectrum function used the selected roots to get the optimal of direction of arrival estimations according to the spectrum values. Theoretical analysis shows that the proposed algorithm can reduce the time complexity compared with the root minimum norm algorithm. And simulation experiments indicate that the proposed algorithm has lower root mean square error when signals and noise are not relevant, while signals and noise are relevant root mean square error of proposed algorithm becomes lower gradually with growth of the signal-to-noise ratio, compared with root minimum norm algorithm.