This paper focuses on optimal beamforming to maximize the mean signal-to-noise ratio (SNR) for a reconfigurable intelligent surface (RIS)-aided MISO downlink system under correlated Rician fading. The beamforming problem becomes non-convex because of the unit modulus constraint of passive RIS elements. To tackle this, we propose a semidefinite relaxation-based iterative algorithm for obtaining statistically optimal transmit beamforming vector and RIS-phase shift matrix. Further, we analyze the outage probability (OP) and ergodic capacity (EC) to measure the performance of the proposed beamforming scheme. Just like the existing works, the OP and EC evaluations rely on the numerical computation of the iterative algorithm, which does not clearly reveal the functional dependence of system performance on key parameters. Therefore, we derive closed-form expressions for the optimal beamforming vector and phase shift matrix along with their OP performance for special cases of the general setup. Our analysis reveals that the i.i.d. fading is more beneficial than the correlated case in the presence of LoS components. This fact is analytically established for the setting in which the LoS is blocked. Furthermore, we demonstrate that the maximum mean SNR improves linearly/quadratically with the number of RIS elements in the absence/presence of LoS component under i.i.d. fading.

翻译：本文针对可重构智能表面（RIS）辅助的MISO下行链路系统，在相关莱斯衰落条件下，研究以最大化平均信噪比（SNR）为目标的波束赋形优化问题。由于无源RIS元素的单位模约束，该波束赋形问题呈现非凸性。为此，我们提出一种基于半定松弛的迭代算法，以获取统计最优的发射波束赋形向量和RIS相位偏移矩阵。进一步，我们通过分析中断概率（OP）和遍历容量（EC）来评估所提波束赋形方案的性能。与现有研究类似，OP与EC的评估依赖于迭代算法的数值计算，未能清晰揭示系统性能与关键参数间的函数关系。因此，针对通用设置的特例情况，我们推导了最优波束赋形向量、相位偏移矩阵及其OP性能的闭式表达式。分析表明，在存在直视（LoS）分量时，独立同分布衰落相比相关衰落更具优势——这一结论在LoS被阻断的场景中通过解析方式得到验证。此外，我们证明在独立同分布衰落条件下，最大平均信噪比随RIS元素数量呈线性/二次方增长（当LoS分量不存在/存在时）。