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

1. Problem Statement

The paper addresses the beamforming gain maximization problem in a downlink wireless communication system assisted by a Fluid Reconfigurable Intelligent Surface (FRIS).

• System Setup: A multi-antenna Base Station (BS) communicates with a single-antenna user via an FRIS. The FRIS consists of a dense grid of candidate ports, but only a subset ($M_o$) can be dynamically selected and activated at any time.
• Constraints:
  • Port Selection: The system must select exactly $M_o$ active ports from $M$ candidates (combinatorial optimization).
  • Phase Quantization: Each active port applies a reflection coefficient with finite-resolution phase shifts chosen from a discrete codebook (unit-modulus).
  • Coupling: The problem involves the joint optimization of the BS transmit beamformer ($f_B$), the set of selected FRIS ports ($\Gamma$), and the discrete phase coefficients ($w_m$).
• Challenge: The strong coupling between the continuous beamformer, the combinatorial port selection, and the discrete phase assignment creates a highly non-convex mixed-discrete optimization problem. Existing methods often rely on unrealistic assumptions (e.g., single-antenna BS, continuous phase shifts, or ignoring the direct link) and fail to provide near-optimal solutions under practical hardware constraints.

2. Methodology

The authors propose an Alternating Optimization (AO) framework that iteratively updates the BS beamformer and the FRIS configuration. The core innovation lies in the geometric reformulation of the FRIS subproblem using Minkowski geometry.

A. Alternating Optimization (AO) Framework

The algorithm alternates between two steps until convergence:

1. BS Beamformer Update: For a fixed FRIS configuration, the optimal beamformer is derived in closed-form as the Maximum Ratio Transmission (MRT) solution.
2. FRIS Configuration Update: For a fixed beamformer, the problem reduces to selecting $M_o$ ports and assigning discrete phases to maximize the effective channel gain.

B. Minkowski Geometry Reformulation (The Core Contribution)

To solve the difficult FRIS subproblem, the authors transform the discrete optimization into a geometric problem:

• Feasible Set Construction: The set of all possible reflected signal sums (Minkowski sum of individual port contributions) is defined.
• Convexification: They prove that maximizing the magnitude of the sum over the discrete set is equivalent to maximizing it over the convex hull of that set (the Minkowski sum of the convex hulls of individual codebooks). This allows the use of convex geometry tools without losing optimality.
• Support Function Maximization: The problem is transformed into maximizing the support function $h(\cdot)$ of the convex feasible set over a directional parameter $\phi$ (angle).
  $$\max |z| = \max_{\phi} h_{PM_o,t}(e^{j\phi})$$
• Constructive Solution: For any given direction $\phi$, the optimal configuration is found constructively:
  1. Scoring: Calculate a "directional score" for every candidate port based on its best possible contribution in direction $\phi$.
  2. Selection: Select the top $M_o$ ports with the highest scores (Top-$M_o$ selection).
  3. Assignment: Assign the discrete phase codeword that maximizes the projection for each selected port.
  4. Search: Perform a 1D search over $\phi$ to find the direction that yields the global maximum.

C. Specialization for Regular $M_p$-gon Codebooks

For the practically important case of regular $M_p$-gon phase codebooks (uniform phase quantization):

• Closed-Form Scores: The authors derive closed-form expressions for the port scores involving cosine functions and phase quantization residuals.
• Finite Search: They prove that the resulting support function is piecewise-smooth. The global maximum can be found by evaluating only a finite set of critical angles (breakpoints where port selection or phase quantization changes), eliminating the need for exhaustive angular sweeping.

3. Key Contributions

1. Problem Formulation: A rigorous formulation of beamforming gain maximization for FRIS under multi-antenna BS and finite-resolution phase constraints, including the direct BS-user link.
2. Geometric Insight: The first work to reveal that FRIS signal combining under discrete constraints induces a convex Minkowski structure, which can be optimally exploited via support functions.
3. Optimal Algorithm: Development of a fully constructive FRIS update subroutine that solves the mixed-discrete problem via direction-dependent scoring and Top-$M_o$ selection.
4. Efficient Search: For regular codebooks, the derivation of a finite candidate-angle search procedure that provably identifies the global optimum without exhaustive search.
5. Convergence: An AO framework with guaranteed monotonic convergence and closed-form updates.

4. Simulation Results

Extensive simulations validate the proposed framework against benchmarks (Random Port Selection and Top-$|h|$ Port Selection):

• Performance: The proposed method consistently outperforms benchmarks across various system parameters (number of ports $M$, selected ports $M_o$, aperture size $W_x$, and phase resolution $M_p$).
• Near-Optimality: Compared to exhaustive search (feasible only for small systems), the proposed method achieves a beamforming gain ratio ($\Delta G$) very close to 1.0, confirming near-optimality.
• Convergence: The AO algorithm converges rapidly, typically saturating within fewer than 10 iterations.
• Robustness: The framework maintains high performance even with coarse phase quantization (low $M_p$) and effectively exploits spatial diversity as the FRIS aperture increases.

5. Significance

This paper provides a fundamental geometric interpretation of FRIS signal combining, bridging the gap between discrete hardware constraints and continuous optimization theory.

• Practicality: It addresses realistic hardware limitations (finite resolution, port selection) that previous theoretical works often ignored.
• Efficiency: By converting a complex combinatorial problem into a 1D search over a finite set of angles, the computational complexity is significantly reduced, making it scalable for 6G systems.
• Foundation: The Minkowski-sum approach offers a new paradigm for optimizing reconfigurable surfaces, suggesting that similar geometric techniques could be applied to other hybrid discrete-continuous wireless optimization problems.

In summary, the paper presents a mathematically rigorous and computationally efficient solution to maximize beamforming gain in FRIS-assisted networks, proving that fluid port selection combined with discrete phase control can achieve near-optimal performance through geometric optimization.