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arxiv: 2606.19102 · v1 · pith:R5TKGGHUnew · submitted 2026-06-17 · 📡 eess.SP

Decentralized Power Control for Over-the-Air Computation with Phase Noise

Martin Dahl , Erik G. Larsson This is my paper

Pith reviewed 2026-06-26 19:22 UTC · model grok-4.3

classification 📡 eess.SP
keywords over-the-air computationphase noisepower controldecentralized precodingmean square errortruncated channel inversionchannel estimationreciprocity
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The pith

Truncated channel inversion for over-the-air computation yields mean-square error independent of receiver antenna count under local channel estimates and phase noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a decentralized power control approach for over-the-air computation when each device knows its channel only locally through calibrated reciprocity. It optimizes parameters of a truncated channel inversion scheme using either an approximate closed-form expression or an exact numerical method. The central result is that the resulting mean-square error does not depend on the number of antennas at the receiver. The analysis further ties this error directly to the expected total phase error summed across all transmitting devices.

Core claim

The proposed TCI scheme for optimizing local power scaling factors in OAC with phase noise achieves an MSE that is independent of the number of receiver antennas. The analysis establishes a direct connection between this MSE and the expected aggregate phase error across devices, providing insight into scalability.

What carries the argument

Truncated channel inversion (TCI), a local precoding method that inverts channel gains up to a truncation threshold and optimizes the threshold and scaling to minimize MSE using only device-local estimates.

If this is right

  • MSE performance remains unchanged as the number of receiver antennas grows.
  • Error scales directly with expected aggregate phase error summed over devices.
  • The scheme performs close to or better than centralized methods that assume perfect global channel knowledge in some tested conditions.
  • Power control can be solved approximately in closed form or exactly by numerical search at each device without central coordination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The antenna-count independence may allow system designers to add receive antennas for other benefits such as interference rejection without trading off computation accuracy.
  • The phase-error connection suggests that reducing phase noise variance at individual devices could be more effective for scaling than increasing antenna count.
  • The local TCI approach could extend to scenarios with multiple computation tasks or time-varying phase statistics if the same isolation property holds.

Load-bearing premise

Channel estimates are available only locally at each device via calibrated reciprocity, and the phase noise model isolates its aggregate effect on MSE from the power-control parameters.

What would settle it

A simulation or measurement in which the MSE of the optimized TCI scheme changes when the number of receiver antennas is varied while holding all other parameters fixed.

Figures

Figures reproduced from arXiv: 2606.19102 by Erik G. Larsson, Martin Dahl.

Figure 1
Figure 1. Figure 1: The upper figure shows T against K for various β. The lower figure shows the maximum β such that t fulfills (23) up to the corresponding T specified by the legend. 10 1 10 0 10 1 MSE 1 10 25 50 75 100 M 10 3 10 2 10 1 10 0 10 1 MSE TCI Approx TCI GS SUM PGDM [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MSE(η, γ) vs M. The upper figure has P = 1, the lower P = 100. K = 10, σ2 = 1, Σ = −K. In [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: MSE(η, γ) vs P. M = 10, K = 10, σ2 = 1, Σ = −K. 10 1 10 0 10 1 10 2 10 3 2 10 2 10 1 10 0 10 1 MSE TCI Approx TCI GS SUM PGDM [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MSE(η, γ) vs σ 2 . M = 10, P = 1, K = 10, Σ = −K. 10 7.5 5 2.5 0.1 | | 0 2 4 6 8 10 MSE TCI Approx, P=1 TCI GS, P=1 TCI Approx, P=10 5 TCI GS, P=10 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: MSE(η, γ) vs Σ < 0. M = 1, σ2 = 1, K = 10. VII. CONCLUSION A TCI scheme for OAC was investigated, where chan￾nel estimates are only available at transmitters. We derived a closed-form and numerical solver to optimize our TCI scheme, which demonstrated competitive performance to refer￾ences with globally available error-free channel estimates. We showed that increasing the number of antennas gives no benefi… view at source ↗
read the original abstract

Estimation of uplink channels is required for coherent over-the-air computation (OAC). When channel estimation is done using calibrated reciprocity, the estimates are only available locally to the devices. This poses a challenge for precoding and decoding, which cannot be coordinated centrally. To this end we use truncated channel inversion (TCI) and propose an approximate closed form solution and an exact numerical solver to optimize the TCI parameters. Importantly, we prove that the proposed TCI scheme is independent of the number of receiver antennas in terms of mean-square-error (MSE). Furthermore, our analysis reveals a clear connection between the MSE and expected aggregate phase error across devices which gives insight to the scalability of OAC. Finally, simulations with comparisons to reference methods from prior work with globally available error-free channel estimates show that proposed is close, even outperforming these references in MSE under some conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a decentralized truncated channel inversion (TCI) power control scheme for over-the-air computation (OAC) in the presence of phase noise, where channel estimates are available only locally at each device due to calibrated reciprocity. It develops an approximate closed-form optimization and an exact numerical solver for the TCI truncation threshold, proves that the mean-square error (MSE) of the scheme is independent of the number of receive antennas M, and derives a connection between the MSE and the expected aggregate phase error across devices. Simulations demonstrate that the proposed method performs close to or better than centralized reference methods that assume error-free global channel state information.

Significance. If the claimed MSE independence from M holds under the given phase noise model, the result would be significant for scalable decentralized OAC, indicating that receive antenna count does not degrade performance when using local TCI. The explicit connection to aggregate phase error offers design insight. Credit is due for providing both the closed-form approximation and numerical solver, plus the mathematical proof of independence and the simulation comparisons.

major comments (3)
  1. [§4] §4 (MSE derivation and independence proof): The central claim that optimized MSE is independent of M requires that the expected aggregate phase error enters the MSE expression in a form fully separable from the local-channel-dependent TCI optimization; the manuscript must explicitly substitute the optimal truncation threshold (closed-form or numerical) and show that no M-dependent cross-terms remain after optimization.
  2. [§3.2, §4] §3.2 and §4: The phase noise model is stated to allow isolation of its aggregate effect, but the proof of independence is load-bearing only if the effective inverted channel after TCI does not re-couple phase noise with the local estimates; a concrete expansion of the MSE expression post-optimization is needed to confirm absence of residual M dependence.
  3. [§5] §5 (Simulation setup): The reported outperformance versus centralized references under some conditions should be accompanied by explicit verification that the simulated MSE remains flat across varying M, together with error-bar details and precise baseline definitions for the reference methods.
minor comments (2)
  1. [Abstract] Abstract: the phrasing 'proposed is close' is grammatically incomplete and should read 'the proposed scheme is close'.
  2. Notation: the definition of the TCI truncation threshold and its relation to the free parameter should be stated once in a dedicated subsection for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications and agreeing to revisions where they strengthen the presentation of our results on the MSE independence from M and the connection to aggregate phase error.

read point-by-point responses

  1. Referee: [§4] §4 (MSE derivation and independence proof): The central claim that optimized MSE is independent of M requires that the expected aggregate phase error enters the MSE expression in a form fully separable from the local-channel-dependent TCI optimization; the manuscript must explicitly substitute the optimal truncation threshold (closed-form or numerical) and show that no M-dependent cross-terms remain after optimization.

    Authors: We agree that an explicit substitution step would make the independence clearer. In Section 4 the MSE is expressed as a product of the optimized TCI term (dependent on local channel statistics) and the expected aggregate phase error term. The latter is independent of M by the phase noise model, and the optimization of the truncation threshold is performed solely over local statistics. In the revised manuscript we will insert the explicit substitution of both the closed-form approximation and the numerical solver into the MSE expression, confirming that all M-dependent factors cancel after optimization. revision: yes

  2. Referee: [§3.2, §4] §3.2 and §4: The phase noise model is stated to allow isolation of its aggregate effect, but the proof of independence is load-bearing only if the effective inverted channel after TCI does not re-couple phase noise with the local estimates; a concrete expansion of the MSE expression post-optimization is needed to confirm absence of residual M dependence.

    Authors: The phase noise enters only through the aggregate term across devices and is not present in the local channel estimates used for TCI. Consequently the post-TCI effective channel separates the phase-noise contribution from the local inversion. To address the request we will add a concrete term-by-term expansion of the optimized MSE in the revised Section 4, explicitly showing that no residual M dependence remains after the substitution of the optimal threshold. revision: yes

  3. Referee: [§5] §5 (Simulation setup): The reported outperformance versus centralized references under some conditions should be accompanied by explicit verification that the simulated MSE remains flat across varying M, together with error-bar details and precise baseline definitions for the reference methods.

    Authors: We agree that these details will improve reproducibility and clarity. In the revised Section 5 we will add (i) explicit simulation results or a dedicated plot confirming that MSE remains constant across a range of M values, (ii) error bars on all reported curves, and (iii) precise parameter settings and implementation details for the centralized reference methods that assume error-free global CSI. revision: yes

Circularity Check

0 steps flagged

No circularity: MSE independence derived from separable phase-noise model and local TCI optimization

full rationale

The paper's central claim is a mathematical proof that optimized TCI yields MSE independent of receiver antenna count M, obtained by isolating the expected aggregate phase error term from the decentralized power-control coefficients. This separation follows directly from the stated channel estimation model (calibrated reciprocity, local estimates only) and the phase-noise model that permits the aggregate effect to enter the MSE expression without re-coupling to the TCI parameters. No fitted input is renamed as a prediction, no self-citation is invoked to justify uniqueness or an ansatz, and the derivation does not reduce to its own inputs by construction. The result is therefore self-contained against the paper's own equations and assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The work rests on standard wireless assumptions rather than new postulates; the central claims depend on the local-estimate model and phase-noise aggregation.

free parameters (1)
  • TCI truncation threshold
    Chosen via approximate closed-form or numerical solver from local channel statistics
axioms (2)
  • domain assumption Calibrated reciprocity provides only local channel estimates at each device
    Invoked to justify decentralized operation
  • domain assumption Phase noise permits isolation of aggregate error from power-control MSE
    Required for the scalability insight

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discussion (0)

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Reference graph

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