  Reconfigurable intelligent surface (RIS) is an emerging technology to enhance
wireless communication in terms of energy cost and system performance by
equipping a considerable quantity of nearly passive reflecting elements. This
study focuses on a downlink RIS-assisted multiple-input multiple-output (MIMO)
wireless communication system that comprises three communication links of
Rician channel, including base station (BS) to RIS, RIS to user, and BS to
user. The objective is to design an optimal transmit covariance matrix at BS
and diagonal phase-shifting matrix at RIS to maximize the achievable ergodic
rate by exploiting the statistical channel state information at BS. Therefore,
a large-system approximation of the achievable ergodic rate is derived using
the replica method in large dimension random matrix theory. This large-system
approximation enables the identification of asymptotic-optimal transmit
covariance and diagonal phase-shifting matrices using an alternating
optimization algorithm. Simulation results show that the large-system results
are consistent with the achievable ergodic rate calculated by Monte Carlo
averaging. The results verify that the proposed algorithm can significantly
enhance the RIS-assisted MIMO system performance.
