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

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.

• The paper introduces JPNCE-SBL, a joint estimation framework that overcomes phase noise and off-grid grid mismatch in AFDM, achieving robust channel recovery.
• It employs reduced-rank subspace tracking and dynamic grid evolution to accurately track phase noise and mitigate basis mismatch without excessive computational cost.
• An EM-based unified estimation prevents error propagation, preserves AFDM’s diversity gain, and closely approaches the performance of ideal CSI.

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

Problem Formulation and Motivation

Affine Frequency Division Multiplexing (AFDM) provides enhanced robustness to delay and Doppler spreads compared to OFDM. However, its practical realization in high-mobility, high-frequency (e.g., mmWave) systems is severely challenged by hardware-induced phase noise (PN) and the presence of channel parameters existing off the predefined discretization grid. These effects jointly induce a severe model mismatch that compromises the effectiveness of traditional channel estimators, resulting in persistent error floors, particularly in high SNR regimes. The interdependence between phase noise and the continuous nature of channel delays and Dopplers leads to a bilinear, nonlinear, and high-dimensional estimation problem.

Sparse Bayesian Learning (SBL) has shown promise as a framework for exploiting the inherent sparsity of delay-Doppler channels. Nevertheless, canonical SBL approaches assume perfect hardware and fixed grid dictionaries, which are impractical for realistic AFDM receivers due to off-grid errors and PN-induced dictionary mismatches. While grid densification improves accuracy, it renders real-time operation infeasible due to the associated computational complexity. Furthermore, decoupled PN and channel estimation leads to mutual error propagation, compounding estimation failures.

Contributions and Methodological Innovations

The paper proposes a joint estimation framework, JPNCE-SBL, structured around three principal innovations:

1. Reduced-Rank Subspace Phase Noise Tracking: Rather than estimating time-domain PN coefficients directly, the method projects the Wiener phase noise process onto a dominant low-dimensional subspace extracted from its covariance structure, thus drastically reducing the effective PN parameter dimension while preserving essential time correlation.
2. Dynamic Grid Evolution: The algorithm constructs an initial delay-Doppler grid and iteratively adjusts virtual grid points toward the true off-grid values using first-order Taylor expansions. This avoids the computational infeasibility of global grid densification and mitigates basis mismatch, yielding improved numerical stability and robustness to coarse grid initializations.
3. Unified EM-Based Estimation: The channel vector, phase noise coefficients, and grid positions are jointly estimated within a consistent EM framework. At each iteration, the E-step computes the posterior distribution of the sparse channel, while the M-step sequentially updates channel hyperparameters, phase noise coefficients, and grid positions. This prevents the error propagation endemic to decoupled approaches and maintains theoretical consistency of the posterior.

The signal model incorporates both the sparse multipath structure and the bilinear PN-channel coupling. SBL priors are specified by a hierarchical Gaussian-Gamma model, with hyperparameters updated via closed-form expressions or quadratic subproblems. All algorithmic steps (hyperparameter, PN subspace, and grid evolution) admit efficient computational forms, notably with complexity per-iteration dominated by $O(N^3 + N^2 M_S)$ due to the adoption of Woodbury matrix identities in the E-step.

Empirical Analysis and Results

Extensive simulation results are provided for a high-frequency ($f_c = 30$ GHz), high-mobility AFDM system under various PN and channel configurations. Key findings include:

• Error Floors Eliminated: Competing estimators—including Newton-OMP, OffGrid-SBL, and GridFission-SBL—exhibit severe high-SNR error floors due to grid mismatch and inability to compensate for multiplicative PN distortions. JPNCE-SBL, in contrast, maintains continuously decreasing NMSE and BER curves, closely tracking the perfect CSI bound even under substantial PN.
• Phase Noise Tracking: The reduced-rank subspace model provides MSE-optimal phase tracking with small subspace sizes for moderate PN, with flexible trade-offs for more severe PN regimes. Accurate trajectory tracking of the PN process is achieved even with rapid temporal evolution.
• Robustness to Grid Resolution: Dynamic grid evolution ensures resilience to initial grid resolution, with performance degradation under coarser grids limited to minor increments—contrasting with traditional SBL methods, which are highly sensitive to grid granularity.
• Preservation of Diversity Gain: With JPNCE-SBL providing accurate CSI, the inherent delay-Doppler diversity benefits of AFDM over OFDM are preserved, while uncompensated PN entirely erases this advantage in baseline estimators.

Theoretical and Practical Implications

The JPNCE-SBL framework demonstrates that fully coupled bilinear inference of PN and off-grid sparse channels in AFDM is computationally feasible—achieving practical time complexity while closing the gap to ideal CSI. The method substantially advances the state of the art in robust, hardware-impaired channel estimation for next-generation wireless waveforms, combining theoretical rigor with implementation awareness.

By jointly addressing both nonlinear PN effects and off-grid dictionary mismatch within a single EM framework, the approach prevents the error propagation found in sequential or decoupled methods. The use of reduced-rank models and grid evolution generalizes beyond AFDM, suggesting broader applicability to other bilinear channel/hardware impairment estimation problems.

Future Directions

Potential extensions include direct generalization to massive MIMO-AFDM, further algorithmic optimizations for extreme real-time constraints, and incorporating additional transceiver impairments such as IQ imbalance or amplifier nonlinearities within the same SBL-based estimation paradigm. The demonstrated resilience to high mobility and oscillator imperfections positions this approach as essential for ultra-reliable, low-latency 6G communications and beyond.

Conclusion

JPNCE-SBL provides a unified, computationally tractable solution to the joint estimation of phase noise and off-grid channel parameters in AFDM systems. Through reduced-rank PN modeling, dynamic grid evolution, and a consistent EM framework, the method achieves robust channel and PN recovery, eliminates high-SNR error floors, and is poised for practical deployment in high-mobility, hardware-impaired communication scenarios.