We consider a fading channel in which a multi-antenna transmitter communicates with a multi-antenna receiver through a reconfigurable intelligent surface (RIS) that is made of ${N}$ reconfigurable passive scatterers impaired by phase noise. The beamforming vector at the transmitter, the combining vector at the receiver, and the phase shifts of the ${N}$ scatterers are optimized in order to maximize the signal-to-noise-ratio (SNR) at the receiver. By assuming Rayleigh fading (or line-of-sight propagation) on the transmitter-RIS link and Rayleigh fading on the RIS-receiver link, we prove that the SNR is a random variable that is equivalent in distribution to the product of three (or two) independent random variables whose distributions are approximated by two (or one) gamma random variables and the sum of two scaled non-central chi-square random variables. The proposed analytical framework allows us to quantify the robustness of RIS-aided transmission to fading channels. For example, we prove that the amount of fading experienced on the transmitter-RIS-receiver channel linearly decreases with ${N}~\gg ~1.$ This proves that RISs of large size can be effectively employed to make fading less severe and wireless channels more reliable.

中文翻译：

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在单用户 MIMO 系统中通过可重构智能表面进行波束成形：存在信道衰落和相位噪声时的 SNR 分布和缩放定律

我们考虑一个衰落信道，其中多天线发射机通过可重构智能表面 (RIS) 与多天线接收机通信，RIS 由 ${N}$ 受相位噪声影响的可重构无源散射体。发射端的波束形成向量，接收端的组合向量，以及相移 ${N}$ 对散射体进行优化以最大化接收器的信噪比 (SNR)。通过假设发射机-RIS 链路上的瑞利衰落（或视距传播）和 RIS-接收机链路上的瑞利衰落，我们证明 SNR 是一个随机变量，它在分布上等价于三个（或二）独立随机变量，其分布由两个（或一个）伽马随机变量和两个缩放的非中心卡方随机变量之和近似。建议的分析框架使我们能够量化 RIS 辅助传输到衰落信道的稳健性。例如，我们证明在发射机-RIS-接收机信道上经历的衰落量随着 ${N}~\gg ~1.$ 这证明大尺寸的RIS可以有效地使衰落不那么严重并使无线信道更可靠。