  Downlink reconfigurable intelligent surface (RIS)-assisted
multi-input-multi-output (MIMO) systems are considered with far-field,
near-field, and hybrid-far-near-field channels. According to the angular or
distance information contained in the received signals, 1) a distance-based
codebook is designed for near-field MIMO channels, based on which a
hierarchical beam training scheme is proposed to reduce the training overhead;
2) a combined angular-distance codebook is designed for mixed-far-near-field
MIMO channels, based on which a two-stage beam training scheme is proposed to
achieve alignment in the angular and distance domains separately. For
maximizing the achievable rate while reducing the complexity, an alternating
optimization algorithm is proposed to carry out the joint optimization
iteratively. Specifically, the RIS coefficient matrix is optimized through the
beam training process, the optimal combining matrix is obtained from the
closed-form solution for the mean square error (MSE) minimization problem, and
the active beamforming matrix is optimized by exploiting the relationship
between the achievable rate and MSE. Numerical results reveal that: 1) the
proposed beam training schemes achieve near-optimal performance with a
significantly decreased training overhead; 2) compared to the angular-only
far-field channel model, taking the additional distance information into
consideration will effectively improve the achievable rate when carrying out
beam design for near-field communications.
