Hierarchical Beam Training for Extremely Large-Scale MIMO: From Far-Field to Near-Field

Extremely large-scale MIMO (XL-MIMO) is a promising technique for future 6G communications. The sharp increase in the number of antennas results in a transition of electromagnetic propagation from the far-field to the near-field. Due to the near-field effect, the exhaustive near-field beam training at all angles and distances requires very high overhead. The improved fast near-field beam training scheme based on time-delay structure can reduce the overhead, but it suffers from very high hardware costs and energy consumption caused by time-delay circuits. In this paper, we propose a near-field two dimension (2D) hierarchical beam training scheme to reduce the overhead without the need for extra hardware circuits. Specifically, we first formulate the multi-resolution near-field codewords design problem covering different angle and distance coverages. Next, inspired by phase retrieval problems in digital holography imaging technology, we propose a Gerchberg-Saxton (GS)-based algorithm to acquire the theoretical codeword by considering the fully digital architecture. Based on the theoretical codeword, an alternating optimization algorithm is proposed to acquire the practical codeword considering the hybrid digital-analog architecture. Finally, with the designed multi-resolution codebooks, we propose a near-field 2D hierarchical beam training scheme to significantly reduce the training overhead, which is verified by extensive simulation results.