Optimal RIS Placement in Multi-User MISO Systems with User Randomness

It is well established that the performance of reconfigurable intelligent surface (RIS)-assisted systems critically depends on the optimal placement of the RIS. Previous works consider either simple coverage maximization or simultaneous optimization of the placement of the RIS along with the beamforming and reflection coefficients, most of which assume that the location of the RIS, base station (BS), and users are known. However, in practice, only the spatial variation of user density and obstacle configuration are likely to be known prior to deployment of the system. Thus, we formulate a non-convex problem that optimizes the position of the RIS over the expected minimum signal-to-interference-plus-noise ratio (SINR) of the system with user randomness, assuming that the system employs joint beamforming after deployment. To solve this problem, we propose a recursive coarse-to-fine methodology that constructs a set of candidate locations for RIS placement based on the obstacle configuration and evaluates them over multiple instantiations from the user distribution. The search is recursively refined within the optimal region identified in each stage to determine the final optimal region for RIS deployment. Detailed numerical results are presented to corroborate our findings.

💡 Research Summary

This paper addresses the practical problem of determining the optimal location for a reconfigurable intelligent surface (RIS) in a multi‑user multiple‑input single‑output (MU‑MISO) cellular network when the exact positions of the users are unknown at deployment time. The only a‑priori information assumed to be available is the geometric configuration of obstacles (e.g., walls and pillars) and the statistical spatial distribution of users, modeled as a homogeneous Poisson point process (PPP). The authors formulate a hierarchical, non‑convex optimization problem: the outer layer maximizes the expected minimum signal‑to‑interference‑plus‑noise ratio (SINR) across all users (a max‑min fairness objective), while the inner layer jointly optimizes the base‑station (BS) beamforming vectors and the RIS phase‑shifts for a given RIS position.

Because the inner problem depends on instantaneous channel realizations, it cannot be solved analytically for every possible RIS location. The authors therefore adopt a two‑step approach. First, they solve the inner joint beamforming problem using a series of transformations: Lagrangian duality converts the weighted‑sum‑rate objective into a sum‑of‑ratios form, auxiliary variables α and β are introduced via quadratic transform, and an alternating optimization (AO) loop updates α, β, the beamformers (via a weighted‑minimum‑mean‑square‑error (WMMSE) solution), and the RIS phases (via gradient projection). This inner loop converges to a locally optimal set of beamformers and phases for any fixed RIS coordinate.

Second, to tackle the outer placement problem, the authors generate a discrete candidate set Q of RIS locations by random sampling within the cell while enforcing two physical constraints: (i) a clear line‑of‑sight (LoS) from the BS to the RIS, and (ii) a minimum distance equal to the Fraunhofer distance of the BS antenna array, ensuring far‑field operation. For each candidate location, many independent PPP user realizations are drawn; for each realization the inner AO algorithm is executed, and the candidate’s performance is measured by the minimum SINR among the users in that realization. The candidate that yields the highest worst‑user SINR for a given realization is added to a solution set S.

Since S is not convex and its elements cannot be averaged meaningfully, the authors propose a recursive coarse‑to‑fine clustering algorithm (Algorithm 3). The solution set is quantized with an initial step size d_start; the quantization cell with the highest count (the mode) defines an “optimal cluster.” A new, smaller search region centered at this mode is created, and the candidate generation/evaluation process is repeated with a halved step size. Recursion continues until a predefined precision d_p is reached, at which point the final RIS deployment region is reported as a square of side d_p centered at the mode.

Complexity analysis shows that the dominant operations are the beamformer update O(K M²) and the phase‑update O(K N² · min{M,K}), leading to an overall per‑iteration cost of O(I_iter · (K M² + K N² · min{M,K})). This is substantially lower than prior works that either jointly optimize placement, beamforming, and phases in a single monolithic problem or rely on exhaustive grid searches.

Simulation results, conducted over a variety of obstacle layouts and user density profiles, confirm the effectiveness of the proposed method. Compared with random RIS placement and with placement strategies that maximize average SINR, the recursive algorithm improves the expected minimum SINR by 15–30 %, raises the average SINR by 10–20 %, and expands the coverage (fraction of users above a target SINR) by 5–12 %. Moreover, the fairness benefit—guaranteeing a higher SINR for the most disadvantaged user—is clearly demonstrated.

In conclusion, the paper delivers a realistic, implementable framework for RIS deployment under user randomness, integrating obstacle awareness, statistical user modeling, and a scalable hierarchical optimization. Future extensions could address multi‑BS/multi‑RIS networks, mobility‑aware user models, and data‑driven candidate generation to further bridge the gap between theory and real‑world deployments.