# Near Field Communications for
DMA-NOMA Networks

Zheng Zhang, , Yuanwei Liu, ,
Zhaolin Wang, , Jian Chen, , and Dong In Kim Zheng Zhang and Jian Chen are
with the School of Telecommunications Engineering, Xidian University, Xi’an
710071, China (e-mail<EMAIL_ADDRESS>jianchen@mail.xidian.edu.cn).Yuanwei Liu and Zhaolin Wang are with the School
of Electronic Engineering and Computer Science, Queen Mary University of
London, London E1 4NS, U.K. (e-mail<EMAIL_ADDRESS>zhaolin.wang@qmul.ac.uk;).Dong In Kim is with the Department of Electrical and
Computer Engineering, Sungkyunkwan University, Suwon 16419, South Korea
(e-mail: dikim@skku.ac.kr).

###### Abstract

A novel near-field transmission framework is proposed for dynamic metasurface
antenna (DMA)-enabled non-orthogonal multiple access (NOMA) networks. The base
station (BS) exploits the hybrid beamforming to communicate with multiple near
users (NUs) and far users (FUs) using the NOMA principle. Based on this
framework, two novel beamforming schemes are proposed. 1) For the case of the
grouped users distributed in the same direction, a beam-steering scheme is
developed. The metric of beam pattern error (BPE) is introduced for the
characterization of the gap between the hybrid beamformers and the desired
ideal beamformers, where a two-layer algorithm is proposed to minimize BPE by
optimizing hybrid beamformers. Then, the optimal power allocation strategy is
obtained to maximize the sum achievable rate of the network. 2) For the case
of users randomly distributed, a beam-splitting scheme is proposed, where two
sub-beamformers are extracted from the single beamformer to serve different
users in the same group. An alternating optimization (AO) algorithm is
proposed for hybrid beamformer optimization, and the optimal power allocation
is also derived. Numerical results validate that: 1) the proposed beamforming
schemes exhibit superior performance compared with the existing imperfect-
resolution-based beamforming scheme; 2) the communication rate of the proposed
transmission framework is sensitive to the imperfect distance knowledge of NUs
but not to that of FUs.

###### Index Terms:

Beamforming optimization, NOMA, near-field communications.

##  I Introduction

Fuelled by the explosive growth of ubiquitous wireless communications and
various intelligent applications, such as extended reality (XR), auto-driving,
and Internet-of-Everything (IoE), the development of the next generation
multiple access (NGMA) techniques for future wireless networks becomes
imminent [1, 2, 3]. To enable flexible and reliable access to the network for
a massive amount of users, each generation of multiple-access technique is
committed to exploiting the new multiplexing domain schemes, such as the
frequency-domain-based first-generation (1G) frequency division multiple
access (FDMA), time-domain-based second-generation (2G) time division multiple
access (TDMA) [4], code-domain-based third-generation (3G) code division
multiple access (CDMA) [5], and orthogonal-subcarrier-based forth-generation
(4G) orthogonal frequency division multiple access (OFDMA) [6]. However, due
to the limited spectrum resources, the aforementioned orthogonal multiple
access (OMA) schemes are struggling to accommodate massive wireless
connectivity. To deal with this challenge, the power-domain multiplexing-based
non-orthogonal multiple access (NOMA) technique has drawn extensive attention
in recent years [7]. By leveraging the superposition coding (SC) and
successive interference cancellation (SIC) at the transmitters and receivers
respectively, NOMA allows to serve multiple users within the same spectrum
resource. The superiority of NOMA technology lies not only in its higher
spectrum and energy efficiency gains compared to OMA schemes [8], but also in
its compatibility, which can be flexibly integrated into existing OMA
communication systems. Especially in recent research, it has been claimed that
NOMA can be utilized as an add-on to the conventional space division multiple
access (SDMA) further to enhance the connectivity and spectral efficiency of
multi-antenna networks [9, 10].

Courtesy of the rapid development of metamaterials, a new antenna paradigm,
namely dynamic metasurface antenna (DMA), has been proposed. Depending on the
Lorentz resonance response characteristics of each element, it can be
classified into two categories, i.e., amplitude-control DMA versus Lorentzian-
constrained phase-shift control DMA [11]. For ease of hardware implementation,
the amplitude-control DMA (also referred to as reconfigurable holographic
surface [12]) is considered in this paper. To elaborate, the DMA utilizes
reference electromagnetic (EM) waves generated by feeds, which propagate along
a metasurface inscribed with the beam pattern and radiate from a radiating
element into free space [13]. By recording the interference between the
reference wave and the desired wave, the amplitude of the reference wave can
be precisely controlled to generate the reconfigurable beam pattern, thus
realizing the spatial beamformer [14]. Compared to the conventional phased
array antennas, DMA does not rely on the active amplifier and the phase-shift
circuits, thus having lower energy consumption and hardware implementation
cost. As an emerging antenna solution for wireless communications, a few works
have been devoted to the beam pattern design in DMA-aided wireless networks
[15, 16, 17, 18]. Specifically, the authors of [15] proposed a DMA-enabled
downlink multi-user transmission framework, where a hybrid beamforming design
was devised to realize accurate multi-beam-steering. Followed by this, the
authors of [15] proposed a new multiple access scheme, namely holographic-
pattern division multiple access (HDMA), which was demonstrated to exhibit a
higher network capacity than the conventional SDMA. In [17], a DMA-empowered
holographic radar was developed for target sensing, which consumed less power
than the phased array-based radar under the same sensing accuracy requirement.
Moreover, the authors of [18] proposed to exploit the multi-band DMA for user
positioning, where the federated learning (FL) framework was adopted to
improve sensing adaptability while guaranteeing privacy.

Inspired by the advantages of DMA, it is natural to focus on the investigation
of DMA-enabled multi-user networks from the multiple access perspective, where
the power-domain-based NOMA is considered to be integrated into DMA
transmission to further improve spectral efficiency. Nevertheless, the low
hardware cost benefit of DMA also results in the fact that antenna arrays tend
to be extremely large in DMA networks, e.g., hundreds or even thousands of
antennas could be deployed at the base station (BS). Such an enlargement of
the array antenna scale leads to a fundamental propagation characteristic
shift of the EM wave, which might change the communication range from the far-
field EM radiated region to the near-field EM radiated region. Specifically,
in far-field regions, the EM propagation can be approximated as the planar
wave, where the planar-wave-based linear phase response should be adopted. In
near-field regions, the EM propagation follows the more complex spherical
wave, where the spherical-wave-based non-linear phase response with respect to
both angle and distance knowledge is required to characterize the signal
propagation [19]. Compared with the far-field channels, the near-field channel
model introduces an extra distance dimension knowledge to favor wireless
transmission design [20, 21, 22]. To elaborate, the authors of [20] proposed
to exploit the distance information to achieve the signal power focusing on
the desired location of free space, (referred as to beamfocusing), which
reduced the leakage of beam energy at uninterested locations and improved
spectral efficiency. The authors of [21] revealed the distance-domain secrecy
gain brought by the near-field channels. In the work [22], the coupled angle
and distance information implicit in the spherical-wave channels was used to
achieve simultaneous angle and distance sensing. More recently, several
preliminary studies have been devoted to exploring the possibilities of NOMA
in near-field transmission by using it as an add-on to SDMA [24, 9, 23, 10].
In particular, in the work [10], the authors unveiled that the near-field
beamfocusing resolution is always imperfect even though the number of antennas
at the BS end tends to infinity, which indicates that any single near-field
beamfocusing beamformer leaks power to other users as well, which provides
theoretical backing for the application of NOMA in the near field.

###  I-A Motivations and Contributions

Although there have been some preliminary studies oriented towards the design
of SDMA/HDMA-based DMA multi-user transmission [15, 16, 25], research on
exploiting NOMA in DMA networks is in its infancy. Actually, since active
modules at the BS, e.g., radio frequency (RF) chains and digital baseband
processing modules, still face expensive hardware costs (especially in the
millimeter wave or terahertz bands), the active module cannot achieve a one-
to-one antenna match (i.e., the fully-digital architecture) in practice, which
limits the capacity of DMA to serve multiple users. Generally, the number of
users that the BS can support in the spatial domain is constrained by the
number of RF chains. Fortunately, integrating NOMA into the SDMA technique
provides a new solution for serving more users with limited RF chains. On the
other hand, the spherical-wave-based near-field channels caused by the large-
scale array of the DMA also pose new design challenges for applying NOMA in
DMA networks. To elaborate, distance-domain knowledge contained in near-field
channels introduces a new beam characteristic, i.e., beamfocusing [20], which
implies that the near-field beam width is narrower than that of the far-field
beam. In particular, even the users located in the same direction cannot be
covered by a single beam as in the far-field case. Although it has been
rigorously proved that beam focussing is unlikely to be of perfect resolution
[24, 9, 23, 10], the fact that far-field users receive much less power than
near-field users makes it difficult to ensure fairness in NOMA transmissions.

Against the above discussion, this paper focuses on an overloaded
communication scenario (with much more communication users than RF chains,),
and proposes a DMA-enabled near-field NOMA framework. Our goal is to maximize
network capacity while guaranteeing fairness between near and far users
through dedicated near-field beamformer design. The main contributions of this
work are summarized below.

  * •

We propose a DMA-enabled near-field NOMA communication framework, where a BS
exploits the hybrid DMA architecture to send signals to multiple near users
(NUs) and far users (FUs) in a NOMA principle. Considering the limitation of
the number of RF chains, each NU is associated with an FU to form a NOMA
group, where a shared hybrid beamformer is designed for each group. Based on
the designed beamformers, a power allocation optimization problem is
formulated to maximize the sum achievable rate under the QoS requirement and
the SIC decoding constraint.

  * •

We first consider a special user location topology in which two users
belonging to the same group are located in the same direction but at different
distances. A beam-steering beamforming scheme is proposed, which aims to
generate large beam-depth beamformers to radiate the same signal power at the
locations of the NUs and FUs for fairness. Specifically, we introduce the
metric of the beam pattern error (BPE) to evaluate the gap between the
practical beamformers and the desired ideal beamformers. A BPE minimization
problem is formulated. Then, we propose a two-layer algorithm to iteratively
optimize the hybrid beamformers. On this basis, the optimal power allocation
strategy is derived to enhance the spectral efficiency of the network.

  * •

We further consider a general case with randomly distributed users, where a
beam-splitting beamforming scheme is proposed. To elaborate, we decompose the
original beamformer into two sub-beamformers, which are used to serve the
different users within the same group, respectively. To guarantee user
fairness, a minimum channel gain maximization is maximized under the amplitude
constraint of the DMA elements and inter-group interference limitation. To
this end, an alternating optimization (AO) algorithm is proposed to optimize
the hybrid beamformers. Then, the optimal power allocation is obtained for
sum-rate maximization.

  * •

Simulation results verify the convergence of the proposed algorithms. It is
also found that: 1) the proposed beam-steering and -splitting schemes
outperform the existing imperfect-resolution-based single beam scheme in near-
field NOMA transmission; 2) the communication performance of the proposed DMA-
enabled near-field NOMA transmission framework is sensitive to the imperfect
distance knowledge of NU, but virtually unaffected by that of the FU.

###  I-B Organization and Notations

The remainder of this paper is as follows. Section II introduces the network
setup and signal model. Section III proposes a two-layer algorithm for the
beam-steering beamformer design. Section IV conceives an AO algorithm for the
beam-splitting beamformer design. We provide the numerical results in Section
V. The conclusion is drawn in Section VI.

Notations: The scalar, vector, and matrix are represented by the lower-case
letter, boldface lower-case letter, and boldface capital, respectively. The
transpose and Hermitian conjugate operations of matrix $\mathbf{X}$ are
denoted by $\mathbf{X}^{T}$ and $\mathbf{X}^{H}$. The $i$-th row and $j$-th
column element of the matrix $\mathbf{X}$ is denoted by $\mathbf{X}^{[i,j]}$.
The Euclidean norm of the vector $\mathbf{x}$ is denoted by $\|\mathbf{x}\|$.
The circularly symmetric complex Gaussian (CSCG) distributed random variable
with zero mean and covariance matrix $a$ is denoted by
$x\sim\mathcal{CN}(0,a)$. $\text{Tr}(\mathbf{X})$, $\text{rank}(\mathbf{X})$,
and $(\mathbf{X})^{-1}$ denote the trace, rank and inverse-matrix operations.
$\mathbf{X}\succeq\mathbf{0}$ represents that $\mathbf{X}$ is a semi-definite
matrix. $\Re({\cdot})$ denotes the real component of the corresponding complex
value. $\jmath$ denotes the unit imaginary number.

##  II System Model

###  II-A Network Description

Figure 1: Hybrid DMA-enabled near-field NOMA communications.

In this paper, we consider a downlink multi-user network, where a BS
communicates with $2K$ users by utilizing the radiation pattern of the DMA.
The DMA is equipped with $M_{\text{t}}=M_{\text{t},v}M_{\text{t},h}$ elements,
where $M_{\text{t},v}$ and $M_{\text{t},h}$ denote the number of elements
located in the vertical and horizontal directions of the DMA, respectively. As
shown in Fig. 1, two categories of users are considered in the network, where
$K$ near users (denoted by $\\{\text{NU}_{1},\cdots,\text{NU}_{K}\\}$) are
located in the vicinity of the BS while the far $K$ users (denoted by
$\\{\text{FU}_{1},\cdots,\text{FU}_{K}\\}$) lie relatively far away from the
BS. All the users are assumed to be single-antenna nodes, each of which only
requires a single data stream from the BS. Under the extremely large-scale DMA
setup, we assume all the users are located in the Fresnel (near-field) region
of the BS, which implies that the distances between the users and the BS are
shorter than the Rayleigh distance $\frac{2D^{2}}{\lambda}$, with $D$
representing the aperture of the BS.

This paper focuses on a connectivity-overloaded network. To elaborate,
$M_{\text{RF}}$ ($K\leq M_{\text{RF}}<2K$) RF chains are integrated into the
BS to connect with $M_{\text{RF}}$ independent feeds of the DMA, each of which
is capable of generating EM waves as the incident signals. For simplification,
we assume that $M_{\text{RF}}=K$ in the following, i.e., the BS allows the
generation of up to $K$ independent digital beams to serve different users. To
further increase the network connectivity with the limited hardware overhead
of the RF chains, the NOMA technique is exploited to serve more users with the
same spatial DoF. In particular, $2K$ users are clustered as $K$ NOMA groups,
each of which consists of one NU and one FU. By employing the superposition
coding (SC) at the BS and the successive interference cancellation (SIC) at
users, the NOMA protocol is adopted within each group for multiple data
transmission.

###  II-B DMA Architecture

Unlike the conventional phase-array-based antennas, DMA is a category of
planar antenna, which avoids the deployment of the power amplifiers and phase
shifters, thus resulting in low energy costs. From the hardware architecture
perspective, DMA is generally composed of three parts, i.e., feeds,
waveguides, and metamaterial radiation elements. Specifically, the feeds are
deployed at the bottom layer of the DMA, which are responsible for receiving
the incident signals up-converted by the RF chains and generating the
reference EM wave. To guide the wave propagation, the waveguide is distributed
along the DMA surface, which serves as the propagation medium of the reference
EM waves and radiates reference EM waves into free space. On the top layer of
the DMA, the metamaterial radiation elements are mounted, which can
intelligently control the radiation pattern of the reference EM waves by
altering its EM response at each element.

The construct of the beamformer at the DMA relies on the physical interference
principle of the EM wave. To elaborate, by recording the interference pattern
between the desired wave and the reference EM wave, the DMA can generate the
radiation pattern that orientates towards the direction of interest. Let
$\mathbf{x}_{v,h}$ denotes the location vector of the $v$-th row and the
$h$-th column element, the reference wave activated by the feed $k$ with
respect to the $v$-th row and the $h$-th column element is given by

$\displaystyle\Gamma_{\text{r}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{s}})=e^{-\jmath\mathbf{r}_{\text{s}}\mathbf{x}_{v,h}^{k}},$
(1)

where $\mathbf{r}_{\text{s}}$ denotes the propagation vector of the reference
EM wave and $\mathbf{x}_{v,h}^{k}$ denotes the location information vector
between the feed $k$ and the $v$-th row and the $h$-th column element.
Similarly, the objective wave with respect to the $v$-th row and the $h$-th
column element is given by

$\displaystyle\Gamma_{\text{o}}(\mathbf{x}_{v,h},\mathbf{r}_{\text{f}})=e^{-\jmath\mathbf{r}_{\text{f}}\mathbf{x}_{v,h}}.$
(2)

Here, $\mathbf{r}_{\text{f}}$ denotes the propagation vector from the origin
of the coordinate system to the objective location $(\phi,\varphi,r)$, in
which $\phi$, $\varphi$, and $r$ denote the corresponding azimuth angle,
elevation angle, and the distance information of the objective position.
Therefore, the interference pattern between
$\Gamma_{\text{r}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{s}})$ and
$\Gamma_{\text{o}}(\mathbf{x}_{v,h},\mathbf{r}_{\text{f}})$ can be expressed
as

$\displaystyle\Gamma_{\text{i}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{f}})=\Gamma_{\text{r}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{s}})\Gamma_{\text{o}}^{*}(\mathbf{x}_{v,h},\mathbf{r}_{\text{f}}),$
(3)

where it is readily verified that the radiation pattern excited by the
reference EM wave orientates the direction of the objective position, i.e.,
$\Gamma_{\text{i}}(\mathbf{x}_{v,h},\mathbf{r}_{\text{f}})\Gamma_{\text{r}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{s}})\propto\Gamma_{\text{o}}(\mathbf{x}_{v,h},\mathbf{r}_{\text{f}})C_{v,h}^{k}$
($C_{v,h}^{k}$ is a constant equals
$C_{v,h}^{k}=|\Gamma_{\text{r}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{s}})|^{2}$).
To adjust the interference radiation pattern, an amplitude-variation-based
controlling approach is adopted at the DMA, where the normalized radiation
amplitude for the feed $k$ at the $v$-th row and the $h$-th column element is
given by

$\displaystyle\mathbf{W}^{[(v-1)h+h,k]}=\frac{\Re{(\Gamma_{\text{i}}(\mathbf{x}_{v,h}^{k},\mathbf{r}_{\text{f}})})+1}{2}e^{-\jmath\mathbf{r}_{\text{s}}\mathbf{x}_{v,h}^{k}}.$
(4)

Based on the above, we consider a hybrid DMA-assisted multiuser transmission
architecture, where the baseband digital module emits signals intended for
multiple users through the baseband digital beamformer. Then, $K$ data streams
are sent to the feeds of the DMA via $K$ RF chains in parallel, which are
further manipulated by the beamformer
$\mathbf{W}\in\mathbb{C}^{M_{\text{t}}\times K}$ for signal broadcasting.

###  II-C Near-Field Channel Model

Figure 2: Near-field channel model.

As all the users are located in the near-field region of the network, the
accurate spherical-wave channel model is required. Note that as the high-
frequency (e.g., millimeter wave or THz) channel is generally predominated by
the LoS component, we only consider the LoS channel in this paper. As depicted
in Fig. 2, consider a three-dimensional (3D) coordinate system, with the DMA
being located in the y-o-z plane. With the assumption that the central element
of the DMA is located at the origin of the coordinate system, the $v$-th row
and the $h$-th column element is located in the coordinate of
$\mathbf{x}_{v,h}=(0,\tilde{v}d,\tilde{h}d)$, where
$\tilde{v}=v-\frac{M_{\text{t},v}+1}{2}$,
$\tilde{h}=h-\frac{M_{\text{t},h}+1}{2}$, $d=\frac{\lambda}{2}$ denotes the
inter-distance between the adjacent elements, and $\lambda$ denotes the
wavelength of carrier wave. Thus, the Euclidean distance between the the
$v$-th row and the $h$-th column element of the DMA and the
$\text{NU}_{i}$/$\text{FU}_{i}$ is given by

$\displaystyle\|\mathbf{x}_{v,h}-\mathbf{s}_{i}^{\varsigma}\|=\big{[}(r_{i}^{\varsigma})^{2}+\tilde{v}^{2}d^{2}+\tilde{h}^{2}d^{2}-2r_{i}^{\varsigma}$
$\displaystyle\qquad\tilde{v}d\sin\phi_{i}^{\varsigma}\sin\varphi_{i}^{\varsigma}-2r_{i}^{\varsigma}\tilde{h}d\cos\varphi_{i}^{\varsigma}\big{]}^{\frac{1}{2}},\
\varsigma\in\\{\text{N},\text{F}\\},$ (5)

where
$\mathbf{s}_{i}^{\varsigma}=(r_{i}^{\varsigma}\cos\phi_{i}^{\varsigma}\sin\varphi_{i}^{\varsigma},r_{i}^{\varsigma}\sin\phi_{i}^{\varsigma}\sin\varphi_{i}^{\varsigma},r_{i}^{\varsigma}\cos\varphi_{i}^{\varsigma})$
denotes the coordinate of the user. Here, the distance from the origin of the
coordinate system to the $\text{NU}_{i}$/$\text{FU}_{i}$ is denoted by
$r_{i}^{\varsigma}$, the azimuth and elevation angles are represented as
$\phi_{i}^{\varsigma}$ and $\varphi_{i}^{\varsigma}$, respectively.
Accordingly, the array response vector from the BS to the
$\text{NU}_{i}$/$\text{FU}_{i}$ can be expressed as

$\displaystyle\mathbf{a}^{\varsigma}_{i}=\big{[}e^{-\jmath\frac{2\pi}{\lambda}(\|\mathbf{x}_{1,1}-\mathbf{s}_{i}^{\varsigma}\|)},\cdots,e^{-\jmath\frac{2\pi}{\lambda}(\|\mathbf{x}_{M_{\text{t},\text{v}},M_{\text{t},\text{h}}}-\mathbf{s}_{i}^{\varsigma}\|)}\big{]}^{T}.$
(6)

Thus, the channel between the BS and the $\text{NU}_{i}$/$\text{FU}_{i}$ can
be expressed as
$\mathbf{h}^{\varsigma}_{\text{LoS},i}=\beta_{i}e^{-\jmath\frac{2\pi}{\lambda}r_{i}^{\varsigma}}\mathbf{a}^{\varsigma}_{i}$,
where $\beta_{i}^{\varsigma}$ denotes the complex gain defined in [26].
Furthermore, we assume that the perfect CSI of near-field users is known at
the BS.

###  II-D Signal Model for NOMA Transmission

To enhance the connectivity of the DMA-enabled network, a NOMA-empowered
transmission scheme is proposed. To elaborate, we consider cluster $2K$ users
as $K$ NOMA pairs, where each NOMA pair is associated with one RF chain and
consists of a NU and a FU. Within a NOMA pair, the joint design of the hybrid
beamformers and the power allocation are performed, where the hybrid
beamformers are required to be deliberately generated to align the
superimposed NOMA signals to the locations of users. For simplification, we
assume that the $i$-th NOMA group is composed of $\text{NU}_{i}$ and
$\text{FU}_{i}$ ($1\leq i\leq K$). Accordingly, the emitted signals from the
BS can be expressed by

$\displaystyle\mathbf{x}=\sum_{i=1}^{K}\mathbf{W}\mathbf{v}_{i}\left(\sqrt{P_{1,i}}s_{i}^{\text{N}}+\sqrt{P_{2,i}}s_{i}^{\text{F}}\right),$
(7)

where $\mathbf{v}_{i}$ is the baseband digital beamforming vector allocated to
$i$-th NOMA group, $s_{i}^{\text{N}}$ and $s_{i}^{\text{F}}$ denote the
signals intended for the $\text{NU}_{i}$ and $\text{FU}_{i}$, respectively,
with satisfying
$\mathbb{E}\\{|s_{i}^{\text{N}}|^{2}\\}=\mathbb{E}\\{|s_{i}^{\text{F}}|^{2}\\}=1$.
Note that $P_{1,i}$ and $P_{2,i}$ are the transmit power allocated to the
$\text{NU}_{i}$ and $\text{FU}_{i}$. Thus, the received signals at the
$\text{NU}_{i}$ and $\text{FU}_{i}$ are given by

$\displaystyle\mathbf{y}_{i}^{\text{N}}=$
$\displaystyle\underbrace{\mathbf{h}^{\text{N}}_{i}\mathbf{W}\mathbf{v}_{i}\sqrt{P_{1,i}}s_{i}^{\text{N}}}_{\text{desired
signal}}+\underbrace{\mathbf{h}^{\text{N}}_{i}\mathbf{W}\mathbf{v}_{i}\sqrt{P_{2,i}}s_{i}^{\text{F}}}_{\text{intra-
group interference}}+$
$\displaystyle\underbrace{\mathbf{h}^{\text{N}}_{i}\sum_{t=1,t\neq
i}^{K}\mathbf{W}\mathbf{v}_{t}\left(\sqrt{P_{1,t}}s_{t}^{\text{N}}+\sqrt{P_{2,t}}s_{t}^{\text{F}}\right)}_{\text{inter-
group interference}}+n_{i}^{\text{N}},$ (8)
$\displaystyle\mathbf{y}_{i}^{\text{F}}=$
$\displaystyle\underbrace{\mathbf{h}^{\text{F}}_{i}\mathbf{W}\mathbf{v}_{i}\sqrt{P_{2,i}}s_{i}^{\text{F}}}_{\text{desired
signal}}+\underbrace{\mathbf{h}^{\text{F}}_{i}\mathbf{W}\mathbf{v}_{i}\sqrt{P_{1,i}}s_{i}^{\text{N}}}_{\text{intra-
group interference}}+$
$\displaystyle\underbrace{\mathbf{h}^{\text{F}}_{i}\sum_{t=1,t\neq
i}^{K}\mathbf{W}\mathbf{v}_{t}(\sqrt{P_{1,t}}s_{t}^{\text{N}}+\sqrt{P_{2,t}}s_{t}^{\text{F}})}_{\text{inter-
group interference}}+n_{i}^{\text{F}}.$ (9)

Note that $n_{i}^{\text{N}},n_{i}^{\text{F}}\sim\mathcal{CN}(0,\sigma^{2})$
denote the additive white Gaussian noise (AWGN) at the $\text{NU}_{i}$ and
$\text{FU}_{i}$, respectively. On receiving the superimposed signals, the SIC
technique is employed at each NOMA group. To elaborate, the user with a strong
channel condition first decodes the signal of the weak-channel user, and
removes it from the received signal observation. Then, the strong-channel user
decodes its own signal without suffering intra-group interference. For the
user with a weak channel condition, it directly decodes its own signal by
treating the strong-channel user signal as the intra-group interference. For
our considered network, the NU is naturally treated as the strong-channel user
and the FU is the weak-channel user. Therefore, the received signal-to-
interference-plus-noise ratio (SINR) at $\text{NU}_{i}$ and $\text{FU}_{i}$ is
given by

$\displaystyle\gamma_{\text{N}_{i}\rightarrow\text{N}_{i}}=\frac{P_{1,i}|(\mathbf{h}^{\text{N}}_{i})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}}{I_{i,\text{inter}}^{\text{N}}+\sigma^{2}},$
(10)
$\displaystyle\gamma_{\text{F}_{i}\rightarrow\text{F}_{i}}=\frac{P_{2,i}|(\mathbf{h}_{i}^{\text{F}})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}}{P_{1,i}|(\mathbf{h}_{i}^{\text{F}})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}+I_{i,\text{inter}}^{\text{F}}+\sigma^{2}},$
(11)

where $I_{i,\text{inter}}^{\text{N}}=\sum_{t=1,t\neq
i}^{K}(P_{1,t}|(\mathbf{h}^{\text{N}}_{i})^{H}\mathbf{W}\mathbf{v}_{t}|^{2}+P_{2,t}|(\mathbf{h}^{\text{N}}_{i})^{H}\mathbf{W}\\\
\mathbf{v}_{t}|^{2})$ and $I_{i,\text{inter}}^{\text{F}}=\sum_{t=1,t\neq
i}^{K}(P_{1,t}|(\mathbf{h}^{\text{F}}_{i})^{H}\mathbf{W}\mathbf{v}_{t}|^{2}+P_{2,t}|(\mathbf{h}^{\text{F}}_{i})^{H}\\\
\mathbf{W}\mathbf{v}_{t}|^{2})$. To ensure the successful SIC decoding
procedure, it is also required that the achievable rate of $\text{NU}_{i}$ to
decode $s_{i}^{\text{F}}$ should be no less than the achievable rate of
$\text{FU}_{i}$ to decode its own signal, i.e.,
$\log_{2}(1+\gamma_{\text{N}_{i}\rightarrow\text{F}_{i}})\geq\log_{2}(1+\gamma_{\text{F}_{i}\rightarrow\text{F}_{i}})$,
where $\gamma_{\text{N}_{i}\rightarrow\text{F}_{i}}$ is given by

$\displaystyle\gamma_{\text{N}_{i}\rightarrow\text{F}_{i}}=\frac{P_{2,i}|(\mathbf{h}^{\text{N}}_{i})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}}{P_{1,i}|(\mathbf{h}_{i}^{\text{N}})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}+I_{i,\text{inter}}^{\text{N}}+\sigma^{2}}.$
(12)

##  III beam-steering beamformer Design

In this section, we consider a special user location topology, i.e., each FU
is located in the same direction as its paired NU. For this scenario, we
propose a beam-steering-based hybrid beamforming scheme, which simultaneously
aligns with the NU and the FU in one NOMA group using the large beam-depth
beamformer (see Fig. 3). Then, an optimal power allocation algorithm is
proposed to maximize the network spectral efficiency.

Figure 3: beam-steering scheme for near-field NOMA transmission.

###  III-A Large Beam-Depth Beamformer Design

To restrict the beam-steering characteristics of the designed beamformer, we
introduce a new performance metric, namely BPE, which aims to characterize the
error degree of the designed beamformer compared with the required beam
pattern [27]. Based on the near-field array response vector
$\mathbf{a}(\phi_{i},\varphi_{i},r_{i})$, the BPE for the beamformer
$\\{\mathbf{v}_{i},\mathbf{W}\\}$ is defined by

$\displaystyle\chi(\mathbf{v}_{i},\mathbf{W})$
$\displaystyle\triangleq\int_{r\in[r^{\text{N}}_{i},r^{\text{F}}_{i}]}\left|t-\left|\mathbf{a}(\phi_{i},\varphi_{i},r)^{H}\mathbf{W}\mathbf{v}_{i}\right|\right|^{2}dr+$
$\displaystyle\int_{r\in[0,r^{\text{N}}_{i}]\cup[r^{\text{F}}_{i},\infty]}\left|\mathbf{a}(\phi_{i},\varphi_{i},r)^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2}dr+$
$\displaystyle\iiint\limits_{\begin{subarray}{c}{\phi\in[-\frac{\pi}{2},\phi_{i})\cup(\phi_{i},\frac{\pi}{2}],}\\\
{\varphi\in[-\frac{\pi}{2},\varphi_{i})\cup(\varphi_{i},\frac{\pi}{2}]}\\\
{r\in[0,\infty]}\end{subarray}}\\!\\!\\!\\!\left|\mathbf{a}(\phi,\varphi,r)^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2}d\phi
d\varphi dr,$ (13)

where $t$ denotes the desired strength of the ideal beam-steering vector. With
the above definition, the beam-steering beamformer design problem can be
formulated as

$\displaystyle(\text{P}1)\ \min\limits_{\mathbf{W},\mathbf{v}_{i}}\quad$
$\displaystyle\sum_{i=1}^{K}\chi(\mathbf{v}_{i},\mathbf{W})$ (14a) s.t.
$\displaystyle\|\mathbf{v}_{i}\|^{2}=1,\ 1\leq i\leq K,$ (14b)
$\displaystyle\mathbf{W}^{[m,n]}\in[0,1],\ 1\leq m\leq M_{\text{t}},\ 1\leq
n\leq K,$ (14c)
$\displaystyle|(\mathbf{a}_{j}^{\varsigma})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}\leq\epsilon,\
i\neq j,$ (14d)

where (14b) denotes the normalized power constraint of the transmit baseband
digital beamformer, (14c) represents the amplitude-control constraint at the
DMA, and (14d) is to limit inter-group interference below a negligible level
$\epsilon\rightarrow 0$. Note that the problem (P1) is intractable to solve
from the perspective of convex optimization due to the integral operations and
infinite integral interval. To facilitate the solving of the problem, we
consider transforming the problem (P1) into a discrete form. Specifically, we
first limit the distance interval from $[0,\infty]$ to a finite interval
$[0,r_{\text{max}}]$. Then, the continuous BPE function are quantized into $Q$
discrete values with $Q_{1}$ azimuth angle samples, $Q_{2}$ elevation angle
samples, and $Q_{3}$ distance samples ($Q_{1}Q_{2}Q_{3}=Q$), where the $q$-th
discrete location information is given by

$\displaystyle\begin{cases}\phi_{q_{1}}=-\frac{\pi}{2}+\frac{(q_{1}-1)\pi}{Q_{1}-1},\
q_{1}\in\\{1,\cdots,Q_{1}\\},\\\
\varphi_{q_{2}}=-\frac{\pi}{2}+\frac{(q_{2}-1)\pi}{Q_{2}-1},\
q_{2}\in\\{1,\cdots,Q_{2}\\},\\\
r_{q_{3}}=\frac{(q_{3}-1)r_{\text{max}}}{Q_{3}-1},\
q_{3}\in\\{1,\cdots,Q_{3}\\},\\\ \end{cases},$ (15)

where there exists a unique mapping rule between the scalar $q$ and the vector
$[q_{1},q_{2},q_{3}]$, i.e., $q=q_{3}+(q_{2}-1)Q_{3}+(q_{1}-1)Q_{3}Q_{2}$.
Thus, we approximate the BPE function for $\\{\mathbf{v}_{i},\mathbf{W}\\}$ as

$\displaystyle\chi(\mathbf{v}_{i},\mathbf{W})\overset{(a)}{\approx}$
$\displaystyle\chi_{1}(\mathbf{v}_{i},\mathbf{W})$ $\displaystyle=$
$\displaystyle\sum_{r_{q_{3}}\in[r^{\text{N}}_{i},r^{\text{F}}_{i}]}\left|t-\left|\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|\right|^{2}\Delta
r+$
$\displaystyle\sum_{r_{q_{3}}\in[0,r^{\text{N}}_{i}]\cup[r^{\text{F}}_{i},r_{\text{max}}]}\\!\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2}\Delta
r+$
$\displaystyle\\!\\!\\!\sum_{\begin{subarray}{c}{\phi_{q_{1}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\phi_{q_{1}}\neq\phi_{i}}\\\
{\varphi_{q_{2}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\varphi_{q_{2}}\neq\varphi_{i}}\\\
{r_{q_{3}}\in[0,r_{\text{max}}]}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{a}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2}\Delta\phi\Delta\varphi\Delta
r,$ (16)

where the approximate equality sign $a$ strictly takes equality when
$\Delta\phi\rightarrow 0$, $\Delta\varphi\rightarrow 0$, and $\Delta
r\rightarrow 0$ satisfy. Accordingly, we can convert the objective function
$\chi_{1}(\mathbf{v}_{i},\mathbf{W})$ as the form as follows.

$\displaystyle\chi_{2}(\mathbf{v}_{i},\mathbf{W})$
$\displaystyle=\frac{\chi_{1}(\mathbf{v}_{i},\mathbf{W})}{\Delta r}$
$\displaystyle=\sum_{r_{q_{3}}\in[r^{\text{N}}_{i},r^{\text{F}}_{i}]}\left|t-\left|\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|\right|^{2}$
$\displaystyle+\sum_{r_{q_{3}}\in[0,r^{\text{N}}_{i}]\cup[r^{\text{F}}_{i},r_{\text{max}}]}\left|\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2}+$
$\displaystyle\sum_{\begin{subarray}{c}{\phi_{q_{1}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\phi_{q_{1}}\neq\phi_{i}}\\\
{\varphi_{q_{2}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\varphi_{q_{2}}\neq\varphi_{i}}\\\
{r_{q_{3}}\in[0,r_{\text{max}}]}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{\bar{a}}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}})^{H}\mathbf{W}\mathbf{v}_{i}\right|^{2},$
(17)

where
$\mathbf{\bar{a}}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}})=\sqrt{\Delta\phi\Delta\varphi\mathbf{a}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}})}$.
To deal with the coupling between $\mathbf{v}_{i}$ and $\mathbf{W}$, we
introduce an auxiliary variable $\mathbf{\bar{v}}_{i}$ with satisfying
$\mathbf{\bar{v}}_{i}=\mathbf{W}\mathbf{v}_{i}$. We consider adopting the
penalty-based optimization framework, where the equality constraint
$\mathbf{\bar{v}}_{i}=\mathbf{v}_{i}\mathbf{W}$ is moved to the objective
function as a penalty term. Hence, the problem (P2) is transformed into

$\displaystyle(\text{P}2)\
\min\limits_{\mathbf{W},\mathbf{v}_{i},\mathbf{\bar{v}}_{i}}\quad$
$\displaystyle\sum_{i=1}^{K}\chi_{2}(\mathbf{\bar{v}}_{i})+\frac{1}{2\rho}\sum_{i=1}^{K}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$
(18a) s.t. $\displaystyle\eqref{P1b},\eqref{P1c},\eqref{P1d},$ (18b)

where the penalty term
$\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}\rightarrow 0$ when
$\rho\rightarrow 0$. To handle the double modulus operation in
$\chi_{2}(\mathbf{\bar{v}}_{i})$, we introduce [27, Lemma 1], which
equivalently converts $\chi_{2}(\mathbf{\bar{v}}_{i})$ as following tractable
form

$\displaystyle\chi_{2}(\mathbf{\bar{v}}_{i})=\chi_{3}(\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i})=\\!\\!\\!\sum_{r_{q_{3}}\in[r^{\text{N}}_{i},r^{\text{F}}_{i}]}\\!\\!\\!\left|te^{\jmath\vartheta_{q_{3}}^{i}}-\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{\bar{v}}_{i}\right|^{2}$
$\displaystyle+\\!\\!\\!\\!\\!\\!\\!\sum_{r_{q_{3}}\in[0,r^{\text{N}}_{i}]\cup\atop[r^{\text{F}}_{i},r_{\text{max}}]}\\!\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{a}(\phi_{i},\varphi_{i},r_{q_{3}})^{H}\mathbf{\bar{v}}_{i}\right|^{2}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}{\phi_{q_{1}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\phi_{q_{1}}\neq\phi_{i}}\\\
{\varphi_{q_{2}}\in[-\frac{\pi}{2},\frac{\pi}{2}],\varphi_{q_{2}}\neq\varphi_{i}}\\\
{r_{q_{3}}\in[0,r_{\text{max}}]}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\mathbf{\bar{a}}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}})^{H}\mathbf{\bar{v}}_{i}\right|^{2}.$
(19)

With these transformations, we reformulate the optimization problem (P2) as

$\displaystyle(\text{P}3)\
\min\limits_{\mathbf{W},\mathbf{v}_{i},\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i}}\quad$
$\displaystyle\sum_{i=1}^{K}\chi_{3}(\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i})+\frac{1}{2\rho}\sum_{i=1}^{K}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$
(20a) s.t. $\displaystyle\eqref{P1b},\eqref{P1c},\eqref{P1d}.$ (20b)

Notably, the optimization variables are separated in the constraints, which
motivates us to employ the two-layer optimization framework to iteratively
solve the problem (P3), where the BCD method is adopted in the inner layer for
variable optimization and $\rho$ is updated in the outer layer.

####  III-A1 Inner layer: subproblem with respect to
$\\{\mathbf{\bar{v}}_{i}\\}$

With the fixed $\\{\mathbf{W},\mathbf{v}_{i},\vartheta_{q_{3}}^{i}\\}$, we can
observe that $\mathbf{\bar{v}}_{i}$ and $\mathbf{\bar{v}}_{j}$ ($i\neq j$) are
fully independent in the objective function (20a), which indicates that the
problem (P3) can be transformed into $K$ independent subproblems without loss
of equivalence. The $i$-th subproblem is given by

$\displaystyle(\text{P}4\text{-}1)\
\min\limits_{\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i}}\quad$
$\displaystyle\chi_{3}(\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i})+\frac{1}{2\rho}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$
(21a) s.t.
$\displaystyle|(\mathbf{a}_{j}^{\varsigma})^{H}\mathbf{\bar{v}}_{i}|^{2}\leq\epsilon,\
i\neq j,$ (21b)

To facilitate solving the unconstrained optimization problem (P4-1), we define
a new constant matrix, which consists of $Q$ array response vectors, i.e.,
$\mathbf{A}\triangleq[\mathbf{\bar{a}}(\phi_{1},\varphi_{1},r_{1}),\cdots,\mathbf{a}(\phi_{q_{1}},\varphi_{q_{2}},r_{q_{3}}),\cdots,\mathbf{\bar{a}}(\phi_{Q_{1}},\varphi_{Q_{2}},r_{Q_{3}})]$.
With the arbitrary $\vartheta_{q_{3}}^{i}$, the problem (P4-1) with respect to
$\mathbf{\bar{v}}_{i}$ can be formulated as

$\displaystyle(\text{P}4\text{-}2)\ \min\limits_{\mathbf{\bar{v}}_{i}}\quad$
$\displaystyle\|\mathbf{t}_{i}-\mathbf{A}\mathbf{\bar{v}}_{i}\|^{2}+\frac{1}{2\rho}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$
(22a) s.t. $\displaystyle\eqref{P4-1b},$ (22b)

where the $q$-th element of $\mathbf{t}_{i}$ is given by

$\displaystyle\mathbf{t}_{i}^{[q]}=\begin{cases}te^{\jmath\vartheta_{q_{3}}^{i}},\
&\text{if}\ \phi_{q_{1}}=\phi_{i},\ \varphi_{q_{2}}=\varphi_{i},\
r_{q_{3}}\in[r^{\text{N}}_{i},r^{\text{F}}_{i}],\\\
0&\text{otherwise},\end{cases}$ (23)

Problem (P4-2) is a convex programming, where the optimal
$\mathbf{\bar{v}}_{i}$ can be directly obtained via the standard convex
solver, such as CVX.

####  III-A2 Inner layer: subproblem with respect to
$\\{\vartheta_{q_{3}}^{i}\\}$

Substituting the optimized result of $\mathbf{\bar{v}}_{i}$ into the problem
(P4-2), we have

$\displaystyle(\text{P}4\text{-}3)\ $
$\displaystyle\min\limits_{\vartheta_{q_{3}}^{i}}\quad\mathbf{t}_{i}^{H}\mathbf{t}_{i}-2\Re\left\\{\mathbf{t}_{i}^{H}\mathbf{A}^{H}\mathbf{\bar{v}}_{i}\right\\}$
(24a) $\displaystyle\text{s.t.}\quad\eqref{dec_vec}.$ (24b)

Note that due to the existence of the equality constraint (23), the optimal
$\vartheta_{q_{3}}^{i}$ cannot be directly derived through the first-order
optimality condition. Instead, we can observe $\mathbf{t}_{i}$ is a sparse
vector, which motivates us to reformulate the problem (P4-3) by shortening the
$\mathbf{t}_{i}$ without the loss of equivalence. Specifically, let
$\mathbf{\bar{t}}_{i}$ denote the subvector of $\mathbf{t}_{i}$ consisting of
all the non-zero elements, it can be expressed as

$\displaystyle\mathbf{\bar{t}}_{i}=\left[\mathbf{t}_{i}^{[q_{i}^{-}]},\mathbf{t}_{i}^{[q_{i}^{-}+1]},\cdots,\mathbf{t}_{i}^{[q_{i}^{+}-1]},\mathbf{t}_{i}^{[q_{i}^{+}]}\right],$
(25)

where $q_{i}^{-}$ and $q_{i}^{+}$ denote the indexes of the first and last
non-zero elements of $\mathbf{t}_{i}$. Thus, we rewrite the objective (24a) as

$\displaystyle\eqref{P4-3a}=\mathbf{\bar{t}}_{i}^{H}\mathbf{\bar{B}}_{i}\mathbf{\bar{t}}_{i}-2\Re\left\\{\mathbf{\bar{t}}_{i}^{H}\mathbf{\bar{c}}_{i}\right\\},$
(26)

where $\mathbf{B}=\mathbf{I}$ and
$\mathbf{c}_{i}=\mathbf{A}^{H}\mathbf{\bar{v}}_{i}$. Here, we extract a sub-
matrix $\mathbf{\bar{B}}_{i}$ from $\mathbf{B}$, which contains the entries
whose column and row index range from $q_{i}^{-}$ to $q_{i}^{+}$. Similarly,
$\mathbf{\bar{c}}_{i}$ denotes a sub-vector of $\mathbf{c}_{i}$ with the
element index ranging from $q_{i}^{-}$ to $q_{i}^{+}$. To proceed, the problem
(P4-3) can be reformulated as the semidefinite relaxation (SDR) form

$\displaystyle(\text{P}4\text{-}4)\ \min\limits_{\mathbf{\tilde{T}}_{i}}\quad$
$\displaystyle\text{Tr}(\mathbf{\bar{T}}_{i}\mathbf{D}_{i})$ (27a) s.t.
$\displaystyle\mathbf{\tilde{T}}_{i}^{[j,j]}=t^{2},\
j\in\\{1,\cdots,q_{i}^{+}-q_{i}^{-}+1\\},$ (27b)
$\displaystyle\text{rank}(\mathbf{\tilde{T}}_{i})=1,$ (27c)
$\displaystyle\mathbf{\tilde{T}}_{i}\succeq\mathbf{0},$ (27d)

where $\mathbf{D}_{i}$ and $\mathbf{\tilde{T}}_{i}$ are given by

$\displaystyle\mathbf{D}_{i}=\begin{bmatrix}\mathbf{\bar{B}}_{i}&-\mathbf{\bar{c}}_{i}\\\
-\mathbf{\bar{c}}_{i}^{H}&0\\\
\end{bmatrix},\quad\mathbf{\tilde{T}}_{i}=\begin{bmatrix}\,\mathbf{\bar{t}}_{i}\\\
1\\\ \end{bmatrix}\cdot[\,\mathbf{\bar{t}}_{i}^{H},1].$ (28)

By removing the rank-one constraint (27c), the problem is degenerated into a
standard semidefinite program (SDP), which can be optimally solved by the CVX.
Then, the rank-one solution $[\,\mathbf{\bar{t}}_{i}^{H},1]$ can be obtained
by the Gaussian randomization procedure with at-least
$\frac{\pi}{4}$-approximation accuracy [28].

####  III-A3 Inner layer: subproblem with respect to $\\{\mathbf{v}_{i}\\}$

With the fixed $\\{\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i},\mathbf{W}\\}$,
the problem (P3) can be divided into $K$ subproblems, where the $i$-th
subproblem with respect to $\\{\mathbf{v}_{i}\\}$ is given by

$\displaystyle(\text{P}5\text{-}1)\ \min\limits_{\mathbf{v}_{i}}\quad$
$\displaystyle\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$ (29a)
s.t. $\displaystyle\eqref{P1b},\eqref{P1d}.$ (29b)

To tackle the non-convex constraint (14b), we rewrite the problem (P5-1) as
the SDR form

$\displaystyle(\text{P}5\text{-}2)\ \min\limits_{\mathbf{\tilde{V}}_{i}}\quad$
$\displaystyle\text{Tr}(\mathbf{\tilde{V}}_{i}\mathbf{E}_{i})$ (30a) s.t.
$\displaystyle\sum_{j=1}^{K}\mathbf{\tilde{V}}_{i}^{[j,j]}=1,$ (30b)
$\displaystyle\mathbf{\tilde{V}}_{i}\succeq\mathbf{0},$ (30c)
$\displaystyle\text{rank}(\mathbf{\tilde{V}}_{i})=1,$ (30d)

where
$\mathbf{E}_{i}=\begin{bmatrix}\mathbf{W}^{H}\mathbf{W}&-\mathbf{W}^{H}\mathbf{\bar{v}}_{i}\\\
-\mathbf{\bar{v}}_{i}^{H}\mathbf{W}&0\\\ \end{bmatrix}$,
$\mathbf{\tilde{V}}_{i}=\begin{bmatrix}\,\mathbf{v}_{i}\\\ 1\\\
\end{bmatrix}\cdot[\,\mathbf{v}_{i}^{H},1]$, and $\mathbf{V}_{i}$ denotes the
submatrix of $\mathbf{\tilde{V}}_{i}$ that consisting of the former $K$-rows
and -columns elements. By ignoring the rank-one solution, the optimal
$\mathbf{\tilde{V}}_{i}$ with general rank can be directly obtained by solving
the SDP problem. Then, we utilize the Gaussian randomization procedure to
extract the rank-one solution from the high-rank $\mathbf{\tilde{V}}_{i}$.

####  III-A4 Inner layer: subproblem with respect to $\\{\mathbf{W}\\}$

With the fixed
$\\{\mathbf{\bar{v}}_{i},\vartheta_{q_{3}}^{i},\mathbf{v}_{i}\\}$, the problem
(P3) is converted to

$\displaystyle(\text{P}6)\ \min\limits_{\mathbf{v}_{i}}\quad$
$\displaystyle\sum_{i=1}^{K}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$
(31a) s.t. $\displaystyle\eqref{P1c},$ (31b)

which is a convex problem and can be directly solved.

####  III-A5 Outer layer: penalty factor update

When the inner layer iteration converges, we update $\rho$ for obtaining the
feasible solution of the original problem (P1), which is updated by

$\displaystyle\rho=\frac{\rho}{\tilde{c}},$ (32)

where $\tilde{c}>1$ denotes the constant update coefficient.

The specific algorithm details are summarized in the Algorithm 1. Let
$f(\mathbf{W}^{l},\mathbf{v}_{i}^{l},\mathbf{\bar{v}}_{i}^{l},(\vartheta_{q_{3}}^{i})^{l})$
denote the objective value at the $l$-th inner-layer iteration, it must hold
that

$\displaystyle
f(\mathbf{W}^{l},\mathbf{v}_{i}^{l},\mathbf{\bar{v}}_{i}^{l},(\vartheta_{q_{3}}^{i})^{l})\overset{a}{\geq}f(\mathbf{W}^{l},\mathbf{v}_{i}^{l},\mathbf{\bar{v}}_{i}^{l+1},(\vartheta_{q_{3}}^{i})^{l+1})\overset{b}{\geq}$
$\displaystyle\\!\\!\\!f(\mathbf{W}^{l},\mathbf{v}_{i}^{l+1},\mathbf{\bar{v}}_{i}^{l+1},(\vartheta_{q_{3}}^{i})^{l+1})\overset{c}{\geq}f(\mathbf{W}^{l+1},\mathbf{v}_{i}^{l+1},\mathbf{\bar{v}}_{i}^{l+1},(\vartheta_{q_{3}}^{i})^{l+1}),$
(33)

where the inequality holds as the suboptimal or optimal solutions are
guaranteed at steps 4-7. Thus, the proposed two-layer algorithm remains mono-
increased over the inner-layer iterations. For the outer layer, when the
penalty factor approaches $0$, the equality constraint
$\mathbf{\bar{v}}_{i}=\mathbf{W}\mathbf{v}_{i}$ can be satisfied and the
feasible solutions of the problem (P1) can be returned.

The complexity of Algorithm 1 is generated by the steps 4 to 7. Specifically,
in step 4, we update the optimal $\mathbf{\bar{v}}_{i}$ by solving the second-
order cone programming (SOCP) program, which causes the complexity of
$\mathcal{O}\big{(}(M_{\text{t}})^{3.5}\big{)}$. In steps 5 and 6, we solve
the standard SDP problem to obtain $\vartheta_{q_{3}}^{i}$ and
$\mathbf{v}_{i}$, which suffers the complexity of
$\mathcal{O}\big{(}(q_{+}-q_{-}+1)^{3.5}\big{)}$ and
$\mathcal{O}\big{(}(K+1)^{3.5}\big{)}$, respectively. The problem (P6) is a
second-order cone programming (SOCP) program, which can be optimally solved
with the complexity of $\mathcal{O}\big{(}(KM_{\text{t}})^{3.5}\big{)}$. Thus,
the whole computational complexity to design the beam-steering beamformers for
$K$ NOMA groups is given by
$\mathcal{O}\big{(}l_{\text{out}}l_{\text{inner}}(KM_{\text{t}}^{3.5}+K(q_{+}-q_{-}+1)^{3.5}+K(K+1)^{3.5}+(KM_{\text{t}})^{3.5})\big{)}$,
where $l_{\text{out}}$ and $l_{\text{inner}}$ denote the number of outer and
inner iterations.

Algorithm 1 Two-layer algorithm.

1: Initialize $\\{\mathbf{W},\mathbf{v}_{i}\\}$ and set the convergence
accuracy $\varepsilon_{1}$ and $\varepsilon_{2}$.

2: repeat

3: repeat

4: update $\mathbf{\bar{v}}_{i}$ $(1\leq i\leq K)$ by solving the problem
(P4-2).

5: update $\vartheta_{q_{3}}^{i}$ $(1\leq i\leq K)$ by solving the problem
(P4-4).

6: update $\mathbf{v}_{i}$ $(1\leq i\leq K)$ by solving the problem (P5-2).

7: update $\mathbf{W}$ by solving the problem (P6).

8: until the objective function converges with the accuracy $\varepsilon_{1}$.

9: update $\rho$ according to (32).

10: until the penalty term
$\sum_{i=1}^{K}\|\mathbf{\bar{v}}_{i}-\mathbf{W}\mathbf{v}_{i}\|^{2}$ falls
below $\varepsilon_{2}$.

###  III-B Optimal Power Allocation Strategy

It readily knows that the optimized large beam-depth beamformer
$\mathbf{W}\mathbf{v}_{i}$ radiates almost no power at the location of the
$j$-th ($j\neq i$) NOMA group. Thus, the inter-interference between different
NOMA groups is efficiently eliminated, which yields the following SINR/signal-
to-noise ratio (SNR) expressions, i.e.,

$\displaystyle\gamma_{\text{N}_{i}\rightarrow\text{N}_{i}}=\frac{P_{1,i}g^{\text{N}}_{i}}{\sigma^{2}},\quad\gamma_{\text{N}_{i}\rightarrow\text{F}_{i}}=\frac{P_{2,i}g^{\text{N}}_{i}}{P_{1,i}g^{\text{N}}_{i}+\sigma^{2}},$
(34)
$\displaystyle\gamma_{\text{F}_{i}\rightarrow\text{F}_{i}}=\frac{P_{2,i}g^{\text{F}}_{i}}{P_{1,i}g^{\text{F}}_{i}+\sigma^{2}}.$
(35)

Here,
$g^{\text{N}}_{i}=|(\mathbf{h}^{\text{N}}_{i})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}$
and
$g^{\text{F}}_{i}=|(\mathbf{h}_{i}^{\text{F}})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}$
denote the channel gains of $\text{NU}_{i}$ and $\text{FU}_{i}$, respectively.

Then, we aim to maximize the spectral efficiency of the network, where an
optimization of maximizing the sum achievable rate of the NU and FU is
formulated, subject to the constraints of total transmit power budget at the
BS, QoS constraint at the users, and SIC decoding constraint. It is given by

$\displaystyle(\text{P}7)\ \max\limits_{P_{q,i}}\quad$
$\displaystyle\sum_{i}\sum_{\varsigma\in\\{\text{N},\text{F}\\}}R_{\varsigma_{i}\rightarrow\varsigma_{i}}$
(36a) s.t.
$\displaystyle\sum\nolimits_{i=1}^{K}\sum\nolimits_{q=1}^{2}P_{q,i}\leq
P_{\text{max}},$ (36b) $\displaystyle
R_{\varsigma_{i}\rightarrow\varsigma_{i}}\geq R_{\text{QoS}}^{\varsigma,i},\
1\leq i\leq K,$ (36c)
$\displaystyle\gamma_{\text{N}_{i}\rightarrow\text{F}_{i}}\geq\gamma_{\text{F}_{i}\rightarrow\text{F}_{i}},\
1\leq i\leq K,$ (36d)

where
$R_{\varsigma_{i}\rightarrow\varsigma_{i}}=\log_{2}(1+\gamma_{\varsigma_{i}\rightarrow\varsigma_{i}})$,
(36b) limits the transmit power lower than the maximal transmit power budget
$P_{\text{max}}$; (36c) guarantees the achievable rate
$R_{\varsigma_{i}\rightarrow\varsigma_{i}}$ is no less than the QoS
requirement $R_{\text{QoS}}^{\varsigma,i}$; (36d) accounts for the successful
SIC decoding constraint; In the following, we consider deriving the optimal
power allocation strategy for the problem (P7).

To elaborate, from the expressions of (34) and (35), we can observe that the
users in each NOMA group can be regarded to perform the single-input single-
output (SISO) transmission, with the determined channel gains
$g^{\text{N}}_{i}$ and $g^{\text{F}}_{i}$. Thus, the SIC decoding constraint
(36d) is equivalent to $g^{\text{N}}_{i}\geq g^{\text{F}}_{i}$ [29]. According
to the definition of the BPE, the designed beamformers have the relatively
same radiated power at the location of the NU and FU in one group, whereas FU
suffers a higher large-scale path loss. It indicates $g^{\text{N}}_{i}\geq
g^{\text{F}}_{i}$ always holds for the considered network, i.e., the
constraint (36d) can be neglected in the problem (P7). Then, the problem (P7)
can be converted to

$\displaystyle(\text{P}8\text{-}1)\ \max\limits_{P_{q,i}}\quad$
$\displaystyle\sum_{i}\sum_{\varsigma\in\\{\text{N},\text{F}\\}}R_{\varsigma_{i}\rightarrow\varsigma_{i}}$
(37a) s.t. $\displaystyle\eqref{P7b},\eqref{P7c}.$ (37b)

To facilitate the optimization of the problem (P8-1), we consider combining
the constraint (36b) and (36c). Let $P_{i}^{\text{g}}$ denotes the practical
power allocated to the $i$-th NOMA group, we can obtain the feasible set of
$P_{i}^{\text{g}}$, i.e.,
$P_{i}^{\text{g}}\in[P_{\text{min},i},P_{\text{max},i}]$, where
$P_{\text{max},i}>0$ and $P_{\text{min},i}>0$ denote maximum and minimum power
allocated to the $i$-th NOMA group, respectively. From the constraint (36c),
it readily knows that the minimum power allocated to the $i$-th NOMA group
should at least guarantee the QoS requirement of each user, this means that

$\displaystyle
P_{1,i}=\frac{\gamma_{\text{QoS}}^{\text{N},i}\sigma^{2}}{g^{\text{N}}_{i}}$
(38) $\displaystyle
P_{2,i}=\frac{\gamma_{\text{QoS}}^{\text{F},i}\sigma^{2}}{g^{\text{F}}_{i}}+\gamma_{\text{QoS}}^{\text{F},i}P_{1,i},$
(39)

where $\gamma_{\text{QoS}}^{\varsigma,i}=2^{R_{\text{QoS}}^{\varsigma,i}}-1$.
Substituting (38) into (39), we can obtain

$\displaystyle
P_{\text{min},i}=\gamma_{\text{QoS}}^{\text{F},i}\sigma^{2}\left(\frac{1}{g^{\text{F}}_{i}}+\frac{\gamma_{\text{QoS}}^{\text{N},i}}{g^{\text{N}}_{i}}\right)+\frac{\gamma_{\text{QoS}}^{\text{N},i}\sigma^{2}}{g^{\text{N}}_{i}}.$
(40)

Due to the maximum transmit power constraint of
$\sum_{i=1}^{K}P_{i}^{\text{g}}\leq P_{\text{max}}$, it holds that

$\displaystyle P_{\text{max},i}=$ $\displaystyle
P_{\text{max}}-\sum_{t=1,t\neq i}^{K}P_{\text{min},t},$ $\displaystyle
P_{\text{max}}-\gamma_{\text{QoS}}^{\text{F},i}\sigma^{2}\left(\frac{1}{g^{\text{F}}_{i}}+\frac{\gamma_{\text{QoS}}^{\text{N},i}}{g^{\text{N}}_{i}}\right)-\frac{\gamma_{\text{QoS}}^{\text{N},i}\sigma^{2}}{g^{\text{N}}_{i}}.$
(41)

With the derived expressions above, we can equivalently convert the problem
(P8-1) to the following form with the feasible set as the intersection of
closed boxes [30]

$\displaystyle(\text{P}8\text{-}2)\
\max\limits_{P_{q,i},P_{i}^{\text{g}}}\quad$
$\displaystyle\sum_{i}\sum_{\varsigma\in\\{\text{N},\text{F}\\}}R_{\varsigma_{i}\rightarrow\varsigma_{i}}$
(42a) s.t. $\displaystyle\sum_{q=1}^{2}P_{q,i}=P_{i}^{\text{g}},\ 1\leq i\leq
K,$ (42b) $\displaystyle
P_{i}^{\text{g}}\in[P_{\text{min},i},P_{\text{max},i}],\ 1\leq i\leq K,$ (42c)
$\displaystyle\sum_{i=1}^{K}P_{i}^{\text{g}}\leq P_{\text{max}},$ (42d)

For the problem (P8-2), we propose a two-stage power allocation algorithm. The
optimal intra-group power allocation $\\{P_{q,i}\\}$ is derived under the
given $\\{P_{i}^{\text{g}}\\}$ in the first stage, and the optimal inter-group
power allocation $\\{P_{i}^{\text{g}}\\}$ is obtained via the bisection method
in the second stage.

####  III-B1 Problem with respect to $\\{P_{q,i}\\}$

Given any feasible $\\{P_{i}^{\text{g}}\\}$, the problem (P8-2) is converted
to

$\displaystyle(\text{P}8\text{-}3)\ \max\limits_{P_{q,i}}\quad$
$\displaystyle\sum_{i}\sum_{\varsigma\in\\{\text{N},\text{F}\\}}R_{\varsigma_{i}\rightarrow\varsigma_{i}}$
(43a) s.t. $\displaystyle\eqref{P8-2b}.$ (43b)

The objective function (43a) and the power constraint (42b) for different NOMA
groups are fully separated, which motivates us to divide the problem (P8-3)
into $K$ subproblems. In each subproblem, we only focus on the intra-group
power allocation for two-user NOMA transmission, where the $i$-th subproblem
is given by

$\displaystyle(\text{P}8\text{-}4)\ \max\limits_{P_{q,i}}\quad$
$\displaystyle\sum_{\varsigma\in\\{\text{N},\text{F}\\}}R_{\varsigma_{i}\rightarrow\varsigma_{i}}$
(44a) s.t. $\displaystyle\sum_{q=1}^{2}P_{q,i}=P_{i}^{\text{g}}.$ (44b)

Note that the problem (P8-4) is a typical sum rate maximization optimization
problem for a single-carrier SISO NOMA network, where the optimal power
allocation is to allocate the extra power to the best-channel user while
maintaining the QoS requirement of the other users. Thus, the optimal power
allocation for the $i$-th NOMA group can be determined as follows.

$\displaystyle
P_{2,i}=\frac{\gamma_{\text{QoS}}^{\text{F},i}(\sigma^{2}+P_{i}^{\text{g}}g^{\text{F}}_{i})}{g^{\text{F}}_{i}+\gamma_{\text{QoS}}^{\text{F},i}g^{\text{F}}_{i}},$
(45) $\displaystyle P_{1,i}=P_{i}^{\text{g}}-P_{2,i}.$ (46)

Algorithm 2 Bisection algorithm for optimal power allocation.

1: Initialize initial $\mu_{\text{lower}}$ and $\mu_{\text{upper}}$. Set a
convergence accuracy $\varepsilon_{3}$.

2: repeat

3: $\mu=\frac{\mu_{\text{lower}}+\mu_{\text{upper}}}{2}$.

4: update $\tilde{P}_{i}^{\text{g}}$ according to (51).

5: if $\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}\geq a_{i}P_{\text{max}}-b_{i}$

6: $\mu_{\text{lower}}=\mu$.

7: else

8: $\mu_{\text{upper}}=\mu$.

9: end

10: until the
$|\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}-a_{i}P_{\text{max}}+b_{i}|\leq\varepsilon_{3}$.

####  III-B2 Problem with respect to $\\{P_{i}^{\text{g}}\\}$

Substituting (45) and (46) into the problem (P8-2), then we can obtain

$\displaystyle(\text{P}8\text{-}5)\ \max\limits_{P_{i}^{\text{g}}}\ $
$\displaystyle\sum_{i=1}^{K}\log_{2}\Bigg{(}1+\frac{g^{\text{N}}_{i}}{\sigma^{2}}\Bigg{(}a_{i}P_{i}^{\text{g}}-b_{i}\Bigg{)}\Bigg{)}$
(47a) s.t. $\displaystyle\eqref{P8-2c},$ (47b)
$\displaystyle\sum_{i=1}^{K}P_{i}^{\text{g}}=P_{\text{max}},$ (47c)

where $a_{i}=\frac{1}{1+\gamma_{\text{QoS}}^{\text{F},i}}$ and
$b_{i}=\frac{\gamma_{\text{QoS}}^{\text{F},i}\sigma^{2}}{g^{\text{F}}_{i}+\gamma_{\text{QoS}}^{\text{F},i}g^{\text{F}}_{i}}$.
Note that we neglect the constant term
$\sum_{i=1}^{K}R_{\text{QoS}}^{\text{F},i}$ in the objective (47a) as it does
not affect the optimization of the solutions. The problem (P8-5) is a standard
convex problem with the affine power constraint, which can be optimally solved
by the Lagrange dual approach. Let
$\tilde{P}_{i}^{\text{g}}=a_{i}P_{i}^{\text{g}}-b_{i}$, the problem (P8-5) can
be reformulated as

$\displaystyle(\text{P}8\text{-}6)\ \max\limits_{\tilde{P}_{i}^{\text{g}}}\ $
$\displaystyle\sum_{i=1}^{K}\log_{2}\Bigg{(}1+\frac{g^{\text{N}}_{i}\tilde{P}_{i}^{\text{g}}}{\sigma^{2}}\Bigg{)}$
(48a) s.t.
$\displaystyle\tilde{P}_{i}^{\text{g}}\in[\tilde{P}_{\text{min},i},\tilde{P}_{\text{max},i}],\
1\leq i\leq K,$ (48b)
$\displaystyle\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}=a_{i}P_{\text{max}}-b_{i},$
(48c)

where $\tilde{P}_{\text{min},i}=a_{i}P_{\text{min},i}-b_{i}$ and
$\tilde{P}_{\text{max},i}=a_{i}P_{\text{max},i}-b_{i}$. The Lagrange dual
function of the problem (P8-6) is given by

$\displaystyle\mathcal{L}(\tilde{P}_{i}^{\text{g}},\mu)=$
$\displaystyle\sum_{i=1}^{K}\log_{2}\Bigg{(}1+\frac{g^{\text{N}}_{i}\tilde{P}_{i}^{\text{g}}}{\sigma^{2}}\Bigg{)}+\mu\Big{(}a_{i}P_{\text{max}}-b_{i}-$
$\displaystyle\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}\Big{)},$ (49)

where $\mu\geq 0$ denotes the Lagrange multiplier for the constraint (48c).
Reviewing the Karush-Kuhn-Tucker (KKT) conditions below

$\displaystyle\text{K1}:\quad$
$\displaystyle\frac{\partial\mathcal{L}(\tilde{P}_{i}^{\text{g}},\mu)}{\partial\tilde{P}_{i}^{\text{g}}}=0,\
\forall i,$ (50a) $\displaystyle\text{K2}:\quad$
$\displaystyle\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}=a_{i}P_{\text{max}}-b_{i},$
(50b) $\displaystyle\text{K3}:\quad$
$\displaystyle\tilde{P}_{i}^{\text{g}}\in[\tilde{P}_{\text{min},i},\tilde{P}_{\text{max},i}],\
1\leq i\leq K,$ (50c)

we can obtain the optimal solutions of $\\{\tilde{P}_{i}^{\text{g}}\\}$ via
the one-dimension search for the Lagrange multiplier $\mu$. In particular,
with any given $\mu$, the optimal inter-group power allocation is given by

$\displaystyle\tilde{P}_{i}^{\text{g}}=\begin{cases}\left(\frac{g^{\text{N}}_{i}}{\mu\sigma^{2}\ln
2}-1\right)\frac{\sigma^{2}}{g^{\text{N}}_{i}},\ \text{if}\
\tilde{P}_{i}^{\text{g}}\in[\tilde{P}_{\text{min},i},\tilde{P}_{\text{max},i}],\\\
0,\ \text{otherwise}.\end{cases}$ (51)

Here, we use the Bisection algorithm to search for the optimal $\mu$ with the
convergence condition of
$\sum_{i=1}^{K}\tilde{P}_{i}^{\text{g}}=a_{i}P_{\text{max}}-b_{i}$. The
detailed pseudo code is summarized in the Algorithm 2. As the intra-group
power allocation is derived in the closed-form expression, the main
computational complexity of optimal power allocation calculation relies on the
Bisection algorithm, which suffers a complexity of
$\mathcal{O}\Big{(}\log_{2}(\frac{\mu_{\text{upper}}-\mu_{\text{lower}}}{\varepsilon_{3}})\Big{)}$.

##  IV beam-splitting beamformer Scheme

Figure 4: beam-splitting scheme for near-field NOMA transmission.

In this section, we consider the random user location distribution. As shown
in Fig. 4, a beam-splitting-based hybrid beamformer strategy is proposed for
NOMA transmission, in which each individual user in a single NOMA group is
served by a sub-beamformer. Then, the optimal power allocation is derived for
the proposed beam-splitting scheme.

###  IV-A beam-splitting Beamformer Design

The idea of beam-splitting is to construct multiple sub-beamformers using one
hybrid beamformer, i.e.,
$\mathbf{W}\mathbf{v}_{i}=\mathbf{W}\mathbf{v}_{i,\text{N}}+\mathbf{W}\mathbf{v}_{i,\text{F}}$,
where the sub-beamformers $\mathbf{W}\mathbf{v}_{i,\text{N}}$ and
$\mathbf{W}\mathbf{v}_{i,\text{F}}$ are designed to serve the NU and FU of the
$i$-th group. The optimization problem can be formulated by

$\displaystyle(\text{P}9\text{-}1)\
\max\limits_{\mathbf{W},\mathbf{v}_{i,\text{N}},\mathbf{v}_{i,\text{F}}}\ $
$\displaystyle\sum_{i=1}^{K}\left(\min\limits_{\varsigma\in\\{\text{N},\text{F}\\}}|(\mathbf{a}_{i}^{\varsigma})^{H}\mathbf{W}\mathbf{v}_{i,\varsigma}|^{2}\right)$
(52a) s.t.
$\displaystyle\|\mathbf{v}_{i,\text{N}}+\mathbf{v}_{i,\text{F}}\|^{2}=1,\
\forall i,$ (52b) $\displaystyle\mathbf{W}^{[m,n]}\in[0,1],\ \forall m,n,$
(52c)
$\displaystyle|(\mathbf{a}_{i}^{\varsigma})^{H}\mathbf{W}\mathbf{v}_{j,\varsigma}|^{2}\leq\epsilon,\
i\neq j,$ (52d)

For fairness guarantee, we focus on a max-min objective in the formulated
problem (P9-1), which aims to radiate the same power on the locations of the
NU and the FU in the common group. However, the problem (P9-1) is intractable
to solve due to the coupled objective function (52a) and the equation
constraint (52b). In the following, we propose an AO algorithm to solve it.

####  IV-A1 Subproblem with respect to
$\\{\mathbf{v}_{i,\text{N}},\mathbf{v}_{i,\text{F}}\\}$

With the fixed $\mathbf{W}$, the problem (P9-1) is reduced to

$\displaystyle(\text{P}9\text{-}2)\
\max\limits_{\mathbf{v}_{i,\text{N}},\mathbf{v}_{i,\text{F}}}\ $
$\displaystyle\sum_{i=1}^{K}\left(\min\limits_{\varsigma\in\\{\text{N},\text{F}\\}}|(\mathbf{a}_{i}^{\varsigma})^{H}\mathbf{W}\mathbf{v}_{i,\varsigma}|^{2}\right)$
(53a) s.t. $\displaystyle\eqref{P9-1b},\eqref{P9-1d}.$ (53b)

From the problem (P9-2), it readily knows that $\mathbf{v}_{i,\varsigma}$ and
$\mathbf{v}_{j,\varsigma}$ ($i\neq j$) are uncoupled, which implies that the
problem (P9-2)can be reformulated as $K$ subproblems without loss of
equivalence. For the subproblem of the $i$-th NOMA group, the optimal
$\mathbf{v}_{j,\varsigma}$ that can maximize the minimum objective function
(53a) is derived by

$\displaystyle\mathbf{v}_{i,\varsigma}=\alpha_{i,\varsigma}\mathbf{W}^{H}\mathbf{a}_{i}^{\varsigma},$
(54)

where the unit-modules constraint is neglected in (53a). Note that
$\alpha_{i,\text{N}}$ and $\alpha_{i,\text{F}}$ are required to satisfy

$\displaystyle\alpha_{i,\text{N}}=\alpha_{i,\text{F}}\frac{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\|^{2}}{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}\|^{2}}.$
(55)

Recalling the unit-modules constraint, we also have
$\|\alpha_{i,\text{N}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}+\alpha_{i,\text{F}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\|=\left\|\alpha_{i,\text{F}}\frac{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\|^{2}}{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}\|^{2}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}+\alpha_{i,\text{F}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\right\|=1$
Thus, the optimal $\alpha_{\text{F}}$ is given by

$\displaystyle\alpha_{i,\text{F}}=\frac{1}{\left\|\alpha_{i,\text{F}}\frac{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\|^{2}}{\|\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}\|^{2}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{N}}+\alpha_{i,\text{F}}\mathbf{W}^{H}\mathbf{a}_{i}^{\text{F}}\right\|}.$
(56)

Substituting (56) into (55), we can obtain the optimal $\alpha_{i,\text{N}}$.

####  IV-A2 Subproblem with respect to $\\{\mathbf{W}\\}$

Under the given $\mathbf{v}_{i,\varsigma}$, the problem (P9-1) can be
transformed into

$\displaystyle(\text{P}9\text{-}3)\ \max\limits_{\mathbf{W}}\ $
$\displaystyle\sum_{i=1}^{K}\left(\min\limits_{\varsigma\in\\{\text{N},\text{F}\\}}\text{Tr}(\mathbf{A}_{i}^{\varsigma}\mathbf{W}\mathbf{V}_{i,\varsigma}\mathbf{W}^{H})\right)$
(57a) s.t. $\displaystyle\eqref{P9-1c},\eqref{P9-1d},$ (57b)

where
$\mathbf{A}_{i}^{\varsigma}=\mathbf{a}_{i}^{\varsigma}(\mathbf{a}_{i}^{\varsigma})^{H}$
and
$\mathbf{V}_{i,\varsigma}=\mathbf{v}_{i,\varsigma}\mathbf{v}_{i,\varsigma}^{H}$.
To tackle the non-convex objective function (57a), we consider constructing
the linear lower-bound function by using the first-order Taylor expansion,
which is given by

$\displaystyle\mathfrak{L}_{i,\varsigma}(\mathbf{W})=$
$\displaystyle-2\text{Tr}((\mathbf{\bar{W}}^{H}-\mathbf{W}^{H})\mathbf{A}_{i}^{\varsigma}\mathbf{W}\mathbf{V}_{i,\varsigma})+$
$\displaystyle\text{Tr}(\mathbf{A}_{i}^{\varsigma}\mathbf{\bar{W}}\mathbf{V}_{i,\varsigma}\mathbf{\bar{W}}^{H})\leq\text{Tr}(\mathbf{A}_{i}^{\varsigma}\mathbf{W}\mathbf{V}_{i,\varsigma}\mathbf{W}^{H}),$
(58)

where $\mathbf{\bar{W}}$ denotes the value of $\mathbf{W}$ optimized in the
previous iteration. Thus, the problem (P9-3) can be efficiently solved by
utilizing the SCA technique. The convex subproblem of each SCA iteration is
given by

$\displaystyle(\text{P}9\text{-}4)\ \max\limits_{\mathbf{W}}\ $
$\displaystyle\sum_{i=1}^{K}\left(\min\limits_{\varsigma\in\\{\text{N},\text{F}\\}}\mathfrak{L}_{i,\varsigma}(\mathbf{W})\right)$
(59a) s.t. $\displaystyle\eqref{P9-1c},\eqref{P9-1d},$ (59b)

which can be directly solved by the CVX toolbox.

Algorithm 3 AO algorithm for beam-splitting beamformer design.

1: Initialize initial $\mathbf{\bar{W}}$. Set the convergence accuracy
$\varepsilon_{4}$ and $\varepsilon_{5}$.

2: repeat

3: update $\mathbf{v}_{i,\text{N}}$ and $\mathbf{v}_{i,\text{F}}$ according to
(54)-(56).

4: repeat

5: optimize $\mathbf{W}$ by solving the problem (P9-4).

6: update $\mathbf{\bar{W}}=\mathbf{W}$.

7: until the objective value converges with an accuracy of $\varepsilon_{4}$.

8: until the objective value converges with an accuracy of $\varepsilon_{5}$.

For the power allocation optimization, it is known that the expressions of the
achievable rate of each user are the same as that of the beam-steering case
due to the existence of the constraint (52d). Thus, the Algorithm 2 can be
employed to obtain the optimal power allocation strategy, and we neglected
here for brevity.

##  V Numerical Results

Figure 5: Simulation setup of the considered network. TABLE I: Simulation parameters | Operating carrier frequency
---
$f=28$ GHz
| Number of transmit/receive antennas
---
$M_{\text{t}}=1024$
| Number of equipped RF chains
---
at the BS
$M_{\text{t}}^{\text{RF}}=K=3$
| Antenna space
---
$d=\frac{\lambda}{2}=\frac{c}{2f}$ m
| Noise power at receivers
---
$\sigma^{2}=-75$ dBm
| QoS requirement
---
$R_{\text{QoS}}^{\text{N},i}=R_{\text{QoS}}^{\text{F},j}=R_{\text{QoS}}$
| Constant scaling coefficients
---
$\bar{c}=1.1$
| Convergence accuracy
---
|  $\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{4}=\varepsilon_{5}=10^{-2}$,
---
$\varepsilon_{3}=10^{-6}$
| Inter-group interference level
---
| $\epsilon=10^{-2}$
---

The numerical simulation results are provided to validate the effectiveness of
the proposed joint beamforming design and power allocation strategies in this
section. The simulation setup is depicted in Fig. 5. We assume that the
central element of the DMA is located at the origin of the coordinate. The NUs
and FUs are assumed to be randomly located on the circular rings of 10 meter
(m) and 15 m, respectively, where the ranges of the azimuth and elevation
angles are from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. The main simulation
parameters are set as Table I. Moreover, each figure is the average result of
$100$ Monte Carle experiments.

Four baseline schemes are considered in this paper:

  * •

beam-steering/splitting-based FDMA: In the beam-steering/splitting-based
frequency division multiple access (FDMA) scheme, we utilize the proposed
beam-steering and beam-splitting schemes to design the spatial beamformers for
multiple user groups, where each user group is served by two orthogonal
frequency bands of equal size [31]. The achievable rate at the NU/FU of the
$i$-th group is given by
$R_{\varsigma_{i}}=\frac{1}{2}\log_{2}(1+\frac{P_{q,i}|(\mathbf{h}^{\varsigma}_{i})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}}{\frac{1}{2}\sigma^{2}})$
($\varsigma_{i}\in\\{\text{N}_{i},\text{F}_{i}\\}$), where $q=1$ for the NU
and $q=2$ for the FU.

  * •

beam-steering/splitting-based TDMA: In the beam-steering/splitting-based time
division multiple access (TDMA) scheme, the BS serves two users belonging to
each group through two equal time slots. Specifically, the BS transmits
signals to the NU in the first slot and then the BS communicates with the FU
in the second slot. In each time slot, the BS applies the total power of the
group to maximize the achievable rate, i.e.,
$R_{\varsigma_{i}}=\frac{1}{2}\log_{2}(1+\frac{P_{i}^{\text{g}}|(\mathbf{h}^{\varsigma}_{i})^{H}\mathbf{W}\mathbf{v}_{i}|^{2}}{\sigma^{2}})$.

  * •

Far-field channel model: In this scheme, the planar-wave-based far-field
channel model is adopted to design the beamformers. Then, we substitute the
optimized beamformers into the practical spherical-wave channels to
characterize the communication performance of the network.

  * •

Zero-forcing scheme: This scheme is a conventional beamforming scheme for the
near-field NOMA transmission [24, 9], where the zero-forcing (ZF) beamformers
are designed only relying on the CSI of NUs, and the FUs are served by the
leaked power of the beamformers oriented to the NUs. For simplifying the
optimization and without losing conviction, the fully-digital ZF beamformers
are considered.

(a) Normalized beamforming gain spectrum of the proposed beam-steering scheme.

(b) Normalized beamforming gain spectrum of the proposed beam-splitting
scheme.

Figure 6: Radiation pattern of the proposed schemes.

To intuitively illustrate the radiation attributes of the proposed beamforming
design schemes in near-field communications, the normalized radiation power
spectrums over the free-space location are drawn in Fig. 6. We consider the
single-group scenario (i.e., $K=1$) with location topology that both the NU
and the FU NU and FU lie in the common spatial plane with the same elevation
angle $\phi_{\text{N}}=\phi_{\text{F}}=0^{\circ}$. It can be observed from
Fig. 6(a) that the proposed beam-steering scheme can achieve signal power
strengthening in the area between the NU and the FU while suppressing signal
leakage in other regions of no interest. Meanwhile, Fig. 6(b) shows that the
proposed beam-splitting scheme is able to achieve the signal power focus on
the multiple locations (also referred as to multi-focus) of the NU and the FU.
Both results demonstrate the effectiveness of the proposed beamforming design
schemes.

Figure 7: Convergence performance of the proposed algorithms with $t=10^{3}$,
$r_{\text{N},i}=10$ m, and $r_{\text{F},i}=15$ m.

Fig. 7 shows the convergence performance of the two-layer and AO algorithms
versus the number of iterations. It can be observed that the BPE value
monotonically decreases over the iterations and converges to a stable solution
in around 10 steps. It is worth noting that the value of BPE does not fall
below a very small value at convergence, such as a value close to 0. However,
this is to be expected because: 1) BPE incorporates all the error accumulation
between the designed beam and the ideal beam characterized by the 3D codebook;
and 2) the practical beam needs to approach the ideal beam with a main lobe
magnitude of $t^{2}=10^{6}$, which inevitably leads to a high radiated power
leakage in other regions, e.g., the beam pattern also produces a high radiated
power in the vicinity of a $90^{\circ}$ azimuth angle in Fig. 6(a). Also can
be seen, the proposed AO algorithm can converge within the finite iterations,
which guarantees a high channel gain even at the far users.

Figure 8: Sum achievable rate versus the transmit power at the BS with
$t=10^{6}$, $r_{\text{N},i}=10$ m, $r_{\text{F},i}=15$ m, and
$R_{\text{QoS}}=1$ bps/Hz. Figure 9: Sum achievable rate versus the average
distance from the NU/FU to the BS with $R_{\text{QoS}}=1$ bps/Hz and
$P_{\text{max}}=15$ dBm.

In Fig. 8 and Fig. 9, we compare the communication performance of the proposed
scheme with the other four baseline schemes. Thereinto, the variations of the
transmit power and the average distance from the NU and FU to the BS are
considered, respectively, where we assume that the NU and FU are apart by a
fixed distance of 5 m, and we consider simultaneously changing their positions
to adjust average distance of the NU and FU from the BS. It is observed that
the proposed beam-steering-based NOMA scheme realizes the highest sum
achievable rate among all the schemes. This can be explained by the fact that:
1) NOMA allows the BS to serve the NU and FU in the same time-frequency
resource block by flexible power control, which is capacity-achieving and
consequently enables better performance than the OMA schemes (i.e., FDMA and
TDMA); 2) due to the mismatch between the beam pattern based on the far-field
planar-wave channel model and the practical near-field spherical-wave
channels, the signal power received at the users are significantly degraded,
which deteriorates communication performance of the network; and 3) since the
conventional ZF scheme is designed based only on the CSI of the NU, it
inevitably leads to an extremely weak channel for FU. Thus, more power should
be allocated to the FU to satisfy its QoS requirement, while NU will obtain
less power, which limits the communication rate of the network. Moreover, it
can be found that the sum achievable rate shows a downward trend with the
increasing transmission distance, which is because that a larger transmission
distance brings a larger path loss, which acquires more transmit power to
maintain the same rate, otherwise the rate will decrease.

Figure 10: Sum achievable rate versus the distance estimation error of the CSI
of the NU and FU with $R_{\text{QoS}}=1$ bps/Hz, $t=10^{3}$, and
$P_{\text{max}}=15$ dBm.

Fig. 10 illustrates the impact of the estimation error of the distance
knowledge of the NU and FU on the communication performance of the NOMA
network. We assume that all the NUs and FUs are estimated to be located in the
distances of 10 m and 15 m, where the realistic distances of NUs and FUs vary
from 8 m to 12 m and from 13 m to 17 m, respectively. The distance estimation
error for the NU/FU of the $i$-th group is defined by $\Delta
d_{\varsigma,i}=d_{\varsigma,i}^{\text{est}}-d_{\varsigma,i}^{\text{rea}}$,
$\varsigma\in\\{\text{N},\text{F}\\}$, where $d_{\varsigma,i}^{\text{est}}$ is
the estimated distance and$d_{\varsigma,i}^{\text{rea}}$ denotes the realistic
distance. Then, the average distance estimation error is given by
$\Delta\bar{d}=\frac{\sum_{i=1}^{K}\sum_{\varsigma\in\\{\text{N},\text{F}\\}}\Delta
d_{\varsigma,i}}{2K}$. It can be observed that the sum achievable rate of the
NOMA network decreases with an increase in $|\Delta\bar{d}|$, which can be
expected as the imperfect distance knowledge will cause the mismatch between
the beam pattern and the practical spherical-wave channels, thus leading to
reduced network performance. Also, we can find an interesting result that both
the proposed beamforming design schemes are sensitive to the distance
estimation error of the NU while not affected by the imperfect distance
information of the FU. This is because, under the optimal power allocation
policy, the FU only needs the power to satisfy its own QoS requirement, while
all remaining power is allocated to the NU to maximize the rate of that NOMA
group, i.e., the achievable rate of the NU dominates the total rate of its
NOMA group. Therefore, the degradation of the channel gains of NUs due to
imperfect distance information can significantly affect the achievable rate of
the network.

##  VI Conclusion

A DMA-enabled near-field NOMA transmission framework was proposed, where NOMA
is exploited to enhance the transmission connectivity of the overloaded
network. A beam-steering beamforming scheme was proposed for the case of same-
direction user distribution, where the BPE metric was introduced to
characterize the gap between the hybrid beamformers and desired perfect
beamformers. A two-layer algorithm was proposed to minimize the BPE by jointly
optimizing the amplitude coefficients of DMA elements and baseband digital
beamformers. On this basis, the globally optimal power allocation strategy was
obtained according to the KKT conditions. Then, a beam-splitting scheme was
proposed for the case of randomly distributed users. An AO algorithm was
proposed to generate the sub-beamformers to serve multiple users, where the
optimal power allocation is derived for the preconfigured sub-beamformers. It
was unveiled that: 1) the proposed beam design schemes show better
communication performance than other baseline schemes; 2) the proposed DMA-
enabled near-field NOMA transmission framework is sensitive to the distance
estimation error of CSI of NUs while insensitive to that of FUs.

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