Precise Near-Field Beam Training with DFT Codebook based on Amplitude-only Measurement

Extremely large antenna arrays (ELAAs) operating in high-frequency bands have spurred the development of near-field communication, driving advancements in beam training and signal processing design. In this work, we present a low-complexity near-field beam training scheme that fully utilizes the conventional discrete Fourier transform (DFT) codebook designed for far-field users. We begin by analyzing the received beam pattern in the near field and derive closed-form expressions for the beam width and central gain. These analytical results enable the definition of an angle-dependent, modified Rayleigh distance, which effectively distinguishes near-field and far-field user regimes. Building on the analysis, we develop a direct and computationally efficient method to estimate user distance, with a complexity of O(1), and further improve its accuracy through a simple refinement. Simulation results demonstrate significant gains in both single- and multi-user settings, with up to 2.38 dB SNR improvement over exhaustive search. To further enhance estimation accuracy, we additionally propose a maximum likelihood estimation (MLE) based refinement method, leveraging the Rician distribution of signal amplitudes and achieving accuracy close to the Cramer--Rao bound (CRB). Simulation shows the single-user and multi-user achievable rates can both approach those obtained with ideal channel state information.