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arxiv: 2508.01201 · v1 · submitted 2025-08-02 · 💻 cs.IT · math.IT

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

Shicong Liu , Xianghao Yu , Jie Xu , Rui Zhang This is my paper

Pith reviewed 2026-05-19 01:55 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords movable antennasnear-field communicationantenna position optimizationachievable ratefunctional optimizationmassive MIMOline-of-sight channels
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The pith

Reformulating antenna positions as a continuous density function maximizes achievable rate in near-field massive MIMO.

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

The paper investigates how to place a large number of movable antennas at a transmitter to increase data rate over a point-to-point near-field channel. It converts the usual discrete choice of exact positions into a continuous problem by defining an antenna position function and its associated density function. Variational methods then produce the condition that the optimal density must satisfy, along with a practical gradient algorithm for general channels. In the simpler line-of-sight near-field case the same approach yields an explicit density formula that places more antennas near the edges. A flexible-array construction is also given so the optimized density can be realized without severe mutual coupling.

Core claim

The achievable rate is expressed as a functional of the antenna density function obtained from a continuous position mapping; variational analysis supplies the stationarity condition whose solution maximizes the functional, and this condition admits a closed-form solution in the near-field line-of-sight setting that concentrates density at the array edges.

What carries the argument

The antenna density function (ADF), which recasts discrete antenna placement as the variable of a rate functional that is then optimized by variational calculus.

If this is right

  • A gradient-based algorithm computes the optimal density numerically for arbitrary channel models.
  • In near-field line-of-sight links the closed-form density places higher concentration at the physical edges of the deployment region.
  • Flexible antenna arrays can realize the density pattern while controlling mutual coupling.
  • Uniform circular arrays offer a practical geometry that retains most of the rate gain.

Where Pith is reading between the lines

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

  • The same density-function approach could be tested on multi-user or time-varying near-field channels.
  • Hardware that locally adjusts element spacing according to the derived density would provide a direct experimental check.
  • Analogous continuous reformulations may simplify design of irregular or sparse arrays in other propagation regimes.

Load-bearing premise

Any discrete set of antenna positions can be represented and improved by a continuous density function without substantial loss of performance.

What would settle it

A direct numerical comparison in which the rate obtained by sampling the optimal continuous density is lower than the rate from jointly optimizing the same number of discrete positions under identical channel and power constraints.

Figures

Figures reproduced from arXiv: 2508.01201 by Jie Xu, Rui Zhang, 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 APF p = f(m) (in blue) and ADF w(p) (in orange) for an M = 16-element movable antenna array. A. Antenna Density Function We first extend the domain of APF f(m) from m ∈ M to m ∈ Mc = [1, M] to facilitate a more tractable analysis. By extending the domain of function f(m), we assume that the antenna elements are continuously distributed over p ∈ [−1, 1], i.e., the aperture area. Therefore, for an arbitr… view at source ↗
Figure 3
Figure 3. Figure 3: The comparison of the exact log-determinant and the asymptotic [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustrations of ADF w(p) with z0 = 10 m and θT = π 2 . expression (53). Furthermore, as the system dimension tends to infinity as N → ∞, the asymptotic behavior of (54) is dominated by the log N term, which leads to a negative first￾order derivative. Consequently, the asymptotic upper bound of the achievable rate is attained with the specific choice of parameters α1 = α2 ≜ α = −0.5. Next, we derive the as… view at source ↗
Figure 5
Figure 5. Figure 5: Antenna positions on a linear array with normalized aperture [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: ADF and corresponding discrete antenna positions at [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: ADF and corresponding discrete antenna positions at [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) The achievable rate performance with different numbers of an [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

The advent of massive multiple-input multiple-output (MIMO) technology has provided new opportunities for capacity improvement via strategic antenna deployment, especially when the near-field effect is pronounced due to antenna proliferation. In this paper, we investigate the optimal antenna placement for maximizing the achievable rate of a point-to-point near-field channel, where the transmitter is deployed with massive movable antennas. First, we propose a novel design framework to explore the relationship between antenna positions and achievable data rate. By introducing the continuous antenna position function (APF) and antenna density function (ADF), we reformulate the antenna position design problem from the discrete to the continuous domain, which maximizes the achievable rate functional with respect to ADF. Leveraging functional analysis and variational methods, we derive the optimal ADF condition and propose a gradient-based algorithm for numerical solutions under general channel conditions. Furthermore, for the near-field line-of-sight (LoS) scenario, we present a closed-form solution for the optimal ADF, revealing the critical role of edge antenna density in enhancing the achievable rate. Finally, we propose a flexible antenna array-based deployment method that ensures practical implementation while mitigating mutual coupling issues. Simulation results demonstrate the effectiveness of the proposed framework, with uniform circular arrays emerging as a promising geometry for balancing performance and deployment feasibility in near-field communications.

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

2 major / 3 minor

Summary. The paper presents a functional perspective for designing antenna positions in massive movable antenna systems for near-field communications. It introduces the antenna position function (APF) and antenna density function (ADF) to reformulate the discrete antenna placement problem as a continuous optimization of the achievable rate functional. Using variational methods, an optimal ADF condition is derived, a closed-form solution is provided for the line-of-sight (LoS) case, and a flexible antenna array deployment method is proposed to address practical issues like mutual coupling. Simulations illustrate the benefits, particularly for uniform circular arrays.

Significance. If the continuous ADF relaxation provides a close approximation to the discrete optimum, this work offers a tractable analytical tool for near-field antenna placement optimization, which is otherwise combinatorial. The closed-form LoS solution highlighting edge antenna density could guide practical designs in scenarios where near-field effects dominate.

major comments (2)
  1. [§3] §3 (functional reformulation): the central step of relaxing discrete antenna positions to a continuous ADF via the rate functional assumes the continuous optimum can be discretized back to finite N with negligible rate loss, but no explicit approximation error bound, convergence rate, or discretization analysis is provided. This is load-bearing for the claim that the derived ADF yields a practical near-optimal deployment.
  2. [§4] §4 (LoS closed-form): the optimal ADF expression is derived under continuous density assumptions; without a follow-on result quantifying the rate gap upon sampling to a finite number of positions (or an explicit discretization procedure), it is unclear whether the edge-density emphasis survives in implementable arrays.
minor comments (3)
  1. [§5] The simulation section would benefit from an explicit statement of the number of antennas N, carrier frequency, and SNR range used when comparing the proposed ADF-based designs against uniform linear and circular baselines.
  2. Notation: ensure that the functional derivative notation for the rate with respect to ADF is introduced once and used consistently; minor inconsistencies appear in the transition from the variational condition to the gradient algorithm.
  3. [§5] Figure 3 (or equivalent rate curves): add a legend entry or caption note clarifying whether the 'optimal ADF' curve is the continuous functional value or the rate after discretization and flexible-array mapping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments identify important aspects of the continuous relaxation that warrant further clarification. We address each major comment point by point below, indicating where revisions will be made to strengthen the theoretical and practical claims.

read point-by-point responses

  1. Referee: [§3] §3 (functional reformulation): the central step of relaxing discrete antenna positions to a continuous ADF via the rate functional assumes the continuous optimum can be discretized back to finite N with negligible rate loss, but no explicit approximation error bound, convergence rate, or discretization analysis is provided. This is load-bearing for the claim that the derived ADF yields a practical near-optimal deployment.

    Authors: We agree that an explicit approximation analysis would strengthen the link between the continuous optimum and finite-N implementations. The manuscript already introduces a flexible antenna array-based deployment method (Section 5) that discretizes the ADF by non-uniform sampling while enforcing minimum spacing to mitigate mutual coupling. Simulations (Section 6) compare the resulting finite arrays against uniform and random placements, showing that the rate loss relative to the continuous functional is small for large N. To address the referee's concern directly, we will add a new subsection (or appendix) in the revision that formalizes the discretization procedure and provides a first-order bound on the rate gap, leveraging the Lipschitz continuity of the rate functional with respect to antenna positions. revision: yes

  2. Referee: [§4] §4 (LoS closed-form): the optimal ADF expression is derived under continuous density assumptions; without a follow-on result quantifying the rate gap upon sampling to a finite number of positions (or an explicit discretization procedure), it is unclear whether the edge-density emphasis survives in implementable arrays.

    Authors: The closed-form LoS ADF is presented primarily to obtain analytical insight into the optimal density distribution, particularly the elevated density near the array edges. The same flexible deployment method discretizes this ADF to obtain concrete positions for finite N. Our numerical results already illustrate that arrays constructed this way outperform uniform circular arrays and retain the performance advantage predicted by the continuous solution. We concur that an explicit quantification of the discretization gap would be beneficial and will incorporate additional analysis and plots in the revised manuscript showing the rate gap as a function of N for the LoS case, confirming that the edge-density emphasis persists in practical arrays. revision: yes

Circularity Check

0 steps flagged

No circularity: standard variational derivation on continuous relaxation

full rationale

The paper reformulates discrete antenna positions via continuous APF/ADF and applies functional analysis plus variational methods to maximize the rate functional, deriving an optimal ADF condition and a closed-form LoS solution. These steps follow directly from the channel model and rate expression using standard calculus of variations; no quantity is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The continuous-domain optimum is an independent mathematical object, with discretization and flexible-array implementation treated as downstream practical steps rather than tautological recoveries.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard MIMO channel modeling assumptions and the validity of the continuous approximation; no explicit free parameters or invented entities are detailed in the abstract.

axioms (1)
  • domain assumption The achievable rate of the near-field channel can be expressed as a functional of the continuous antenna density function (ADF).
    This assumption enables the reformulation of the discrete placement problem into a continuous variational optimization problem.

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Forward citations

Cited by 2 Pith papers

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    Optimal positions for massive movable antennas in near-field channels are the roots of polynomials solving specific ODEs, with a closed-form asymptotic solution that improves spectral efficiency.

  2. A General EM-Based Channel Model for Reconfigurable Antenna Systems

    eess.SP 2026-04 unverdicted novelty 5.0

    A spherical vector wave expansion-based channel model for reconfigurable antennas incorporates position and orientation effects, validated against simulations with up to 70% rate improvement from dynamic orientation a...

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