Title：Fast Near-Field Beam Training for Extremely Large-Scale Array

Background

Compared with the conventional far-field beam training method that searches for the best beam angular only, the near-field beam training is more challenging since it requires a beam search over both the angular and distance domains due to the spherical wavefront propagation model.

The existing near-field training methods for the narrow-band require a two-dimensional exhaustive search for all possible beam angular and distances, thus leading to prohibitively high training overhead.

Method

1、System model (N-antenna BS LOS channel)

A narrow-band XL-array communication system

Channel coefficient

$\mathbf{h}_{\mathrm{near}}^H=\sqrt{N}h\mathbf{b}^H(\theta,r)$

where $h=\frac{\sqrt{\beta}}{r}e^{-\frac{j 2\pi r}{\lambda}}$

Near-field steering vector

$\mathbf{b}^{H}\left(\theta,r\right)=\frac{1}{\sqrt{N}}\left[e^{-j2\pi(r^{(0)}-r)/\lambda},\cdots,e^{-j2\pi(r^{(N-1)}-r)/\lambda}\right]$

Far-field steering vector

$\mathbf{a}^H(\theta)\triangleq\frac{1}{\sqrt{N}}\left[1,e^{-j\pi\theta},\cdots,e^{-j\pi(N-1)\theta}\right]$

notice $ \mathbf{a}^H(\theta) {a}(\theta)=1$

The received signal at the user is given by

$y_\mathrm{near}=\mathbf{h}_\mathrm{near}^H\mathbf{v}x+z_0=\sqrt{N}h\mathbf{b}^H(\theta,r)\mathbf{v}x+z_0$

where $v$ represents the transmit beamforming vector

2、 Two-phase near-field beam training method

Defination of the normalized beam gain

$\mathbf{A}(\mathbf{u}^H,\mathbf{w})=|\mathbf{u}^H\mathbf{w}|$

where the beamforming vector $w$ and the channel steering vector $\mathbf{u}^H$

对于MISO far field LOS channel，beamtraining就是将360度离散细分以后，将每个steer vector的共轭转置作为transmit beamforming vector代入，找个功率最大值，因为远场只与angular有关，所以beam gain通常只有一个极窄角度处功率很高。而对于near field，steer vector同时与distance和angular有关就会有角度的扩散。

First phase

we aim to estimate the spatial angle of the user in the first phase based on the far-field beam training method.
we propose a new middle-K angle selection scheme that selects K candidate spatial angles in the middle of the quantized dominant-angle region rather than selecting one spatial angle only.

Second phase
a customized polar-domain beam training method is proposed for the second phase to estimate the effective user distance based on the non-uniform distance sampling method.

Conclusion

Last, it is worth mentioning that there exists a fundamental tradeoff between the beam training performance versus the number of candidate angles, K. Specifically, when K is larger, it incurs a larger training overhead, while leading to a higher beamforming gain due to the more accurate angle estimation.

Summary

能将近场beamtraining分两步来节省开销的主要原因在于：将远场的beamforming vector代入近场以后，虽然角度有扩散，但用户真实角度是近似位于角度扩散的中心区域，相当于确定了用户在一个窄角度内，方便后续距离处理。

其次，这样做可以区分近场与远场，如果接收端功率比较集中，则位于远场，便不需要第二步了。