  Extremely large-scale multiple-input multiple-output (XL-MIMO) promises to
provide ultrahigh data rates in millimeter-wave (mmWave) and Terahertz (THz)
spectrum. However, the spherical-wavefront wireless transmission caused by
large aperture array presents huge challenges for channel state information
(CSI) acquisition and beamforming. Two independent parameters (physical angles
and transmission distance) should be simultaneously considered in XL-MIMO
beamforming, which brings severe overhead consumption and beamforming
degradation. To address this problem, we exploit the near-field channel
characteristic and propose two low-overhead hierarchical beam training schemes
for near-field XL-MIMO system. Firstly, we project near-field channel into
spatial-angular domain and slope-intercept domain to capture detailed
representations. Then we point out three critical criteria for XL-MIMO
hierarchical beam training. Secondly, a novel spatial-chirp beam-aided codebook
and corresponding hierarchical update policy are proposed. Thirdly, given the
imperfect coverage and overlapping of spatial-chirp beams, we further design an
enhanced hierarchical training codebook via manifold optimization and
alternative minimization. Theoretical analyses and numerical simulations are
also displayed to verify the superior performances on beamforming and training
overhead.
