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Movable Antenna-Enhanced Multiuser Communication: Optimal Discrete Antenna Positioning and Beamforming

Yifei Wu, Dongfang Xu, Derrick Wing Kwan Ng, Wolfgang Gerstacker, Robert Schober

arXiv:2308.02304v1cs.ITeess.SPmath.IT

TL;DR

Fixed antenna locations limit the exploitation of spatial channel variation, while earlier movable-antenna designs assumed continuous motion or lacked joint optimality guarantees. This paper models discrete antenna movement and jointly optimizes antenna positions and beamforming with a globally convergent generalized Bender’s decomposition algorithm. In simulations, the proposed scheme outperforms three baseline schemes across the considered minimum-SINR range.

  • Problem

    Fixed-position antennas cannot fully exploit channel variation across the transmitter area, and earlier movable-antenna designs assumed unrestricted motion or could not guarantee joint optimality.

    Source: §I · p. 2

  • Method

    The paper models movable-antenna positions as discrete choices and jointly optimizes those positions and base-station beamforming using generalized Bender’s decomposition.

    Source: §I · p. 3

  • Results

    The proposed scheme outperforms the three baseline schemes across the considered range of minimum required SINR values.

    Source: §IV · pp. 15–16

  • Takeaways & Limitations

    The GBD-based algorithm obtains a globally optimal solution for the formulated discrete-position resource-allocation problem and can benchmark suboptimal designs.

    Source: §V · p. 17

Abstract

from arXiv · show

Movable antennas (MAs) are a promising paradigm to enhance the spatial degrees of freedom of conventional multi-antenna systems by flexibly adapting the positions of the antenna elements within a given transmit area. In this paper, we model the motion of the MA elements as discrete movements and study the corresponding resource allocation problem for MA-enabled multiuser multiple-input single-output (MISO) communication systems. Specifically, we jointly optimize the beamforming and the MA positions at the base station (BS) for the minimization of the total transmit power while guaranteeing the minimum required signal-to-interference-plus-noise ratio (SINR) of each individual user. To obtain the globally optimal solution to the formulated resource allocation problem, we develop an iterative algorithm capitalizing on the generalized Bender's decomposition with guaranteed convergence. Our numerical results demonstrate that the proposed MA-enabled communication system can significantly reduce the BS transmit power and the number of antenna elements needed to achieve a desired performance compared to state-of-the-art techniques, such as antenna selection. Furthermore, we observe that refining the step size of the MA motion driver improves performance at the expense of a higher computational complexity.

I. INTRODUCTION

Fixed antenna positions limit how well conventional MIMO can exploit spatial channel variation, while holographic MIMO’s large element count challenges practical implementation. This paper studies discrete movable-antenna positioning and beamforming jointly, seeking a globally optimal design for a more practical spatially adaptive system.

I. INTRODUCTION

  • Fixed antenna positions cannot fully exploit channel variation across the transmitter area, limiting system performance despite antenna selection’s reduction in RF-chain requirements.Holographic MIMO can exploit spatial degrees of freedom with densely spaced elements, but its large element count challenges channel estimation and data processing.Source: §I · p. 2
  • Movable antennas reposition a small number of elements to exploit spatial degrees of freedom, bridging holographic and conventional MIMO systems.Their reduced element count also lowers signal-processing complexity compared with holographic MIMO.Source: §I · pp. 2–3
  • Prior joint designs assumed freely adjustable positions and used AO or gradient descent, which cannot guarantee joint optimality and may depend on initialization.Practical electromechanical controls instead move antennas in discrete increments, giving the transmitter area finite spatial resolution.Source: §I · p. 3
  • The paper introduces a discrete movable-antenna position model and an iterative generalized Bender’s decomposition algorithm to obtain a globally optimal joint design.The algorithm is intended to provide a performance benchmark for suboptimal designs.Source: §I · p. 3

A. Channel Model

The channel model represents each movable antenna’s channel as a choice among responses at discrete candidate locations. It combines those channel responses with binary position selections and models the received signal using the resulting channel matrix and linear beamforming.

A. Channel Model

  • The model assumes perfect channel state information at the base station and restricts each antenna to one of N candidate locations on a grid with spacing d.The channel from each antenna to the users depends on its selected position; d reflects the motion increment of the electromechanical device.Source: A. Channel Model · p. 5, H “ ˆHB, (2) · p. 5

H “ ˆHB, (2)

  • The stacked candidate-location channel matrix and binary selection variables form the channel matrix for the chosen antenna positions.Each binary vector selects one candidate location for its corresponding movable antenna.Source: A. Channel Model · p. 5, H “ ˆHB, (2) · p. 6
  • Robust design for imperfect channel state information is left as future work.The stated model assumes the base station has perfect channel state information over the transmitter area.Source: H “ ˆHB, (2) · p. 5, A. Channel Model · p. 5
  • The received signal for user k is modeled as its channel row applied to the beamformed symbols, plus additive noise.The beamforming matrix contains one linear beamforming vector per user.Source: H “ ˆHB, (2) · p. 6

B. Resource Allocation Problem

The resource allocation problem minimizes BS transmit power while meeting every user’s minimum SINR. It models antenna-position choices as discrete assignments and enforces minimum spacing between MA elements.

B. Resource Allocation Problem

III. SOLUTION OF OPTIMIZATION PROBLEM

The solution section reformulates the joint design problem to make generalized Benders decomposition applicable. Its first reformulation addresses a coupled constraint that otherwise prevents direct application of GBD.

III. SOLUTION OF OPTIMIZATION PROBLEM

  • The solution approach first transforms problem (9) into an equivalent MINLP as the foundation for the GBD algorithm.Source: §III · p. 7

A. Problem Reformulation

  • The coupled constraint C2 prevents direct application of GBD, so the authors reformulate it to enable the method.Source: A. Problem Reformulation · p. 7
  • Lemma 1 replaces equality constraint C2 with equivalent LMI constraints using positive semidefinite auxiliary variables U and V.Source: A. Problem Reformulation · p. 8

U ´ BBH˘

The reformulation converts key constraints into forms suitable for generalized Benders decomposition. With discrete variables fixed the resulting MINLP is convex in its continuous variables, supporting guaranteed global convergence of GBD.

U ´ BBH˘

  • The minimum-distance constraint C5 is reformulated as three linear inequalities using binary auxiliary variables.Source: U ´ BBH˘ · p. 8
  • A phase rotation preserves feasibility and objective value, allowing the nonconvex SINR constraint to be replaced by two equivalent convex constraints.Source: U ´ BBH˘ · p. 9
  • The reformulated resource allocation problem is stated with constraints C1a, C1b, C2a, C2b, C3, C4, C5a, C5b, and C5c.Source: U ´ BBH˘ · p. 9
  • GBD is guaranteed to converge globally because the reformulated MINLP is convex in continuous variables when discrete variables are fixed, and linear in discrete variables when continuous variables are fixed.Source: U ´ BBH˘ · p. 9

B. GBD Procedure

The GBD procedure splits the MINLP into primal and master problems, updating objective bounds iteratively.

B. GBD Procedure

  • Each GBD iteration solves a primal problem with discrete variables fixed to update the upper bound, then a master problem with continuous variables fixed to update the lower bound.Source: B. GBD Procedure · p. 9

1) Primal Problem:

The primal problem evaluates each candidate discrete antenna configuration through a convex optimization problem, with a separate feasibility check for configurations that fail the original problem.

1) Primal Problem:

  • For each iteration, the primal problem uses the master problem’s discrete variables to evaluate the candidate antenna configuration.Its decision variables include X, W, U, and V.Source: 1) Primal Problem · p. 10
  • When the primal problem is feasible, its convex formulation can be solved with a standard convex solver such as CVX.The optimal solution and associated Lagrangian multipliers are used in the algorithm.Source: 1) Primal Problem · p. 10
  • If the primal problem is infeasible, an always-feasible convex check generates a cut that excludes the infeasible discrete configuration from later master problems.The check uses auxiliary variable λ.Source: 1) Primal Problem · pp. 10–11

2) Master Problem:

The master problem uses nonlinear convex duality and cuts derived from feasible and infeasible primal iterations to select discrete antenna variables.

2) Master Problem:

  • The master problem is formulated using nonlinear convex duality and an epigraph variable η.The epigraph form includes the problem constraints and the cuts accumulated across iterations.Source: 2) Master Problem · p. 11
  • Optimality and feasibility cuts are formed from the multipliers of feasible primal problems and infeasible feasibility checks, respectively.These cuts constrain the master problem across the corresponding iterations.Source: 2) Master Problem · p. 11
  • The master problem’s cut inequalities can be recast as linear inequalities, yielding a mixed integer linear program solvable by standard MILP solvers.The optimal master solution supplies the next discrete variables.Source: 2) Master Problem · p. 12

3) Overall Algorithm:

The algorithm alternates between primal evaluation and a master problem, using optimality or feasibility cuts to update bounds until the global-optimality gap meets the convergence tolerance.

3) Overall Algorithm:

  • Feasible primal iterations generate optimality cuts and update the upper bound, while infeasible iterations use a feasibility check to generate feasibility cuts.The master problem then updates the discrete variables and the lower bound.Source: 3) Overall Algorithm · pp. 12–13
  • The algorithm reduces the gap between its upper and lower bounds and is guaranteed to reach the global optimum in finitely many iterations for a given tolerance.The lower bound comes from the master problem’s objective value.Source: 3) Overall Algorithm · p. 13
  • Although worst-case complexity scales exponentially with the number of MA elements, simulations converged in significantly fewer iterations than exhaustive search.Source: 3) Overall Algorithm · p. 13

IV. NUMERICAL RESULTS

The simulations compare the proposed design with three baselines in a four-antenna, four-user system. The proposed design uses less transmit power across the reported comparisons, while performance depends on antenna-positioning precision and transmitter-area size.

IV. NUMERICAL RESULTS

  • The simulations use four BS MA elements and four single-antenna users at 5 GHz, with a square transmitter area quantized into discrete positions.The tested step sizes are 0.01 m and λ/2 = 0.03 m.Source: §IV · p. 14
  • Across the full SINR range, the proposed scheme outperforms all three baselines; its optimal solution has a 4 dB advantage over AO, which gains 5 dB over fixed-array baseline 1.Increasing the MA step size to λ/2 incurs roughly 2 dB performance loss.Source: Figure 2 · p. 15, §IV · pp. 15–16
  • The baselines are randomly fixed antenna positions, antenna selection over a fixed UPA, and an alternating-optimization method that updates beamforming and antenna positions.Source: §IV · p. 15
  • As transmitter area grows, power falls for the proposed and AO schemes but not the two fixed-position baselines; the proposed design saturates at normalized area size 3.For larger areas, the gap between the proposed design and AO widens.Source: Figure 3 · p. 16, §IV · pp. 16–17

V. CONCLUSION

The paper formulates optimal resource allocation for a multiuser MA-enabled downlink MISO system with discrete antenna motion and reports a globally optimal iterative solution. Simulations show the system’s superiority and the algorithm’s optimality; future work will address computational efficiency under imperfect CSI.

  • The iterative GBD-based algorithm obtains a globally optimal solution to the formulated resource-allocation problem.Source: §V · p. 17
  • The work presents the first optimal resource-allocation design for multiuser MA-enabled downlink MISO, modeling antenna movement as discrete to reflect hardware limits.Source: §V · p. 17
  • The optimization minimizes BS transmit power while guaranteeing each user’s minimum SINR, and simulations show system superiority and algorithm optimality.Source: §V · p. 17
  • Future work will develop a computationally efficient suboptimal scheme for practical MA-enabled systems with imperfect CSI.Source: §V · p. 17
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