Movable-Antenna Array Enhanced Downlink NOMA

Derrick Wing Kwan Ng; Lipeng Zhu; Nianzu Li; Peiran Wu

REVIEW 3 major objections 6 minor 14 references

Movable-Antenna Array Enhanced Downlink NOMA

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Movable antennas at the base station can lift NOMA downlink sum rates.

desk verdict Competent incremental MA-NOMA downlink extension whose SDR rank-one step is a cited gap rather than a proof, and which skips the closest baseline, but still deserves referee time. read the letter →

arxiv 2506.11438 v1 pith:MRFLVWGM submitted 2025-06-13 eess.SP

Nianzu Li , Peiran Wu , Lipeng Zhu , Derrick Wing Kwan Ng This is my paper · ORCID

classification eess.SP

keywords movableantennaarraydownlinkNOMAsumratemaximizationbeamformingoptimizationpositionsuccessiveconvexapproximationfield-responsechannelmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that letting a base station's antennas move within a local region, and optimizing their positions together with the transmit beamforming, raises the sum rate of a downlink non-orthogonal multiple access (NOMA) system above both fixed-position-antenna (FPA) NOMA and orthogonal multiple access (OMA). The setting is a base station with $M$ movable antennas serving $K$ single-antenna users under a total power budget, a minimum antenna separation, and per-user rate constraints. The authors formulate the joint sum-rate maximization over antenna positions, beamforming vectors, and successive-interference-cancellation order, then solve it with an alternating optimization scheme in which each block is handled by successive convex approximation. Their simulations show the proposed scheme consistently outperforms the benchmarks and needs less transmit power to reach a given sum rate. The paper is an algorithmic and numerical demonstration; it does not claim a new theoretical capacity result.

What carries the argument

The central machinery is the field-response channel model combined with an alternating optimization (AO) loop. In this model the channel is the superposition of $L_k$ multipath components whose phases depend linearly on each antenna's coordinates, making antenna position a continuous optimization variable rather than a discrete choice. The beamforming subproblem is relaxed to a semidefinite program by dropping the rank-one constraint, justified by a theorem from [10] that the authors carry over from IRS-NOMA; the antenna-position subproblem is made convex by lower-bounding the desired signal power and upper-bounding the interference plus noise via quadratic Taylor surrogates. The AO loop alternates these two steps, and the objective is non-decreasing, which the paper cites as the convergence guarantee.

What would settle it

Solve the original rank-constrained beamforming subproblem for a small instance (e.g., $M=2$, $K=2$) by global or exhaustive search; if any solution with a beamforming matrix of rank greater than one achieves a strictly higher sum rate than the relaxed solution, or if the relaxed solution's eigenvalue decomposition violates the SIC rate constraints, the rank-one carryover assumption fails.

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Extended reading notes

Core claim

The central claim is that the extra spatial degrees of freedom from repositioning a base-station antenna array materially improve the achievable downlink NOMA sum rate. In the field-response channel model, each user's channel vector is $\mathbf{h}_k(\tilde{\mathbf{u}}) = \mathbf{G}_k^H(\tilde{\mathbf{u}})\mathbf{f}_k$, so moving the antennas changes the phase of every multipath term and thereby the whole channel. The proposed algorithm alternates between optimizing the beamforming matrices $\{\mathbf{W}_k\}$ for fixed antenna positions and optimizing each antenna position $\mathbf{u}_m$ for fixed beamforming, with convex surrogates built from Taylor expansions of the signal-power functions. The reported result is that this joint design outperforms both NOMA with fixed antenna positions and OMA with fixed positions, and that the MA gain grows with the number of antennas and users. The paper attributes the NOMA-specific benefit to antenna movement magnifying channel disparities among users, which is precisely the condition power-domain NOMA exploits.

Load-bearing premise

The load-bearing premise is that the semidefinite-relaxation rank-one result proved for IRS-NOMA beamforming still holds under the movable-antenna field-response channel model, and that the base station has perfect channel state information and ideal successive interference cancellation; if that premise fails, the beamforming from the relaxed problem may not be feasible for the original problem.

Editorial extensions

If this is right

If the central claim holds, MA-enhanced NOMA becomes a practical route to higher spectral efficiency in downlink multi-user cells without additional bandwidth, RF chains, or transmit power.
The same alternating SCA structure can be applied to other MA-array problems where the objective is a sum of logarithmic SINR terms, such as weighted sum-rate or max-min fairness.
Because the gain is partly attributed to enlarging channel disparities, the scheme should be most effective when users share similar average path loss but have resolvable multipath angular structure.
The reported transmit-power savings for a fixed target sum rate imply an explicit trade-off between antenna movement and energy consumption in system design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit extension the paper does not pursue is robustness to imperfect CSI: the Taylor surrogates and the rank-one relaxation would need to be re-derived for stochastic or estimated channels before the reported gains can be trusted outside simulation.
The rank-one carryover is the most fragile link; a targeted search for small counterexamples in the MA field-response model would either certify or break the relaxation's validity.
A harder benchmark than the equal-length-slot OMA used here would be OMA with optimized time allocation or FPA-NOMA with user ordering optimized separately; the gap reported in Figures 3 and 4 might shrink under those comparisons.
The paper moves only the base-station array; extending the same joint position-beamforming design to movable antennas at the user side, or to uplink, is a direct next step the authors do not address.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a competent incremental extension of MA-NOMA to a downlink BS array, with a real algorithmic contribution, but the SDR rank-one argument is imported rather than proved, and the closest baselines are skipped. It deserves referee time, but it needs a tightened proof or numerical rank check before I'd trust the beamforming extraction step.

What's actually new: prior MA-NOMA works put a single MA at the user side or handled uplink; here they put M antennas at the BS, jointly optimizing beamforming and antenna positions in a downlink NOMA system. The problem is formulated cleanly and the SCA/AO decomposition is standard but carefully executed—the gradient/Hessian derivations in Appendix A are genuine work and look correct. The convergence argument (monotone non-decreasing, compact feasible set) is fine. Simulations over 1000 realizations show gains over FPA-NOMA and OMA-FPA; that central claim is plausible and not circular.

Soft spots, in order. First, the rank-one SDR step is load-bearing. They drop (8f) citing Theorem 1 of [10], which was proved for an IRS-NOMA problem with fixed channels. The current problem has a different objective (slack variables α, β), different SIC constraints, and the linearization in (10) is introduced after the relaxation. No argument is given that the rank-one property survives. If a higher-rank optimum occurs, the eigenvalue decomposition yields a beamforming set that isn't feasible for (8), and the simulation curves lose their grounding. I don't see a counterexample, and I suspect the theorem can be adapted, but currently it's a gap, not a proof. The authors should either provide the adaptation or add a numerical check on the rank of W_k for every iteration. Second, the paper cites [13] (arXiv 2412.12531) as an existing MA-NOMA work but never compares against it. Since [13] appears to be very close, the lack of comparison leaves the actual increment unclear. That's a citable omission, not a fatal one. Third, the perfect-CSI and ideal-SIC assumptions are stated without discussion of robustness; minor for an optimization paper.

Who benefits: people working on MA-assisted NOMA or on SDR-based beamforming. I'd bring it to a reading group to discuss the rank-one issue, but I wouldn't change my research plans based on it. Recommendation: send to peer review. It deserves a serious referee; the rank-one gap and the missing [13] comparison should be the main review points.

Referee Report

3 major / 6 minor

Summary. The paper studies a downlink MISO NOMA system in which the base station is equipped with multiple movable antennas (MAs) whose two-dimensional positions can be adjusted within a finite region. The authors formulate a sum-rate maximization problem that jointly optimizes the MA positions, the transmit beamforming vectors, and the SIC decoding order, subject to power, minimum-rate, inter-antenna distance, and SIC decodability constraints. Because the problem is highly non-convex, they propose an alternating optimization algorithm: for fixed antenna positions, the beamforming subproblem is relaxed to an SDP after dropping rank-one constraints, using a cited rank-one tightness theorem; for fixed beamforming, each MA position is updated by successive convex approximation using quadratic lower/upper bounds. The algorithm is shown numerically to converge monotonically, and simulations report that the proposed MA-NOMA design outperforms FPA-based NOMA and OMA over a range of transmit powers and antenna numbers.

Significance. The paper addresses a timely and relevant problem at the intersection of two active research areas, movable antennas and NOMA, and it is one of the first to place the MA array at the base station rather than at the user side. The system model and problem formulation are clear, and the alternating SCA framework is a reasonable and standard approach to a difficult non-convex problem. The numerical study is substantial (1000 channel realizations, multiple M and K) and the reported gains over FPA-NOMA and OMA are consistent across the power and antenna dimensions. The main weaknesses are the unverified transfer of a rank-one SDR result from an IRS-NOMA setting and the absence of comparisons with prior MA-NOMA schemes. If the rank-one issue is resolved, the contribution is useful and publishable, but its significance is moderate and incremental rather than groundbreaking.

major comments (3)

[III.B.1, Eqs. (8)–(11)] The removal of the rank-one constraint (8f) is load-bearing: it is what allows the SDP solution to be converted by eigenvalue decomposition into feasible beamforming vectors for problem (8). The paper asserts, citing Theorem 1 of [10], that the SDR of problem (8) always has an optimal solution with rank(W_k)=1 for all k. However, [10] proves that theorem for an IRS-aided NOMA beamforming problem with fixed channels, whereas the present SDP has the objective sum_k R_{k→k} together with slack variables (α,β), the SIC constraints (7e), and the linearized constraints (10). These differences are not discussed. If a higher-rank optimum occurs for some channel realization, eigenvalue decomposition of W_k^* will not produce an exact w_k, and the beamforming update is not feasible for problem (8); the rates in Figs. 3 and 4 would then not be achieved by the claimed algorithm. Please provide an adapted proof, or a precise statement of how problem (11) falls within the scope of [10, Theorem 1], or numerical verification of the rank-one property over all tested channel realizations and iterations.
[Section IV, Figs. 3 and 4] The numerical evaluation compares the proposed design only with NOMA-FPA and OMA-FPA baselines. Since the introduction identifies [11]–[13] as prior MA-NOMA systems, the reader cannot tell whether the proposed BS-side MA array improves on the existing MA-NOMA state of the art, or only on FPA/OMA baselines. Please add a benchmark from the most comparable prior MA-NOMA design, or explicitly justify why the different antenna deployment (single MA at the user side versus an MA array at the BS) makes a quantitative comparison inappropriate. Without this, the claim of 'significantly improve' is established only against FPA and OMA, not against prior MA-NOMA work.
[Section III.A and Algorithm 1] The paper states that the optimal decoding order is found by solving problem (6) for all possible orders and selecting the maximum, but the algorithm and complexity analysis in Section III.B are written for a single fixed order, and Section IV does not say which order was used to generate Figs. 3 and 4. For K=3 there are six possible orders and for K=2 there are two, so full enumeration is plausible; nevertheless, the manuscript should state explicitly whether Figures 3 and 4 use full enumeration and should give the corresponding complexity. Without this information the reported curves are not reproducible and the phrase 'optimal sum rate' is not fully justified.

minor comments (6)

[III.A, Eq. (7e)] Constraint (7e) lower-bounds R_{k→i} using α_{k,k} and β_{k,k}, not α_{k,i} and β_{k,i}. This may be intentional, but it is easy to misread as a typo; please add a sentence explaining the role of this constraint (it enforces R_{k→i} ≥ R_{k→k} through the lower bound on R_{k→k}).
[Appendix A, Eq. (21c)] In the expression for ∂²Γ_{k,i}(u_m)/∂y_m², the second summation should contain cos(κ̄_{k,i,m,ℓ_3}(u_m)), not sin(κ̄_{k,i,m,ℓ_3}(u_m)), since it is the derivative of the sin term in (20b). Please correct this formula and verify that the Hessian bound δ_{k,i} is computed consistently.
[III.B.2, after Eq. (13)] The inequality justifying δ_{k,i} is written as ||∇²Γ_{k,i}(u_m)||_F I ⪰ ∇²Γ_{k,i}(u_m); the identity matrix should be dimension 2, i.e., ||∇²Γ_{k,i}(u_m)||_F I_2, to avoid ambiguity about the matrix size.
[Section IV] The paper reports averages over 1000 channel realizations but does not provide error bars or confidence intervals. Since the comparisons in Figs. 3 and 4 are the main evidence for the central claim, please report standard errors or confidence intervals, or at least state the dispersion of the results.
[III.B.3, Algorithm 1] The algorithm requires an initial feasible point ({w^0_k},{u^0_m}) for problem (7), but no procedure for constructing such a point is given. Please describe how feasibility is achieved, especially with respect to the minimum-rate constraints (6d) and the SIC constraints.
[Throughout] There are several language and typographical issues, for example 'donwlink' in the Introduction and 'All points in our simulation curves is averaged' in Section IV. These should be corrected in a careful revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; minor self-citations are not load-bearing.

full rationale

The paper's central derivation is a non-convex optimization reformulation (SDR+SCA+AO) evaluated against FPA/OMA benchmarks. No parameter is fitted to the reported curves; the sum-rate curves are produced by running the algorithm on random channels. The rank-one SDR guarantee is imported from [10], which is an external IRS-NOMA paper (Mu et al.), not from the present authors; whether the theorem transfers to the MA channel model is a derivation gap, not a circular reduction. Self-citations [1], [2], [7], [12] appear in motivation/channel-model/intro contexts and do not carry the central claim. The problem (7) equivalence is credited to [10, Prop. 1] as a known transformation, not derived from the paper's own output. Thus no step reduces, by construction, to its own input; the modest score reflects only minor self-citation presence, with no load-bearing circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new physical entities, fitted constants, or ad hoc parameters are introduced. All simulation constants are standard channel parameters from the prior MA literature. The main assumptions are inherited from prior MA and NOMA work.

assumptions (4)

domain assumption The field-response channel model (3) exactly describes the channel from each movable antenna to each user under far-field conditions.
Invoked in Section II-A; the algorithm optimizes positions inside this model, so model error directly changes the claimed gains.
domain assumption Perfect CSI is available at the base station.
Assumed in Section II-B after (6); no estimation error or feedback delay is modeled.
standard math The slack-variable transformation in (7) is exactly equivalent to the original NOMA sum-rate problem (6).
Used to convert the problem, cited from [10, Proposition 1]; the equivalence is not re-derived for the MA channel.
domain assumption The SDR rank-one result in [10, Theorem 1] applies to the beamforming subproblem (8) with MA channels.
Used to drop constraint (8f); the theorem's conditions are not verified in this paper.

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Cite this review

Pith. "Pith review of Movable-Antenna Array Enhanced Downlink NOMA." pith.science (2026). https://pith.science/paper/MRFLVWGM

@misc{pith2026250611438,
  author       = {Pith},
  title        = {Pith review of: Movable-Antenna Array Enhanced Downlink NOMA},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MRFLVWGM}},
  note         = {Machine review of arXiv:2506.11438}
}

read the original abstract

Movable antenna (MA) has gained increasing attention in the field of wireless communications due to its exceptional capability to proactively reconfigure wireless channels via localized antenna movements. In this paper, we investigate the resource allocation design for an MA array-enabled base station serving multiple single-antenna users in a downlink non-orthogonal multiple access (NOMA) system. We aim to maximize the sum rate of all users by jointly optimizing the transmit beamforming and the positions of all MAs at the BS, subject to the constraints of transmit power budget, finite antenna moving region, and the conditions for successive interference cancellation decoding rate. The formulated problem, inherently highly non-convex, is addressed by successive convex approximation (SCA) and alternating optimization methods to obtain a high-quality suboptimal solution. Simulation results unveil that the proposed MA-enhanced downlink NOMA system can significantly improve the sum rate performance compared to both the fixed-position antenna (FPA) system and the traditional orthogonal multiple access (OMA) system.

Figures

Figures reproduced from arXiv: 2506.11438 by the authors.

**Figure 2.** Evaluation of the convergence of our proposed algorithm for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Average sum rate of different schemes versus the maximum transmit [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Reviewed August 7, 2026 · model on record in the stance chip above.

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Y . Zhouet al., “Movable antenna empowered downlink NOMA systems: Power allocation and antenna position optimization,”IEEE Wireless Commun. Lett., vol. 13, no. 10, pp. 2772–2776, 2024

work page 2024

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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, Jun. 2024

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work page Pith review arXiv 2024

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