Weighted Sum-Rate Enhancement for Flexible Intelligent Metasurface-Assisted Multicell Systems

Abstract: Flexible intelligent metasurface (FIM) technology has emerged as a promising technology for enhancing wireless communication performance by dynamically reshaping the propagation environment. Compared with conventional rigid reconfigurable intelligent surfaces (RIS), an FIM is composed of multiple electromagnetic (EM) scattering units, each of which can flexibly modify its displacement in the direction normal to the surface, thereby cooperatively morphing the overall surface shape. This additional degree of freedom (DoF) enables improved beamforming and interference mitigation, particularly in complex multicell scenarios. In this paper, an optimization problem for maximizing the weighted sum-rate (WSR) in a multicell multi-user multiple-input single-output (MU-MISO) system assisted by an FIM deployed at the cell boundary is investigated. We jointly optimize the transmit beamforming at the base station (BS), the phase shift matrix, and the FIM surface shape, subject to constraints on the transmit power budget, unit-modulus reflection coefficients, and surface shape morphing range. Due to the non-convex objective function with highly coupled variables, solving the formulated optimization problem is challenging. To tackle this challenge, we propose an efficient alternating optimization framework that leverages the weighted minimum mean square error (WMMSE) method to reformulate the problem and the block coordinate descent (BCD) algorithm to iteratively update the variables. Specifically, the Riemannian conjugate gradient (RCG) algorithm is leveraged to optimize the phase shift matrix, while the projected gradient descent (PGD) method is adopted to optimize the surface shape of the FIM. Additionally, the optimal beamforming vectors are obtained in closed form.

• The paper proposes a joint optimization of beamforming, phase shift, and surface morphing that yields around a 33% improvement in weighted sum-rate.
• It employs a block coordinate descent framework with closed-form WMMSE updates for beamforming, Riemannian gradient methods for phase shifts, and projected gradient descent for surface deformation.
• Extensive simulations confirm scalability, rapid convergence, and robust performance even under significant CSI imperfections in dense, multipath-rich environments.

Weighted Sum-Rate Enhancement for Flexible Intelligent Metasurface-Assisted Multicell Systems

Introduction and Motivation

The paper addresses the maximization of the weighted sum-rate (WSR) in multicell multi-user multiple-input single-output (MU-MISO) downlink systems augmented by flexible intelligent metasurfaces (FIMs). Unlike conventional rigid reconfigurable intelligent surfaces (RIS), FIMs introduce additional degrees of freedom by allowing independent displacement of their electromagnetic (EM) scattering units perpendicular to the surface. This morphing capability enhances EM field manipulation, thereby providing spatial adaptability, improved beamforming, and more effective interference management in dense, multipath-rich wireless environments.

By placing an FIM at the cell boundary in a multicell setting, the system not only benefits from phase shift reconfigurations (as in conventional RIS) but also actively reshapes the wireless propagation environment via 3D surface morphing. This setting leads to complex, non-convex optimization challenges due to the high degree of coupling among transmitter beamforming, FIM phase shift, and surface morphology parameters.

System Architecture and Channel Model

The considered scenario features $L$ microcells, each with a base station (BS) utilizing $M$ antennas (ULA), serving $K$ single-antenna UEs. The FIM, deployed at the cell boundary, comprises $N = N_y N_z$ EM units, where each unit's displacement $d_n$ is bounded by a maximum morphing range. The 3D morphing of the FIM enables the system to orchestrate the constructive superposition of multipath components while simultaneously providing spatial, phase, and amplitude control.

A narrowband multipath channel model is adopted, capturing both line-of-sight and non-line-of-sight behaviors for BS-FIM and FIM-UE links. The channel incorporates the morphable steering vector, parametrized by both the phase shift matrix $\Phi$ and the surface displacement vector $\mathbf{d}$, resulting in an overall channel representation sensitive to the EM environment's flexible geometry.

Optimization Problem and Mathematical Framework

The core optimization problem targets joint maximization of the system WSR by optimizing:

1. The transmit beamforming matrix at each BS,
2. The unit-modulus FIM phase shift matrix,
3. The vector of FIM element displacements (surface shape).

Formally, the problem is to maximize

$$\sum_{l=1}^L \sum_{k=1}^K \omega_{l,k} R_{l,k}(\mathbf{W}, \mathbf{\Phi}, \mathbf{d})$$

subject to per-BS transmit power, unit-modulus FIM phase constraints, and surface morphing bounds.

The primary technical challenges arise from the non-convexity due to coupled optimization variables and the nonlinear interaction between phase, geometry, and multipath structure. The problem is rigorously recast as a weighted minimum mean square error (WMMSE) minimization, enabling tractable alternating updates.

Alternating Optimization Strategy

The solution framework decomposes the parameter space into four blocks:

• Decoding weights and auxiliary variables ($\mathbf{u}$, $\mathbf{v}$),
• Transmit beamforming ($M$0),
• FIM phase shift matrix ($M$1),
• FIM surface shape ($M$2).

WMMSE and Closed-Form Steps: The methodology yields closed-form updates for $M$3, $M$4, and the beamforming matrix. The beamforming update is a convex quadratic subproblem with a Lagrange-dual approach for satisfying the transmit power constraint.

Phase Shift Optimization: The phase shift matrix update is performed on the complex circle manifold using a Riemannian conjugate gradient method, efficiently handling the unit-modulus constraint and exploiting the problem's structure.

Surface Shape Optimization: The surface morphing variables are optimized via projected gradient descent. The authors introduce a “FIM kernel” which analytically connects surface deformation and multipath response. The gradient is computed using chain rule expansions over the kernel structure, and the step size determination leverages a quadratic approximation for low-complexity updates.

This block coordinate descent (BCD) strategy iteratively updates each group of variables while fixing the others, guaranteeing objective non-decrease and convergence to stationary points due to bounded, continuous domains.

Numerical Analysis and Results

The simulation study demonstrates the superiority of FIM-assisted multicell systems over several baseline schemes: rigid RIS, random phase/morphing, and conventional systems without FIM/RIS. Notable findings include:

• WSR Gain: The optimized FIM-assisted system achieves on average a 33% WSR improvement over conventional RIS-based systems across a variety of practical propagation and user priority configurations.
• Scalability: The algorithm scales linearly with the number of FIM elements ($M$5), rendering it computationally tractable for large arrays.
• Convergence: Convergence is rapid and robust, and the algorithm remains efficient across SNR regimes.
• Robustness to CSI Imperfections: Even under significant channel state information (CSI) estimation errors (with normalized MSE $M$6 up to 0.8), the FIM-based scheme maintains marked WSR superiority over rigid RIS.
• Surface Morphing Range: System performance exhibits diminishing returns as the morphing range surpasses the EM wavelength. Optimal shaping is achievable within practical, sub-wavelength deformation bounds.
• Effect of Multipath: Performance advantage over RIS grows as the environmental multipath richness increases, underscoring the unique capability of FIM to exploit spatial and multipath diversity.
• Holistic Shape Optimization: Optimization of the entire FIM surface yields significantly higher rates than optimizing displacements of only a subset of elements; even single-element optimization surpasses rigid RIS.

Practical and Theoretical Implications

Practical Deployment: The findings imply that physically realizable FIMs with fast morphing actuators and standard EM units can offer substantial, scalable improvements in future multicell heterogeneous networks, particularly for 6G and beyond. The algorithmic design accommodates real-time adaptation, with dual time-scale operation (phase shifts in microseconds, shape updates in milliseconds) enabled by current hardware capabilities.

Algorithmic Generality: The proposed alternating framework is extensible to more complex multi-antenna, multi-frequency, and secure communications settings, as well as to systems integrating advanced machine learning for further CSI acquisition efficiency.

Theoretical Impact: The work establishes that the canonical RIS limitation—fixed geometry—can be fundamentally overcome by surface morphing, pushing the limits of EM environment control and achievable rates. By introducing the “FIM kernel,” the analysis provides tools adaptable to broader classes of metasurface-based optimization problems.

Future Directions: Several open challenges are highlighted, such as dynamic misalignment arising from disparate time scales of phase shift and morphing actuation, and the necessity for integrated power consumption and energy efficiency modeling in FIM architectures.

Conclusion

This paper makes a strong argument, supported by both algorithmic innovation and comprehensive evaluation, for the deployment of flexible intelligent metasurfaces in multicell, multiuser MISO networks. The joint optimization of active (beamforming), passive (phase shift), and geometric (surface shape) parameters provides a new paradigm for wireless environment reconfiguration. The demonstrated WSR gains, scalability, and practical feasibility position FIMs as a promising technological direction for addressing the capacity and quality-of-service demands of next-generation wireless systems.