Energy Efficiency Maximization for Movable Antenna Communication Systems

Bingli Jiao; Jingze Ding; Lipeng Zhu; Rui Zhang; Yuping Zhao; Zijian Zhou

REVIEW 3 major objections 6 minor 44 references

Energy Efficiency Maximization for Movable Antenna Communication Systems

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

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Pith's one-line read Accounting for the energy and delay of antenna movement in the optimization objective, rather than after the fact, is what makes movable-antenna uplinks energy-efficient.

desk verdict Solid single-user analysis; the multi-user max-min EE algorithm optimizes a different objective than the one defined in (19)-(21), so the central claim needs a major correction. read the letter →

arxiv 2506.07129 v2 pith:6UC3CCQU submitted 2025-06-08 cs.IT eess.SPmath.IT

Jingze Ding , Zijian Zhou , Lipeng Zhu , Yuping Zhao , Bingli Jiao , Rui Zhang This is my paper · ORCID

classification cs.ITeess.SPmath.IT MSC 94A05

keywords movableantennaenergyefficiencysteppermotormulti-useruplinkmax-minfairnesssuccessiveconvexapproximationfield-responsechannelmodelpositionoptimization

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 movable antennas are worth their flexibility only when the energy and time spent physically moving them are put inside the optimization objective. The authors model each user's stepper motor as consuming a fixed energy per meter of movement, and the transmission block as pausing data while antennas travel, then maximize the minimum user energy efficiency by jointly choosing receive combining, transmit powers, and antenna positions. For a single user they propose a one-dimensional exhaustive search with a closed-form upper bound; for multiple users they give an alternating successive-convex-approximation algorithm whose objective provably converges. Simulations show the resulting design beats both movable-antenna schemes that ignore movement costs and conventional fixed-position antennas, and is robust to coarse angle information.

What carries the argument

The load-bearing object is the energy-efficiency ratio $EE_k = R_k/E_k$ with $R_k = (T - \max_j |x_j - x_0^j|/v_j)\log_2(1+\Gamma_k)$ and $E_k = \bar{E}_k |x_k - x_0^k| + (p_k/\eta_{c,k})(T - \max_j |x_j - x_0^j|/v_j)$. Two components feed it: Proposition 1's linear stepper-motor energy model, derived from constant-torque constant-current motor equations, and the field-response channel model $\mathbf{h}_k = \mathbf{G}_k^H \mathbf{f}_k(x_k)$, which expresses channel gain as a sum of cosines in position. The optimization is carried by a fractional-programming transform that linearizes the ratio, then by successive convex approximation: quadratic surrogate functions bound the cosine terms, converting the power and position subproblems into convex programs that can be iterated.

What would settle it

Measure the actual drawn energy of a two-phase stepper motor moving the antenna over distances from 0.1λ to 2λ at speeds from 0.005 m/s to 0.2 m/s with realistic acceleration ramps; if the per-meter energy is not approximately constant across speed and direction, Proposition 1 fails and the optimized energy-efficiency trade-offs shift.

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Extended reading notes

Core claim

The paper's central claim is that energy efficiency, not throughput or signal-to-interference-plus-noise ratio, should drive antenna movement decisions. Proposition 1 reduces the motor cost to $E_{m,k} = \bar{E}_k |x_k - x_0^k|$, so each user's total energy is movement energy plus transmission energy over the remaining block, and the throughput per block shrinks by the longest user's movement delay. From this, the single-user analysis shows that a one-path channel leaves the antenna at its starting point, while with multiple paths the attainable performance depends on whether the starting position already sits at a channel-gain peak. The multi-user algorithm alternates a closed-form MMSE combining update, a convex power allocation, and a convexified position update, with monotone convergence to a suboptimal solution of the max-min problem. The authors' stated conclusion is that the movement-aware design outperforms throughput-only or SNR-only MA designs and fixed-position antennas in energy efficiency, especially when moving regions are large or motor energy rates are high.

Load-bearing premise

The results depend on the premise that moving an antenna costs energy exactly proportional to distance at a fixed per-meter rate, regardless of speed, acceleration, stopping accuracy, or load; if real stepper motors charge more for fast or jerky moves, the optimized trade-offs shift.

Editorial extensions

If this is right

When moving regions are large, throughput-maximizing or SNR-maximizing MA schemes can consume so much movement energy and time that their energy efficiency falls below fixed-position antennas; the proposed EE-maximizing scheme keeps movement distances stable.
In a single-user system with one channel path, the optimal antenna position is the initial position, because movement only subtracts from the communication phase.
The multi-user algorithm converges monotonically, reaching stability within about fifteen iterations, so the max-min EE design is computationally feasible for small to moderate networks.
The design tolerates coarse angle information: quantizing virtual angles at resolutions around 10 to 15 levels preserves most of the energy-efficiency gain.

Reading between the lines

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

If a stepper motor's real energy draw has a speed-dependent component, the same alternating framework could absorb it by making the per-meter energy rate a function of speed, though the simple distance-only structure and the closed-form insights would need revision.
The pause-transmission assumption makes the design most attractive for slowly varying channels; for fast-moving terminals the movement delay dominates, and a natural extension is to schedule repositioning less often across blocks.
The simulation pattern of staying near the initial position when movement costs are high suggests a deployable heuristic: move an antenna only when the predicted channel-gain gain exceeds the movement energy plus the lost transmission time, which the proposed algorithms already implement implicitly.

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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: the single-user half of this paper is clean and worth reading; the multi-user half, which is the headline contribution, has a reformulation error that means Algorithm 2 is not actually maximizing the EE defined in (19)-(21).

What is genuinely new and good: the movement delay and energy cost model is a reasonable extension of prior MA work, and the single-user Dinkelbach update, the power-clamping formula (31), and the upper bound in Proposition 2 all check out. The 1D exhaustive search is simple but appropriate, and the performance analysis for one-, two-, and multi-path cases is competently done. If the paper stopped there, it would be a modest but sound contribution.

The soft spot is the load-bearing one. In the position subproblem (52), constraint C10 forces xi1 to be at least every user's movement delay, so at optimum xi1 equals the maximum delay. Constraint C11 forces xi2 to be at most every user's delay, so xi2 equals the minimum delay. But the surrogate objective in C13 charges movement energy using xi1 and communication energy using (T - xi2). That means the movement-energy term is scaled by the global max delay times each user's speed, and the communication-energy term is charged against T minus the min delay rather than T minus the max delay. For users whose delay is neither the max nor the min, the surrogate EE in (63) does not match the actual EE in (21). The convergence proof makes this worse: equality (a2) claims xi1 = xi2 = max delay, which violates C11 unless every user happens to have the same movement delay. As written, the converged value is a lower bound on a different objective, and the paper's claim that Algorithm 2 maximizes the minimum EE in (19)-(21) is not established.

Secondary issues: the stepper-motor model in Proposition 1 assumes constant torque and speed-independent energy per meter, with no experimental validation; the simulations have no error bars or released code; and the closest prior work on EE for MA systems, [34], is never benchmarked. These are addressable and less serious than the multi-user reformulation error.

My recommendation: send it to a serious referee, but direct the referee to the multi-user position subproblem. The single-user part plus the upper bound could be a publishable short paper on its own; the multi-user claim needs substantial revision before the results can be trusted.

Referee Report

3 major / 6 minor

Summary. The paper studies energy-efficiency maximization in an uplink multi-user system where each user has a single movable antenna (MA) moving on a 1D track. The system model includes the time and energy cost of antenna movement through a stepper-motor model, and the objective is to maximize the minimum user energy efficiency by jointly optimizing the BS receive combining matrix, transmit powers, and MA positions. For the single-user case, the paper proposes a 1D exhaustive-search algorithm with an inner Dinkelbach power update and derives an upper bound on energy efficiency under different numbers of channel paths. For the multi-user case, the paper proposes an SCA-based alternating algorithm that solves power and position subproblems, and reports simulation results showing gains over MA schemes that ignore movement costs and over fixed-position antennas. The central claim is that the proposed algorithms maximize the min-user energy efficiency defined in Eqs. (19)-(21).

Significance. If the central claim were established, the paper would make a useful practical contribution by including movement delay and energy in MA design, an aspect that is often ignored. The single-user analysis is largely sound: the Dinkelbach update in (27), the power formula in (29)-(31), and the upper-bound inequality in Proposition 2 are internally consistent and provide useful insight into when antenna movement is not worthwhile. The multi-user SCA structure is also a reasonable template. However, the position subproblem in Section IV-B maximizes a modified objective that does not match the EE defined in (19)-(21), and the convergence proof in (64) contains a false equality. These issues are load-bearing because they directly affect the paper's headline claim that Algorithm 2 maximizes the minimum energy efficiency of the original problem. The simulation comparisons and the robustness discussion are therefore not sufficient, as submitted, to support the stated conclusions.

major comments (3)

[Section IV-B, Eqs. (52)-(63)] The position subproblem does not optimize the energy efficiency defined in Eqs. (19)-(21). In (52), constraint C10 forces xi1 >= |x_k - x0_k|/v_k for all k, and the objective term -alpha_k Ebar_k xi1 v_k is decreasing in xi1, so at optimum xi1 equals the global maximum movement delay. Constraint C11 forces xi2 <= |x_k - x0_k|/v_k for all k, and the term -alpha_k p_k/eta_c,k (T - xi2) is increasing in xi2, so at optimum xi2 equals the global minimum movement delay. Consequently, for a user whose movement delay is neither the maximum nor the minimum, the movement-energy term in C13 uses v_k times the global maximum delay instead of the user's own distance, and the communication-energy term uses T minus the global minimum delay instead of T minus the global maximum delay. Thus (63) maximizes a lower-bound surrogate of the Dinkelbach objective, not the original EE, and the converged value ς is not the max-min EE of problem (22). The claim holds only when all users' movement delays coincide.
[Section IV-C, Eq. (64)] Equality (a2) in the convergence proof states that constraints C10 and C11 are both active with xi1^{i-1} = xi2^{i-1} = max_k |x_k^{i-1}-x0_k|/v_k. This contradicts C11, which requires xi2 <= |x_k-x0_k|/v_k for every k. If any user's movement delay is smaller than the maximum, xi2 cannot equal the maximum delay without violating C11. The equality can hold only when all users' movement delays are identical. Equality (a5) relies on the same assumption. Therefore the monotonic convergence argument does not establish convergence to a solution of problem (45); it can at most establish monotonicity for the modified surrogate objective introduced in (52)-(63).
[Section III-B3, Eqs. (39)-(41)] The derivation of the period of G_{L>2}(x) is not rigorous. The chain of equivalences in (39)-(40) claims that equality of the sums implies (1 - e^{j 2pi chi theta_ab / lambda}) = 0 for all a,b. This is only a sufficient condition; the double sum can vanish through cancellation of cross terms even when no individual phase difference is an integer multiple of 2pi. Hence the expression chi = Q lambda / (2c) in (41) is not established as the minimum period of the multi-path channel gain, and the subsequent conditions for achieving the upper bound and the 'Quantized' simulation scheme built on this period are not supported as stated.

minor comments (6)

[Equation (26)] Equation (26) is the Dinkelbach-transformed objective, not the energy efficiency in (24); the text 'can be rewritten as' is misleading and should be rephrased, e.g., 'the Dinkelbach surrogate objective is'.
[Algorithm 1] The 1D exhaustive search discretizes the moving region and represents each sub-region by its center, so the statement that 'the optimal solutions to problem (25) can be obtained' is an approximation. A bound on the discretization error or a clear statement that the result is near-optimal would be appropriate.
[Equation (40)] The double arrow '⇔' in (40) should be '⇒' because the condition (1 - e^{j2pi chi theta_ab / lambda}) = 0 is only sufficient for the sum equality.
[Algorithm 1 and Algorithm 2] Algorithm 1 contains duplicated words 'and and' in lines 5 and 8. Also, the initialization of Algorithm 2 specifies xi1^0 but not xi2^0, even though xi2 appears in the iteration and in the convergence proof; this should be fixed.
[Notation] The notation section says 'I_N denotes an identical matrix of size N x N'; this should read 'identity matrix'.
[Proposition 1] The stepper-motor energy model assumes constant torque and constant current, leading to an energy consumption rate Ebar_k that is independent of speed and acceleration. The paper should state explicitly that this is an idealized model and that hardware-dependent losses may make the true movement energy depend on speed and positioning accuracy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the EE objective is a defined quantity, the optimization uses standard fractional-programming and SCA tools, and cited same-group prior work supplies channel-modeling machinery rather than forcing the conclusion.

full rationale

The paper's derivation chain is self-contained. The energy efficiency in (21) is a stated definition built from the stepper-motor cost model (5)-(16), the communication-energy expression (18), the throughput expression (20), and the block-duration split in (19); no parameter is fitted to the EE target. The Dinkelbach transforms (26) and (43) are standard fractional-programming auxiliary constructs, with updates in (27) and (44) being the usual ratio values, not a renamed prediction. The SCA surrogates in (53)-(59) are local convex approximations of the exact terms (33) and (57), so they do not smuggle in the claimed max-min result. Proposition 2's upper bound follows algebraically in Appendix A from dropping the nonnegative movement energy divided by available time in (65), and it is explicitly labeled as an idealized bound requiring x0 = x-bar. The cited overlapping-author works [14]-[17] provide the field-response channel model, the MMSE receiver form, and the quadratic surrogate machinery; these are modeling and algorithmic tools with stated assumptions, not a uniqueness theorem invoked to forbid alternatives or to force the EE-maximization conclusion. The skeptical observation about constraints C10/C11 in problem (52) is a potential fidelity or correctness issue concerning whether the surrogate objective matches the original max-min EE (19)-(21), but it is not a circular reduction: the algorithm's objective is still derived from the same cost and throughput expressions, albeit possibly with mismatched delay variables. Under the hard rule requiring an explicit Eq.-to-Eq. reduction or a fitted parameter renamed as a prediction, no circular step can be exhibited, so the circularity score is 0.

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

The central claim rests on the field-response channel model, a simplified linear stepper-motor cost model, a block schedule with suspended transmission during movement, and perfect CSI. These are assumptions the reader must accept. The algorithmic proofs themselves use standard convex optimization and do not introduce additional fitted quantities or new physical entities.

free parameters (5)

E_bar_k (energy consumption rate) = 0.175 J/m
Hand-selected in Table I based on commercial stepper motor data; it directly sets the movement cost in the EE objective.
eta_c,k (communication power conversion efficiency) = 0.5
Assumed 50 percent efficiency for all users; affects the trade-off between transmit power and EE.
v_k (MA moving speed) = 0.1 m/s
Chosen uniformly for all users; controls movement delay in (19)-(20).
x0_k (initial MA position) = A/2
Initial position set to the region midpoint; the upper bound of Proposition 2 requires x0 = argmax channel gain to be tight.
Q (AoD quantization resolution) = 10 and 15
Resolution for approximating the multi-path period in Section III-B3; chosen by hand for simulation.

assumptions (6)

standard math Dinkelbach fractional programming and SCA surrogate tightness results are used as given.
Applied in Algorithms 1 and 2; standard results from [36]-[38].
domain assumption Field-response channel model h_k = G_k^H f_k(x_k) with phase-only position dependence.
Borrowed from the MA literature [14]; the whole problem is posed in this model.
ad hoc to paper Stepper motor constant-torque and constant-current model in Eqs. (6)-(11).
Used to derive the linear movement cost, but is an idealized physical model with no experimental validation in the paper.
domain assumption Data transmission is suspended while MAs move; communication duration is T minus the maximum movement delay over users.
Operational block structure assumed in (19)-(20); changing this assumption changes the EE objective.
domain assumption Perfect CSI is available at the start of the block and CSI estimation costs are ignored.
Stated in Section II-B; the authors note this makes their results an upper bound for practical CSI.
ad hoc to paper Virtual AoDs are quantized with resolution Q to approximate the period of the multi-path channel gain.
Section III-B3 uses this to state conditions for achieving the upper bound; without quantization the period is not exactly characterized.

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Cite this review

Pith. "Pith review of Energy Efficiency Maximization for Movable Antenna Communication Systems." pith.science (2026). https://pith.science/paper/6UC3CCQU

@misc{pith2026250607129,
  author       = {Pith},
  title        = {Pith review of: Energy Efficiency Maximization for Movable Antenna Communication Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UC3CCQU}},
  note         = {Machine review of arXiv:2506.07129}
}

read the original abstract

This paper investigates energy efficiency maximization for movable antenna (MA)-aided multi-user uplink communication systems by considering the time delay and energy consumption incurred by practical antenna movement. We first examine the special case with a single user and propose an optimization algorithm based on the one-dimensional (1D) exhaustive search to maximize the user's energy efficiency. Moreover, we derive an upper bound on the energy efficiency and analyze the conditions required to achieve this performance bound under different numbers of channel paths. Then, for the general multi-user scenario, we propose an iterative algorithm to fairly maximize the minimum energy efficiency among all users. Simulation results demonstrate the effectiveness of the proposed scheme in improving energy efficiency compared to existing MA schemes that do not account for movement-related costs, as well as the conventional fixed-position antenna (FPA) scheme. In addition, the results show the robustness of the proposed scheme to imperfect channel state information (CSI) and provide valuable insights for practical system deployment.

Figures

Figures reproduced from arXiv: 2506.07129 by the authors.

**Figure 2.** Energy efficiency and normalized MA moving distance v [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 4.** Energy efficiency versus MA moving speed. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Convergence behaviors of Algorithm 2 and benchmark s [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Minimum energy efficiency and normalized average MA m [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

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