A Joint Optimization of Pilot and Phase Shifts in Uplink Channel Estimation for Hybrid RIS-Aided Multi-User Communication Systems

In this paper, we consider the uplink channel estimation problem for multi-user communication systems that utilize a hybrid reconfigurable intelligent surface (HRIS). In particular, we are interested in the HRIS that has the capability of reflecting and sensing the impinging signals. Our objective is to jointly design the pilot sequences transmitted by user equipments (UEs) and the amount of phase shifts applied by the HRIS for accurate channel estimation. We formulate a multi-objective optimization problem in terms of the mean squared errors of UEs-to-RIS and RIS-to-base station (BS) channels, and approximate it by assuming that the BS has a perfect estimate of the UEs-to-RIS channel. As a result, a sufficient condition is derived for a feasible solution to be Pareto-optimal to the approximated problem, which simultaneously achieves the Cramér-Rao bounds. We propose two joint designs that satisfy the sufficient condition when the pilot length is at least as long as the product of the numbers of HRIS elements and UEs. The proposed design solutions are easy to implement and lead to computationally efficient channel estimations. The average performances of the proposed designs are analyzed under Rayleigh fading. Numerical results demonstrate that the proposed joint designs outperform non-joint designs that use randomized phase shifts and conventional channel estimation methods for passive RISs.