Source-linked AI summary
Over-the-Air Computation with Power Amplifier Nonlinearity
David Nordlund, Ayça Özçelikkale, Zheng Chen
TL;DR
AirComp's power-control design is challenged by PA nonlinearity, which can make strategies developed for linear amplifiers degrade as transmit power rises. The paper derives a receiver-MSE expression for coherent AirComp with a memoryless polynomial PA model and finds that accounting for nonlinearity mitigates high-power degradation.
Problem
The impact of PA nonlinearity on AirComp power-control design remains largely unexplored, even though practical PAs can distort signals near their power limits.
Source: §I · p. 1Method
The paper develops a coherent AirComp framework and derives a closed-form receiver MSE for a memoryless polynomial PA model.
Source: §I · p. 1, B. Nonlinear Power Amplifiers · p. 3Results
As transmit power rises, linear-PA-designed power control can show rapidly increasing MSE, whereas nonlinearity-aware control stabilizes MSE at high power.
Source: B. Nonlinearity-Aware Power Control · p. 5Takeaways & Limitations
Accounting for PA nonlinearity can avoid the unbounded degradation seen with linear-PA-designed power control, even without explicitly modeling simulation saturation.
Source: B. Nonlinearity-Aware Power Control · p. 5
Abstract
from arXiv · showhide
Over-the-air computation (AirComp) enables efficient wireless data aggregation by exploiting the superposition of signals transmitted by multiple source nodes, but its performance is sensitive to hardware impairments such as power amplifier (PA) nonlinearity. In this paper, we develop an analytical framework for coherent AirComp with PA nonlinearities and derive a closed-form expression for the receiver mean-squared error (MSE) under a memoryless polynomial PA model. Our numerical results show that power control optimized under the conventional linear-PA assumption can exhibit rapidly increasing MSE as the transmit power budget increases, whereas incorporating PA nonlinearity into the power control design prevents this degradation and maintains low MSE over the considered power range.
I. INTRODUCTION
AirComp aggregates data through wireless signal superposition, but power-amplifier nonlinearity can undermine power control designed for ideal linear amplifiers. The paper develops a coherent AirComp framework that characterizes receiver MSE under nonlinear PAs.
I. INTRODUCTION
- AirComp exploits wireless-channel superposition to compute functions while transmitting data, supporting efficient aggregation largely independent of source-node count.The paper frames AirComp as a joint communication-and-computation approach for massive IoT networks.Source: §I · p. 1
- Power control designed for linear PAs can suffer rapidly increasing MSE as the power budget rises, while nonlinearity-aware design limits that growth and improves high-power performance.The framework derives a closed-form receiver MSE for a memoryless polynomial PA model.Source: §I · p. 1
- PA nonlinearity remains underexplored in AirComp power control, despite prior work on peak-power and clipping effects.PAs may distort signals near their power limits, and source-side power control is an alternative to hardware digital predistortion.Source: §I · p. 1
II. COHERENT OVER-THE-AIR COMPUTATION WITH POWER AMPLIFIER NONLINEARITY
- With known CSI, source nodes precompensate channel phase so their signals combine coherently at the receiver, while practical PAs distort the transmitted symbols.The system targets the sum of normalized complex Gaussian source data under average per-node power constraints.Source: §II · pp. 1–2
- The paper incorporates nonlinear PA distortions into common AirComp processing strategies and characterizes estimation performance using MSE.Conventional schemes are typically derived assuming perfectly linear PAs.Source: §II · p. 2
A. Linear Power Amplifiers
This section outlines linear-PA AirComp processing, including transmit-side and receive-side choices, and uses MSE to measure system performance.
A. Linear Power Amplifiers
- The server post-processes the received signal with a scaling function to form an estimate of the computation result.The supplied passage identifies the receiver scaling as ψ(y) = γy.Source: A. Linear Power Amplifiers · p. 2
- System performance is measured by the mean-squared error of the estimate.The passage introduces MSE as the performance metric.Source: A. Linear Power Amplifiers · p. 2
- The section introduces widely used choices of transmit power control and receiver scaling for AirComp processing.These are described as pre- and post-processing functions under a linear-PA assumption.Source: A. Linear Power Amplifiers · p. 2
1) Full Channel Inversion:
Full channel inversion aligns transmitted amplitudes, while partial inversion relaxes that requirement for nodes with weaker channels. The partial-inversion formulation includes both established variants and an intermediate power-use regime.
1) Full Channel Inversion:
- Full channel inversion imposes a condition on transmit power and receive scaling to align both signal phase and amplitude across nodes.The associated estimate is the desired sum plus zero-mean effective noise inversely proportional to the power scaling factor η.Source: 1) Full Channel Inversion · p. 2
2) Partial Channel Inversion:
- As the number of nodes grows, a poor channel can reduce η and effective SNR under full channel inversion, motivating partial inversion that permits a small bias.Partial inversion is introduced to reduce effective noise variance.Source: 2) Partial Channel Inversion · p. 2
- Partial inversion uses a channel-quality threshold and a power fraction δ for nodes outside the inversion set; δ = 0 and δ = 1 recover truncated and threshold-based power control.The intermediate regime 0 < δ < 1 had not, to the authors' knowledge, been studied, and may be optimal when PA nonlinearity is considered.Source: 2) Partial Channel Inversion · pp. 2–3
B. Nonlinear Power Amplifiers
The section extends coherent AirComp to memoryless polynomial PA distortions and gives an MSE framework applicable to arbitrary power allocation. It then specializes the framework to partial channel inversion and numerically optimizes its parameters despite the resulting problem being generally nonconvex.
B. Nonlinear Power Amplifiers
- The memoryless polynomial PA model extends coherent AirComp analysis to nonlinear distortion.The framework allows arbitrary source-node power allocations and derives receiver MSE for the specified source-data distribution.Source: B. Nonlinear Power Amplifiers · p. 3
- For partial channel inversion, the MSE is expressed as a function of η, δ, and hth.These parameters jointly affect effective noise and nonlinear distortion.Source: B. Nonlinear Power Amplifiers · p. 3
- The parameter-selection problem is generally nonconvex, so the paper uses numerical optimization rather than seeking a globally optimal solution.For fixed (η, δ), the MSE changes with hth only at source-node channel magnitudes, reducing its search to K + 1 intervals and a bounded two-dimensional optimization.Source: B. Nonlinear Power Amplifiers · p. 3
- The polynomial coefficients can be estimated using established PA-modeling techniques.The model is applied to transmitted signals after the PA introduces nonlinear distortion.Source: B. Nonlinear Power Amplifiers · p. 3
III. SIMULATION RESULTS
The simulations use a ten-node AirComp network with Rayleigh-faded links and compare polynomial PA behavior with a saturation-aware model. Results are averaged across network, channel, source-data, and noise realizations.
III. SIMULATION RESULTS
- The system consists of K = 10 source nodes uniformly and randomly distributed in an annulus with radii 1 m and 100 m around a central server; the channel is subject to Rayleigh fading.Source: §III · pp. 3–4
- The simulations use a third-order polynomial PA, with α3 = 0 for the linear case and α3 = 0.0479 e^−j 2.816 for a nonlinear example.Because polynomial models may not represent practical PA saturation at large input amplitudes, the Sat-Poly simulation clips inputs above |xk| = 3.Source: §III · p. 4
- Reported results are averaged over 102 network and channel realizations and 102 source-data and noise realizations.Source: §III · p. 4
A. Effect of Power Amplifier Nonlinearity
The simulations compare processing schemes across PA-nonlinearity and channel-noise settings. With nonlinearity, increasing transmit power can eventually worsen MSE, and the relative performance of power-control schemes changes.
A. Effect of Power Amplifier Nonlinearity
- The comparison evaluates full and partial channel inversion under linear and nonlinear PAs, with and without channel noise.The nonlinear cases use α3 = 0.0479 e^−j 2.816, the noisy cases use σ2 = 10^−8, and the partial-inversion threshold is hth,0 = 10^−4.Source: A. Effect of Power Amplifier… · p. 4
- With a nonlinear PA and channel noise, MSE reaches a minimum at finite transmit power; without noise, MSE increases with power.At low power, noise dominates in the noisy case; beyond the minimum, nonlinear distortion dominates and further power increases degrade performance.Source: Figure 2 · p. 4, A. Effect of Power Amplifier… · p. 4
- PA nonlinearity changes the relative performance of power-control schemes, and MSE can no longer be reduced arbitrarily at high input power.For large P, maximum-power transmission by nodes with insufficient channel quality can add enough distortion to outweigh effective-noise reduction, motivating intermediate 0 < δ < 1.Source: A. Effect of Power Amplifier… · p. 4
B. Nonlinearity-Aware Power Control
The section compares nonlinearity-aware power control with linear-PA baselines and a saturation-aware genie under PA nonlinearity. The proposed scheme stabilizes MSE at high power, while Lin-Opt degrades as nonlinear distortion grows.
B. Nonlinearity-Aware Power Control
- Fig. 3 compares the proposed scheme with Lin-Opt, Truncated, and a saturation-aware genie.The curves report true system performance under nonlinearity, evaluated at parameters selected by the respective optimization problems.Source: B. Nonlinearity-Aware Power Control · pp. 4–5
- At sufficiently large P, Lin-Opt becomes worse than Truncated, while the proposed scheme adapts its power-control parameters and converges to finite MSE.At small P, Lin-Opt and the proposed scheme perform essentially identically because the PA operates predominantly in its linear region.Source: B. Nonlinearity-Aware Power Control · p. 5
- The proposed scheme tracks the genie’s high-P MSE stabilization, although a relatively small performance gap remains across the considered range.At low P the gap primarily reflects the genie’s use of actual source realizations; at high P, its additional knowledge of actual PA saturation also contributes.Source: B. Nonlinearity-Aware Power Control · p. 5
- The proposed optimization uses statistical knowledge of the source distribution and PA model, without explicitly accounting for the saturation region used in simulations.Despite this, it avoids the unbounded degradation observed for power control designed under the linear-PA assumption.Source: B. Nonlinearity-Aware Power Control · p. 5
IV. CONCLUSION
The paper analyzes coherent AirComp with PA nonlinearity and derives receiver MSE under a memoryless polynomial PA model. Its numerical results show that accounting for nonlinearity in power control avoids the degradation associated with linear-PA design as power increases.
IV. CONCLUSION
- The analysis derives a closed-form receiver MSE for coherent AirComp under a memoryless polynomial PA model and circularly symmetric complex Gaussian source data.Fig. 3 reports MSE for the proposed scheme, Lin-Opt, Truncated, and a saturation-aware genie.Source: §IV · p. 5, Figure 3 · p. 5
- Power control designed for linear PAs can show rapidly increasing MSE as transmit power increases when PA nonlinearity is present.Incorporating PA nonlinearity into the design mitigates this degradation and substantially improves high-power performance.Source: §IV · p. 5
- The proposed scheme exhibits the genie’s qualitative high-power behavior while relying on statistical knowledge of the source distribution and polynomial PA model.The genie has knowledge of actual PA saturation and instantaneous source data.Source: §IV · p. 5
APPENDIX A PROOF OF THEOREM 1
The proof uses circular symmetry and independence of the Gaussian source symbols to establish moment relations, then uses the lemma to compute the MSE.
APPENDIX A PROOF OF THEOREM 1
- For circularly symmetric complex Gaussian source symbols, the proof writes each symbol in magnitude-phase form, with phase uniform and independent of magnitude.This representation yields zero means for the symbols and the defined powered symbols.Source: Appendix A · p. 5
- Independence makes cross-moments between powered symbols from distinct nodes zero, and the proof also uses independence of the noise term.The displayed moment relation for a symbol and its powered versions is (l + l′ + 1)!.Source: Appendix A · p. 5
- The proof computes the MSE using Lemma 1 and the stated coefficients d_k,l in the powered-symbol expansion.The supplied passage gives d_k,0 = 1 − c_k,0 and d_k,l = −c_k,l for l ≥ 1.Source: Appendix A · p. 5