Ultra-massive multiple-input multiple-output (UM-MIMO) is the enabler of Terahertz (THz) communications in next-generation wireless networks. In THz UM-MIMO systems, a new paradigm of cross-field communications spanning from near-field to far-field is emerging, since the near-field range expands with higher frequencies and larger array apertures. Precise beam alignment in cross-field is critical but challenging. Specifically, unlike far-field beams that rely only on the angle domain, the incorporation of dual-domain (angle and distance) training significantly increases overhead. A natural question arises of whether far-field beam training can be deployed for crossfield beam alignment. In this paper, this question is answered, by demonstrating that the far-field training enables sufficient signal-to-noise ratio (SNR) in both far-and near-field scenarios, while exciting all channel dimensions. Based on that, we propose a subarray-coordinated hierarchical (SCH) training with greatly reduced overhead. To further obtain high-precision beam designs, we propose a two-phase angle and distance beam estimator (TPBE). Extensive simulations demonstrate the effectiveness of the proposed methods. Compared to near-field exhaustive search, the SCH possesses 0.2% training overhead. The TPBE achieves 0.01 degrees and 0.02 m estimation root-mean-squared errors for angle and distance. Furthermore, with the estimated beam directions, a near-optimal SNR with 0.11 dB deviation is attained after beam alignment.
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