# Near-Field Wideband Beam Training Based on Distance-Dependent Beam Split

Tianyue Zheng, Mingyao Cui, Zidong Wu, and Linglong Dai, Fellow, IEEE This
work was funded in part by the National Science Fund for Distinguished Young
Scholars (Grant No. 62325106), and in part by the National Key R&D Program of
China (No. 2023YFB811503). T. Zheng, Z. Wu, and L. Dai are with the Department
of Electronic Engineering, Tsinghua University, Beijing 100084, China, and
also with the Beijing National Research Center for Information Science and
Technology (BNRist), Beijing 100084, China (e-mails: {zhengty22,
<EMAIL_ADDRESS>{daill}@tsinghua.edu.cn). M. Cui is with the
Department of Electrical and Electronic Engineering, the University of Hong
Kong, Hong Kong (e-mail: cuimy23@connect.hku.hk).

###### Abstract

Near-field beam training is essential for acquiring channel state information
in 6G extremely large-scale multiple input multiple output (XL-MIMO) systems.
To achieve low-overhead beam training, existing method has been proposed to
leverage the near-field beam split effect, which deploys true-time-delay
arrays to simultaneously search multiple angles of the entire angular range in
a distance ring with a single pilot. However, the method still requires
exhaustive search in the distance domain, which limits its efficiency. To
address the problem, we propose a distance-dependent beam-split-based beam
training method to further reduce the training overheads. Specifically, we
first reveal the new phenomenon of distance-dependent beam split, where by
manipulating the configurations of time-delay and phase-shift, beams at
different frequencies can simultaneously scan the angular domain in multiple
distance rings. Leveraging the phenomenon, we propose a near-field beam
training method where both different angles and distances can simultaneously
be searched in one time slot. Thus, a few pilots are capable of covering the
whole angle-distance space for wideband XL-MIMO. Theoretical analysis and
numerical simulations are also displayed to verify the superiority of the
proposed method on beamforming gain and training overhead.

###### Index Terms:

Beam training, extremely large-scale MIMO (XL-MIMO), near-field, wideband.

##  I Introduction

With the dramatically increasing demand for data transmission in future sixth-
generation (6G) communications, a 10-fold increase in spectrum efficiency
compared with 5G is anticipated in 6G [1, 2]. To fulfill this vision, 5G
massive multiple input multiple output (MIMO) systems will evolve into 6G
extremely large-scale MIMO (XL-MIMO) systems [3, 4, 5]. Thanks to the
dramatically increased antenna aperture, the spatial multiplexing and
beamforming gains could be significantly enhanced in XL-MIMO [6]. To harvest
these gains, acquiring accurate channel state information efficiently by beam
training is necessary. Specifically, beam training is realized by selecting an
optimal beam codeword from a predefined beam codebook so as to assign
directional beams to target users, which serves as a dominate CSI acquition
scheme in millimeter wave (mmWave) and terahertz (THz) XL-MIMO systems [7, 8].

###  I-A Prior Works

Compared with 5G massive MIMO, beam training for 6G XL-MIMO will induce
overwhelming training overheads due to fundamental change in electromagnetic
(EM) field property. Specifically, in 5G systems when the antenna number at
the base station (BS) is not very large, channel model mainly considers far-
field propagation where radiated EM waves can be approximated as planar waves.
Thus, the array response vector of the far-field channel is solely related to
the angle. Correspondingly, the orthogonal Discrete Fourier Transform (DFT)
codebook can conduct an efficient scan in the angular domain [9, 10, 11],
where each codeword in the codebook corresponds to a narrow beam directing to
a specific angle.

As for the 6G XL-MIMO systems, however, the increased antenna aperture leads
to more users located in spherical-wave-based near-field regions [12, 13].
Unlike the angle-dependent far-field channel, the angle and distance from BS
to user jointly determine the array response vectors of near-field channels.
Thereby, to perform near-field beam training, a polar-domain codebook was
proposed in [12] to simultaneously obtain accurate angle and distance
information. This codebook partitions the entire “angle-distance” space into
multiple grids, where each codeword is construced by the array response vector
aligned with one sampled grid. During beam training, all codewords are
sequentially tested, where the angle and distance information are estimated
based on the codeword yielding the highest received power. Unfortunately,
although this beam training method can acquire the near-field CSI accurately,
it consumes unacceptable pilot overheads due to the two-dimensional exhaustive
search for all possible BS-to-user angles and distances.

To cope with this problem, two categories of near-field beam training have
been proposed, namely narrowband and wideband methods. Consider the narrowband
methods at first. They are realized either by hierarchical beam training [14,
15, 16, 17] or inference from far-field beam training [18, 19, 20, 21, 22].
The near-field hierarchical beam training methods employ multi-resolution
codebooks with different angle and distance coverages. Specifically, the
spatial region covered by a lower-resolution codeword at any given layer is
subdivided into several higher-resolution and non-overlapping spatial regions
in the subsequent layer. During beam training, one can start from the lowest-
resolution codebook and progress to the highest-resolution one layer by layer,
and thereby gradually reduces the angle and distance estimation granularity by
choosing the spatial region with larger received power in each layer [14, 15,
16, 17]. As for the inference from far-field beam training, these methods
attempt to recover near-field CSI with the aid of far-field codebook via a
two-stage training strategy. In the first stage, the methods estimate all
candidate angles based on the pattern of received power provided by far-field
exhaustive beam training. Then the second stage uses a polar-domain codebook
to search for distance within candidate angles or directly estimates the
distance from far-field received beam pattern with deep learning-based methods
[18, 19, 20, 21, 22].

The above methods can to some extent reduce the training overheads.
Nevertheless, the narrowband methods only measures the received power of one
subcarrier in a single time slot, limiting the information content obtained
per pilot transition. This constrains the efficiency of narrowband near-field
beam training. In contrast, the majority of practical communication systems
are wideband systems and it motivates us to develop wideband near-field beam
training methods, which utilize information in multiple subcarriers [23] to
further reduce the beam training overheads. Unfortunately, to our best
knowledge, few works have focused on wideband near-field beam training.

Excavating frequency resources, in our prior work [24], a wideband near-field
beam training scheme, namely near-field rainbow, was designed based on near-
field beam split. This scheme was built on the true-time-delay (TTD) array. By
manipulating the time-delay (TD) parameters, multiple beams at different
frequencies are focused on multiple angles in a specific distance ring. In
other words, multiple angles can be searched simultaneously in a single time
slot, which significantly improves the training efficiency. However, since the
beam split in [24] is essentially distance-independent beam split, the optimal
distance is obtained by sequentially traversing different distance rings.
Thereby, the method in [24] could not get rid of the additional overheads in
the distance domain and results in relatively high training overheads.

###  I-B Our Contributions

To tackle this problem, by exploiting _distance-dependent_ feature of near-
field beam-split, we propose a near-field beam training method where both
different angles and distances can be searched simultaneously in one time
slot, achieving efficient beam training.111Simulation codes will be provided
to reproduce the results in this paper:
http://oa.ee.tsinghua.edu.cn/dailinglong/publications/publications.html. Our
contributions are summarized as follows.

  * •

Firstly, we reveal the mechanism of near-field distance-dependent beam split.
Specifically, [24] has proved that with the elaborate configuration of TD
parameters, the beams at different frequencies could be made distributed on
multiple angles of _one-distance-ring_. In this paper, we prove that by
introducing phase-shift (PS), the collaboration of TD and PS can spread the
focused points of multi-frequency beams on different angles of _multi-
distance-rings_. In this way, the traditional distance-independent beam split
can be transformed into the distance-dependent beam split.

  * •

Secondly, leveraging the mechanism of distance-dependent beam split, an on-
grid wideband beam training method is proposed to realize efficient near-field
beam training. In the proposed scheme, both the angle and distance domain can
be searched with beams of different subcarriers in a single pilot, where the
beam with the highest power can be efficiently determined. Moreover,
quantitative analysis of the required pilot number to guarantee the full
coverage of the whole angle-distance space is presented.

  * •

To further improve the beam training performance for off-grid users, we
propose two off-grid beam training algorithms: the auxiliary beam pair-
assisted method and match filter-based method. Their principle is to employ
the envelope of received powers over multiple subcarriers, rather than relying
on the single subcarrier with the highest power, to jointly select the optimal
codeword.

  * •

Finally, we present simulation results to validate the efficiency of the
proposed method based on distance-dependent beam split. It shows that the
proposed beam training scheme is capable of achieving near-optimal rate
performance with a lower overhead compared to existing methods. Moreover, the
enhanced algorithms tailored for off-grid users surpass the performance of the
on-grid algorithm, particularly in regions of low signal-to-noise ratio (SNR).

###  I-C Organization and Notation

####  I-C1 Organization

The rest of the paper is organized as follows. Section II introduces the near-
field XL-MIMO wideband channel model, and formulate the near-field beam
training problem. Then the mechanism of existing near-field distance-
independent beam split will be reviewed. In Section III, the mechanism of
distance-dependent beam split is introduced, and based on it we provide the
design of an efficient wideband near-field beam training method. In Section
IV, two improved beam training methods for off-grid users are described.
Simulation results and conclusions are provided in Section V and Section VI,
respectively.

####  I-C2 Notation

${\bf a}^{H}$, ${\bf A}^{H}$ denote the conjugate transpose of vector $\bf a$
and matrix $\bf A$, respectively; $\left\|{\bf a}\right\|$ denotes the $l_{2}$
norm of vector $\bf a$; $\odot$ denotes Hadamard product;
$\mathcal{C}\mathcal{N}({\bf\mu},{\bf\Sigma})$ denotes the probability density
function of complex multivariate Gaussian distribution with mean ${\bf\mu}$
and variance ${\bf\Sigma}$; ${\rm round}(a)$, $\lfloor a\rfloor$ and $\lceil
a\rceil$ denote rounding to, up to, and down to the nearest integer,
respectively.

##  II System Model

In this section, we first introduce the near-field wideband channel model in
XL-MIMO systems and formulate the near-field beam training problem. Then, we
will briefly review the traditional near-field beam split proposed in [24].

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

We examine a millimeter-wave wideband near-field XL-MIMO system. The BS
equipped with an $N_{t}$-antenna uniform linear array serves a single-antenna
user using orthogonal frequency division multiplexing (OFDM) of $M$
subcarriers. The bandwidth, central carrier frequency, speed of light, central
wavelength, and antenna spacing are denoted as $B$, $f_{c}$, $c$,
$\lambda_{c}=c/f_{c}$ and $d=\lambda_{c}/2$, respectively. The lower and upper
subcarrier frequencies can be written as $f_{L}=f_{c}-B/2$ and
$f_{H}=f_{c}+B/2$ and the frequency of the $m$-th subcarrier as
$f_{m}=f_{c}+\frac{B}{M}(m-1-\frac{M-1}{2})$. Besides, the number of BS
antennas is assumed to be odd and is denoted by $N_{t}=2N+1$ for expression
simplicity.

In this paper, we mainly focus on the downlink beam training process. Let
$s_{m}=1$ be the normalized transmitted symbol at $f_{m}$, then the received
signal $y_{m}$ at the user equipment (UE) of the $m$-th subcarrier is given by

$y_{m}=\sqrt{P_{t}}{\bf h}_{m}^{H}{\bm{w}}_{m}s_{m}+n_{m},$ (1)

where $P_{t}>0$ is the transmit power, ${\bf h}_{m}\in\mathbb{C}^{N_{t}}\times
1$ the downlink channel, ${\bm{w}}_{m}\in\mathbb{C}^{N_{t}\times 1}$ the unit-
norm beamformer, and $n_{m}\sim\mathcal{CN}(0,\sigma^{2})$ the complex
circularly-symmetric Gaussian noise with a variance of $\sigma^{2}$. We adopt
the Saleh-Valenzuela channel model in [25] to represent the downlink channel
${\bf h}_{m}\in\mathbb{C}^{N_{t}\times 1}$ at the $m$-th subcarrier as

${\bf h}_{m}=\sqrt{\frac{N_{t}}{L}}\sum_{l=1}^{L}\beta_{m,l}e^{-j2\pi
f_{m}\tau_{l}}{\bf a}_{m}(\theta_{l},r_{l}),$ (2)

where $L$ is the number of paths being either line-of-sight (LoS) or non-line-
of-sight (NLoS) path, and $\beta_{m,l}$, $\tau_{l}$,
$\theta_{l}=\sin\vartheta_{l}$, $r_{l}$ denote the path gain, time delay,
spatial direction, and distance from the UE/scatter to the center of the
antenna array of the $l$-th path, respectively. Due to the significant path
loss experienced by scattered signals, millimeter-wave communications heavily
rely on the LoS path [26], which is our primary focus:

${\bf h}_{m}=\sqrt{N_{t}}\beta_{m}e^{-jk_{m}r}{\bf a}_{m}(\theta,r),$ (3)

where $k_{m}=2\pi f_{m}/c$ denotes the wavenumber at $m$-th subcarrier and the
path gain of the LoS path is modeled as [27]

$\beta_{m}=\frac{\lambda_{m}}{4\pi r}.$ (4)

Therefore, we can obtain the relationship between the path gain at $m$-th
subcarrier and central frequency $f_{c}$ as

$\beta_{m}=\frac{f_{c}}{f_{m}}\beta_{c}.$ (5)

The near-field array response vector ${\bf a}_{m}(\theta,r)$ in (3) is derived
based on spherical wave propagation:

${\bf
a}_{m}(\theta,r)=\frac{1}{\sqrt{N_{t}}}[e^{-jk_{m}(r^{(-N)}-r)},\ldots,e^{-jk_{m}(r^{(N)}-r)}]^{T}$
(6)

where $r^{(n)}=\sqrt{r^{2}+n^{2}d^{2}-2r\theta nd}$ represents the distance
from the $n$-th BS antenna to the UE. By utilizing the Taylor expansion
$\sqrt{1+x}\approx 1+\frac{1}{2}x-\frac{1}{8}x^{2}$ [12], the distance
difference $r^{(n)}-r$ can be approximated by:

$r^{(n)}-r\approx-nd\theta+n^{2}d^{2}\frac{1-\theta^{2}}{2r}.$ (7)

For expression simplicity, we denote $\alpha=\frac{1-\theta^{2}}{2r}$. For
fixed $\alpha$, the curve plotted by $(r=\frac{1-\theta^{2}}{2\alpha},\theta)$
can be viewed as a distance ring in the physical space. In this case, ${\bf
a}_{m}(\theta,r)$ can be approximated with ${\bf b}_{m}(\theta,\alpha)$
written as

$\begin{split}&{\bf a}_{m}(\theta,r)\approx{\bf b}_{m}(\theta,\alpha)=\\\
&\frac{1}{\sqrt{N_{t}}}[e^{jk_{m}((-N)d\theta-(-N)^{2}d^{2}\alpha)},\ldots,e^{jk_{m}(Nd\theta-N^{2}d^{2}\alpha)}].\end{split}$
(8)

Vector ${\bf b}_{m}(\theta,\alpha)$ effectively captures the spherical wave
propagation in wideband scenarios.

###  II-B Formulation of Near-Field Beam Training Problem

The near-field beam training attempts to estimate the location information
$(\theta,\alpha)$ of the user, thereby aligning the beamforming vector with
this location. To adapt to the near-field communication scenario, the optimal
beamforming vector is typically selected from the predefined polar-domain
codebook [12], which can be represented as

${\mathcal{A}}_{m}=[{\bf b}_{m}(\theta_{1},\alpha_{1}^{1}),\cdots,{\bf
b}_{m}(\theta_{1},\alpha_{1}^{S_{1}}),\cdots,{\bf
b}_{m}(\theta_{N_{t}},\alpha_{N_{t}}^{S_{N_{t}}})],$ (9)

where $S_{n}$ is the number of sampled distance grids at $\theta_{n}$. Each
column of ${\mathcal{A}}_{m}$ is a codeword focused on the grid
$(\theta_{n},\alpha_{n}^{s_{n}})$, with $s_{n}=1,2,\cdots,S_{n}$. When
conducting exhaustive beam training, we sequentially employ the codewords to
perform beamforming and select the codeword leading to the largest received
power to estimate the physical location $(\hat{\theta},\hat{\alpha})$, i.e.

$(\hat{\theta},\hat{\alpha})={\rm
arg}\max_{(\theta_{n},\alpha_{n}^{s_{n}})}\sum_{m=1}^{M}\|\sqrt{P_{t}}{\bf
h}_{m}^{H}{\bf b}_{m}(\theta_{n},\alpha_{n}^{s_{n}})+n_{m}\|^{2}.$ (10)

Finally, we set the beamforming vector as
${\bm{w}}_{m}(\hat{\theta},\hat{\alpha})={\bf
b}_{m}(\theta_{n},\alpha_{n}^{s_{n}})$ to serve the user. Consequently, the
size of the polar-domain codebook is determined by the multiplication of the
number of sampled angular grids and sampled distance grids. This results in a
large codebook size, which in turn leads to prohibitive beam training
overheads.

###  II-C Near-Field Distance-Independent Beam Split

In this subsection, we review the mechanism of near-field beam split proposed
in [24] and explain how to employ it to achieve efficient beam training.

Figure 1: TD-PS precoding structure.

We utilize a TD-PS precoding structure, as presented in Fig. 1, to manipulate
the near-field beam split effect. Note that the TD array adopted in [24] is a
special case of TD-PS array by forcing the phase shift to zero. Therefore, the
TD-PS array can achieve much more flexible control of the beam split effect,
particularly in the distance dimension.

As illustrated in Fig. 1, each antenna is sequentially connected to a time-
delay unit and a phase-shift unit. A TD unit is capable of tuning frequency-
dependent phase shift by actively delaying the signal. Accordingly, we can
express the $n$-th element of the TD beamforming vector ${\bf w}_{m}^{\rm TD}$
at $f_{m}$, as

$[{\bf w}_{m}^{\rm TD}]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-j2\pi
f_{m}\tau^{(n)^{\prime}}}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}r_{t}^{(n)^{\prime}}}$
(11)

where $\tau^{(n)^{\prime}}$ is the adjustable delay of the $n$-th TD unit,
while $r_{t}^{(n)^{\prime}}\overset{\Delta}{=}c\tau^{(n)^{\prime}}$. We use
the superscript ′ to indicate the adjustable parameters. Since ${\bf
w}_{m}^{\rm TD}$ has a similar form to the array response vector ${\bf
b}_{m}(\theta,\alpha)$, we can set $r_{t}^{(n)^{\prime}}$ as
$r_{t}^{(n)^{\prime}}=nd\theta_{t}^{\prime}-n^{2}d^{2}\alpha_{t}^{\prime}$,
where $\theta_{t}^{\prime}$ and $\alpha_{t}^{\prime}$ are defined as the
_adjustable TD parameters_. Thereby, the beamforming vector at $f_{m}$ can be
rewritten as

$[{\bf w}_{m}^{\rm
TD}(\theta_{t}^{\prime},\alpha_{t}^{\prime})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}(nd\theta_{t}^{\prime}-n^{2}d^{2}\alpha_{t}^{\prime})}.$
(12)

Furthermore, the $n$-th element of the beamforming vector, ${\bf w}^{\rm
PS}(\theta_{p}^{\prime},\alpha_{p}^{\prime})$, generated by frequency-
independent PSs can be similarly set as

$[{\bf w}^{\rm
PS}(\theta_{p}^{\prime},\alpha_{p}^{\prime})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{c}(nd\theta_{p}^{\prime}-n^{2}d^{2}\alpha_{p}^{\prime})},$
(13)

where $\theta_{p}^{\prime},\alpha_{p}^{\prime}$ are correspondingly
_adjustable PS parameters_. By configuring TD-PS beamforming structure with
parameters
$\theta_{t}^{\prime},\alpha_{t}^{\prime},\theta_{p}^{\prime},\alpha_{p}^{\prime}$,
the array gain at the $m$-th subcarrier on an arbitrary location
$(\theta,\alpha)$ can be presented as

$\displaystyle|({\bf w}_{m}^{\rm
TD}(\theta_{t}^{\prime},\alpha_{t}^{\prime})\odot{\bf w}^{\rm
PS}(\theta_{p}^{\prime},\alpha_{p}^{\prime}))^{T}{\bf b}_{m}(\theta,\alpha)|$
(14) $\displaystyle=$
$\displaystyle\frac{1}{N_{t}}\left\lvert\sum_{n=-N}^{N}e^{jnd(k_{m}(\theta-\theta_{t}^{\prime})-k_{c}\theta_{p}^{\prime})-jn^{2}d^{2}(k_{m}(\alpha-\alpha_{t}^{\prime})-k_{c}\alpha_{p}^{\prime})}\right\rvert$
$\displaystyle=$ $\displaystyle G(k_{m}\theta-
k_{m}\theta_{t}^{\prime}-k_{c}\theta_{p}^{\prime},k_{m}\alpha-
k_{m}\alpha_{t}^{\prime}-k_{c}\alpha_{p}^{\prime}),$

where we define $G(x,y)=\frac{1}{N_{t}}\left\lvert\sum_{n=-N}^{N}e^{jndx-
jn^{2}d^{2}y}\right\rvert$ and it achieves a maximal value at $(x,y)=(0,0)$.
According to [24], the beam at frequency $f_{m}$ is focused on the location
$(\theta_{m},\alpha_{m})$ with the maximum array gain. Thereby we obtain
$k_{m}\theta_{m}-k_{m}\theta_{t}^{\prime}-k_{c}\theta_{p}^{\prime}=0$ and
$k_{m}\alpha_{m}-k_{m}\alpha_{t}^{\prime}-k_{c}\alpha_{p}^{\prime}=0$, or
equivalently:

$\displaystyle\theta_{m}$
$\displaystyle=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}\theta_{p}^{\prime},$
(15) $\displaystyle\alpha_{m}$
$\displaystyle=\alpha_{t}^{\prime}+\frac{f_{c}}{f_{m}}\alpha_{p}^{\prime}.$
(16)

By setting $\theta_{p}^{\prime}\neq 0$, the spatial angle $\theta_{m}$ is
related to the frequency $f_{m}$, making beams at different frequencies split
towards different spatial angles. Moreover, when $\alpha_{p}^{\prime}=0$,
$\alpha_{m}$ becomes independent of $f_{m}$, indicating that all beams are
focused on the same distance ring $\alpha_{m}=\alpha_{t}^{\prime}$. To perform
fast beam training, the authors in [24] propose to distribute multi-frequency
beams over multiple angles in the same distance ring simultaneously by setting
$\theta_{p}^{\prime}\neq 0$ and $\alpha_{p}^{\prime}=0$. Then, it exhaustively
searches different distance rings by configuring the value of
$\alpha_{t}^{\prime}$.

However, it is clear that the near-field beam split in [24] is essentially
distance-independent, where beams can cover multiple angles in only one
specific distance ring. It fails to fully exploit the degree-of-freedoms
provided by the beam split effect, which still requires exhaustive search in
the distance domain and limits its efficiency. To deal with this problem, we
will introduce the distance-dependent beam split phenomena and how we can
search multiple angles as well as distances at the same time with a single
pilot by utilizing this phenomenon.

##  III Distance-Dependent Beam Split and Proposed Near-Field Beam Training
Scheme

In this section, the mechanism of distance-dependent beam split is elaborated.
Thanks to the periodicity of the angle dimension, beams can skip periodically
between $\theta=1$ and $\theta=-1$. Therefore, by elaborately designing the TD
and PS parameters, we can divide the subcarriers over the entire bandwidth
into several groups, and the beams of each group cover the angular range with
different distance ranges. Taking advantage of this mechanism, we propose a
distance-dependent beam-split-aided near-field beam training method. In the
proposed method, multiple distance rings can be covered by the beams at
different frequency groups simultaneously in the near field. Based on the
discovery, both angle and distance dimensions can be searched simultaneously
with different subcarriers in a single pilot during beam training.

###  III-A Distance-Dependent Beam Split

For a better understanding of the mechanism of distance-dependent beam split,
in this subsection, we start from the periodical beam split in the far field.
Then we will illustrate the distance-dependent beam split in near-field
scenarios.

####  III-A1 Mechanism of Far-field Periodical Beam Split

To illustrate the far-field periodical beam split phenomena, we first analyze
the periodicity of the array gain with the TD-PS precoding structure.
Specifically, the distance $r$ is assumed to be larger than the Rayleigh
distance so that distance-related parameters $\alpha$ reliably approach 0.
Then, the array response vector becomes $[{\bf
b}_{m}(\theta,\alpha)]_{n}=[{\bf
b}_{m}(\theta,0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{jk_{m}nd\theta}$. Accordingly,
the TD and PS beamforming vector can be set as $[{\bf w}_{m}^{\rm
TD}(\theta_{t}^{\prime},0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}nd\theta_{t}^{\prime}}$
and $[{\bf w}^{\rm
PS}(\theta_{p}^{\prime},0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{c}nd\theta_{p}^{\prime}}$,
respectively. Therefore, the array gain in (14) can be simplified into

$G(k_{m}\theta-
k_{m}\theta_{t}^{\prime}-k_{c}\theta_{p}^{\prime},0)=\Xi_{N_{t}}\left(\frac{dk_{m}}{\pi}(\theta-\theta_{t}^{\prime}-\frac{f_{c}}{f_{m}}\theta_{p}^{\prime})\right),$
(17)

where $\Xi_{N_{t}}(x)=\sin\frac{N_{t}\pi}{2}x/N_{t}\sin\frac{\pi}{2}x$ is the
Dirchlet sinc function. The far-field array gain $G(x,0)$ is evidently a
periodic function of period $2\pi/d$, i.e.,

$\displaystyle G(x-\frac{2p\pi}{d},0)=G(x,0),\>\>\forall p\in\mathbb{Z}.$ (18)

This periodicity implies that all the points
$x=\frac{2p\pi}{d},p\in\mathbb{Z}$ can maximize the array gain $G(x,0)$. In
this context, the spatial angle $\theta_{m}$ at frequency $f_{m}$ should be
derived from
$k_{m}\theta_{m}-k_{m}\theta_{t}^{\prime}-k_{c}\theta_{p}^{\prime}=\frac{2p_{m}\pi}{d},$
as

$\theta_{m}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}(\theta_{p}^{\prime}+2p_{m}).$
(19)

The physical significance of $p_{m}$ in (19) is interpreted as follows.

###### Remark 1

(Physical significance of integer $p$): Since the spatial angle $\theta_{m}$
corresponds to an actual physical angle $\vartheta$, $\theta_{m}$ has to
satisfy the angle constraint $\theta_{m}\in[-1,1]$. To realize the case of
$p_{m}=0$ as presented in (15), the variables $\theta_{t}^{\prime}$ and
$\theta_{p}^{\prime}$ need to be carefully designed to make
$\theta_{m}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}\theta_{p}^{\prime}$
satisfy the angle constraint $\theta_{m}\in[-1,1]$. However, as long as the
delay circuits allow, the variables $\theta_{t}^{\prime}$ and
$\theta_{p}^{\prime}$ can be arbitrarily adjusted, making it possible to break
the angle constraint:
$\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}\theta_{p}^{\prime}\notin[-1,1]$. In
this context, the integer $p_{m}$ is activated and it will naturally become
non-zero to shift the value of
$\theta_{m}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}(\theta_{p}^{\prime}+2p_{m})$
into the physical angle range, $[-1,1]$. In other words, by adjusting the
delay parameters $\theta_{t}^{\prime}$ and $\theta_{p}^{\prime}$, the integer
$p_{m}$ can be automatically controlled to satisfy the angle constraint.

(19) allows us to illustrate the far-field periodical beam split. To begin
with, we adjust the parameters $\theta_{t}^{\prime}$ and $\theta_{p}^{\prime}$
such that the beam of the first subcarrier is steered to
$\theta_{1}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{1}}\theta_{p}^{\prime}=-1$
with $p_{1}=0$. Without loss of generality, we assume $\theta_{t}^{\prime}>0$
and $\theta_{p}^{\prime}<0$. In this case, as the subcarrier frequency $f_{m}$
grows, the focused direction $\theta_{m}$ gradually increases from $-1$ to $1$
with $p_{m}$ remaining to $0$. Suppose that the value of $\theta_{p}^{\prime}$
is large enough such that the focused direction $\theta_{m}$ can approach
$\theta_{m_{1}}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m_{1}}}\theta_{p}^{\prime}=1$
at the $m_{1}$-th subcarrier where $m_{1}<M$. Subsequently, if we continue to
increase the frequency, then
$\theta_{m_{1}+1}=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m_{1}+1}}\theta_{p}^{\prime}$
definitely exceeds $1$. According to Remark 1, the integer $p_{m_{1}+1}$ will
automatically become $-1$ to shift $\theta_{m_{1}+1}$ from
$\theta_{t}^{\prime}+\frac{f_{c}}{f_{m_{1}+1}}\theta_{p}^{\prime}$ to
$\theta_{t}^{\prime}+\frac{f_{c}}{f_{m_{1}+1}}(\theta_{p}^{\prime}-2)$ for
guaranteeing the angle constraint $\theta_{m_{1}+1}\in[-1,1]$. Besides, the
beam at frequency $f_{m_{1}+1}$ is steered to
$\theta_{t}^{\prime}+\frac{f_{c}}{f_{m_{1}+1}}(\theta_{p}^{\prime}-2)\approx-1$.
That is to say, due to the periodic property of the angle domain, the spatial
direction of the beam “skips” from $1$ to $-1$ when the frequency arises from
$f_{m_{1}}$ to $f_{m_{1}+1}$. In addition, with the continuous increment of
the frequency, the focused direction of the beam will gradually rise until the
focused direction $\theta_{m_{2}}$, with $m_{1}<m_{2}<M$, arrives $1$ for the
second time. Then, the beam will “skip” again to $-1$ by automatically tuning
the integer $p_{m_{2}+1}$ to $-2$. This beam skip process will be repeated
until the last subcarrier $f_{M}$222If TD-PS parameters are initialized as
$\theta_{p}^{\prime}+2p>0$, the focus direction of the beam will decrease
conversely with respect to frequency $f_{m}$. Then the beam will skip from
$-1$ to $1$ when automatically increasing $p$.. This phenomenon is called the
periodical beam split.

###### Remark 2

Utilizing far-field periodical beam split, an important insight is that
several beams of different frequencies $f$ may focus in the same physical
direction. In other words, the beams over the entire bandwidth can
periodically scan the angle range $[-1,1]$.

####  III-A2 Mechanism of Near-field Distance-Dependent Beam Split

Consider the periodic pattern of beam split in the polar domain. Similar to
the periodic far-field array gain in (18), the near-field array gain $G(x,y)$
exhibits a periodicity as well. As proven in [24], $G(x,y)$ is periodic
against the vector variable $(x,y)$ with a period of
$(\frac{2\pi}{d},\frac{2\pi}{d^{2}})$, i.e., $G(x-\frac{2\pi
p}{d},y-\frac{2\pi q}{d^{2}})=G(x,y)$, for $\forall p,q\in\mathbb{Z}$. The
periodicity implies that $(x,y)=(\frac{2\pi p}{d},\frac{2\pi q}{d^{2}})$ are
all optimal solutions to maximize the array gain $G(x,y)$. Then the focused
location $(\theta_{m},\alpha_{m})$ of the beam at frequency $f_{m}$ in (15)
and (16) should be rewritten as

$\displaystyle\theta_{m}$
$\displaystyle=\theta_{t}^{\prime}+\frac{f_{c}}{f_{m}}(\theta_{p}^{\prime}+2p_{m}),$
(20) $\displaystyle\alpha_{m}$
$\displaystyle=\alpha_{t}^{\prime}+\frac{f_{c}}{f_{m}}(\alpha_{p}^{\prime}+\frac{2q_{m}}{d}),$
(21)

which are derived from $k_{m}\theta-
k_{m}\theta_{t}^{\prime}-k_{c}\theta_{p}^{\prime}=\frac{2\pi p_{m}}{d}$ and
$k_{m}\alpha-k_{m}\alpha_{t}^{\prime}-k_{c}\alpha_{p}^{\prime}=\frac{2\pi
q_{m}}{d^{2}}$. Clearly, the periodic beam split pattern in the near field is
a generalization of (19) in the far field. When $B\ll f_{c}$, (20) and (21)
can be approximated by the first-order Taylor expansion as

$\displaystyle\theta_{m}$
$\displaystyle\approx\theta_{t}^{\prime}+2(\theta_{p}^{\prime}+2p)-\frac{\theta_{p}^{\prime}+2p}{f_{c}}f_{m},$
(22) $\displaystyle\alpha_{m}$
$\displaystyle\approx\alpha_{t}^{\prime}+2(\alpha_{p}^{\prime}+\frac{2q_{m}}{d})-\frac{\alpha_{p}^{\prime}+\frac{2q_{m}}{d}}{f_{c}}f_{m}.$
(23)

(22) and (23) suggest that the focused direction/distance of the beam grows
linearly with the frequency $f_{m}$.

We summarize the mechanism of near-field distance-dependent beam split as
follows. Formulas (20) and (21) reveal that TD-PS arrays can control the
coverage range of multi-frequency beams over the _angular_ and _distance_
dimensions simultaneously. Consider the angular dimension at first. The
formula (20) has the same expression as (19) in the far-field case. The beam
split pattern over the angular dimension in near-field scenarios is identical
to that in the far field. Thereafter, the manipulation of parameters
$\theta_{t}^{\prime}$ and $\theta_{p}^{\prime}$ can be the same as what we
have discussed in Section III-A1. We then consider the focused distance of
multi-frequency beams presented in (21). The major difference between the beam
split pattern over the angular and distance dimensions is attributed to the
physical constraints imposed on $\theta_{m}$ and $\alpha_{m}$. The physical
direction $\theta_{m}$ is bounded between $-1$ and $1$ while the distance
ring, $\alpha_{m}$, can grow from 0 to infinity. This difference implies that,
as opposed to the periodic pattern of the angle-dimension beam split, the
focused distance ring $\alpha_{m}$ increases/decreases monotonically with the
frequency $f_{m}$, as long as $\alpha_{p}^{\prime}+\frac{2q_{m}}{d}\neq 0$ and
$\alpha_{t}^{\prime}+\frac{f_{c}}{f_{m}}(\alpha_{p}^{\prime}+\frac{2q_{m}}{d})\geq
0$. Moreover, we can just ignore the subscript $m$ of $q_{m}$ because $q_{m}$
remains the same when changing the frequency $f_{m}$. For example, if the
multi-frequency beams are desired to cover the entire distance range
$[\alpha_{p},\alpha_{q}]$, we can simply set

$\displaystyle\alpha_{L}$
$\displaystyle=\alpha_{t}^{\prime}+\frac{f_{c}}{f_{L}}(\alpha_{p}^{\prime}+\frac{2q}{d})\geqslant\alpha_{q},$
(24) $\displaystyle\alpha_{H}$
$\displaystyle=\alpha_{t}^{\prime}+\frac{f_{c}}{f_{H}}(\alpha_{p}^{\prime}+\frac{2q}{d})\leqslant\alpha_{p},$
(25)

where $\alpha_{p}^{\prime}+\frac{2q}{d}$ is assumed to be greater than 0
without loss of generality.

In summary, by integrating the periodic pattern of the angle-dimension and the
monotonous pattern of the distance-dimension, we arrive at the distance-
dependent beam split effect. As presented in Fig. 2(a), the focused directions
of the multi-frequency beams fluctuate periodically between $[-1,1]$ while the
focused distance ring increases monotonically with the frequency $f_{m}$. In
such cases, the focused locations $(\theta_{m},\alpha_{m}),m=1,2\ldots M$ are
distributed over several inclined strips. Each strip covers the entire angle
range $[-1,1]$ while different strips occupy different distance ranges.

Figure 2: Comparison of distance-dependent beam split with traditional beam
split with different settings: (a) Beams of different subcarriers in proposed
distance-dependent beam split; (b) Beams of different subcarriers in
traditional beam split with $\alpha_{p}^{\prime}+\frac{2q}{d}=0$; (c) Beams of
different subcarriers in traditional beam split with
$\alpha_{p}^{\prime}+\frac{2q}{d}\neq 0$.

Distance-dependent beam split versus traditional near-field beam split: To
create the multi-strip distribution of beam split, it is necessary to make
$\theta_{p}^{\prime}+2p$ large enough for triggering the rapid fluctuation in
the angular domain and make $\alpha_{p}^{\prime}+\frac{2q}{d}$ non-zero for
changing the distance ring monotonously. These requirements differ the
distance-dependent beam split from the traditional counterpart in [24], which
are compared in Fig. 2. In [24], by setting $\theta_{p}^{\prime}+2p$ with a
_non-zero but relatively small_ value, the focused direction of the beams is
linearly related to the frequency $f_{m}$ in _a single period_ ; while by
setting $\alpha_{p}^{\prime}+\frac{2q}{d}=0$, the focused distance ring
remains unchanged. Thus, beams of different frequencies focus in different
angles in a single distance ring $\alpha=\alpha_{t}^{\prime}$, or equivalently
the single horizontal strip in Fig. 2(b). Additionally, in Fig. 2(c), we make
$\alpha_{p}^{\prime}+\frac{2q}{d}$ non-zero but keep the same
$\theta_{p}^{\prime}+2p$ as that of Fig. 2(b). This beam-split pattern
suggests that both the focused direction and distance ring increase
monotonically with the frequency $f_{m}$. However, without exploiting the
periodicity of the angle-dimension beam split, the beams can only occupy one
inclined strip while the multi-strip distribution is still concealed.

###  III-B Distance-Dependent Beam Split based Near-field Beam Training

Applying the discovered distance-dependent beam split phenomenon, it is
promising that during beam training, different angles and distances can be
searched simultaneously using one pilot. Thereby, a few pilots are capable of
covering the whole two-dimensional space. This observation motivates us to
design a new near-field beam training method based on this phenomenon. The
design principles, the configuration of TD-PS parameters, and detailed beam
training procedures are presented one by one in the sequel.

Figure 3: Add another pilot to guarantee coverage of possible regions.

####  III-B1 Basic design principles

The basic principles for designing near-field beam training aided by distance-
dependent beam split involve the consideration of reliability and efficiency.

  * •

Reliability. To support reliable beam training, each location inside the
communication region must be covered by one training beam. To quantify this
metric, we require that each location within the “angle-distance” region lies
in the $3~{}{\rm dB}$-beam width of a certain beam.

  * •

Efficiency. In addition to reliability, we aim to use as few pilots as
possible to improve training efficiency.

####  III-B2 Configuration of TD-PS parameters

Building on the above principles, we elaborate on the manipulation of TD-PS
parameters to create multi-strip beam pattern for beam training. We start from
the manipulation of a single pilot to guarantee the $3~{}{\rm dB}$ coverage in
the angle domain. Then we will explain how to utilize interleaved pilots to
guarantee the $3~{}{\rm dB}$ coverage in the distance domain.

For the first pilot, let’s consider the angle-related parameters
$\theta_{t}^{\prime}$ and $\theta_{p}^{\prime}$, as well as the integer
$p_{m}$ first. Based on (20), the focused direction difference of adjacent
subcarriers on the same strip with $p_{m}=p_{m+1}$ is

$|\theta_{m+1}-\theta_{m}|=|f_{c}(\theta_{p}^{\prime}+2p_{m})(\frac{1}{f_{m+1}}-\frac{1}{f_{m}})|.$
(26)

For ease of discussion, we assume $\theta_{p}^{\prime}+2p_{m}$ is always
greater than 0 in the sequel, while all our conclusions can be
straightforwardly generalized to negative $\theta_{p}^{\prime}+2p_{m}$
situations. Formula (26) suggests that a larger $\theta_{p}^{\prime}+2p_{m}$
leads to a faster change in the focused direction versus frequency $f_{m}$.
Thus, a relatively large $\theta_{p}^{\prime}+2p_{m}$ can broaden the coverage
area of beams, thereby improving beam training efficiency. However, since the
direction difference in (26) grows linearly with $\theta_{p}^{\prime}+2p_{m}$,
excessively large $\theta_{p}^{\prime}+2p_{m}$ will lead to uncovered angular
range between adjacent subcarriers. Taking these two factors into
consideration, Lemma 1 provides an upper limit of $\theta_{p}^{\prime}+2p_{m}$
that can avoid angular coverage hole.

###### Lemma 1

Consider an arbitrary physical location $(\bar{\theta},\bar{\alpha})$. If
$\theta_{p}^{\prime}+2p_{m}>0$, to make the direction $\bar{\theta}$ lie in
the angle-domain $3~{}{\rm dB}$-beamwidth of a certain beam,
$\theta_{p}^{\prime}$ should satisfy

$\theta_{p}^{\prime}\in\left\\{\theta_{p}^{\prime}:\theta_{p}^{\prime}+2p_{m}\leqslant\frac{0.88}{N_{t}}(f_{m}+f_{m+1})\frac{M}{B},\forall
p_{m}\in\mathbb{Z}\right\\}.$ (27)

###### Proof:

(See Appendix A).

It is easy to prove that when $\theta_{p}^{\prime}+2p_{m}>0$, $p_{m}$ is a
non-decreasing function with frequency $f_{m}$ (i.e.
$\theta_{p}^{\prime}+2p_{m}\leqslant\theta_{p}^{\prime}+2p_{M}$). Besides,
$\frac{0.88}{N_{t}}(f_{m}+f_{m+1})\frac{M}{B}$ has minimum value at
$f_{m}=f_{L}$. Thus, by setting

$\theta_{p}^{\prime}\in\left\\{\theta_{p}^{\prime}:\theta_{p}^{\prime}+2p_{M}\leqslant\frac{1.76f_{L}M}{N_{t}B},\forall
p_{M}\in\mathbb{Z}\right\\},$ (28)

(27) holds for all subcarriers. Therefore, by setting
$p_{M}=\lfloor\frac{0.88\gamma f_{L}M}{N_{t}B}\rfloor$, $\theta_{p}^{\prime}$
can be determined by

$\theta_{p}^{\prime}=\frac{1.76\gamma f_{L}M}{N_{t}B}-2p_{M},$ (29)

where $\gamma\in[0,1]$ is a hyperparameter. When $\gamma=1$, the equality in
(28) holds, while otherwise, it denotes an oversampling strategy in the angle
domain.

While the PS parameter $\theta_{p}^{\prime}$ influences the direction
difference, the TD parameter $\theta_{t}^{\prime}$ determines the intercept of
the strip according to (22). In other words, it affects the starting/ending
direction (i.e. $\theta_{1}$ or $\theta_{M}$) of this pilot. Without loss of
generality, we expect that the ending direction of the beam at frequency
$f_{M}$ is $\theta_{M}=1$. Accordingly, the parameter $\theta_{t}^{\prime}$
can be calculated by

$\theta_{t}^{\prime}=1-\frac{f_{c}}{f_{M}}(\theta_{p}^{\prime}+2p_{M}).$ (30)

For future usage, here we also calculate the integer $p_{1}$ of the first
subcarrier. $p_{1}$ should be automatically adjusted to satisfy the angle
constraint
$\theta_{t}^{\prime}+\frac{f_{c}}{f_{L}}(\theta_{p}^{\prime}+2p_{1})\in[-1,1]$,
so it can be presented by

$p_{1}={\rm
round}\left(\frac{-\theta_{t}^{\prime}\frac{f_{L}}{f_{c}}-\theta_{p}^{\prime}}{2}\right).$
(31)

Next, we continue to discuss the manipulation of distance-related parameters
$\alpha_{t}^{\prime}$ and $\alpha_{p}^{\prime}$, and the corresponding integer
$q$. To guarantee that beams can cover the desired distance range,
$\alpha_{t}^{\prime}$ and $\alpha_{p}^{\prime}$ should meet the requirements
in (24) and (25). We assume that the possible user distribution distance is
$[\alpha_{\rm min},\alpha_{\rm max}]$. By setting
$(\alpha_{p},\alpha_{q})=(\alpha_{\rm min},\alpha_{\rm max})$ and subtracting
(25) from (24), $\alpha_{p}^{\prime}+\frac{2q}{d}$ is supposed to satisfy

$\alpha_{p}^{\prime}\in\left\\{\alpha_{p}^{\prime}:\alpha_{p}^{\prime}+\frac{2q}{d}\geq\frac{\alpha_{\rm
max}-\alpha_{\rm min}}{\frac{f_{c}}{f_{L}}-\frac{f_{c}}{f_{H}}},\forall
q\in\mathbb{Z}\right\\}.$ (32)

Therefore, by setting $q=\lfloor\frac{\alpha_{\rm max}-\alpha_{\rm
min}}{\frac{f_{c}}{f_{L}}-\frac{f_{c}}{f_{H}}}\frac{d}{2}\rfloor$,
$\alpha_{p}^{\prime}$ can be set as

$\alpha_{p}^{\prime}=\frac{\alpha_{\rm max}-\alpha_{\rm
min}}{\frac{f_{c}}{f_{L}}-\frac{f_{c}}{f_{H}}}-\frac{2q}{d},$ (33)

After obtaining $\alpha_{p}^{\prime}$, the parameter $\alpha_{t}^{\prime}$ can
be arbitrarily selected from the range

$\displaystyle\alpha_{t}^{\prime}\in[\alpha_{\rm
max}-\frac{f_{c}}{f_{L}}(\alpha_{p}^{\prime}+\frac{2q}{d}),\alpha_{\rm
min}-\frac{f_{c}}{f_{H}}(\alpha_{p}^{\prime}+\frac{2q}{d})],$ (34)

to meet the constraints (24) and (25).

###### Remark 3

In summary, the parameters designed in (29), (30), (33), and (34) allow us to
create the multi-strip beam pattern in Fig. 2(a) using one single pilot. These
beams can guarantee the 3 dB coverage of the entire angular range in each
strip.

However, one remaining issue is that the $3~{}{\rm dB}$ coverage in the
distance domain may not be satisfied, since there exists coverage hole between
adjacent strips as shown in Fig. 2(a). Specifically, consider two searched
locations at the same angle but adjacent strips as illustrated by the circled
blue points in Fig. 3. Based on (22) and (23), the angle and distance
differences, $\Delta\theta$ and $\Delta\alpha$, of these two locations can be
calculated as

$\displaystyle\Delta\theta$
$\displaystyle=\frac{\theta_{p}^{\prime}+2p_{m}}{f_{c}}\Delta f=2,$ (35)
$\displaystyle\Delta\alpha$
$\displaystyle=\frac{\alpha_{p}^{\prime}+\frac{2q}{d}}{f_{c}}\Delta f,$ (36)

where $\Delta f$ denotes the frequency difference. From equations (35) and
(36), the distance difference $\Delta\alpha$ can be expressed as

$\Delta\alpha=\frac{2(\alpha_{p}^{\prime}+\frac{2q}{d})}{\theta_{p}^{\prime}+2p_{m}}\leqslant\frac{2(\alpha_{p}^{\prime}+\frac{2q}{d})}{\theta_{p}^{\prime}+2p_{1}}\overset{\Delta}{=}\Delta\alpha^{\rm
upper}.$ (37)

To realize the coverage over the entire distance domain, we expect that the
distance difference $\Delta\alpha$ is smaller than the sum of the distance-
domain $3~{}{\rm dB}$-beam width of the two beams, which can be presented as

$\Delta\alpha\leqslant\eta_{\alpha}^{f_{a}}+\eta_{\alpha}^{f_{b}}.$ (38)

Here, $f_{a}$ and $f_{b}$ refer to the frequency of the corresponding beams
and $\eta_{\alpha}^{f}$ is the distance-domain $3~{}{\rm dB}$-beam width at
frequency $f$. As proven in Appendix B, the beam width $\eta_{\alpha}^{f}$ has
an analytical expression:
$\eta_{\alpha}^{f}=\frac{4\beta^{2}f_{c}^{2}}{N_{t}^{2}cf}$, where
$\beta=1.318$.

In practice, however, the constraint (38) is hard to satisfy, indicating that
a single pilot is not sufficient to cover all distances. To fill up the
coverage hole of a single pilot, we plug in more pilots to create _interleaved
multi-strip beam pattern_ , as illustrated in Fig. 3. To this aim, recall that
the intercept of a strip is dependent on the TD parameter
$\theta_{t}^{\prime}$ according to (22). By changing the value of
$\theta_{t}^{\prime}$, the ending direction $\theta_{M}$ is modified and thus
the strips of one pilot are shifted. This observation motivates us to retain
the parameters $\theta_{p}^{\prime},\alpha_{t}^{\prime}$, and
$\alpha_{p}^{\prime}$ but set $\theta_{t}^{\prime}$ as different values for
different pilots to create the interleaved multi-strip beam pattern. We
disperse different pilots to create uniformly spaced strips (the circled red
points in Fig. 3) to efficiently fill in the uncovered ranges.

To elaborate, we denote the total pilot number as $K$ and introduce the
superscript $k$ to index the TD parameter $\theta_{t}^{\prime k}$ and the
ending direction $\theta_{M}^{k}$ of the $k$-th pilot. The minimal pilot
number required for distance coverage is given as follows.

###### Remark 4

The $K$ pilots need to fill up the distance-domain coverage hole with an upper
length of $\Delta\alpha^{\rm upper}$. Given that the distance-domain beam
width has a lower bound $\eta^{f}_{\alpha}\geq\eta^{f_{H}}_{\alpha}$, the
minimal pilot number is determined by

$\displaystyle K\geq\frac{\Delta\alpha^{\rm
upper}}{2\eta^{f_{H}}_{\alpha}}=\frac{(\alpha_{p}^{\prime}+\frac{2q}{d})N_{t}^{2}cf_{H}}{4(\theta_{p}^{\prime}+2p_{1})\beta^{2}f_{c}^{2}}.$
(39)

After obtaining the pilot number $K$, we can uniformly sample the ending
directions from the range $[-1,1]$ such that
$\theta_{M}^{k}=1-\frac{2}{K}(k-1)$ for $k\in\\{1,2,\cdots,K\\}$ to ensure
uniformly spaced strips. Therefore, $\\{\theta_{t}^{\prime k}\\}$ for each
pilot are given as

$\theta_{t}^{\prime
k}=\theta_{M}^{k}-\frac{f_{c}}{f_{M}}(\theta_{p}^{\prime}+2p_{M}),\quad
k\in\\{1,2\cdots,K\\}.$ (40)

It is notable that the parameter $\theta_{t}^{\prime}$ designed in (30) is
exactly the first parameter $\theta_{t}^{\prime 1}$ designed in (40) since
$\theta_{M}^{1}=1$.

Based on the discussions above, we summarize the TD-PS parameter manipulation
procedures in Algorithm 1. For Step 1-3, by setting the angle-related
parameters of the first pilot, we manipulate the beams to periodically cover
the angular range while ensuring the $3$-dB coverage in the angle domain. Then
based on Step 4, the beams are set to cover the desired distance range.
Finally, we utilize interleaved pilots to ensure $3$-dB coverage in distance
domain, as shown in Step 5-6.

Algorithm 1 TD-PS parameter manipulation for distance-dependent beam-split-
based beam training.

0: $N_{t}$, $f_{c}$, $B$, $M$, $\alpha_{\rm max}$, $\alpha_{\rm min}$, and
$\gamma$.

1: Set $p_{M}=\lfloor\frac{0.88\gamma f_{L}M}{N_{t}B}\rfloor$ and obtain
$\theta_{p}^{\prime}$ by (29);

2: Set $\theta_{M}^{1}=1$ and obtain $\theta_{t}^{\prime 1}$ for the first
pilot by (30);

3: Set $p_{1}$ as (31);

4: Set $q=\lfloor\frac{\alpha_{\rm max}-\alpha_{\rm
min}}{\frac{f_{c}}{f_{L}}-\frac{f_{c}}{f_{H}}}\frac{d}{2}\rfloor$ and obtain
$\alpha_{p}^{\prime}$ and $\alpha_{t}^{\prime}$ based on (33) and (34);

5: Calculate the required pilot number $K$ based on (39);

6: Obtain $\\{\theta_{t}^{\prime k}\\}$ for the remaining $K-1$ pilots based
on (40).

6: Designed parameters $\\{\theta_{t}^{\prime k}\\}$, $\theta_{p}^{\prime}$,
$\alpha_{t}^{\prime}$ and $\alpha_{p}^{\prime}$.

####  III-B3 Beam training process

In the beam training process, the BS sequentially transmits $K$ pilots to the
user. The received array gains over all subcarriers and pilots are written as

$\begin{split}{\bm{g}}=&[g(1,1),g(2,1),\cdots,g(M,1),\\\
&\cdots,g(1,K),g(2,K),\cdots,g(M,K)],\end{split}$ (41)

where $g(m,k)$ is the array gain at the $m$-th subcarrier of the $k$-th pilot.
The user selects the subcarrier with the highest received power (i.e. the
strongest element of ${\bm{g}}(\hat{m},\hat{k})$) and the estimated physical
location can be calculated based on (20) and (21).

###  III-C Hardware Implementations of Proposed Method

This sub-section introduces the hardware implementation of the TD-PS structure
to realize the proposed beam training algorithm. In particular, we mainly
focus on the expensive TD circuits. The function of _time delay_ can be
implemented mainly by two kinds of approaches, namely the adjustable true-
time-delay-based methods [28, 29] and fixed true-time-delay-based methods [30,
31]. The former one can provide continuously adjustable time delay over the
entire range with significantly high power consumption and hardware complexity
[31]. In contrast, the available time-delays of fixed true-time-delay-based
method are restricted to several fixed values. Thus, this architecture has
much lower power consumption and hardware complexity, with the sacrifice of TD
resolution.

Figure 4: Hardware implementations of TD by fixed time delay network.

Fortunately, the low-cost fixed true-time-delay-based methods could adequately
satisfy the requirements for realizing the proposed beam training algorithm.
To be specific, the scanned positions of beams are fixed and the corresponding
time delay values are elaborately designed and can be determined and
calculated in advance. Besides, only $K$ (usually a small number such as 2,3)
time delay values need to be realized by TD. Therefore, as shown in Fig. 4, we
propose to deploy a nonuniform-quantization TD with fixed time delay unit and
a switch network, which can be implemented by an extensively-utilized
microstrip line or waveguide of fixed length [30]. For the $n$-th TD, we
calculate the $K$ desired time delay value as
$\tau^{(n)^{\prime}k},k=1,2,\cdots,K$:

$\tau^{(n)^{\prime}}=\frac{r_{1}^{(n)^{\prime}}}{c}=\frac{nd\theta_{t}^{\prime}-n^{2}d^{2}\alpha_{t}^{\prime}}{c}.$
(42)

Then we construct the TD with the fixed TD comprising $K$ fixed time delay
units and a switch, indicating the selected fixed TD in different time slots.
In this context, only $\log_{2}K$ bits are required to represent the selected
time delay.

Note that the incorporation of TDs may introduce additional Insertion Loss
(IL) into the link budget, indicating the signal attenuation experienced as it
traverses the hardware device. For the proposed implementation, the IL
associated with the TD is the cumulative effect of the insertion losses from
one fixed time delay unit and one switch, i.e., $IL_{FTD}+IL_{switch}$, which
is significantly lower than adjustable TDs [30]. Besides, as discussed in
[32], additional IL introduced by fixed TD may not have significant influence
on beamforming gain and it can be compensated by improving the amplification
factor of the power amplifiers.

##  IV Improved Beam Training for Off-grid Users

Compared to the method in [24], the sampled density of direction within
$[-1,1]$ of the proposed method in Section III is relatively low, which limits
its performance for off-grid users (i.e. users not on the searched locations).
In this section, we propose two algorithms, namely the auxiliary beam pair-
assisted method and the match filter-based method, to solve this problem. The
core idea of the proposed methods is to employ the received power of multiple
subcarriers, rather than only the one with the highest received power to
jointly estimate the users’ locations.

###  IV-A Auxiliary beam pair-assisted method

Figure 5: Auxiliary beam pair-assisted method.

As shown in Fig. 5, when the user is located between two beams of frequency
$f_{a}$ and $f_{b}$, the scheme in Section III, which chooses the codeword
having the highest received power at frequency $f_{a}$, is suboptimal.
Inspired by [33, 34], we design an auxiliary beam pair-assisted beam training
method based on near-field distance-dependent beam split. The method utilizes
the amplitude distribution of the auxiliary beam pair, $f_{a}$ and $f_{b}$, to
accurately estimate the angle and distance of off-grid users. To this aim, we
will first analyze the amplitude distribution of near-field beams, and then
the beam training method is elaborated.

####  IV-A1 Analysis of amplitude distribution of near-field beams

The equal-power contour of near-field beams can be clarified by the following
Proposition 1.

###### Proposition 1

The equal-power contour of near-field beams can be approximated as an ellipse
equation:

$G(k(\theta-\bar{\theta}),k(\alpha-\bar{\alpha}))=1-\sigma_{1}(\theta-\bar{\theta})^{2}-\sigma_{2}(\alpha-\bar{\alpha})^{2},$
(43)

where $(\theta,\alpha)$ is the focused location of the beam,
$(\bar{\theta},\bar{\alpha})$ is an arbitrary physical location within the
curve of $3~{}{\rm dB}$-beam width. Parameters $\sigma_{1},\sigma_{2}$ are
presented as

$\displaystyle\sigma_{1}=\frac{1}{24}N_{t}^{2}\pi^{2}\frac{f^{2}}{f_{c}}\quad{\rm
and}\quad\sigma_{2}=\frac{\pi^{2}}{90}\frac{N_{t}^{4}d^{4}}{\lambda^{2}}.$
(44)

###### Proof:

(See Appendix C).

####  IV-A2 Procedures of auxiliary beam pair-assisted method

According to the analysis above, the received array gain of the beam at
frequency $f_{a}$ and $f_{b}$ each defines an ellipse in which the user is
located. The angle and distance to be estimated can be obtained through the
intersection of the ellipses. This is the basic idea of the auxiliary beam
pair-assisted method. Then we will illustrate the procedures step by step as
follows.

  * •

Firstly, we perform the beam training proposed in Section III to obtain
subcarrier $f_{a}$ with the highest received power. Suppose we can obtain the
value of the highest power $P_{a}$, we calculate the array gain as $G_{a}$.
Besides, we denote the focus location of the beam as
$(\theta_{a},\alpha_{a})$.

  * •

Then we compare the received power at subcarriers $f_{a-1}$ and $f_{a+1}$, and
select the one with higher power, denoted as $f_{b}$. Similarly, record the
focus location $(\theta_{b},\alpha_{b})$ and array gain as $G_{b}$.

  * •

The corresponding equal-amplitude ellipse defined by subcarrier $f_{a}$ and
$f_{b}$ can be written as

$\left\\{\begin{array}[]{cc}\sigma_{1}(\theta_{a}-\theta)^{2}+\sigma_{2}(\alpha_{a}-\alpha)^{2}&=1-G_{a}\\\
\sigma_{1}(\theta_{b}-\theta)^{2}+\sigma_{2}(\alpha_{b}-\alpha)^{2}&=1-G_{b}\end{array}\right.,$
(45)

based on which the estimated $\hat{\theta}$ and $\hat{\alpha}$ can be obtained
using the Newton method.

###  IV-B Match filter-based method

The auxiliary beam pair-assisted beam training utilizes a pair of beams to
improve the training performance for off-grid users. In this subsection, we
generalize it to a match filter-based beam training method, which advocates
the use of array gains over all subcarriers and pilots for accurate beam
training.

In this method, a substantial number of grids with different angles and
distances are sampled offline. For each grid, the distribution of array gain
over different subcarriers and pilots can be calculated by the location of the
grid. Then, in the online stage, we select the grid whose array gain
distribution matches the best with the received array gain distribution of the
user. Since the sampled grids can be much denser than the searched locations,
the performance for off-grid users is improved. Moreover, the characteristics
of the match filter also protect training from noise. The framework of this
method is elaborated below.

  * •

Offline stage: Sample the possible user locations by $L$ grids on the angle
domain and $S$ grids on the distance domain. The sampled grids are supposed to
be denser than the searched locations by the proposed beam training method.
For the sampled location $(\theta_{l},\alpha_{s})$, the array gain at the
$m$-th subcarrier of the $k$-th pilot is presented as

$\displaystyle g_{l,s}(m,k)$ $\displaystyle=|({\bf w}_{m}^{\rm
TD}(\theta_{t}^{\prime k},\alpha_{t}^{\prime})\odot{\bf w}^{\rm
PS}(\theta_{p}^{\prime},\alpha_{p}^{\prime}))^{T}{\bf
b}_{m}(\theta_{l},\alpha_{s})|.$ (46)

Then, the corresponding array gain vector, ${\bm{g}}_{l,s}$, is

$\displaystyle{\bm{g}}_{l,s}=[g_{l,s}(1,1),g_{l,s}(2,1),\cdots,g_{l,s}(M,1),$
$\displaystyle\>\>\cdots,g_{l,s}(1,K),g_{l,s}(2,K),\cdots,g_{l,s}(M,K)].$ (47)

  * •

Online stage: For each of the $L\times S$ grids, the similarity between the
array gain vector of location $(\theta_{l},\alpha_{s})$ and the received array
gain vector of the user is calculated as

$r_{l,s}={\bm{g}}^{T}{\bm{g}}_{l,s},s=1,2,\cdots,S,l=1,2,\cdots,L.$ (48)

Finally, we select the grid with largest similarity $r_{l,s}$ and the
estimated location of the user is $\hat{\theta}=\theta_{l}$ and
$\hat{\alpha}=\alpha_{s}$.

To reduce feedback overhead, the users are supposed to estimate $\hat{\theta}$
and $\hat{\alpha}$ locally and feed back the results to the BS, rather than
feeding back all the received power to the BS.

##  V Simulation Results

In this section, we present simulations to validate the proposed distance-
dependent beam split phenomenon. Subsequently, the performance of the proposed
distance-dependent beam split-based beam training schemes, encompassing both
on-grid algorithms and off-grid algorithms, is assessed.

###  V-A The Demonstration of Distance-dependent Beam Split

(a) Distance-dependent beam split with a single pilot. (b) Distance-dependent
beam split with two pilots.

Figure 6: Illustration of near field near-field distance-dependent beam split.

We first verify that the designed TD-PS parameters in Algorithm 1 are capable
of producing distance-dependent near-field beam split. As an example, we
suppose the BS is equipped with $N_{t}=128$ antennas. The carrier frequency is
$f_{c}=10~{}{\rm GHz}$, the number of subcarriers is $M=512$, and the
bandwidth is $B=2~{}{\rm GHz}$. The users are uniformly distributed within
angle range $[-\pi/3,\pi/3]$, and distance range $[\rho_{\rm min},\rho_{\rm
max}]=[5~{}{\rm m},200~{}{\rm m}]$, corresponding to a distance ring range
$[\alpha_{\rm min},\alpha_{\rm max}]=[1/400,1/10]$. According to Algorithm 1,
the TD-PS parameters are defined as follows. Based on (28),
$\theta_{p}^{\prime}+2p_{M}$ should satisfies
$\theta_{p}^{\prime}+2p_{M}\leqslant\frac{1.76f_{L}}{N_{t}}\frac{M}{B}=31.68$
and we set $\theta_{p}^{\prime}=1.68$, $p_{M}=15$ with $\gamma=1$. Then (30)
and (31) determine that $\theta_{t}^{\prime 1}-27.8$ and $p_{1}=12$. Then
based on (32), $\alpha_{p}^{\prime}+\frac{2q}{d}\geqslant\frac{\alpha_{\rm
max}-\alpha_{\rm min}}{\frac{f_{c}}{f_{L}}-\frac{f_{c}}{f_{H}}}=0.483$, and
thus $\alpha_{p}^{\prime}$ is set as $0.5$ with $q=0$ here. Accordingly,
$\alpha_{t}^{\prime}$ is manipulated as
$\alpha_{t}^{\prime}=-0.454\in[\alpha_{\rm
max}-\frac{f_{c}}{f_{L}}(\alpha_{p}^{\prime}+\frac{2q}{d}),\alpha_{\rm
min}-\frac{f_{c}}{f_{H}}(\alpha_{p}^{\prime}+\frac{2q}{d})]=[-0.456,-0.452]$.

We depict the distribution of beams over $M$ subcarriers generated by the TD-
PS beamformer in Fig. 6 (a). As we expect, the focused points of beams create
a multi-strip beam pattern, validating the effectiveness of distance-dependent
beam split. It is also observable that the beams of a single pilot in Fig. 6
(a) cannot cover the entire user region, leaving a coverage hole between
strips. According to (39), the minimum pilot number required is calculated as
$K=2$. Thereby, we can add another pilot with the parameter
$\theta_{t}^{\prime 2}=-28.8$ to make the second ending direction be
$\theta_{M}^{2}=0$ based on (40). The beam patterns of the two pilots are
plotted together in Fig. 6 (b). In this context, the entire possible region is
thoroughly searched, guaranteeing the reliability of beam training.

###  V-B Beam Training performance

We then evaluate the performance of the proposed distance-dependent beam-
split-based near-field beam training. The default simulation settings are as
follows. The BS equips an $N_{t}=256$-element ULA while the carrier frequency
is $f_{c}=30~{}{\rm GHz}$. The bandwidth is $B=5~{}{\rm GHz}$ with subcarrier
number $M=1024$. The users are uniformly distributed within angle range
$[-\pi/3,\pi/3]$, and distance range $[\rho_{\rm min},\rho_{\rm
max}]=[5~{}{\rm m},200~{}{\rm m}]$, corresponding to a distance ring range
$[\alpha_{\rm min},\alpha_{\rm max}]=[1/400,1/10]$. According to Algorithm 1,
we employ $K=3$ pilots for beam training. The TD-PS parameters are set as
$\theta_{p}^{\prime}=0.784$, $\\{\theta_{t}^{\prime
k}\\}=\\{-32.95,-33.62,-34.29\\}$, $\alpha_{p}^{\prime}=0.58$, and
$\alpha_{t}^{\prime}=-0.53$, while the corresponding $\gamma=0.95$,
$p_{1}=15$, $p_{M}=18$ and $q=0$. To evaluate the effectiveness of beam
training, we employ the average rate performance as a metric, which is
presented as

$\displaystyle R$
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\log_{2}\left(1+\frac{P_{t}}{\sigma^{2}}\left\|\mathbf{h}_{m}^{T}\mathbf{w}_{m}\right\|^{2}\right)$
(49)
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\log_{2}\left(1+\frac{P_{t}N_{t}\beta_{m}^{2}}{\sigma^{2}}\left\|\mathbf{b}_{m}^{T}\left(\theta_{0},r_{0}\right)\mathbf{w}_{m}\right\|^{2}\right)$

where ${\rm SNR}=\frac{P_{t}N_{t}\beta_{m}^{2}}{\sigma^{2}}$. The simulation
results below are obtained by $1000$ realizations of Monte Carlo trails.

Figure 7: Average rate performance vs. SNR.

To begin with, Fig. 7 presents the average rate performance against SNR
ranging from $5~{}{\rm dB}$ to $20~{}{\rm dB}$. The proposed beam training
algorithms for both on-grid and off-grid cases are included. The training
overhead is $K=3$. The sampled grids in the angle and distance domain are
$L=1024$ and $S=10$. When the SNR is relatively high (e.g. ${\rm
SNR}>10~{}{\rm dB}$), both the proposed off-grid algorithms can improve the
performance of distance-dependent beam split-based beam training performance.
While at low SNR scenarios, the performance of the auxiliary beam pair-
assisted method is almost the same as the on-grid method. This may result from
the fact that an inaccurate beam gains $G_{a}$ and $G_{b}$ with noise poses
the “error accumulation” phenomena when calculating the estimated
$\hat{\theta}$ and $\hat{\alpha}$. Therefore, the auxiliary beam pair-assisted
method might be vulnerable to noise. In contrast, thanks to the capability of
restraining noise interference, match-filter-based method can ensure
significant performance improvement for off-grid users even in low SNR
regimes.

Then we compare the proposed distance-dependent beam-split-based near-field
beam training enhanced by match filter with the following benchmarks:

  * •

Perfect CSI: BS has access to the perfect channel vector $\mathbf{h}_{m}$,
which establishes the upper bound for performance.

  * •

Near-field exhaustive beam training: This method conducts an exhaustive search
across all potential directions and distances to identify the optimal
codeword.

  * •

Near-field rainbow based beam training: As mentioned above, the near-field
rainbow-based method [24] searches multiple angles in a single distance ring
with different subcarriers concurrently and sequentially searches through
various distance rings exhaustively.

  * •

Far-field rainbow-based beam training: Proposed in [35], this method employs
TD beamforming to concurrently search for the optimal angle across different
subcarriers while disregarding the distance information.

TABLE I: Comparisons of overheads for different Schemes Schemes | Training Overheads | Values
---|---|---
Near-field exhaustive beam training | $TS$ | $10240$
Near-field rainbow based beam training | $S$ | $10$
Far-field rainbow based beam training | $1$ | $1$
Proposed distance-dependent beam split based beam training | $K$ | $3$

Table I summarizes the required pilot overheads for different techniques. We
sample $L=1024$ grids for the angle domain and $S=10$ grids for the distance
ring. Therefore, the pilot overheads of the near-field rainbow [24] are
$S=10$, while for the near-field exhaustive method, the training overhead is
up to $LS=10240$. Besides, the pilot number is $K=3$ for proposed distance-
dependent beam split-based beam training. As for the far-field rainbow
technique, the optimal angle can be obtained with only a single pilot through
TD beamforming, while the distance information is neglected.

Figure 8: Average rate performance vs. training overhead.

Fig. 8 presents the average rate performance against the training overhead.
The SNR is set as $15~{}{\rm dB}$. The training overhead ranges from $0$ to
$8000$. In each time slot, we employ the optimal beamforming vector identified
in prior time slots to serve the user. The far-field rainbow-based technique
requires only one pilot. However, since the far-field approach focuses solely
on angle information and disregards distance information, its average rate
performance is suboptimal. Furthermore, near-field exhaustive scheme requires
more than $2000$ pilots to attain a satisfactory performance, which is
impractical for real-world systems. Then we focus on the comparison of near-
field rainbow-based beam training [24] and the proposed method. For fair
comparison, the number of sampled directions for the match filter is set as
$L=1024$ to guarantee the same sample rate in the angle domain with near field
rainbow [24] (the number of sampled directions is $M=1024$) while the number
of sampled distances is both $S=10$. As depicted in Fig. 8, the proposed
method employs only $3$ pilots to achieve comparable average rate performance
while near-field rainbow-based method [24] requires $10$. Therefore, since
both the angle and distance domain can be searched with different subcarriers
simultaneously, the proposed method further improves the near-field beam
training efficiency.

Figure 9: Average rate performance vs. distance.

Fig. 9 illustrates the average rate performance versus the distance. We have
capped the maximum training overhead for all the methods under consideration
as $256$ and set SNR as $15~{}{\rm dB}$. Additionally, the distance between
the user and BS increases from $10$ to $50$ meters. As illustrated in Fig. 9,
the proposed method with match filter achieves near-optimal rate performance
with lower training overheads compared to near-field rainbow-based method.
Regarding the far-field scheme, as the distance decreases, near-field
propagation becomes more prominent, leading to a rapid degradation in the
average rate for the far-field scheme. Moreover, due to limited maximum
training overhead, the exhaustive scheme faces difficulties in searching for
locations within the near field.

##  VI Conclusions

In this paper, we demonstrate a phenomenon called distance-dependent beam
split where we can divide the subcarriers into different groups which
periodically cover the angular range with different distance ranges by
elaborately manipulation of the TD-PS parameters. Then a wideband beam
training method has been proposed based on the new discovery and we further
improve the performance for off-grid users. The simulation results have
confirmed the efficacy of the proposed method, which emerges as a promising
approach for achieving efficient beam training in XL-MIMO systems. The paper
also presents the first attempt to fully unleash the capability of near-field
beam split. Future research could concentrate on extending the distance-
dependent beam-split to uniform planar array configurations. In addition,
investigation of the low-hardware-complexity TD-PS structure with a small
number of TDs [36] is also left for future research.

## Appendix

###  VI-A Proof of Lemma 1

Consider a beam focused on the location $(\theta,\alpha)$ at frequency $f$. To
evaluate the beam coverage in the angle domain, we can assume that
$\alpha=\bar{\alpha}$. Then, the achieved beamforming gain on the location
$(\bar{\theta},\bar{\alpha})$ is expressed as

$G(k(\theta-\bar{\theta}),0)=\left\lvert\frac{\sin\frac{N_{t}\pi(\theta-\bar{\theta})f}{2f_{c}}}{N_{t}\sin\frac{\pi(\theta-\bar{\theta})f}{2f_{c}}}\right\rvert\overset{(a)}{\approx}\left\lvert\frac{\sin\frac{N_{t}\pi(\theta-\bar{\theta})f}{2f_{c}}}{\frac{N_{t}\pi(\theta-\bar{\theta})f}{2f_{c}}}\right\rvert,$
(50)

where (a) holds because $N_{t}$ is very large. In this case, $3~{}{\rm
dB}$-beam width in the angle domain can be numerically solved from
$|G(k(\theta-\bar{\theta}),0)|=\frac{1}{\sqrt{2}}$:

$|\theta-\bar{\theta}|\approx\frac{0.88f_{c}}{N_{t}f}.$ (51)

To guarantee the entire coverage in the angle domain according to the
reliability criterion, the angle difference of $\theta_{m}$ and $\theta_{m+1}$
should be smaller than the sum of the $3~{}{\rm dB}$-beam width of the beam at
the $m$-th and $m+1$-th subcarrier, i.e.,

$|\theta_{m+1}-\theta_{m}|\leqslant\eta_{\theta}^{f_{m}}+\eta_{\theta}^{f_{m+1}}=\frac{0.88}{N_{t}}(\frac{f_{c}}{f_{m+1}}+\frac{f_{c}}{f_{m}}),$
(52)

where $\eta_{\theta}^{f}$ is the angle-domain $3~{}{\rm dB}$-beam width of the
beam at frequency $f$. Combining the above equation and (26) we obtain

$|f_{c}(\theta_{p}^{\prime}+2p_{m})(\frac{1}{f_{m+1}}-\frac{1}{f_{m}})|\leq\frac{0.88}{N_{t}}(\frac{f_{c}}{f_{m+1}}+\frac{f_{c}}{f_{m}}),$
(53)

which gives rise to the results of (27) in Lemma 1.

###  VI-B Derivation of the distance-domain 3 dB-beam width

In (14), when $\theta=\bar{\theta}$, the array gain on an arbitrary physical
distance $\bar{\alpha}$ is

$\displaystyle G(0,k(\alpha-\bar{\alpha}))$
$\displaystyle=\frac{1}{N_{t}}\left\lvert\sum_{n=-N}^{N}e^{-jn^{2}d^{2}k(\alpha-\bar{\alpha})}\right\rvert=F(\beta)$
(54)

where $F(\beta)\overset{\Delta}{=}|C(\beta)+jS(\beta)|/\beta$, and
$C(\beta)=\int_{0}^{\beta}\cos(\frac{\pi}{2}t^{2}){\rm d}t$ and
$S(\beta)=\int_{0}^{\beta}\sin(\frac{\pi}{2}t^{2}){\rm d}t$ are Fresnel
functions. The variable $\beta$ is written as
$\beta=\sqrt{\frac{N_{t}^{2}d^{2}}{2\lambda}\left(\alpha-\bar{\alpha}\right)}$
[12]. For $3~{}{\rm dB}$-beam width, it can be numerically solved from
$|F(\beta)|=\frac{1}{\sqrt{2}}$ that $\beta\approx 1.318$. Thus, the distance-
domain $3~{}{\rm dB}$-beam width at frequency $f$ can be denoted as

$\eta_{\alpha}^{f}=|\alpha-\bar{\alpha}|=\frac{\beta^{2}\lambda}{N_{t}^{2}d^{2}}=\frac{4\beta^{2}f_{c}^{2}}{N_{t}^{2}cf}.$
(55)

###  VI-C Proof of Proposition 1

We use the product the angle-domain array gain in (50) and distance-domain
array gain in (54) to approximate the array gain (14) on an arbitrary physical
location $(\bar{\theta},\bar{\alpha})$ as

$G(k(\theta-\bar{\theta}),k(\alpha-\bar{\alpha}))\approx\Xi_{N_{t}}\left(\frac{(\theta-\bar{\theta})f}{f_{c}}\right)F(\beta).$
(56)

Based on Taylor expansion, if $\bar{\theta},\bar{\alpha}$ is within $3~{}{\rm
dB}$-beam width of mainlobe (i.e. $\theta-\bar{\theta}\rightarrow
0,\alpha-\bar{\alpha}\rightarrow 0$), two parts of $G$ can be written as

$\displaystyle\Xi_{N_{t}}\left(\frac{(\theta-\bar{\theta})f}{f_{c}}\right)$
$\displaystyle\approx
1-\frac{1}{24}N_{t}^{2}\pi^{2}\frac{f^{2}}{f_{c}}(\theta-\bar{\theta})^{2}$
(57) $\displaystyle F(\beta)$ $\displaystyle\approx
1-\frac{\pi^{2}}{90}\frac{N_{t}^{4}d^{4}}{\lambda^{2}}(\alpha-\bar{\alpha})^{2}.$
(58)

Combining (56)-(58) and neglecting the high-order small numbers, we arrive at
(43).

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Microresonator-enhanced, Waveguide-coupled Emission from Silicon Defect
Centers for Superconducting Optoelectronic Networks

A. N. Tait∗, S. M. Buckley, A. N. McCaughan, J. T. Chiles,

S. Nam, R. P. Mirin and J. M. Shainline,

National Institute of Standards and Technology, Boulder, CO 80305, USA
<EMAIL_ADDRESS>
###### Abstract

Superconducting optoelectronic networks could achieve scales unmatched in
hardware-based neuromorphic computing. After summarizing recent progress in
this area, we report new results in cryogenic silicon photonic light sources,
components central to these architectures.

##  1 Superconducting optoelectronic neural networks

Artificial neural networks are capable of a wide variety of tasks in control,
simulation, and recognition. Specialized CMOS hardware has been developed to
vastly reduce the power consumption of neuromorphic computations as compared
to conventional, centralized architectures. CMOS neuromorphic hardware can
outperform in certain application areas, but they are then primarily limited
by interconnect performance. Extending the application areas of hardware
neuromorphics will require new physical platforms, such as those based on
photonics, analog electronics, or Josephson junctions. Superconducting
optoelectronic neural networks (SOENs) offer a route to hardware neuromorphic
systems with the largest scales considered to date [1]. In the SOEN
architecture, light is used for communication in order to avoid
bandwidth/distance/dissipation tradeoffs affecting both analog and digital
electronic interconnects. Signaling at the lowest possible light levels is
achieved using single-photon detectors. Since these detectors must operate at
cryogenic temperatures, other unusual physical phenomena can be employed. In
particular, at these temperatures, crystal defect centers in silicon can emit
light. W centers are ideal candidates because they emit at 1220 nm, a
wavelength that can be guided by silicon-on-insulator (SOI) waveguides, and
because they can be fabricated with a simple ion implantation. Cryogenic
silicon photonic platforms to support SOENs are nascent but rapidly
developing.

Fig. 1: Recent results on Superconducting Optoelectronic Networks (SOENs). a)
Latest conception of the SOEN neuron, adapted from [2]. “T” is the optical
transmitter consisting of a silicon LED. b) Part of a SOEN neuron: a
thresholding, amplification, transmission, and detection drive train, adapted
from [3]. c) Demonstration of the drive chain in (b). d) Optimization of
photoluminescent brightness of W-center emitters in silicon-on-insulator
substrates, adapted from [4].

Fig. 1 represents selected results from the past 6 months of SOEN research.
The latest conception of the neuron signal pathway [2] is shown in Fig. 1(a).
Incoming photons are detected by superconducting nanowire single-photon
detectors (SNSPDs). Induced currents are then summed and integrated in a
magnetic flux loop. The resulting singal is thresholded and amplified to
ultimately drive a W-center-based silicon light emitting diode (LED): the
transmitter (T). A key part of this drive train was demonstrated in the
circuit of Fig. 1(b) from [3]. The circuit consisted of a high-impedance
superconducting switch driving a silicon LED whose output is, in turn,
detected in an SNSPD. The essential role of the switch is to transduce a
superconductor-compatible (50 mV) pulse into a semiconductor-compatible (1 V)
pulse that then drives the LED. The LED-to-SNSPD optical link was first
demonstrated in [5]. Fig. 1(d) shows that, when 50 mV is applied to the
switch, photon clicks are detected at the end of the link with a mean interval
of 4.6 $\mu$s.

The low heat dissipation density promised by SOENs depends strongly on a
mastery of silicon photonic light sources: increasing brightness, radiative
efficiency, waveguide coupling, etc. The properties of silicon emissive
defects in photonics-compatible SOI have not been studied or optimized until
recently. Figure 1(d) shows an optimization of implant parameters performed in
[4]. Unlike in bulk Si (blue triangles), W-center brightness in SOI is
strongly dependent on implant energy. Compared to prior work on waveguide-
coupling, which used an 80 keV implant [5, 3], photoluminescence (PL)
brightness has been increased by two orders-of-magnitude.

##  2 Silicon photoluminescence coupled to a waveguide and microdisk resonator

In this work, we demonstrate coupling of optimized W centers to a silicon
photonic circuit. The circuit – consisting of a microdisk resonator,
waveguide, and grating coupler – enables spatial differentiation of free-space
and waveguide-coupled emission. We show that waveguide-coupled emission is
significantly enhanced at wavelengths corresponding to the modes of the
resonator. Substantial research has been devoted to SOI-compatible emitters
using III-V semiconductors, germanium, and rare-earths; however these each
complicate fabrication with, respectively, epitaxy or wafer bonding, quantum
wells, and Al2O3 optics. W centers are created simply by damaging the silicon
crystal though silicon ion bombardment and then annealing – both of these
steps are CMOS-compatible at the wafer scale. Embedding the emitters directly
in the crystal presents a great simplification for applications below 40 K.
W-center emission into a silicon-on-insulator waveguide has been observed
before in an electrically injected LED [5]. While electrical pumping is the
eventual direction of our research, the optically pumped setup presents
several advantages for research: 1) many devices can be probed in an array,
meaning that the physics of the emitters can be better studied and device
parameters better optimized; 2) photoluminescent properties can be studied
independent of electrical factors (e.g. ohmic heating); 3) fabrication is
2-step (Si rib, Si+ implant) instead of 7-step (Si rib, Si pedestal, Si+
implant, heavy/light boron, heavy/light phosphorus). Coupling between W
centers and resonator modes has not been shown before, although copper-based
centers have been coupled to resonances of a suspended photonic crystal in
[6]. Coupling resonance-enhanced silicon emission to a photonic integrated
circuit has not been shown with any defect center.

The substrate used is a 76.2 mm SOI wafer with 220 nm silicon device layer on
2 $\mu$m buried oxide. The wafer is patterned and implanted with 28Si+ at an
energy of 25 keV and fluence of 5$\times 10^{12}$/cm2. The first mask is
stripped. The wafer is patterned again and fully etched through the silicon
device layer. Finally, the whole wafer is encapulated by a 1.5 $\mu$m PECVD
oxide. More detail on fabrication can be found in [4]. Waveguides are designed
to be 350 nm wide such that they are single-mode at 1220 nm. Cutback tests
indicate a single-mode waveguide loss of 18.9 dB/cm, which is significantly
higher than other runs, we expect due to a new etch recipe.

Fig. 2: Setup and circuit. a) Simplified experimental setup for distinguishing
waveguide-coupled photoluminescence (PL). b) Visible image of the circuit. c)
Infrared image of PL from the circuit when the microdisk is pumped. Some PL
exits directly into free-space. Other PL is waveguide-coupled to the GC. d)
Zoomed in IR image of the microdisk.

We introduce an offset collection technique that allows for observation of
waveguide-coupled PL. In the silicon photonic circuit, shown in Fig. 2(b), a
microdisk resonator with 10 $\mu$m radius is implanted with W centers. It is
coupled to a waveguide with a coupling gap of 100 nm. The waveguide is routed
to a grating coupler (GC), designed to emit normal to the sample. When the
microdisk is pumped, it emits upward into free-space with a spectrum similar
to that of unpatterned SOI. Waveguide-coupled light, on the other hand, is
routed to the GC. Since this GC is offset from the free-space emission source,
the waveguide-coupled component is spatially discernable.

A simplified experimental setup is shown in Fig. 2(a). An 11 mW laser pump at
635 nm (red) is focused onto a device with W centers. The long working
distance objective has a numerical aperture of 0.42. Radiative recombination
in W centers exceeds non-radiative pathways only below 45 K, so the sample is
placed in an optical cryostat. Infrared PL around 1218 nm (depicted as blue)
returns from the pumped microdisk itself and from the offset GC. When the
returning light is imaged with a lens, the two emission sources are spatially
separated upon reaching a slit entering the spectrometer. Rotating a mirror
before the spectrometer allows selection between centered and offset
collection. We verify this spatial resolving technique with an InGaAs camera,
images from which are shown in Fig. 2(c, d). The perimeter of the disk appears
brighter due to scattering of the microdisk modes off of the sidewalls.
Additional scattering occurs at the coupling region leading to a brighter spot
on the left of the disk.

Fig. 3: Comparison of spectra when collecting from directly over the top of
the pumped microdisk (red), the offset GC (blue), and an unpatterned control
region (black). The zero-phonon line amplitude and upper bound of full-width
half maximum (FWHM) are indicated.

Results indicate profound differences in the PL spectra of direct collection
from the microdisk center (i.e., free-space coupled) and offset collection
from the GC (i.e. microresonator- and waveguide-coupled). Fig. 3 compares
these with a control that is a large unpatterned, implanted region. The zero-
phonon line (ZPL) at 1218 nm is brighter over the center of the microdisk than
unpatterned due to scattering off of the device sidewalls. Otherwise the
microdisk and control are similar. Collecting from the GC yields several
high-Q peaks corresponding to the modes of the microdisk. That means
waveguide-coupled emission is enhanced on resonance, even compared to the
brighter controls. The ZPL from the GC is weak because the cavity is not on
resonance. The quality factor of some peaks was measured up to 7,160, which is
a lower bound as limited by the resolution of the spectrometer. Amplitudes are
affected by the efficiency of the GC, but this was not characterized at the
temperatures used. Within the same cooldown, stadard deviation is estimated to
be 10%, like in [4].

The demonstration of coupling W-center emission to a silicon photonic circuit
represents progress in light sources for silicon photonics. Due to their ease
of fabrication, they could play a significant role in large-scale photonic
systems. Further work could also investigate Purcell enhancement, nonlinear
phenomena, optical amplifiers, or electrical pumping.

This is a contribution of NIST, an agency of the U.S. government, not subject
to copyright.

## References

  * [1] J. M. Shainline _et al._ , Phys. Rev. Applied 7, 034013 (2017).
  * [2] J. M. Shainline _et al._ , Journal of Applied Physics 126, 044902 (2019).
  * [3] A. N. McCaughan _et al._ , Nature Electronics 2, 451–456 (2019).
  * [4] S. M. Buckley _et al._ , arXiv eprint, arXiv:1911.01317 (2019).
  * [5] S. Buckley _et al._ , Applied Physics Letters 111, 141101 (2017).
  * [6] H. Sumikura _et al._ , Scientific Reports 4, 5040 (2014).