Analytical Optimization for Antenna Placement in Pinching-Antenna Systems

H. Vincent Poor; Zhiguo Ding

REVIEW 3 major objections 5 minor 1 cited by

Analytical Optimization for Antenna Placement in Pinching-Antenna Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For pinching-antenna systems, this paper proves closed-form placement rules: fairness OMA uses the users' mean position, while greedy and NOMA setups favor users nearest the waveguide.

desk verdict Internally correct closed-form placement rules for pinching antennas, built on an unvalidated point-source model; deserves peer review but not uncritical adoption. read the letter →

arxiv 2507.13307 v1 pith:QXYRPCGF submitted 2025-07-17 cs.IT math.IT

Zhiguo Ding , H. Vincent Poor This is my paper · ORCID

classification cs.ITmath.IT

keywords pinchingantennasantennalocationoptimizationclosed-formorthogonalmultipleaccessnon-orthogonalpathlosssuccessiveinterferencecancellationuserfairness

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

The paper asks a simple question that previous pinching-antenna work left to numerical search: where on a straight dielectric waveguide should the single active antenna be placed to serve several users? Using a path-loss model in which the channel depends only on the squared Euclidean distance from the antenna to each user, it derives exact closed-form rules for three access schemes. For fairness-oriented OMA, the optimal position is the arithmetic mean of the users' x-coordinates, $x^*=\frac{1}{M}\sum_m x_m$, and the users' distances to the waveguide play no role. For two-user greedy OMA at high SNR, the antenna moves toward the user with smaller $|y_m|$; for two-user NOMA with target rate $R$, the optimum is $x^*=\frac{x_2+e^{R}x_1}{e^{R}+1}$, which approaches the strong user's coordinate as $R$ grows. A sympathetic reader would care because these closed forms turn antenna placement into a design rule rather than a black-box optimization, and they clarify when user clustering near the waveguide matters.

What carries the argument

The load-bearing object is the squared-distance path-loss term $\tau_{mx}=(x-x_m)^2+y_m^2+d^2$, the distance factor between user $m$ and an antenna at $(x,0,d)$ on a waveguide at height $d$. All three placement results reduce to algebraic facts about these quadratic terms. Minimizing $\sum_m\tau_{mx}$ over $x$ gives the arithmetic mean, which is why fair OMA lands at $x^*=\bar{x}$; the high-SNR greedy-OMA objective collapses to minimizing the product $\tau_{1x}\tau_{2x}$, whose derivative is a cubic that Lemma 4 analyzes by sign; and the NOMA problem, after imposing successive-interference-cancellation ordering, becomes a small KKT system whose stationarity condition is linear in $x$ and yields the weighted mean $x^*=\frac{x_2+e^{R}x_1}{e^{R}+1}$. KKT conditions of convex reformulations are the proof machinery that converts each resource-allocation problem into a one-dimensional closed form.

What would settle it

On a real dielectric-waveguide testbed, fix one user and measure received power at a dense set of pinch positions, then fit the model $(x-x_m)^2+y_m^2+d^2$; a decisive check is whether the fairness-OMA optimum for several users, found by exhaustive search over measured channels, equals $\frac{1}{M}\sum_m x_m$ to within the measurement error. If the fitted model deviates substantially, the derived formulas fail.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that pinching-antenna placement in this geometric channel model has simple closed-form optima whose qualitative behavior depends on the multiple-access strategy. For max-min fair OMA (problem (P1)) and for total-power minimization with rate targets (problem (P2)), the optimal antenna coordinate is $x^*=\frac{1}{M}\sum_{m=1}^{M}x_m$, with optimal powers proportional to the resulting distances: $P_m^*=\frac{\tau_{mx^*}}{\sum_i\tau_{ix^*}}P$, where $\tau_{mx}=(x-x_m)^2+y_m^2+d^2$. This contradicts the intuitive equal-distance placement: the users' waveguide distances $y_m$ drop out of the location decision. For throughput-maximizing greedy OMA with two users, a high-SNR analysis shows the optimum lies closer to the user with smaller $|y_m|$. For NOMA with users ordered so that $y_1^2\le y_2^2$ and target rate $R$, the optimal location is $x^*=\frac{x_2+e^{R}x_1}{e^{R}+1}$, so the antenna is pulled toward the user nearest the waveguide and approaches $x_1$ as $R$ increases. The paper also proves that pinching antennas strictly reduce transmit power versus a fixed center antenna, with the saving proportional to $\frac{1}{M}\epsilon\left(\sum_i x_i\right)^2$.

Load-bearing premise

The argument assumes that every user's received power depends only on the squared straight-line distance to the antenna, as if the antenna radiated equally in every direction; if the real radiation pattern changes with the pinch position or frequency, the closed-form optima are not the true optima of the physical system.

Editorial extensions

If this is right

For fairness-oriented OMA, the pinching antenna can be placed at the users' mean x-coordinate without knowing their distances to the waveguide, and the optimal power split follows the resulting distances.
The transmit-power saving over a fixed antenna at the service-area center is strictly positive and equals $\frac{1}{M}\epsilon\left(\sum_i x_i\right)^2$, so clustering users on one side of the service area increases the pinching-antenna gain.
In two-user greedy OMA at high SNR, the optimal antenna is closer to the user with smaller $|y_m|$, and the optimum can be found from the roots of a cubic equation instead of a dense one-dimensional search.
In two-user NOMA with target rate $R$, the optimal antenna position is a convex combination of the two users' x-coordinates with weight $e^R$, trending to the waveguide-nearest user as $R$ grows.
At large target rates, the NOMA scheme's power requirement is lower than the corresponding OMA scheme's, because the exponential factor $e^R$ multiplies the distance reduction for the strong user.

Reading between the lines

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

Editorial extension: if measured pinching-antenna radiation turns out to be directional, the same proof pipeline can be rerun with $\tau_{mx}$ replaced by a measured pattern factor; the fairness-OMA mean rule survives only for patterns symmetric in $x$ about the waveguide.
Editorial extension: the $y$-independence of the fair-OMA location suggests a testable scheduling heuristic: group users by x-coordinate rather than by proximity to the waveguide when maximizing the worst-case rate.
Editorial extension: because the NOMA location is a rate-controlled convex combination, the target rate $R$ can be treated as a tuning knob that continuously sweeps the antenna from the midpoint toward the strong user, which may be useful for dynamic reconfiguration.

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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: the math is right and the closed-form placement rules are genuinely new, but the whole edifice sits on a channel model that is borrowed from prior work and never validated. Worth refereeing, but not for engineering claims as-is.

What's new and good: for user-fairness OMA, the optimal location is the mean of the users' x-coordinates, independent of their y-coordinates; for greedy OMA at high SNR, the optimum moves toward the user with smaller |y|; for NOMA, the optimum is a weighted average x* = (x2 + e^R x1)/(e^R + 1). I rechecked the KKT steps in Appendices A, C, E and the high-SNR reduction; all consistent. The exhaustive-search comparisons in the figures support the optimality claims. The two-user outage probability in Lemma 2 is also a solid closed-form result. This is a real step beyond the numerical-optimization papers in the reference list.

Soft spots, in proportion: the load-bearing assumption is Eq. (1), an isotropic point source with path loss 1/|ψ - ψ_m|^2. Real pinching antennas are leaky-wave structures, and the radiation pattern, leakage rate, and effective phase center likely depend on the pinch position, frequency, and waveguide mode. If so, every location rule is only approximate. The paper gives no experimental or full-wave validation; Section V just re-exercises the same model. This is the main thing a referee should push on. Second, Corollary 2's power-reduction claim over conventional antennas is an artifact of the baseline: the conventional antenna is fixed at the service-area center, not at its own optimal location. Put a conventional antenna at x* = mean(xm) and it achieves exactly what the pinching antenna achieves. The statement is technically true given the stated baseline, but the broader takeaway is overstated. Third, the greedy and NOMA results are only for M=2, and there is no code or seeds. These are minor for an analytical paper.

All in all, the central optimization results hold under the stated model. This paper will be useful to researchers who want design insight rather than black-box numerical optimization. I'd send it to review, with the expectation that the authors validate Eq. (1) or clearly frame the results as model-dependent. I wouldn't cite it as an empirical fact until that is done.

Recommendation: engage with it; it's a legitimate analytical contribution, but handle the model caveat.

Referee Report

3 major / 5 minor

Summary. The paper studies pinching-antenna placement under a distance-dependent path-loss model and derives closed-form antenna locations for three multiple-access schemes. For user-fairness-oriented OMA, Lemma 1 and Corollary 1 give x* = (1/M) Σ_m x_m and show that user y-coordinates do not affect the optimal location. For two-user greedy OMA at high SNR, Lemma 4 states that the antenna should be placed closer to the user with smaller |y_m|. For two-user NOMA with target rate R and y_1^2 ≤ y_2^2, Lemma 5 gives x* = (x_2 + e^R x_1)/(e^R + 1), which approaches x_1 as R grows. The paper also derives optimal power allocations, an outage-probability expression for two users, and numerical comparisons with exhaustive search. The algebraic derivations in Appendices A, C, and E are internally consistent under the stated model, and the numerical sections verify the closed forms against exhaustive search.

Significance. If the results are valid, the paper provides simple, interpretable design rules for pinching-antenna placement: for fairness-oriented OMA the optimal location is the x-mean of the users, while for NOMA and greedy OMA the y-coordinates matter and the antenna moves toward the user closest to the waveguide. These closed forms are more insightful than purely numerical optimization and are backed by KKT analysis and numerical verification. The main caveats are that the entire analysis inherits the unvalidated isotropic point-source model of Eq. (1), that Corollary 2 compares against a suboptimally located conventional antenna, and that the claimed global optimality of the NOMA solution is not fully proved. With those points addressed, the paper would be a useful contribution to the emerging pinching-antenna literature.

major comments (3)

[§4-B and Appendix F] Lemma 6 does not establish global optimality for problem (P9). The proof shows that the candidate solution (22) satisfies the U1-strong SIC assumption and the omitted box constraints, but this only proves feasibility and optimality within the branch where U1 is the strong user. Since the original problem is the union of two possible SIC orders (U1 strong or U2 strong), the author must either prove that the U2-strong branch cannot yield a lower total transmit power or explicitly state the result as 'optimal among NOMA schemes with U1 as the strong user.' This gap is load-bearing for the claim that a low-complexity optimal solution has been obtained.
[§2, Corollary 2 and Eq. (9)] The comparison to conventional fixed-location antennas is unfair. Eq. (9) evaluates the conventional antenna at x = 0 (the center of the service area), while the pinching antenna is optimized at x* = (1/M) Σ_m x_m. A conventional antenna placed at the same optimized location x* would achieve exactly the same total transmit power P_pin, so the strictly positive ΔP in Eq. (11) is an artifact of the suboptimal baseline, not a benefit of pinching antennas per se. The corollary and Remark 3 should be revised to compare against an optimized conventional antenna or to explicitly state that the benchmark is a center-fixed antenna.
[§2, Eq. (1)] All closed-form results rest on Eq. (1), which models the pinching antenna as an isotropic point source whose path loss depends only on the Euclidean distance |ψ_m − ψ_OMA|^2. This model is taken from [9] and is not validated against measurements or electromagnetic simulation of actual leaky-wave pinching antennas. If the radiation pattern, leakage rate, or effective phase center varies with the pinch position, frequency, or waveguide mode, then the objective functions in (P1), (P2), (P4), and (P9) change and the claimed optimal locations (Lemmas 1, 4, and 5) would no longer be correct. This is a correctness-risk concern rather than an internal inconsistency; the authors should either provide validation that Eq. (1) is adequate for the studied regimes or explicitly discuss how the results depend on this modeling assumption.

minor comments (5)

[Appendix D, Eq. (52)] The displayed expression for f((x1+x2)/2) is dimensionally inconsistent: the left-hand side has units of distance^4 while the right-hand side (x1-x2)(y1^2-y2^2) has units of distance^3. Based on the argument that follows, the quantity is presumably the derivative f'((x1+x2)/2) = (x1-x2)(y1^2-y2^2) ≥ 0 (up to sign convention). Please correct this equation and its surrounding text.
[§4-B, Lemma 6] The condition R ≥ 1/2 is unnecessary for the inequality in Appendix F: for any R > 0, e^{2R} > 1, so the first term in Eq. (62) is negative and y1^2 - y2^2 ≤ 0 by the user ordering. The proof works for all R > 0, and the stated threshold should be removed or justified.
[§4-C, Eq. (23)] The approximation x* ≈ x2/(e^R + 1) + x1 is correct for large R, but the notation could be confusing because the e^R factor multiplying x1 has been absorbed into the approximation. Please write it explicitly as x* ≈ x2/(e^R + 1) + e^R x1/(e^R + 1) → x1.
[§3] The greedy-allocation OMA results are explicitly restricted to the two-user case and to a high-SNR approximation. The abstract and introduction could more prominently state this limitation, because the phrase 'greedy-allocation-based OMA transmission' in the abstract may suggest a general M-user result.
[Throughout] There are several typographical errors that should be corrected in revision: 'countintuitive' (Remark 2), 'unformlly' (Section V), 'otpimal' (Section V), 'tranmit' (page 4), 'expoential' (Section IV-C2), 'satisifed' (Appendix F), and 'poisition' in the Fig. 1 caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the placement rules are derived from the stated channel model via KKT analysis, not assumed or fitted.

full rationale

The paper's central results—Lemma 1 and Corollary 1 (x* = (1/M) Σ x_m), Lemma 4 (high-SNR greedy OMA places the antenna nearer the user with smaller |y_m|), and Lemma 5 (x* = (x_2 + e^R x_1)/(e^R + 1) for NOMA)—are obtained by explicitly solving the optimization problems (P1), (P2), (P4), and (P9). The inputs are the channel rate expression in Eq. (1), the power constraints, and the feasible interval for x. None of the stated conclusions appears as an assumption in these inputs; each is a consequence of the KKT conditions or of the high-SNR asymptotic approximation. There is no fitted parameter that is later renamed as a prediction, and the numerical sections verify the analytical solutions against exhaustive search rather than tuning them. Eq. (1) is borrowed from the authors' earlier work [9], and that path-loss model is physically unvalidated for real pinching antennas; however, this is a modeling-premise risk, not a circularity, because the paper does not claim to derive Eq. (1) from its placement results and does not use the placement results to justify Eq. (1). The comparison in Corollary 2 against a center-fixed conventional antenna may be an unfair baseline, but that is a fairness/correctness concern, not an input–output circularity. Overall, the derivation chain is self-contained: assumptions in, closed-form optima out, with standard convex-optimization algebra in between.

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

No parameters are fitted to data; all constants are physical channel parameters or design targets. The central claim rests on the point-source path-loss model in equation (1), the single-waveguide line constraint, and standard KKT optimality conditions. No new entities are introduced beyond the already-proposed pinching antenna itself.

assumptions (5)

domain assumption Channel model: received SNR is eta P_m / (sigma^2 |psi_m - psi_OMA|^2), i.e., free-space path loss from a point source at the pinch point; waveguide and antenna radiation details are ignored.
Invoked in equation (1) and throughout; all optima depend on this distance-only model. Unverified against any measurement; real pinching antennas may have non-isotropic patterns.
domain assumption A single pinching antenna is activated on one waveguide and its location is fixed over M time slots (or over OFDMA symbols).
Section II after equation (1). Underlies the need for a single x; dynamic activation would change the problem.
domain assumption Users lie in a rectangle with -DL/2 <= x_m <= DL/2 and -DW/2 <= y_m <= DW/2, the waveguide is at y=0 and height d, and x is constrained to [-DL/2, DL/2].
System model setup in Section II; results such as x* = mean x_m depend on the interval being symmetric and on the line constraint y=0.
standard math Convex optimization KKT conditions are necessary and sufficient for the formulated problems.
Used in proofs of Lemmas 1, 3, and 5; Slater's condition is satisfied because the feasible sets have nonempty relative interiors for the assumed parameter ranges.
domain assumption For NOMA, the power-domain SIC model requires the strong user to decode the weak user's message, with rate constraints formulated as in (P10).
Section IV; standard NOMA model from the cited literature, used to build the constraints in problem (P9).

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Cite this review

Pith. "Pith review of Analytical Optimization for Antenna Placement in Pinching-Antenna Systems." pith.science (2026). https://pith.science/paper/QXYRPCGF

@misc{pith2026250713307,
  author       = {Pith},
  title        = {Pith review of: Analytical Optimization for Antenna Placement in Pinching-Antenna Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXYRPCGF}},
  note         = {Machine review of arXiv:2507.13307}
}

read the original abstract

As the main issue in pinching-antenna system design, antenna location optimization is key to realizing channel reconfigurability and system flexibility. Most existing works in this area adopt sophisticated optimization and learning tools to identify the optimal antenna locations in a numerical manner, where insightful understandings of the pinching antenna placement are still missing. Motivated by this research gap, this paper aims to carry out analytical optimization for pinching antenna placement, where closed-form solutions for the optimal antenna locations are obtained to reveal the impact of antenna placement on the system performance. In particular, for the user-fairness-oriented orthogonal multiple access (OMA) based transmission, analytical results are obtained to reveal that the pinching antenna needs to be activated at the place that would be beneficial to all served users; however, the users' distances to the waveguide have no impact on the location selection. For the greedy-allocation-based OMA transmission, an asymptotic study based on a high signal-to-noise ratio approximation is carried out to show that the optimal antenna location is in close proximity to the user who is nearest to the waveguide. For non-orthogonal multiple access (NOMA) based transmission, even with a user-fairness-oriented objective, the obtained analytical results show that the optimal antenna location is not the position that can benefit all users, but rather is near the user positioned closest to the waveguide.

Figures

Figures reproduced from arXiv: 2507.13307 by the authors.

**Figure 2.** The convexity of the objective function of problem (P [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The user-fairness performance, measured by [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Outage data rates, i.e., (1 − P om)R, achieved by the considered OMA transmission schemes, for the case that x ∗ = 1 M PM m=1 xm, where DL = 40 m, and R = 2.5 bits per channel use (BPCU) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 4.** Figure 4: The total transmit power, i.e., PM m=1 Pm which is the objective function of problem (P2), required by the considered OMA transmission schemes, where DL = 40 m. optimal solution of problem (P1) can be obtained, as shown in Lemma 1. To evaluate the accuracy of the obtai…

**Figure 6.** Figure 6: The system throughput, measured by PM m=1 ROMA m which is the objective function of problem (P4), achieved by the considered OMA transmission schemes, where DL = 40 m and M = 2. which is the objective function of problem (P9), required by the considered NOMA transmissi…

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Reference graph

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