  In this paper, we investigate the channel estimation problem for extremely
large-scale multi-input and multi-output (XL-MIMO) systems, considering the
spherical wavefront effect, spatially non-stationary (SnS) property, and
dual-wideband effects. To accurately characterize the XL-MIMO channel, we first
derive a novel spatial-and-frequency-domain channel model for XL-MIMO systems
and carefully examine the channel characteristics in the angular-and-delay
domain. Based on the obtained channel representation, we formulate XL-MIMO
channel estimation as a Bayesian inference problem. To fully exploit the
clustered sparsity of angular-and-delay channels and capture the inter-antenna
and inter-subcarrier correlations, a Markov random field (MRF)-based
hierarchical prior model is adopted. Meanwhile, to facilitate efficient channel
reconstruction, we propose a sparse Bayesian learning (SBL) algorithm based on
approximate message passing (AMP) with a unitary transformation. Tailored to
the MRF-based hierarchical prior model, the message passing equations are
reformulated using structured variational inference, belief propagation, and
mean-field rules. Finally, simulation results validate the convergence and
superiority of the proposed algorithm over existing methods.
