Channel Estimation with Hierarchical Sparse Bayesian Learning for ODDM Systems

Orthogonal delay-Doppler division multiplexing (ODDM) is a promising modulation technique for reliable communications in high-mobility scenarios. However, the existing channel estimation frameworks for ODDM systems cannot achieve both high accuracy and low complexity simultaneously, due to the inherent coupling of delay and Doppler parameters. To address this problem, a two-dimensional (2D) hierarchical sparse Bayesian learning (HSBL) based channel estimation framework is proposed in this paper. Specifically, we address the inherent coupling between delay and Doppler dimensions in ODDM by developing a partially-decoupled 2D sparse signal recovery (SSR) formulation on a virtual sampling grid defined in the delay-Doppler (DD) domain. With the help of the partially-decoupled formulation, the proposed 2D HSBL framework first performs low-complexity coarse on-grid 2D sparse Bayesian learning (SBL) estimation to identify potential channel paths. Then, high-resolution fine grids are constructed around these regions, where an off-grid 2D SBL estimation is applied to achieve accurate channel estimation. Simulation results demonstrate that the proposed framework achieves performance superior to conventional off-grid 2D SBL with significantly reduced computational complexity.

💡 Research Summary

This paper addresses the challenging problem of channel estimation for Orthogonal Delay‑Doppler Division Multiplexing (ODDM) in high‑mobility scenarios, where the coupling between delay and Doppler parameters makes it difficult to achieve both high accuracy and low computational complexity. Existing approaches either use one‑dimensional (1D) off‑grid Sparse Bayesian Learning (SBL), which yields accurate estimates but suffers from prohibitive complexity, or two‑dimensional (2D) SBL that assumes a bi‑orthogonal (fully decoupled) delay‑Doppler structure. The latter assumption does not hold for practical ODDM systems, leading to performance degradation.

To overcome these limitations, the authors propose a novel two‑dimensional hierarchical sparse Bayesian learning (HSBL) framework that is built on a partially‑decoupled 2D sparse signal recovery (SSR) formulation. The key ideas are:

1. Virtual Sampling Grid – A grid is defined over the delay domain (M₀ points) and the Doppler domain (N₀ points). The true channel parameters (delay lₚ, Doppler kₚ) are generally off‑grid; the authors model the off‑grid offsets Δl and Δk using first‑order Taylor expansions of the steering vectors. This yields a linearized model that captures the coupling between delay and Doppler while still allowing a sparse representation.

2. Tensor‑Matrix Operation (⊛) – The coupling is expressed through a specialized tensor‑matrix product that partially separates the two dimensions, resulting in the compact matrix equation
   **Y = K(Δk) ×