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Fixed pilot sequences can overstate the gains of learned MIMO-OFDM receivers under power amplifier distortion

Mohammad Tasin, Al Imran

arXiv:2610.33663v1eess.SP

TL;DR

Pilot distortion can make a channel estimate depend on the exact pilot sequence, so a receiver trained with one fixed pilot may learn that sequence-specific error. The paper tests two learned receivers on trained and unseen pilots in simulated MIMO-OFDM links and compares them with training-free estimates. Both learned receivers gain more on training pilots, while smoothing lowers BER across tested sequences when delay spread is short.

  • Problem

    Prior evaluations examined other changes in training and testing, but the studies reviewed did not report varying pilot sequences, despite pilot-dependent channel-estimation error.

    Source: §1 · pp. 1–2

  • Method

    The paper simulates two- and four-antenna MIMO-OFDM links, tests two learned receivers on trained and unseen pilots, and compares them with training-free channel estimates.

    Source: §1 · p. 2

  • Results

    37 points separated the neural receiver’s training-pilot gain from its unseen-pilot gain at the highest SNR after short training; both learned receivers gained more on training pilots.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Evaluate learned receivers on unseen pilot sequences and compare them with training-free estimates, including smoothing where delay spread is short.

    Source: §1 · p. 2

    The clean-pilot control is simulation-only, and the PA-aware check needs a PA model or loopback measurement.

    Source: §6.3 · p. 14

Abstract

from arXiv · show

Neural-network receivers are proposed to undo the distortion of a power amplifier driven close to saturation, where it is most efficient. They estimate the channel from pilots, known symbols sent in every frame. The pilots pass through the same amplifier as the data, so the channel estimate carries a small error that depends on the pilot sequence. If one sequence is always sent, as by default in the Sionna simulator, the error never changes, and a network can learn it instead of undoing the amplifier. We tested this in simulated two- and four-antenna orthogonal frequency-division multiplexing links; results are for binary pilots at the highest signal-to-noise ratio tested. A network with separate weights for each subcarrier lowered the bit error rate by 14 to 22% with its training pilot but raised it by 2 to 18% with every other pilot tested, and longer training widened this gap. A convolutional neural receiver following an open-source tutorial gained 37 percentage points more on its training pilot than on unseen pilots after short training, and 9 points more after nine times longer training, much of which also appeared with a linear amplifier. The gap shrank with amplifier back-off but persisted where the in-band distortion meets the 5G limit. On a channel with short delay spread, smoothing the channel estimate across subcarriers without training lowered the error rate on all 107 pilot sequences tested, by 30% on average. Learned receivers should therefore be tested on unseen pilots and against such simple baselines.

1. Introduction

A power amplifier is most efficient near saturation, but its distortion harms OFDM signals; pilot-based channel estimates can carry a sequence-specific error that a receiver trained on a fixed pilot may learn. The paper tests learned receivers on trained and unseen pilots and compares them with training-free estimates.

1. Introduction

  • Near saturation, a power amplifier is efficient but compresses and phase-shifts signal peaks, which spread distortion across OFDM subcarriers.The paper positions neural receivers among remedies that avoid some costs of back-off, peak reduction, predistortion, and receiver-side clipping cancellation.Source: §1 · p. 1
  • A fixed pilot makes the distortion-induced channel-estimation error repeat across frames, allowing a network to learn a sequence-specific pattern rather than the amplifier distortion.This is a concern because 5G NR demodulation reference signals vary with slot, OFDM symbol, and scrambling identity.Source: §1 · p. 1
  • The simulations compare two- and four-antenna MIMO-OFDM links using a per-subcarrier network and a Sionna-tutorial convolutional receiver, each tested on trained and unseen pilots against two training-free estimates.The per-subcarrier network is a positive control because its separate subcarrier weights can store a pattern across subcarriers.Source: §1 · p. 2
  • With binary pilots, both learned receivers gained more on their training pilot than on unseen pilots; smoothing lowered BER on every tested sequence, averaging about 30% at the highest SNR when delay spread was short.Only the neural receiver after long training outperformed smoothing on unseen pilots, and only at high SNR.Source: §1 · p. 2
  • The paper frames its contribution as measuring pilot-error behavior, testing whether learned receivers memorize it, and proposing evaluation with unseen pilots, undistorted-pilot controls, and training-free estimates.The stated scope includes limits of training-free estimates at long delay spreads and with an inexact amplifier model.Source: §1 · p. 2

2. Related work

Related work covers robustness tests for learned receivers, channel estimation under pilot distortion, and the pilots used during training. The authors note that prior studies they examined did not report varying the pilot sequence between training and testing.

2. Related work

  • Studies of learned receivers under hardware impairments vary factors such as amplifier models, channels, impairment levels, users, or modulation, but the examined studies did not report changing pilots between training and testing.Pihlajasalo et al. train on 30 dithered variants of a measured amplifier and validate on 10 others.Source: §2 · p. 2
  • Work on distorted pilots includes distortion-aware and blind channel estimators, neural handling of least-squares mismatch [18], smoothing across subcarriers, and constant-envelope Zadoff–Chu pilots.The authors found neither PA-aware nor smoothed estimates used as learned-receiver references under PA nonlinearity.Source: §2 · p. 2
  • Other work supplies pilot patterns or reference-signal estimates to neural receivers [31], adapts receivers online [30], or randomizes nuisance factors [41], but does not consider a nonlinear power amplifier.Testing on unseen pilots is described as grouped validation and a stress test, against warnings that learned physical-layer models can lose benefits outside training conditions [44].Source: §2 · p. 2

3. System model and receivers

The section specifies simulated MIMO-OFDM links with nonlinear power amplifiers and phase noise, then describes how pilot-based estimates feed training-free and learned receivers. Its receiver designs expose how pilot-dependent channel-estimation error can interact with learned processing.

3.1. Link and power amplifier

  • The simulated links use two or four antennas, 64 subcarriers, 16-QAM data, a Rapp amplifier at 3 dB input back-off by default, and TDL-A fading.The model also includes independent Wiener phase noise; the channel is constant over each frame.Source: §3.1 · pp. 2–3

3.2. What the PA does to data and to pilots

  • Pilot-gain ripple averaged 13.8% across 200 random sequences versus 13.7% data EVM, while two example patterns correlated only 0.24.For the two example sequences, the gain standard deviations were 14% and 15% of their means.Source: §3.2 · p. 4

3.3. Training-free receivers

  • The LMMSE baseline linearly interpolates plain LS pilot estimates, while training-free smoothers filter those estimates across subcarriers before equalization.The smoother uses either a local linear fit or an LMMSE frequency-correlation model, with parameters fixed before experiments.Source: §3.3 · pp. 4–5

3.4.1. Learned stages after the tracker

  • The per-subcarrier network uses pilot-specific channel-estimate error to refine symbols, whereas a shared-matrix network can correct only the error’s near-unity mean.The compared learned stages also include a per-stream gain and a shared gain-plus-cubic model.Source: §3.4.1 · p. 5

3.4.2. Neural receiver

  • The Sionna-tutorial convolutional receiver processes the full resource grid and is given LS estimates and tracked equalizer outputs so it can be tested with different pilots.Its convolutions share weights across the grid, but many parameters are tied to individual resource elements.Source: §3.4.2 · p. 6

3.5. A training-free model of memorisation

  • A training-free memoriser divides a received pilot by a stored pilot-specific gain, matching the trained per-subcarrier network’s gain on its training pilot within 1 percentage point in specified cases.It loses more on unseen pilots and is presented as a pessimistic stand-in for a memorising stage, not as a universally bounded model.Source: §3.5 · p. 6

4. Evaluation protocol and simulation setup

The evaluation tests whether learned receivers generalize across pilot sequences, whether pilot distortion accounts for their gains, and how they compare with training-free estimates. Simulations use shared frames for method comparisons and report seed-level uncertainty.

4.1. Check 1: unseen pilot sequences

  • Check 1 compares a receiver trained on sequence 42 with unseen fixed and random pilots, measuring the gap between training-pilot and unseen-pilot BER reductions.A receiver compensating only for the PA should have a gap near zero.Source: §4.1 · p. 6

4.2. Check 2: controls without pilot distortion

  • Check 2 repeats tests with clean pilots, the true channel, or a linear PA to assess whether pilot distortion is associated with the advantage.The clean-pilot and true-channel tests apply to the PSN and SMN; the linear-PA test applies to the neural receiver.Source: §4.2 · p. 6

4.3. Check 3: simple references

  • Check 3 compares learned receivers with smoothed and PA-aware estimates, and tests learned stages against minimal gain-based stages.Results include the absolute BER of the reference.Source: §4.3 · p. 6

4.4. Simulation setup and statistics

  • Results are means over seeds with Student-t 95% confidence intervals; intervals are descriptive because there is no multiple-comparison correction.Same-frame runs isolate training variability, while other runs also include test-set variability; conclusions rely on several-pp differences with the same sign in every seed.Source: §4.4 · pp. 6–7
  • The simulated links use 2 × 2 or 4 × 4 antennas, 64 subcarriers, 16-QAM data, and binary pilots, with specified alternate pilot types and channel settings.The listed setup includes TDL-A and other channel conditions, phase noise, and several SNRs.Source: §4.4 · p. 7

5. Results

The per-subcarrier network performs better on its training pilot but worse on other tested pilots. Its gap grows with longer training, while shared-weight and training-free comparisons help distinguish pilot-specific behavior from broader correction.

5.1. A per-subcarrier network memorises the pilot

  • At 30 dB, increasing PSN training from 500 to 5,000 iterations raises the gap from 27.2 to 37.2 pp.Training-pilot loss falls while held-out-pilot loss rises, consistent with memorisation.Source: §5.1 · p. 7
  • Sharing PSN matrices across blocks of 2, 4, 8, and 16 subcarriers lowers its gain from 22.0% to 12.2%, 6.0%, 3.7%, and 2.1%, respectively.The fully shared matrix gains 1.4%; the SMN has a gap of 1.5 pp or less on every sequence.Source: §5.1 · p. 7
  • Across 107 pilot sequences, the memoriser loses on 92 to 100% of unseen sequences, while LIN-9 smoothing and the PA-aware estimate help on all sequences by 29.7% and 23.5% on average.The memoriser’s gap ranges from 23 to 86 pp across five training pilots.Source: §5.1 · p. 7, Table 3 · p. 8
  • At 30 dB, the PSN reduces LMMSE-baseline BER by 22% (2 × 2) and 14% (4 × 4) on sequence 42, but raises it by 2 to 15% on other sequences.In the 10-seed 2 × 2 run, the training-pilot reduction is +21.9%, while the reductions on sequences 123, 7, 2024, and random pilots are −2.6, −2.7, −9.7, and −15.0%.Source: §5.1 · p. 7, Figure 3 · p. 8

5.2. A neural receiver of established architecture

The tutorial-style neural receiver also has a larger gain on its training pilot than on unseen pilots after short fixed-pilot training. Longer and random-pilot training reduce this gap, though a residual gap can remain and is not wholly attributable to PA distortion.

5.2. A neural receiver of established architecture

  • At 30 dB, the gap falls from 37.7 pp after 4,000 iterations to 9.0 pp after 36,000, while unseen fixed pilots gain 39.1% after the longer training.After 36,000 iterations, the gap remains positive in all nine seed-and-SNR combinations, although the 30 dB confidence interval includes zero.Source: §5.2 · p. 8
  • At 30 dB after 36,000 iterations, random-pilot training gains 48.7% on unseen fixed pilots versus 39.1% after fixed-pilot training, and its own gap is near zero.The difference in gains is 9.6 pp.Source: §5.2 · p. 8
  • With a linear PA, the fixed-pilot receiver still has a 6.4 pp gap after 36,000 iterations, and the estimated PA-attributable gap is 2.7 pp at 30 dB.The residual gap after long training therefore mostly does not come from the PA, according to the authors.Source: §5.2 · p. 8
  • After 4,000 iterations, the neural receiver reduces LMMSE-baseline BER by 28.7% on its training pilot but raises it by 7.9% on unseen fixed pilots, a 36.6 pp gap.With random-pilot training, the gap is −0.7 pp.Source: §5.2 · p. 8, Table 4 · p. 9
  • After 36,000 random-pilot iterations, the receiver’s unseen-pilot BER is 25.8% below LIN-9 at 30 dB, but 32.6% worse at 18 dB.At 24 dB it is near LIN-9 (+1.7%).Source: §5.2 · p. 9

5.3. The gap at lower amplifier distortion

The pilot-specific gap shrinks as amplifier back-off increases but remains at an operating point where EVM meets the stated 5G limit. Controls show the PSN advantage depends on distorted pilots; true-channel results are also affected by phase tracking.

5.3. The gap at lower amplifier distortion

  • As IBO rises from 3 to 5 dB, the PSN gap falls from 27.2 to 7.5 pp and the short-trained neural receiver gap falls from 36.6 to 16.4 pp.At 4 dB, data EVM is 11.2%, meeting the stated 12.5% 3GPP allowance for a base station using 16-QAM.Source: §5.3 · p. 9, Table 5 · p. 9

5.4. Controls without pilot distortion

  • With PA-distorted pilots, the PSN beats the SMN by 20.7 pp at 30 dB, but the advantage is −0.1 pp with clean pilots or the true channel.The authors attribute the disappearance with clean pilots to the absence of a pilot-specific b_k pattern to store.Source: §5.4 · p. 9
  • The ideal-channel baseline BER is 0.105 because of phase tracking; sequential tracking lowers genie-receiver BER from 0.084 to 0.026, still above the estimated-channel value of 0.019.Without phase noise, the true-channel receiver reaches 0.0075.Source: §5.4 · p. 10

5.5. Training-free estimates and what is left for learning

On short-delay channels, smoothing a plain channel estimate lowers BER across pilot sequences and can outperform learned stages. At longer delay spreads, fixed smoothing windows fail, while the PA-aware estimate retains gains.

5.5. Training-free estimates and what is left for learning

  • Smoothing’s gains combine removal of pilot ripple and noise averaging, and the effects are not additive.In the phase-noise-limited link, smoothing gains 28.5% and the PA-aware estimate gains 20.6%; smoothing the PA-aware estimate adds 11%, falling to 2.4% without thermal noise and vanishing without thermal or phase noise.Source: §5.5 · p. 10, Table 7 · p. 11
  • After smoothing, the PSN’s unseen-pilot gap falls from 28.1 to 2.8 pp with LIN-9 and 1.3 pp with LM-100 at 2 × 2.With 5,000 rather than 500 iterations, the LIN-9 gap remains 2.9 pp; coded transmission also preserves the ordering of plain, smoothed, and PA-aware estimates.Source: §5.5 · p. 10

5.6. Delay spread: where smoothing stops working

  • At 300 ns and beyond, fixed windows fail: LIN-9 raises TDL-C BER by 27% at 300 ns and at least fivefold at 1000 ns, while LM-100 collapses at 300 ns.A smoother matched to the true delay spread degrades more gradually but loses most of its benefit at 30 dB; at 1000 ns, the last TDL taps exceed the cyclic prefix, so those cases test estimation only.Source: §5.6 · pp. 10–11

5.5. Training-free estimates and what is left for learning

  • 30–36% (2 × 2) and 27–31% (4 × 4): LIN-9 or LM-100 smoothing lowers LMMSE-baseline BER on every test sequence at 30 dB.The reductions exceed the PSN gains on its training sequence and the PA-aware estimate in the same runs.Source: §5.5 · p. 10, Table 6 · p. 11

5.6. Delay spread: where smoothing stops working

  • The PA-aware estimate, which does not average across subcarriers, gains 22–23% at every tested delay spread.Where smoothing fails, the pilot fingerprint returns, and the PSN has negative unseen-pilot results in all five 1000 ns seeds.Source: §5.6 · p. 11

5.7. Pilot design, PA sampling and back-off

Pilot design and PA simulation affect the pilot-specific channel-estimation ripple. The ripple tracks data EVM, while its impact on learned receivers varies across pilot families and operating points.

5.7. Pilot design, PA sampling and back-off

  • QPSK pilots produce a larger learned-receiver gap than BPSK: the PSN’s gap is 71.5 pp, while the SMN shows −0.2 pp.The QPSK ripple has a phase component; for constant-envelope Zadoff–Chu pilots, ripple vanishes at critical sampling and the trained PSN gap is 0.1 pp [−1.5, 1.7].Source: §5.7 · pp. 11–12
  • The pilot ripple matches data EVM within 10% across the tested PA models and sampling setups, and LIN-9 reduces it to 4.9% as predicted for independent subcarrier ripples.With oversampling and a guard band, random BPSK ripple falls from 0.138 to 0.110; critical sampling overstates it by about a quarter but does not create it.Source: §5.7 · p. 11, Table 9 · p. 12
  • In an NR-like configuration, whole-symbol QPSK pilots have 12.1% deterministic ripple, while comb-2 reference signals with data have 5.7% deterministic plus 10.0% data-varying ripple.When data share the pilot symbol, the PA-aware estimate can no longer be computed.Source: §5.7 · p. 12
  • At 3 dB IBO, the Rapp PA’s 13.7% data EVM is slightly above the 12.5% 3GPP base-station limit for 16-QAM, and the pilot-specific gap persists near strong compression.For the Rapp PA, the PA-aware estimate gains 34% and 25% at 1 and 3 dB but zero at 6 and 9 dB; the Saleh PA gains 22–37% across tested back-offs.Source: §5.7 · p. 12

5.8. Limits of the PA-aware estimate

The PA-aware estimate removes the pilot ripple exactly only when supplied the true PA model. Its practical gains depend on model or calibration accuracy.

5.8. Limits of the PA-aware estimate

  • The PA-aware estimate removes ripple exactly only because it is given the true PA model.A wrong PA family is critical: a Saleh reference for a Rapp PA doubles baseline BER, while a Rapp reference for a Saleh PA retains 48% of the gain.Source: §5.8 · p. 12
  • A back-off error of up to 1 dB retains 89–100% of the gain, but a wrong PA family can sharply reduce or reverse it.Loopback calibration retains 95–97% of the gain at 30 dB SNR but only 25–28% for Rapp at 20 dB; these simulated results indicate trends, not hardware tolerances.Source: §5.8 · p. 12

6. Discussion

A fixed pilot can let learned receivers exploit a repeatable, sequence-specific channel-estimation error, overstating their gain. The discussion compares receivers with smoothing, proposes evaluation checks, and notes where the findings apply.

6.1. Why a fixed pilot overstates the gain

  • A fixed pilot makes PA-induced channel-estimation error sequence-specific and repeatable, letting a sufficiently flexible receiver learn that error rather than only undo the amplifier.The error is largest where pilot distortion is strong; transmitter EVM indicates its size but does not rule out a gap, including with a linear PA.Source: §6.1 · p. 12
  • The PSN can store the pattern with per-subcarrier matrices, while the convolutional receiver can locate subcarriers through per-resource-element normalisation and zero-padded grid edges.After long training, the neural receiver’s gap fell to 3 to 9 pp but did not vanish within 36,000 iterations; much of the remaining gap at 30 dB also appeared with a linear PA.Source: §6.1 · pp. 12–13

6.2. Recommended evaluation protocol

  • The proposed protocol holds out deployment-varying pilots, trains with changing pilots, reports the gain’s spread, and includes undistorted-pilot controls and simple channel-estimation references.It also reports training length, held-out-pilot validation loss, and how pilots were generated.Source: §6.2 · pp. 13–14
  • Using a suggested 1 pp equivalence margin, the SMN on LIN-9 passed, while PSN on LIN-9 and the neural receiver after 36,000 iterations failed; the Zadoff–Chu result was inconclusive.The SMN on plain LS narrowly failed, and the suggested margin depends on the baseline and SNR.Source: §6.2 · p. 14
  • Against plain LS, PSN appeared to gain 14 to 22%; against smoothing, the long-trained neural receiver beat it by 26% at 30 dB, matched it at 24 dB, and lost at 18 dB.A comparison with plain LS alone or testing only on the training pilot would hide the baseline and pilot dependencies.Source: §6.2 · p. 14

6.3. Applicability

  • The fixed-pilot effect is strongest where PA distortion of pilots is large, and the pattern learned from one simulated PA should be tested across PA models and back-offs.Fixed training sequences also occur in deployment, but their pattern depends on the transmitter’s PA, back-off, and AM/PM.Source: §6.1 · p. 12, §6.3 · p. 14

A. List of experiments

The experiment list records configurations and seed counts, with a default Rapp amplifier setting unless otherwise stated.

B. Conclusion

The conclusion reports that fixed-pilot training can inflate learned-receiver gains, while ordinary channel-estimate smoothing can reduce error without training. It recommends unseen-pilot tests and simple baselines, while noting important evaluation boundaries.

  • A separate-weight network lowered BER by 14 to 22% on its training pilot but raised it by 2 to 18% on every other tested pilot; longer training widened the gap.The tutorial-style neural receiver gained 37 percentage points more on its training pilot after short training, and 9 points more after nine times longer training.Source: B. Conclusion · p. 14
  • On short-delay-spread channels, ordinary subcarrier smoothing without training or PA knowledge lowered BER by about 30% at the highest tested SNR; long-delay spreading made smoothing stop working.Source: B. Conclusion · p. 14
  • Evaluation should use changing and unseen pilots, undistorted-pilot controls, and smoothed estimates and other simple references; hardware tests with PA memory and full-link NR signals remain to be done.Source: B. Conclusion · p. 14

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