Joint Phase Noise and Off-Grid Channel Estimation for AFDM Systems via Sparse Bayesian Learning

Guanghua Liu; Huaijin Zhang; Lixia Xiao; You Xu; Zilong Liu

arxiv: 2604.17858 · v1 · submitted 2026-04-20 · 📡 eess.SP

Joint Phase Noise and Off-Grid Channel Estimation for AFDM Systems via Sparse Bayesian Learning

You Xu , Huaijin Zhang , Lixia Xiao , Guanghua Liu , Zilong Liu This is my paper

Pith reviewed 2026-05-10 04:32 UTC · model grok-4.3

classification 📡 eess.SP

keywords AFDMphase noise estimationoff-grid channel estimationsparse Bayesian learningexpectation maximizationWiener phase noise

0 comments

The pith

A sparse Bayesian learning method jointly estimates phase noise and off-grid effects in AFDM to approach perfect channel state information.

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

This paper aims to solve the problem of coupled phase noise and off-grid shifts in affine frequency division multiplexing systems, which cause conventional estimators to hit a high signal-to-noise ratio error floor. It introduces JPNCE-SBL, which uses a reduced-rank subspace projection to model the main part of the Wiener phase noise and a dynamic grid evolution to fix off-grid channel errors. These are combined in an expectation-maximization framework for joint updates. If successful, this would allow better performance in NMSE and BER, nearly matching ideal conditions without perfect prior knowledge.

Core claim

The paper establishes that by projecting the phase noise onto a reduced-rank subspace and evolving the grid dynamically to remove off-grid mismatches, all within a sparse Bayesian learning EM loop that jointly refines the channel and noise estimates, the estimation error floor is lifted and performance approaches that of perfect channel state information.

What carries the argument

JPNCE-SBL, an algorithm that integrates reduced-rank subspace projection for Wiener phase noise and iterative dynamic grid evolution for fractional shifts inside an EM-based sparse Bayesian learning procedure for joint estimation.

If this is right

The joint updates prevent error propagation between phase noise and channel estimates.
Dynamic grid evolution avoids the need for dense global grids, reducing computation.
Performance in normalized mean square error and bit error rate significantly exceeds that of existing methods.
Under practical phase noise, results nearly match the perfect channel state information benchmark.

Where Pith is reading between the lines

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

The technique may extend to other OFDM variants or systems with similar phase noise impairments.
Real-world deployment could benefit from reduced pilot overhead due to better estimation accuracy.
Further analysis might explore the method's robustness when the Wiener process assumption is violated.

Load-bearing premise

The reduced-rank subspace projection captures enough of the phase noise energy and the dynamic grid updates remove off-grid errors without creating new inaccuracies or requiring full densification.

What would settle it

Experimental results where JPNCE-SBL shows no improvement in NMSE or BER over benchmarks, or fails to approach perfect CSI performance in the presence of practical phase noise, would disprove the effectiveness of the approach.

Figures

Figures reproduced from arXiv: 2604.17858 by Guanghua Liu, Huaijin Zhang, Lixia Xiao, You Xu, Zilong Liu.

**Figure 1.** Figure 1: The graphical model for the JPNCE-SBL method. γm and the noise precision β = σ −2 w are independently governed by Gamma distributions with small constants: p(γ|a, b) = M YS−1 m=0 Γ(γm|a, b), p(w|β) = CN (w|0, β−1 IN ). (9) The corresponding graphical model for the joint estimation process is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 3.** Figure 3: and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: PN trajectory tracking performance of the proposed subspace projection method. approximation. It can be observed that a compact subspace of L = 16 is sufficient under mild PN conditions, while severe PN requires an expanded dimension to suppress premature saturation, indicating a fundamental trade-off between accuracy and complexity. Furthermore, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: NMSE performance comparison of different estimators versus SNR in the presence of transceiver phase noise. 0 5 10 15 20 25 30 SNR (dB) 10-5 10-4 10-3 10-2 10-1 100 BER AFDM-ideal (Perfect CSI) OFDM-ideal (Perfect CSI) AFDM-PN Proposed (No PN Est.) OFDM-PN (No PN Est.) AFDM-PN Proposed (With PN Est.) OFDM-PN (With PN Est.) Channel Estimation With PN [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: BER performance comparison of different estimators versus SNR in the presence of transceiver phase noise. AFDM outperforms OFDM by fully exploiting the delayDoppler diversity in doubly-dispersive channels. However, when PN is uncompensated, both waveforms suffer from severe BER saturation, erasing the performance gain of AFDM. In contrast, the proposed JPNCE-SBL recovers this gain by providing high-preci… view at source ↗

read the original abstract

In practical affine frequency division multiplexing (AFDM) systems, the intricate coupling of oscillator phase noise (PN) and off-grid fractional shifts traps conventional estimators in a severe high-SNR error floor. To address these challenges, we propose a joint PN and channel estimation method based on sparse Bayesian learning (JPNCE-SBL). Specifically, a reduced-rank subspace projection is first introduced to capture the dominant eigen-energy of the Wiener PN process. Concurrently, a dynamic grid evolution strategy is designed to iteratively eliminate off-grid errors without requiring computationally prohibitive global grid densification. Both components are integrated into a unified Expectation-Maximization (EM) framework, where the channel and PN estimates are jointly updated at each iteration to prevent error propagation. Simulation results demonstrate that JPNCE-SBL significantly outperforms existing benchmarks in both NMSE and BER, closely approaching the perfect channel state information case under practical PN conditions.

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.

Desk Editor's Note private letter to a colleague

The paper integrates reduced-rank PN projection, dynamic grid evolution, and joint EM-SBL for AFDM, but the truncation approximation for Wiener phase noise looks unverified at the SNRs where error floors matter.

read the letter

The main takeaway is that this work puts together a joint estimator for phase noise and off-grid channel effects in AFDM using sparse Bayesian learning, with a reduced-rank subspace step for the Wiener PN and an iterative grid adjustment to handle fractional shifts, all inside one EM loop. The integration is the concrete step forward; it lets the channel and PN estimates refine each other instead of handing errors back and forth between separate stages. That framing matches the practical coupling described in the abstract and avoids the usual overhead of a globally dense grid. The reported simulation gains in NMSE and BER, getting close to perfect CSI, are the part that would interest people working on AFDM links under mobility or mmWave conditions. The method description itself is clear enough on paper to follow the flow from subspace projection through the dynamic grid to the joint updates. The soft spot is exactly the one flagged in the stress-test note. Wiener phase-noise covariance grows linearly and its eigenvalues decay slowly, so a fixed low-rank truncation can leave a residual whose effect grows with SNR. The abstract gives no captured-energy fraction, no truncation-error norm, and no operating-SNR details, so there is no way to check whether the approximation stays harmless precisely where the claimed floor removal is supposed to occur. Simulation parameters are also missing, which makes the benchmark comparisons hard to weigh. This is aimed at the small group already working on AFDM or SBL-based impairment mitigation; a reader who already knows the separate PN and off-grid literature would see the value in the unified EM treatment. It deserves peer review because the problem is real, the proposed combination is specific, and the gaps are fixable with added numbers and checks rather than fundamental flaws in the setup.

Referee Report

2 major / 2 minor

Summary. The paper proposes JPNCE-SBL, a joint phase noise (PN) and off-grid channel estimation algorithm for affine frequency division multiplexing (AFDM) systems. It combines sparse Bayesian learning with a reduced-rank subspace projection to capture dominant eigen-energy of the Wiener PN process, a dynamic grid evolution strategy to mitigate off-grid errors without global densification, and an EM framework for iterative joint updates of channel and PN estimates. Simulations are reported to show substantial gains in NMSE and BER over benchmarks, approaching perfect-CSI performance under practical PN conditions.

Significance. If the central claims hold after verification, the work addresses a practically relevant coupling of impairments in AFDM that limits high-SNR performance in high-mobility or mmWave scenarios. The combination of reduced-rank PN modeling with adaptive SBL grid refinement offers a structured way to avoid both error floors and excessive complexity, which could inform estimator design in related multicarrier and OTFS-like systems.

major comments (2)

[Abstract and §3 (method description)] The performance claim that the method eliminates the high-SNR error floor rests on the reduced-rank subspace projection capturing 'dominant eigen-energy' of the Wiener PN process (Abstract). For a Wiener phase-noise covariance (Toeplitz with linearly growing diagonals), eigenvalue decay is slow; a fixed low-rank truncation therefore leaves a residual variance that grows with SNR. The manuscript reports neither the captured-energy fraction nor the truncation-error norm at the operating SNRs used in the simulations, so it is impossible to confirm that the approximation remains harmless precisely where the error floor would otherwise appear.
[Abstract and simulation results section] The abstract asserts that JPNCE-SBL 'significantly outperforms existing benchmarks in both NMSE and BER' and 'closely approaches the perfect channel state information case,' yet supplies no details on channel models, PN variance parameters, grid sizes, number of Monte Carlo runs, or statistical significance. Without these, the empirical support for the central claim cannot be verified or reproduced.

minor comments (2)

[§3] Clarify the exact rank selection criterion for the subspace projection and whether it is fixed or SNR-dependent.
[§4] Add a brief complexity analysis (flops per iteration) to quantify the benefit of avoiding global grid densification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and §3 (method description)] The performance claim that the method eliminates the high-SNR error floor rests on the reduced-rank subspace projection capturing 'dominant eigen-energy' of the Wiener PN process (Abstract). For a Wiener phase-noise covariance (Toeplitz with linearly growing diagonals), eigenvalue decay is slow; a fixed low-rank truncation therefore leaves a residual variance that grows with SNR. The manuscript reports neither the captured-energy fraction nor the truncation-error norm at the operating SNRs used in the simulations, so it is impossible to confirm that the approximation remains harmless precisely where the error floor would otherwise appear.

Authors: We acknowledge that explicitly reporting the captured eigen-energy fraction and the norm of the truncation error at the simulated SNRs would provide stronger justification for the reduced-rank approximation, particularly given the known slow decay of eigenvalues in the Wiener process covariance. Although our numerical results demonstrate that JPNCE-SBL eliminates the high-SNR error floor and approaches perfect-CSI performance, we will add these quantitative metrics (e.g., percentage of captured energy and residual norm) for the relevant SNR range in the revised Section III and/or simulation results to allow direct verification of the approximation quality. revision: yes
Referee: [Abstract and simulation results section] The abstract asserts that JPNCE-SBL 'significantly outperforms existing benchmarks in both NMSE and BER' and 'closely approaches the perfect channel state information case,' yet supplies no details on channel models, PN variance parameters, grid sizes, number of Monte Carlo runs, or statistical significance. Without these, the empirical support for the central claim cannot be verified or reproduced.

Authors: The simulation parameters—including the specific channel model (e.g., number of paths and delay-Doppler spreads for AFDM), PN variance values, dynamic grid sizes, number of Monte Carlo realizations, and other settings—are fully specified in the Simulation Results section of the manuscript. To address the concern about verifiability, we will revise the abstract to include a concise summary of the key parameters (PN variance range, Monte Carlo count, and grid evolution details) while respecting length limits. If statistical significance measures (e.g., error bars) are not already shown, we will add them to the revised figures. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript proposes JPNCE-SBL by introducing a reduced-rank subspace projection to capture dominant Wiener PN eigen-energy and a dynamic grid evolution within an EM framework for joint estimation. No equations, derivations, or self-citations are exhibited in the provided text that reduce any claimed prediction or result to its own inputs by construction. The central claims rest on the proposed algorithmic integration and simulation validation against benchmarks, which remain independent of tautological redefinitions or fitted-input renamings. This is the normal case of a method paper whose performance claims are externally falsifiable via the reported NMSE/BER curves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; full paper likely contains additional modeling assumptions and parameter choices not visible here.

axioms (2)

domain assumption Phase noise follows a Wiener process whose energy is concentrated in a low-dimensional subspace
Invoked to justify reduced-rank projection; standard but unverified in abstract.
domain assumption Channel and PN effects admit sparse representations amenable to SBL
Core modeling choice enabling the Bayesian framework.

pith-pipeline@v0.9.0 · 5465 in / 1261 out tokens · 46911 ms · 2026-05-10T04:32:15.595201+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A vision of 6G wireless sy stems: Applications, trends, technologies, and open research pro blems,

W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless sy stems: Applications, trends, technologies, and open research pro blems,” IEEE Network, vol. 34, no. 3, pp. 134–142, 2020

2020

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Non-ortho gonal AFDM: A promising spectrum-efﬁcient waveform for 6G high-m obility communications,

Y . Zhang, Q. Yi, L. Musavian, T. Xu, and Z. Liu, “Non-ortho gonal AFDM: A promising spectrum-efﬁcient waveform for 6G high-m obility communications,” in Proc. IEEE 36th Int. Symp. Pers., Indoor Mobile Radio Commun. (PIMRC) , Istanbul, Turkiye, 2025, pp. 1–6

2025

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Z. Sui, Z. Liu, L. Musavian, L.-L. Y ang, and L. Hanzo, “Gen - eralized spatial modulation aided afﬁne frequency divisio n multi- plexing,” IEEE Trans. Wireless Commun. , early access, 2025, doi: 10.1109/TWC.2025.3613062

work page doi:10.1109/twc.2025.3613062 2025

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X. Zhang, H. Yin, Y . Tang, Y . Ge, Y . Zeng, M. Wen, Z. Liu, et al. , “A uniﬁed multicarrier waveform framework for next-generati on wireless networks: Principles, performance, and challenges,” IEEE Commun. Surveys Tuts., early access, 2026, doi: 10.1109/COMST.2026.3672602

work page doi:10.1109/comst.2026.3672602 2026

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H. S. Rou et al. , “From orthogonal time-frequency space to afﬁne frequency-division multiplexing: A comparative study of n ext-generation waveforms for integrated sensing and communications in dou bly disper- sive channels,” IEEE Signal Process. Mag. , vol. 41, no. 5, pp. 71–86, Sep. 2024

2024

[6] [6]

Afﬁne frequency division multiplexing with index modulation: Full diversity condition, performance analys is, and low- complexity detection,

Y . Tao et al. , “Afﬁne frequency division multiplexing with index modulation: Full diversity condition, performance analys is, and low- complexity detection,” IEEE J. Sel. Areas Commun. , vol. 43, no. 4, pp. 1041–1055, Apr. 2025

2025

[7] [7]

MIMO-AFDM outperforms MIMO-OFDM in the face of hardware impairments,

Z. Sui et al., “MIMO-AFDM outperforms MIMO-OFDM in the face of hardware impairments,” arXiv preprint arXiv:2601.00502 , 2026

work page arXiv 2026

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S. Wu and Y . Bar-Ness, “OFDM systems in the presence of pha se noise: Consequences and solutions,” IEEE Trans. Commun. , vol. 52, no. 11, pp. 1988–1996, Nov. 2004

1988

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R. Wang, H. Mehrpouyan, M. Tao, and Y . Hua, “Channel estim ation and carrier recovery in the presence of phase noise in OFDM relay systems,” in Proc. IEEE Global Commun. Conf. (GLOBECOM) , Austin, TX, USA, Dec. 2015, pp. 1–6

2015

[10] [10]

Matched ﬁltering-bas ed channel estimation for AFDM systems in doubly selective cha n- nels,

X. Li, Z. Liu, Z. Zhou, and P . Fan, “Matched ﬁltering-bas ed channel estimation for AFDM systems in doubly selective cha n- nels,” IEEE Trans. Wireless Commun. , 2026. [Online]. Available: https://arxiv.org/abs/2507.0926

work page arXiv 2026

[11] [11]

Channel es timation for AFDM with superimposed pilots,

K. Zheng, M. Wen, T. Mao, L. Xiao, and Z. Wang, “Channel es timation for AFDM with superimposed pilots,” IEEE Trans. V eh. Technol., vol. 74, no. 2, pp. 3389–3394, Feb. 2025

2025

[12] [12]

Off-grid channel estimation with sparse Bayesian learni ng for OTFS systems,

Z. Wei et al., “Off-grid channel estimation with sparse Bayesian learni ng for OTFS systems,” IEEE Trans. Wireless Commun. , vol. 21, no. 9, pp. 7407–7426, Sep. 2022

2022

[13] [13]

Joint sparse graph for enhanced MIMO-AFDM receiver design,

Q. Luo, J. Zhu, Z. Liu, Y . Tang, P . Xiao, G. Chen, and J. Shi , “Joint sparse graph for enhanced MIMO-AFDM receiver design,” IEEE Trans. Wireless Commun., vol. 25, pp. 3272–3286, 2026

2026

[14] [14]

Grid evolution for dou bly fractional channel estimation in OTFS systems,

X. Li, P . Fan, Q. Wang, and Z. Liu, “Grid evolution for dou bly fractional channel estimation in OTFS systems,” IEEE Trans. V eh. Technol., vol. 74, no. 2, pp. 3486–3490, Feb. 2025

2025

[15] [15]

The variational inference appro ach to joint data detection and phase noise estimation in OFDM,

D. D. Lin and T. J. Lim, “The variational inference appro ach to joint data detection and phase noise estimation in OFDM,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 1862–1874, May 2007

2007

[16] [16]

Joint estimation of channel and oscillator phase noise in MIMO systems,

H. Mehrpouyan et al. , “Joint estimation of channel and oscillator phase noise in MIMO systems,” IEEE Trans. Signal Process. , vol. 60, no. 9, pp. 4790–4807, Sep. 2012

2012

[17] [17]

S ubspace- based phase noise estimation in OFDM receivers,

P . Mathecken, S. Werner, T. Riihonen, and R. Wichman, “S ubspace- based phase noise estimation in OFDM receivers,” in Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP) , Apr. 2015, pp. 3227– 3231

2015

[18] [18]

Sparse Bayesian learnin g of delay- Doppler channel for OTFS system,

L. Zhao, W.-J. Gao, and W. Guo, “Sparse Bayesian learnin g of delay- Doppler channel for OTFS system,” IEEE Commun. Lett. , vol. 24, no. 12, pp. 2766–2769, Dec. 2020

2020

[19] [19]

Bayesian compressive sensing for NLOS mmWave imaging under imprecisely multiangle surfaces,

Y . Xu et al. , “Bayesian compressive sensing for NLOS mmWave imaging under imprecisely multiangle surfaces,” IEEE Signal Process. Lett., vol. 32, pp. 2075–2079, 2025

2075

[20] [20]

Maximum likelihood-based grid less DoA estimation using structured covariance matrix recovery an d SBL with grid reﬁnement,

R. R. Pote and B. D. Rao, “Maximum likelihood-based grid less DoA estimation using structured covariance matrix recovery an d SBL with grid reﬁnement,” IEEE Trans. Signal Process. , vol. 71, pp. 802–815, Mar. 2023

2023