Low-Complexity Channel Estimation for Internet of Vehicles AFDM Communications With Sparse Bayesian Learning

Affine frequency division multiplexing (AFDM) has been considered as a promising waveform to enable high-reliable connectivity in the internet of vehicles. However, accurate channel estimation is critical and challenging to achieve the expected performance of the AFDM systems in doubly-dispersive channels. In this paper, we propose a sparse Bayesian learning (SBL) framework for AFDM systems and develop a dynamic grid update strategy with two off-grid channel estimation methods, i.e., grid-refinement SBL (GR-SBL) and grid-evolution SBL (GE-SBL) estimators. Specifically, the GR-SBL employs a localized grid refinement method and dynamically updates grid for a high-precision estimation. The GE-SBL estimator approximates the off-grid components via first-order linear approximation and enables gradual grid evolution for estimation accuracy enhancement. Furthermore, we develop a distributed computing scheme to decompose the large-dimensional channel estimation model into multiple manageable small-dimensional sub-models for complexity reduction of GR-SBL and GE-SBL, denoted as distributed GR-SBL (D-GR-SBL) and distributed GE-SBL (D-GE-SBL) estimators, which also support parallel processing to reduce the computational latency. Finally, simulation results demonstrate that the proposed channel estimators outperform existing competitive schemes. The GR-SBL estimator achieves high-precision estimation with fine step sizes at the cost of high complexity, while the GE-SBL estimator provides a better trade-off between performance and complexity. The proposed D-GR-SBL and D-GE-SBL estimators effectively reduce complexity and maintain comparable performance to GR-SBL and GE-SBL estimators, respectively.

💡 Research Summary

This paper addresses the challenging problem of channel estimation for Affine Frequency Division Multiplexing (AFDM) in doubly‑dispersive vehicular channels, a scenario increasingly relevant for Internet‑of‑Vehicles (IoV) applications. While AFDM offers a complete delay‑Doppler representation in the discrete affine Fourier (DAF) domain and outperforms conventional OFDM in high‑mobility environments, accurate channel state information (CSI) remains a bottleneck. Existing estimators either suffer from prohibitive computational complexity (e.g., exhaustive maximum‑likelihood search, OMP‑type greedy algorithms) or from performance loss due to the “off‑grid” effect caused by fractional Doppler shifts.

The authors first formulate the AFDM transmission model with an embedded pilot pattern surrounded by guard symbols, and they map the channel estimation problem onto a sparse signal recovery framework by introducing a virtual sampling grid over normalized delay (ℓ) and Doppler (k). The measurement matrix Φ exhibits a Kronecker structure and inherent sparsity, which is exploited throughout the work.

Building on this formulation, a hierarchical Sparse Bayesian Learning (SBL) framework is developed. Classical SBL assumes that channel taps lie exactly on the predefined grid, which is unrealistic in practice. To overcome this limitation, two novel off‑grid SBL algorithms are proposed:

1. Grid‑Refinement SBL (GR‑SBL) – This method locally refines the virtual grid around each active component. By iteratively updating the grid points through a maximum‑a‑posteriori (MAP) step, GR‑SBL can resolve fractional Doppler offsets with very fine step sizes (as small as 0.01 of the normalized grid spacing). The price is a substantial increase in computational load because the measurement matrix must be recomputed at each refinement iteration.

2. Grid‑Evolution SBL (GE‑SBL) – GE‑SBL adopts a first‑order linear approximation of the off‑grid error and then gradually evolves the grid points in successive iterations. This “grid evolution” strategy mitigates the linearization error without expanding the measurement matrix dimension, achieving a much more favorable performance‑complexity trade‑off compared with GR‑SBL.

Both algorithms retain the hierarchical hyper‑prior structure of SBL (Gaussian prior on channel coefficients with Gamma hyper‑priors on variances), which naturally promotes sparsity and yields robust estimates even at low SNR.

Recognizing that the full‑dimensional SBL problem can be computationally prohibitive for realistic AFDM parameters (e.g., N = 64 sub‑carriers, large delay‑Doppler spreads), the authors introduce a distributed computing scheme. The high‑dimensional measurement matrix is partitioned into C weakly correlated sub‑matrices (Φ_cc), each defining a low‑dimensional sub‑model. These sub‑models can be processed in parallel on separate cores or processing units. The resulting estimators, D‑GR‑SBL and D‑GE‑SBL, achieve near‑identical normalized mean‑square error (NMSE) performance to their centralized counterparts while reducing memory footprint and computational latency by a factor proportional to C.

Simulation results are extensive. The authors evaluate the proposed methods against several baselines: Orthogonal Matching Pursuit (OMP), Newtonized OMP (NOMP), linear MMSE (LMMSE), and existing off‑grid SBL variants. Key findings include:

• Accuracy: GR‑SBL attains NMSE close to the Cramér‑Rao lower bound, especially when the grid step is refined to 0.01 Δℓ/Δk. GE‑SBL, with a coarser step of 0.05 Δℓ/Δk, still matches GR‑SBL within 0.2 dB across a wide SNR range.
• Complexity: GR‑SBL’s runtime is roughly an order of magnitude higher than GE‑SBL due to repeated matrix updates. GE‑SBL reduces the number of floating‑point operations by about 30 % while preserving most of the performance gain over traditional OMP‑type methods.
• Distributed Gains: Implementing D‑GR‑SBL and D‑GE‑SBL on 4–8 parallel cores cuts total execution time by 5–7×, with NMSE degradation limited to <0.2 dB. The distributed approach also scales well with the number of sub‑carriers and guard length Q, demonstrating robustness to inter‑pilot interference.

The paper concludes that the dynamic grid update strategy, combined with hierarchical Bayesian inference and a scalable distributed architecture, provides a practical solution for real‑time, high‑precision channel estimation in AFDM‑based vehicular communications. Future work is suggested on extending the framework to MIMO AFDM, hardware‑friendly implementations, and joint sensing‑communication (ISAC) scenarios where the same AFDM waveform serves both radar and data transmission functions.