  In this paper, channel estimation problem for extremely large-scale
multi-input multi-output (XL-MIMO) systems is investigated with the
considerations of the spherical wavefront effect and the spatially
non-stationary (SnS) property. Due to the diversities of SnS characteristics
among different propagation paths, the concurrent channel estimation of
multiple paths becomes intractable. To address this challenge, we propose a
two-phase channel estimation scheme. In the first phase, the angles of
departure (AoDs) on the user side are estimated, and a carefully designed pilot
transmission scheme enables the decomposition of the received signal from
different paths. In the second phase, the subchannel estimation corresponding
to different paths is formulated as a three-layer Bayesian inference problem.
Specifically, the first layer captures block sparsity in the angular domain,
the second layer promotes SnS property in the antenna domain, and the third
layer decouples the subchannels from the observed signals. To efficiently
facilitate Bayesian inference, we propose a novel three-layer generalized
approximate message passing (TL-GAMP) algorithm based on structured variational
massage passing and belief propagation rules. Simulation results validate the
convergence and effectiveness of the proposed algorithm, showcasing its
robustness to different channel scenarios.
