While iterative matrix inversion methods excel in computational efficiency, memory optimization, and support for parallel and distributed computing when managing large matrices, their limitations are also evident in multiple-input multiple-output (MIMO) fading channels. These methods encounter challenges related to slow convergence and diminished accuracy, especially in ill-conditioned scenarios, hindering their application in future MIMO networks such as extra-large aperture array (ELAA). To address these challenges, this paper proposes a novel matrix regularization method termed symmetric rank-$1$ regularization (SR-$1$R). The proposed method functions by augmenting the channel matrix with a symmetric rank-$1$ matrix, with the primary goal of minimizing the condition number of the resultant regularized matrix. This significantly improves the matrix condition, enabling fast and accurate iterative inversion of the regularized matrix. Then, the inverse of the original channel matrix is obtained by applying the Sherman-Morrison transform on the outcome of iterative inversions. Our eigenvalue analysis unveils the best channel condition that can be achieved by an optimized SR-$1$R matrix. Moreover, a power iteration-assisted (PIA) approach is proposed to find the optimum SR-$1$R matrix without need of eigenvalue decomposition. The proposed approach exhibits logarithmic algorithm-depth in parallel computing for MIMO precoding. Finally, computer simulations demonstrate that SR-$1$R has the potential to reduce iterative iterations by up to $33\%$, while also significantly improve symbol error probability by approximately an order of magnitude.

翻译：迭代矩阵求逆方法在管理大矩阵时虽具有计算效率高、内存优化好、支持并行与分布式计算等优势，但在多输入多输出（MIMO）衰落信道中其局限性也显而易见。这些方法面临收敛缓慢和精度下降的挑战，尤其在病态场景中更为突出，阻碍了其在未来MIMO网络（如超大规模阵列天线ELAA）中的应用。针对这些问题，本文提出一种新型矩阵正则化方法——对称秩-$1$ 正则化（SR-$1$R）。该方法通过向信道矩阵添加一个对称秩-$1$ 矩阵来实现正则化，主要目标是使正则化后矩阵的条件数最小化。这能显著改善矩阵条件，从而实现正则化矩阵的快速、精确迭代求逆。随后，通过对迭代求逆结果应用Sherman-Morrison变换，即可得到原始信道矩阵的逆矩阵。我们的特征值分析揭示了优化SR-$1$R矩阵所能达到的最佳信道条件。此外，本文提出一种功率迭代辅助（PIA）方法，可在无需特征值分解的情况下求解最优SR-$1$R矩阵。该方法在MIMO预编码的并行计算中具有对数级算法深度。最后，计算机仿真表明，SR-$1$R可将迭代次数最多减少33%，同时将符号误码率显著降低约一个数量级。