Lovász条件下LLL算法最简复Givens矩阵形式的研究

Research on Complex Givens Matrix in LLL AlgorithmUnder the Lovász Condition

摘要:

为了解决复数域下基于QR分解的LLL(A. K. Lenstra, H. W. Lenstra and L. Lovász)算法中复Givens旋转矩形式不统一的问题,文章从复数域下原始LLL算法中Gram-Schmidt系数与QR分解的上三角矩阵R中元素之间的关系出发,证明了上三角矩阵R的元素与Gram-Schmidt系数以及Lovász条件之间的等价的关系；从复数的指数形式出发,推导出2种适合LLL算法的复Givens旋转矩阵形式,并证明只有其中一种符合Lovász条件下复Givens旋转矩阵形式。仿真结果表明,采用基于QR分解的复数域LLL算法的MIMO系统相比采用基于Gram-Schmidt正交化LLL算法的MIMO系统具有更好的误比特率性能。

Abstract:

In order to solve inconsistency of the complex Givens rotation matrix in LLL (A. K. Lenstra, H. W. Lenstra, and L. Lovász) algorithm based on the QR decomposition, from the equivalence relations between Gram-Schmidt coefficients of original LLL algorithm and elements in the upper triangular matrix R of QR decomposition, two complex Givens rotation matrices which were suitable for LLL algorithm were deduced and the Givens rotation matrix was proved meeting the Lovász conditions in terms of complex exponential expression. Simulation results showed that MIMO systems with complex field LLL algorithm based on QR decomposition had better BER (bit Error Rate) performance than the MIMO systems employing the existing complex LLL algorithm.