Source-linked AI summary
Flexible Beam Coverage Optimization for Movable-Antenna Array
Dong Wang, Weidong Mei, Boyu Ning, Zhi Chen
TL;DR
Fixed-position array geometry limits flexible beamforming, while uniform coverage across multiple regions of varying number and width calls for jointly designed antenna weights and positions. The paper jointly optimizes movable-antenna weights and positions using alternating optimization with successive convex approximation. At R=[0,180°], AO has higher max-min gain than both benchmarks for N=6 and N=8; separately, multi-region R2 minimum is −1 dB versus FPA’s −19 dB, with MA-FAB slightly higher.
Problem
Fixed-position array geometry limits flexible beamforming, while uniform coverage across multiple regions of varying number and width calls for jointly designed antenna weights and positions.
Source: §I · p. 1Method
The paper jointly optimizes movable-antenna weights and positions using alternating optimization with successive convex approximation.
Source: §I · p. 1, §II · p. 2Results
At R=[0,180°], AO has higher max-min gain than both benchmarks for N=6 and N=8; separately, multi-region R2 minimum is −1 dB versus FPA’s −19 dB, with MA-FAB slightly higher.
Source: §IV · p. 5Takeaways & Limitations
The proposed MA-enabled design significantly outperforms conventional FPAs by more flexibly catering to subregion geometry, and antenna weights may matter more as region width increases.
Source: §V · p. 6
Abstract
from arXiv · showhide
Fluid antennas (FAs) and movable antennas (MAs) have attracted increasing attention in wireless communications recently. As compared to the conventional fixed-position antennas (FPAs), their geometry can be dynamically reconfigured, such that more flexible beamforming can be achieved for signal coverage and/or interference nulling. In this paper, we investigate the use of MAs to achieve uniform coverage for multiple regions with arbitrary number and width in the spatial domain. In particular, we aim to jointly optimize the MAs weights and positions within a linear array to maximize the minimum beam gain over the desired spatial regions. However, the resulting problem is non-convex and difficult to be optimally solved. To tackle this difficulty, we propose an alternating optimization (AO) algorithm to obtain a high-quality suboptimal solution, where the MAs weights and positions are alternately optimized by applying successive convex approximation (SCA) technique. Numerical results show that our proposed MAbased beam coverage scheme can achieve much better performance than conventional FPAs.
I. INTRODUCTION
Fixed-position array geometry limits flexible beamforming, while movable antennas offer a way to vary array geometry. This paper targets uniform beam coverage across spatial regions of varying number and width by jointly designing antenna weights and positions.
- Fixed FPA geometry creates steering-vector correlation or orthogonality across angles, making flexible beam coverage and interference nulling difficult.Source: §I · p. 1
- Prior FA/MA studies considered flexible interference nulling, whereas this work focuses on uniform beam coverage across one or multiple spatial regions.Source: §I · p. 1
- The design jointly optimizes MA weights and positions to maximize the minimum gain across regions with varying number and width.The paper motivates this setting by user mobility and changing traffic demands.Source: §I · p. 1
- The proposed AO algorithm alternates weight and position updates using SCA, and numerical results report better coverage than conventional FPAs.Source: §I · p. 1
II. SYSTEM MODEL
The system uses a linear array of movable antennas whose positions can vary within a one-dimensional transmit region. The design jointly selects positions and analog beamforming weights to maximize worst-case gain across disjoint angular regions.
II. SYSTEM MODEL
- The transmitter has N movable antennas in a linear array, with positions adjustable within a one-dimensional region of length D.Source: §II · p. 2
- Analog beamforming uses constant-magnitude weights, with the phase shift specified for each antenna.Source: §II · p. 2
- The desired angular coverage comprises K disjoint subregions, and changing their number and widths supports flexible coverage.Source: §II · p. 2
- The objective is to maximize minimum beam gain over the desired regions, subject to position, antenna-spacing, and beamformer-modulus constraints.The spacing constraint avoids antenna coupling; the problem is non-convex because of region continuity, coupled weights and positions, and unit-modulus constraints.Source: §II · p. 2
III. PROPOSED ALGORITHM
The algorithm makes the continuous-region optimization tractable by sampling each subregion and expressing the max-min objective in epigraph form. It then alternates between optimizing weights and positions using SCA.
III. PROPOSED ALGORITHM
- Each continuous angular subregion is discretized into a set of sampling points.Source: §III · p. 2
- An auxiliary variable t converts the max-min beam-gain objective into epigraph constraints over the sampled points.Source: §III · p. 2
- The resulting problem remains non-convex in weights and positions, so the algorithm optimizes them alternately with SCA.Source: §III · p. 2
A. Optimization of ω for Given x
With antenna positions fixed, the algorithm optimizes the analog beamforming vector through a lifted semidefinite formulation. A penalty-based SCA step addresses the relaxed rank-one condition before recovering the beamformer.
A. Optimization of ω for Given x
- With the antenna position vector fixed, the algorithm optimizes the analog beamforming vector using a lifted matrix formulation.Source: A. Optimization of ω for Given x · p. 2
- Relaxing the rank-one constraint yields an SDP, but its solution may not be rank one and Gaussian randomization can have a large optimality gap.Source: A. Optimization of ω for Given x · p. 2
- The method penalizes Tr(V) − σ(V), which is zero exactly for rank-one V, and uses a first-order SCA approximation of the non-convex penalty.The penalty parameter controls the penalty term.Source: A. Optimization of ω for Given x · p. 3
- The convexified subproblem is solved iteratively; the paper reports that the converged V is rank one in its simulations, enabling SVD-based recovery of ω.Source: A. Optimization of ω for Given x · p. 3
B. Optimization of x for Given ω
With the beamformer fixed, the antenna positions are optimized by approximating a cosine-dependent gain constraint with an SCA surrogate. The resulting position subproblem is a QCQP solved by an interior-point algorithm.
B. Optimization of x for Given ω
- SCA replaces the cosine in the gain constraint with a quadratic surrogate around the current antenna positions.The surrogate is used to relax constraint (7a).Source: B. Optimization of x for Given ω · p. 3
- The relaxed gain constraint is a convex quadratic constraint, yielding a QCQP for optimizing the antenna positions.The paper states that the resulting QCQP can be solved by an interior-point algorithm.Source: B. Optimization of x for Given ω · pp. 3–4
- The position optimization is repeated across SCA iterations, updating x from one iteration to the next.Source: B. Optimization of x for Given ω · p. 4
C. Overall Algorithm and Complexity Analysis
The algorithm alternates SCA updates of the beamformer and antenna positions, holding one fixed while optimizing the other. The authors state that the resulting AO algorithm converges.
C. Overall Algorithm and Complexity Analysis
- Each AO iteration optimizes the beamformer with x fixed, then optimizes x with the updated beamformer fixed.The algorithm initializes each inner SCA with the preceding AO iterate.Source: C. Overall Algorithm and Complexity… · p. 4
- The objective sequence for the V update is non-decreasing, and the authors state that both SCA updates and the overall AO algorithm converge.The proof for the x update is omitted for brevity.Source: C. Overall Algorithm and Complexity… · p. 4
- The complexity analysis defines L(k) as the total number of sampling points.Source: C. Overall Algorithm and Complexity… · p. 4
D. Initialization
Because AO performance depends critically on initialization, the authors initialize antenna positions uniformly and derive the initial beamformer from an SDR solution using Gaussian randomization.
D. Initialization
- The initial antenna positions are uniformly spaced across the linear array.Source: D. Initialization · p. 4
- With those positions fixed, the method solves the SDR problem and uses Gaussian randomization based on its solution to generate the initial beamformer.Source: D. Initialization · p. 4
- Among randomization realizations, the initial beamformer is chosen to maximize the SDR problem's objective value.Source: D. Initialization · p. 4
IV. NUMERICAL RESULTS
Across single-region and multi-region tests, the proposed MA scheme achieves higher max-min beam gain than the benchmarks in the reported overall comparisons. Its advantage over fixed analog beamforming grows for wider coverage, while optimized MA positions use larger apertures and nonuniform spacing.
Evaluation setup
- The simulations use fc = 1 GHz and array length D = 8λ, comparing optimized FPA weights at half-wavelength spacing with MA-FAB, which fixes the beamformer and optimizes positions.The AO and SCA convergence thresholds are 10^-5 and 0.01, respectively.Source: §IV · p. 4
Single-region coverage
- For the 180° region, the proposed AO scheme achieves a considerably higher max-min beam gain than MA-FAB and FPA for both N = 6 and N = 8.The optimized MA schemes use larger apertures and nonuniform spacing than the FPA benchmark; MA-FAB also outperforms FPA.Source: Figure 2 · p. 5, §IV · p. 5
- With N = 6 and N = 8, the MA schemes use larger apertures and nonuniform spacing than FPA, and MA-FAB outperforms FPA in beam coverage.For the proposed scheme, aperture increases from 1.14 at N = 6 to 1.2 at N = 8.Source: §IV · p. 5
Coverage-width comparison
- The max-min beam gains of all considered schemes decrease monotonically as coverage width increases, consistent with the stated constant total beam gain over the spatial domain.The section attributes the decrease to the wider region distributing that total gain over more directions.Source: §IV · pp. 5–6
Multi-region coverage
- In multi-region coverage, the proposed scheme's minimum gain exceeds FPA's maximum in R1 and R3; in R2, FPA's minimum is −19 dB versus −1 dB for the proposed scheme, though MA-FAB is slightly higher.The proposed positions form three subarrays aligned with the three coverage regions; aperture x8 rises from 1.2 in single-region coverage to 1.65 in multi-region coverage.Source: §IV · p. 5, Figure 3 · p. 6
Coverage-width comparison
- For narrow coverage up to 50°, the proposed scheme and MA-FAB perform similarly and both significantly outperform FPA; as the region widens, the proposed scheme's advantage over MA-FAB grows.The reported trend indicates that analog beamforming matters more for challenging wide-region coverage.Source: §IV · p. 6
V. CONCLUSION
The paper jointly optimizes movable-antenna weights and positions for uniform beam coverage, using an alternating-optimization algorithm and reporting stronger performance than conventional FPAs.
- The proposed MA-enabled design significantly outperforms conventional FPAs by more flexibly catering to the geometry of the subregions.Source: §V · p. 6
- The design jointly optimizes antenna weights and positions to maximize minimum beam gain over the desired spatial regions.Source: §V · p. 6
- The AO algorithm provides a high-quality suboptimal solution, and antenna weights may matter more as region width increases.Source: §V · p. 6
- For K = 1, Fig. 4 plots max-min beam gain versus coverage width.Source: Figure 4 · p. 6