Energy Efficient Multi-User Beamforming and 3D Position Optimization for SIM-Assisted UAVs

Technical Summary: Energy Efficient Multi-User Beamforming and 3D Position Optimization for SIM-Assisted UAVs

Problem Statement
This paper addresses the challenge of energy-efficient (EE) downlink multi-user communications in Unmanned Aerial Vehicle (UAV) systems. While UAVs offer flexible deployment and strong line-of-sight connectivity, equipping them with fully digital beamforming architectures is often impractical due to the high hardware complexity, power consumption, and payload weight associated with requiring a dedicated radio-frequency (RF) chain per antenna element. To mitigate these constraints, the authors investigate a hybrid architecture where the UAV is equipped with Stacked Intelligent Metasurfaces (SIM). Unlike single-layer Reconfigurable Intelligent Surfaces (RIS) that only apply element-wise phase shifts, SIMs utilize multiple cascaded metasurface layers to perform wave-domain analog beamforming. This allows for complex signal processing directly in the electromagnetic domain, drastically reducing the number of required RF chains.

The core problem formulated is the joint optimization of three coupled variables to maximize system Energy Efficiency (defined as the ratio of achievable sum-rate to total power consumption):

1. Digital Precoder: The low-dimensional digital beamforming matrix at the RF chains.
2. SIM Phase Shifts: The controllable phase shifts across all layers of the stacked metasurface.
3. UAV 3D Position: The spatial coordinates of the UAV.

The optimization is subject to transmit power constraints, unit-modulus constraints on the passive metasurface elements, and UAV deployment regions. The problem is highly non-convex due to the fractional objective function, the cascaded nature of the SIM channel, the non-linear dependence of the channel on UAV position, and the non-convex unit-modulus constraints.

Methodology
To solve this challenging non-convex problem, the authors propose a transform-based alternating optimization framework. The methodology proceeds as follows:

1. Fractional Programming Reformulation: The fractional EE objective is converted into a subtractive form using Dinkelbach's method. An auxiliary parameter is introduced to iteratively update the EE value.
2. Auxiliary Variable Transformation: To handle the non-concave logarithmic sum-rate and the signal-to-interference-plus-noise ratio (SINR) terms, the authors employ dual and quadratic transforms. This introduces auxiliary variables ($\alpha_k$ and $y_k$) that decouple the SINR terms, transforming the objective into a form amenable to block-wise optimization.
3. Digital Beamformer Update: With SIM phases and UAV position fixed, the digital precoder is updated by solving a convex quadratic program. The optimal solution is derived in closed form using Karush-Kuhn-Tucker (KKT) conditions, with a bisection search employed to satisfy the transmit power constraint.
4. SIM Phase Optimization: The optimization of SIM phase shifts, subject to unit-modulus constraints, is performed using Riemannian manifold ascent. The feasible set is treated as a product manifold of complex unit circles. The algorithm computes the Euclidean gradient, projects it onto the tangent space of the manifold, and updates the phases via a retraction operator to maintain unit magnitude.
5. UAV Position Optimization: The UAV position is updated using Successive Convex Approximation (SCA). A concave surrogate function is constructed around the current position based on analytically derived channel gradients. The position is then updated via a projected gradient ascent step within the feasible deployment region.

The overall algorithm iterates between these blocks until convergence, with theoretical analysis confirming monotonic improvement of the transformed objective and convergence to a stationary solution.

Key Contributions

• Unified Framework: The paper develops a unified optimization framework specifically for SIM-assisted UAV downlink systems, jointly optimizing digital precoding, cascaded SIM phase shifts, and 3D UAV positioning under a hardware-aware power model.
• Hardware-Aware EE Maximization: Unlike prior works focusing solely on spectral efficiency or sum-rate, this work formulates an EE maximization problem that explicitly accounts for RF chain power, baseband processing, element-wise SIM control power, and UAV propulsion constraints.
• Algorithmic Innovation: The authors propose a novel algorithmic approach combining Dinkelbach's method, dual/quadratic transforms, Riemannian manifold optimization, and SCA to tackle the specific non-convexities introduced by the SIM architecture and UAV mobility.
• Performance Analysis: The study provides convergence and complexity analyses, characterizing the computational cost of the proposed algorithm.

Numerical Results
Simulation results, conducted at 28 GHz with Rician fading channels, compare the proposed joint design against two benchmarks: a Fully Digital (FD) scheme and a SIM-assisted Maximum Ratio Transmission (SIM-MRT) scheme.

• Energy Efficiency: The proposed scheme significantly outperforms both benchmarks. For a system with $M=36$ elements and 4 users, the proposed method achieves approximately 58% higher EE than the FD scheme and over 700% higher than the SIM-MRT scheme at 30 dBm transmit power.
• Sum-Rate Trade-offs: While the FD scheme achieves the highest absolute sum-rate, the proposed SIM-based architecture offers a favorable trade-off, achieving competitive sum-rates with significantly lower hardware complexity and power consumption. The SIM-MRT scheme performs poorly in multi-user scenarios due to its inability to manage inter-user interference.
• Parameter Sensitivity:
  • Transmit Power: EE increases with power up to a saturation point (around 30 dBm), after which it declines due to diminishing rate gains relative to power consumption.
  • SIM Size ($M$): Increasing the number of elements improves beamforming gain but increases circuit power. The proposed scheme scales more favorably than the FD scheme as $M$ increases, whereas the FD scheme's EE degrades sharply due to the linear increase in RF chain power.
  • Number of Layers ($L$): Increasing the number of SIM layers does not always improve EE. Beyond a certain point (e.g., $L > 2$), the additional wave-domain processing gains are offset by increased hardware/control power and propagation losses across the stack.

Significance and Claims
The paper claims that the proposed joint design effectively bridges the gap between low-complexity designs and fully digital systems. By leveraging the wave-domain processing capabilities of SIMs, the architecture reduces the number of required RF chains while maintaining high beamforming flexibility. The results demonstrate that achieving high energy efficiency in UAV systems requires a careful balance between spatial degrees of freedom, hardware complexity, and power consumption. The study highlights that simply increasing transmit power or stacking more metasurface layers does not guarantee improved EE; instead, a holistic optimization of the digital precoder, SIM configuration, and UAV trajectory is essential. The work establishes that SIM-based architectures are particularly suitable for energy-constrained aerial platforms where size, weight, and power are critical constraints.