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arxiv: 2512.21660 · v1 · submitted 2025-12-25 · 💻 cs.IT · eess.SP· math.IT

Near-Field Communication with Massive Movable Antennas: An Electrostatic Equilibrium Perspective

Shicong Liu , Xianghao Yu , Shenghui Song , Khaled B. Letaief This is my paper

Pith reviewed 2026-05-16 19:24 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords movable antennasnear-field communicationantenna placementmassive MIMOweighted Fekete problemelectrostatic equilibriumODE solutionsspectral efficiency
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The pith

Optimal positions for massive movable antennas in near-field systems are the roots of polynomial solutions to ODEs from an electrostatic equilibrium reformulation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that antenna placement in large-scale near-field MIMO can be recast as a weighted Fekete problem in the normalized angular domain. This reformulation turns the optimization into an electrostatic equilibrium problem whose solutions are characterized by the roots of polynomials that satisfy specific ordinary differential equations. An efficient two-step eigenvalue decomposition computes these positions, and an asymptotic analysis supplies a closed-form expression that remains close to the optimum for large arrays. The approach yields higher spectral efficiency by better exploiting spatial degrees of freedom while remaining robust to parameter variations.

Core claim

The central claim is that the near-field antenna placement problem, once expressed in the angular domain, becomes a weighted Fekete problem whose optimality condition is exactly the electrostatic equilibrium condition. The optimal positions are therefore the roots of the polynomial solutions to the associated ordinary differential equations in the normalized angular variable. A two-step eigenvalue decomposition extracts these roots directly, and the limiting case of infinite array size produces an explicit closed-form placement that approaches the theoretical optimum.

What carries the argument

The weighted Fekete problem in the normalized angular domain, whose equilibrium condition is solved by the roots of polynomials satisfying specific ODEs.

If this is right

  • The scheme efficiently harnesses the spatial DoFs of near-field channels and produces prominent gains in spectral efficiency.
  • The method maintains robustness against mismatches in system parameters such as distance or frequency.
  • The asymptotic closed-form solution closely approaches the theoretical optimum across a wide range of practical scenarios.
  • Optimal positions can be obtained at low complexity via a two-step eigenvalue decomposition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The electrostatic-equilibrium view may extend to placement problems in other wave-propagation domains such as acoustics or optics.
  • The ODE characterization could enable closed-form sensitivity analysis of optimal positions with respect to carrier frequency or user distance.
  • The efficient solver might support real-time reconfiguration when users or scatterers move.
  • The link to Fekete points suggests possible connections to optimal sampling or quadrature designs in signal processing.

Load-bearing premise

The reformulation of the near-field antenna placement problem into a weighted Fekete problem in the angular domain accurately captures the spatial degrees of freedom without material approximation error for practical channel models and array sizes.

What would settle it

For a moderate-sized array and a concrete near-field channel realization, compare the spectral efficiency achieved by the ODE-root positions against the efficiency obtained from direct numerical maximization of the mutual information; any consistent gap would falsify the claim that the equilibrium solution is optimal.

Figures

Figures reproduced from arXiv: 2512.21660 by Khaled B. Letaief, Shenghui Song, Shicong Liu, Xianghao Yu.

Figure 1
Figure 1. Figure 1: (a) The considered near-field communication scenario under the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The relationships among antenna positions [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cases of UE positions in the near-field region: Case I: The position ˜˜ [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The approximation error between Hx in (5) and the truncated approximation Hx = PMDT in (22). • Random: In each simulation step, 1, 000 antenna place￾ment patterns are randomly generated at the BS for SE evaluation. • Greedy Selection [42]: The antenna positions are suc￾cessively selected from GAS = 2M uniform grids by orthogonal matching pursuit. • Gradient Descent (GD) [15]: The antenna positions are obta… view at source ↗
Figure 7
Figure 7. Figure 7: The illustration of the asymptotic behavior with respect to (a) centroid [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The antenna position patterns of different antenna placement schemes [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

Recent advancements in large-scale position-reconfigurable antennas have opened up new dimensions to effectively utilize the spatial degrees of freedom (DoFs) of wireless channels. However, the deployment of existing antenna placement schemes is primarily hindered by their limited scalability and frequently overlooked near-field effects in large-scale antenna systems. In this paper, we propose a novel antenna placement approach tailored for near-field massive multiple-input multiple-output systems, which effectively exploits the spatial DoFs to enhance spectral efficiency. For that purpose, we first reformulate the antenna placement problem in the angular domain, resulting in a weighted Fekete problem. We then derive the optimality condition and reveal that the {optimal} antenna placement is in principle an electrostatic equilibrium problem. To further reduce the computational complexity of numerical optimization, we propose an ordinary differential equation (ODE)-based framework to efficiently solve the equilibrium problem. In particular, the optimal antenna positions are characterized by the roots of the polynomial solutions to specific ODEs in the normalized angular domain. By simply adopting a two-step eigenvalue decomposition (EVD) approach, the optimal antenna positions can be efficiently obtained. Furthermore, we perform an asymptotic analysis when the antenna size tends to infinity, which yields a closed-form solution. Simulation results demonstrate that the proposed scheme efficiently harnesses the spatial DoFs of near-field channels with prominent gains in spectral efficiency and maintains robustness against system parameter mismatches. In addition, the derived asymptotic closed-form {solution} closely approaches the theoretical optimum across a wide range of practical scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that the near-field massive movable-antenna placement problem can be reformulated in the normalized angular domain as a weighted Fekete problem whose optimality condition is an electrostatic equilibrium; the equilibrium positions are the roots of polynomials satisfying specific ODEs, which are obtained via a two-step EVD procedure, and an asymptotic closed-form solution is derived for large arrays, yielding spectral-efficiency gains over conventional placements as confirmed by simulations.

Significance. If the angular-domain reformulation holds without material error, the work supplies a principled, low-complexity method for exploiting spatial DoFs in near-field mMIMO together with an explicit ODE characterization and a parameter-free asymptotic limit; these elements would constitute a concrete advance in scalable antenna-position optimization.

major comments (3)
  1. [reformulation and optimality-condition derivation] The central mapping of the near-field channel matrix to a weighted Fekete problem in the normalized angular domain (abstract and the reformulation section) implicitly discards residual spherical-wave curvature and Fresnel-phase terms; for the array sizes where gains are claimed (N ≈ 64–256), even modest radial dependence can displace equilibrium roots by fractions of a wavelength and thereby alter the reported spectral-efficiency improvement.
  2. [ODE framework and EVD method] The ODE characterization and the subsequent two-step EVD procedure are presented as exact solutions to the equilibrium problem, yet the manuscript provides neither the full derivation of the ODE coefficients from the Fekete potential nor a direct numerical check that the obtained roots satisfy the original optimality condition for finite-N near-field channels.
  3. [asymptotic analysis] The asymptotic closed-form solution is asserted to approach the theoretical optimum across practical scenarios, but no quantitative error bound or convergence rate with respect to array size is supplied, leaving the practical utility of the closed form unsubstantiated beyond the reported simulations.
minor comments (2)
  1. [system model and reformulation] Notation for the normalized angular variable and the weighting function in the Fekete problem should be introduced with explicit definitions before the optimality condition is stated.
  2. [numerical results] Simulation figures would benefit from error bars or multiple random channel realizations to demonstrate robustness against the claimed parameter mismatches.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the insightful and constructive comments on our manuscript. We address each of the major comments point by point below, indicating the revisions we intend to make to improve the clarity and rigor of the work.

read point-by-point responses

  1. Referee: The central mapping of the near-field channel matrix to a weighted Fekete problem in the normalized angular domain (abstract and the reformulation section) implicitly discards residual spherical-wave curvature and Fresnel-phase terms; for the array sizes where gains are claimed (N ≈ 64–256), even modest radial dependence can displace equilibrium roots by fractions of a wavelength and thereby alter the reported spectral-efficiency improvement.

    Authors: We recognize that the reformulation in the normalized angular domain involves approximating the near-field effects by emphasizing angular components. This mapping is derived under the assumption that the dominant phase variations are captured in the angular domain for large arrays. To strengthen the manuscript, we will add a dedicated subsection analyzing the approximation error due to residual spherical-wave curvature for N ranging from 64 to 256, including comparisons with exact near-field models and its impact on spectral efficiency gains. revision: yes

  2. Referee: The ODE characterization and the subsequent two-step EVD procedure are presented as exact solutions to the equilibrium problem, yet the manuscript provides neither the full derivation of the ODE coefficients from the Fekete potential nor a direct numerical check that the obtained roots satisfy the original optimality condition for finite-N near-field channels.

    Authors: We will provide the full derivation of the ODE coefficients from the weighted Fekete potential in an appendix of the revised manuscript. Additionally, we will include numerical results demonstrating that the polynomial roots obtained via the two-step EVD procedure satisfy the electrostatic equilibrium condition (i.e., the optimality condition) for finite-N near-field channels across the simulated scenarios. revision: yes

  3. Referee: The asymptotic closed-form solution is asserted to approach the theoretical optimum across practical scenarios, but no quantitative error bound or convergence rate with respect to array size is supplied, leaving the practical utility of the closed form unsubstantiated beyond the reported simulations.

    Authors: We agree that providing quantitative error bounds would better substantiate the asymptotic analysis. In the revised version, we will derive and present an error bound and convergence rate for the closed-form solution as the array size tends to infinity, supported by additional analytical and numerical results. revision: yes

Circularity Check

0 steps flagged

No circularity: reformulation to weighted Fekete yields independent equilibrium characterization and separate asymptotic solution

full rationale

The derivation begins with a reformulation of the antenna placement objective into a weighted Fekete problem in the normalized angular domain. From this known potential-theoretic object the paper derives the electrostatic equilibrium optimality condition and then constructs an ODE whose polynomial roots give the positions. The asymptotic closed-form is obtained by a separate limiting argument as array size tends to infinity. None of these steps reduces to a fitted parameter, a self-citation chain, or a renaming of the input; each is a standard mathematical consequence of the preceding reformulation. The provided text contains no load-bearing self-citations or fitted-input predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the near-field placement problem can be exactly recast as a weighted Fekete problem in the angular domain; no free parameters or invented physical entities are introduced.

axioms (1)
  • domain assumption The antenna placement problem can be reformulated in the angular domain as a weighted Fekete problem.
    This reformulation is invoked to derive the optimality condition that reveals the electrostatic equilibrium.

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 1 internal anchor

  1. [1]

    A tutorial on extremely large-scale MIMO for 6G: Fundamentals, signal processing, and applications,

    Z. Wanget al., “A tutorial on extremely large-scale MIMO for 6G: Fundamentals, signal processing, and applications,”IEEE Commun. Surveys Tuts., vol. 26, no. 3, pp. 1560–1605, thirdquarter 2024

    work page 2024

  2. [2]

    Alternating minimization algorithms for hybrid precoding in millimeter wave MIMO systems,

    X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaief, “Alternating minimization algorithms for hybrid precoding in millimeter wave MIMO systems,” IEEE J. Sel. Topics Signal Process., vol. 10, no. 3, pp. 485–500, Apr. 2016

    work page 2016

  3. [3]

    Intelligent surfaces empowered wireless network: Recent advances and the road to 6G,

    Q. Wuet al., “Intelligent surfaces empowered wireless network: Recent advances and the road to 6G,”Proc. IEEE, vol. 112, no. 7, pp. 724–763, July 2024

    work page 2024

  4. [4]

    Reconfig- urable holographic surface-enabled multi-user wireless communications: Amplitude-controlled holographic beamforming,

    R. Deng, B. Di, H. Zhang, Y . Tan, and L. Song, “Reconfig- urable holographic surface-enabled multi-user wireless communications: Amplitude-controlled holographic beamforming,”IEEE Trans. Wireless Commun., vol. 21, no. 8, pp. 6003–6017, Aug. 2022

    work page 2022

  5. [5]

    Tse and P

    D. Tse and P. Viswanath,Fundamentals of Wireless Communication. USA: Cambridge University Press, 2005

    work page 2005

  6. [6]

    Near-field channel estimation in mixed LoS/NLoS environments for extremely large-scale MIMO systems,

    Y . Lu and L. Dai, “Near-field channel estimation in mixed LoS/NLoS environments for extremely large-scale MIMO systems,”IEEE Trans. Commun., vol. 71, no. 6, pp. 3694–3707, June 2023

    work page 2023

  7. [7]

    Non-uniform array design for robust LoS MIMO via convex optimization,

    M. Palaiologos, M. H. C. Garc ´ıa, A. Kakkavas, R. A. Stirling-Gallacher, and G. Caire, “Non-uniform array design for robust LoS MIMO via convex optimization,” inProc. IEEE 34th Int. Symp. Pers. Indoor Mob. (PIMRC), Toronto, ON, Canada, Sept. 2023, pp. 1–6

    work page 2023

  8. [8]

    Array geometry optimization for region-of-interest near-field beamforming,

    R. Moisseev, G. Itzhak, and I. Cohen, “Array geometry optimization for region-of-interest near-field beamforming,” inProc. IEEE Int. Conf. Acoustics, Speech Signal Process. (ICASSP), Seoul, South Korea, Apr. 2024, pp. 576–580

    work page 2024

  9. [9]

    Movable antenna enabled near-field communications: Channel modeling and performance opti- mization,

    L. Zhu, W. Ma, Z. Xiao, and R. Zhang, “Movable antenna enabled near-field communications: Channel modeling and performance opti- mization,”IEEE Trans. Commun., pp. 1–1, 2025,early access

    work page 2025

  10. [10]

    Movable antennas for wireless commu- nication: Opportunities and challenges,

    L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless commu- nication: Opportunities and challenges,”IEEE Commun. Mag., vol. 62, no. 6, pp. 114–120, June 2024

    work page 2024

  11. [11]

    Fluid antenna systems,

    K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y . Zhang, “Fluid antenna systems,”IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950–1962, Mar. 2021

    work page 1950

  12. [12]

    Pinching- antenna systems: Architecture designs, opportunities, and outlook,

    Y . Liu, Z. Wang, X. Mu, C. Ouyang, X. Xu, and Z. Ding, “Pinching- antenna systems: Architecture designs, opportunities, and outlook,” IEEE Commun. Mag., pp. 1–7, 2025,early access

    work page 2025

  13. [13]

    Movable antenna-aided hybrid beamforming for multi- user communications,

    Y . Zhanget al., “Movable antenna-aided hybrid beamforming for multi- user communications,”IEEE Trans. Veh. Technol., pp. 1–6, 2025,early access

    work page 2025

  14. [14]

    A pattern- reconfigurable antenna for single-RF 5G millimeter-wave communica- tions,

    S. Tang, Y . Zhang, Z. Han, C.-Y . Chiu, and R. Murch, “A pattern- reconfigurable antenna for single-RF 5G millimeter-wave communica- tions,”IEEE Antennas Wireless Propag. Lett., vol. 20, no. 12, pp. 2344– 2348, Dec. 2021

    work page 2021

  15. [15]

    Movable-antenna enhanced multiuser communication via antenna position optimization,

    L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,”IEEE Trans. Wireless Commun., vol. 23, no. 7, pp. 7214–7229, July 2024

    work page 2024

  16. [16]

    Weighted sum-rate maximiza- tion for movable antenna-enhanced wireless networks,

    B. Feng, Y . Wu, X.-G. Xia, and C. Xiao, “Weighted sum-rate maximiza- tion for movable antenna-enhanced wireless networks,”IEEE Wireless Commun. Lett., vol. 13, no. 6, pp. 1770–1774, June 2024

    work page 2024

  17. [17]

    Movable-antenna po- sition optimization: A graph-based approach,

    W. Mei, X. Wei, B. Ning, Z. Chen, and R. Zhang, “Movable-antenna po- sition optimization: A graph-based approach,”IEEE Wireless Commun. Lett., vol. 13, no. 7, pp. 1853–1857, July 2024

    work page 2024

  18. [18]

    Capacity maximization for FAS-assisted multiple access channels,

    H. Xuet al., “Capacity maximization for FAS-assisted multiple access channels,”IEEE Trans. Commun., vol. 73, no. 7, pp. 4713–4731, July 2025

    work page 2025

  19. [19]

    Movable antenna enabled integrated sensing and communication,

    W. Lyu, S. Yang, Y . Xiu, Z. Zhang, C. Assi, and C. Yuen, “Movable antenna enabled integrated sensing and communication,”IEEE Trans. Wireless Commun., vol. 24, no. 4, pp. 2862–2875, Apr. 2025

    work page 2025

  20. [20]

    Movable antenna enhanced wireless sensing via antenna position optimization,

    W. Ma, L. Zhu, and R. Zhang, “Movable antenna enhanced wireless sensing via antenna position optimization,”IEEE Trans. Wireless Com- mun., vol. 23, no. 11, pp. 16 575–16 589, Nov. 2024

    work page 2024

  21. [21]

    Array design based on the worst-case Cram ´er-Rao bound to account for multiple targets,

    C. A. Kokke, M. Coutino, R. Heusdens, and G. Leus, “Array design based on the worst-case Cram ´er-Rao bound to account for multiple targets,” inProc. Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, USA, Oct. 2023, pp. 1174–1178

    work page 2023

  22. [22]

    Fluid antenna enabling secret communications,

    B. Tang, H. Xu, K.-K. Wong, K.-F. Tong, Y . Zhang, and C.-B. Chae, “Fluid antenna enabling secret communications,”IEEE Commun. Lett., vol. 27, no. 6, pp. 1491–1495, June 2023

    work page 2023

  23. [23]

    Secure wireless communication via movable-antenna array,

    G. Hu, Q. Wu, K. Xu, J. Si, and N. Al-Dhahir, “Secure wireless communication via movable-antenna array,”IEEE Signal Process. Lett., vol. 31, pp. 516–520, Jan. 2024

    work page 2024

  24. [24]

    Movable-antenna position optimization for physical-layer security via discrete sampling,

    W. Mei, X. Wei, Y . Liu, B. Ning, and Z. Chen, “Movable-antenna position optimization for physical-layer security via discrete sampling,” inProc. IEEE Global Commun. Conf. (GLOBECOM), Dec. 2024, pp. 4750–4755

    work page 2024

  25. [25]

    Near-field multiuser communications aided by movable antennas,

    J. Ding, L. Zhu, Z. Zhou, B. Jiao, and R. Zhang, “Near-field multiuser communications aided by movable antennas,”IEEE Wireless Commun. Lett., vol. 14, no. 1, pp. 138–142, Jan. 2025

    work page 2025

  26. [26]

    Joint beamforming and antenna design for near-field fluid antenna system,

    Y . Chen, M. Chen, H. Xu, Z. Yang, K.-K. Wong, and Z. Zhang, “Joint beamforming and antenna design for near-field fluid antenna system,” IEEE Wireless Commun. Lett., vol. 14, no. 2, pp. 415–419, Feb. 2025

    work page 2025

  27. [27]

    Degrees of freedom of holographic MIMO in multi-user near-field channels,

    H. Chen, S. Yue, M. Di Renzo, and H. Zhang, “Degrees of freedom of holographic MIMO in multi-user near-field channels,”IEEE Commun. Lett., vol. 29, no. 6, pp. 1186–1190, June 2025

    work page 2025

  28. [28]

    Channel estimation for extremely large-scale MIMO: Far-field or near-field?

    M. Cui and L. Dai, “Channel estimation for extremely large-scale MIMO: Far-field or near-field?”IEEE Trans. Commun., vol. 70, no. 4, pp. 2663–2677, Apr. 2022

    work page 2022

  29. [29]

    Spatial multiplexing in near-field line-of-sight MIMO communi- cations: Paraxial and non-paraxial deployments,

    J. C. Ruiz-Sicilia, M. D. Renzo, P. Mursia, A. Kaushik, and V . Sciancale- pore, “Spatial multiplexing in near-field line-of-sight MIMO communi- cations: Paraxial and non-paraxial deployments,”IEEE Trans. Green Commun. Netw., vol. 9, no. 1, pp. 338–353, Mar. 2025

    work page 2025

  30. [30]

    Sensing- enhanced channel estimation for near-field XL-MIMO systems,

    S. Liu, X. Yu, Z. Gao, J. Xu, D. W. K. Ng, and S. Cui, “Sensing- enhanced channel estimation for near-field XL-MIMO systems,”IEEE J. Sel. Areas Commun., vol. 43, no. 3, pp. 628–643, Mar. 2025

    work page 2025

  31. [31]

    Using a movable RFID antenna to automatically determine the position and orientation of objects on a tabletop,

    S. Hinske and M. Langheinrich, “Using a movable RFID antenna to automatically determine the position and orientation of objects on a tabletop,” inSmart Sensing and Context. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, pp. 14–26

    work page 2008

  32. [32]

    Non-uniform linear antenna array design and optimization for millimeter-wave communica- tions,

    P. Wang, Y . Li, Y . Peng, S. C. Liew, and B. Vucetic, “Non-uniform linear antenna array design and optimization for millimeter-wave communica- tions,”IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7343–7356, Nov. 2016

    work page 2016

  33. [33]

    E. B. Saff and V . Totik,Logarithmic Potentials with External Fields. Springer Cham, 2024

    work page 2024

  34. [34]

    CVX: Matlab software for disciplined convex programming, version 2.1,

    I. CVX Research, “CVX: Matlab software for disciplined convex programming, version 2.1,” https://cvxr.com/cvx, Aug. 2014

    work page 2014

  35. [35]

    On Stieltjes polynomials,

    M. Marden, “On Stieltjes polynomials,”Trans. Am. Math. Soc., vol. 33, no. 4, pp. 934–944, Oct. 1931

    work page 1931

  36. [36]

    On the polynomial of stieltjes,

    E. B. V . Vleck, “On the polynomial of stieltjes,”Bull. Am. Math. Soc., pp. 426–438, June 1898

    work page

  37. [37]

    Ronveaux,Heun’s Differential Equations

    A. Ronveaux,Heun’s Differential Equations. Oxford University Press, Oct. 1995

    work page 1995

  38. [38]

    M. A. Al-Gwaiz,Sturm-Liouville Theory and its Applications, ser. Springer Undergraduate Mathematics Series. Springer London, 2008

    work page 2008

  39. [39]

    Zwillinger,Handbook of Differential Equations

    D. Zwillinger,Handbook of Differential Equations. Elsevier Science, 1998, no. 1

    work page 1998

  40. [40]

    Fraunhofer and Fresnel distances: Unified derivation for aperture antennas,

    K. T. Selvan and R. Janaswamy, “Fraunhofer and Fresnel distances: Unified derivation for aperture antennas,”IEEE Antennas Propag. Mag., vol. 59, no. 4, pp. 12–15, Aug. 2017

    work page 2017

  41. [41]

    Szeg ˝o,Orthogonal Polynomials, ser

    G. Szeg ˝o,Orthogonal Polynomials, ser. American Mathematical Society colloquium publications. American mathematical society, 1939

    work page 1939

  42. [42]

    Flexible precoding for multi-user movable antenna communications,

    S. Yang, W. Lyu, B. Ning, Z. Zhang, and C. Yuen, “Flexible precoding for multi-user movable antenna communications,”IEEE Wireless Com- mun. Lett., vol. 13, no. 5, pp. 1404–1408, May 2024

    work page 2024

  43. [43]

    Near-Field Communication with Massive Movable Antennas: A Functional Perspective

    S. Liu, X. Yu, J. Xu, and R. Zhang, “Near-field communication with massive movable antennas: A functional perspective,” 2025. [Online]. Available: https://arxiv.org/abs/2508.01201

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  44. [44]

    Movable antenna-enabled co-frequency co-time full-duplex wireless communica- tion,

    J. Ding, Z. Zhou, W. Li, C. Wang, L. Lin, and B. Jiao, “Movable antenna-enabled co-frequency co-time full-duplex wireless communica- tion,”IEEE Commun. Lett., vol. 28, no. 10, pp. 2412–2416, Oct. 2024

    work page 2024