A Secure Beamforming Design: When Fluid Antenna Meets NOMA

Fumiyuki Adachi; Hyundong Shin; Jie Tang; Junteng Yao; Kai-Kit Wong; Lifeng Mai; Tuo Wu

REVIEW 3 major objections 4 minor 17 references

A Secure Beamforming Design: When Fluid Antenna Meets NOMA

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Jointly optimizing secure beamforming and fluid antenna positions is claimed to maximize the achievable secrecy rate in a downlink NOMA system, outperforming fixed-antenna NOMA and OMA baselines in simulations.

desk verdict The FAS+NOMA secure beamforming idea is new and the beamforming optimization is sound, but the position subproblem reverses the NOMA rate constraint, so the reported secrecy gains likely come from infeasible antenna positions. read the letter →

arxiv 2411.08386 v1 pith:L5GRLM42 submitted 2024-11-13 eess.SP

Lifeng Mai , Junteng Yao , Jie Tang , Tuo Wu , Kai-Kit Wong , Hyundong Shin , Fumiyuki Adachi This is my paper · ORCID

classification eess.SP

keywords fluidantennasystemsNOMAsecurebeamformingphysicallayersecuritysecrecyratealternatingoptimizationmajorization-minimizationMISO

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 the base station's antennas move, not just steering beams, can substantially improve physical-layer secrecy in a downlink non-orthogonal multiple access system. Treating the cell-edge user as a potential eavesdropper, it maximizes the achievable secrecy rate by jointly optimizing transmit beamforming vectors and the positions of M fluid antennas. The authors propose alternating optimization that iterates between a convex beamforming subproblem and a convex per-antenna position subproblem, using majorization-minimization and Taylor expansions to handle nonconvex terms. Numerical simulations show secrecy-rate gains over fixed-position antenna NOMA, random-position antenna NOMA, and fluid-antenna OMA baselines.

What carries the argument

The central machinery is alternating optimization combined with majorization-minimization surrogates. Slack variables $\tau$ and $\epsilon_e$ split the fractional secrecy objective, the bilinear product $\tau\epsilon_e$ is bounded by a first-order Taylor surrogate, convex quadratic terms in $w_1$ are lower-bounded by tangent lines, and the quadratic term $g_e^\dagger(t_m)V_e g_e(t_m)$ is upper-bounded using the maximum eigenvalue of $V_e$. Each subproblem is thereby recast as a convex program, solved with standard convex solvers, and the antenna positions are updated one at a time while the others stay fixed.

What would settle it

Directly evaluate the returned $w_1^*, w_2^*, t^*$ by computing the original constraints (9b)-(9e) and the secrecy rate (8). If the point violates $R_{k,2} \ge r$ or $\|t_m - t_k\|_2 \ge D$, or if running the alternating algorithm from many random initializations yields materially different secrecy rates, the claim of a locally optimal secure design would be contradicted.

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Extended reading notes

Core claim

The paper claims that moving the transmit antennas is itself a degree of freedom for secrecy: by reconfiguring antenna positions, the base station shapes the channel to the intended cell-center user more favorably than to the eavesdropping cell-edge user, and it does so jointly with secure beamforming. Specifically, the achievable secrecy rate $R_s = R_{c,1} - R_{e,1}$ is maximized under a cell-edge rate requirement, a total power constraint, and a minimum antenna separation $D$; the resulting alternating optimization yields a locally optimal secure beamforming pair and fluid-antenna position set, with computed secrecy rates above the tested benchmarks.

Load-bearing premise

The load-bearing premise is that the successive Taylor approximations and the alternating procedure preserve feasibility of the original constraints and converge to a meaningful local optimum; if the surrogate bounds are not globally valid, the final point may violate the rate, power, or antenna-separation constraints.

Editorial extensions

If this is right

Secrecy in a downlink NOMA system can be improved without extra transmit power, purely by re-positioning the base station's fluid antennas alongside the beamforming design.
The alternating algorithm is computationally tractable, since each subproblem is convex and can be solved with standard convex optimization tools.
Simulated secrecy rate increases with the number of fluid antennas $M$, and the proposed design consistently beats fixed-position NOMA, random-position NOMA, and fluid-antenna OMA across the tested power levels and antenna counts.
The approach combines NOMA's spectral efficiency with FAS's spatial flexibility, reducing the channel-similarity problem that normally weakens successive interference cancellation.
The secrecy-rate objective (8) and the rate-constraint reformulations extend directly to other secure multi-user settings, as long as the eavesdropper's channel model is available.

Reading between the lines

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

A natural testable extension is to replace the perfect-channel assumption with a robust design over channel uncertainty, since position optimization would likely need to hedge against estimation error.
Because the algorithm optimizes one fluid-antenna position at a time with the others fixed, a fully joint position update might reach comparable secrecy rates in fewer outer iterations.
The same MM and Taylor machinery transfers to multiple cell-edge users or multiple eavesdroppers by summing the eavesdropper rate terms, suggesting the formulation is not limited to one potential eavesdropper.
If the convergence and feasibility concerns are resolved, the paper's approach would establish antenna position as a spatial resource on par with beamforming power for physical-layer security.

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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

Colleague — quick take: the problem formulation is new and the beamforming half is competent, but the position subproblem flips the sign of the NOMA rate constraint, so the reported secrecy rates probably rest on infeasible antenna positions.

What's new and good. This is the first combination I've seen of secure beamforming, fluid antenna systems, and NOMA in a downlink with the cell-edge user treated as an eavesdropper. The AO decomposition — optimize beamformers with positions fixed, then positions with beamformers fixed — is a natural fit, and the MM/Taylor machinery in the beamforming subproblem is standard and largely correct. The numerical comparisons against fixed-position, random-position, and OMA baselines are reasonable. The citation pattern looks normal for a systems letter; the relevant FAS-NOMA and physical-layer-security references are there.

Where it falls apart. In Sec III-B, the authors rewrite (9b) using μ_{i,j} = w'_{1,i}w_{1,j} − L_r w'_{2,i}w_{2,j}. But (9b), via (10), is equivalent to d_{k,2} − L_r d_{k,1} ≥ L_r σ². Their μ produces d_{k,1} − L_r d_{k,2} ≥ L_r σ² — the opposite inequality. So (25) does not enforce the SIC rate requirement; it enforces its negation. The stress-test note is correct. This is a concrete algebraic error in the core derivation, not a missing convergence proof. If the simulations used (25) as written, the secrecy rates in Figs. 1–2 correspond to positions that violate the minimum rate r. The results may be salvageable if this is a typo and the sign in μ is swapped, but the manuscript as written does not support the central claim.

Other soft spots are secondary: (18) and (31) are called convex 'recasts' but are really inner approximations via Taylor bounds, with no proof of global validity or AO convergence. The abstract says users each have a FA while the system model gives them fixed-position antennas. Simulation details (FA region size, position sampling) are thin. These are normal for a letter, but they compound the main problem.

Recommendation: I wouldn't cite this in its current state. The idea is worth pursuing, and the fix is straightforward, so I'd send it to peer review rather than desk reject — a competent referee will catch the sign flip and the authors can rerun the simulations. For now, treat the numerical claims as unverified.

Referee Report

3 major / 4 minor

Summary. The letter studies a downlink MISO NOMA system in which the base station has M fluid antennas and serves a cell-center user (CU) and a cell-edge user (CEU), with the CEU treated as a potential eavesdropper. The authors formulate a secrecy-rate maximization problem over transmit beamforming vectors and fluid-antenna positions, and propose an alternating optimization (AO) algorithm. The beamforming subproblem is handled by majorization-minimization, and the antenna-position subproblem is handled by Taylor-based approximations that are claimed to yield convex subproblems solvable by CVX. Numerical results compare the proposed scheme against fixed-position antenna, random-position antenna, and OMA baselines, and report secrecy-rate gains.

Significance. If the derivation were correct, the paper would provide a useful design for combining fluid-antenna position optimization with secure NOMA beamforming, with a concrete geometric channel model and a benchmark comparison including FPA, RPA, and OMA. The problem formulation is relevant to physical-layer security for FAS-assisted NOMA, and the AO/MM structure is a standard and plausible approach. However, the paper provides no proofs, no reproducibility artifacts, and, as detailed below, contains a sign error and a convexity error in the position subproblem that directly affect the validity of the reported results. The central claim is therefore not currently supported by the manuscript.

major comments (3)

[Section III-B, Eq. (25)] The definition μ_{i,j}=w'_{1,i}w_{1,j}−L_r w'_{2,i}w_{2,j} makes the left side of the transformed constraint equal d_{k,1}(t)−L_r d_{k,2}(t). Constraint (9b), as correctly rewritten in Eqs. (10) and (17), is equivalent to d_{k,2}(t)−L_r d_{k,1}(t) ≥ L_r σ². Thus Eq. (25) enforces the opposite inequality: it requires the s1 signal power to dominate the s2 signal power, rather than the SIC-compatible condition that s2 be decodable with the required rate r. As written, the position subproblem can return FA placements that violate (9b), so the secrecy rates in Figs. 1–2 cannot be attributed to feasible solutions of problem (9). The coefficient should be μ_{i,j}=w'_{2,i}w_{2,j}−L_r w'_{1,i}w_{1,j}, and if the simulations were run using the reversed expression, they need to be rerun.
[Section III-B, Eqs. (23)–(24) and (31)] The paper states that (23) is convex with respect to t_m and that problem (31) is convex. However, b^l_{k,q}(t_m) in (23) contains a negative quadratic term −(β_{k,m}/2)||t_m−t^{(l)}_m||², so it is concave, not convex. Meanwhile ψ(τ,ε_e;τ^{(l)},ε^{(l)}_e) defined before Eq. (16) is convex in (τ,ε_e). Constraint (24) therefore takes the form concave ≥ convex, which is not a convex constraint in general. Consequently, problem (31) is not obviously convex and cannot be solved by CVX as claimed. The authors need either a valid convex surrogate for (24) or a different algorithm with a proof that the feasible set is convex or that the iterative procedure retains feasibility.
[Section III, overall AO convergence] No convergence or feasibility guarantee is provided for the alternating procedure. The manuscript labels (18) and (31) as convex recasts, but they are inner approximations via Taylor bounds; the paper does not prove that the bounds are globally valid or that the AO sequence converges to a locally optimal or even feasible point of the original problem. In particular, the lower bound in (23) requires a Lipschitz constant β_{k,m} that is stated but not derived, and no monotonicity or fixed-point argument is given for the alternating updates. Even if the sign error and convexity issue are corrected, this missing analysis leaves the algorithm's output without a formal claim to optimality or feasibility.

minor comments (4)

[Abstract and Section II] The abstract says the CU and CEU are each equipped with a fluid antenna, while Section II states that both users are equipped with fixed-position antennas and Section II-B takes the receive field response vectors f_c and f_e to be all-ones. Please correct this inconsistency and align the abstract with the actual system model.
[Section III-B, Eq. (22) and surrounding text] The notation w'_{q,i} is not defined; it presumably denotes the complex conjugate of w_{q,i}. Please define it explicitly, since the subsequent quadratic-form manipulations depend on it.
[Section III-B] The derivation of d_{k,q}(t_m) as Σ_{n=1}^M ξ_{n,m} g†_k(t^{(l)}_n)V_k g_k(t_m) is written as an equality, but the actual quadratic form contains additional terms with indices i=m, j≠m; these are later accounted for through the 2Re and W^m_k,q terms in the lower bound. This is confusing and should be rewritten as a lower-bound construction rather than an equality.
[Throughout] There are several typos and notation issues: the duplicated 'h_c and h_c' in Section II-A, the subscript/superscript q appearing inconsistently in μ^q_{1,m}, and the misspelling 'assited' in the conclusion. The names 'Vandenberghe' and the indexing of L_r^{(k)} vs. r in (10)–(17) should also be checked for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the secrecy-rate claim is optimized directly from the geometric channel model, and no fitted quantity is repackaged as a prediction.

full rationale

The paper's central claim is that jointly optimizing secure beamforming vectors and fluid antenna positions maximizes achievable secrecy rate. The objective and constraints are derived from the stated geometric channel model h_k(t) = G_k(t)^† Σ f_k and the secrecy-rate expression (8), and the numerical curves follow from directly optimizing that objective under the given rate, power, and antenna-spacing constraints. No parameter is fitted to the reported secrecy-rate curves, and no target quantity is used as an input to force the comparison results. The MM and Taylor-based transformations in Section III are standard inner/outer approximations whose validity is a convergence and feasibility question, not a case of defining the output in terms of the input. The baselines (FPA, RPA, OMA) are benchmarking choices rather than circular evidence. Citations to prior FAS work and to reference [17] supply modeling assumptions and a standard quadratic upper bound, but the paper's secrecy-rate result does not reduce to any self-citation. A possible sign error in the position subproblem would be a correctness issue, not a circularity issue, because it does not make the derived result equivalent to its inputs by construction.

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

The paper introduces no invented entities and fits no parameters to data. The central derivation relies on the geometric far-field channel model, ideal SIC, and the validity of successive convex approximations. Simulation inputs such as r, Pmax, alpha, and D are conventional choices, not fitted values used to force the claimed outcome.

assumptions (4)

domain assumption The far-field geometric channel model holds: antennas move within a region much smaller than the link distance, and AoDs/AoAs remain constant across positions.
Used in Section II-B to define phase shifts via rho_k^p(t_m), which is the basis for h_c(t) and h_e(t). If near-field effects or angle variations occur when the FA moves, the derived channels are inaccurate.
domain assumption Perfect successive interference cancellation at the CU and CEU, and the CEU can decode the CU signal after removing its own signal.
Equations (6) and (7) define rates R_{k,1} without SIC error, and the secure rate Rs=R_{c,1}-R_{e,1} in (8) depends on this idealization.
ad hoc to paper The Taylor expansions used in the MM and FA position updates produce valid bounds for the original nonconvex functions.
The paper asserts the transformed constraints in (18) and (31) are convex and equivalent without proving the bounds are global. This is an ad hoc premise for the algorithm's validity.
domain assumption The simulation channel model, with i.i.d. complex Gaussian path responses and uniform AoDs, is representative of a real FAS-NOMA channel.
Section IV uses this model to generate Figures 1 and 2; if real channels differ substantially, the reported secrecy rate gains may not generalize.

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Pith. "Pith review of A Secure Beamforming Design: When Fluid Antenna Meets NOMA." pith.science (2026). https://pith.science/paper/L5GRLM42

@misc{pith2026241108386,
  author       = {Pith},
  title        = {Pith review of: A Secure Beamforming Design: When Fluid Antenna Meets NOMA},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5GRLM42}},
  note         = {Machine review of arXiv:2411.08386}
}

abstract

This letter proposes a secure beamforming design for downlink non-orthogonal multiple access (NOMA) systems utilizing fluid antenna systems (FAS). We consider a setup where a base station (BS) with $M$ fluid antennas (FAs) communicates to a cell-center user (CU) and a cell-edge user (CEU), each with a FA. The CU is the intended recipient while the CEU is regarded as a potential eavesdropper. Our aim is to maximize the achievable secrecy rate by jointly optimizing the secure beamforming vectors and the positions of FAs. To tackle this, we adopt an alternating optimization (AO) algorithm that optimizes secure beamforming and the positions of the FAs iteratively while keeping the other variables fixed. Numerical results illustrate that when FAs meet NOMA, the proposed scheme greatly enhances the secrecy rate compared to conventional multiple-input single-output (MISO) fixed antenna NOMA systems and other benchmark schemes.

Figures

Figures reproduced from arXiv: 2411.08386 by the authors.

**Figure 2.** The secrecy rate versus antenna M, where the constraint r = 2 bps/Hz and pmax/σ2 = 10 dB. Therefore, the optimization problem of tm is rewritten as max tm,τ,ǫe τ (31a) s.t. (9d), (24), (25), (29) & (30). (31b) Note that the above problem (31) is convex with respect to tm, which can be solved using CVX [16]. We sequentially optimize each transmit FA position tm until the algorithm converges to a fixed t. Accordingly,… view at source ↗

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Reference graph

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Reviewed August 12, 2026 · model on record in the stance chip above.

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Nonorthogonal multiple access for 5G and beyond,

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work page 2017

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work page 2022

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

work page 1950

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A tutorial on ﬂuid antenna system for 6G networks: En- compassing communication theory, optimization methods an d hardware designs,

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Fluid antenna system enhancing orthogonal and non- orthogonal multiple access,

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