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6D Movable Antenna Enhanced Wireless Network Via Discrete Position and Rotation Optimization

Xiaodan Shao, Rui Zhang, Qijun Jiang, Robert Schober

arXiv:2403.17122v1cs.ITeess.SP

TL;DR

Discrete movement constraints make continuously adjustable 6DMA surfaces difficult to implement, motivating capacity optimization over finite positions and rotations. The paper jointly optimizes multiple 6DMA surfaces using offline and online methods for known and unknown channel statistics, and simulations show significant capacity gains over fixed-position antennas and limited-mobility 6DMAs.

  • Problem

    Discrete position and rotation constraints limit practical 6DMA deployment, requiring joint optimization of finite surface configurations to maximize average network capacity.

  • Method

    The paper jointly selects discrete 3D positions and rotations for multiple 6DMA surfaces, using Monte Carlo offline optimization with known channel statistics and CSM-based online optimization when statistics are unavailable.

  • Results

    Simulations show that the proposed offline and online algorithms significantly improve network capacity over fixed-position antennas and 6DMAs with limited movability under discrete movement constraints.

  • Takeaways & Limitations

    Discrete 6DMA movement can still provide substantial capacity gains by adapting surface positions and rotations to users’ spatial distribution.

Abstract

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Six-dimensional movable antenna (6DMA) is an effective approach to improve wireless network capacity by adjusting the 3D positions and 3D rotations of distributed antenna surfaces based on the users' spatial distribution and statistical channel information. Although continuously positioning/rotating 6DMA surfaces can achieve the greatest flexibility and thus the highest capacity improvement, it is difficult to implement due to the discrete movement constraints of practical stepper motors. Thus, in this paper, we consider a 6DMA-aided base station (BS) with only a finite number of possible discrete positions and rotations for the 6DMA surfaces. We aim to maximize the average network capacity for random numbers of users at random locations by jointly optimizing the 3D positions and 3D rotations of multiple 6DMA surfaces at the BS subject to discrete movement constraints. In particular, we consider the practical cases with and without statistical channel knowledge of the users, and propose corresponding offline and online optimization algorithms, by leveraging the Monte Carlo and conditional sample mean (CSM) methods, respectively. Simulation results verify the effectiveness of our proposed offline and online algorithms for discrete position/rotation optimization of 6DMA surfaces as compared to various benchmark schemes with fixed-position antennas (FPAs) and 6DMAs with limited movability. It is shown that 6DMA-BS can significantly enhance wireless network capacity, even under discrete position/rotation constraints, by exploiting the spatial distribution characteristics of the users.

I. INTRODUCTION

The paper adapts 6DMA base stations to practical discrete position and rotation constraints, jointly optimizing multiple surfaces to maximize average network capacity with or without statistical channel knowledge. It develops offline and online optimization methods and evaluates them against fixed-position and limited-mobility architectures.

  • Fixed-position antennas cannot fully adapt to non-uniform user distributions without adding antennas, while added antennas increase hardware, energy, and processing costs.
  • 6DMA surfaces independently adjust their 3D positions and rotations according to users’ spatial distribution and statistical channel information.Each surface connects to a base-station CPU through an extendable, rotatable rod controlled by motors.
  • Discrete stepper-motor movement limits spatial degrees of freedom and can reduce capacity relative to continuously adjustable 6DMA surfaces.This constraint motivates joint optimization of surface positions and rotations in the discrete case.
  • The proposed system selects each surface’s position and rotation from predefined finite sets while maximizing average network capacity under discrete movement constraints.The BS contains multiple finite-size antenna surfaces, such as uniform planar arrays, that can be adjusted independently.
  • The offline algorithm uses statistical channel knowledge and Monte Carlo samples, whereas the online algorithm uses measured achievable sum-rates and conditional sample mean optimization without prior channel statistics.The online method generates feasible discrete position-rotation combinations and optimizes directly from measured data.
  • Simulations show that the proposed 6DMA-BS significantly improves network capacity over fixed-position antennas and 6DMAs with limited movability, even under discrete constraints.The reported gains exploit users’ spatial distribution characteristics.

1) Discrete Rotation Constraints to Avoid Signal Reflection:

The paper imposes rotation constraints on 6DMA surfaces to avoid mutual signal reflections, expressing these constraints through surface normal vectors and indicator variables.

  • 1) Discrete Rotation Constraints to Avoid Signal Reflection:: Rotation constraints are imposed to avoid mutual signal reflections between any two 6DMA surfaces.
  • 1) Discrete Rotation Constraints to Avoid Signal Reflection:: The outward normal of each surface is obtained by rotating its local normal vector into the global coordinate system.
  • 1) Discrete Rotation Constraints to Avoid Signal Reflection:: The rotation constraint is reformulated in terms of position and rotation indicator vectors for discrete optimization.

2) Discrete Rotation Constraints to Avoid Signal Blockage:

The channel model incorporates discrete 6DMA positions and rotations while accounting for surface placement, multiuser channels, and multipath propagation. It also specifies geometric direction variables used to characterize each channel path.

  • A minimum distance dmin separates 6DMA surfaces at different discrete positions to avoid overlap and mutual coupling.
  • The system models uplink transmission from a random number of spatially distributed single-antenna users to the 6DMA-BS.
  • Each 6DMA surface selects a discrete position and rotation, and the selected vectors describe the configuration of all B surfaces.
  • The multiple-access channel matrix aggregates the channels from all K users to all antennas on the B 6DMA surfaces.
  • Each user channel is modeled as a multipath channel whose path coefficients, directions, steering vectors, and effective antenna gains depend on the selected surface configuration.
  • For each path, azimuth and elevation angles define a pointing vector that characterizes its direction relative to the BS.

1) 6D Steering Vector:

The 6D steering vector describes how a signal arriving along a given path couples to a 6DMA surface. It depends on the surface’s selected position and rotation and on the carrier wavelength.

  • The path pointing vector is defined from the path’s elevation and azimuth angles.
  • The 6D steering vector combines the surface geometry with the incoming signal direction for each user and channel path.
  • The steering vector varies with the 6DMA surface position q_i_b and rotation u_j_b, with λ denoting the carrier wavelength.

2) Effective Antenna Gain:

The effective antenna gain is obtained by expressing each path’s arrival direction in the local coordinate system of a 6DMA surface. The resulting gain depends on the adopted antenna radiation pattern.

  • The incoming direction is projected onto each 6DMA surface’s local coordinate system to determine its local arrival direction.
  • The local elevation and azimuth angles of the signal direction are derived from that projection.
  • The effective antenna gain is then defined for each surface and channel-path direction in linear scale.
  • The gain is also represented in dBi and depends on the radiation pattern of the adopted antenna.

III. PROBLEM FORMULATION

The paper formulates discrete 6DMA configuration as an average-capacity optimization over users and channels. It represents candidate surface channels, enforces practical selection constraints, and proposes separate solutions depending on statistical channel knowledge availability.

  • Candidate multiple-access channels are defined for every discrete position and rotation available to a 6DMA surface.
  • The received-signal model uses user transmit signals, common transmit power p, and additive white Gaussian noise at the BS.
  • Average network capacity is used because both the number of users and their locations are random.
  • The optimization selects position and rotation indicator vectors for all B surfaces to maximize average network capacity under discrete movement constraints.
  • Binary indicator constraints encode whether each discrete position or rotation is selected, while geometric constraints prevent reflections, blockage, overlap, and mutual coupling.
  • When statistical channel knowledge is available, the average capacity is approximated offline using Monte Carlo simulation; otherwise, an online method uses measured achievable sum rates for discrete configurations.

IV. OFFLINE OPTIMIZATION WITH STATISTICAL CHANNEL KNOWLEDGE

The offline method approximates average network capacity from Monte Carlo channel realizations under known statistical channel information, then solves the resulting discrete position/rotation optimization through reformulation and relaxation. The problem is a difficult non-convex integer program because multiple surfaces require jointly selecting constrained positions and rotations.

  • Offline optimization with statistical channel knowledge: Monte Carlo realizations approximate average network capacity when users’ statistical channel knowledge is available.The method generates independent realizations of user numbers and locations, then averages the corresponding achievable rates.
  • Offline optimization with statistical channel knowledge: The resulting formulation is a non-convex integer program that becomes difficult to solve efficiently as the numbers of candidate positions, rotations, or surfaces increase.Its objective and constraints are non-convex in binary position and rotation indicators.
  • Offline optimization with statistical channel knowledge: The optimization jointly selects discrete 3D positions and rotations for multiple 6DMA surfaces under practical constraints.Unlike fixed-position antenna selection, both surface positions and rotations must be selected for several surfaces.
  • Offline optimization with statistical channel knowledge: The method reformulates non-convex quadratic constraints as convex linear inequalities using binary auxiliary variables.The reformulation introduces auxiliary variables to represent products of binary indicators.
  • Offline optimization with statistical channel knowledge: Linear programming relaxation and a conditional-gradient solution provide a sub-optimal procedure for the reformulated problem.Binary variables are relaxed to [0,1], after which the relaxed problem is solved by the conditional gradient method.
  • Offline optimization with statistical channel knowledge: The proposed offline algorithm has complexity order O(TfM 2N 2L2 ¯KΩ) and is convergent because its gradient-based search does not decrease the objective.Tf is the maximum iteration count, while ¯K is the largest user count among Monte Carlo realizations.

V. ONLINE OPTIMIZATION WITHOUT STATISTICAL CHANNEL KNOWLEDGE

The online method optimizes discrete 6DMA positions and rotations without prior statistical channel knowledge. It uses achievable sum-rate measurements at the base station for different discrete configurations.

  • Online optimization without statistical channel knowledge: The online algorithm solves discrete position and rotation optimization without any a priori statistical channel knowledge.It bases the optimization on measured achievable sum-rate values for different combinations of surface configurations.

A. Generation of Discrete Positions and Rotations

The paper generates feasible discrete position-rotation pairs for 6DMA surfaces using points on a spherical surface and associated orientation constructions. The generated set is designed to respect minimum-distance and practical movement constraints.

  • Generation of Discrete Positions and Rotations: The generated position-rotation set contains |V|c = ML pairs, where M is the number of positions and L is the number of rotations per position.Each pair combines one discrete position with one corresponding discrete rotation.
  • Generation of Discrete Positions and Rotations: The generated position-rotation pairs satisfy the practical discrete position and rotation constraints required by the 6DMA-BS.The construction is used as the feasible configuration set for subsequent optimization.
  • Generation of Discrete Positions and Rotations: Discrete positions are uniformly generated on a spherical surface using the Fibonacci Sphere scheme.The sphere is centered at the CPU and uses the largest possible radius within the site space.
  • Generation of Discrete Positions and Rotations: The number of generated positions can be selected to guarantee a minimum distance between any two positions.This construction supports the required spatial separation constraint.
  • Generation of Discrete Positions and Rotations: For each discrete position, rotations are constructed by aligning local axes with spherical-coordinate basis directions or with a spatial-surface normal.These alternatives define feasible orientation frames for the surface at the selected position.

B. CSM-Based Online Algorithm

The CSM-based online algorithm estimates the performance of discrete position-rotation pairs from measured sum-rates and selects configurations using conditional sample means. Its computational complexity is dominated by these CSM computations.

  • CSM-based online algorithm: The online algorithm randomly samples sets of B discrete position-rotation pairs and measures each set’s achievable sum-rate.Positions and rotations are selected with equal probability from the offline-generated discrete sets.
  • CSM-based online algorithm: Conditional sample means estimate average sum-rate performance for each position-rotation pair conditioned on that pair appearing in sampled sets.If no sampled set contains a pair, its conditional sample mean is set to zero.
  • CSM-based online algorithm: Position and rotation utility values derived from the conditional sample means determine the selected indices for all 6DMA surfaces.The algorithm separately determines optimized position and rotation indices from these utilities.
  • CSM-based online algorithm: The CSM-based online algorithm has complexity order O(TB), dominated by computing conditional sample means for T sampled configurations and B surfaces.This complexity is stated for the sum-rate-based implementation.

VI. SIMULATION RESULTS

Simulations evaluate the proposed offline and CSM-based online 6DMA optimization under discrete movement constraints across user distributions and system settings. Network capacity generally improves with greater movement flexibility and exceeds limited-movability benchmarks, especially for spatially non-uniform users.

  • Simulation setup: The simulations use LoS channels, randomly generated users from an NHPP spatial model, and hotspot regions within a 3D spherical-annulus coverage area.The offline method uses 100 Monte Carlo realizations, while the online method uses sample size T = M^2L^2.
  • Offline optimization: The proposed offline algorithm’s average network capacity increases with transmit power and with the numbers of discrete positions M and rotations L.More choices provide additional spatial degrees of freedom for adapting directional antennas to non-uniform user distributions.
  • Online optimization: The CSM-based online algorithm also benefits from larger M and L, but using more movable surfaces can improve flexibility while increasing movement-hardware and power-consumption costs.With NB = 64 total antennas, the cited comparison reports N = 4 performing worse than N = 1 because the latter provides more surfaces for movement.
  • Algorithm comparison: The offline algorithm outperforms the online algorithm because it uses more discrete movement options and statistical channel information, but it has higher computational complexity.The compared settings are M = 600, L = 27 offline and M = 100, L = 2 online.
  • Benchmark comparison: Both proposed algorithms significantly outperform benchmark schemes with limited position or rotation adaptability, with larger gaps as the average user count increases.The larger gaps are attributed to improved multiuser-interference mitigation in more interference-limited systems.
  • User spatial distribution: All schemes deteriorate as the regular-user proportion ξ increases, while the proposed algorithms retain an advantage whose gap grows for more spatially non-uniform user distributions.The results indicate stronger effectiveness when users are spatially clustered or diverse rather than uniformly distributed.

VII. CONCLUSION

The paper jointly optimizes discrete 3D positions and rotations of 6DMA surfaces to maximize average network capacity under practical movement constraints. Monte Carlo offline and CSM-based online algorithms achieve significant gains over fixed-position and limited-movability benchmarks by exploiting user spatial distributions.

  • Conclusion: The study jointly optimizes 6DMA-surface positions and rotations under discrete movement constraints for cases with known and unknown channel statistics.It applies Monte Carlo and conditional sample mean methods to the corresponding offline and online problems.
  • Conclusion: Simulations show significant capacity gains over fixed-position antennas and 6DMAs with limited movability.The gains arise from selecting antenna positions and rotations according to spatial user distributions, explicitly or implicitly depending on channel knowledge.
  • Conclusion: The results present 6DMA-BS as a practical way to enhance MIMO capacity without increasing the number of antennas.This conclusion is stated for future-generation wireless networks under the studied discrete position and rotation constraints.
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