In this paper, a deterministic equivalent of ergodic sum rate and an
algorithm for evaluating the capacity-achieving input covariance matrices for
the uplink large-scale multiple-input multiple-output (MIMO) antenna channels
are proposed. We consider a large-scale MIMO system consisting of multiple
users and one base station with several distributed antenna sets. Each link
between a user and an antenna set forms a two-sided spatially correlated MIMO
channel with line-of-sight (LOS) components. Our derivations are based on novel
techniques from large dimensional random matrix theory (RMT) under the
assumption that the numbers of antennas at the terminals approach to infinity
with a fixed ratio. The deterministic equivalent results (the deterministic
equivalent of ergodic sum rate and the capacity-achieving input covariance
matrices) are easy to compute and shown to be accurate for realistic system
dimensions. In addition, they are shown to be invariant to several types of
fading distribution.

本文提出了大规模 MIMO 天线通道的确定性等效和评估容量实现输入协方差矩阵的算法。研究基于大维随机矩阵理论，在终端天线数量无限接近于无穷大的假设下做了相关推导。结果易于计算，精确性较高，并且不受多种衰落分布影响。

大规模 MIMO 多用户接入信道的容量与具有相关天线的分布式集合

On Capacity of Large-Scale MIMO Multiple Access Channels with  Distributed Sets of Correlated Antennas

Consider an $N\times n$ random matrix $Y_n=(Y^n_{ij})$ where the entries are
given by $Y^n_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^n_{ij}$, the $X^n_{ij}$
being independent and identically distributed, centered with unit variance and
satisfying some mild moment assumption. Consider now a deterministic $N\times
n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean
norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a
deterministic $N\times N$ matrix-valued function T_n(z) analytic in
$\mathbb{C}-\mathbb{R}^+$ such that, almost surely, \[\lim_{n\to+\infty,N/n\to
c}\biggl(\frac{1}{N}\operatorname
{Trace}(\Sigma_n\Sigma_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname
{Trace}T_n(z)\biggr)=0.\] Otherwise stated, there exists a deterministic
equivalent to the empirical Stieltjes transform of the distribution of the
eigenvalues of $\Sigma_n\Sigma_n^T$. For each n, the entries of matrix T_n(z)
are defined as the unique solutions of a certain system of nonlinear functional
equations. It is also proved that $\frac{1}{N}\operatorname {Trace} T_n(z)$ is
the Stieltjes transform of a probability measure $\pi_n(d\lambda)$, and that
for every bounded continuous function f, the following convergence holds almost
surely \[\frac{1}{N}\sum_{k=1}^Nf(\lambda_k)-\int_0^{\infty}f(\lambda)\pi
_n(d\lambda)\mathop {\longrightarrow}_{n\to\infty}0,\] where the
$(\lambda_k)_{1\le k\le N}$ are the eigenvalues of $\Sigma_n\Sigma_n^T$. This
work is motivated by the context of performance evaluation of multiple
inputs/multiple output (MIMO) wireless digital communication channels. As an
application, we derive a deterministic equivalent to the mutual information:
\[C_n(\sigma^2)=\frac{1}{N}\mathbb{E}\log
\det\biggl(I_N+\frac{\Sigma_n\Sigma_n^T}{\sigma^2}\biggr),\] where $\sigma^2$
is a known parameter.

本研究介绍了一种用于评估多输入 / 多输出无线数字通信渠道性能的理论方法，通过引入随机矩阵和确定性等效概念以及研究其特征值的 Stieltjes 变换来实现。