Fast matrix inversion methods based on Chebyshev and Newton iterations for zero forcing precoding in massive MIMO systems

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.