Beamforming Gain Maximization for Fluid Reconfigurable Intelligent Surface: A Minkowski Geometry Approach

This paper investigates beamforming-gain maximization for a fluid reconfigurable intelligent surface (FRIS)-assisted downlink system, where each active port applies a finite-resolution unit-modulus phase selected from a discrete codebook. The resulting design couples the multi-antenna base-station (BS) beamformer with combinatorial FRIS port selection and discrete phase assignment, leading to a highly nonconvex mixed discrete optimization. To address this challenge, we develop an alternating-optimization (AO) framework that alternates between a closed-form maximum-ratio-transmission (MRT) update at the BS and an {optimal} FRIS-configuration update. The key step of the proposed FRIS configuration is a Minkowski-geometry reformulation of the FRIS codebook superposition: by convexifying the feasible reflected-sum set and exploiting support-function identities, we convert the FRIS subproblem into a one-dimensional maximization over a directional parameter. For each direction, the optimal configuration is obtained constructively via per-port directional scoring, Top-$M_o$ port selection, and optimal codeword assignment. For the practically important regular $M_p$-gon phase-shifter codebook, we further derive closed-form score expressions and establish a piecewise-smooth structure of the resulting support function, which leads to a finite critical-angle search that provably identifies the global optimum without exhaustive angular sweeping. Simulation results demonstrate that the proposed framework consistently outperforms benchmarks, achieves near-optimal beamforming gains in exhaustive-search validations, accurately identifies the optimal direction via support-function maximization, and converges rapidly within a few AO iterations.

💡 Research Summary

This paper addresses the beamforming‑gain maximization problem in a downlink system where a multi‑antenna base station (BS) is assisted by a fluid reconfigurable intelligent surface (FRIS). Unlike conventional RIS, FRIS can dynamically select a subset of its densely packed candidate ports and each active port can only apply a finite‑resolution, unit‑modulus phase chosen from a discrete codebook. The joint optimization of the BS transmit beamformer, the set of selected ports, and the discrete phase values leads to a highly non‑convex mixed‑integer problem that is intractable by brute‑force methods.

The authors propose an alternating‑optimization (AO) framework that decouples the problem into two sub‑problems. With a fixed FRIS configuration, the optimal BS beamformer is obtained in closed form as a maximum‑ratio‑transmission (MRT) vector aligned with the effective channel (a^{\star}=h_{d}+ \sum_{m\in\Gamma} w_{m}h_{m}). Conversely, with a fixed beamformer, the FRIS sub‑problem is transformed using convex‑geometry tools. The reflected‑sum set generated by the selected ports is expressed as a Minkowski sum of finite sets ({h_{m}W_{m}}). By taking the convex hull of this union, the authors introduce the support function (\sigma_{Z(\Gamma)}(u)=\max_{z\in Z(\Gamma)}\Re{u^{H}z}), where (u=e^{j\theta}) denotes a unit‑norm directional vector. Maximizing the support function over (\theta) yields the direction that gives the largest projection of the feasible reflected‑sum set, and for any fixed (\theta) the optimal FRIS configuration can be constructed explicitly.

For each candidate direction (\theta), a per‑port score is defined as
\