Phase Selection and Analysis for Multi-Frequency Multi-User RIS Systems Employing Subsurfaces in Correlated Ricean and Rayleigh Environments

Phase selection design for reconfigurable intelligent surfaces (RISs) is a significant research challenge, as a closed-form optimal solution for a multi-user (MU) system is believed to be intractable. While existing methods achieve strong near-optimal performance, they typically entail high computational complexity. In this work, we take a different approach and propose a practical method that achieves competitive performance while substantially reducing computational complexity. To do so, we consider a RIS divided into subsurfaces. Each subsurface is designed specifically for one user, who is served on their own frequency band. The other subsurfaces (those not designed for this user) provide additional uncontrolled scattering. We derive the exact closed-form expression for the mean signal-to-noise ratio (SNR) for the proposed subsurface design (SD) when all channels experience correlated Ricean fading. We simplify this to find the mean SNR for line-of-sight (LoS) channels and channels experiencing correlated Rayleigh fading. An iterative SD (ISD) process is proposed, where subsurfaces are designed sequentially, and the phases that are already set are used to enhance the design of the remaining subsurfaces. This is extended to a converged ISD (CISD), where the ISD process is repeated multiple times until the SNR increases by less than a specified tolerance. The ISD and CISD both provide a performance improvement over the SD, which increases as the number of RIS elements increases. The SD is significantly simpler than the lowest complexity MU method we know of, and despite each user having less bandwidth, the SD outperforms the existing method in some key scenarios. The SD is more robust to strongly LoS channels and clustered users, as it does not rely on spatial multiplexing like other MU methods. Combined with the complexity reduction, this makes the SD an attractive phase selection method.