  Orthogonal delay-Doppler (DD) division multiplexing (ODDM) has been recently
proposed as a promising multicarrier modulation scheme to tackle Doppler spread
in high-mobility environments. Accurate channel estimation is of paramount
importance to guarantee reliable communication for the ODDM, especially when
the delays and Dopplers of the propagation paths are off-grid. In this paper,
we propose a novel grid refinement and adjustment-based sparse Bayesian
inference (GRASBI) scheme for DD domain channel estimation. The GRASBI involves
first formulating the channel estimation problem as a sparse signal recovery
through the introduction of a virtual DD grid. Then, an iterative process is
proposed that involves (i) sparse Bayesian learning to estimate the channel
parameters and (ii) a novel grid refinement and adjustment process to adjust
the virtual grid points. The grid adjustment in GRASBI relies on the maximum
likelihood principle to attain the adjustment and utilizes refined grids that
have much higher resolution than the virtual grid. Moreover, a low-complexity
grid refinement and adjustment-based channel estimation scheme is proposed,
that can provides a good tradeoff between the estimation accuracy and the
complexity. Finally, numerical results are provided to demonstrate the accuracy
and efficiency of the proposed channel estimation schemes.
