Multi-antenna relays and intelligent reflecting surfaces (IRSs) have been utilized to construct favorable channels to improve the performance of wireless systems. A common feature between relay systems and IRS-aided systems is the two-hop multiple-input multiple-output (MIMO) channel. As a result, the mutual information (MI) of two-hop MIMO channels has been widely investigated with very engaging results. However, a rigorous investigation on the fundamental limits of two-hop MIMO channels, i.e., the first and second-order analysis, is not yet available in the literature, due to the difficulties caused by the two-hop (product) channel and the noise introduced by the relay (active IRS). In this paper, we employ large-scale random matrix theory (RMT), specifically Gaussian tools, to derive the closed-form deterministic approximation for the mean and variance of the MI. Additionally, we determine the convergence rate for the mean, variance and the characteristic function of the MI, and prove the asymptotic Gaussianity. Furthermore, we also investigate the analytical properties of the fundamental equations that describe the closed-form approximation and prove the existence and uniqueness of the solution. An iterative algorithm is then proposed to obtain the solution for the fundamental equations. Numerical results validate the accuracy of the theoretical analysis.

翻译：多天线中继和智能反射面已被用于构建有利信道以提升无线系统性能。中继系统与智能反射面辅助系统的一个共同特征在于两跳多输入多输出信道。因此，两跳MIMO信道的互信息已得到广泛研究并取得了引人关注的成果。然而，由于两跳（乘积）信道以及中继（有源智能反射面）引入噪声带来的困难，目前文献中尚未出现对两跳MIMO信道基本极限（即一阶和二阶分析）的严格研究。本文利用大规模随机矩阵理论（特别是高斯工具）推导了MI均值和方差的闭式确定性近似。此外，我们确定了MI均值、方差及特征函数的收敛速率，并证明了其渐近高斯性。进一步地，我们研究了描述闭式近似的基本方程的分析性质，证明了解的存在唯一性，并提出了求解该基本方程的迭代算法。数值结果验证了理论分析的准确性。