An Approximate Wave-Number Domain Expression for Near-Field XL-Array Channel

As Extremely large-scale array (XL-array) technology advances and carrier frequency rises, the near-field effects in communication are intensifying. In near-field conditions, channels exhibit a diffusion phenomenon in the angular domain, existing research indicates that this phenomenon can be leveraged for efficient parameter estimation and beam training. However, the channel model in angular domain lacks closed-form analysis, making the time complexity of the corresponding algorithm high. To address this issue, this paper analyzes the near-field diffusion effect in the wave-number domain, where the wave-number domain can be viewed as the continuous form of the angular domain. A closed-form approximate wave-number domain expression is proposed, based on the Principle of Stationary Phase. Subsequently, we derive a simplified expression for the case where the user's distance is much larger than the array aperture, which is more concise. Subsequently, we verify the accuracy of the proposed approximate expression through simulations and demonstrate its effectiveness using a beam training example. Results indicate that the beam training scheme, improved by the wave-number domain approximation model, can effectively estimate near-field user parameters and perform beam training using far-field DFT codebooks. Moreover, its performance surpasses that of existing DFT codebook-based beam training methods.