Optimal Time and Power Allocation for Phase-Shift Configuration and Downlink Channel Estimation in RIS-Aided Systems

A fundamental trade-off exists between the time and energy allocated to pilots and data, accuracy of channel estimates, and data rate in a reconfigurable intelligent surface (RIS)-aided system. We optimize the above trade-off for a two-phase training scheme. In the first phase, the base station (BS) estimates the channel from the uplink pilots and configures the RIS. In the second phase, the user equipment estimates the channel from the downlink pilots and coherently demodulates the data. We derive an expression for the achievable rate that accounts for the impact of the channel estimation errors on the RIS phase-shift configuration and data demodulation. Our analysis uses a novel tractable approximation for the effective downlink channel gain and a novel proof that it is asymptotically Gaussian even in the presence of spatial correlation. Our analysis applies to the scenario where enough pilots are sent to estimate the cascaded channels and the cascaded channel grouping scenario that uses fewer pilots. We also study two channel models that depend on the location of the RIS relative to the BS. We derive insightful, closed-form expressions for the optimal powers and training durations. The optimal solution highlights the importance of boosting the pilot powers.