PaperlayerView exact revision on arXiv

Source-linked AI summary

User Association and Transmit Beamforming for Rotatable Antenna-Enabled LAE Networks

Bowen Yao, Chao Zhang, Ying-Chang Liang

arXiv:2609.35010v1eess.SP

TL;DR

Fixed antenna tilt schemes use a common boresight that cannot align with users across diverse azimuth and elevation angles. This paper jointly optimizes user association, transmit beamforming, and rotatable antenna orientations, and reports the highest sum-rate among evaluated schemes across examined settings.

  • Problem

    Fixed antenna tilt schemes share a boresight that cannot align with distributed users across diverse azimuth and elevation angles, reducing gain toward intended users.

    Source: §I · p. 1

  • Method

    An alternating optimization algorithm combines projected gradient ascent for user association, WMMSE for beamforming, and Frank-Wolfe updates for antenna orientations.

    Source: §VII · p. 11

  • Results

    The proposed AO algorithm achieves the highest sum-rate among evaluated schemes across examined settings and improves over fixed-orientation and nearest-BS-association benchmarks.

    Source: §I · p. 2, §VII · p. 11

  • Takeaways & Limitations

    The candidate-scanning and antenna block-based algorithms retain most of the achievable gain while reducing orientation-optimization computational cost.

    Source: §I · p. 2, §VII · p. 11

Abstract

from arXiv · show

This paper investigates rotatable antenna (RA)-enabled low-altitude economy (LAE) networks, where multiple base stations (BSs) are deployed to serve both terrestrial and aerial users. Unlike conventional fixed orientation antennas, RAs can adjust their boresights to serve users at diverse azimuth and elevation angles, thereby providing three-dimensional (3D) coverage. To improve the performance of LAE networks, we formulate a network sum-rate maximization problem that jointly optimizes user association, transmit beamforming, and RA orientations. However, in such LAE networks, aerial users are often visible to multiple BSs through line-of-sight (LoS) links, making user association and interference management particularly challenging. Moreover, user association and RA orientations are closely coupled: user association affects the RA pointing directions, while the RA orientations in turn reshape the channel conditions for user association. To solve this problem, we propose an alternating optimization (AO) algorithm that integrates weighted minimum mean-square error (WMMSE) for transmit beamforming, projected gradient ascent updates for user association, and Frank-Wolfe updates for RA orientations. Furthermore, we develop a discrete candidate-scanning scheme that replaces continuous orientation optimization with a finite search to reduce computational complexity. We also propose an antenna block-based scheme to reduce the number of orientation variables. Simulation results show that the proposed joint design achieves a higher sum-rate than the fixed antenna orientation and nearest BS association benchmarks. Moreover, the candidate-scanning and block-based schemes achieve approximately 89.9% and 94.3% of the performance with high transmit power, respectively, while reducing the complexity of orientation updates.

I. INTRODUCTION

The paper targets the coupled choices of serving base station and antenna orientation in LAE networks, then jointly optimizes user association, beamforming, and rotatable-antenna orientations. It also introduces lower-complexity orientation updates for practical deployment.

I. INTRODUCTION

  • Aerial users can have LoS links to multiple BSs, creating both candidate serving choices and severe interference; association and RA orientations also reshape each other’s conditions.The prior schemes discussed use predetermined association, while user association affects RA pointing directions and RA directions alter channel conditions for BS selection.Source: §I · p. 2
  • The proposed network jointly optimizes user association, transmit beamforming, and RA orientations to serve terrestrial and aerial users using their 3D locations.The design coordinates these variables to enhance desired signals, mitigate multi-user interference, and reduce gain loss from boresight misalignment.Source: §I · p. 2
  • An alternating optimization algorithm updates user association with projected gradient ascent, beamforming with WMMSE, and RA orientations with Frank-Wolfe.The problem is first reformulated as an equivalent WMMSE problem.Source: §I · p. 2
  • Candidate scanning restricts each RA boresight to finitely many feasible directions based on user locations, while block-based optimization gives all RAs in a group a shared boresight.Both schemes are proposed to reduce the computational complexity of RA orientation optimization.Source: §I · p. 2
  • The joint design achieves higher sum-rate than fixed nearest-BS association and fixed-orientation benchmarks; its low-complexity algorithms achieve approximately 89.9% and 94.3% of performance.The introduction reports these percentages without assigning one to each algorithm in this passage.Source: §I · p. 2

A. RA Configuration

  • The network comprises multiple BSs serving terrestrial and aerial users, with independently adjustable RA boresights characterized by zenith and azimuth angles.Each BS uses a planar array; the zenith angle is bounded to account for the limited rotation range.Source: §II · p. 2, A. RA Configuration · pp. 2–3

B. Channel Model

The channel model combines RA geometry and directional gain with free-space propagation. The signal model incorporates single-BS user association, transmission, interference, and achievable-rate calculations.

B. Channel Model

  • The channel model uses RA-to-user distance and propagation direction to determine directional antenna gain, then applies the Friis transmission model to form the channel.The RA locations are defined relative to each BS array center, and the channel vector collects the antenna-level channels.Source: B. Channel Model · p. 3

C. Signal Model

  • Each user is associated with only one BS, and binary association variables identify the serving BS; transmit beamformers are set to zero for unassociated users.The model defines transmit data and beamforming vectors before expressing the received signal at each user.Source: C. Signal Model · p. 3
  • The signal model defines each served user’s SINR and uses it to express the achievable rate.Source: C. Signal Model · p. 4

III. PROBLEM FORMULATION

The paper formulates network sum-rate maximization that jointly chooses user association, transmit beamforming, and RA orientations, which are coupled through the effective channels. It recasts the problem in an equivalent WMMSE form for joint optimization.

III. PROBLEM FORMULATION

  • The network sum-rate objective jointly optimizes user association, transmit beamforming, and RA orientations because serving-BS choices affect the optimal antenna directions.The effective channels also depend on RA orientations, making the joint problem non-convex.Source: §III · p. 4, §IV · p. 4
  • Each user is associated with one BS, while each RA has a normalized orientation vector limited to a maximum rotation angle from its initial direction.The objective is the sum of user rates weighted by their association variables, subject to individual BS transmit-power budgets.Source: §III · p. 4
  • Omitting the constraint that a beamformer must be zero for an unassociated BS does not change the optimal objective value.Setting such a beamformer to zero saves power and cannot increase interference to served users.Source: §III · p. 4

A. Problem Reformulation

  • The non-convex sum-rate objective is reformulated into an equivalent WMMSE problem using auxiliary equalizers and weights.For fixed association, beamforming, and orientations, the optimal equalizer and weight variables are specified by the reformulation.Source: A. Problem Reformulation · pp. 4–5
  • The received-power term used in the reformulation denotes total received power at user k.The achievable-rate expression is also represented through a positive weight and an identity involving the mean-square error.Source: A. Problem Reformulation · pp. 4–5
  • For fixed beamforming, minimizing the mean-square error with respect to the equalizer gives the MMSE equalizer, and the corresponding MMSE yields an equivalent rate expression.The achievable rate is expressed as the negative base-2 logarithm of the MMSE.Source: A. Problem Reformulation · p. 4

B. Optimization for User Association

The user-association subproblem relaxes binary association variables to continuous values, then uses projected gradient ascent and recovers a binary association after AO iterations.

B. Optimization for User Association

  • Relaxing each binary association variable to [0, 1] yields a problem equivalent to the original association problem in optimal objective value.An optimal relaxed solution assigns each user’s entire association weight to a single BS, although solving directly to optimality may yield a low-quality AO initialization.Source: B. Optimization for User Association · p. 5
  • Projected gradient ascent updates the relaxed association variables, with a stepsize and Euclidean projection onto the feasible set.The relaxed variables satisfy the per-user association feasible-set constraint.Source: B. Optimization for User Association · p. 5
  • After AO iterations, binary recovery assigns each user to the BS with the largest association value.Source: B. Optimization for User Association · p. 5

C. Optimization for Transmit Beamforming

With other variables fixed, the method updates transmit beamforming through per-BS subproblems and updates RA orientations using Frank-Wolfe steps with Armijo backtracking. The overall algorithm alternates updates of the auxiliary variables, association, beamforming, and orientations.

C. Optimization for Transmit Beamforming

  • For fixed auxiliary variables, association, and orientations, transmit beamforming separates into independent subproblems for each BS.The KKT stationarity condition gives the beamforming vector, with each BS’s Lagrange multiplier found by bisection to satisfy its power constraint.Source: C. Optimization for Transmit Beamforming · p. 5

D. Optimization for the Orientations of RAs

  • The RA-orientation subproblem is handled by Frank-Wolfe updates that approximate the objective at current feasible orientations and use Armijo backtracking for the step size.The method projects gradients onto the tangent space because each orientation vector is constrained to a unit sphere.Source: D. Optimization for the Orientations… · p. 6
  • The RA-orientation objective is monotonically non-decreasing across updates because the aggregate improvement is nonnegative and an Armijo-compatible step size exists.Source: D. Optimization for the Orientations… · p. 6
  • The overall AO algorithm updates auxiliary variables, user association, transmit beamforming, and RA orientations in sequence until its stopping condition or iteration limit.Source: D. Optimization for the Orientations… · pp. 6–7

1) Convergence Analysis:

The AO objective is monotonically non-decreasing and converges because it is bounded above. The paper also analyzes complexity separately for association, beamforming, and orientation updates.

1) Convergence Analysis:

  • The objective sequence is monotonically non-decreasing and converges because the objective is bounded above under the gain bound and transmit-power constraint.The association, beamforming, and Frank-Wolfe orientation updates each preserve or increase the objective.Source: 1) Convergence Analysis · p. 7

2) Computational Complexity:

  • Association updates cost O(KB log B), beamforming updates cost O(BKM^2 + BM^3), and orientation-gradient updates cost O(BMK(K + B)).The total complexity is analyzed over tAO AO iterations and tFW Frank-Wolfe iterations per AO step.Source: 2) Computational Complexity · p. 7

V. LOW-COMPLEXITY JOINT OPTIMIZATION ALGORITHMS

The orientation updates become more costly as the number of RAs grows, motivating two low-complexity alternatives. Candidate scanning searches a finite set of feasible directions, while block-based optimization shares one boresight among RAs in each block.

V. LOW-COMPLEXITY JOINT OPTIMIZATION ALGORITHMS

  • Orientation-update complexity increases with the number of RAs, motivating two methods that simplify orientation optimization while leaving other variable updates unchanged.The methods are discrete candidate scanning and antenna block-based optimization.Source: §V · p. 7

A. Discrete Candidate-Scanning Algorithm

  • Candidate scanning sequentially selects, for each RA, the objective-maximizing direction from a finite candidate set.User directions motivate the candidates, and infeasible directions are projected onto the feasible region.Source: A. Discrete Candidate-Scanning Algorithm · p. 7
  • Including the current orientation among the candidates ensures the objective value cannot decrease during an orientation update.The candidate set is evaluated with the other optimization variables fixed.Source: A. Discrete Candidate-Scanning Algorithm · p. 8

B. Antenna Block-Based Algorithm

The antenna block-based method reduces orientation variables by assigning a shared boresight to each block of RAs. Simulations report that the proposed joint design achieves the highest sum-rate among the compared schemes across the reported convergence, power, rotation-angle, directivity, and array-size evaluations.

B. Antenna Block-Based Algorithm

  • Each antenna block shares one boresight direction, reducing the number of orientation variables optimized.The method partitions the RAs at each BS into multiple blocks and optimizes each block's shared orientation.Source: B. Antenna Block-Based Algorithm · p. 8

VI. SIMULATION RESULTS

  • The simulations use six BSs in a regular hexagonal layout and 16 users, split evenly between terrestrial and aerial users.The terrestrial users are at 1.5 m; aerial-user heights are uniformly distributed over [40, 60] m.Source: §VI · p. 8

A. Performance of the Proposed Algorithm

  • The proposed algorithm achieves the highest sum-rate at convergence and consistently leads the benchmarks as transmit power increases.The convergence comparison includes Baselines 1–3; the transmit-power comparison includes all schemes.Source: A. Performance of the Proposed Algorithm · p. 9, Figure 4 · p. 9
  • As transmit power increases, the gap between the proposed algorithm and fixed-orientation Baseline 1 widens.The passage attributes this trend to orientation optimization becoming increasingly beneficial in the interference-limited scenario.Source: A. Performance of the Proposed Algorithm · p. 9
  • As the maximum rotation angle grows, optimized-orientation schemes increase in sum-rate and approach plateaus at approximately 1.25 rad.The proposed algorithm has the highest sum-rate among schemes with optimized RA orientations; fixed-orientation baselines remain invariant.Source: A. Performance of the Proposed Algorithm · pp. 9–10, Figure 5 · p. 9
  • As directivity increases, the proposed algorithm and Baselines 2, 4, and 5 improve, while fixed-orientation schemes decline after a slight gain at p = 1.At p = 0, orientation optimization provides no additional benefit; increasing directivity aggravates misalignment loss for fixed orientations.Source: A. Performance of the Proposed Algorithm · p. 10, Figure 6 · p. 10
  • Sum-rates rise with UPA dimension N, and the proposed algorithm remains highest across the considered array configurations.Its performance gap relative to Baseline 2 narrows as N increases, indicating a smaller user-association optimization benefit for larger arrays.Source: A. Performance of the Proposed Algorithm · p. 10, Figure 7 · p. 10

B. Performance of the Low-Complexity Algorithms

The low-complexity orientation schemes trade sum-rate for simpler orientation updates, while the proposed AO algorithm achieves the highest sum-rate among evaluated schemes across examined settings.

B. Performance of the Low-Complexity Algorithms

  • The evaluated low-complexity schemes share the same updates for U, W, A, and V, differing only in the update of F.They include antenna block-based, discrete candidate-scanning, and combined block-based candidate-scanning schemes.Source: B. Performance of the Low-Complexity… · p. 10
  • For a 4 × 4 UPA, the 1 × 1 scheme achieves the highest sum-rate, and performance loss increases with block size.The 2 × 2 scheme is closer to 1 × 1 than the 4 × 4 scheme.Source: B. Performance of the Low-Complexity… · pp. 10–11
  • Among the low-complexity schemes, antenna block-based orientation incurs the smallest loss, while discrete candidate scanning loses more than continuous adjustment.The comparison uses a 2 × 2 UPA; the block-based scheme uses 1 × 2 blocks.Source: B. Performance of the Low-Complexity… · p. 11, Figure 9 · p. 11

VII. CONCLUSION

  • The proposed AO algorithm achieves the highest sum-rate among evaluated schemes across the examined settings, while two lower-complexity algorithms retain most of the gain.Both reduce the computational cost of orientation optimization.Source: §VII · p. 11

APPENDIX A DERIVATION OF THE USER-ASSOCIATION PROJECTION

Appendix A derives the Euclidean projection used in the user-association update, whose subproblem is strictly convex and has a unique globally optimal solution.

APPENDIX A DERIVATION OF THE USER-ASSOCIATION PROJECTION

  • The projection subproblem P7 is strictly convex and has a unique globally optimal solution.The appendix derives the Euclidean projection presented in Eq. (28).Source: Appendix A · p. 11
  • The derivation uses KKT conditions and a closed-form expression for the updated association vector.It introduces Lagrange multipliers for constraints (53b) and (53c).Source: Appendix A · p. 11
  • The association update does not decrease the objective.Source: Appendix A · p. 12

APPENDIX B DERIVATION OF THE RA ORIENTATION GRADIENT

Appendix B derives the gradient of the WMMSE objective with respect to RA orientations, accounting for how an orientation changes the effective channel and received power.

APPENDIX B DERIVATION OF THE RA ORIENTATION GRADIENT

  • With U, W, A, and V fixed, only the MSE term depends on F, and the chain rule yields the orientation gradient of the WMMSE objective.Source: Appendix B · p. 12
  • An RA orientation affects only its associated channel entry, and only channels from BS b depend on f_b,m.The derivation uses the resulting received-power derivative and effective-channel gradient.Source: Appendix B · p. 12
  • Differentiating the channel expression with respect to f_b,m gives the orientation-dependent channel derivative used in the gradient.At the visibility boundary, the zero Clarke subgradient is selected when [x]+ = 0.Source: Appendix B · p. 12
Loading 2609.35010v1…