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arxiv: 2310.20504 · v4 · pith:AKVWRDNPnew · submitted 2023-10-31 · 💻 cs.IT · math.IT

SumComp: Coding for Digital Over-the-Air Computation via the Ring of Integers

Saeed Razavikia , Jos\'e Mairton Barros Da Silva J\'unior , Carlo Fischione This is my paper

Pith reviewed 2026-05-24 06:02 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords over-the-air computationdigital codingmultiple access channelQAM modulationPAM modulationarithmetic meannomographic functionsMSE analysis
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The pith

SumComp coding aligns QAM and PAM symbols in the ring of integers to enable lower-error digital over-the-air computation of arithmetic and geometric means.

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

The paper introduces SumComp as a coding scheme tailored to QAM and PAM modulations for digital over-the-air computation in multiple access channels. It derives an MSE expression for the arithmetic mean and an upper bound on maximum absolute error for nomographic functions. Simulations show roughly 10 dB gains in normalized MSE over both analog over-the-air methods and the prior ChannelComp approach, specifically in low-noise regimes. A sympathetic reader would care because the scheme turns channel superposition into a computational resource rather than interference, lowering error for tasks where only the result of a function matters.

Core claim

SumComp coding operates by mapping data to integer-ring representations of QAM and PAM symbols so that their superposition at the receiver directly yields the desired function value plus noise, with explicit MSE analysis for the arithmetic mean and an MAE upper bound that holds across a class of nomographic functions.

What carries the argument

SumComp coding scheme that places digitally modulated QAM/PAM symbols into the ring of integers to achieve aligned superposition for function computation.

If this is right

  • MSE for arithmetic-mean computation is explicitly characterized and lower than prior digital and analog baselines in low noise.
  • MAE for a family of nomographic functions is bounded above, independent of the specific function chosen within the class.
  • Computational complexity drops relative to the general ChannelComp scheme because SumComp is specialized to QAM and PAM.
  • Performance advantage appears most clearly when noise is low enough that alignment errors dominate over additive noise.

Where Pith is reading between the lines

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

  • The integer-ring alignment may extend naturally to other lattice-based modulations if the same superposition property can be preserved.
  • In networks where many sensors compute the same aggregate function, the scheme could reduce total transmit energy by allowing each node to send fewer bits while still achieving the target accuracy.
  • If channel estimation is imperfect, the method might still tolerate small phase offsets better than analog over-the-air approaches because the discrete integer structure provides some quantization margin.

Load-bearing premise

The multiple access channel produces ideal superposition of the modulated symbols with only additive noise and no phase or amplitude distortions that would break the integer alignment.

What would settle it

A measurement of normalized MSE for arithmetic-mean computation over a real wireless channel that includes phase noise or amplitude variation, showing whether the reported 10 dB gain disappears.

Figures

Figures reproduced from arXiv: 2310.20504 by Carlo Fischione, Jos\'e Mairton Barros Da Silva J\'unior, Saeed Razavikia.

Figure 1
Figure 1. Figure 1: Block diagram illustrating the complete transmis [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Constellation diagram showcasing the transmissi [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The complete encoding procedure. The output of pre [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SumComp coded Hexagonal QAM of order 8 for two different choices of pq1, q2q. In [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Gray code vs SumComp code for QAM 16 modulation with pq1, q2q “ p1, 4q. same integer grid. To map r P C back to the grid Zrρs, we use the associated quantizer operator to the ring of integers Zrρs denoted by Qρ : C ÞÑ Zrρs and defined as Qρpµq “ argmin xPZrρs }µ ´ x}2, (13) where Qρ assigns every µ P C to the nearest point, with respect to the Euclidean distance, the ring of integers Zrρs. Hence, using Qρ,… view at source ↗
Figure 6
Figure 6. Figure 6: Monte Carlo numerical evaluation of the MSE of comp [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Monte Carlo evaluation of MAE of computing the mean [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance comparison between SumComp coding, c [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Computational complexity for both encoder and dec [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

Communication and computation are traditionally treated as separate entities, allowing for individual optimizations. However, many applications focus on local information's functionality rather than the information itself. For such cases, harnessing interference for computation in a multiple access channel through digital over-the-air computation can notably increase the computation, as established by the ChannelComp method. However, the coding scheme originally proposed in ChannelComp may suffer from high computational complexity because it is general and is not optimized for specific modulation categories. Therefore, this study considers a specific category of digital modulations for over-the-air computations, QAM and PAM, for which we introduce a novel coding scheme called SumComp. Furthermore, we derive an MSE analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the MAE for a set of nomographic functions. Simulation results affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches regarding both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows approximately $10$ dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.

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

1 major / 1 minor

Summary. The manuscript proposes SumComp, a specialized coding scheme for digital over-the-air computation of nomographic functions over the ring of integers using QAM and PAM constellations. It derives a closed-form MSE expression for the arithmetic mean and an MAE upper bound for a class of nomographic functions, then reports simulation results claiming approximately 10 dB normalized-MSE improvement over analog OTA computation and the general ChannelComp scheme in low-noise regimes.

Significance. If the derivations and gains hold under the stated channel model, the specialization to QAM/PAM reduces complexity relative to general ChannelComp while providing analytical performance guarantees; this could be useful for practical wireless distributed computation tasks such as mean estimation in sensor networks. The explicit MSE analysis and MAE bound constitute a clear technical contribution beyond pure simulation.

major comments (1)
  1. [System Model / Simulation Results] The MSE analysis for the arithmetic mean and the reported 10 dB gain both presuppose that the received signal equals the exact integer sum of the transmitted constellation points plus noise, with no residual phase rotation or per-user amplitude scaling. This modeling assumption is load-bearing for the central claims; the paper should either derive a robustness bound or provide simulations that include such impairments (System Model and Simulation sections).
minor comments (1)
  1. [Abstract] The abstract states the 10 dB gain but does not reference the specific MSE expression or simulation parameters (e.g., constellation size, number of users, SNR range); adding these would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the system model assumptions. We address the point below.

read point-by-point responses

  1. Referee: [System Model / Simulation Results] The MSE analysis for the arithmetic mean and the reported 10 dB gain both presuppose that the received signal equals the exact integer sum of the transmitted constellation points plus noise, with no residual phase rotation or per-user amplitude scaling. This modeling assumption is load-bearing for the central claims; the paper should either derive a robustness bound or provide simulations that include such impairments (System Model and Simulation sections).

    Authors: The manuscript develops the SumComp scheme and its performance analysis under the standard effective-channel model for digital over-the-air computation in which perfect synchronization and per-user channel compensation are assumed, so that the received signal is exactly the integer sum of the transmitted symbols plus noise. This modeling choice is made to isolate the contribution of the coding scheme over the ring of integers and is consistent with the ChannelComp framework the paper builds upon. We acknowledge that residual phase or amplitude mismatches would affect the integer-sum property. In the revision we will (i) explicitly state the assumption and its justification in the System Model section and (ii) add a short set of simulations that inject small residual phase rotations and per-user amplitude scalings to illustrate sensitivity under mild impairments. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation and claims are self-contained

full rationale

The paper introduces SumComp as a new coding scheme for QAM/PAM in digital over-the-air computation over the integer ring, derives an MSE expression for the arithmetic mean and an MAE upper bound for nomographic functions from the superposition model plus noise, and reports simulation gains versus ChannelComp and analog baselines. No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in; the performance numbers are obtained from explicit simulation of the stated channel model rather than by algebraic identity with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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