  Reconfigurable intelligent surface (RIS) is a newly-emerged technology that
might fundamentally change how wireless networks are operated. Though
extensively studied in recent years, the practical limitations of RIS are often
neglected when assessing the performance of RIS-assisted communication
networks. One of these limitations is that each RIS element is restricted to
incur a controllable phase shift to the reflected signal from a predefined
discrete set. This paper studies an RIS-assisted multi-user multiple-input
multiple-output (MIMO) system, where an RIS with discrete phase shifts assists
in simultaneous uplink data transmission from multiple user equipments (UEs) to
a base station (BS). We aim to maximize the sum rate by optimizing the receive
beamforming vectors and RIS phase shift configuration. To this end, we
transform the original sum-rate maximization problem into a minimum mean square
error (MMSE) minimization problem and employ the block coordinate descent (BCD)
technique for iterative optimization of the variables until convergence. We
formulate the discrete RIS phase shift optimization problem as a mixed-integer
least squares problem and propose a novel method based on sphere decoding (SD)
to solve it. Through numerical evaluation, we show that the proposed discrete
phase shift design outperforms the conventional nearest point mapping method,
which is prevalently used in previous works.
