Energy Efficiency Maximization for Movable Antenna-Enhanced System Based on Statistical CSI

Biqian Feng; Wenjun Zhang; Xintai Chen; Yongpeng Wu

REVIEW 3 major objections 4 minor 3 cited by

Energy Efficiency Maximization for Movable Antenna-Enhanced System Based on Statistical CSI

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Movable antennas under statistical CSI can raise MIMO energy efficiency by about 7–15% over fixed arrays, with a finite movement region sufficient for near-optimal performance.

desk verdict Solid incremental MA-EE design under statistical CSI, but the 7-15% gains rest on an unvalidated large-system approximation at N=M=4; needs a Monte Carlo check before I'd trust the ranking. read the letter →

arxiv 2501.10694 v1 pith:HHKCBVFR submitted 2025-01-18 cs.IT eess.SPmath.IT

Xintai Chen , Biqian Feng , Yongpeng Wu , Wenjun Zhang This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords movableantennasfluidenergyefficiencystatisticalCSIMIMOdeterministicequivalentalternatingoptimizationsuccessiveconvexapproximation

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 tries to show that a MIMO link whose antennas can be repositioned inside small regions can be made more energy-efficient than a fixed-array link even when the transmitter knows only statistical channel state information, not the instantaneous channel. The authors formulate energy efficiency as the ratio of the average achievable rate to an affine power-consumption model, and propose an alternating-optimization algorithm that updates the transmit covariance matrix and the antenna position vectors in turn. Because the expectation in the rate is intractable, they approximate it with a deterministic equivalent derived for large antenna arrays. Their numerical study reports gains of 7.3%, 7.1%, and 14.6% in energy efficiency over one-sided movable-antenna and fixed-uniform-array benchmarks, and shows that the gain saturates once the antenna movement region reaches about 1.4 wavelengths.

What carries the argument

The load-bearing object is the deterministic equivalent of the average achievable rate, Eqs. (10)–(15), which replaces the expectation over the random scattered path matrix with fixed-point equations for $\Phi$, $\Theta$, $\tilde{\Phi}$, $\tilde{\Theta}$ and the matrix-valued functions $\eta$, $\tilde{\eta}$. This converts the stochastic rate into a closed-form function of the transmit covariance matrix and of the antenna position vectors separately, making the EE ratio tractable inside an alternating-optimization loop; the inner problems are then handled by Dinkelbach's method, water-filling, and successive convex approximation with numerical gradients.

What would settle it

Run a Monte Carlo estimate of the true average rate in Eq. (7) at $N=M=4$ for the same channels and compare the resulting EE landscape with the DE-based objective used in Figs. 1–2; a systematic gap or a shift of the saturation point would show that the reported gains are artifacts of the approximation.

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Extended reading notes

Core claim

The central claim is that energy-efficiency maximization for a movable-antenna MIMO system under statistical CSI can be reformulated into deterministic subproblems and solved by an alternating-optimization algorithm that generates a non-decreasing sequence of EE values. On the transmit side, the algorithm alternates between a Dinkelbach/water-filling update of the covariance matrix and an SCA update of the transmit antenna positions; on the receive side, it runs an SCA update of the receive positions against a quadratic surrogate. With $N=M=4$ antennas and a Rician channel, the optimized system is reported to outperform the benchmarks at every power level and region size, reaching 7.3%, 7.1%, and 14.6% higher EE than the TMA, RMA, and UPA schemes at region size $X=2.2\lambda$, while EE saturates for region sizes above roughly $1.4\lambda$.

Load-bearing premise

The assumption that the deterministic-equivalent reformulation accurately matches the true average achievable rate at the small antenna count simulated ($N=M=4$), where the technique's large-system justification is not guaranteed to hold.

Editorial extensions

If this is right

A practical MA transceiver can reap EE gains using slowly varying statistical CSI, avoiding the delay and power cost of instantaneous feedback and antenna movement per coherence block.
The saturation of EE with region size means the antennas do not need large travel ranges; a movement region of roughly $1.4\lambda$ captures nearly all of the gain.
Because the transmit covariance update has closed-form Dinkelbach/water-filling structure, the AO algorithm scales to larger antenna counts at polynomial cost, with the dominant term $O(L_{\mathrm{ao}}\hat{L}_t N^4)$ coming from the SCA position update.
Deploying MAs on only one link side already beats fixed uniform planar arrays, so one-sided retrofits are a viable intermediate step.

Reading between the lines

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

If the DE approximation is tight only asymptotically, the exact small-system gains may differ from the reported percentages; a Monte Carlo check at $N=M=4$ would settle which part of the claimed 7–15% is real.
The same AO/DE machinery should extend to multiuser or secure MIMO, where the average rate is replaced by a weighted sum rate or secrecy rate, and the saturation-with-region-size behavior would suggest how large movement regions need to be in those settings.
As $N$ and $M$ grow, the finite-region saturation hints at a diminishing-returns law in which the optimal region diameter stays bounded in wavelengths rather than growing with aperture; this is a testable prediction the paper does not make.

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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

Quick take: this is a legitimate incremental result—the first EE-maximizing MA-MIMO design under statistical CSI—but the numerical claims lean on a deterministic equivalent (DE) approximation at N=M=4 with no Monte Carlo check. The DE error could be comparable to the 7.3–14.6% gains, so the ranking and the saturation behavior might not survive a finite-size check.

What's new: the problem formulation (EE with S-CSI for MA-MIMO) and the AO algorithm combining DE, Dinkelbach, water-filling, and SCA. The transmit-side Dinkelbach and water-filling are standard, but the EE formulation with APV optimization appears new. The paper is well grounded in prior work and doesn't overclaim conceptual novelty.

What's done well: the DE-based reformulation is clearly presented, the algorithms are specified enough to reproduce, complexity analysis is included, and the benchmark comparisons are meaningful. As an engineering design procedure, it's coherent.

Main soft spot: the DE is derived from large-system asymptotics, but the simulations use N=M=4, L=5. The objective being maximized is the approximate DE, not the true average rate, and Figs. 1–2 plot that approximation. The gains are small enough that a few percent DE error could flip the ranking. I'd want a Monte Carlo validation of the DE at those sizes, or at least a quantitative statement of agreement, before trusting the numbers.

Second: the convergence analysis only shows objective-value non-decreasing, not stationarity of the limit point. The contribution section claims a stationary point is "efficiently developed," but that isn't proven. This is a weaker claim than stated.

Third: the receive surrogate (Eq. 34) is asserted without derivation; there is no proof it's a valid concave lower bound. SCA convergence depends on that property. The transmit side is clearer.

Fourth: the simulation setup draws random AoDs/AoAs, but the text doesn't say whether the plotted curves are single realizations or averaged over many. No error bars, no trial count. That should be explicit.

Overall: the core idea is sensible and the paper is a careful incremental contribution. The gaps are fixable. I'd send it to review, but the referees should insist on a finite-size DE check and a proper convergence statement. If the DE error is small, this is a solid engineering result; if not, the main numerical conclusions lack evidence.

Referee Report

3 major / 4 minor

Summary. The paper studies a single-user MIMO system with movable antennas at both transmitter and receiver, where only statistical CSI is available. The authors formulate the maximization of the long-term energy efficiency over the transmit covariance matrix and the antenna position vectors, subject to minimum-distance and power constraints. Since the expected rate is difficult to evaluate, they adopt deterministic equivalent (DE) approximations from earlier work to obtain surrogate objectives for the transmit and receive variables, and then propose an alternating optimization (AO) algorithm that updates the transmit covariance and positions via Dinkelbach/water-filling and successive convex approximation (SCA). Numerical results with N=M=4 report energy-efficiency gains of 7.3%, 7.1%, and 14.6% over TMA, RMA, and UPA benchmarks at a movement-region size X=2.2λ, and show saturation of the EE as the region size grows.

Significance. If the numerical claims are correct, the paper offers a tractable S-CSI-based design for MA-enhanced MIMO and demonstrates a useful, if modest, energy-efficiency advantage over fixed arrays, including the practically relevant observation that a finite movement region suffices. The paper builds on established DE results rather than re-deriving them, and the algorithmic structure is reasonable. However, the central numerical evidence currently rests on an unvalidated large-system approximation at small system dimensions and on an incomplete convergence argument. The contribution is therefore promising but not yet convincingly established.

major comments (3)

[Sec. III-A, Eqs. (10)-(15), and Sec. IV, Table I, Figs. 1-2] The DE reformulation in Eqs. (10)-(15) is derived under large-system asymptotics in refs. [16] and [17], yet all numerical results use N=M=4 with L=5 paths (Table I). The paper optimizes and plots the DE objectives R̄_t and R̄_r, not the true expectation R in Eq. (7), and no Monte Carlo validation, finite-size error bound, or sensitivity analysis is reported. Because the claimed improvements over benchmarks are only 7.3%, 7.1%, and 14.6% at X=2.2λ (Sec. IV.B), a DE error of a few percent could change the ranking of schemes or make the saturation in Fig. 2 an artifact of the approximation. The authors should add a finite-size validation of the DE against Monte Carlo rates for the simulated N=M=4 setting and clearly state whether the plotted curves are the DE objectives or the true average rates.
[Sec. III-E, Eqs. (22)-(24), (34)-(36)] The convergence analysis in Sec. III-E only shows that the AO sequence has a non-decreasing objective value and that the feasible set is compact; this does not imply convergence to a stationary point of the original problem (9) or even of the DE-reformulated problem. In particular, the transmit update uses a finite-difference gradient in Eq. (20) and an Armijo line search in Eq. (24), but the receive surrogate in Eq. (34) is introduced without a proof that it is a valid concave lower bound with the gradient-matching property, and the statement that the receive update is 'similar' to the transmit case is not a substitute. The authors should either provide a formal SCA convergence argument (sufficient decrease and gradient consistency) or explicitly state the weaker convergence guarantee that is actually established.
[Sec. IV, Figs. 1-2] The simulation setup states that distances and AoDs/AoAs are random, but the paper never specifies how many random realizations are averaged or whether confidence intervals are used. Without this information, the smooth curves in Figs. 1-2 could correspond to a single channel realization, in which case the reported gains are not statistically meaningful, or to an unspecified averaging procedure that should be described. The EE values and the comparison at X=2.2λ need error bars or at least an explicit statement of the number of trials and the averaging method.

minor comments (4)

[Algorithm 1, Step 5] The instruction 'Construct R̄_r(r) by iterative process (12)' is ambiguous because R̄_r in Eq. (14) depends on t and Q through Γ, Θ, and Φ as well as on r; the step should state explicitly that t and Q are fixed at their most recent values before solving problem (17).
[Eq. (20) and Table I] The finite-difference step ε1=10^-3 is used to compute numerical gradients for the transmit SCA, but no sensitivity study with respect to ε1 is reported; the accuracy of the optimized positions may depend on this parameter, especially because Q(t) in Eq. (18) is itself evaluated by an iterative inner procedure.
[Eqs. (10)-(15)] The notation for the auxiliary matrices Φ, ~Φ, Θ, and ~Θ is dense and their dimensions are not stated in the main text; adding a short paragraph that defines the dimensions and the fixed variables in each DE expression would improve readability.
[Fig. 2] The axis label of Fig. 2 shows only 'Energy Efficiency (bps/Hz)'; since the denominator in Eq. (8) includes power, the unit should be bits/s/Hz/W (or per Joule), and this should be stated consistently in the text.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: DE machinery comes from independent prior work; minor self-citations are background only.

full rationale

The load-bearing derivation chain is: start from the stochastic average rate R(t,r,Q) in Eq. (7); invoke the deterministic equivalents from refs. [16] and [17] (by other author groups) to obtain Eqs. (10)-(15); then optimize the resulting approximate objectives by AO/SCA/Dinkelbach. None of these steps defines the target quantity in terms of the fitted output: the DE expressions are not fitted to the EE curves, no parameter is calibrated to the gains in Figs. 1-2, and all benchmarks are evaluated under the same objective. The only self-citations are refs. [6] and [9], which are used as background for planar movement and for the S-CSI MA setup; neither carries a load-bearing theorem or substitutes for the derivation here. The main caveat is a correctness/robustness risk, not circularity: the DE is a large-system asymptotic approximation, and the paper uses it at N=M=4 without Monte Carlo validation, so the 7-15% EE gains could in principle be affected by DE error. But approximating an expectation by a known asymptotic equivalent is not a self-referential reduction; the approximation is imported from independent work and is not equivalent to the paper's conclusions by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on two imported assumptions: the accuracy of the deterministic equivalent at small antenna numbers, and the validity of the SCA surrogates. The former is inherited from asymptotic theory without finite-size verification; the latter are asserted with only a monotonicity argument. Algorithm hyperparameters (δt, δr, step sizes, Armijo parameters) are chosen by hand and are not derived from the problem.

free parameters (6)

Strong concavity parameter for transmit SCA (δt) = 0.02
Chosen by hand (Table I) to ensure strong concavity of the surrogate (22); no theoretical guidance for selection.
Strong concavity parameter for receive SCA (δr) = 0.02
Set to 0.02 in Table I, same role as δt in surrogate (34).
Initial step size (τ0) for backtracking line search = 1
Set in Table I; controls convergence speed but not part of the physical model.
Step size reduction factor (τ) = 0.5
Geometric reduction factor in the line search (Sec. III-B).
Armijo control parameter (ξ) = 0.6
Parameter in the Armijo-Goldstein condition (24); chosen by hand.
Finite difference step (ε1) = 1e-3
Used in numerical gradient (20); affects accuracy of the gradient estimate.

assumptions (5)

domain assumption Deterministic equivalent formulas (10)-(15) are valid for the considered channel model and system sizes.
The DE is derived from asymptotic analysis (large N,M) in refs. [16],[17]; the paper applies it to N=M=4 without verification against Monte Carlo.
domain assumption The Rician channel model with far-field plane waves, fixed AoDs/AoAs, and i.i.d. Gaussian scattering paths holds.
Assumed in Sec. II-A; standard in MA literature but restricts applicability to rich-scattering environments with sufficient angular spread.
domain assumption Minimum antenna spacing D ≥ λ/2 avoids mutual coupling.
Invoked in Sec. II (C1, C2) following ref. [12]; this is a conventional engineering approximation.
domain assumption The affine power consumption model Ptot = ω tr(Q) + N Pc + Ps is accurate.
Used in Sec. II-B based on ref. [15]; ignores rate-dependent and position-dependent power variations.
ad hoc to paper The SCA surrogate (22) with the chosen δt, combined with the Armijo line search, yields monotone improvement of the true EE.
The paper does not prove that the surrogate is a lower bound of EE(t); it asserts strong concavity and relies on the Armijo condition (24) to ensure actual objective increase. The same holds for the receive surrogate (34).

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Cite this review

Pith. "Pith review of Energy Efficiency Maximization for Movable Antenna-Enhanced System Based on Statistical CSI." pith.science (2026). https://pith.science/paper/HHKCBVFR

@misc{pith2026250110694,
  author       = {Pith},
  title        = {Pith review of: Energy Efficiency Maximization for Movable Antenna-Enhanced System Based on Statistical CSI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHKCBVFR}},
  note         = {Machine review of arXiv:2501.10694}
}

read the original abstract

This paper investigates an innovative movable antenna (MA)-enhanced multiple-input multiple-output (MIMO) system designed to enhance communication performance. We aim to maximize the energy efficiency (EE) under statistical channel state information (S-CSI) through a joint optimization of the transmit covariance matrix and the antenna position vectors (APVs). To solve the stochastic problem, we consider the large number of antennas scenario and resort to deterministic equivalent (DE) technology to reformulate the system EE w.r.t. the transmit variables, i.e., the transmit covariance matrix and APV, and the receive variables, i.e., the receive APV, respectively. Then, we propose an alternative optimization (AO) algorithm to update the transmit variables and the receive variables to maximize the system EE, respectively. Our numerical results reveal that, the proposed MA-enhanced system can significantly improve EE compared to several benchmark schemes and the optimal performance can be achieved with a finite size of movement regions for MAs.

Figures

Figures reproduced from arXiv: 2501.10694 by the authors.

**Figure 1.** EE versus the maximum transmit power. 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 Energy Efficiency (bps/Hz) MA TMA RMA UPA [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** EE versus the size of the movement region. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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Reference graph

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Reviewed August 10, 2026 · model on record in the stance chip above.

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