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

TL;DR

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 is developed, which consistently outperforms benchmarks, achieves near-optimal beamforming gains in exhaustive-search validations, and converges rapidly within a few AO iterations.

Abstract

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.

Figures (13)

• Figure 1: FRIS-assisted downlink system.
• Figure 2: Geometrical illustration of $h_{\mathcal{C}}(u)$ with respect to (a) general and (b) polygon type of $\mathcal{C}$.
• Figure 3: Geometric illustration and the resulting Top-$M_o$ port selection based on arc dominance in the Thales-circle representation with $M=3$ and $M_0=2$.
• Figure 4: Geometric interpretation of $h_{\mathrm{Conv}(\mathcal{W}_{M_p})}(e^{j\phi})$ and $s_m(\phi)$ for a regular $M_p$-gon phase codebook and the resulting per-port score with $M_p=5$. The left figure depicts $\mathrm{Conv}(\mathcal{W}_{M_p})$ of the unit-modulus $M_p$-ary phase-shifter codebook and the evaluation of $h_{\mathrm{Conv}(\mathcal{W}_{M_p})}(e^{j\phi})=\cos(\delta_{M_p}(\phi))$ along direction $e^{j\phi}$, while the right figure illustrates the scaled and rotated convex set $\mathrm{Conv}(h_m\mathcal{W}_{M_p})$, where multiplication by $h_m=a_m e^{j\alpha_m}$ induces a rotation by $\alpha_m$ and a radial scaling by $a_m$, resulting in $s_m(\phi)=a_m\cos(\delta_{M_p}(\phi-\alpha_m))$.
• Figure 5: Rate $R^{(t)}$ according to iteration $t$ under general codebook.

Theorems & Definitions (15)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Corollary 3.1
• proof
• Lemma 4
• proof