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Gain Before Tracking: Compensating Power-Amplifier Nonlinearity and Phase Noise in AFDM Uplinks

Mohammad Tasin, Sadman Sakib Rahman

arXiv:2610.33672v1eess.SP

TL;DR

AFDM is a candidate for fast-changing channels, but prior AFDM receiver work had not jointly corrected PA compression and phase noise. This paper uses the transform’s uniform component magnitudes to motivate scalar gain correction before phase tracking; in the reported realistic setting, gain-first improves error rates, while the effect is not significant at light back-off.

  • Problem

    Prior AFDM receiver fixes addressed individual impairments, and none of the work known to the authors corrected PA nonlinearity and phase noise together.

    Source: §I · p. 1

  • Method

    The receiver corrects the PA’s scalar gain before applying a decision-directed phase tracker, whose symbol decisions compression can corrupt.

    Source: §I · p. 1

  • Results

    Gain-first lowers uncoded BER by 7.5–12% for AFDM and about 5% for OFDM in the realistic regime, but the effect is not significant at 8 dB back-off.

    Source: B. Gain Before Tracking · p. 5, §VI · p. 7

  • Takeaways & Limitations

    A closed-form gain without pilots performs about as well as a pilot-based joint complex gain that spends 6.25% of the frame.

    Source: §VI · p. 7

    The results are simulated, use one convolutional-code family, and model a memoryless symbol-rate PA without oversampling, spectral regrowth, or memory.

    Source: §VI · p. 7

Abstract

from arXiv · show

Affine frequency division multiplexing (AFDM) is a candidate waveform for fast-moving vehicles and satellites, but real transmitters add power-amplifier (PA) compression and oscillator phase noise, and no AFDM receiver corrects both. We show that AFDM's transform treats every component equally in magnitude, so the PA's effect reduces to one gain shared by all components and phase noise to a common rotation that OFDM's standard tracker removes. The order matters: the tracker relies on symbol decisions that PA compression corrupts, so the gain should be corrected first. On a six-path, fractional-Doppler channel with 256 subcarriers and a standard-compliant PA back-off, doing so lowers the uncoded bit-error rate by 7.5-12% for AFDM and about 5% for OFDM, and saves 0.5 dB at 10% block error with a rate-3/4 code; the effect is not significant at light back-off. A closed-form gain matches a blind estimate and, without pilots, performs comparably to a pilot-based joint gain estimate that costs 6% overhead. Controls show that AFDM's edge over OFDM is mostly spreading across the channel, not Doppler robustness: single-carrier DFT-s-OFDM matches it at equal back-off.

I. INTRODUCTION

AFDM targets rapidly changing channels, but prior receiver work had not jointly corrected PA nonlinearity and phase noise. The paper motivates gain-first correction and tests it alongside waveform comparisons.

I. INTRODUCTION

  • AFDM was proposed for doubly-dispersive channels, but prior AFDM fixes addressed individual impairments rather than correcting PA nonlinearity and phase noise together.The paper describes prior work as mostly measuring hardware damage, with fixes for phase noise or receiver IQ imbalance.Source: §I · p. 1
  • Equimodularity makes PA gain and distortion approximately uniform across AFDM components, while phase noise becomes a common rotation plus leakage that an OFDM-style tracker can remove.The paper identifies correcting gain before tracking as necessary because compression corrupts the tracker’s symbol decisions.Source: A. Related Work · p. 1, §I · p. 1

A. Related Work

  • Gain-first correction lowers uncoded BER by 7.5–12% for AFDM and about 5% for OFDM relative to the reverse order in the realistic regime.The effect is not significant at 8 dB back-off; at rate 3/4 it gives 0.5 dB at 10% BLER at the NR-compliant back-off.Source: A. Related Work · p. 2
  • AFDM’s 14–31% lower uncoded BER than OFDM over νmax = 0.1–4 is attributed mostly to spreading across channel delay diversity, not Doppler robustness; DFT-s-OFDM matches it at equal back-off.The edge nearly vanishes with a rate-1/2 code.Source: A. Related Work · p. 2

A. DAFT Signal Model

  • The DAFT signal model uses a chirp, FFT, and second chirp, with the OFDM reference obtained by setting c1 = c2 = 0.Both matrices are unitary and equimodular; AFDM frames use 16-QAM symbols and a chirp-periodic prefix.Source: A. DAFT Signal Model · p. 2

B. Doubly-Dispersive Channel

The paper models a six-path channel with fractional Doppler, PA compression, and Wiener phase noise. Its receiver follows channel sounding and LMMSE equalization with scalar gain correction before decision-directed tracking.

B. Doubly-Dispersive Channel

  • The channel model uses six paths with fractional Doppler, and fractional Doppler spreads each effective-channel path across neighboring bins.The default maximum Doppler is νmax = 1, with a sweep from 0.1 to 4.Source: B. Doubly-Dispersive Channel · p. 2

C. Power Amplifier and Phase Noise

  • The memoryless Rapp PA is operated at 5 dB input back-off by default, where its 9.1% EVM is within the 12.5% 16-QAM uplink limit.The 3 dB stress case has 14.0% EVM, while 8 dB gives 3.9%; the PA is applied at symbol rate without oversampling.Source: C. Power Amplifier and Phase Noise · p. 2

D. Channel Sounding and Baseline Receiver √

  • Channel sounding sends one known-symbol pilot frame per chirp-subcarrier through the full transmit chain, and the resulting estimated channel feeds a dense O(N^3) LMMSE equalizer.The sounding frames share one phase-noise realization and the preamble overhead is not charged.Source: D. Channel Sounding and Baseline… · p. 2

III. RECEIVER

  • After LMMSE equalization, the receiver applies an untrained scalar gain correction followed by a decision-directed phase tracker.The gain is applied before the tracker and may come from the PA model, blind moments, or a known-symbol preamble.Source: §III · p. 2, Figure 1 · p. 3

A. Stage 1: Scalar Gain Correction

  • The scalar gain estimates a per-component Bussgang correction; its closed-form version uses the IBO and PA model, while blind moments need neither model nor pilots.Preamble calibration uses known-symbol frames and also removes the static phase offset left by channel sounding.Source: A. Stage 1: Scalar Gain Correction · p. 3

B. Stage 2: Decision-Directed CPE Tracker

A decision-directed tracker estimates common phase from hard symbol decisions. Because PA compression can bias those decisions, the paper argues for restoring gain before tracking.

B. Stage 2: Decision-Directed CPE Tracker

  • The tracker estimates common phase from hard decisions on nearest 16-QAM points and refreshes those decisions over three iterations.The tracker-first order applies gain after tracking; gain-first applies Stage 1 before the tracker.Source: B. Stage 2: Decision-Directed CPE… · p. 3

C. Why the Gain Must Come First

  • At 5 dB back-off, compression shrinks the outer 16-QAM level toward the decision threshold, so erroneous decisions bias the phase estimate and later gain restoration cannot undo that phase error.The predicted ordering effect should fade as back-off grows and vanish for a linear PA.Source: C. Why the Gain Must Come First · p. 3

D. Cost

  • Both correction stages have O(N) cost, while the shared dense LMMSE equalizer dominates at O(N^3).Closed-form gain requires the PA model and IBO; blind moments use one unlabeled block, while preamble gain uses K known-symbol frames.Source: D. Cost · p. 3

IV. THEORY

The theory explains why AFDM can use a shared PA gain and OFDM-style phase-noise tracking, while identifying the approximations and limits of those results. It does not predict the measured correction-order effect or AFDM-versus-OFDM BER difference.

A. Equimodularity and the Gaussian Approximation √

  • For AFDM, equal-magnitude DAFT basis entries make time-domain samples approximately Gaussian at large N, so the Bussgang constants are approximately those of OFDM.The distribution-function approximation error is bounded at O(N^-1/2).Source: A. Equimodularity and the Gaussian… · p. 3

B. Bussgang Constants for the Rapp PA

  • At 5 dB IBO, the Rapp PA has |B| = 0.914 and EVM = 9.1% under the Gaussian-input Bussgang model.The model specifies a unit-power Gaussian input and defines the PA back-off parameter κ.Source: B. Bussgang Constants for the Rapp PA · p. 3

C. Subcarrier Uniformity

  • Under the Gaussian approximation, AFDM PA distortion is approximately uniform across chirp-subcarriers, with one shared gain and distortion power.This result also applies to OFDM but not to transforms whose samples are far from Gaussian, such as the identity transform.Source: C. Subcarrier Uniformity · p. 4

D. Phase-Noise Kernel Equivalence

  • The phase-noise characterization makes equimodularity necessary as well as sufficient for the stated kernel properties, and identifies the affine–Fourier family by its DFT-like magnitude pattern.The paper describes this converse characterization as new; the receiver and experiments do not use it.Source: D. Phase-Noise Kernel Equivalence · p. 4
  • For equimodular bases, the CPE is the same J0 on every subcarrier and a decision-directed tracker works identically in DFT and DAFT domains; leakage is not reduced versus OFDM.Total phase-noise leakage power is 1 − |J0|^2 in every row.Source: D. Phase-Noise Kernel Equivalence · p. 4

E. Linewidth Threshold

The measurements show a gain-first BER advantage under PA compression, while the linewidth rule and experimental setup bound the conditions represented.

E. Linewidth Threshold

  • At 30 kHz spacing, the heuristic linewidth threshold is 550, 234, and 44 Hz at 3, 5, and 8 dB IBO, respectively.It is an operating rule, not a fundamental limit, and is the same for every equimodular waveform at equal N and Ts; the MIMO extension is derived but unevaluated.Source: E. Linewidth Threshold · p. 4

A. Setup

  • The experiments use N = 256, a six-path fractional-Doppler channel, 5 dB IBO, 100 Hz phase noise, and idealized sounding with uncharged overhead.The sounding assumes quasi-static channel conditions and shared phase noise.Source: A. Setup · p. 4

B. Gain Before Tracking

  • At 24 dB and default νmax = 1, the tracker alone reduces uncoded BER from 2.68 × 10−2 to 1.36 × 10−2.Table I reports uncoded BER for AFDM / OFDM at N = 256 and IBO = 5 dB, as a mean over 30 channels.Source: B. Gain Before Tracking · p. 5, Table I · p. 5
  • Across Doppler, AFDM’s order effect falls from 12.1% to 7.5%, is not significant at 8 dB back-off, and gain-first lowers first-decision errors by 8–10%.The first-decision improvement is modest and holds on only 19–20 of 30 channels; the authors regard the proposed mechanism as supported, not proven.Source: B. Gain Before Tracking · p. 5

C. Where the Gain Comes From

The section finds that gain-first receiver performance can be matched without pilots, while waveform controls attribute AFDM’s edge largely to spreading rather than Doppler robustness.

C. Where the Gain Comes From

  • The blind M2M4 gain estimate matches the closed-form gain within noise, while the preamble-calibrated gain improves BER by 9.3% at the default point in idealized sounding.The preamble uses 4096 known symbols, and both its advantage and the phase offset depend on how sounding is done.Source: C. Where the Gain Comes From · p. 5
  • A pilot-free closed-form gain ties a pilot-based joint complex gain estimate, whose 16 in-frame pilots cost 6.25% overhead.The closed-form gain with pilot phase is the best practical variant reported; adding the tracker after pilot gain or using closed-form gain with pilot phase lowers BER by 4–22% relative to gain-first, mostly with intervals containing zero.Source: C. Where the Gain Comes From · p. 5
  • The theory’s contribution is the structure of one scalar gain on all components, applied before tracking, rather than a unique closed-form gain value.The control results attribute AFDM’s edge mostly to spreading across the channel.Source: C. Where the Gain Comes From · p. 5, D. AFDM versus OFDM · p. 6
  • A correctly specified hardware-aware LMMSE receiver comes within about 2% of the closed-form receiver, so the paper does not claim a gain over that baseline.The stated added value of tracking and ordering is handling phase noise.Source: C. Where the Gain Comes From · pp. 5–6

D. AFDM versus OFDM

  • At νmax = 1, AFDM’s BER edge over OFDM rises from 4.7% with plain LMMSE to 22.0% with a tracker, 30.8% gain-first, and 33.1% with preamble gain.With ideal hardware, AFDM’s BER is 77–99.6% below OFDM’s at every Doppler; the edge does not grow with Doppler, falling from 30% at νmax = 0.1 to 14% at νmax = 4.Source: D. AFDM versus OFDM · p. 6
  • AFDM’s edge is not significant at N = 64, reaches 39.9% at N = 512, and varies with PA back-off, including 6.1% at 3 dB and 50.8% at 8 dB.At 3 dB the PA’s EVM violates the NR limit; the edge is already 30% at νmax = 0.1 and falls to 14% at νmax = 4.Source: D. AFDM versus OFDM · p. 6
  • Under the authors’ PA model, DFT-s-OFDM matches AFDM at equal IBO; using preamble gain, its mean BER is 19% lower at νmax = 0.1 and 2.5% lower at νmax = 1.The authors attribute spreading-waveform gains to delay diversity across six delayed paths even at negligible Doppler, while noting AFDM’s chirp rate may help once Doppler is large.Source: D. AFDM versus OFDM · pp. 6–7

E. Coded Performance

At rate 3/4, AFDM needs less SNR than OFDM to reach 10% BLER in the reported coded settings, and gain-first correction improves coded performance. The study also qualifies this edge by code rate and idealized sounding, and describes its supplementary diagnostics.

E. Coded Performance

  • At rate 1/2, AFDM’s 10% BLER SNR advantage over OFDM is only 0–0.5 dB across configurations, as strong coding recovers much of OFDM’s diversity.The preamble gain improves AFDM BLER further than the closed-form gain, but its apparent removal of the rate-1/2 error floor may reflect the idealized sounding.Source: E. Coded Performance · p. 7

VI. CONCLUSION AND LIMITATIONS

  • The study’s coded results are simulated with one convolutional-code family, 64 codewords per channel, and νmax = 1, N = 256; the PA is memoryless and symbol-rate.Sounding is quasi-static, shares one phase-noise realization across pilot frames, and does not charge overhead; the preamble-gain advantage and some error-floor observations may partly be artifacts.Source: §VI · pp. 7–8

E. Coded Performance

  • At rate 3/4 and IBO = 5 dB, gain-first reduces AFDM BLER by 9.2% at 16 dB and 24.4% at 20 dB, saving 0.5 dB at 10% BLER.At 3 dB back-off, it saves 1.6 dB; at 8 dB, the 0.2 dB gain is not significant at higher SNR. Its effect is smaller for OFDM in every reported row.Source: E. Coded Performance · p. 7, Table VI · p. 8
  • At rate 3/4 and IBO = 5 dB, 10% BLER requires 18.3 dB for AFDM and 21.0 dB for OFDM with gain-first correction.At 8 dB back-off, the corresponding SNRs are 17.1 dB and 19.3 dB; at 3 dB, OFDM does not reach 10% BLER by 24 dB while AFDM reaches it at 21.3 dB.Source: E. Coded Performance · p. 7, Table VII · p. 8

Supplementary Material

  • The supplement organizes proofs in Appendices A–C, the linewidth-threshold derivation in D, experiment protocols and controls in E, and diagnostics in F.An extended supplement is also described as available from the authors.Source: Supplementary Material · p. 10

APPENDIX A PROOF OF LEMMA 1: EQUIMODULARITY OF THE DAFT BASIS

The DAFT basis is equimodular, giving AFDM time-domain samples an equal-weight structure that supports large-N approximations to OFDM-like signal statistics. The appendix also characterizes which unitary modulators reproduce OFDM’s phase-noise leakage magnitudes.

APPENDIX A PROOF OF LEMMA 1: EQUIMODULARITY OF THE DAFT BASIS

  • Every DAFT basis entry has modulus 1/N, so each AFDM time-domain sample is a sum of N equal-weight independent terms.For large N, this supports an approximately complex-Gaussian sample distribution and approximately OFDM-like PAPR and Bussgang accuracy; the Gaussian and Bussgang claims are approximations.Source: Appendix A · p. 10

APPENDIX B PROOF OF THEOREM 1(C): CONVERSES OF THE KERNEL CHARACTERISATION

  • The affine-Fourier family is the unique class of unitary modulators whose full phase-noise leakage-magnitude pattern matches OFDM’s for every diagonal unitary phase impairment.The DAFT and OCDM belong to this class; OTFS in the symplectic domain does not in general.Source: Appendix B · p. 11

APPENDIX C PROOF OF PROPOSITION 1: SUBCARRIER-UNIFORMITY OF DAFT-DOMAIN DISTORTION

Under a Gaussian approximation, the PA’s Bussgang gain and distortion power are approximately uniform across DAFT-domain components. Simulations report nearly flat per-component distortion power and no measured per-subcarrier variability advantage over a shared gain.

APPENDIX C PROOF OF PROPOSITION 1: SUBCARRIER-UNIFORMITY OF DAFT-DOMAIN DISTORTION

  • Under the Gaussian approximation, a memoryless PA produces uncorrelated Bussgang residuals, so unitary DAFT processing leaves distortion covariance approximately white and the gain identical across chirp-subcarriers.AFDM samples are only approximately Gaussian, so the uniformity claim is approximate rather than exact.Source: Appendix C · p. 12
  • Measured distortion power varied by 4.7% across chirp-subcarriers, while per-subcarrier and shared-Bussgang-weight results differed by 0.1 percentage points (p = 0.715).A TOST test confirmed equivalence within 1 percentage point (pTOST = 0.0032).Source: Appendix C · p. 12

APPENDIX D DERIVATION OF THE LINEWIDTH THRESHOLD ∆f ∗

The receiver removes PA gain and common phase error but leaves structured ICI as residual noise. The appendix defines a heuristic linewidth threshold by balancing ICI against the PA distortion that scalar correction cannot remove.

APPENDIX D DERIVATION OF THE LINEWIDTH THRESHOLD ∆f ∗

  • The receiver’s scalar and local stages leave structured ICI unsuppressed, and at large linewidth ICI can make the tracker’s hard decisions unreliable.Suppressing more ICI requires estimating additional spectral components of the phase-noise process.Source: Appendix D · p. 12
  • The heuristic ∆f ∗ sets residual ICI power equal to the PA distortion power that the scalar stage cannot remove; it is an operating rule, not a fundamental limit.The derivation uses the small-phase-variance approximation to the ICI power.Source: Appendix D · p. 12
  • At equal N and symbol period, ∆f ∗ is the same for AFDM, OFDM, and OCDM; it scales as 1/Ts and approximately as 1/N at fixed sampling rate.In the main experiments, fixed subcarrier spacing makes ∆f ∗ nearly independent of N.Source: Appendix D · p. 13

APPENDIX E EXPERIMENTAL PROTOCOL AND SOUNDING CONTROLS

The appendix specifies the statistical protocol and tests whether the gain-first advantage persists across sounding models and gain estimators. In the toy setup, the effect is not a sounding artifact, depends on PA compression, and does not require the closed-form gain.

A. Protocol

  • Uncoded effects are mean per-channel relative reductions and coded effects are one minus the ratio of mean BLERs, with 95% bootstrap intervals over 30 channels.BLER wins count only informative channels; main-text p-values use two-sided tests without multiplicity correction and are descriptive.Source: A. Protocol · p. 13

B. Sounding Models, Gain Estimators and the HWI Baseline (Toy Setup)

  • The toy rerun tests whether the effect comes from sounding assumptions, gain choice, or an HWI baseline across five PA-on sounding models and a linear-PA control.The setup uses 64 subcarriers, two paths, 30 channels, and 640 frames; Tables S2 and S3 report mean BERs and paired effects.Source: B. Sounding Models, Gain Estimators… · p. 13, Table S2 · p. 14, Table S3 · p. 14
  • Gain-first beats tracker-first by 19–39% on all 30 channels under every PA-on sounding model in the toy setup.With a linear PA, the closed-form gain is approximately 1 and the orders coincide; a blind M2M4 gain is within 0.3–2.1% of the closed form.Source: B. Sounding Models, Gain Estimators… · pp. 13–14, Table S3 · p. 14
  • With corrected covariance and bias removal, the HWI baseline is within about 2% of the closed-form gain because it is itself gain-first.As implemented with regularizer σ^2_d I, the baseline loses by 21–78%; the blind M2M4 gain needs no PA model or IBO, but requires one block of unlabeled symbols.Source: B. Sounding Models, Gain Estimators… · p. 14

APPENDIX F ADDITIONAL DIAGNOSTICS

The additional diagnostics examine gain calibration, the gain-first mechanism, pilot-based alternatives, and waveform spreading controls. They show that the gain sweep’s optimum is at the nominal multiplier, while the spreading comparisons distinguish waveform edges from correction-order effects.

APPENDIX F ADDITIONAL DIAGNOSTICS

  • The appendix’s realistic-regime diagnostic uses N = 256, six paths, IBO = 5 dB, 100 Hz phase noise, and fulldrive sounding.It reports closed-form scalar |W| = 1.0857 with phase −1.92°, and pilot-compression-corrected |Wp| = 1.0601 with phase −0.55°.Source: Appendix F · p. 15

A. Gain Sweep

  • The gain sweep’s optimum multiplier is m = 1.00 in all four rows, and the pilot-compression-corrected scalar is 1.5% from the genie value.The uncorrected closed form is 3.8% from the genie value; the BER basin is flat enough that the corrected form never helped.Source: A. Gain Sweep · p. 15, Table S4 · p. 15

B. Mechanism

  • Compared with tracker-first, gain-first lowers first-decision SER on 19–20 of 30 channels, phase error on 19–21, and final SER on 28–29.Source: B. Mechanism · p. 15

C. PT-RS-Style Baseline

  • Sixteen known QPSK pilots impose 6.25% overhead for the PT-RS-style alternatives, while the genie gain is an unrealizable per-frame scalar from true data symbols.Table S5 reports gain-first relative BER reduction against each alternative; positive values mean gain-first is better, and ties are included in win counts.Source: C. PT-RS-Style Baseline · p. 15, Table S5 · p. 15

D. Spreading Controls

  • With real hardware, AFDM’s and DFT-s-OFDM’s edges over OFDM differ across Doppler: at νmax = 0.1 they are 21.3% and 45.2%, respectively.At νmax = 1 the corresponding edges are 22.0% and 36.7%; no paired intervals were computed for AFDM versus DFT-s-OFDM.Source: D. Spreading Controls · p. 16, Table S6 · p. 16
  • DFT-s-OFDM’s gain-first order effect is 4.7% [−2.0, 10.3] and 4.6% [2.4, 6.7] at the two reported Doppler settings.The closed-form scalar is unsuitable for DFT-s-OFDM because its lower PAPR means less compression at equal IBO; the preamble column is the fair comparison.Source: D. Spreading Controls · p. 16, Table S6 · p. 16
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