  Extremely large-scale multiple-input multiple-output (XL-MIMO) is crucial for
satisfying the high data rate requirements of the sixth-generation (6G)
wireless networks. In this context, ensuring accurate acquisition of channel
state information (CSI) with low complexity becomes imperative. Moreover,
deploying an extremely large antenna array at the base station (BS) might
result in some scatterers being located in near-field, while others are
situated in far-field, leading to a hybrid-field communication scenario. To
address these challenges, this paper introduces two stochastic gradient pursuit
(SGP)-based schemes for the hybrid-field channel estimation in two scenarios.
For the first scenario in which the prior knowledge of the specific proportion
of the number of near-field and far-field channel paths is known, the scheme
can effectively leverage the angular-domain sparsity of the far-field channels
and the polar-domain sparsity of the near-field channels such that the channel
estimation in these two fields can be performed separately. For the second
scenario which the proportion is not available, we propose an off-grid
SGP-based channel estimation scheme, which iterates through the values of the
proportion parameter based on a criterion before performing the hybrid-field
channel estimation. We demonstrate numerically that both of the proposed
channel estimation schemes achieve superior performance in terms of both
estimation accuracy and achievable rates while enjoying lower computational
complexity compared with existing schemes. Additionally, we reveal that as the
number of antennas at the UE increases, the normalized mean square error (NMSE)
performances of the proposed schemes remain basically unchanged, while the NMSE
performances of existing ones improve. Remarkably, even in this scenario, the
proposed schemes continue to outperform the existing ones.
