  This paper investigates the uplink channel estimation of the millimeter-wave
(mmWave) extremely large-scale multiple-input-multiple-output (XL-MIMO)
communication system in the beam-delay domain, taking into account the
near-field and beam-squint effects due to the transmission bandwidth and array
aperture growth. Specifically, we model the sparsity in the delay domain to
explore inter-subcarrier correlations and propose the beam-delay domain sparse
representation of spatial-frequency domain channels. The independent and
non-identically distributed Bernoulli-Gaussian models with unknown prior
hyperparameters are employed to capture the sparsity in the beam-delay domain,
posing a challenge for channel estimation. Under the constrained Bethe free
energy minimization framework, we design different structures on the beliefs to
develop hybrid message passing (HMP) algorithms, thus achieving efficient joint
estimation of beam-delay domain channel and prior hyperparameters. To further
improve the model accuracy, the multidimensional grid point perturbation
(MDGPP)-based representation is presented, which assigns individual
perturbation parameters to each multidimensional discrete grid. By treating the
MDGPP parameters as unknown hyperparameters, we propose the two-stage HMP
algorithm for MDGPP-based channel estimation, where the output of the initial
estimation stage is pruned for the refinement stage for the computational
complexity reduction. Numerical simulations demonstrate the significant
superiority of the proposed algorithms over benchmarks with both near-field and
beam-squint effects.
