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REVIEW 5 major objections 4 minor 1 cited by

On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Digital over-the-air computation can be made noise-aware by replacing Euclidean constellation distances with noise-tailored metrics, yielding a max-min design that reduces computation error.

desk verdict Plausible noise-aware extension of ChannelComp, but the central derivation has an algebraic gap and the paper's own sum-function result undercuts the broad MSE claim; worth a careful revise. read the letter →

arxiv 2506.15950 v2 pith:5LBRBB52 submitted 2025-06-19 eess.SP

Saeed Razavikia , Carlo Fischione This is my paper · ORCID
classification eess.SP
keywords over-the-aircomputationdigitalmodulationconstellationdesignnoise-awaremax-minoptimizationdistancemetricheavy-tailednoiseChannelComp
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 claims that the constellation diagram used for digital over-the-air computation (OAC) should be designed with the channel noise distribution in mind, and that doing so lowers the computation error. The proposed method replaces the Euclidean distance in the ChannelComp encoder-design constraint with noise-tailored metrics: an exponential of squared distance for Gaussian noise, an exponential of distance for Laplace noise, and a power-law distance for heavy-tailed noise. Constellation design then becomes a max-min optimization that minimizes the worst-case pairwise computation error, and the paper proves that under sub-exponential noise with low variance this is asymptotically equivalent to minimizing the mean-square error. The paper also shows that the original Euclidean ChannelComp choice emerges naturally for heavy-tailed noise, and that stochastic fading can be handled by averaging the pairwise distances over the channel covariance. If correct, this gives a principled, distribution-aware way to tune digital OAC modulations for applications such as distributed learning, wireless control, and IoT.

What carries the argument

The central object is the distance metric $D_C$ inside the ChannelComp framework (a method for computing finite-valued functions over the MAC by designing digital modulations so that superimposed constellation points remain distinct). The optimization problem P2 turns the requirement that distinct function outputs have distinct superimposed points into a Lipschitz-style separation constraint. The paper's machinery is to replace the intractable true misdetection probability with an upper bound that depends only on the Euclidean separation $|r_i-r_j|$ and the noise tail: the Chernoff bound for sub-exponential noise, an arctangent tail bound for Cauchy noise, and a power-law tail for stable distributions. These bounds convert the min-max error problem into a quadratically constrained program in the modulation vector $x$, solvable by semidefinite relaxation. The load-bearing identity is the log-sum-exp bound used in Proposition 1: it shows that $\min_x \sum_{i,j}\exp(-\tilde A_{i,j}(x)/\nu)$ and $\min_x \max_{i,j} \tilde A_{i,j}(x)$ have the same minimizers as $\nu\to 0^+$, which is what justifies the max-min criterion as an MSE proxy.

What would settle it

Run the paper's sum-function experiment (K=12 nodes, q=16, AWGN, σ<1) and compute the empirical maximum pairwise error and MSE for the exponential-metric and Euclidean constellations; if the exponential-metric design does not beat Euclidean, the central claim fails in that regime.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the distance function $D_C$ in the ChannelComp design problem—which enforces $D_C(r_i,r_j)\ge \lambda |f(i)-f(j)|^2$ for every pair of distinct function outputs $i,j$—is not a free choice but should be derived from the channel noise statistics. For AWGN, using the Chernoff bound on the ML misdetection probability $Pr(r_i\to r_j)$ leads to $D_C(r_i,r_j)=\exp(|r_i-r_j|^2/(4\sigma))$; for generalized normal noise (which includes Laplace) it becomes $|r_i-r_j|^{\beta-1}\exp(|r_i-r_j|^\beta/\alpha^\beta)$; and for heavy-tailed Cauchy noise it becomes a power law $|r_i-r_j|^{2/\eta}$, recovering the Euclidean $|r_i-r_j|^2$ as $\eta\to 1$. The paper further proves (Proposition 1) that for sub-exponential noise with vanishing variance, the minimizer of the mean-square computation error converges to a minimizer of the maximum pairwise error, so the max-min formulation is asymptotically MSE-optimal. Under stochastic Rician or Rayleigh fading, the same framework applies by replacing each pairwise separation with its expectation under the channel covariance matrix.

Load-bearing premise

The derivation assumes the probability of confusing two constellation points can be bounded using only the Euclidean distance between them, with the decision boundary halfway between, ignoring the full shape of the superimposed constellation.

Editorial extensions

If this is right

  • If the noise-aware metrics deliver what the paper claims, OAC constellations can be tuned to the channel's noise distribution, lowering mean-square computation error relative to the original Euclidean ChannelComp design in high-noise regimes.
  • Proposition 1 implies that in low-variance sub-exponential noise, an MSE-optimal constellation is also worst-case optimal, so practitioners may optimize either objective.
  • For heavy-tailed (Cauchy-like) noise, the design reduces to the Euclidean metric, explaining why ChannelComp's original choice is well suited to impulsive interference.
  • Under stochastic fading, the same max-min design can be optimized in expectation using only the channel covariance matrix, avoiding per-realization channel estimation.
  • The paper's own experiments show a boundary: for the sum function under low-noise Gaussian, the exponential AWGN metric underperforms Euclidean ChannelComp because the superimposed decision regions are nearly vertical; this indicates the metric should be aligned with the true decision-boundary orientation.

Reading between the lines

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

  • A natural extension is to replace the pairwise-distance bound with a bound that accounts for the actual Voronoi (decision) geometry of the superimposed constellation; for sum-type functions this would likely favor axis-aligned or function-aware metrics, and could explain the low-noise sum-function reversal the paper reports.
  • The equivalence result suggests a more general principle: for any symmetric unimodal noise with a sub-exponential tail, the max-min and MSE objectives coincide at low variance, so one could derive noise-tailored metrics for other distributions (Gamma, Weibull, log-normal) and test them numerically in the same framework.
  • The power-law metric for heavy-tailed noise implies that as the tail becomes heavier (smaller stability index α), the required separation between constellation points grows without bound, so there is a fundamental limit to reliable OAC under very heavy-tailed noise; a testable prediction is that the achievable MSE degrades as a power of the scale parameter.
  • The fading extension via covariance expectation could be combined with non-Gaussian noise models to design constellations under correlated fading plus impulsive interference, a regime that current numerical experiments do not cover.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. This paper proposes a noise-aware constellation design for digital over-the-air computation (OAC), extending the authors' earlier ChannelComp framework. The encoder design is formulated as a max-min optimization problem (Problem P2) in which the distance metric between superimposed constellation points is tailored to the channel noise distribution: an exponential metric for Gaussian noise, an exponential metric for Laplace noise, and a power-law metric for heavy-tailed noise. The paper also claims that, under low-variance sub-exponential noise, minimizing the mean-square computation error is asymptotically equivalent to minimizing the maximum pairwise computation error (Proposition 1), and it reports simulations for sum, product, max, and mean/geometric functions over Gaussian and Cauchy noise. The central claim is that the noise-tailored distances yield substantially lower MSE than leading digital OAC baselines.

Significance. If the claims were fully established, this would be a useful contribution to digital OAC: it gives a concrete, optimization-based way to adapt constellation design to the noise tail, and it connects the max-min design criterion to the channel statistics in a manner that extends the ChannelComp framework. The intended distance metrics are simple and amenable to the existing SDP relaxation approach, and the numerical experiments cover several functions and noise models. However, the main theoretical derivation contains a concrete algebraic error, the heavy-tail section is internally inconsistent about the error exponent, and the paper's own Section V-B shows that the pairwise-distance surrogate can misrank designs. These issues bear directly on the paper's central claim, so the current version is not yet suitable for publication without substantial revision.

major comments (5)
  1. [Section IV-B, Eqs. (18)-(19)] The change of variables does not transform (18) into (19). Taking the logarithm of the constraint e^{|r_i-r_j|^2/(4σ)} ≥ λ|f_i-f_j| gives |r_i-r_j|^2 ≥ 4σ ln λ + 4σ ln|f_i-f_j|, whereas (18) requires |r_i-r_j|^2 ≥ 8σ ln|f_i-f_j| + t with t = e^{4σλ}. These constraints are not equivalent, so the exponential distance metric in (20) does not follow from the Chernoff-bound derivation as written. Since this metric is the main product of Section IV-B, the derivation needs to be corrected or replaced by a direct argument from the Chernoff bound.
  2. [Section IV-B, Remark 4, and Section V-B] The derivation replaces the misdetection probability Pr(r_i → r_j) by a bound that depends only on the pairwise Euclidean distance |r_i - r_j|, with the ML decision boundary implicitly assumed to be the midpoint. In the full superimposed constellation, the ML decision regions are determined by all constellation points, so pairwise Euclidean separation does not control the true error event. The paper's own Figure 6(a) is a falsifying instance: for the sum function under low-variance Gaussian noise, the AWGN exponential metric is worse than the Euclidean ChannelComp metric because the decision regions are nearly vertical. Thus the abstract's broad claim that the proposed noise-aware design 'achieves a substantially lower MSE' is not established; at best the metrics minimize a pairwise-distance surrogate.
  3. [Sections IV-D and V-C] The heavy-tail derivation is internally inconsistent. Section IV-D defines the computation error with exponent η < 1 and states that MSE is not suitable for heavy-tailed noise, yet Table I uses the metric |r_1-r_2| and Section V-A says 'Cauchy noise with η = 2 (since the metric is MSE here)', and Figure 7 is labeled with η = 2 while the axes read 'MAE (dB)' despite the caption saying 'MSE performance'. Moreover, in the paper's discrete-decoder setting the computation error is bounded because the output of the tabular mapper is finite, so the claim that MSE is undefined is not justified. The reported heavy-tail gains therefore appear to be obtained under a different objective than the one derived in Section IV-D, and the heavy-tail numerical results need to be reinterpreted or rerun with a consistent error criterion.
  4. [Section IV-D, Eq. (29)] The inequality arctan^{-1}(1/|x|) ≥ |x| is false in general; for example, x = 2 gives arctan(1/2) ≈ 0.46 < 2. Consequently, replacing the inverse-arctangent constraint in (28) by the linear constraint |r_i-r_j|/(2γ) ≥ t|f_i-f_j|^η is not a valid lower-bound relaxation, and the power-law metric (30) is not justified by this argument. A correct bound, or a direct derivation of the power-law metric, is needed.
  5. [Proposition 1 and Appendix A] The proof of Proposition 1 is not sufficient for the stated claim. The proof minimizes an upper bound on Ξ_{i,j} obtained from the exponential tail bound in (36); without a matching lower bound, a minimizer of the upper bound need not be a minimizer of J_MSE. The log-sum-exp argument in (39)-(42) only shows that the surrogate objective converges to J_max, not that the MSE minimizers converge to J_max minimizers. In addition, the proof uses a specific exponential tail bound and does not establish the 'every accumulation point' statement for general sub-exponential noise as stated in the proposition.
minor comments (4)
  1. [Section IV-C, Eq. (21)] The stated integral inequality ∫_z^∞ e^{-u^β} du ≤ (1/β) z^{β-1} e^{-z^β} is incorrect; the correct exponent is z^{1-β}. The subsequent expression for the tail probability appears to use the correct exponent, so this is likely a typo, but it should be fixed for clarity.
  2. [Figure 7 and Section V-C] The caption of Figure 7 says 'MSE performance' while the axis labels read 'MAE (dB)', and the text refers to both MSE and MAE. These labels should be harmonized so the reader knows which quantity is plotted.
  3. [Section V-A, paragraph after Figure 5] The sentence 'Cauchy noise with η = 2 (since the metric is MSE here)' is confusing because Section IV-D explicitly argues that MSE is not suitable for heavy-tailed noise and defines η < 1. Please clarify which objective is actually used.
  4. [Notation, Eq. (29)] The notation arctan^{-1}(·) is ambiguous: it could mean the inverse function or the reciprocal. Please define it explicitly at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the noise-tailored distance metrics are derived from noise-tail upper bounds and evaluated by independent Monte-Carlo MSE, not fitted to the target curves.

full rationale

The paper's central derivation chain is not circular. The proposed distance metrics in Table I are obtained from upper bounds on pairwise misdetection: the Chernoff/Q-function bound for Gaussian noise (Eq. (15) to Eq. (20)), the generalized-normal tail integral (Eq. (21) to Eq. (23)), and the Cauchy tail bound with the arctan inequality (Eq. (27) to Eq. (30)). These derivations use the assumed noise parameters (σ, α, γ) as inputs; they do not fit the metrics to the MSE curves reported in Section V, so the later Monte-Carlo evaluation is an independent check rather than a re-statement of the design criterion. The heavy-tail result that |r_i - r_j| recovers the ChannelComp Euclidean metric as η→1 is presented as a consistency check, not as a prediction derived from the paper's own outputs. Self-citations to ChannelComp [26], the relaxation gap [30], and the Part II sampling scheme [42] supply framework and implementation tools; the noise-to-metric mapping itself is not reduced to those citations. Two substantial correctness risks remain, but they are not circularity: (i) the rewrite from Eq. (18) to Eq. (19) is algebraically invalid—with λ = ln(t)/(4σ), the Chernoff constraint does not become e^{|r_i-r_j|^2/(4σ)} ≥ λ |f(i)-f(j)|—so the AWGN exponential metric is not rigorously derived from (18); and (ii) the pairwise Euclidean upper bound in Eq. (15)/(21) ignores the full ML decision-region geometry of the superimposed constellation, a limitation the paper itself concedes in Section V-B and Section VI. These flaws affect whether the abstract's broad performance claim is established, but they are not cases where a prediction is equivalent to its inputs by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The only hand-chosen quantity is eta; the other inputs such as sigma, gamma, beta, and alpha are channel or noise parameters declared by the model. The distances are surrogates, not fitted constants. The circularity burden is low because the design criterion is derived from the noise distribution rather than from the target error curves.

free parameters (1)
  • eta (heavy-tail error exponent) = eta=2 in simulations, eta<1 in the derivation
    Introduced to make the heavy-tail error criterion finite under infinite-variance noise, but experiments set eta=2 with only 'since the metric is MSE here' as justification. The resulting distance D_C=|ri-rj|^(2/eta) depends directly on this hand-chosen exponent.
assumptions (5)
  • domain assumption The channel noise distribution is known to the transmitter and is symmetric (Gaussian, generalized normal, or Cauchy).
    Section IV derives distance metrics that require the noise distribution and symmetry for the midpoint decision rule; in practice the noise may be unknown or non-symmetric.
  • domain assumption The maximum-likelihood decision boundary between any two constellation points is the Euclidean midpoint.
    Remark 4 states this for symmetric GND noise; it underlies Eq. (15) and the heavy-tail bound, but is not generally valid for arbitrary multidimensional decision regions.
  • standard math Pairwise error probability can be replaced by a tail or Chernoff upper bound depending only on Euclidean distance.
    Used in Eq. (15), Eq. (21), and Eq. (36) to make the max-min problem tractable; the bounds are standard but are not exact objectives.
  • standard math The solution of Problem P1 (overlap avoidance) equals the solution of Problem P2 (distance separation) from prior ChannelComp work.
    Inherited from reference [26]; the paper relies on this equivalence to justify the max-min formulation without reproving it.
  • domain assumption Uniform prior over function outputs and finite range M.
    Used in the MSE bounds and Proposition 1, Eq. (13) and Appendix A.

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Cite this review

Pith. "Pith review of On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design." pith.science (2026). https://pith.science/paper/5LBRBB52

@misc{pith2026250615950,
  author       = {Pith},
  title        = {Pith review of: On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LBRBB52}},
  note         = {Machine review of arXiv:2506.15950}
}
read the original abstract

Over-the-air computation (OAC) leverages the physical superposition property of wireless multiple access channels (MACs) to compute functions while communication occurs, enabling scalable and low-latency processing in distributed networks. While analog OAC methods suffer from noise sensitivity and hardware constraints, existing digital approaches are often limited in design complexity, which may hinder scalability and fail to exploit spectral efficiency fully. This two-part paper revisits and extends the ChannelComp framework, a general methodology for computing arbitrary finite-valued functions using digital modulation. In Part I, we develop a novel constellation design approach that is aware of the noise distribution and formulates the encoder design as a max-min optimization problem using noise-tailored distance metrics. Our design supports noise models, including Gaussian, Laplace, and heavy-tailed distributions. We further demonstrate that, for heavy-tailed noise, the optimal ChannelComp setup coincides with the solution to the corresponding max-min criterion for the channel noise with heavy-tailed distributions. Numerical experiments confirm that our noise-aware design achieves a substantially lower mean-square error than leading digital OAC methods over noisy MACs. In Part II, we consider a constellation design with a quantization-based sampling scheme to enhance modulation scalability and computational accuracy for large-scale digital OAC.

Figures

Figures reproduced from arXiv: 2506.15950 by the authors.

Figure 1
Figure 1. Block diagram of the communication model. Each node [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2(a) shows the max function computation using the BPSK modulation. The correspondence between the constellation points and the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Superimposed constellation diagrams for two modulation formats. Subfigure (a) shows the superposition of two PAM- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Comparison of the functions y “ |x| (blue) and y “ arctan´1 ` 1{|x| ˘ (red) on the interval r´20, 20s. Substituting into (14) and following similar steps, the objective becomes: min x max i,j |f piq ´ f pjq | 2 |ri ´ rj | β´1 e ´ |ri´rj | β αβ , s.t. }x} 2 2 ď 1, (22) …
Figure 5
Figure 5. Figure 5: Figure5(a)-5(d) depict constellation diagrams for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: MSE performance for distribution-aware constellation designs over a noisy MAC with Gaussian noise. Figure 6(a) and 6(b) present results for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: MSE performance for distribution-aware constellation designs over a noisy MAC with Cauchy distribution. Figure 7(a) presents results for the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Forward citations

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Reference graph

Works this paper leans on

54 extracted references · 49 canonical work pages · cited by 1 Pith paper

  1. [1]

    Computation over multiple-access channels,

    B. Nazer and M. Gastpar, “Computation over multiple-access channels,” IEEE Trans. Info. Theo. , vol. 53, no. 10, pp. 3498–3516, 2007

    work page 2007

  2. [2]

    A survey on over-the-air computation,

    A. S ¸ahin and R. Yang, “A survey on over-the-air computation,” IEEE Communications Surveys & Tutorials , pp. 1–1, 2023

    work page 2023

  3. [3]

    To repeat or not to repeat: Insights from scaling LLM under token-crisis,

    F. Xue, Y . Fu, W. Zhou, Z. Zheng, and Y . You, “To repeat or not to repeat: Insights from scaling LLM under token-crisis,” Advances in Neural Information Processing Systems , vol. 36, pp. 59 304–59 322, 2023

    work page 2023

  4. [4]

    Scaling laws for neural language models,

    J. Kaplan, S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu, and D. Amodei, “Scaling laws for neural language models,” arXiv preprint arXiv:2001.08361 , 2020. 13

    arXiv 2001

  5. [5]

    Bigscience large open-science open-access multilingual lan- guage model,

    M. AI, “Bigscience large open-science open-access multilingual lan- guage model,” BigScience, 2022

    work page 2022

  6. [6]

    Estimating the carbon footprint of BLOOM, a 176B parameter language model,

    A. S. Luccioni, S. Viguier, and A.-L. Ligozat, “Estimating the carbon footprint of BLOOM, a 176B parameter language model,” Journal of Machine Learning Research , vol. 24, no. 253, pp. 1–15, 2023

    work page 2023

  7. [7]

    Distributed training of large language models,

    F. Zeng, W. Gan, Y . Wang, and P. S. Yu, “Distributed training of large language models,” in IEEE International Conference on Parallel and Distributed Systems (ICPADS), 2023, pp. 840–847

    work page 2023

  8. [8]

    Over-the-air computation for distributed systems: Something old and something new,

    Z. Chen, E. G. Larsson, C. Fischione, M. Johansson, and Y . Malitsky, “Over-the-air computation for distributed systems: Something old and something new,” IEEE Network, vol. 37, no. 5, pp. 240–246, 2023

    work page 2023

Show all 54 references
  1. [9]

    Accelerating distributed optimization via over-the- air computing,

    N. A. Mitsiou, P. S. Bouzinis, P. D. Diamantoulakis, R. Schober, and G. K. Karagiannidis, “Accelerating distributed optimization via over-the- air computing,” IEEE Trans. on Commun., vol. 71, no. 9, pp. 5565–5579, 2023

    2023

  2. [10]

    Wireless for machine learning: A survey,

    H. Hellstr ¨om, J. M. B. da Silva Jr, M. M. Amiri, M. Chen, V . Fodor, H. V . Poor, and C. Fischione, “Wireless for machine learning: A survey,” Foundations and Trends® in Signal Processing, vol. 15, no. 4, pp. 290– 399, 2022

    2022

  3. [11]

    Federated learning over wireless fading channels,

    M. M. Amiri and D. G ¨und¨uz, “Federated learning over wireless fading channels,” IEEE Trans. Wireless Commun. , vol. 19, no. 5, pp. 3546– 3557, 2020

    2020

  4. [12]

    Modulation-free M2M communications for mission-critical applications,

    S. Cai and V . K. Lau, “Modulation-free M2M communications for mission-critical applications,” IEEE Transactions on Signal and Infor- mation Processing over Networks , vol. 4, no. 2, pp. 248–263, 2017

    2017

  5. [13]

    WHYPE: A scale-out architecture with wireless over-the-air majority for scalable in-memory hyperdimensional computing,

    R. Guirado, A. Rahimi, G. Karunaratne, E. Alarc ´on, A. Sebastian, and S. Abadal, “WHYPE: A scale-out architecture with wireless over-the-air majority for scalable in-memory hyperdimensional computing,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems , vol...

    2023

  6. [14]

    Compute-and-forward: Harnessing interfer- ence through structured codes,

    B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interfer- ence through structured codes,” IEEE Trans. on Info. Theo. , vol. 57, no. 10, pp. 6463–6486, 2011

    2011

  7. [15]

    Harnessing interference for analog function computation in wireless sensor networks,

    M. Goldenbaum, H. Boche, and S. Sta ´nczak, “Harnessing interference for analog function computation in wireless sensor networks,” IEEE Trans. Sig. Proc., vol. 61, no. 20, pp. 4893–4906, 2013

    2013

  8. [16]

    Nomographic functions: Efficient computation in clustered Gaussian sensor networks,

    M. Goldenbaum, H. Boche, and S. Sta ´nczak, “Nomographic functions: Efficient computation in clustered Gaussian sensor networks,” IEEE Trans. Wireless Commun., vol. 14, no. 4, pp. 2093–2105, 2014

    2014

  9. [17]

    Over-the-air computation for IoT networks: Computing multiple functions with antenna arrays,

    L. Chen, N. Zhao, Y . Chen, F. R. Yu, and G. Wei, “Over-the-air computation for IoT networks: Computing multiple functions with antenna arrays,” IEEE Internet of Things J. , vol. 5, no. 6, pp. 5296– 5306, 2018

    2018

  10. [18]

    Robust design for massive CSI acquisition in analog function computation networks,

    F. Ang, L. Chen, N. Zhao, Y . Chen, and F. R. Yu, “Robust design for massive CSI acquisition in analog function computation networks,”IEEE Trans. Veh. Tech., vol. 68, no. 3, pp. 2361–2373, 2019

    2019

  11. [19]

    Robust analog function computa- tion via wireless multiple-access channels,

    M. Goldenbaum and S. Stanczak, “Robust analog function computa- tion via wireless multiple-access channels,” IEEE Trans. on Commun. , vol. 61, no. 9, pp. 3863–3877, 2013

    2013

  12. [20]

    Over-the-air computation over balanced numer- als,

    A. S ¸ahin and R. Yang, “Over-the-air computation over balanced numer- als,” in IEEE Globecom Workshops, 2022

    2022

  13. [21]

    Computing functions over-the-air using digital modulations,

    S. Razavikia, J. M. B. Da Silva J ´unior, and C. Fischione, “Computing functions over-the-air using digital modulations,” in IEEE ICC , 2023, pp. 5780–5786

    2023

  14. [22]

    signSGD: Compressed optimisation for non-convex problems,

    J. Bernstein, Y .-X. Wang, K. Azizzadenesheli, and A. Anandkumar, “signSGD: Compressed optimisation for non-convex problems,” in Int. Conf. Mach. Learn. PMLR, 2018, pp. 560–569

    2018

  15. [23]

    Distributed learning over a wireless network with non- coherent majority vote computation,

    A. S ¸ahin, “Distributed learning over a wireless network with non- coherent majority vote computation,” IEEE Trans. Wireless Commun. , 2023

    2023

  16. [24]

    Over-the-air majority vote computation with modulation on conjugate-reciprocal zeros,

    ——, “Over-the-air majority vote computation with modulation on conjugate-reciprocal zeros,” IEEE Trans. Wireless Commun. , 2024

    2024

  17. [25]

    Waveforms for computing over the air,

    A. P ´erez-Neira, M. Martinez-Gost, A. S ¸ahin, S. Razavikia, C. Fischione, and K. Huang, “Waveforms for computing over the air,” arXiv preprint arXiv:2405.17007, 2024

    2024

  18. [26]

    ChannelComp: A general method for computation by communications,

    S. Razavikia, J. M. B. Da Silva J ´unior, and C. Fischione, “ChannelComp: A general method for computation by communications,” IEEE Trans. on Commun., vol. 72, no. 2, pp. 692–706, 2023

    2023

  19. [27]

    SumComp coding: Digital over-the-air computation via the ring of integers,

    ——, “SumComp coding: Digital over-the-air computation via the ring of integers,” IEEE Trans. on Commun. , 2024

    2024

  20. [28]

    Blind federated learning via over-the-air q-QAM,

    ——, “Blind federated learning via over-the-air q-QAM,” IEEE Trans. Wireless Commun., pp. 1–1, 2024

    2024

  21. [29]

    A novel channel coding scheme for digital multiple access computing,

    X. Yan, S. Razavikia, and C. Fischione, “A novel channel coding scheme for digital multiple access computing,” inIEEE International Conference on Communications, 2024, pp. 3851–3857

    2024

  22. [30]

    ReMAC: Digital multiple access computing by repeated trans- missions,

    ——, “ReMAC: Digital multiple access computing by repeated trans- missions,” IEEE Trans. on Commun. , pp. 1–1, 2025

    2025

  23. [31]

    Channel-aware constellation design for digital OTA computation,

    Z. Li, C. Chen, and C. Fischione, “Channel-aware constellation design for digital OTA computation,” arXiv preprint arXiv:2501.14675 , 2025

    2025 arXiv

  24. [32]

    Modulation and coding for linear Gaussian channels,

    G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Trans. Info. Theo. , vol. 44, no. 6, pp. 2384– 2415, 1998

    1998

  25. [33]

    On the optimality of uniform pulse amplitude modulation,

    A. M. Makowski, “On the optimality of uniform pulse amplitude modulation,” IEEE Trans. Info. Theo. , vol. 52, no. 12, pp. 5546–5549, 2006

    2006

  26. [34]

    Constellation-optimal beam- formers for multiuser MISO broadcast visible light communications,

    Y .-Y . Zhang, H.-Y . Yu, and J.-K. Zhang, “Constellation-optimal beam- formers for multiuser MISO broadcast visible light communications,” IEEE Trans. Info. Theo. , vol. 65, no. 6, pp. 3848–3870, 2019

    2019

  27. [35]

    Signal processing and detection,

    J. M. Cioffi, “Signal processing and detection,” Lect. notes course EE379A Digit. Commun.-Signal Process , pp. 1–101, 2014

    2014

  28. [36]

    Performance analysis of APSK modulation for DVB-S2 transmission over nonlin- ear channels,

    W. Sung, S. Kang, P. Kim, D.-I. Chang, and D.-J. Shin, “Performance analysis of APSK modulation for DVB-S2 transmission over nonlin- ear channels,” International Journal of Satellite Communications and Networking, vol. 27, no. 6, pp. 295–311, 2009

    2009

  29. [37]

    Approaching capacity by equiproba- ble signaling on the Gaussian channel,

    F.-W. Sun and H. C. Van Tilborg, “Approaching capacity by equiproba- ble signaling on the Gaussian channel,” IEEE Trans. Info. Theo., vol. 39, no. 5, pp. 1714–1716, 1993

    1993

  30. [38]

    Constellation design for memoryless phase noise channels,

    F. Kayhan and G. Montorsi, “Constellation design for memoryless phase noise channels,” IEEE Trans. Wireless Commun. , vol. 13, no. 5, pp. 2874–2883, 2014

    2014

  31. [39]

    Optimization of two- dimensional signal constellations in the presence of Gaussian noise,

    G. Foschini, R. Gitlin, and S. Weinstein, “Optimization of two- dimensional signal constellations in the presence of Gaussian noise,” IEEE Trans. on Commun. , vol. 22, no. 1, pp. 28–38, 2003

    2003

  32. [40]

    Performance of nonuniform constellations on the Gaussian channel,

    W. Betts, A. R. Calderbank, and R. Laroia, “Performance of nonuniform constellations on the Gaussian channel,” IEEE Trans. Info. Theo. , vol. 40, no. 5, pp. 1633–1638, 1994

    1994

  33. [41]

    Space modulation with CSI: Constellation design and performance evaluation,

    M. Maleki, H. R. Bahrami, S. Beygi, M. Kafashan, and N. H. Tran, “Space modulation with CSI: Constellation design and performance evaluation,” IEEE Trans. Veh. Tech., vol. 62, no. 4, pp. 1623–1634, 2012

    2012

  34. [42]

    On designing modulation for over- the-air computation — Part II: Pyramid sampling,

    S. Razavikia and C. Fischione, “On designing modulation for over- the-air computation — Part II: Pyramid sampling,” arXiv preprint arXiv:2506.16208, 2025

    2025 arXiv

  35. [43]

    Blind asynchronous over-the-air federated edge learning,

    S. Razavikia, J. A. Peris, J. M. B. Da Silva, and C. Fischione, “Blind asynchronous over-the-air federated edge learning,” in IEEE Globecom workshop, 2022, pp. 1834–1839

    2022

  36. [44]

    Optimal receive filter design for misaligned over-the-air computation,

    H. Hellstr ¨om, S. Razavikia, V . Fodor, and C. Fischione, “Optimal receive filter design for misaligned over-the-air computation,” in 2023 IEEE Globecom Workshops (GC Wkshps) , 2023, pp. 1529–1535

    2023

  37. [45]

    Timely and painless breakups: Off-the-grid blind message recovery and users’ demixing,

    S. Daei, S. Razavikia, M. Skoglund, G. Fodor, and C. Fischione, “Timely and painless breakups: Off-the-grid blind message recovery and users’ demixing,” IEEE Trans. Info. Theo., vol. 71, no. 7, pp. 5226–5257, 2025

    2025

  38. [46]

    One-bit over-the-air aggregation for communication-efficient federated edge learning: Design and convergence analysis,

    G. Zhu, Y . Du, D. G ¨und¨uz, and K. Huang, “One-bit over-the-air aggregation for communication-efficient federated edge learning: Design and convergence analysis,” IEEE Trans. Wireless Commun. , vol. 20, no. 3, pp. 2120–2135, 2020

    2020

  39. [47]

    Broadband analog aggregation for low-latency federated edge learning,

    G. Zhu, Y . Wang, and K. Huang, “Broadband analog aggregation for low-latency federated edge learning,” IEEE Trans. Wireless Commun. , vol. 19, no. 1, pp. 491–506, 2019

    2019

  40. [48]

    Digital over-the-air computation: Achieving high reliability via bit-slicing,

    J. Liu, Y . Gong, and K. Huang, “Digital over-the-air computation: Achieving high reliability via bit-slicing,” IEEE Trans. Wireless Com- mun., 2025

    2025

  41. [49]

    Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers,

    M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers,” IEEE Trans. Info. Theo. , vol. 47, no. 6, pp. 2383–2399, 2002

    2002

  42. [50]

    Space-time codes for high data rate wireless communication: performance criterion and code construction,

    V . Tarokh, N. Seshadri, and A. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Info. Theo. , vol. 44, no. 2, pp. 744–765, 1998

    1998

  43. [51]

    New exponential bounds and approximations for the computation of error probability in fading channels,

    M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. , vol. 2, no. 4, pp. 840–845, 2003

    2003

  44. [52]

    Transmit beamforming for physical-layer multicasting,

    N. Sidiropoulos, T. Davidson, and Z.-Q. Luo, “Transmit beamforming for physical-layer multicasting,” IEEE Trans. Sig. Proc. , vol. 54, no. 6, pp. 2239–2251, 2006

    2006

  45. [53]

    Ex- perimental evidence for heavy tailed interference in the IoT,

    L. Clavier, T. Pedersen, I. Larrad, M. Lauridsen, and M. Egan, “Ex- perimental evidence for heavy tailed interference in the IoT,” IEEE Communications Letters, vol. 25, no. 3, pp. 692–695, 2020

    2020

  46. [54]

    A universal approximation result for difference of log-sum-exp neural networks,

    G. C. Calafiore, S. Gaubert, and C. Possieri, “A universal approximation result for difference of log-sum-exp neural networks,” IEEE Trans. on neural net. and learning systems , vol. 31, no. 12, pp. 5603–5612, 2020

    2020

Pith tools

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