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

A Near Field Low Time Complexity Beam Training Scheme Based on Spatial Orthogonal Decomposition