# Near-Field Rainbow: Wideband Beam Training for XL-MIMO

Mingyao Cui, Linglong Dai, Zhaocheng Wang, Shidong Zhou, and Ning Ge  All
authors are with the Beijing National Research Center for Information Science
and Technology (BNRist) as well as the Department of Electronic Engineering,
Tsinghua University, Beijing 100084, China (e-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>gening@tsinghua.edu.cn). This work was supported in
part by the National Key Research and Development Program of China (Grant
No.2020YFB1807201), in part by the National Natural Science Foundation of
China (Grant No. 62031019), and in part by the European Commission through the
H2020-MSCA-ITN META WIRELESS Research Project under Grant 956256.

###### Abstract

Wideband extremely large-scale multiple-input-multiple-output (XL-MIMO) is a
promising technique to achieve Tbps data rates in future 6G systems through
beamforming and spatial multiplexing. Due to the extensive bandwidth and the
huge number of antennas for wideband XL-MIMO, a significant near-field beam
split effect will be induced, where beams at different frequencies are focused
on different locations. The near-field beam split effect results in a severe
array gain loss, so existing works mainly focus on compensating for this loss
by utilizing the time delay (TD) beamformer. By contrast, this paper
demonstrates that although the near-field beam split effect degrades the array
gain, it also provides a new possibility to realize fast near-field beam
training. Specifically, we first reveal the mechanism of the near-field
controllable beam split effect. This effect indicates that, by dedicatedly
designing the delay parameters, a TD beamformer is able to control the degree
of the near-field beam split effect, i.e., beams at different frequencies can
flexibly occupy the desired location range. Due to the similarity with the
dispersion of natural light caused by a prism, this effect is also termed as
the near-field rainbow in this paper. Then, taking advantage of the near-field
rainbow effect, a fast wideband beam training scheme is proposed. In our
scheme, the close form of the beamforming vector is elaborately derived to
enable beams at different frequencies to be focused on different desired
locations. By this means, the optimal beamforming vector with the largest
array gain can be rapidly searched out by generating multiple beams focused on
multiple locations simultaneously through only one radio-frequency (RF) chain.
Finally, simulation results demonstrate the proposed scheme is able to realize
near-optimal near-field beam training with a very low training overhead.

###### Index Terms:

XL-MIMO, near-field, wideband, beam training.

##  I Introduction

Wideband extremely large-scale multiple-input-multiple-output (XL-MIMO) has
been regarded as a promising technology to meet the capacity requirement for
future 6G communications [1, 2]. Benefiting from the huge spatial multiplexing
gain provided by a very large number of antennas, XL-MIMO is able to
significantly increase the spectrum efficiency [3]. Moreover, XL-MIMO can also
provide very high beamforming gain to compensate for the severe path loss at
millimeter-wave (mmWave) or terahertz (THz) band, which may provide tens of
GHz-wide bandwidth to enable Tbps data rates for future 6G networks [4].

Nevertheless, due to the very large array aperture and the extensive bandwidth
at high frequencies, a significant near-field beam split effect will be
introduced [5]. Firstly, compared to the far-field propagation in conventional
5G systems where electromagnetic (EM) wavefronts can be approximated as
_planar_ , the deployment of XL-MIMO, especially at high-frequency, indicates
that the near-field propagation will become essential in 6G networks, where
the EM wavefronts have to be accurately modeled as _spherical_ [6, 7]. The
boundary between the far-field and near-field regions is determined by the
Rayleigh distance, which is proportional to the square of the array aperture
and the signal frequency [8]. With the increased array aperture and frequency
of XL-MIMO, its near-field range can be several dozens and even hundreds of
meters [9]. For example, for a 1-meter diameter array operating at 30 GHz, the
near-field region extends to distances of 200 meters, dominating the typical
outdoor communication environments. As a result, spherical wavefronts should
be exploited to realize near-field beamfocusing (near-field beamforming) in
XL-MIMO systems to allow focusing signals at a _specific location_ , rather
than the conventional far-field beamsteering (far-field beamforming) that
steers signals towards a _specific angle_ [9].

Secondly, the very large bandwidth results in the near-field beam split
effect. In XL-MIMO systems, phase-shifter (PS) based beamformer is widely
considered to generate focused beams aligned with certain locations to provide
beamfocusing gain [10]. Such a beamformer works well for narrowband systems.
However, for wideband systems, the beams at different frequencies with
spherical wavefronts will be focused on different physical locations due to
the use of nearly _frequency-independent_ PSs, which is referred to as the
near-field beam split effect [5].  This effect results in a severe array gain
loss since the beams over different frequencies cannot be aligned with the
target user in a certain location, which should be carefully addressed.

###  I-A Prior Works

As the antenna number is not very large for current 5G massive MIMO
communications [11], existing works mainly focus on a simplified _far-field
beam split_ effect. To be specific, the channels are modeled under the planar
wavefronts, and the beams over different frequencies are split into different
spatial angles, where the distance information is ignored. These works can be
generally classified into two categories, i.e., techniques overcoming the far-
field beam split effect [12, 13, 14, 15, 16] and those taking advantage of it
[17, 18, 19]. The first category desires to mitigate the array gain loss
caused by the far-field beam split. To realize this target, introducing time-
delay circuits into beamforming structures, such as true-time-delay array [12,
13] or delay-phase precoding structure [15, 16], is considered to be
promising. Thanks to the _frequency-dependent_ phase shift provided by the
time-delay circuit, the generated beams over the entire bandwidth can be
aligned with a certain spatial angle, and thus the far-field beam split effect
can be alleviated. For the second category of taking advantage of the far-
field beam split, it has been proved that time-delay circuits can not only
mitigate the far-field beam split, but also flexibly control its degree [17,
19, 18]. By carefully designing the delay parameters, the covered angular
range of the beams over different frequencies can be controlled. Benefiting
from this fact, very fast channel station information (CSI) acquisition in the
far-field, such as fast beam training or beam tracking, can be realized. In
[17, 18], based on the true-time-delay array architecture, fast beam training
schemes with only one pilot overhead were studied by generating frequency-
dependent beams to simultaneously search multiple angles. In [19], a fast beam
tracking scheme based on the delay-phase precoding architecture was proposed
by adaptively adjusting the degree of the far-field beam split according to
the user mobility. From the discussion above, we can find that although far-
field beam split results in a severe beamsteering gain loss that should be
addressed, it can also benefit the fast far-field CSI acquisition in massive
MIMO systems.

When it comes to the XL-MIMO systems, the more realistic near-field beam split
effect should be considered, since the antenna number is huge. Recently,
researchers in [5] have tried to compensate for the corresponding array gain
loss in the near-field range. Specifically, the near-field beam split effect
was defined and analyzed in our previous work [5], and the time-delay (TD)
based beamformer was also utilized to overcome this effect. We have proposed
to partition the entire array into multiple sub-arrays, and then the user can
be assumed to be located within the near-field range of the entire array but
in the far-field range of each sub-array. Based on this partition, time-delay
circuits can also be utilized to compensate for the group delays across
different sub-arrays induced by near-field spherical wavefronts. As a result,
the beams over the entire bandwidth can be focused on the desired spatial
angle and distance, and the near-field beam split effect is alleviated
accordingly.

Efficient design of wideband XL-MIMO beamfocusing to alleviate the near-field
beam split effect requires accurate near-field CSI. To meet this requirement,
an intuitive way is to directly utilize the existing wideband far-field CSI
acquisition schemes [17, 19, 18] to estimate the near-field wideband channel.
However, since the far-field planar wavefront mismatches the near-field
spherical wavefront, these methods [17, 19, 18] may be not valid in the near-
field range. To cope with this problem, inspired by the classical far-field
hierarchical beam training scheme [20], a near-field hierarchical beam
training method is proposed in [21], which uniformly searches multiple angles
and distances to obtain the near-field CSI.  Moreover, from the perspective of
near-field array gain, [6] proved that the distances should be non-uniformly
searched for improving channel estimation accuracy. Nevertheless, existing
near-field CSI acquisition methods assume that, the bandwidth is not very
large, so the near-field beam split effect is not considered. Moreover, unlike
the angle-dependent far-field CSI, to obtain the near-field CSI, the angle and
distance information should be estimated simultaneously, which may result in
unacceptable pilot overhead for near-field beam training [21]. Unfortunately,
to the best of our knowledge, determining how to obtain accurate wideband XL-
MIMO near-field CSI with acceptable pilot overhead has not been studied in the
literature.

###  I-B Our Contributions

To fill in this gap, inspired by the two categories of research on the far-
field beam split, i.e., overcoming and taking advantage of it, we unveil that
although the near-field beam split effect degrades the array gain, it can also
provide a new possibility to benefit the near-field CSI acquisition. Based on
this new finding, we propose a fast wideband near-field beam training scheme
by taking advantage of the near-field beam split effect. Specifically, our
contributions are summarized as follows.

  * •

Firstly, we prove the effect of near-field controllable beam split, i.e.,
time-delay circuits can flexibly control the degree of near-field beam split.
Specifically, it has been proved in far-field scenarios that, time-delay
circuits can control the covered angular range of the beams over different
frequencies when the distance is very large [17, 19, 18]. By contrast, we will
reveal that not only the covered angular range, but also the covered distance
range of the beams over different frequencies can be controlled by the
elaborate design of time delays, and thus the control of the near-field beam
split is achievable. A simple analogy of the near-field controllable beam
split effect is the dispersion of white light with a large bandwidth caused by
a prism, thus in this paper, this effect is also termed as “near-field
rainbow”. This near-field rainbow effect allows us to generate multiple beams
focusing on multiple locations simultaneously by only one radio-frequency (RF)
chain, which is not achievable for existing schemes.

  * •

Then, based on the mechanism of the near-field rainbow, a wideband near-field
beam training scheme is proposed to realize fast near-field CSI acquisition.
In the proposed scheme, multiple beams focusing on multiple angles in a given
distance range are generated by time-delay circuits in each time slot. Next,
for different time slots, different distance ranges are measured by fine-
tuning the time-delay parameters. By this means, the optimal spatial angle can
be searched in a frequency-division manner, while the optimal distance is
obtained in a time-division manner. Unlike the exhaustive near-field beam
training scheme that searches only one location in each time slot, the
proposed scheme is able to search multiple locations in each time slot, thus
the beam training overhead can be significantly reduced.

  * •

Finally, we provide simulation results to demonstrate the mechanism of the
near-field rainbow and verify the advantages of the proposed beam training
scheme. We demonstrate that our scheme is able to achieve a satisfactory
average rate performance at a significantly reduced training overhead. We also
show that our beam training scheme outperforms the existing far-field schemes
in the near-field range. Additionally, our scheme also works well in the far-
field range, since the proposed near-field beam training scheme is able to
automatically decay to the far-field beam training in the far-field scenarios.

###  I-C Organization and Notation

_Organization_ : The rest of this paper is organized as below. In Section II,
the system model is introduced. We first introduce the near-field beam split
effect in wideband XL-MIMO systems and then discuss how to mitigate it with a
time-delay beamformer. Then in Section III, we prove the mechanism of the
near-field controllable beam split. In Section IV, the exhaustive near-field
beam training is defined and the near-field rainbow based beam training is
proposed. Section V provides the simulation results. Finally, conclusions are
drawn in Section VI.

_Notation_ : Lower-case and upper-case boldface letters represent vectors and
matrices, respectively; $[{\mathbf{x}}]_{n}$ denotes the $n$-th element of the
vector ${\mathbf{x}}$; $(\cdot)^{*}$, $(\cdot)^{T}$, and $(\cdot)^{H}$ denote
the conjugate, transpose, and conjugate transpose, respectively; $|\cdot|$ and
$\text{Tr}(\cdot)$ denote the absolute and trace operator; $\lceil x\rceil$
denotes the nearest integer greater than or equal to $x$;
$\mathcal{CN}(\mu,\Sigma)$ and $\mathcal{U}(a,b)$ denote the Gaussian
distribution with mean $\mu$ and covariance $\Sigma$, and the uniform
distribution between $a$ and $b$, respectively.

##  II System Model

In this section, we first introduce the near-field channel model and the near-
field beam split effect in wideband XL-MIMO, then we discuss how to mitigate
it by time-delay circuits.

###  II-A Near-Field Wideband Channel Model

Figure 1:  The near-field spherical wavefront model.

In this paper, a wideband XL-MIMO system is considered. We assume the base
station (BS) is equipped with a uniform linear array (ULA) to serve a single
omnidirectional antenna user using orthogonal frequency division multiplexing
(OFDM) with $M$ subcarriers. For expression simplicity, we assume the number
of BS antennas is $N_{t}=2N+1$. We denote $B$, $c$, $f_{c}$,
$\lambda_{c}=\frac{c}{f_{c}}$ as the bandwidth, the light speed, the central
carrier frequency, and the central wavelength, respectively. The antenna
spacing is presented as $d=\frac{\lambda_{c}}{2}$.

Due to the severe path loss incurred by the scatters, mmWave and THz
communications heavily rely on the line-of-sight (LoS) path [4]. Therefore, we
mainly focus on the near-field LoS channel, while the discussions in this
paper can be straightforwardly extended to the non-LoS (NLoS) scenarios. As
shown in Fig. 1, the user is located at
$(r_{0}\cos\vartheta_{0},r_{0}\sin\vartheta_{0})$ with
$\theta_{0}=\sin\vartheta_{0}$. Taking into account the near-field spherical
wave characteristic [22], the channel between the $n$-th BS antenna and the
user at the $m$-th subcarrier with $n=-N,\cdots,0,\cdots,N$ and
$m=1,2,\cdots,M$ can be represented as

$\displaystyle[{\mathbf{h}}_{m}]_{n}=\beta_{m}^{(n)}e^{-j{k_{m}r_{0}^{(n)}}},$
(1)

where $k_{m}=\frac{2\pi f_{m}}{c}$ denotes the wavenumber at subcarrier
frequency $f_{m}=f_{c}+\frac{B}{M}\left(m-1-\frac{M-1}{2}\right)$ and
$r^{(n)}_{0}$ represents the distance between the user and the $n$-th BS
antenna. As the coordinate of the $n$-th BS antenna is $(0,nd)$, $r_{0}^{(n)}$
can be derived from the geometry as
$r_{0}^{(n)}=\sqrt{(r_{0}\cos\vartheta_{0})^{2}+(r_{0}\sin\vartheta_{0}-nd)^{2}}=\sqrt{r_{0}^{2}+n^{2}d^{2}-2r_{l}\theta_{0}nd}$.
The path gain $\beta^{(n)}_{m}$ can be modeled as [23]

$\displaystyle\beta^{(n)}_{m}=\sqrt{G_{t}F\left(\vartheta_{0}^{(n)}\right)}\frac{\lambda_{m}}{4\pi
r_{0}^{(n)}},$ (2)

where wavelength $\lambda_{m}=\frac{f_{m}}{c}$ and $\vartheta^{(n)}_{0}$
denotes the angle between the user and the $n$-th BS antenna. Moreover,
$G_{t}$ is the antenna gain and $F\left(\vartheta\right)$ denotes normalized
power radiation pattern, satisfying
$\int_{\vartheta=0}^{\pi/2}G_{t}F(\vartheta)\sin\vartheta d\vartheta=1$. An
example of normalized power radiation pattern is
$F(\vartheta)=\cos^{3}\vartheta,\vartheta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
[23]. Generally, the distance $r_{0}$ between the user and BS is larger than
the array aperture $D=N_{t}d$. For example, for a 256-element ULA working at
30 GHz, $r_{0}$ is very likely to be larger than $D=256\times 0.5\times
10^{-3}=1.28$ meters. With $r_{0}>D$, we can assume
$\beta^{(-N)}_{m}\approx\cdots\approx\beta^{(N)}_{m}\approx\beta_{m}=\sqrt{G_{t}F\left(\vartheta_{0}\right)}\frac{\lambda_{m}}{4\pi
r_{0}}$ based on the Fresnel approximation [24]. As a result, the near-field
LoS channel ${\mathbf{h}}_{m}\in\mathbb{C}^{N_{t}\times 1}$ can be represented
as

$\displaystyle{\mathbf{h}}_{m}$
$\displaystyle=\beta_{m}\left[e^{-j{k_{m}r_{0}^{(-N)}}},\cdots,e^{-j{k_{m}r_{0}^{(N)}}}\right]^{T}=\sqrt{N_{t}}\beta_{m}{\mathbf{a}}_{m}(\theta_{0},r_{0}),$
(3)

where ${\mathbf{a}}_{m}(\theta_{0},r_{0})$ is the near-field array response
vector.

Figure 2:  The schematic diagrams of different distance rings
$\alpha=\frac{1-\theta^{2}}{2r}$.

Notice that since the near-field spherical wave characteristic is considered,
the array response vector ${\mathbf{a}}_{m}(\theta_{0},r_{0})$ is
significantly different from the classical far-field array response vector
[25], where the latter is based on the planar wave assumption and ignores the
influence of distance $r_{0}$. The radius of the near-field region is
determined by the Rayleigh distance $\text{RD}=\frac{2D^{2}}{\lambda_{c}}$
[8]. For a 1-meter diameter array operating at 30 GHz, its Rayleigh distance
reaches up to 200 meters. Therefore, in such a mmWave XL-MIMO system, the
near-field spherical wave characteristic becomes essential and the distance
information cannot be ignored.

However, since the distance $r_{0}^{(n)}$ is a complicated radical function
with respect to the antenna index $n$, it is difficult to analyze the property
of the near-field spherical wavefront directly from (3). To deal with this
problem, the Fresnel approximation [24] can be adopted to approximate
$r_{0}^{(n)}$ as

$\displaystyle
r_{0}^{(n)}\overset{(a)}{\approx}r_{0}-nd\theta_{0}+n^{2}d^{2}\frac{1-\theta_{0}^{2}}{2r_{0}},$
(4)

where $(a)$ is based on the second-order Taylor expansion $\sqrt{1+x}\approx
1+\frac{1}{2}x-\frac{1}{8}x^{2}$. For expression simplicity, we denote
$\alpha_{0}=\frac{1-\theta_{0}^{2}}{2r_{0}}$ and
$\alpha=\frac{1-\theta^{2}}{2r}$. As shown in Fig. 2, the curve
$\alpha=\frac{1-\theta^{2}}{2r}$ corresponds to a ring in the physical space,
and thus the curve $\alpha=\frac{1-\theta^{2}}{2r}$ is termed as _distance
ring_ $\alpha$. Then, according to (4), the $n$-th element of
${\mathbf{a}}_{m}(\theta_{0},r_{0})$ can be approximated as

$\displaystyle[{\mathbf{a}}_{m}(\theta_{0},r_{0})]_{n}$
$\displaystyle\approx\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}(r_{0}-nd\theta_{0}+n^{2}d^{2}\alpha_{0})}=e^{-jk_{m}r_{0}}[{\mathbf{b}}_{m}(\theta_{0},\alpha_{0})]_{n},$
(5)

where
$[{\mathbf{b}}_{m}(\theta_{0},\alpha_{0})]_{n}=\frac{1}{\sqrt{N}}e^{jk_{m}(nd\theta_{0}-n^{2}d^{2}\alpha_{0})}$
denotes the $n$-th element of vector
${\mathbf{b}}_{m}(\theta_{0},\alpha_{0})$. Since the residue phase
$e^{-jk_{m}r_{0}}$ in (5) has no relationship with the antenna index $n$, we
only need to focus on the vector ${\mathbf{b}}_{m}(\theta_{0},\alpha_{0})$.

###  II-B Near-Field Beam Split

At mmWave or THz band, _frequency-independent_ PS-based beamformer is widely
considered to serve the user by generating a focused beam aligned with
location $(r_{0},\theta_{0})$ in the polar coordinate. Generally, the
beamfocusing vector ${\mathbf{w}}_{\text{PS}}$ generated by frequency-
independent PSs is usually set as
${\mathbf{w}}_{\text{PS}}={\mathbf{b}}_{c}^{*}(\theta_{0},\alpha_{0})$ to
focus the beam energy on the location $(r_{0},\theta_{0})$ [26], where
${\mathbf{b}}_{c}(\theta_{0},\alpha_{0})$ represents the array response vector
(5) at frequency $f_{c}$.

Figure 3:  The near-field beam split effect.

However, the array response vector (5) of the wideband channel is _frequency-
dependent_ , which mismatches the _frequency-independent_ beamfocusing vector
${\mathbf{w}}_{\text{PS}}={\mathbf{b}}_{c}^{*}(\theta_{0},\alpha_{0})$. This
mismatch leads to the fact that, the beams generated by
${\mathbf{w}}_{\text{PS}}$ at different frequencies will be focused on
different locations, which is detailed in the following Lemma 1.

######  Lemma 1.

For near-field wideband communications, the beam at frequency $f_{m}$
generated by
${\mathbf{w}}_{\text{PS}}={\mathbf{b}}_{c}^{*}(\theta_{0},\alpha_{0})$ will be
focused on the location $(\theta_{m},r_{m})$, satisfying

$\displaystyle\theta_{m}$
$\displaystyle=(k_{c}/k_{m})\theta_{0}=(f_{c}/f_{m})\theta_{0}=\theta_{0}/\eta_{m},$
(6) $\displaystyle\alpha_{m}$
$\displaystyle=(k_{c}/k_{m})\alpha_{0}=(f_{c}/f_{m})\alpha_{0}=\alpha_{0}/\eta_{m},$
(7)

where we define $\alpha_{m}=\frac{1-\theta_{m}^{2}}{2r_{m}}$,
$\eta_{m}=f_{m}/f_{c}$, and $k_{c}=\frac{2\pi f_{c}}{c}$.

_Proof_ : At frequency $f_{m}$, the array gain achieved by
${\mathbf{w}}_{\text{PS}}$ on an arbitrary user location $(r,\theta)$ with
$\alpha=\frac{1-\theta^{2}}{2r}$ is

$\displaystyle|{\mathbf{w}}_{\text{PS}}^{T}{\mathbf{b}}_{m}(\theta,\alpha)|$
$\displaystyle=\frac{1}{N_{t}}\left|\sum_{n=-N}^{N}e^{jnd(k_{m}\theta-
k_{c}\theta_{0})-jn^{2}d^{2}(k_{m}\alpha-k_{c}\alpha_{0})}\right|$
$\displaystyle=G(k_{m}\theta-k_{c}\theta_{0},k_{m}\alpha-k_{c}\alpha_{0}),$
(8)

where we define $G(x,y)=\frac{1}{N_{t}}\left|\sum_{n}e^{jndx-
jn^{2}d^{2}y}\right|$. Apparently, the beam at frequency $f_{m}$ is focused on
the location $(r_{m},\theta_{m})$ corresponding to the maximum array gain
$G(k_{m}\theta_{m}-k_{c}\theta_{0},k_{m}\alpha_{m}-k_{c}\alpha_{0})$, i.e.,

$\displaystyle(r_{m},\theta_{m})=\arg\max_{r,\theta}G(k_{m}\theta-
k_{c}\theta_{0},k_{m}\alpha-k_{c}\alpha_{0}),$ (9)

where $\alpha_{m}=\frac{1-\theta_{m}^{2}}{2r_{m}}$. Since $G(0,0)=1$ and
$|G(x,y)|\leq 1$, it is obvious that $(x,y)=(0,0)$ is an optimal solution to
maximize the function $G(x,y)$. Therefore, we have
$k_{m}\theta_{m}-k_{c}\theta_{0}=0$ and $k_{m}\alpha_{m}-k_{c}\alpha_{0}=0$,
which leads to the results of (6) and (7). $\hfill\blacksquare$

According to Lemma 1, the beam at $f_{m}$ is focused on the location
$(\theta_{m},r_{m})=(\theta_{m},\frac{1-\theta_{m}^{2}}{2\alpha_{m}})=(\frac{\theta_{0}}{\eta_{m}},r_{0}\frac{\eta_{m}-\frac{1}{\eta_{m}}\theta_{0}^{2}}{1-\theta_{0}^{2}})$.
As shown in Fig. 3, since the desired user is located at
$(r_{0},\theta_{0})\neq(r_{m},\theta_{m})$, the beam at frequency $f_{m}$
cannot be aligned with the desired user location.  This misalignment is termed
as the “near-field beam split” effect [5], which will result in a severe array
gain loss when $f_{m}$ is far away from $f_{c}$ in wideband XL-MIMO systems.
For example, if the carrier frequency and bandwidth are $30$ GHz and $1$ GHz,
and the BS is equipped with a 256-element ULA, then around 50% subcarriers
will suffer from more than 50% array gain loss [5]. Therefore, the near-field
beam split effect should be elaborately addressed, especially when the
bandwidth is very large.

###  II-C Time-Delay Based Beamformer

To mitigate the near-field beam split effect, a method is to utilize TD based
beamformer rather than PS based beamformer to generate frequency-dependent
beams to match the frequency-dependent channels. Notice that the TD based
beamsteering has been studied in the far-field wideband systems [12, 13, 14,
15, 16, 17, 18, 19], while in this study, we consider utilizing it to overcome
and control the near-field beam split effect.

Figure 4:  The TD beamformer architecture.

As shown in Fig. 4, we assume the BS is equipped with an $N_{t}$-element TD
beamformer111The main purpose of this paper is to derive that time-delay
circuits are able to control the near-field beam split effect. Therefore, we
directly utilize the time-delay beamformer as an example, while the discussion
on some other low-power consumption architectures, such as delay-phase
precoding, is left for future works.. Then the frequency response at frequency
$f_{m}$ of the $n$-th antenna is
$[{\mathbf{w}}_{m}]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-j2\pi
f_{m}\tau^{(n)^{\prime}}}$, where ${\mathbf{w}}_{m}$ denotes the frequency-
dependent beamforming vector, and $\tau^{(n)^{\prime}}$ denotes the adjustable
time-delay parameter of the $n$-th time-delay circuit. For expression
simplicity, we denote $r^{(n)^{\prime}}=c\tau^{(n)^{\prime}}$ as the
adjustable distance, and then $[{\mathbf{w}}_{m}]_{n}$ becomes
$\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}r^{(n)^{\prime}}}$. Notice that
$[{\mathbf{w}}_{m}]_{n}$ has a similar form with the array response vector
$[{\mathbf{b}}_{m}(\theta_{0},r_{0})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{jk_{m}(nd\theta_{0}-n^{2}d^{2}\alpha_{0})}$.
Thus, the $n$-th adjustable distance $r^{(n)^{\prime}}$ can be set as
$r^{(n)^{\prime}}=nd\theta^{\prime}-n^{2}d^{2}\alpha^{\prime}$, while
$\theta^{\prime}$ and $\alpha^{\prime}$ are adjustable delay parameters of a
TD beamformer. Then, the corresponding beamfocusing vector at $f_{m}$ can be
presented as

$\displaystyle[{\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}(nd\theta^{\prime}-n^{2}d^{2}\alpha^{\prime})}.$
(10)

In this case, at frequency $f_{m}$, the array gain on an arbitrary location
$(r,\theta)$ with $\alpha=\frac{1-\theta^{2}}{2r}$ is given by

$\displaystyle|{\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime})^{T}{\mathbf{b}}_{m}(\theta,\alpha)|$
$\displaystyle=\frac{1}{N_{t}}\left|\sum_{n=-N}^{N}e^{jnd(k_{m}\theta-
k_{m}\theta^{\prime})-jn^{2}d^{2}(k_{m}\alpha-k_{m}\alpha^{\prime})}\right|$
$\displaystyle=G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime})).$
(11)

Obviously, the beam at $f_{m}$ is focused on
$(r_{m},\theta_{m})=\arg\max_{r,\theta}G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$.
Similar to the derivation of the optimal solution of (9),
$G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$ approaches
its maximum value when $k_{m}(\theta-\theta^{\prime})=0$ and
$k_{m}(\alpha-\alpha^{\prime})=0$. Therefore, the beam at frequency $f_{m}$ is
focused on $\theta_{m}=\theta^{\prime}$ and
$r_{m}=r^{\prime}=\frac{1-\theta^{\prime 2}}{2\alpha^{\prime}}$, which is not
relevant to $f_{m}$. Then, once the LoS path information $(r_{0},\theta_{0})$
is available at the BS, and the delay parameters are set as
$\theta^{\prime}=\theta_{0}$ and
$\alpha^{\prime}=\frac{1-\theta_{0}^{2}}{2r_{0}}$, then the beams across the
entire bandwidth are able to be focused on the location
$\theta_{m}=\theta^{\prime}=\theta_{0}$ and $r_{m}=\frac{1-\theta^{\prime
2}}{2\alpha^{\prime}}=r_{0}$. As a consequence, the near-field beam split
effect can be mitigated by the TD beamformer.

On the other hand, efficient wideband beamfocusing requires that the LoS path
information $(r_{0},\theta_{0})$ is available at the BS side. To realize this
requirement, in the following discussions, we prove that a TD beamformer can
not only mitigate the near-field beam split effect, but also flexibly control
its degree. Then, we will further utilize this property to achieve efficient
near-field beam training to obtain $(r_{0},\theta_{0})$.

##  III Mechanism of Near-Field Controllable Beam Split

In [17, 19, 18], the researchers proved the mechanism of far-field
controllable beam split, i.e., by elaborately designing the time-delay
parameters, the beams over the entire bandwidth are able to cover a desired
angular range in the far-field. Taking advantage of this mechanism, multiple
beams aligned with multiple angles can be generated by only one RF chain to
acquire the far-field CSI rapidly. Similarly, to obtain the near-field CSI, we
surprisingly find that time-delay circuits can not only control the angular
coverage range but also control the distance coverage range of the beams,
i.e., the near-field controllable beam split is achievable. In this section,
we will prove the mechanism of this effect, and then in the next section
utilize this mechanism to achieve fast near-field beam training. For readers
better understand our ideas, we first prove the far-field controllable beam
split and then extend it to the near-field scenarios.

###  III-A Far-Field Controllable Beam Split

For the far-field scenarios, the distances $r$ and $r^{\prime}$ are assumed to
be larger than the Rayleigh distance RD, so that the spherical wavefront can
be approximated as a planar wavefront. In this case, all of the distance-
related parameters $\alpha=\frac{1-\theta^{2}}{2r}$ and
$\alpha^{\prime}=\frac{1-\theta^{\prime 2}}{2r^{\prime}}$ reliably approach 0.
Then, the array response vector becomes
$[{\mathbf{b}}_{m}(\theta,\alpha)]_{n}=[{\mathbf{b}}_{m}(\theta,0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{jk_{m}nd\theta}$,
and the beamfocusing vector realized by a TD beamformer becomes
$[{\mathbf{w}}_{m}(\theta^{\prime},0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}nd\theta^{\prime}}$.
Therefore, the array gain in (II-C) can be simplified as

$\displaystyle
G\left(k_{m}(\theta-\theta^{\prime}),0\right)=\left|\frac{\sin\left(\frac{N_{t}}{2}dk_{m}(\theta-\theta^{\prime})\right)}{N_{t}\sin\left(\frac{1}{2}dk_{m}(\theta-\theta^{\prime})\right)}\right|,$
(12)

where
$G(x,0)=\left|\frac{\sin\left(\frac{N_{t}}{2}dx\right)}{N_{t}\sin\left(\frac{1}{2}dx\right)}\right|$.
It has been previously proved that the beams over the entire bandwidth
generated by ${\mathbf{w}}_{m}(\theta^{\prime},0)$ are focused on the spatial
angle $\theta_{m}=\theta^{\prime}$, with $m=1,2,\cdots,M$. However, notice
that the spatial angle $\theta_{m}$ is corresponding to an actual physical
angle $\vartheta_{m}=\arcsin\theta_{m}$ as shown in Fig. 1, which implies the
value range of $\theta_{m}$ is restrained by $\theta_{m}\in[-1,1]$. By
contrast, $\theta^{\prime}$ is an adjustable parameter of a TD beamformer. A
question naturally arises that, if the adjustable parameter is set as
$\theta^{\prime}\notin[-1,1]$, then what it has to do with the actual spatial
angle $\theta_{m}$? Actually, the answer to this question is exactly the
mechanism of the far-field controllable beam split.

For expression simplicity, parameter $\theta^{\prime}\notin[-1,1]$ is termed
as the abnormal value, while parameter $\theta^{\prime}\in[-1,1]$ is termed as
the actual value. Restrained by the value range of $\theta_{m}$, if
$\theta^{\prime}$ is an abnormal value, it is obvious that
$\theta_{m}\neq\theta^{\prime}$. To acquire the relationship between
$\theta_{m}$ and $\theta^{\prime}$, we observe that the far-field array gain
$G(x,0)$ is a periodic function against $x$ with a period $\frac{2\pi}{d}$.
For any integer $p\in\mathbb{Z}$, we have

$\displaystyle G\left(x-\frac{2p\pi}{d},0\right)$
$\displaystyle=\left|\frac{\sin\left(\frac{N}{2}dx-
Np\pi\right)}{N\sin\left(\frac{1}{2}dx-p\pi\right)}\right|=\left|\frac{\sin\left(\frac{N}{2}dx\right)}{N\sin\left(\frac{1}{2}dx\right)}\right|=G(x,0).$
(13)

In Section II, we have indicated that the optimal solution $x^{\text{opt}}$ to
maximize $G(x,0)$ is $(x^{\text{opt}},0)=(0,0)$. However, according to (13),
we find that $(0,0)$ is just one of the optimal solutions for maximizing
$G(x,0)$. The periodicity of $G(x,0)$ implies that the optimal solutions
should satisfy $(x^{\text{opt}},0)=(\frac{2p\pi}{d},0)$, $p\in\mathbb{Z}$. As
a consequence, by solving
$\theta_{m}=\arg\max_{\theta}G(k_{m}(\theta-\theta^{\prime}),0)$, the focused
spatial angle $\theta_{m}$ at frequency $f_{m}$ is derived as

$\displaystyle\theta_{m}=\theta^{\prime}+\frac{2p\pi}{dk_{m}}=\theta^{\prime}+\frac{2p}{\eta_{m}},$
(14)

where the function of $p\in\mathbb{Z}$ is to ensure $\theta_{m}\in[-1,1]$
becomes an actual spatial angle. From (14), the mechanism of the far-field
controllable beam split is acquired, which has the following features.

Feature 1: Since $p$ is an integer, if $p\neq 0$, the spatial angle
$\theta_{m}$ is related to the frequency $f_{m}$, which means beams at
different frequencies will split towards different spatial angles. Therefore,
despite the TD based beamsteering architecture being utilized, the far-field
beam split effect can also be induced.

Feature 2: The specific value of $p\in\mathbb{Z}$ is determined by
$\theta^{\prime}$. If $\theta^{\prime}\in[-1,1]$ belongs to the actual value
range of $\theta_{m}$, then $p$ can be zero and $\theta_{m}$ exactly equals to
$\theta^{\prime}$. That is to say, with $\theta^{\prime}\in[-1,1]$, the far-
field beam split effect is eliminated. While if $\theta^{\prime}\notin[-1,1]$
becomes an abnormal value, then to guarantee that $\theta_{m}$ is an actual
spatial angle, it is obvious that $p$ cannot be zero. For instance, if
$\theta^{\prime}$ is set as $1.5$, at the central frequency $f_{c}$ with
$\eta_{c}=\frac{f_{c}}{f_{c}}=1$, the corresponding spatial angle is
$\theta_{c}=1.5+2p/\eta_{c}=1.5+2p$, $p\in\mathbb{Z}$. Only if $p=-1$ can
$\theta_{c}=-0.5\in[-1,1]$ be an actual spatial angle. Therefore, with
$\theta^{\prime}\notin[-1,1]$, $p$ is not zero and thus $\theta_{m}$ is a
function of the frequency $f_{m}$, where the beam split effect is induced
again.

Feature 3: Notice that the beam split pattern (14) of a TD beamformer is
different from (6) of a PS beamformer. For a PS beamformer, the degree of beam
split is fixed and uncontrollable. However, for a TD beamformer, the degree of
beam split is controllable by adjusting $\theta^{\prime}$. For instance,
considering the central frequency $f_{c}$, if $\theta^{\prime}$ is set as
$1.5$, then $p$ should be $-1$ and $\theta_{c}=1.5-2=-0.5$. On the other hand,
if $\theta^{\prime}$ is set as $3.5$, then $p$ should be $-2$ and
$\theta_{c}=3.5-4=-0.5$. For the above two examples, although the spatial
angles $\theta_{c}$ are the same, the integers $p$ are different, which
indicates the degree of the beam split
$\theta_{m}=\theta^{\prime}+2p/\eta_{m}$ is different. Therefore, by adjusting
the delay parameter $\theta^{\prime}$, the integer $p$ is indirectly
controlled, then the degree of beam split is adjusted. This property is termed
as the far-field controllable beam split. Benefiting from this property, the
adjustment on the angular coverage range of the beams is achieved, which can
be utilized to realize fast far-field CSI acquisition [17, 19, 18].

In the next sub-section, we will extend this conclusion to a more general
near-field scenario.

###  III-B Near-Field Controllable Beam Split

For the near-field scenario, the distances $r$ and $r^{\prime}$ are lower than
the Rayleigh distance. Then $\alpha$ and $\alpha^{\prime}$ can not be zero,
and the near-field spherical wave characteristics should be considered. To
prove the mechanism of the near-field controllable beam split, analog to the
derivation in the far-field scenarios, we consider the abnormal values for the
adjustable parameters $\theta^{\prime}$ and $\alpha^{\prime}$ in (10)
simultaneously.

Specifically, it has been discussed before $\theta^{\prime}\notin[-1,1]$ is an
abnormal value for the angle. As for the distance-related parameter
$\alpha=\frac{1-\theta^{2}}{2r}$, since $1>\theta^{2}$ and $r>0$, a realistic
$\alpha$ should be larger than zero. On this condition, $\alpha^{\prime}<0$ is
obvious the abnormal value for the distance ring. If
$\theta^{\prime}\in[-1,1]$ and $\alpha^{\prime}>0$ are actual values, it was
proved in Section II-C that the near-field beam split effect is eliminated,
i.e., beams are focused on $\theta_{m}=\theta^{\prime}$ and
$\alpha_{m}=\alpha^{\prime}$ for all frequencies $f_{m}$. On the other hand,
for the abnormal value $\theta^{\prime}\notin[-1,1]$ or $\alpha^{\prime}<0$,
the following Lemma 2 explains the behaviors of the beams generated by
${\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime})$.

######  Lemma 2.

If $\theta^{\prime}\notin[-1,1]$ or $\alpha^{\prime}<0$, the beam at frequency
$f_{m}$ generated by ${\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime})$
according to (10) will be focused on the location $(\theta_{m},r_{m})$,
satisfying

$\displaystyle\theta_{m}$
$\displaystyle=\theta^{\prime}+\frac{2p\pi}{dk_{m}}=\theta^{\prime}+\frac{2p}{\eta_{m}},$
(15) $\displaystyle\alpha_{m}$
$\displaystyle=\alpha^{\prime}+\frac{2q\pi}{d^{2}k_{m}}=\alpha^{\prime}+\frac{2q}{d\eta_{m}},$
(16)

where $r_{m}=\frac{1-\theta_{m}^{2}}{2\alpha_{m}}$ and $p,q\in\mathbb{Z}$ are
integers to guarantee $\theta_{m}\in[-1,1]$ and $\alpha_{m}>0$.

_Proof_ : Similar to the derivation of (14), the periodicity of the near-field
array gain function $G(x,y)$ should be considered. It is proved in Appendix A
that $G(x,y)$ is a periodic function against the vector variable $(x,y)$ with
a period $(\frac{2\pi}{d},\frac{2\pi}{d^{2}})$. Therefore, for any integers
$p\in\mathbb{Z}$ and $q\in\mathbb{Z}$, $G(x,y)$ can be rewritten as

$\displaystyle G(x,y)=G(x-\frac{2p\pi}{d},y-\frac{2q\pi}{d^{2}}).$ (17)

It has been indicated in Section II that the optimal solution
$(x^{\text{opt}},y^{\text{opt}})$ to maximize the array gain $G(x,y)$ is
$(x^{\text{opt}},y^{\text{opt}})=(0,0)$. The periodicity in (17) implies
$(0,0)$ is just one of the optimal solutions for maximizing $G(x,y)$. Besides,
the optimal solutions should satisfy
$(x^{\text{opt}},y^{\text{opt}})=(\frac{2p\pi}{d},\frac{2q\pi}{d^{2}})$ with
$p\in\mathbb{Z}$ and $q\in\mathbb{Z}$. Accordingly, to obtain the focused
location of the beam at frequency $f_{m}$ by solving
$(\theta_{m},r_{m})=\arg\max_{\theta,r}G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$,
we have $k_{m}(\theta_{m}-\theta^{\prime})=\frac{2p\pi}{d}$ and
$k_{m}(\alpha_{m}-\alpha^{\prime})=\frac{2q\pi}{d^{2}}$. As a consequence,
$\theta_{m}$ and $\alpha_{m}$ become $\theta^{\prime}+\frac{2p}{\eta_{m}}$ and
$\alpha^{\prime}+\frac{2q}{d\eta_{m}}$, respectively, where the function of
integers $p$ and $q$ is to ensure that $\theta_{m}\in[-1,1]$ and
$\alpha_{m}>0$ are an actual spatial angle and an actual distance ring.
$\hfill\blacksquare$

According to Lemma 2, the mechanism of the near-field controllable beam split
is acquired, which has the following features.

Figure 5:  The schematic diagrams of the near-field rainbow on (a) the angle
dimension and (b) the distance dimension.

Feature 1: If $p\neq 0$ or $q\neq 0$, then $\theta_{m}$ or $\alpha_{m}$ is
related to the frequency $f_{m}$, which means beams at different frequencies
will be focused on different locations. Therefore, despite the TD beamformer
being utilized, the near-field beam split effect can also be induced.

Feature 2: As the function of the abnormal value of $\theta^{\prime}$ has been
discussed in the previous section, we mainly explain the impact of the
abnormal value of $\alpha^{\prime}$ on the integer $q$ here. If
$\alpha^{\prime}>0$ is an actual value, then $q$ could be zero, and for all
beams, we have $\alpha_{m}=\alpha^{\prime}$. However, if $\alpha^{\prime}<0$
is an abnormal value, to guarantee $\alpha_{m}>0$, the value of $q$ must be
larger than 0, and thus the beams at different subcarriers will be focused on
different distances $r_{m}=\frac{1-\theta_{m}^{2}}{2\alpha_{m}}$. For
instance, let the delay parameter $\alpha^{\prime}$ be $-\frac{2}{d}$, then we
have $\alpha_{m}=\frac{2}{d}(\frac{q}{\eta_{m}}-1)$. To guarantee
$\alpha_{m}\geq 0$ is an actual distance ring, the integer $q$ should be
larger than $\eta_{m}$. Accordingly, when $f_{m}\leq f_{c}$ and
$\eta_{m}=\frac{f_{m}}{f_{c}}\leq 1$, then the integer $q=1$ is enough to make
$\alpha_{m}>0$ and the beam at $f_{m}$ is focused on the distance ring
$\alpha_{m}=\frac{2}{d}(\frac{1}{\eta_{m}}-1)$. On the other hand, when
$f_{m}>f_{c}$ and $\eta_{m}=\frac{f_{m}}{f_{c}}>1$, then the integer $q=2$ is
enough to make $\alpha_{m}>0$ and the beam at $f_{m}$ is focused on the
distance ring $\alpha_{m}=\frac{2}{d}(\frac{2}{\eta_{m}}-1)$. As a
consequence, the abnormal value of $\alpha^{\prime}$ is able to introduce and
control the beam split effect on the distance dimension.

Feature 3: The far-field scenario is a special case of the near-field
controllable beam split by setting $\alpha_{m}=\alpha^{\prime}=0$. The far-
field scenario assumes the distance is very large, and it can only control the
beam split effect on the angle $\theta_{m}$. By contrast, the near-field
controllable beam split effect can adjust the degree of the beam split effect
on the angle $\theta_{m}$ and distance $r_{m}$ jointly. For instance, one can
set $\theta^{\prime}$ as an abnormal value and $\alpha^{\prime}$ as an actual
value, then we have $\theta_{m}=\theta^{\prime}+2p/\eta_{m}$,
$\alpha_{m}=\alpha^{\prime}$ and
$r_{m}=\frac{1-\theta_{m}^{2}}{2\alpha_{m}}=\frac{1-(\theta^{\prime}+2p/\eta_{m})^{2}}{2\alpha^{\prime}}$.
Therefore, the beam at frequency $f_{m}$ is focused on a location satisfying
$\left(\frac{1-\theta_{m}^{2}}{2r_{m}},\theta_{m}\right)=\left(\alpha^{\prime},\theta^{\prime}+\frac{2p}{\eta_{m}}\right)$,
which indicates that the beams are focused on multiple angles in the distance
ring $\alpha^{\prime}$ in the near-field, as shown in Fig. 5 (a). Moreover, we
can set $\theta^{\prime}$ as an actual value and $\alpha^{\prime}$ as an
abnormal value, then the beams are focused on multiple distances at the same
angle as shown in Fig. 5 (b), satisfying
$\left(\frac{1-\theta_{m}^{2}}{2r_{m}},\theta_{m}\right)=\left(\alpha^{\prime}+\frac{2q}{d\eta_{m}},\theta^{\prime}\right)$.
As a result, by carefully designing the parameters $\alpha^{\prime}$ and
$\theta^{\prime}$, the beams across the entire band could occupy multiple
angles and distances simultaneously, and this property is more flexible than
that in the far-field scenario.

In conclusion, the mechanism of near-field controllable beam split indicates
that a TD beamformer can not only mitigate the near-field beam split effect,
but also control its degree, i.e., one can flexibly adjust the covered angular
range and distance range of the beams generated by TD beamformer over the
entire bandwidth. A simple analogy of this effect is the dispersion of white
light caused by a prism. Since a prism has different refractive indices for
the wideband white light, the different frequency components of the pure light
will disperse and eventually produce the rainbow. In our discussion, the
function of time-delay circuits is similar to that of the prism. Therefore,
the near-field controllable beam split effect is also termed as “near-field
rainbow” in this paper.

By elaborately designing the pattern of the near-field rainbow as shown in
Fig. 5, efficient near-field beam management can be realized, which will be
discussed in the next section.

##  IV Proposed Beam Training Scheme

In this section, we first introduce the concept of near-field beam training
and indicate the overhead for exhaustive near-field beam training is much
larger than the far-field beam training, which is unacceptable in XL-MIMO
systems. To solve this problem, we propose a wideband beam training algorithm
by taking advantage of the near-field rainbow.

###  IV-A Exhaustive Near-Field Beam Training

Beam training is a well-adopted scheme to obtain CSI for current 5G systems
[27]. The concept of near-field beam training can be derived from the
classical far-field beam training. For the classical far-field beam training,
the spatial angle information $\theta_{0}$ is desired. The optimal
beamsteering vector for a user is selected from a predefined far-field beam
codebook through a training procedure between the BS and the user. Generally,
each codeword in the far-field codebook determines a unique spatial angle,
where the distance is assumed to be very large so that the near-field property
is ignored. Therefore, the entire far-field codebook occupies all of the
potential angles in the far-field.

Similarly, the near-field beam training desires to obtain the location
information $(r_{0},\theta_{0})$ of the dominant path between the BS and the
user. The optimal beamfocusing vector is selected from a predefined near-field
beam codebook through a training procedure between the BS and the user. Each
near-field codeword determines a unique location, and the entire near-field
codebook occupies all of the desired angles and distances.

To be more specific, we introduce the procedure of exhaustive near-field beam
training realized by a TD beamformer. Notice that the procedure below is also
valid for the PS beamformer when the bandwidth is not very large. We define
$[\theta_{\min},\theta_{\max}]$ as the potential range of spatial angle,
satisfying $-1\leq\theta_{\min}\leq\theta_{\max}\leq 1$. Moreover, we assume
the minimum distance between the BS and the user is $\rho_{\min}$, so that the
potential range of distance is $r\in[\rho_{\min},+\infty]$ or
$\alpha=\frac{1-\theta^{2}}{2r}\in[0,\alpha_{\max}]$ with
$\alpha_{\max}=\frac{1}{2\rho_{\min}}$. Then, multiple angles from
$\theta\in[\theta_{\min},\theta_{\max}]$ and distances from
$r\in[\rho_{\min},+\infty]$ are sampled simultaneously to construct the near-
field codebook. As proved in [6], the sampled locations could satisfy

$\displaystyle\theta_{u}$
$\displaystyle=\theta_{\min}+\frac{u}{U}(\theta_{\max}-\theta_{\min}),$ (18)
$\displaystyle\alpha_{s}$ $\displaystyle=\frac{s}{S}\alpha_{\max},$ (19)

where $u=0,1,\cdots,U-1$ and $s=0,1,\cdots,S-1$. $U$ denotes the number of
sampled angles and $S$ denotes the number of sampled distance rings. Notice
that when $S=1$, we have $\alpha_{s}\equiv 0$, and then the near-field
codebook is naturally decayed to the far-field codebook. The exhaustive near-
field beam training scheme searches the entire codebook $\\{\theta_{u}\\}$ and
$\\{\alpha_{s}\\}$ to obtain the optimal beamfocusing vector. Apparently, the
overhead for exhaustive near-field beam training, i.e., the number of time
slots used for beam training, is $T_{1}=US$.

In the $t$-th time slot, where $t=sU+u$ with $t=0,1,\cdots,T_{1}-1$,
$u=0,1,\cdots,U-1$, and $s=0,1,\cdots,S-1$, we set the parameters of the TD
beamformer as
$\theta_{t}^{\prime}=\theta_{u}=\theta_{\min}+\frac{u}{U}(\theta_{\max}-\theta_{\min})$
and $\alpha_{t}^{\prime}=\alpha_{s}=\frac{s}{S}\alpha_{\max}$, and then
generate the beamfocusing vector
${\mathbf{w}}_{m}(\theta_{t}^{\prime},\alpha_{t}^{\prime})$ according to (10).
Since $\theta_{t}^{\prime}$ and $\alpha_{t}^{\prime}$ are actual values, the
BS is able to transmit the pilot sequence to the user by the beam focused on
the location
$(r_{t}^{\prime},\theta_{t}^{\prime})=\left(\frac{1-\theta_{u}^{2}}{2\alpha_{s}},\theta_{u}\right)$.
So the received signal $y_{m,t}$ in the $t$-th time slot at $f_{m}$ is

$\displaystyle
y_{m,t}=\sqrt{P_{t}}{\mathbf{h}}_{m}^{T}{\mathbf{w}}_{m}(\theta_{t}^{\prime},\alpha_{t}^{\prime})x_{m}+n_{m},$
(20)

where $n_{m}\sim\mathcal{CN}(0,\sigma^{2})$ denotes the Gaussian noise,
$P_{t}$ denotes the transmit power, and $x_{m}$ denotes the transmit pilot
satisfying $\|x_{m}\|^{2}=1$. After $T_{1}$ time slots, the estimated physical
location
$(\hat{r},\hat{\theta})=(\frac{1-\hat{\theta}^{2}}{2\hat{\alpha}},\hat{\theta})$
corresponding to the largest user received power can be selected from the
$T_{1}$ measured locations, where,

$\displaystyle(\hat{\theta},\hat{\alpha})=\arg\max_{\theta_{t}^{\prime},\alpha_{t}^{\prime}}\sum_{m=1}^{M}\|y_{m,t}\|^{2}.$
(21)

Finally, a near-field beam ${\mathbf{w}}_{m}(\hat{\theta},\hat{\alpha})$
aligned with the location $(\hat{r},\hat{\theta})$ can be generated to serve
the user, and a near-optimal beamfocusing gain can be achieved.

Nevertheless, since the exhaustive near-field beam training scheme has to
search the entire near-field codebook exhaustively, and the scale of the near-
field codebook is much larger than that of the far-field, i.e. $US\gg U$, the
training overhead is unacceptable in practice. Therefore, a near-field beam
training scheme with low pilot overhead is essential for XL-MIMO systems.

###  IV-B Proposed Near-Field Beam Training Scheme

The main reason for the high training overhead of exhaustive near-field beam
training is that only one physical location can be measured in each time slot.
By contrast, as we discussed in Section III-B, a TD beamformer is able to
generate multiple beams focusing on multiple locations by only one RF chain.
Taking advantage of the near-field rainbow, multiple physical locations can be
measured simultaneously in each time slot. Inspired by this observation, we
propose a near-field rainbow-based beam training method to significantly
reduce the training overhead.

Generally, since the spatial resolution of an antenna array on the angle is
much higher than that on the distance, the number of sampled angles $U$ is
usually much larger than the number of sampled distances $S$ [6]. Thus, the
training overhead is mainly determined by the search of angle $\theta_{0}$.
Therefore, the near-field rainbow on the angle dimension, as shown in Fig. 5
(a), is utilized to avoid the exhaustive search of angle, i.e., the angle-
related parameters $\theta^{\prime}$ are set as abnormal values, while the
distance-related parameters $\alpha^{\prime}$ can be set as actual values. In
other words, the proposed scheme searches the optimal angle in a frequency
division manner, and searches the optimal distance ring in a time division
manner. The specific procedure of the proposed beam training scheme is
illustrated in Algorithm 1.

Algorithm 1 : Proposed near-field rainbow based beam training scheme

0: Potential angular range $[\theta_{\min},\theta_{\max}]$, focused angle
$\theta_{c}$ at the central frequency, potential distance range
$[0,\alpha_{\max}]$, the number of sampled distances $S$, central frequency
$f_{c}$, bandwidth $B$.

0: Estimated physical location $(\hat{r},\hat{\theta})$

1: $f_{L}=f_{c}-\frac{B}{2}$, $f_{H}=f_{c}+\frac{B}{2}$

2: Calculate the abnormal delay parameter:
$\theta^{\prime}=\theta_{c}-2\left\lceil\max\left\\{f_{L}(\theta_{\max}-\theta_{c}),f_{H}(\theta_{c}-\theta_{\min})\right\\}/B\right\rceil$

3: for $t\in\\{0,1,\cdots,{S-1}\\}$ do

4: Calculate the normal delay parameter:
$\alpha_{t}^{\prime}=\frac{t}{S}\alpha_{\max}$

5: Obtain the beamfocusing vector:
$[{\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime}_{t})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}(nd\theta^{\prime}-n^{2}d^{2}\alpha^{\prime}_{t})}$

6: Received signal:
$y_{m,t}=\sqrt{P_{t}}{\mathbf{h}}_{m}^{T}{\mathbf{w}}_{m}(\theta^{\prime},\alpha_{t}^{\prime})x_{m}+n_{m}$

7: end for

8: $(\hat{m},\hat{t})=\arg\max_{m,t}\|f_{m}y_{m,t}\|^{2},$

9:
$\hat{\theta}=\theta^{\prime}+(\theta_{c}-\theta^{\prime})f_{c}/f_{\hat{m}}$

10: $\hat{\alpha}=\alpha^{\prime}_{\hat{t}}$

11: return
$(\hat{r},\hat{\theta})=\left(\frac{1-\hat{\theta}^{2}}{2\hat{\alpha}},\hat{\theta}\right)$.

Firstly, in steps 1-2, we desire to design the abnormal parameter
$\theta^{\prime}\notin[-1,1]$, so that the frequency-dependent angles
$\theta_{m}=\theta^{\prime}+(2pf_{c})/f_{m}$ across the entire band is able to
cover the entire potential angle range $[\theta_{\min},\theta_{\max}]$, as
shown in Fig. 5 (a). To realize this target, we first assume that, at the
central frequency $f_{c}$, the beam is aligned with the angle
$\theta_{c}=\theta^{\prime}+2p$, where $\theta_{c}$ is a predefined spatial
angle satisfying $-1\leq\theta_{\min}<\theta_{c}<\theta_{\max}\leq 1$. Then we
have

$\displaystyle\theta^{\prime}=\theta_{c}-2p,\quad\>p\in\mathbb{Z}.$ (22)

Without loss of generality, we assume the delay parameter
$\theta^{\prime}<-1$, then
$p=\frac{\theta_{c}-\theta^{\prime}}{2}>\frac{-1-\theta^{\prime}}{2}>0$ is a
positive integer. In this case, the frequency-dependent angle
$\theta_{m}=\theta^{\prime}+(2pf_{c})/f_{m}$ is monotonically decreasing with
respect to the frequency $f_{m}$.

Moreover, since the available bandwidth is $B$, the lowest frequency and the
highest frequency are $f_{L}=f_{c}-\frac{B}{2}$ and $f_{H}=f_{c}+\frac{B}{2}$,
respectively. Therefore, the minimum and maximum values of $\theta_{m}$ are
$\theta^{\prime}+\frac{2pf_{c}}{f_{H}}$ and
$\theta^{\prime}+\frac{2pf_{c}}{f_{L}}$, respectively. To cover the entire
potential angles $[\theta_{\min},\theta_{\max}]$, it should satisfy

$\displaystyle\theta_{\min}$
$\displaystyle\geq\theta^{\prime}+(2pf_{c})/f_{H},$ (23)
$\displaystyle\theta_{\max}$
$\displaystyle\leq\theta^{\prime}+(2pf_{c})/f_{L}.$ (24)

By solving (22), (23), and (24), one of the solutions to $\theta^{\prime}$ is

$\displaystyle\theta^{\prime}=\theta_{c}-2\left\lceil\max\left\\{f_{L}(\theta_{\max}-\theta_{c}),f_{H}(\theta_{c}-\theta_{\min})\right\\}/B\right\rceil,$
(25)

where $\lceil x\rceil$ denotes the nearest integer greater than or equal to
$x$, and $p$ is given by
$p=\frac{\theta_{c}-\theta^{\prime}}{2}=\left\lceil\frac{1}{B}\max\left\\{f_{L}(\theta_{\max}-\theta_{c}),f_{H}(\theta_{c}-\theta_{\min})\right\\}\right\rceil$.

Then, in step 4, different distance rings are searched in different time slots
by adjusting the normal delay parameter $\alpha^{\prime}_{t}$. Similar to the
exhaustive near-field beam training scheme, the potential range of distance
ring is $\alpha\in[0,\alpha_{\max}]$, and the number of distance rings to be
measured is $S$. Therefore, in the $t$-th time slot, $\alpha_{t}^{\prime}$ is
set as $\frac{t}{S}\alpha_{\max}$ with $t=0,1,\cdots,S-1$ to measure the
beamfocusing gains on the distance ring
$\frac{1-\theta^{2}}{2r}=\alpha_{t}^{\prime}$.

As a result, with the parameters $\alpha_{t}^{\prime}$ and $\theta^{\prime}$,
as shown in Fig. 5 (a), the beams
${\mathbf{w}}_{m}(\theta^{\prime},\alpha_{t}^{\prime})$ generated in step 5 by
the TD beamformer is able to occupy the entire angular range in the distance
ring $\alpha_{t}^{\prime}$. Utilizing these beams, in step 6, the received
signal $y_{m,t}$ in the $t$-th time slot at frequency $f_{m}$ is denoted as

$\displaystyle y_{m,t}$
$\displaystyle=\sqrt{P_{t}}{\mathbf{h}}_{m}^{T}{\mathbf{w}}_{m}(\theta^{\prime},\alpha_{t}^{\prime})x_{m}+n_{m}$
$\displaystyle=\sqrt{P_{t}}\sqrt{N_{t}}\beta_{m}{\mathbf{a}}_{m}^{T}(\theta_{0},r_{0}){\mathbf{w}}_{m}(\theta^{\prime},\alpha_{t}^{\prime})x_{m}+n_{m}$
$\displaystyle=\sqrt{P_{t}N_{t}}\beta_{m}g_{m,t}x_{m}+n_{m},$ (26)

where
$g_{m,t}={\mathbf{a}}_{m}^{T}(\theta_{0},r_{0}){\mathbf{w}}_{m}(\theta^{\prime},\alpha_{t}^{\prime})$.
The optimal beamfocusing vector is corresponding to the maximum beamfocusing
gain $\|g_{m,t}\|^{2}$. Since $\|x_{m}\|^{2}=1$ and
$n_{m}\sim\mathcal{CN}(0,\sigma^{2})$, the maximum likelihood estimation of
$g_{m,t}$ is

$\displaystyle\hat{g}_{m,t}=\frac{y_{m,t}}{\sqrt{P_{t}N_{t}}\beta_{m}x_{m}}.$
(27)

Thus, the beamfocusing gain $\|\hat{g}_{m,t}\|^{2}$ can be estimated as
$\|\hat{g}_{m,t}\|^{2}=\frac{1}{P_{t}N_{t}}\frac{\|y_{m,t}\|^{2}}{\|\beta_{m}\|^{2}}$.
Based on the definition of $\|\beta_{m}\|^{2}$ in Section II, we have
$\|\hat{g}_{m,t}\|^{2}=C\|f_{m}y_{m,t}\|^{2}$, where $C$ is a constant that is
not relevant to $f_{m}$. Notice that the proposed beam training scheme does
not require the specific value of ${\|\beta_{m}\|^{2}}$. As long as the
relationship between $\beta_{m}$ and $f_{m}$ is available, whether quadratic
function, exponential function, and so on, the proposed scheme is always
valid.

Besides, in step 8, based on the formulation of beamfocusing gain
$\|\hat{g}_{m,t}\|^{2}=C\|f_{m}y_{m,t}\|^{2}$, the label of the optimal beam
$\hat{t}$ and $\hat{m}$ is acquired by maximizing $\|\hat{g}_{m,t}\|^{2}$ over
the total $S$ measured distance rings and $M$ subcarriers, where

$\displaystyle(\hat{m},\hat{t})=\arg\max_{m,t}\|f_{m}y_{m,t}\|^{2}.$ (28)

Eventually, in steps 9-10, based on the mechanism of the near-field rainbow
(15), the estimated spatial location is

$\displaystyle\hat{\theta}$
$\displaystyle=\theta^{\prime}+2p/\eta_{\hat{m}}=\theta^{\prime}+{(\theta_{c}-\theta^{\prime})f_{c}}/{f_{\hat{m}}},$
(29) $\displaystyle\hat{\alpha}$ $\displaystyle=\alpha_{\hat{t}}^{\prime}.$
(30)

After that, the beam training is completed, and the BS can generate the
beamfocusing vector ${\mathbf{w}}_{m}(\hat{\theta},\hat{\alpha})$ to serve the
user for data transmission.

The advantage of the proposed near-field rainbow based beam training method is
that the optimal angle is searched out in a frequency division manner.
Therefore, the training overhead is only determined by the overhead for
searching the optimal distance ring, which is significantly reduced. In the
next sub-section, we quantitatively analyze the training overheads of the two
near-field beam training schemes above.

###  IV-C Comparison on the Beam Training Overhead

Beam training overhead refers to the number of time slots used for beam
training. It is obvious that, the training overhead of the exhaustive near-
field beam training scheme is $T_{1}=US$, while the training overhead of the
proposed near-field rainbow based scheme is $T_{2}=S$. As discussed in [6],
the number of distance rings is generally much less than the number of angles.
For instance, if the BS antenna number is $N=256$, the carrier is $f_{c}=60$
GHz, and the minimum distance is $\rho_{\min}=2$ m, then $U$ is usually set as
$U=N=256$ while $S$ is generally set as 10 [6]. In this case, the training
overhead $T_{2}=S=10$ is much less than $T_{1}=US=2560$. Therefore, benefiting
from the near-field rainbow, the proposed scheme is able to significantly
reduce the pilot overhead for near-field beam training, which will be further
verified by simulation results in Section V.

##  V Simulation Results

In this section, simulations are provided to verify the effect of the near-
field rainbow and demonstrate the performance of the proposed near-field
rainbow based beam training scheme. We consider a wideband XL-MIMO system,
where the BS equips an $N_{t}=256$-element ULA with a TD beamformer. The
carrier frequency is $f_{c}=60$ GHz, the number of sub-carriers is $M=2048$,
and the bandwidth is $B=3$ GHz.

###  V-A The Demonstration of Near-Field Controllable Beam Split

Figure 6:  Near-field rainbow achieved by a TD beamformer. In (a), we set
$\theta^{\prime}=-6\notin[-1,1]$ and
$\alpha^{\prime}=\frac{1}{20}\text{m}^{-1}>0$. The array gains at multiple
frequencies w.r.t the angle on distance ring $\alpha^{\prime}$ are evaluated.
In (b), we set $\theta^{\prime}=\frac{\pi}{8}\in[-1,1]$ and
$\alpha^{\prime}=-\frac{2}{d}<0$. The array gains at multiple frequencies
w.r.t the distance on angle $\theta^{\prime}$ are evaluated.

Firstly, in Fig. 6, we verify that time-delay circuits are able to produce
near-field controllable beam split. Specifically, in Fig. 6 (a), the near-
field rainbow on the dimension of angle is evaluated. We set the delay
parameter $\theta^{\prime}$ as $\theta^{\prime}=-6\notin[-1,1]$ and the
parameter $\alpha^{\prime}$ as $\alpha^{\prime}=\frac{1}{2r^{\prime}}$ with
$r^{\prime}=10$ m. On the distance ring
$\alpha^{\prime}=\frac{1-\theta^{2}}{2r}$, the array gains at different
frequencies with respect to (w.r.t) the angle are shown in Fig. 6 (a). For
clarity, only a few frequencies are plotted. The beams over different
frequencies are focused on multiple angles in the distance ring
$\alpha^{\prime}=\frac{1-\theta^{2}}{2r}$, and cover a given angular range
$[-0.2,0.2]$. Therefore, the near-field rainbow on the angle dimension is
verified.

Then, in Fig. 6 (b), the near-field rainbow on the dimension of distance is
evaluated. We set the angle $\theta^{\prime}$ as
$\theta^{\prime}=\sin\frac{\pi}{8}\in[-1,1]$ and the distance ring
$\alpha^{\prime}$ as $\alpha^{\prime}=-\frac{2}{d}<0$. In the physical angle
$\theta^{\prime}$, the array gains at different frequencies w.s.t the distance
are shown in Fig. 6(b). For clarity, only a few frequencies are plotted. The
beams over different frequencies are focused on multiple distances in the
angle $\theta^{\prime}$, which covers the entire distance range. Therefore,
the near-field rainbow on the distance dimension is also achievable.

###  V-B Beam Training Performance

In this subsection, the performance of the proposed near-field rainbow based
beam training scheme is evaluated. The potential spatial angle range of the
user is set as
$[\theta_{\min},\theta_{\max}]=[-\sin\frac{\pi}{3},\sin\frac{\pi}{3}]$, while
the potential distance of the user is set as
$[\rho_{\min},+\infty]=[3\>\text{m},+\infty]$ and thus the range of $\alpha$
is $\alpha\in[0,\frac{1}{6}\text{m}^{-1}]$. The spatial angle at the center
frequency is fixed to $\theta_{c}=0$. Then, for the proposed near-field
rainbow based scheme, according to (25), the delay parameter $\theta^{\prime}$
is acquired as $\theta^{\prime}=\xi=-36$ with $p=18$. Moreover, we set the
number of distance rings to be searched as $S=10$. For the exhaustive near-
field beam training, the number of angles to be searched is $U=N=256$.
Finally, we use the average rate performance to quantify the beam training
performance, which is mathematically defined as

$\displaystyle R$
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\log_{2}\left(1+\frac{P_{t}}{\sigma^{2}}\|{\mathbf{h}}_{m}^{T}{\mathbf{w}}_{m}\|^{2}\right)=\frac{1}{M}\sum_{m=1}^{M}\log_{2}\left(1+\frac{P_{t}N_{t}\beta_{m}^{2}}{\sigma^{2}}\|{\mathbf{a}}_{m}^{T}(\theta_{0},r_{0}){\mathbf{w}}_{m}\|^{2}\right),$
(31)

where ${\mathbf{w}}_{m}$ denotes the beamfocusing vector for data transmission
searched by beam training and SNR
$=\frac{P_{t}N_{t}\beta_{m}^{2}}{\sigma^{2}}$ denotes the signal-to-power
ratio. The compared benchmarks are shown below.

  * •

Perfect CSI: The perfect channel ${\mathbf{h}}_{m}$ is available at the BS,
which is served as the benchmark for the upper bound of average rate
performance.

  * •

Far-field rainbow based beam training: The classical wideband far-field beam
training scheme achieved by TD beamformer in [18]. This method can be regarded
as the far-field rainbow-based beam training scheme, where only the optimal
angle is searched in a frequency-division manner and the distance ring
$\alpha$ is assumed to be zero.

  * •

Near-field hierarchical beam training: This method is the hierarchical beam
scheme training designed for reconfigurable intelligence surface (RIS) aided
near-field communications [21]. We transform this method to XL-MIMO scenarios
for comparison.

  * •

Far-field hierarchical beam training: This method is the classical
hierarchical beam training designed for the far-field scenarios [20].

  * •

Exhaustive search: This method is the exhaustive near-field beam training
scheme provided in Section IV-A. In the training procedure, we first fix the
distance ring $\alpha_{s}$ and exhaustively search all of the angles
$\theta_{u}$. Then, we change the distance ring $\alpha_{s}$ and repeat the
above steps until all of the distance rings and angles are measured.

Figure 7:  Average rate performance vs. training overhead.

Fig. 7 illustrates the average rate performance against the training overhead.
The training overhead is increasing from 0 to $US=2560$. The SNR is set as 10
dB. $10^{3}$ realizations of the user location are generated for Monte Carlo
simulations, where
$\theta_{0}\sim\mathcal{U}(-\sin\frac{\pi}{3},\sin\frac{\pi}{3})$ and
$r_{0}\sim\mathcal{U}(3\>\text{meters},30\>\text{meters})$. In the $t$-th time
slot, we utilize the optimal beamforming vector searched during the time slots
$1\sim t$ to serve the user. Specifically, the training overhead of the two
far-field schemes are both very low, e.g., 22 overhead for the hierarchical
far-field scheme and only one overhead for the far-field rainbow based scheme.
However, since the far-field schemes only consider the angle information while
ignoring the distance information, their average rate performance is not
satisfactory. Besides, to achieve a satisfactory average rate, the training
overhead of the near-field hierarchical scheme and the exhaustive scheme will
be very high, e.g., 200 overhead for the near-field hierarchical scheme and
more than 1000 for the exhaustive scheme. By contrast, the proposed near-field
rainbow based beam training scheme enjoys a much higher average rate
performance with very low overhead. This is thanks to two factors: 1) both the
angle and distance information are considered, 2) the near-field rainbow
effect is exploited to avoid the exhaustive search for the angle information.
Actually, the proposed scheme has already achieved 96% of the average rate
benchmark with only 8 training overhead.

Figure 8:  Average rate performance vs. SNR.  Figure 9:  Average rate
performance vs. distance.

Fig. 8 shows the impact of SNR on different beam training schemes. Here, the
SNR is increasing from -5 dB to 15 dB. The maximum training overhead
$T_{\max}$ for all considered methods is set as $T_{\max}=256$. Notice that in
our simulations, only the overhead for exhaustive beam training is
$T_{\max}=256$, while the overhead for other schemes is set according to their
overhead requirements as illustrated in Fig. 7, which are always less than
$T_{\max}$. The other simulation settings are the same as those in Fig. 7. It
is clear from Fig. 8 that, the proposed scheme outperforms all existing far-
field and near-field beam training schemes, and is able to achieve the near-
optimal achievable average rate performance compared with the benchmark. In
addition, we can observe that the exhaustive beam training scheme suffers from
severe degradation of the average rate performance. This is mainly because the
maximum training overhead (256) is much less than the required training
overhead (more than 1000) for the exhaustive scheme to achieve satisfactory
average rate performance.

In Fig. 9, we illustrate the average rate performance against the distance.
Here, the distance $r_{0}$ between the user and BS is gradually increasing
from 3 meters to 60 meters. The parameters are set as follows: $\text{SNR}=10$
dB, $T_{\max}=256$,
$\theta_{0}\in\mathcal{U}(-\sin\frac{\pi}{6},\sin\frac{\pi}{6})$. The other
simulation settings are the same as those in Fig. 8. The impact of near-field
propagation on the beam training performance is clear in Fig. 9. For the far-
field schemes, with the decrease of distance, near-field propagation becomes
dominant and thus the average rate of these far-field schemes rapidly
deteriorates. As for the degradation of the exhaustive scheme, since the
maximum training overhead is limited to 256, this scheme is hard to search the
locations in the near-field. Moreover, although the near-field hierarchical
beam training scheme is able to slightly alleviate the average rate loss, its
performance is unacceptable when the distance is less than 20 meters. By
contrast, since the proposed scheme is able to search the optimal angle and
distance with very low pilot overhead by exploiting the near-field rainbow,
its performance is robust to all considered distances, whether in the near-
field or in the far-field.

Figure 10:  Average rate performance vs. angle.

Fig. 10 shows the average rate performance against the angle $\theta_{0}$.
Here, the angle $\theta_{0}$ is gradually increasing from $-\sin\frac{\pi}{3}$
to $\sin\frac{\pi}{3}$ and the distance $r_{0}$ is randomly generated from
$r_{0}\sim\mathcal{U}(3\>\text{meters},30\>\text{meters})$. The other
simulation settings are the same as those in Fig. 9. Notice that although the
distance $r_{0}$ is fixed, the distance ring
$\alpha_{0}=\frac{1-\theta_{0}^{2}}{2r_{0}}$ still varies with the angle
$\theta_{0}$. For the far-field schemes, their performance is severely
degraded when the angle $\theta_{0}$ is around zero. This is because the near-
field property is more significant when the angle is around zero, which has
been proved in [5]. Moreover, there exists severe fluctuation for the near-
field hierarchical beam training scheme. This is because the near-field
hierarchical scheme proposed in [21] creates the near-field codebook by
uniformly sampling the codeword in the cartesian coordinates. It has been
indicated in [6] that this kind of codebook cannot realize satisfactory beam
training performance in the entire near-field environment. By contrast, the
proposed scheme achieves a near-optimal average rate for all considered
angles.

##  VI Conclusions

This paper investigated the wideband near-field beam training for XL-MIMO
systems. The mechanism of the near-field rainbow is revealed, i.e., time-delay
circuits can flexibly control the degree of the near-field beam split effect.
Then, a near-field rainbow based beam training scheme was proposed to search
the optimal angle in a frequency-division manner, and the optimal distance is
searched in a time-division manner. The simulation results verified that: i)
the beams generated by a TD beamformer over the entire bandwidth can cover
multiple angles and distances, ii) our near-field rainbow based scheme can
achieve a near-optimal average rate with much-reduced training overhead, iii)
the performance of our scheme is robust to the distance and angle. This paper
unveiled that although the near-field beam split effect induces a severe
beamfocusing gain loss, it also provides a new possibility to realize fast
near-field CSI acquisition. In our future work, we will extend the mechanism
of the near-field rainbow to other beamforming architectures, such as the
delay-phase precoding architecture [15].

## Appendix A. The Periodicity of $G(x,y)$

$G(x,y)$ is a periodic function against the vector variable $(x,y)$ with
period $(\frac{2\pi}{d},\frac{2\pi}{d^{2}})$. Specifically, for any integers
$p\in\mathbb{Z}$ and $q\in\mathbb{Z}$, we have

$\displaystyle G\left(x-\frac{2p\pi}{d},y-\frac{2q\pi}{d^{2}}\right)$
$\displaystyle=\frac{1}{N_{t}}\left|\sum_{n=-N}^{N}e^{jnd(x-\frac{2p\pi}{d})-jn^{2}d^{2}(y-\frac{2q\pi}{d^{2}})}\right|$
$\displaystyle=\frac{1}{N_{t}}\left|\sum_{n=-N}^{N}e^{jndx-
jn^{2}d^{2}y}e^{j2\pi n(qn-p)}\right|.$ (32)

Since $2\pi n(qn-p)$ is an integer multiple of $2\pi$, we have $e^{j2\pi
n(qn-p)}=1$. Therefore,

$\displaystyle
G\left(x-\frac{2p\pi}{d},y-\frac{2q\pi}{d^{2}}\right)=\frac{1}{N_{t}}\left|\sum_{n=-N}^{N}e^{jndx-
jn^{2}d^{2}y}\right|=G(x,y).$ (33)

As a result, the periodicity of function $G(x,y)$ is proved.

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