  With the application of high-frequency communication and extremely large MIMO
(XL-MIMO), the near-field effect has become increasingly apparent. The
near-field beam design now requires consideration not only of the angle of
arrival (AoA) information but also the curvature of arrival (CoA) information.
However, due to their mutual coupling, orthogonally decomposing the near-field
space becomes challenging. In this paper, we propose a Joint Autocorrelation
and Cross-correlation (JAC) scheme to address the coupling information between
near-field CoA and AoA. First, we analyze the similarity between the near-field
problem and the Doppler problem in digital signal processing, revealing that
the autocorrelation function can effectively extract CoA information.
Subsequently, utilizing the obtained CoA, we transform the near-field problem
into a far-field form, enabling the direct application of beam training schemes
designed for the far-field in the near-field scenario. Finally, we analyze the
characteristics of the far and near-field signal subspaces from the perspective
of matrix theory and discuss how the JAC algorithm handles them. Numerical
results demonstrate that the JAC scheme outperforms traditional methods in the
high signal-to-noise ratio (SNR) regime. Moreover, the time complexity of the
JAC algorithm is $\mathcal O(N+1)$, significantly smaller than existing
near-field beam training algorithms.
