Optimizing Fluid Antenna Configurations for Constructive Interference Precoding

Luteng Zhu; Mingjie Shao; Tong Zhang; Wenxuan Sun; Yao Ge; Zhi Liu

REVIEW 5 major objections 5 minor 31 references

Optimizing Fluid Antenna Configurations for Constructive Interference Precoding

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that jointly optimizing fluid-antenna positions and constructive-interference precoding via a smoothed block-coordinate algorithm lowers bit error rate versus fixed arrays and PSO, at low cost.

desk verdict A credible, modest paper: new FAS-plus-CI formulation, sensible BCD solver, but the BER claims rest on an unvalidated SEP surrogate and missing error bars. read the letter →

arxiv 2507.11093 v1 pith:27FWGMS5 submitted 2025-07-15 eess.SP cs.ITmath.IT

Wenxuan Sun , Mingjie Shao , Luteng Zhu , Yao Ge , Tong Zhang , Zhi Liu This is my paper · ORCID

classification eess.SPcs.ITmath.IT

keywords fluidantennasystemsconstructiveinterferenceprecodingsymbolerrorprobabilityMPSKsignalingblockcoordinatedescentsafetymarginmultiuserMISOpositionoptimization

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 asks whether letting each base-station antenna slide within a small local interval, and designing the precoder and the antenna positions together, improves reliability in a multiuser downlink with MPSK signaling. It tries to establish that the two degrees of freedom reinforce each other: constructive-interference precoding benefits from positions that widen each user's safety margin, and the resulting nonconvex, nonsmooth problem can be solved cheaply by smoothing the max objective and alternating between precoder and position updates. If the claim is right, fluid antennas become a practical reliability upgrade rather than a heavy optimization burden, delivering lower bit error rate than fixed antenna arrays and lower runtime than particle-swarm position tuning.

What carries the argument

The load-bearing object is the safety margin $\alpha_{i,t}$, the distance of the noiseless received symbol from the boundary of its MPSK decision region, defined as $\alpha_{i,t} = \Re\{h(z)_i^\top x_t s_{i,t}^*\} - |\Im\{h(z)_i^\top x_t s_{i,t}^*\}|\cot(\pi/M)$. Maximizing the smallest safety margin is justified by a Q-function bound on symbol error probability. To make the problem smooth, the max over users and time slots is replaced by $\psi(\bar X,z)$, a smoothed max with a simplex weight vector and a quadratic penalty, whose optimal weights have a closed form. Block coordinate descent then alternates an accelerated projected gradient update for the precoders and a projected gradient update with backtracking for the positions; the disjoint interval constraints make the position projection an element-wise thresholding, which keeps the per-iteration cost low.

What would settle it

Simulate the optimized CIAP, fixed-array, and PSO-based designs under the paper's settings and compare empirical bit error rates at SNRs in the 20–30 dB range; if CIAP does not beat both baselines there, the central claim fails. A second check is to measure, for CIAP's optimized solutions, the gap between the Q-function bound in inequality (7) and the simulated symbol error rate: if the bound is much looser for CIAP solutions than for baseline solutions, the reported gain may come from optimizing the surrogate rather than from true SEP reduction.

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Extended reading notes

Core claim

The paper's central claim is that worst-case symbol error probability minimization for a fluid-antenna MISO downlink can be reformulated as maximizing the smallest constructive-interference safety margin, and that this reformulation admits a low-complexity solver. The authors convert the max-of-margins objective into a smoothed surrogate, then apply block coordinate descent: an accelerated projected gradient step for the precoding vectors and a projected gradient step with backtracking line search for the antenna positions, with element-wise projection onto disjoint intervals. Simulations are presented showing that this joint design reduces bit error rate below both CI precoding with fixed antenna positions and CI precoding with PSO-optimized positions under QPSK, 8PSK, and 16PSK, with per-iteration complexity $O(KNT)$ for precoding and $O(KN^2T)$ for positions.

Load-bearing premise

The optimization maximizes a stand-in quantity, the smallest safety margin, that is connected to the true symbol error probability only through a Q-function bound, and the paper never checks how tight that bound is at the signal-to-noise ratios where the algorithm is evaluated.

Editorial extensions

If this is right

If the claim holds, constructive-interference precoding and fluid antenna positioning are complementary: a better antenna geometry widens the safety margins that symbol-level precoding can exploit.
The reported per-iteration costs of $O(KNT)$ and $O(KN^2T)$ mean antenna-position optimization adds only modest overhead on top of precoding, making fluid-antenna gains accessible where particle-swarm optimization is too expensive.
Larger position adjustment ranges, such as $\delta = 0.5$, improve BER for both fluid-antenna designs while CIAP keeps the edge, so the benefit grows with antenna mobility.
The BER gains widen as modulation order rises from QPSK to 16PSK, so the method is most relevant when high-order MPSK is used for spectral efficiency.

Reading between the lines

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

Editorial inference: the same smoothing-plus-BCD template could be carried to QAM constellations, where the constructive-interference region is a polygon rather than a cone; the safety-margin geometry would change but the outer loop would not.
Editorial inference: a sensitivity analysis of the smoothing parameter $\mu = 0.3 + \log(1+\sigma)$ is missing from the paper; testing several values across SNR would show whether the reported BER gains are robust.
Editorial inference: since the SEP bound in inequality (7) has a factor-two gap, an independent SEP evaluation would settle whether CIAP is truly minimizing symbol error probability or only maximizing its surrogate.
Editorial inference: the runtime comparison fixes block length at $T=5$; scaling $T$ and the number of users while holding antennas fixed would test whether the complexity advantage persists in longer blocks.

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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, the short version: this is a legitimate, incremental paper. It formulates a genuinely new problem—jointly optimizing nonoverlapping fluid antenna position intervals and constructive-interference precoding under MPSK—and solves it with a sensible BCD scheme: APG for the precoder, PGD for antenna positions, Nesterov smoothing. The simulations are real Monte Carlo runs and show consistent BER gains over fixed arrays and a PSO baseline, plus much lower runtime. If you work in FAS or symbol-level precoding, worth a skim.

What it does well: the interval-constrained antenna model makes the projection trivial (element-wise thresholding), and the complexity table is honest. The SEP bound conversion in (7)-(8) is standard; replacing SEP minimization with worst-case safety-margin maximization is reasonable, and the closed-form inner λ in (17) is a nice implementation detail. The plots cover several MPSK orders and a δ sensitivity check. The citation pattern looks fine—mostly on-topic prior work, no obvious self-citation inflation.

Soft spots, in proportion. The load-bearing weakness is exactly what the stress test flags: the paper never verifies that the Q-function bound in (7) is tight at the optimized operating points. Maximizing the worst-case margin is only a proxy for minimizing worst-case SEP, and the constant factor 2 could in principle reorder solutions. It probably does not matter much in practice, but since the headline claim is BER reduction, a simple plot of true SEP versus the bound for the optimized solution would settle it. Second, no confidence intervals or statistical significance over 500 Monte Carlo trials. At moderate SNR the CIAP and BCD-PSO curves are close, and without error bars the claimed advantage is partly unproven. Third, the smoothing parameter μ = 0.3 + log(1+σ) is ad hoc, with no sensitivity study—minor if the results are robust, but unverified. Fourth, PSO hyperparameters are not disclosed, so the runtime comparison may partly reflect a weak PSO implementation rather than algorithmic superiority. Finally, the BCD loop has no convergence guarantee for the nonconvex antenna block; the APG convergence is block-level only. None of these are fatal, but together they make the empirical claims provisional.

Who it is for: FAS and symbol-level precoding researchers wanting a low-complexity baseline for joint position-precoding design. It deserves serious peer review—the formulation is new and the method is clean—but a referee should ask for error bars, a surrogate-tightness check, and PSO details. I would send it out, with revision expected.

Referee Report

5 major / 5 minor

Summary. This paper studies a downlink multiuser MISO system with fluid antennas whose positions are individually adjustable within disjoint intervals, under MPSK signaling. The authors formulate a worst-case symbol error probability (SEP) minimization problem, replace it with a safety-margin maximization based on a Q-function bound, smooth the nonsmooth objective, and propose a block coordinate descent (BCD) algorithm that alternates between an accelerated projected gradient (APG) precoding update and a projected gradient antenna-position update. Simulations compare the proposed CIAP algorithm against fixed-position arrays and a BCD-PSO baseline, reporting BER gains and lower runtime, with a per-iteration complexity analysis.

Significance. The paper addresses a previously unstudied combination of fluid antenna systems and constructive interference precoding, and the proposed BCD framework is a sensible starting point. The convex precoding subproblem is clearly formulated and solved by APG with an explicit per-iteration complexity table, and the disjoint-interval antenna model leads to a cheap element-wise projection. The simulation study uses a standard LoS channel model and reports results over 500 Monte-Carlo trials. If the surrogate fidelity and convergence issues raised below are resolved, the approach could offer a practical low-complexity FAS design.

major comments (5)

[Section III, Eqs. (7)-(8)] The paper replaces SEP minimization with safety-margin maximization, justified by the bound Q(√2 sin(π/M)α/σ) ≤ SEP ≤ 2Q(√2 sin(π/M)α/σ). However, no evidence is given that maximizing α is equivalent to minimizing SEP at the solutions found. The upper and lower bounds differ by a factor of 2, so the ordering of candidate solutions by α need not match the ordering by true SEP. Please report the worst-case SEP or BER under the proposed solution and compare it with the value predicted by the bound for the operating SNRs in Figs. 2-4. Without this, the central claim that CIAP reduces BER is not established by the optimization objective.
[Section IV-B, Algorithm 1] The overall BCD loop for the nonconvex problem (13) has no convergence guarantee. The text states that APG 'guarantee[s] global convergence,' but the cited result applies only to the convex subproblem (18) with fixed z; the position subproblem (22) is nonconvex, and BCD with an inexact inner solver may not converge to a stationary point. The authors either need to provide a convergence analysis of Algorithm 1 or state it as a heuristic and include a numerical convergence check (e.g., objective and worst-case margin versus iteration) in the simulations.
[Eq. (19), Algorithm 2 line 7] The APG update in Eq. (19) uses the gradient ∇_{x_t}ψ(¯X_k,z_ℓ) evaluated at the current iterate ¯X_k, not at the extrapolated point y_{t,k} defined in (20). Standard accelerated projected gradient methods evaluate the gradient at y_k. The algorithm as written (with gradient at x_k) does not match the convergence guarantee from [29]; it is a projected gradient step with momentum, and the claimed 'guarantee global convergence' is not applicable. Please either evaluate the gradient at y_{t,k} (i.e., use ∇_{x_t}ψ(¯Y_k,z_ℓ) in (19) and Algorithm 2) or revise the algorithm and its justification.
[Section V, Figs. 2-4] The average BER curves are reported over 500 Monte-Carlo trials without confidence intervals or error bars. The visible gap between CIAP and BCD-PSO at moderate SNR (e.g., Fig. 2 around 20-25 dB) appears small relative to the random variation expected from 500 trials. Please provide error bars or a statistical significance test (e.g., a paired t-test) to support the claim that CIAP outperforms BCD-PSO.
[Section V, Fig. 6 and parameter settings] The smoothing parameter μ = 0.3 + log(1+σ) in Section V is introduced without sensitivity analysis. Since ψ depends on μ, the reported BER improvements may depend on this ad hoc choice; please provide results for a range of μ values and justify the formula. Also, the BCD-PSO baseline does not report its hyperparameters (population size, maximum iterations, inertia weights, etc.), which makes the runtime comparison in Fig. 6 difficult to interpret as a fair algorithmic comparison. Please disclose the PSO settings and, ideally, provide a comparison with a well-tuned PSO.

minor comments (5)

[Section III, after Eq. (10)] The text says 'Problem (9) can be transformed into a minmax problem as (12)', but (9) is a definition of vectors, not a problem; the reference should be to (10).
[Section II, Fig. 1] The figure is referenced but not included in the text; please ensure it is inserted in the final version.
[Section V, legends] The legend uses 'FP A' in Fig. 2 and 'FPA' in Fig. 5; please keep the baseline name consistent.
[Algorithm 3] The backtracking line search step length η_B is not defined; please specify the backtracking rule or cite a reference for the implementation.
[Section V, first paragraph] The sentence 'In this setting, the parameter δ ∈ [0, 0.5] is set to satisfy the minimum spacing requirement λ/2 between antennas' is not fully clear; for δ=0.1 the effective minimum separation is different, so please state how δ is chosen for a given minimum spacing.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity; the SEP-to-safety-margin surrogate is an external analytic bound, and the reported BER gains come from genuine Monte Carlo simulations.

full rationale

The derivation chain is problem (6) -> Q-function bound (7) -> safety-margin maximization (8) -> smoothed BCD (13). None of these steps is circular. The bound (7) is an external analytic result cited to [27] and [28], not a quantity fitted to this paper's outputs, and [28] is an independent group's work, so the self-citation in [27] is not load-bearing. The algorithm optimizes the safety margin, while the claimed gains are evaluated by actual bit error rate simulations over 500 Monte-Carlo trials; the BER curves are not mathematical consequences of the optimization objective. A mismatch between the surrogate and true SEP would be a correctness or approximation-fidelity concern, not a circularity. Several references are by members of the same research group (e.g., [2], [10], [19], [20], [27]), but they supply standard bounds, algorithmic tools, or background results and are not used to define the target quantity in terms of itself. The smoothing parameter mu = 0.3 + log(1+sigma) is chosen heuristically without sensitivity analysis, which is a robustness concern, but it is not fitted to the data being predicted. No prediction in the paper reduces by construction to its inputs, and no self-citation chain forces the reported result. Therefore the circularity score is low.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard optimization tools and a prior CI bound. The main ad hoc element is the smoothing parameter formula, and the main domain risk is the LoS channel model together with the SEP proxy.

free parameters (2)

Smoothing parameter mu = 0.3 + log(1 + sigma)
Controls how closely psi approximates the nonsmooth max objective; the formula is chosen by hand without a sensitivity analysis. It appears in Section IV.A and the simulation setup.
Position adjustment factor delta = 0.1 default, 0.5 in Fig. 5
Defines the feasible interval size for each antenna position; set by hand per experiment, not optimized, and it determines the flexibility of the antenna placement.

assumptions (4)

domain assumption The SEP can be bounded by the Q-function of the safety margin alpha as in Eq. (7)
The bound is standard in CI precoding literature, but the paper relies on it to replace SEP minimization with margin maximization.
domain assumption The downlink channel is a far-field LoS model h_i = alpha_i a(z, beta_i)
Used throughout the system model; ignores multipath and assumes known departure angles and propagation gains.
ad hoc to paper The smoothing function psi with mu = 0.3 + log(1 + sigma) is a valid surrogate for the nonsmooth objective
The formula is not derived and the approximation accuracy at operating SNR is unquantified.
domain assumption Antenna positions move within disjoint intervals that preserve a minimum spacing of lambda/2
The interval model (2n-1)D/2N plus or minus delta D/2N is the paper's design choice to decouple the position constraints and enable element-wise projection.

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Pith. "Pith review of Optimizing Fluid Antenna Configurations for Constructive Interference Precoding." pith.science (2026). https://pith.science/paper/27FWGMS5

@misc{pith2026250711093,
  author       = {Pith},
  title        = {Pith review of: Optimizing Fluid Antenna Configurations for Constructive Interference Precoding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27FWGMS5}},
  note         = {Machine review of arXiv:2507.11093}
}

abstract

The fluid antenna system (FAS) has emerged as a new physical-layer concept to provide enhanced propagation conditions for multiuser multiple-input multiple-output (MIMO) communications over conventional fixed arrays. This work focuses on minimizing the maximum symbol error probability (SEP) under $M$-ary phase shift keying (MPSK) signaling in a multiuser downlink equipped with FAS, where each antenna moves within nonoverlapping intervals. This specific problem of joint SEP minimization with FAS and constructive interference (CI) precoding has not been previously addressed. The resulting problem turns out to be a nonconvex and nonsmooth optimization challenge. We transform the SEP minimization problem into a safety margin maximization problem in constructive interference precoding. Then, we customize a smoothing technique and a block coordinate descent (BCD) algorithm, with emphasis on low computational complexity. Simulation results show that our approach can reduce bit error rate (BER) compared to both the fixed arrays and FAS designed by existing particle swarm optimization (PSO). Also, our approach shows attractively low computational complexity compared to PSO benchmarks.

Figures

Figures reproduced from arXiv: 2507.11093 by the authors.

**Figure 1.** Illustration of system model We consider a far-field communication scenario. The downlink channel from the BS to the ith user follows a line-of-sight (LOS) model, which can be modeled as hi = αia(z, βi), i = 1, . . . , K, (1) where αi denotes the propagation gain and βi ∈ [0, π] is the departure angle for the ith user. The steering vector a(z, βi) ∈ C N is defined by a(z, βi) =      e −j 2π λ cos(βi)z1 e −j 2π… view at source ↗

**Figure 4.** Average BER performance versus SNR; 16PSK [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Average BER performance for different δ computationally efficient than BCD-PSO. Next, we investigate the impact of the size of the antenna position region. We fix the total range D unchanged and alter the inter antenna spacing controlled by the parameter δ. We simulate the algorithms for N = 10, K = 10 with 16PSK modulation, comparing the BER performance for δ = 0.1 and δ = 0.5. As shown in [PITH_FULL_IMAGE:figures… view at source ↗

Figures from the paper (1 more)

**Figure 6.** Figure 6: Average runtime in different problem sizes [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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