We provide accurate approximations of the sum-rate capacity of an opportunistic time-sharing downlink, when a reconfigurable intelligent surface (RIS) assists the transmission from a single-antenna base station (BS) to single-antenna user equipments (UEs). We consider the fading effects of both the direct (i.e., BS-to-UEs) and reflection (i.e, BS-to-RIS-to-UEs) links, by developing two approximations: the former one is based on hardening of the reflection channel for large values of the number of meta-atoms; the latter one relies on the distribution of the sum of Nakagami variates and does not require channel hardening. Our derivations show the dependence of the sum-rate capacity as a function of both the number of users and the number of meta-atoms, as well as to establish a comparison with a downlink without an RIS. Numerical results corroborate the accuracy and validity of the mathematical analysis.

翻译：我们针对可重构智能表面（RIS）辅助单天线基站（BS）向单天线用户设备（UE）传输的场景，提供了机会性时分下行链路总速率容量的精确近似。我们同时考虑了直射链路（即BS-UE）和反射链路（即BS-RIS-UE）的衰落效应，并提出了两种近似方法：第一种基于反射信道在超原子数量较大时的硬化特性；第二种基于Nakagami变量之和的分布，无需依赖信道硬化。推导结果揭示了总速率容量随用户数量和超原子数量的变化关系，并与未部署RIS的下行链路进行了对比。数值仿真验证了数学分析的准确性和有效性。