On the capacity of TDMA downlink with a reconfigurable intelligent surface

We provide accurate approximations of the sum-rate capacity of a time-division multiple access (TDMA) down-link, when a reconfigurable intelligent surface (RIS) assists the transmission from a single-antenna base station (BS) to K 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 Q of meta-atoms at the RIS; 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 K and Q, as well as to establish a comparison with a TDMA downlink without an RIS. Numerical results corroborate the accuracy of the proposed approximations and the validity of the mathematical analysis.