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 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.

💡 Research Summary

The paper tackles the challenging problem of phase‑selection for reconfigurable intelligent surfaces (RIS) in multi‑user (MU) wireless systems. While optimal closed‑form solutions for MU RIS are considered intractable, existing approaches rely on iterative optimization or AI/ML techniques that, although near‑optimal, impose heavy computational and channel‑estimation burdens—particularly problematic for passive RIS that lack onboard processing.

The authors propose a fundamentally different, low‑complexity strategy based on partitioning the RIS into K equal “subsurfaces.” Each subsurface is dedicated to a single user equipment (UE) that operates on its own orthogonal frequency band (OFDM sub‑carrier set). Consequently, inter‑user interference is eliminated in the frequency domain, and a simple matched‑filter (MF) receiver suffices at the base station (BS). For the k‑th user, only the N/K RIS elements belonging to its subsurface need to be optimized, reducing the required CSI by a factor of K.

The core technical contribution is a closed‑form expression for the mean signal‑to‑noise ratio (SNR) of this Subsurface Design (SD) under correlated Rician fading for all links (UE‑BS direct, UE‑RIS, RIS‑BS). The channel model combines a rank‑1 line‑of‑sight (LoS) component with a Kronecker‑structured correlated Rayleigh component, parameterized by per‑link Ricean K‑factors and spatial correlation matrices. By exploiting the statistical independence of the LoS and scattered parts, the authors derive an exact mean‑SNR formula involving confluent hypergeometric and Laguerre functions. They then present two important simplifications: (i) pure LoS (K→∞) where the expression collapses to a simple product of array gains, and (ii) pure Rayleigh (K→0) yielding a form that depends only on the trace of the correlation matrices. These analytical results provide direct insight into how RIS size, user count, correlation strength, and K‑factor affect performance.

Building on SD, the paper introduces two iterative enhancements. The Iterative Subsurface Design (ISD) sequentially designs subsurfaces: when designing the k‑th subsurface, the phases already set for users 1…k‑1 are treated as fixed and incorporated into the expectation of the total SNR, leading to a refined phase choice for the current subsurface. The Converged ISD (CISD) repeats the ISD cycle until the incremental SNR gain falls below a predefined tolerance, guaranteeing convergence to a locally optimal configuration. Both ISD and CISD retain the low‑complexity spirit: ISD requires O(K·N/K) operations, while CISD adds only a modest multiplicative factor equal to the number of outer iterations. In contrast, the lowest‑complexity full‑MU method known to the authors (reference