Abstract

In massive MIMO (mMIMO) systems, large matrix inversion is a challenging problem due to the huge volume of users and antennas. Neumann series (NS) and successive over relaxation (SOR) are two typical methods that solve such a problem in linear precoding. NS expands the inverse of a matrix into a series of matrix vector multiplications, while SOR deals with the same problem as a system of linear equations and iteratively solves it. However, the required complexities for both methods are still high. In this paper, four new joint methods are presented to achieve faster convergence and lower complexity in matrix inversion to determine linear precoding weights for mMIMO systems, where both Chebyshev iteration (ChebI) and Newton iteration (NI) are investigated separately to speed up the convergence of NS and SOR. Firstly, joint Chebyshev and NS method (ChebI-NS) is proposed not only to accelerate the convergence in NS but also to achieve more accurate inversion. Secondly, new SOR-based approximate matrix inversion (SOR-AMI) is proposed to achieve a direct simplified matrix inversion with similar convergence characteristics to the conventional SOR. Finally, two improved SOR-AMI methods, NI-SOR-AMI and ChebI-SOR-AMI, are investigated for further convergence acceleration, where NI and ChebI approaches are combined with the SOR-AMI, respectively. These four proposed inversion methods provide near optimal bit error rate (BER) performance of zero forcing (ZF) case under uncorrelated and correlated mMIMO channel conditions. Simulation results verify that the proposed ChebI-NS has the highest convergence rate compared to the conventional NS with similar complexity. Similarly, ChebI-SOR-AMI and NI-SOR-AMI achieve faster convergence than the conventional SOR method. The order of the proposed methods according to the convergence speed are ChebI-SOR-AMI, NI-SOR-AMI, SOR-AMI, then ChebI-NS, respectively. ChebI-NS has a low convergence because NS has lower convergence than SOR. Although ChebI-SOR-AMI has the fastest convergence rate, NI-SOR-AMI is preferable than ChebI-SOR-AMI due to its lower complexity and close inversion result.

中文翻译：

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基于Chebyshev和Newton迭代的快速矩阵求逆方法，用于大规模MIMO系统中的零迫迫预编码

摘要

在大规模MIMO（mMIMO）系统中，由于用户和天线的数量巨大，大型矩阵求逆是一个具有挑战性的问题。Neumann级数（NS）和连续过度松弛（SOR）是解决线性预编码中此类问题的两种典型方法。NS将矩阵的逆扩展为一系列矩阵向量乘法，而SOR处理与线性方程组相同的问题并迭代求解。但是，两种方法所需的复杂度仍然很高。本文提出了四种新的联合方法以实现更快的收敛速度和更低的矩阵求逆复杂度，从而确定mMIMO系统的线性预编码权重，其中分别研究了切比雪夫迭代（ChebI）和牛顿迭代（NI）以加快收敛速度NS和SOR。首先，提出了切比雪夫和NS联合方法（ChebI-NS），不仅可以加速NS的收敛，而且可以实现更准确的反演。其次，提出了一种新的基于SOR的近似矩阵求逆（SOR-AMI），以实现具有与传统SOR相似的收敛特性的直接简化的矩阵求逆。最后，研究了两种改进的SOR-AMI方法，即NI-SOR-AMI和ChebI-SOR-AMI，以进一步提高收敛速度，其中将NI和ChebI方法分别与SOR-AMI相结合。在不相关和相关的mMIMO信道条件下，这四种拟议的反演方法提供了接近最佳的零强迫（ZF）误码率（BER）性能。仿真结果验证了所提出的ChebI-NS与具有相似复杂度的传统NS相比具有最高的收敛速度。同样，与传统的SOR方法相比，ChebI-SOR-AMI和NI-SOR-AMI的收敛速度更快。所提出的方法根据收敛速度的顺序分别是ChebI-SOR-AMI，NI-SOR-AMI，SOR-AMI，然后是ChebI-NS。ChebI-NS具有较低的收敛性，因为NS的收敛性低于SOR。尽管ChebI-SOR-AMI具有最快的收敛速度，但NI-SOR-AMI比ChebI-SOR-AMI更可取，因为它具有较低的复杂度和近似的反转结果。