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.

翻译：随着高频通信与超大规模MIMO（XL-MIMO）技术的应用，近场效应日益显著。近场波束设计不仅需要考虑到达角（AoA）信息，还需要考虑到达曲率（CoA）信息。然而，由于两者的相互耦合，对近场空间进行正交分解面临挑战。本文提出一种联合自相关与互相关（JAC）方案以解决近场CoA与AoA之间的耦合信息。首先，我们分析了近场问题与数字信号处理中多普勒问题的相似性，揭示自相关函数可有效提取CoA信息。随后，利用获取的CoA将近场问题转换为远场形式，使得适用于远场的波束训练方案可直接应用于近场场景。最后，我们从矩阵理论角度分析远场与近场信号子空间的特征，并讨论JAC算法如何处理这些子空间。数值结果表明，在高信噪比（SNR）条件下，JAC方案优于传统方法。此外，JAC算法的时间复杂度为$\mathcal O(N+1)$，显著低于现有近场波束训练算法。