# Multi-User Rate Splitting in Optical Wireless Networks

Ahmad Adnan Qidan, Khulood Alazwary, Taisir El-Gorashi, Majid Safari, Senior
Member, IEEE, Harald Haas, Fellow, IEEE, Richard V. Penty, Fellow, IEEE, Ian
H. White, Fellow, IEEE, and Jaafar M. H. Elmirghani, Fellow, IEEE This work
has been supported by the Engineering and Physical Sciences Research Council
(EPSRC), in part by the INTERNET project under Grant EP/H040536/1, and in part
by the STAR project under Grant EP/K016873/1 and in part by the TOWS project
under Grant EP/S016570/1. All data are provided in full in the results section
of this paper.

###### Abstract

Optical wireless communication (OWC) has recently received massive interest as
a new technology that can support the enormous data traffic increasing on
daily basis. Laser-based OWC networks can provide terabits per second (Tbps)
aggregate data rates. However, the emerging OWC networks require clusters of
optical transmitters to provide uniform coverage for multiple users. In this
context, multi-user interference (MUI) is a crucial issue that must be managed
efficiently to provide high spectral efficiency. Rate splitting (RS) is
proposed as a transmission scheme to serve multiple users simultaneously by
splitting the message of a given user into common and private messages, and
then, each user decodes the desired message following a certain procedure. In
radio frequency (RF) networks, RS provides higher spectral efficiency compared
with orthogonal and non-orthogonal transmission schemes. Considering the high
density of OWC networks, the performance of RS is limited by the cost of
providing channel state information (CSI) at transmitters and by the noise
resulting from interference cancellation. In this work, a user-grouping
algorithm is proposed and used to form multiple groups, each group contains
users spatially clustered. Then, an outer precoder is designed to manage
inter-group interference following the methodology of blind interference
alignment (BIA), which reduces the requirements of CSI at RF or optical
transmitters. For intra-group interference, RS is applied within each group
where the users belonging to a given group receive a unique common message on
which their private messages are superimposed. Furthermore, an optimization
problem is formulated to allocate the power among the private messages
intended to all users such that the sum rate of the network is maximized. To
relax the complexity of the optimization problem, multiple multipliers are
used where an algorithm iterates to determine a sub-optimal solution. The
results show the effectiveness of the proposed scheme called limited CSI-RS
compared to other counterpart schemes.

###### Index Terms:

Optical wireless communications, user-groping, interference management, power
allocation

##  I Introduction

The use of the Internet has massively increased in recent years due to
emerging technologies such as Internet of Things (IoT), robotics, video
streaming, 3D printing and virtual reality (VR), etc. This unprecedented
growth in data traffic causes several challenges on current radio frequency
(RF) networks including lack of resources, power consumption and secrecy.
Therefore, researchers in both industrial and academic communities have
investigated the possibility to define new technologies that can work with RF
networks to support high-user demands and relax traffic-congestion in the next
generation (6G) of wireless communications [1, 2, 3]. Optical wireless
communication (OWC) has received attention as a promising technology with the
potential to overcome the drawbacks of RF networks where the optical band
offers license-free bandwidth, improved secrecy and usually optical access
points (APs) provide high energy efficiency compared with RF APs. An OWC
network can provide an aggregate data rate in a range of gigabit per second
(Gbps) using conventional light emitting diodes (LEDs) [4, 5, 6]. Like other
technologies, OWC faces several challenges such as light blockage, the
confined converge area of the optical AP, and the low modulation speed of LED
light sources [7, 8]. It is worth mentioning that increasing the number of
LED-based optical APs in an indoor environment can provide a wide coverage
area and seamless user-transition. However, LEDs usually are installed
primarily for illumination, and therefore, increasing the number of LEDs is
limited by the recommended illumination levels in such indoor environments.
Alternatively, Infrared lasers such as vertical-cavity surface-emitting lasers
(VCSELs) can be used for data transmission since they have high modulation
speeds compared to LEDs, and they are commercially available at low prices. In
an indoor environment, clusters of VCSELs can be deployed on the ceiling to
provide uniform coverage. However, the total radiated power of the VCSELs must
be within eye safety regulations. It was shown that VCSEL-based OWC networks
can provide up to terabits per second (Tbps) aggregate data rates [9, 10, 11].

Interference management is a crucial issue in high density multiple-input and
multiple-output (MIMO) OWC scenarios where multiple users must be served
simultaneously to enhance spectral efficiency. Orthogonal techniques such as
time division multiple access (TDMA) [12], orthogonal frequency division
multiple access (OFDMA)[13], space division multiple access (SDMA)[14], code
division multiple access (CDMA)[15], wavelength division multiple access
(WDMA)[16] can be applied to allocate resources in an orthogonal fashion among
users. Despite the avoidance of interference, users might experience low
quality of service due to lack of resources. Recently, non-orthogonal multiple
access (NOMA) [17, 18] has been proposed for OWC to manage multi-user
interference (MUI) through the power domain where power allocation is
performed considering the channel quality of each user. It was shown that NOMA
provides high spectral efficiency in OWC compared with other orthogonal
transmission schemes due to the fact that the resources of the network can be
reused among users at a given time [18]. However, if users have comparable
channel gains, the application of NOMA becomes a challenge. As such, NOMA
relies on perfect channel state information (CSI) at the transmitters to give
users the opportunity to decode their information correctly in such OWC
scenarios. From the CSI perspective, blind interference alignment (BIA) is
proposed for wireless networks [19] to maximize the multiplexing gain without
the need for CSI at the transmitters. In particular, the precoding matrix of
each user is determined following the construction of the BIA transmission
block. In [20], a reconfigurable optical detector composed of multiple
photodiodes is derived to provide a set of linearly independent channel
responses from transmitters, which are essential to apply BIA in OWC. In [21],
BIA is applied in an OWC network with limited information on user-distribution
and channel coherence time at the transmitters, and its superiority in terms
of the achievable user rate is demonstrated compared with transmit precoding
and orthogonal schemes. It is worth mentioning that the density of the network
determines the size of the BIA transmission block. In other words, BIA suffers
performance-degradation in large-scale OWC networks where the channel
coherence time must be large enough to deliver such a large transmission block
and the noise resulting from interference cancellation increases considerably
with the number of users. In [21, 22, 23], various network topology approaches
including network-cenric and user-centric perspectives are designed to relax
the limitations of BIA in large-scale LED-based OWC networks where multiple
elastic and/or static cells are formed, and then, BIA manages the interference
within each cell regardless of the neighboring cells formed. However, the
complexity of these network topology approaches can be excessively high in the
emerging OWC networks using VCSELs, where the number of transmitters is
extremely high to ensure coverage. Therefore, interference management in such
VCSEL-based OWC scenarios requires further investigation.

For RF networks, a new transmission scheme referred to as rate splitting (RS)
was proposed in [24] to treat the interference and relax the requirements of
CSI at the transmitters. The methodology of RS relies on splitting the message
of a given user into common and private messages. In general, RS allocates a
fraction of the total power to the private messages intended to their
corresponding users, while the remaining power is devoted to deliver the
common message to all users. Basically, the common message given by a public
codebook in the network is superimposed on top of private messages, and must
be decoded by all users with minimum error probability. On the other hand,
each user, after decoding the common message and removing it from the received
signal using successive interference cancellation (SIC), can decode its
private message while treating other private messages as noise. Interestingly,
the levels of power allocated to the common and private messages in RS-based
transmission are highly controlled by the accuracy of CSI at transmitters. In
[25], a closed-form strategy for RS is proposed to serve $K$ users in a time
correlated multiple-input single-output (MISO) broadcast channel scenario
considering imperfect CSI at transmitters. It is shown that by splitting the
message of a user into two parts and treating one part as a super common
message on which the other part of the message and the messages of other users
are superimposed, the achievable sum rate is higher compared with benchmarking
schemes such as zero forcing (ZF) and TDMA at the optimal degree of freedom
(DoF). In [26], a novel scheme called hierarchical rate splitting (HRS) using
two-layers of RS was proposed to manage intra-group and inter-group
interference in MIMO networks. In [27], a topological RS (TRS) scheme with
weighted sum interpretation is derived in $L$-cell MISO interference channel
scenarios considering imperfect CSI at the transmitters. In HRS and TRS, users
are arranged into groups, and multiple common messages are transmitted to the
formed groups where each common message contains the private messages of the
users belonging to a certain group. The superiority of HRS and TRS was
demonstrated compared to traditional RS in terms of the data rate achieved due
to the fact that theses more advanced schemes reduce the noise resulting from
the use of SIC at each user. In [28], the RS strategy was applied in a
downlink multi-user MISO system to achieve high quality of service under the
minimization of power consumption.

In the context of OWC, the application of all the proposed RS schemes in the
literature is not straightforward due to the high density of the emerging OWC
networks compared with RF networks. In [29], RS and HRS schemes were
implemented in Laser-based OWC to manage multi-user interference. It was shown
that the data rate achieved by RS and/or HRS decreases considerably in large
size networks due to the requirements of CSI and noise enhancement. In [30],
an optimization problem was formulated to allocate the power among the various
messages of HRS with the aim of enhancing the sum rate of the network.
However, power allocation in HRS-based OWC networks involves high complexity
due to the transmission of the three different messages, i.e., inner, outer
and private messages. In contrast, in this work, a novel interference
management scheme using RS is proposed in an OWC network to address three key
problems, namely the high number of optical transmitters and users, the
requirements of CSI and the power allocation optimization problem. The main
contributions of the paper are as follows:

  * •

We formulate an optimization problem with an objective function that aims to
partition all users into multiple groups based on the sum rate maximization of
the users belonging to each group using RS. Interestingly, the sum rate of
each group cannot be determined unless all the groups are already formed and
inter-group interference is aligned. Therefore, the optimum user-grouping must
be found through exhaustive search. To avoid complexity, we propose a dynamic
algorithm that arranges users spatially clustered into a specific group. In
this sense, users are divided into two different sets of users, main and edge
users, in accordance to certain conditions. Then, the formation of each group
starts from a unique main user, and around that main user, some of the edge
users located at the closest distance are grouped.

  * •

After the formation of multiple groups, a RS-based interference management
scheme, referred to as Limited CSI-Rate Splitting (limited CSI-RS), is derived
to jointly mange inter-group interference and intra-group interference. In
limited CSI-RS, an outer precoder is designed to mange the interference among
the groups by following the methodology of BIA proposed in [19]. While, the
users belonging to each group receive a common message superimposed on top of
the private messages intended to the users of that group. Mathematical
expressions are derived to explain the proposed scheme in detail.

  * •

Power allocation is crucial in RS-based transmission due to the combination of
different messages. Considering the application of the limited CSI-RS scheme,
a fixed power is allocated to the common message intended to the users of each
group, and an optimization problem is formulated to allocate the power among
the private messages of all the users, maximizing the overall sum rate of the
network. By solving the original optimization problem, an optimum solution can
be determined at high complexity. Therefore, we reformulate the optimization
problem around multiple multipliers, which can be updated separately to
guarantee the provision of sub-optimal solutions with low complexity.

The results demonstrate that our proposed dynamic user grouping algorithm
converges to the optimal solution determined through exhaustive search under
fixed power allocation. Moreover, the limited CSI-RS scheme achieves high
performance in terms of sum data rates, BER and energy efficiency in dense OWC
networks compared with other benchmarking schemes, which include BIA and RS.
Finally, the formulated power allocation problem utilizes the power budget
available in the network effectively, achieving data rates considerably higher
than the fixed power allocation approach.

The remainder of this paper is organized as follows. In Section II, the system
model of the OWC network considered is described. The fundamentals of BIA and
RS used as baseline in this work are defined in Section III. In Section IV,
the user-grouping approach is designed, and the limited CSI-RS schemes is
derived in detail. The formulation and analysis of power allocation for the
proposed scheme are derived in Section V. Finally, Sections VI and VII present
simulation results and provide concluding remarks.

Notation. The notations considered in this work are defined as follows. First,
matrices and vectors are denoted by the bold upper case and lower case
letters, respectively. To represent identity and zero matrices with $M\times
M$ dimension, we call out to $\mathbf{I}_{M}$ and $\mathbf{0}_{M}$ notations,
respectively, while $\mathbf{0}_{M,N}$ denotes a $M\times N$ zero matrix,
$[\,\,]^{T}$ and $[\,\,]^{H}$ indicate the transpose and hermitian transpose
operators, respectively. Finally, $\mathbb{E}$ is the statistical expectation,
and $\mathrm{col}\\{\\}$ is the column operator that stacks the considered
vectors in a column.

Figure 1: An OWC network composed of a number of optical APs serving multiple
users.

##  II System Model

An indoor downlink OWC network is considered as shown in Fig. 1. It consists
of multiple optical APs, $L,l=\big{\\{}1,\dots,L\big{\\}}$, installed on the
ceiling to provide uniform coverage for multiple users,
$K,k=\big{\\{}1,\dots,K\big{\\}}$, on the receiving plane. Each user is
equipped with an optical detector composed of multiple photodiodes,
$M,m=\big{\\{}1,\dots,M\big{\\}}$, providing linearly independent channel
responses [20, 21], as shown in Fig. 2. Note that, each photodiode points to a
distinct direction to guarantee connectivity, and its orientation vector is
determined by the elevation $\theta^{[k,m]}$ and azimuth $\alpha^{[k,m]}$
angles on the x-y plane, as follows

$\begin{split}\hat{\mathbf{n}}^{[k,m]}=\left[\sin\left(\theta^{[k,m]}\right)\cos\left(\alpha^{[k,m]}\right),\,\,\right.\phantom{\\{}\left.\sin\left(\theta^{[k,m]}\right)\sin\left(\alpha^{[k,m]}\right),\,\,\cos\left(\theta^{[k,m]}\right)\right],\end{split}$
(1)

That is, the irradiance and incidence angles are determined by
$\phi_{l}^{[k]}=\arccos\left(\frac{\hat{\mathbf{n}}_{l}\cdot\mathbf{v}_{l}^{[k]}}{\|\hat{\mathbf{n}}_{l}\|\|\mathbf{v}_{l}^{[k]}\|}\right)$
and
$\varphi_{l}^{[k]}(m)=\arccos\left(\frac{\hat{\mathbf{n}}^{[k,m]}\cdot\mathbf{v}_{l}^{[k]}}{\|\hat{\mathbf{n}}^{[k,m]}\|\|\mathbf{v}_{l}^{[k]}\|}\right)$,
respectively, where $\hat{\mathbf{n}}_{l}$ is the normal vector. Considering
full connectivity coordination among the $L$ APs 111Full connectivity is an AP
coordination scheme that enhances the overall SNR of the network [21]. Note
that, the new RS-based scheme derived in this paper is applicable in a multi-
cell scenario. However, fractional frequency reuse must be considered to avoid
inter-cell interference., the transmitted signal can be written in vector form
as

$\mathbf{x}=\begin{bmatrix}x_{1}&x_{2}&\dots&x_{L}\end{bmatrix}^{T}\in\mathbb{R}_{+}^{L\times
1},$ (2)

where $x_{l}$ is the signal transmitted by optical AP $l$. Thus, the signal
received by user $k$ regardless of any multiple access schemes can be written
in general form as

$y^{[k]}=\mathbf{h}^{[k]}\left(m^{[k]}\right)^{T}\mathbf{x}+z^{[k]},$ (3)

where $\mathbf{h}^{[k]}\left(m^{[k]}\right)\in\mathbb{R}_{+}^{L\times 1}$ is
the channel vector between the $L$ optical APs and user $k$ at photodiode $m$,
which is given by

$\mathbf{h}^{[k]}\left(m^{[k]}\right)=\begin{bmatrix}h^{[k]}_{1}(m^{[k]})&h^{[k]}_{2}(m^{[k]})&\dots&h^{[k]}_{L}(m^{[k]})\end{bmatrix}^{T}.$
(4)

Moreover, $z^{[k]}$ is real valued additive white Gaussian noise with zero
mean and variance $\sigma^{2}_{z}$, which is given by the sum of the
contributions from both shot noise and thermal noise [10].

Figure 2: An optical detector with multiple photodiodes connected to a signal
processing chain.

We consider the use of VCSELs as transmitters under eye safety regulations [9,
10]. The beam profile of the VCSEL is Gaussian dictated by the DC bias current
applied, and its transmitted power is determined based on the beam waist
$W_{0}$, the wavelength $\lambda$ and the distance $d$ through which the beam
travels. In this context, the beam radius of the VCSEL at photodiode $m$ of
user $k$ located at the receiving plane is given by

$W_{d}=W_{0}\left(1+\left(\frac{d}{d_{Ra}}\right)^{2}\right)^{1/2},$ (5)

where $d_{Ra}$ is the Rayleigh range determined by $d_{Ra}=\frac{\pi
W^{2}_{0}n}{\lambda},$ where $n$ is the refractive index of the medium, which
is air in this case, i.e., $n=1$. In general, the spatial distribution of the
intensity of VCSEL $l$ over the transverse plane at distance $d$ is given by

$I_{l}(r,d)=\frac{2P_{t,l}}{\pi
W^{2}_{d}}~{}\mathrm{exp}\left(-\frac{2r^{2}}{W^{2}_{d}}\right),$ (6)

where $P_{t,l}$ is the optical power. Considering that the detection area of
the optical detector is $A_{rec}$, the area of each photodiode within that
detector is given by $A_{m}=\frac{A_{rec}}{M}$, $m\in M$. Hence, the power
received at each of the photodiodes of user $k$ from VCSEL $l$ is given by

$\begin{split}P_{m,l}=&\int_{0}^{A_{m}/2\pi}I(r,d)2\pi
rdr=P_{t,l}\left[1-\mathrm{exp}\left(-2\left(\frac{A_{m}}{2\pi
W_{d}}\right)^{2}\right)\right],\end{split}$ (7)

assuming photodiode $m$ of user $k$ is located right under transmitter $l$,
more mathematical expressions and details on the received power calculations
in different scenarios can be found in [9, 10].

In this work, all the optical APs are connected through a central unit (CU)
that has information regarding the distribution of the users and the channel
coherence time. It also controls the resources of the network such that
optimization problems with different contexts can be performed to enhance the
performance of the network. Moreover, a WiFi AP is deployed to provide uplink
transmission where users can forward the information needed for solving the
optimization problems. It is worth mentioning that the need for CSI at the
transmitters in this work varies according to the transmission scheme
considered to provide multiple access service in the network.

##  III  Advanced Multi-user Interference management Schemes

In this section, we introduce the methodologies of RS and BIA in managing
multi-user interference with and without CSI at transmitters, respectively,
highlighting useful notations to derive the novel scheme proposed in Section
IV.

###  III-A Rate Splitting (RS)

For multiple input-single output (MISO) broadcast Channel(BC) scenarios, RS
can be applied to serve $K$ users simultaneously in the presence of accurate
CSI at transmitters [26, 29]. Basically, all the transmitters send a super
common message denoted by $s_{c}$ superimposed on the top of
$\mathbf{s_{p}}=\begin{bmatrix}{s^{[1]}_{p}}&{s^{[2]}_{p}}&\dots&{s^{[K]}_{p}}\end{bmatrix}$
private messages intended to $K$ users, where ${s^{[k]}_{p}}$ is the private
message desired by user $k$. Therefore, the transmitted signal in RS-based
transmission can be expressed as

$\mathbf{x}=\sqrt{P_{c}}\mathbf{w}_{c}s_{c}+\sum_{k=1}^{K}\sqrt{P_{p}}\mathbf{w}^{[k]}_{p}{s^{[k]}_{p}}$
(8)

where $\mathbf{w}_{c}$ is the unit-norm precoding vector of the common
message.
Moreover,
$\mathbf{W_{p}}=\left[\mathbf{w^{[1]}_{p}}~{}\mathbf{w^{[2]}_{p}}~{}\dots~{}\mathbf{w^{[K]}_{p}}\right]$
is a vector that contains the precoding vectors of the $K$ private messages
intended to the users in the network. It is worth mentioning that the power
allocated to the common and private messages, ${P_{c}}$ and ${P_{p}}$,
respectively, is usually determined through a fixed power allocation approach
to avoid complexity, where a fraction of the total power ${P_{T}}$ is divided
uniformly among the private messages, while the residual power is allocated to
the common message. That is, the power allocated to the common message is
$P_{c}={P_{T}}(1-t)$, where $t\in(0,1]$, and therefore, the power allocated to
each private message is $P_{p}=\frac{{P_{T}}t}{K}$. At this point, the
received signal of user $k$ can be expressed as

${{y}}^{[k]}=\sqrt{P_{c}}{\mathbf{h}}^{[k]}\mathbf{w}_{c}{s}_{c}+\sqrt{P_{p}}{\mathbf{h}}^{[k]}\mathbf{w}_{p}^{[k]}{s}_{p}^{[k]}\underbrace{+\sum^{K}_{k^{\prime}\neq
k}\sqrt{P_{p}}{\mathbf{h}}^{[k]}\mathbf{w}_{p}^{[k^{\prime}]}{s}_{p}^{[k^{\prime}]}}_{\text{multi-
user interference}}+{{z}}^{[k]}.$ (9)

To decode the desired information, each user first decodes the common message
with relatively low error probability, while treating all the private messages
as noise. Subsequently, eliminating the common message from the received
signal through applying Successive Interference Cancellation (SIC), and then,
decoding the desired private message. From (9), the SINR equations of the
common and private messages can be derived as

$\gamma^{[k]}_{c}=\frac{P_{c}\left|\mathbf{h}^{[k]^{H}}\mathbf{w}_{c}\right|^{2}}{\sum_{k=1}^{K}P_{p}\left|\mathbf{h}^{[k]^{H}}\mathbf{w}^{[k]}_{p}\right|^{2}+{\sigma_{z}}^{2}}$
(10)
$\gamma_{p}^{[k]}=\frac{P_{p}\left|\mathbf{h}^{[k]^{H}}\mathbf{w}^{[k]}_{p}\right|^{2}}{\sum_{k^{\prime}\neq
k}P_{p}\left|\mathbf{h}^{[k]^{H}}\mathbf{w}^{[k^{\prime}]}_{p}\right|^{2}+{\sigma_{z}}^{2}}$
(11)

Therefore, the achievable data rate of the common message is
$R_{c}=\log_{2}\left(1+\min\limits_{k}\left\\{\gamma_{c}^{[k]}\right\\}\right)$,
where $\min\limits_{k}\left\\{\gamma_{c}^{[k]}\right\\}$ ensures that each
user can decode the common message. While, the sum rate of the private
messages is $R_{p}=\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{p}^{[k]}\right)$. As
a result, the sum rate of the network considering RS is
$R_{\text{RS}}=R_{c}+R_{p}$.

Figure 3: BIA transmission block for an OWC network. Each color represents a
channel state selected by the corresponding user.

###  III-B Blind Interference Alignment (BIA)

According to [19], BIA manages the interference among users with no CSI at
transmitters where the precoding matrix of each user is determined following
the construction of a unique transmission block. In BIA, the transmission
block is divided into two sub-blocks, Block 1 and Block 2, over which users
receive their information. In an OWC network with $L$ transmitters serving $K$
users, the whole transmission block comprises $(L-1)^{K}+K(L-1)^{K-1}$ time
slots. The first $(L-1)^{K}$ time slots belong to Block 1, while the last
$K(L-1)^{K-1}$ time slots belong to Block 2, as shown in Fig. 3. Note that,
the purpose of dividing the time slots in such a way between Block 1 and Block
2 is to allocate $\ell=\\{1,\dots,(L-1)^{K-1}\\}$ orthogonal alignment blocks
to each user, which determines the the precoding matrices of the users. It is
worth mentioning that each alignment block allocated to a given user $k$ must
satisfy the following two conditions:

  * •

Ensuring the decodability of each message. Each alignment bock consists of $L$
time sots to deliver the message
$\mathbf{s}_{\ell}^{[k]}=\begin{bmatrix}{s}_{\ell,1}^{[k]}&{s}_{\ell,2}^{[k]}&\dots&{s}_{\ell,L}^{[k]}\end{bmatrix}$
from $L$ transmitters to user $k$, which switches its channel state over that
alignment block to receive the message.

  * •

Alignment of the interference. During transmission over the $\ell$-th
alignment block allocated to user $k$, all other users, $k^{\prime}\neq k$,
must remain in a fixed channel state such that the interfering signals can be
aligned in at least one dimension less than the desired information.

These conditions can be guaranteed by allocating the first $L-1$ time slots of
each alignment block over Block 1, while the last time slot is provided over
Block 2. In other words, all users receive their information simultaneously
during $(L-1)^{K}$ time slots causing serve interference among them, which can
be measured over Block 2 where users are served in an orthogonal fashion.
Therefore, each user can decode its information after interference
subtraction. The achievable user rate is given by

$R_{\text{BIA}}^{[k]}=\frac{1}{L+K-1}\log_{2}\left(\mathbf{I}+P_{\rm{str}}\mathbf{H}^{[k]}{\mathbf{H}^{[k]}}^{H}{\mathbf{R}_{z}}^{-1}\right),$
(12)

where $P_{\rm{str}}$ is the power allocated to each stream,
$\mathbf{H}^{[k]}=\begin{bmatrix}\mathbf{h}^{[k]}(1)&\dots&\mathbf{h}^{[k]}(L)\end{bmatrix}^{T}\in\mathbb{R}^{L\times
L}$ is the channel matrix of user $k$, and
$\mathbf{R}_{z}=\begin{bmatrix}K\mathbf{I}_{L-1}&0\\\ 0&1\end{bmatrix}$.

##  IV  A Novel Rate Splitting-based Interference Management Scheme

The conventional RS scheme explained in Section III is suitable for RF
networks due to the large coverage area of the RF AP, which makes the
provision of CSI at RF transmitters less expensive. However, RS is subject to
performance-degradation in high density OWC networks in which a high number of
optical APs are deployed to expand the coverage and serve multiple active
users simultaneously. This is because the provision of CSI at optical APs
becomes a challenge where the information of users as well as their data must
be exchanged among the whole set of optical APs. Therefore, the imperfection
of CSI must be taken into consideration in such scenarios. Besides, having a
high number of users means that significant noise enhancement might occur
since each user must cancel the interference received due to serving other
users. One of the solutions to enhance the performance of RS in OWC is to
divide the users into multiple groups from the user centric perspective. In
this context, each user is grouped with users located within a threshold
distance denoted by $d_{th}$. Afterwards, RS can be applied to mange multi-
user interference within each group, forming parallel RS schemes working
simultaneously in multiple groups. In this case, the users of each group are
subject to inter-group interference due to the fact that the same RS precoding
matrices are used within each group. Given this, the bandwidth can be divided
by the number of groups allocating an exclusive frequency or time slot to each
group. Despite the low complexity of this approach, low data rates are
achieved due to insufficient use of the resources. Motivated by the
methodology of BIA in managing mulit-user interference without the need for
CSI, we design a novel and general framework, namely Limited CSI-RS
transmission, which smartly combines the features of both BIA and RS resulting
in a robust interference management scheme. In particular, once the users are
divided into multiple groups, the precoding matrix of BIA can be used as an
outer precoder that manages inter-group interference with limited CSI to the
channel coherence time and the distributions of users [31]. While, the users
belonging to each group receive common and private messages to align intra-
group interference using RS. It is worth pointing out that BIA suffers from
noise enhancement as the number of users increases (see equation (12)), and
that the length of the transmission block increases with the size of the
network, which means that the channel coherence time must be large enough to
ensure the delivery of such a large transmission block with relatively minimum
errors. However, in limited CSI-RS transmission, the number of groups
determines the length of the transmission block instead of the number of
users, and therefore, it is expected to decrease considerably. Furthermore, RS
is subject to the need for CSI, which is relaxed in limited CSI-RS
transmission due to the implementation of BIA. In this section, we first
formulate an optimization problem for user-grouping and provide a practical
solution using a dynamic algorithm, and then, the design of the limited CSI-RS
transmission scheme is reported in detail.

###  IV-A User grouping algorithm

Prior to the application of the limited CSI-RS scheme, users are divided into
a number of groups, $G$, $g=\\{1,\dots,G\\}$, where each group $g$ contains
$K_{g}$ active users. First, users are arranged into two different sets, a
main set $\mathcal{K}_{m}$ and an edge set $\mathcal{K}_{ed}$. In particular,
the main set contains $|\mathcal{K}_{m}|=K_{m}$ main users 222Users are
classified as main users under two conditions; i) higher received power from
most of the APs on the ceiling, ii) the minimum distance between two main
users is larger than $d_{th}$. located at the center of the indoor
environment, while the edge set contains $|\mathcal{K}_{ed}|=K_{ed}$ edge
users located around the main users. It is worth pointing out that arranging
the users into two different sets reduces the complexity of searching for the
optimal set of groups where each main user is selected as the first element in
a formed group. After that, an optimization problem can be formulated to form
$G$ groups. The objective function of the sum rate for two users
$i\in\mathcal{K}_{m}$ and $j\in\mathcal{K}_{ed}$ is given by

$U(R)=\mathit{x^{[i,j]}}\bigg{(}\mathit{w_{i}}R^{[i]}({P_{c}},{P_{p}})+\mathit{w_{j}}R^{[j]}({P_{c}},{P_{p}})\bigg{)},\\\
$ (13)

where $\mathit{x^{[i,j]}}$ is an association variable that indicates
$\mathit{x^{[i,j]}}=1$ if users $i$ and $j$ are grouped together, otherwise it
is $0$. Moreover, $\mathit{w_{k}}$, $k\in K$, is the weight that achieves
balance between the overall sum rate of the users and the system fairness, and
$R^{[k]}({P_{c}},{P_{p}})$ is the user rate as function of the power levels
allocated to the common and private messages. Note that, power allocation
among the messages of RS intended to the users can affect the results of
grouping users. However, solving these two problems, group formation and power
allocation, jointly involves significant complexity, and therefore, a fixed
power allocation approach is considered among the users to form multiple
groups. At this point, user grouping can be performed through maximizing the
following objective function under a set of constraints

$\displaystyle\max_{\bm{x}}\quad$
$\displaystyle\sum\limits^{K_{m}}_{{i=1}}\sum\limits_{{j}\in\mathcal{K}_{ed}}U(R)$
(14a) s.t.
$\displaystyle\sum\limits_{i\in\mathcal{K}_{m}}\mathit{x^{[i,j]}}=i,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
j\in\mathcal{K}_{ed},$ (14b) $\displaystyle\sum\limits_{j\in
K_{ed}}\mathit{x^{[i,j]}}=K_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
i\in\mathcal{K}_{m},$ (14c)
$\displaystyle\mathit{x^{[i,k]}}\in\\{0,1\\},K_{m}\cup K_{ed}=K,$ (14d)

where the first constraint ensures that each main user belongs to one group
only, while the second constraint determines the number of the edge users
$K_{i}$ associated with each main user $i$ forming a group $g$ with $i\cup
K_{i}=K_{g}$ users. Note that, the user rate based on limited CSI-RS
transmission cannot be determined unless the groups are already formed, as
explained in detail in the following sub-section. Therefore, an exhaustive
search method must be adopted to solve the optimization problem in (14),
finding the optimal set of groups at high computational time [23]. Despite the
optimality of exhaustive search, a practical solution is essential and is
formulated as follows:

####  IV-A1 Dynamic user-grouping algorithm

To relax the optimization problem in (14), the groups can be formed separately
from the maximization of the sum rate, considering the distance among the
users. In the context of RS, grouping users located close to each other, i.e.,
users who share the same spatial correlation matrix, maximizes the sum rate
due to the relaxation of the CSI requirements at the transmitters. Therefore,
a five-step dynamic algorithm is proposed to divide users into multiple
groups, while guaranteeing that the users belonging to each group are
spatially clustered:

  1. 1.

Each main user, $i\in\mathcal{K}_{m}$, is grouped with the closest edge user,
$j\in\mathcal{K}_{ed}$, i.e.,

$j=\arg\min\limits_{j\in\mathcal{K}_{ed}}\mathrm{d}(i,j),$ (15)

where $\mathrm{d}(i,j)$ denotes the Euclidean distance between main user $i$
and edge user $j$. Thus,
$\mathrm{d}(i,j)=\sqrt{(\check{x}_{i}-\check{x}_{j})^{2}+(\check{y}_{i}-\check{y}_{j})^{2}}$,
where $(\check{x}_{i},\check{y}_{i})$ and $(\check{x}_{j},\check{y}_{j})$ are
the locations of users $i$ and $j$, respectively.

  2. 2.

The formation of a dynamic group $g$ starts from a random pair given by main
user $i$ and edge user $j$, i.e., $i\rightarrow j$, as described in the first
step. Note that, the pair $i\rightarrow j$ must be unique and must not belong
to any other group. The centroid of group $g$ at the $\kappa$-th iteration of
the proposed dynamic algorithm is defined as
$\zeta_{g}(\kappa)=\left(\check{x}_{\zeta_{g}}(\kappa),\check{y}_{\zeta_{g}}(\kappa)\right)$.
It is worth mentioning that at the first iteration, the centroid of group $g$,
$\zeta_{g}(\kappa=1)$, is given by the coordinates
$(\check{x}_{i},\check{y}_{i})$ of main user $i$. At this point, group $g$ is
composed of $\mathcal{K}_{g}=\\{i,j\\}$ with the centroid $\zeta_{g}$
determined according to the coordinates of users $i$ and $j$.

  3. 3.

The expansion of group $g$ can be done by including the edge users located at
a maximum distance less than $d_{th}$ from
$\zeta_{g}=\left(\check{x}_{\zeta_{g}}(\kappa),\check{y}_{\zeta_{g}}(\kappa)\right)$.
That is, the condition $\mathrm{d}\left(\zeta_{g},\forall
j^{\prime}\in{\mathcal{K}_{ed}}\right)\leq d_{th}$ is satisfied, 333 If one
edge user $j^{\prime}$ satisfies the condition of the threshold distance with
more than one centroid, i.e., with more than one initial group, during a given
iteration, edge user $j$ is assigned randomly to one of them. where $d_{th}$
is the distance that delimits the boundaries of group g. Subsequently, the
center of group $g$ is updated to

$\zeta_{g}(\kappa)=\left(\dfrac{\check{x}_{\zeta_{g}}(\kappa-1)+\check{x}_{j^{{}^{\prime}}}}{2},\dfrac{\check{y}_{\zeta_{g}}(\kappa-1)+\check{y}_{j^{{}^{\prime}}}}{2}\right).$
(16)

The dynamic algorithm continues in the same fashion until there are no more
edge users satisfying the condition of the threshold distance with the given
group $g$, i.e, $\mathcal{K}_{g}$ is determined. Note that, each group is
formed iteratively including one edge user at each iteration, while updating
the centroid of group $g$ according to (16).

  4. 4.

Once the users forming group $g$ are determined, the algorithm considers
grouping other edge users that do not belong to any formed group, and can be
included with the user set $\mathcal{K}_{g}$. Thus, the elements of group $g$
is updated as

$\mathcal{K}_{g}=\mathcal{K}_{g}\cup\\{{j\in\mathcal{K}_{ed}},\mathrm{d}(\zeta_{g}(\kappa),j)\leq
d_{th}\\}.$ (17)

Subsequently, the centroid of group $g$ is updated again by averaging the
locations of the new included edge users according to (16).

  5. 5.

The proposed algorithm ends with arranging $K$ users into $G$ groups, while
each group with a unique main user as well as a unique set of edge users,
i.e.,

$~{}\begin{matrix}\mathcal{K}_{g}=i\cup\mathcal{K}_{i},\mathcal{K}_{i}\cap\mathcal{K}_{i^{{}^{\prime}}}=\emptyset,\mathcal{K}_{g}\cap\mathcal{K}_{g^{{}^{\prime}}}=\emptyset,~{}\\\
(i\neq i^{{}^{\prime}},g\neq
g^{{}^{\prime}}),\big{\\{}{i,i^{{}^{\prime}}}\big{\\}}\in\mathcal{K}_{m},\big{\\{}\mathcal{K}_{g},\mathcal{K}_{g^{{}^{\prime}}}\big{\\}}\in
K.\end{matrix}$ (18)

###  IV-B Limited CSI-RS transmission

After the formation of multiple groups, the Limited CSI-RS scheme is applied
to manage inter-group interference using BIA as an outer precoder among the
groups to relax CSI requirements at transmitters, while the users belonging to
each group receive a common message and private messages for intra-group
interference, as follows

Figure 4: An OWC use case where $L=2$ APs serving $G=2$ groups, each group $g$
with $K_{g}=2$ users. The groups are formed in accordance to the five steps of
the dynamic algorithm. The users of each group follow the same channel state
switching.

####  IV-B1 Motivational example

For illustrative purposes, we first derive the mathematical expressions of the
Limited CSI-RS scheme for a toy OWC example that is composed of $L=2$ optical
AP serving $K=4$ users arranged into $G=2$, each group with $K_{g}$ users, as
shown in Fig. 4. In this context, the methodology of BIA explained in the
previous section is followed to design the precoding matrix of each group
denoted by $\mathbf{B^{[g]}}$, i.e., the users of each group use the same
outer precoding matrix, such that the interference among the groups is
aligned. Therefore, the transmitted signal considering the application of RS
within each group can be expressed as

$\mathbf{X}=\begin{bmatrix}\mathbf{x}[1]\\\ \mathbf{x}[2]\\\ \mathbf{x}[3]\\\
\end{bmatrix}=\mathbf{B}^{[1]}\bigg{(}\sqrt{P_{c}^{[1]}}\mathbf{w}_{c}^{[1]}\mathbf{s}_{c}^{[1]}+\sqrt{P_{p}^{[1,1]}}\mathbf{w}_{p}^{[1,1]}\mathbf{s}_{p}^{[1,1]}+\sqrt{P_{p}^{[2,1]}}\mathbf{w}_{p}^{[2,1]}\mathbf{s}_{p}^{[2,1]}\bigg{)}\\\
+\mathbf{B}^{[2]}\bigg{(}\sqrt{P_{c}^{[2]}}\mathbf{w}_{c}^{[2]}\mathbf{s}_{c}^{[2]}+\sqrt{P_{p}^{[1,2]}}\mathbf{w}_{p}^{[1,2]}\mathbf{s}_{p}^{[1,2]}+\sqrt{P_{p}^{[2,2]}}\mathbf{w}_{p}^{[2,2]}\mathbf{s}_{p}^{[2,2]}\bigg{)},$
(19)

where ${P_{c}^{[g]}}$ is the power of the common message intended to group
$g$, $\mathbf{w}_{c}^{[g]}$ is the unit-norm precoding vector of the common
message intended to group $g$ and $\mathbf{s}_{c}^{[g]}$ denotes the common
message intended to users 1 and 2 belonging to group $g$. Moreover,
$P_{p}^{[k,g]}$, $k\in K_{g}$, is the power allocated to the private message
intended to user $k$ belonging to group $g$, $\mathbf{w}_{p}^{[k,g]}$ is the
unit-norm precoding vector of the private message intended to user $k\in
K_{g}$, and
$\mathbf{s}^{[g]}_{p}=[\mathbf{s}_{p}^{[1,g]},\mathbf{s}_{p}^{[2,g]}]\in\mathbb{C}^{2}$
is the data vector of the private messages intended to the user 1 and 2 of
group $g$. In this example, the transmission to two groups must occur over a
predefined transmission block comprising three time slots in accordance to BIA
methodology to manage inter-group interference as in (19). The users of each
group switch their channel pattern over two time slots to receive information,
while devoting a certain time slot to measure the interference received due to
the transmission to the adjacent group. Therefore, the BIA-based outer
precoding matrices $\mathbf{B}^{[1]}$ and $\mathbf{B}^{[1]}$ can be expressed
as

$\mathbf{B}^{[1]}=\begin{bmatrix}\mathbf{I_{2}}\\\ \mathbf{I_{2}}\\\
\mathbf{0_{2}}\\\
\end{bmatrix},~{}\mathbf{B}^{[2]}=\begin{bmatrix}\mathbf{I_{2}}\\\
\mathbf{0_{2}}\\\ \mathbf{I_{2}}\\\ \end{bmatrix},$ (20)

respectively. It is worth pointing out that these outer precoding matrices are
given by 0 and 1 values and CSI at transmitters is not needed to determine
their structure. Focussing on user 1 belonging to group 1, without loss of
generality, the received signal can be expressed as

$\mathbf{y}^{[1,1]}=\underbrace{\begin{bmatrix}\mathbf{h}^{[1,1]}(1)^{T}\\\
\mathbf{h}^{[1,1]}(2)^{T}\\\ \mathbf{0}_{2,1}^{T}\\\
\end{bmatrix}}_{\text{rank=2}}\bigg{(}\sqrt{P_{c}^{[1]}}\mathbf{w}_{c}^{[1]}\mathbf{s}_{c}^{[1]}+\sqrt{P_{p}^{[1,1]}}\mathbf{w}_{p}^{[1,1]}\mathbf{s}_{p}^{[1,1]}+\sqrt{P_{p}^{[2,1]}}\mathbf{w}_{p}^{[2,1]}\mathbf{s}_{p}^{[2,1]}\bigg{)}+\underbrace{\begin{bmatrix}\mathbf{h}^{[1,1]}(1)^{T}\\\
\mathbf{0}_{2,1}^{T}\\\ \mathbf{h}^{[1,1]}(1)^{T}\\\
\end{bmatrix}}_{\text{rank=1}}\\\
\bigg{(}\sqrt{P_{c}^{[2]}}\mathbf{w}_{c}^{[2]}\mathbf{s}_{c}^{[2]}+\sqrt{P_{p}^{[1,2]}}\mathbf{w}_{p}^{[1,2]}\mathbf{s}_{p}^{[1,2]}+\sqrt{P_{p}^{[2,2]}}\mathbf{w}_{p}^{[2,2]}\mathbf{s}_{p}^{[2,2]}\bigg{)}+\begin{bmatrix}{z}^{[1,1]}(1)\\\
{z}^{[1,1]}(2)\\\ {z}^{1,1}(3)\\\ \end{bmatrix},$ (21)

where $\mathbf{y}^{[1,1]}=\text{col}\\{{y}^{[1,1]}(\kappa)\\}^{3}_{\kappa=1}$.
It can be seen from equation (21) that the desired information of user 1
appears into a full rank matrix $[1~{}1~{}0]^{T}$, while the interference
caused by the transmission to the users belonging to group 2 is received into
one dimension $[1~{}0~{}1]^{T}$. Thus, the desired information is decodable
after measuring and canceling inter-group interference. Note that, user 1
receives the desired information over the first time slot polluted with
interference and the second time slot, which is interference-free time slot.
In other words, the first and second time slots form an alignment block over
which the users of group 1 receive information. As a consequence, user 1
devotes the third time slot to measure and cancel the interference received
over the first time slot of the alignment block at the cost of noise
enhancement. Thus, the received signal of user 1 belonging to group 1 after
inter-group interference cancellation can be written as

$\mathbf{\widehat{y}}^{[1,1]}={\begin{bmatrix}\mathbf{h}^{[1,1]}(1)^{T}\\\
\mathbf{h}^{[1,1]}(2)^{T}\\\
\end{bmatrix}}\bigg{(}\sqrt{P_{c}^{[1]}}\mathbf{w}_{c}^{[1]}\mathbf{s}_{c}^{[1]}+\sqrt{P_{p}^{[1,1]}}\mathbf{w}_{p}^{[1,1]}\mathbf{s}_{p}^{[1,1]}\\\
+\sqrt{P_{p}^{[2,1]}}\mathbf{w}_{p}^{[2,1]}\mathbf{s}_{p}^{[2,1]}\bigg{)}+\begin{bmatrix}{z}^{[1,1]}(1)-{z}^{[1,1]}(3)\\\
{z}^{[1,1]}(2)\\\ \end{bmatrix}.$ (22)

It can be easily seen that the received signal of user 1 is free of inter-
group interference with an increase in noise. Interestingly, $\mathbf{h}(1)$
and $\mathbf{h}(2)$ are linearly independent channel responses, and therefore,
the information of user 1 can be decoded after canceling intra-group
interference by applying the RS technique. In equation (22), the RS-based
transmitted signal to group 1 is

$\mathbf{X}=\bigg{(}\sqrt{P_{c}^{[1]}}\mathbf{w}_{c}^{[1]}\mathbf{s}_{c}^{[1]}+\sqrt{P_{p}^{[1,1]}}\mathbf{w}_{p}^{[1,1]}\mathbf{s}_{p}^{[1,1]}+\sqrt{P_{p}^{[2,1]}}\mathbf{w}_{p}^{[2,1]}\mathbf{s}_{p}^{[2,1]}\bigg{)}.$
(23)

The message of user 1 is split into common and private messages,
$\mathbf{s}_{c}^{[1]}$ and $\mathbf{s}_{p}^{[1,1]}$. In this case, user 1
follows the same procedure of conventional RS to decode its private message by
decoding first the common message $\mathbf{s}_{c}^{[1]}$, while treating the
private messages $\mathbf{s}_{p}^{[1,1]}$ and $\mathbf{s}_{p}^{[2,1]}$ as
noise. After that, the decoded common message is eliminated by applying SIC,
giving user 1 the opportunity to decode its interference-free private message.
Not that, user 2 belonging to group 1 follows the same procedure to decode the
private message $\mathbf{s}_{p}^{[2,1]}$, as well as the other users belonging
to group 2. In the following, the methodology of the limited CSI-RS
transmission scheme is introduced for the general OWC scenario.

####  IV-B2 General case

We define the limited CSI-RS transmission scheme as a general framework that
is applicable for different scenarios. Thus, we consider an OWC system
composed of $L$ optical APs serving $K$ users. The application of the proposed
dynamic user-grouping algorithm arranges the $K$ users into a set of groups
$G$, and each group contains $K_{g}$ users. The proposed limited CSI-RS scheme
has the ability to deliver interference-free signals for the users in the
network as follows: the methodology of BIA is used to create a transmission
block comprising a number of alignment blocks allocated to each group. In
contrast to the conventional BIA schemes, the number of alignment blocks
allocated to each group is given by $\ell=\\{1,\dots,(L-1)^{G-1}\\}$, and
therefore, a total of $G\times(L-1)^{G-1}$ alignment blocks are built for all
the groups over a transmission block. To satisfy the rule of constructing
these alignment blocks while guarantee the ability to measure inter-group
interference by the users belonging to each group, the transmission block must
comprise $(L-1)^{G}+G\times(L-1)^{G-1}$ time slots. Once the construction of
the transmission block is determine, the BIA-based outer precoding matrix of
each group $\mathbf{B}^{[g]}$ can be designed by arranging each alignment
block $\ell$ into a column in the precoding matrix, the rows in that column
are given by $L\times L$ identity matrices corresponding to the time slots
forming that alignment block. For instance, in equation (20), the precoding
matrix $\mathbf{B}^{[1]}$ contains 2 $\times$ 2 identity matrices in the first
and second rows corresponding to the first and second time slots that form the
only alignment block allocated to group 1 given the number of transmitters and
groups in that use case. It is worth mentioning that the outer precoding
matrices for all the groups can be determined in this way following the
structure of the BIA transmission block without the need for CSI at
transmitters. Considering that the users belonging to the group use the same
outer precoding matrix, the transmitted signal for the general case can be
expressed as

$\mathbf{X}=\sum^{G}_{g=1}\mathbf{B}^{[g]}\bigg{(}\sqrt{P_{c}^{[g]}}\mathbf{w}_{c}^{[g]}\mathbf{s}_{c}^{[g]}+\sqrt{P_{p}^{[k,g]}}\mathbf{W}_{p}^{[g]}\mathbf{s}_{p}^{[g]}\bigg{)},$
(24)

Note that,
$\mathbf{X}=\text{col}\\{\mathbf{x}(\kappa)\\}_{\kappa=1}^{\mathcal{V}}$,
where $\mathcal{V}$ is the total number of time slots that form the
transmission block of BIA,
$\mathbf{W}_{p}^{[g]}=\bigg{[}\mathbf{w}_{p}^{[1,g]},\dots,\mathbf{w}_{p}^{[k,g]},\dots,\mathbf{w}_{p}^{[K_{g},g]}\bigg{]}$
is the inner precoding vector that contains the inner precoding matrices of
the private messages intended to the $K_{g}$ users belonging to group $g$, and
$\mathbf{s}_{c}^{[g]}=\text{col}\bigg{\\{}\mathbf{s}_{c,\ell}^{[g]}\bigg{\\}}_{\ell}^{(L-1)^{G-1}}$
contains the common messages transmitted to group $g$, $g\in G$, during the
entire transmission block, where $\mathbf{s}_{c,\ell}^{[g]}$ is the common
message transmitted over a certain alignment block $\ell$. Moreover,
$\mathbf{s}_{p}^{[g]}=\bigg{[}\mathbf{s}_{p}^{[1,g]},\dots,\mathbf{s}_{p}^{[k,g]},\dots,\mathbf{s}_{p}^{[K_{g},g]}\bigg{]}$,
where
$\mathbf{s}_{p}^{[k,g]}=\text{col}\bigg{\\{}\mathbf{s}_{p,\ell}^{[k,g]}\bigg{\\}}_{\ell}^{(L-1)^{G-1}}$
contains the private messages transmitted to user $k$, $k\in K_{g}$, over all
the alignment blocks allocated to group $g$, where
$\mathbf{s}_{p,\ell}^{[k,g]}$ is the private message transmitted over a
certain alignment block $\ell$. For detailed mathematical explanation on the
allocation of alignment blocks to multiple groups over the BIA transmission
block, we refer readers to [19]. Note that, power allocation among multiple
groups as well as the messages intended to the users of each group is
addressed in the next section.

The BIA-based outer precoding matrix of each group gives the opportunity for
$K_{g}$ users to measure and cancel the interference received due to serving
other $K_{g^{\prime}}$ users belonging to adjacent group $g^{\prime}$,
$g^{\prime}\neq g$. Without loss of generality, we consider a specific
alignment block $\ell$ allocated to group $g$, and formulate the received
signal of user $k$ belonging to that group, $k\in K_{g}$, after interference
cancellation, which can be expressed as

$\tilde{\mathbf{y}}^{[k,g]}=\begin{bmatrix}{\mathbf{h}^{[k,g]}(1)}^{T}\\\
\vdots\\\ {\mathbf{h}^{[k,g]}(L-1)}^{T}\\\
{\mathbf{h}^{[k,g]}(L)}^{T}\end{bmatrix}\bigg{(}\sqrt{P_{c}^{[g]}}\mathbf{w}_{c,\ell}^{[g]}\mathbf{s}_{c,\ell}^{[g]}+\sqrt{P_{p}^{[k,g]}}\mathbf{W}_{p,\ell}^{[g]}\mathbf{s}_{p,\ell}^{[g]}\bigg{)}+\begin{bmatrix}z^{[k,g]}[1]-\sum\limits_{g^{\prime}\neq
g}^{G}z^{[k,g]}[\tau_{g}^{\prime}]\\\ \vdots\\\
z^{[k,g]}[L-1]-\sum\limits_{g^{\prime}\neq
g}^{G}z^{[k,g]}[\tau_{g}^{\prime}]\\\ z^{[k,g]}[L]\\\ \end{bmatrix},$ (25)

where $\tilde{\mathbf{y}}^{[k]}\in\mathbb{R}_{+}^{L\times 1}$ and the temporal
index refers to the position in the alignment block rather than the
corresponding temporal index within the BIA transmission block for the sake of
simplicity. At this point, user $k$ belonging to group $g$ is subject to
intra-group interference as in equation (25), which can be measured and
canceled considering RS. The received signal of user $k\in K_{g}$ polluted by
intra-group interference can be rewritten as

$\tilde{\mathbf{y}}^{[k,g]}=\sqrt{P_{c}^{[g]}}{\mathbf{H}}^{[k,g]}\mathbf{w}_{c}^{[g]}\mathbf{s}_{c}^{[g]}+\sqrt{P_{p}^{[k,g]}}{\mathbf{H}}^{[k,g]}\mathbf{w}_{p}^{[k,g]}\mathbf{s}_{p}^{[k,g]}\underbrace{+\sum^{K_{g}}_{k^{\prime}\neq
k}\sqrt{P_{p}^{[k^{\prime},g]}}{\mathbf{H}}^{[k,g]}\mathbf{w}_{p}^{[k^{\prime},g]}\mathbf{s}_{p}^{[k^{\prime},g]}}_{\text{intra-
group interference}}+\tilde{\mathbf{z}}^{[k,g]}.$ (26)

Following the conventional RS procedure of intra-group interference
cancellation, user $k$ can decode its desired information at the cost of noise
enhancement. It is worth mentioning that the proposed limited CSI-RS scheme
results in low noise compared with traditional RS due to the fact that each
user measures and cancels the private messages sent to $K_{g}<K$ users
regardless of all other users in the network.

###  IV-C Achievable data rate

We derive now the achievable sum rate of the proposed limited CSI-RS
transmission scheme for the general case. Focussing on the use of RS within
each group, the message of user $k$ is divided into common and private
messages. These messages are decodable, and user $k$ belonging to group $g$
decodes first the common message treating all the private messages as noise,
and therefore, the SINR of the common message can be expressed as

$\gamma_{c}^{[k,g]}=\frac{P^{[g]}_{c}\left|\mathbf{w}^{[g]}_{c}\right|^{2}}{\sum_{k=1}^{K_{g}}P^{[k,g]}_{p}\left|\mathbf{w}^{[k,g]}_{p}\right|^{2}+{\sigma_{z}}^{2}}.$
(27)

Subsequently, user $k$ decodes its private message treating the private
messages intended to other users belonging to group $g$ as noise. That is, the
SINR of the private message is given by

$\gamma^{[k,g]}_{p}=\frac{P^{[k,g]}_{p}\left|\mathbf{w}^{[k,g]}_{p}\right|^{2}}{\sum^{K_{g}}_{k^{\prime}\neq
k}P^{[k^{\prime},g]}_{p}\left|\mathbf{w}^{[k^{\prime},g]}_{p}\right|^{2}+{\sigma_{z}}^{2}}.$
(28)

Considering the full procedure of the limited CSI-RS scheme in managing
interference among multiple groups in the network, the sum rate of the common
messages is given by

${R_{c}}=\sum_{g=1}^{G}b_{ab}\times\mathbb{E}\left[\log\det\left(\mathbf{I}+\gamma_{c}^{[k,g]}\mathbf{H^{[k,g]}}\mathbf{H^{[k,g]^{H}}}\mathbf{R_{\tilde{z}}}^{-1}\right)\right],$
(29)

where $b_{ab}$ is the ratio of the alignment blocks allocated uniformly among
the groups. For $L$ transmitters serving $G$ groups, it is given by
$\frac{1}{G+L-1}$. Moreover, $\mathbf{R_{\tilde{z}}}$ is the covariance
matrix, i.e.,

$\mathbf{R_{\tilde{z}}}=\begin{bmatrix}(G-1)\mathbf{I}_{{L}-1}&\mathbf{0}\\\
\mathbf{0}&1\\\ \end{bmatrix}.$ (30)

On the other hand, the sum rate of the private messages intended to $K_{g}$
users belonging to each group can be expressed as

$R_{p}^{[g]}=\sum^{K_{g}}_{k=1}b_{ab}\times\mathbb{E}\left[\log\det\left(\mathbf{I}+\gamma_{p}^{[k,g]}\mathbf{H^{[k,g]}}\mathbf{H^{[k,g]^{H}}}\mathbf{R_{\tilde{z}}}^{-1}\right)\right].$
(31)

As a result, the overall sum rate of the proposed scheme is equal to
$R_{\tiny{\text{limited CSI-RS}}}=R_{c}+\sum_{g=1}^{G}R_{p}^{[g]}$.

##  V Power Allocation in limited CSI-RS transmission

The proposed limited CSI-RS scheme reduces the requirements of CSI at
transmitters due to the formation of multiple groups guaranteeing that each
group contains users spatially clustered. In addition, the outer precoding
matrices of the formed groups are deigned without CSI following the structure
of the BIA transmission block. It is worth pointing out that power allocation-
based data rate maximization in the context of RS is a complex task since the
power values of the common and private messages are coupled (see equations
(27) and (28)). By exploiting the robustness of the proposed scheme to any
imperfection in CSI at transmitters, which means intra-group interference can
be relatively low, the power of the private message is defined as a vital
parameter that can highly dictate the sum rate of the users. In this context,
we derive a closed-form sub-optimal power allocation method that ensures data
rate maximization compared to simply allocating the power uniformly among the
messages of the users.

###  V-A Problem formulation

In particular, an objective function is defined to allocate the power among
the private messages intended for the users belonging to each group, while a
fixed power $P^{\dagger}_{c}$ is allocated to the common message. That is

$U(R_{\tiny{\text{limited CSI-
RS}}})=\sum^{G}_{g}\bigg{(}{R^{[g]}_{c}}(P^{\dagger}_{c},P^{[k,g]}_{p})+\sum^{K_{g}}_{k=1}{R^{[k,g]}_{p}}(P^{[k,g]}_{p})\bigg{)},$
(32)

Note that, ${R^{[g]}_{c}}(P^{\dagger}_{c},P^{[k,g]}_{p})$ and
${R^{[k,g]}_{p}}(P^{[k,g]}_{p})$ can be easily derived from equations (29) and
(31), respectively. In equation (18), each group is composed of a unique set
of users, and therefore, the overall data rate of the network can be maximized
by ensuring higher sum rate within each group. In this context, we formulate
an optimization problem that maximizes the minimum sum rate of the $K_{g}$
users belonging to each group through controlling power allocation among their
private messages as follows:

$\displaystyle\max_{p}$ $\displaystyle\bigg{\\{}\min_{g\in
G}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})\bigg{\\}}$ (33) s.t.
$\displaystyle{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})\geq
R^{[g]}_{min},~{}~{}~{}~{}$
$\displaystyle\sum_{k\in\mathcal{K}_{g}}P^{[k,g]}_{p}+P^{\dagger}_{c}\leq
P^{[g]}_{max},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall g\in G,$ $\displaystyle
P^{[k,g]}_{p}\leq
P^{T}_{p},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
k\in\mathcal{K}_{g},$ $\displaystyle P^{[k,g]}_{p}\geq 0,\sum_{g\in
G}P^{[g]}_{max}\leq P_{T},~{}~{}~{}~{}k\in\mathcal{K}_{g},g\in G,$

where
${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})=\bigg{(}{R^{[g]}_{c}}(P^{\dagger}_{c},P^{[k,g]}_{p})+\sum^{K_{g}}_{k=1}{R^{[k,g]}_{p}}(P^{[k,g]}_{p})\bigg{)}$.
The optimization problem in (33) is defined as max-min fractional program
under several constraints with a particular structure. It is classified as
concave-convex fractional program that has high complexity [32]. The first
constraint guarantees that the sum rate of the users belonging to a given
group $g$ is higher than the minimum data rate $R^{[g]}_{min}$ required to
ensure high quality of service. The second constraint controls the total power
allocated to the common and private messages intended to $K_{g}$ users, which
must be less than or equal to the maximum power dedicated to each group
$P^{[g]}_{max}$. The third constraint limits the power allocated to the
private message intended to a certain user, $k\in K_{g}$, where it must be
less than or equal to the total power $P^{T}_{p}$ devoted to send the private
messages of the $K_{g}$ users. The last constraint defines the feasible region
of the optimization problem, as well as ensures that the total power consumed
by $G$ groups is less than or equal to the maximum power consumption $P_{T}$
allowed in the network, which can be calculated according to [9] under eye
safety regulations due to the use of the VCSEL as an optical transmitter.

###  V-B Sub-optimal solution

The optimization problem in (33) can be solved using a parametric approach
[32], where it can be transformed into a convex optimization problem at a
given parameter value. Thus, the new objective function can be written as

$f(\rho)=\max_{p}\bigg{\\{}\min_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}\bigg{\\}},$
(34)

where $\rho=P^{[g]}_{T}\delta$, considering
$P^{[g]}_{T}=P^{\dagger}_{c}+\sum_{k\in\mathcal{K}_{g}}P^{[k,g]}_{p}$ as the
total power consumed by group $g$. Moreover, $\delta=\Big{(}\min\limits_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})/P^{[g]}_{T}\Big{\\}}\Big{)}$
is a non-negative parameter where at its optimal value, the power is allocated
among the private messages intended to $K_{g}$ users maximizing the minimum
sum rate of group $g$. According to [33], equation (34) can be optimally
determined under the constraints in equation (33) through finding a root of
$f(\rho)=0$ using Dinkelbach-type algorithm. From equations (33) and (34), the
optimization problem can be reformulated as

$\displaystyle\max_{p}$ $\displaystyle\bigg{\\{}\min_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}\bigg{\\}}$
(35) s.t. $\displaystyle\min_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho,$
$\displaystyle{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})\geq
R^{[g]}_{min},~{}~{}~{}~{}$
$\displaystyle\sum_{k\in\mathcal{K}_{g}}P^{[k,g]}_{p}+P^{\dagger}_{c}\leq
P^{[g]}_{max},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall g\in G,$ $\displaystyle
P^{[k,g]}_{p}\leq
P^{T}_{p},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
k\in\mathcal{K}_{g},$ $\displaystyle P^{[k,g]}_{p}\geq 0,\sum_{g\in
G}P^{[g]}_{max}\leq P_{T},~{}~{}~{}~{}k\in\mathcal{K}_{g},g\in G.$

Interestingly, the optimization problem in (35) is a convex optimization
problem due to the fact that it has a linear objective function under convex
constraints. Therefore, the optimal value can be equivalently found using a
distributed algorithm via Lagrangian decomposition [34, 35]. Given that, The
Lagrangian function of (35) considering other groups in the network can be
written as

$\mathcal{F}=\min_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}\left(1-\sum_{g\in
G}\lambda_{g}\right)+\sum_{g\in G}\mathcal{F}_{g},$ (36)

where

$\mathcal{F}_{g}=\Big{\\{}(\lambda_{g}+\xi_{g}){R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-(\delta\lambda_{g}+\nu_{g})(P^{\dagger}_{c}+\sum_{k\in\mathcal{K}_{g}}P^{[k,g]}_{p})\\\
+\beta^{[k]}_{g}(P^{T}_{p}-P^{[k,g]}_{p})-\xi_{g}R^{[g]}_{min}+\nu_{g}P^{[g]}_{max}\Big{\\}},$
(37)

and $\lambda_{g}$, $\xi_{g}$, $\nu_{g}$ and $\beta^{[k]}_{g}$ are Lagrangian
multipliers corresponding to the four constraints in equation (35),
respectively. Therefore, the optimal values of the power allocated to the
private messages in the network can be equivalently found through maximizing
$\mathcal{F}$ in equation (36).

Using the Karush-Kuhn-Tucker (KKT) conditions [36, 37], the partial derivative
$\frac{\partial\mathcal{F}_{g}}{\partial P^{[k,g]}_{p}}$ is defined as a
monotonically decreasing function with respect to the power allocated
$P^{[k,g]}_{p}$ to the private message intended to user $k$ belonging to group
$g$. That is, if the partial derivative
$\frac{\partial\mathcal{F}_{g}}{\partial
P^{[k,g]}_{p}}|_{{P^{[k,g]}_{p}}=0}\leq 0$, the optimum power value
$P^{*[k,g]}_{p}$ equals zero. Moreover, if the partial derivative
$\frac{\partial\mathcal{F}_{g}}{\partial
P^{[k,g]}_{p}}|_{{P^{[k,g]}_{p}}=1}\geq 0$, the optimum value $P^{*[k,g]}_{p}$
equals one. Otherwise, the optimum power value $P^{*[k,g]}_{p}$ can be found
as follows

####  V-B1 Optimality at $\beta^{[k]}_{g}$ and $\nu_{g}$ values

at given values for $\rho$, $\lambda_{g}$ and $\xi_{g},$ $\forall g\in G$,
$P^{*[k,g]}_{p}$ can be calculated by solving the following equation for each
user, $k\in K_{g}$, belonging to group $g$

$\frac{\partial\mathcal{F}_{g}}{\partial
P^{[k,g]}_{p}}=(\lambda_{g}+\xi_{g})\frac{\partial{R^{[k,g]}_{p}}(P^{[k,g]}_{p})}{\partial
P^{[k,g]}_{p}}-(\delta\lambda_{g}+\nu_{g})-\beta^{[k]}_{g}+\xi_{g}\frac{\partial{R^{[g]}_{c}}(P^{\dagger}_{c},P^{[k,g]}_{p})}{\partial
P^{[k,g]}_{p}}=0.$ (38)

Therefore, equation (37) can be modified taking into consideration the optimum
value of the private message from (38) as follows

$\mathcal{F}^{*}_{g}=\Big{\\{}(\lambda_{g}+\xi_{g}){R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{*[k,g]}_{p})-(\delta\lambda_{g}+\nu_{g})(P^{\dagger}_{c}+\sum_{k\in\mathcal{K}_{g}}P^{*[k,g]}_{p})\\\
+\beta^{[k]}_{g}(P^{T}_{p}-P^{*[k,g]}_{p})-\xi_{g}R^{[g]}_{min}+\nu_{g}P^{[g]}_{max}\Big{\\}}.$
(39)

At this point, the gradient projection method can be applied to solve the dual
problem, then, updating the Lagrangian multipliers $\beta^{[k]}_{g}$ and
$\nu_{g}$ as follows

$\beta^{[k]}_{g}(\kappa)=\left[\beta^{[k]}_{g}(\kappa-1)-\epsilon_{1}(\kappa-1)\left(P^{T}_{p}-P^{[k,g]}_{p}(\kappa-1)\right)\right]^{+},$
(40)
$\nu_{g}(\kappa)=\bigg{[}\nu_{g}(\kappa-1)-\epsilon_{2}(\kappa-1)\Big{(}P^{[g]}_{max}-\Big{(}P^{\dagger}_{c}+\sum_{k\in\mathcal{K}_{g}}P^{[k,g]}_{p}(\kappa-1)\Big{)}\Big{)}\bigg{]}^{+},$
(41)

where $\kappa$ denotes the iteration of the gradient algorithm, and $[.]^{+}$
is a projection on the positive orthant to take into account the fact that we
have $\beta^{[k]}_{g},\nu_{g}\geq 0$. Moreover, $\epsilon_{1}$ and
$\epsilon_{2}$ are sufficient small step sizes at a given iteration
$(\kappa-1)$ that are taken in the direction of the negative gradient for the
multipliers $\beta^{[k]}_{g}$, $\nu_{g}$, respectively. According to equation
(41), the Lagrangian multiplier $\beta^{[k]}_{g}$ works as a message to ensure
that the power allocated to the private message of user $k$ satisfies the
maximum power allowed for each private message. While in equation (41),
$\nu_{g}$ works to ensure that the overall power allocated to the private
messages intended to $K_{g}$ users satisfies the maximum power constraint of
group $g$. Note that at each iteration, the multipliers $\beta^{[k]}_{g}$ and
$\nu_{g}$ are updated until the optimal value $P^{*[k,g]}_{p}$ is found.

Algorithm 1 Overall power Allocation algorithm

1:Input: $G$ and $\rho$ ;

2:Initialisation: $\lambda_{g}\geq 0$, $\xi_{g}\geq 0$, $\kappa=1$, $I_{1}=1$;

3:All $K_{g}$ users of each group broadcast their demands;

4:All APs set common values for $\lambda_{g}$ and $\xi_{g}$ for each group;

5:while $I_{1}=1$ do

6: Initialisation: $\beta^{[k]}_{g}\geq 0$, $\nu_{g}\geq 0$, $I_{2}=1$;

7: while $I_{2}=1$ do;

8: for Each $g\in G$ do;

9: for Each user $k\in\mathcal{K}_{g}$ do

10: Allocate power according to (38);

11: Update $\beta^{[k]}_{g}$ according to (40) ;

12: end for

13: Update $\nu_{g}$ according to (41) ;

14: end for

15: if $\mid P^{[k,g]}_{p}(\kappa)-P^{[k,g]}_{p}(\kappa-1)\mid\leq\epsilon$
then;

16: $I_{2}=0$;

17: else

18: ${\kappa\longleftarrow\kappa+1}$;

19: end if

20: end while;

21: Initialisation:$I_{3}=1$;

22: while $I_{3}=1$ do;

23: APs determine
$\Gamma(\kappa)=\min\limits_{g}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}$
;

24: if
${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho=\Gamma(\kappa)$ or
${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho=\Gamma(\kappa)$
for every group with $\lambda_{g}=0$ then;

25: $I_{2}=0$;

26: else

27: Update $\lambda_{g}$ for each group ;

28: Update power allocation according to (42) ;

29: end if

30: end while;

31: if
${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})(\kappa)-{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})(\kappa-1)\mid\leq\epsilon$
then;

32: $I_{1}=0$;

33: else

34: $\xi_{g}$ for each group is updated according to (45) ;

35: ${\kappa\longleftarrow\kappa+1}$;

36: end if

37:end while

38:Output:$P^{*[k,g]}_{p}$

####  V-B2 Optimality at $\lambda_{g}$ and $\xi_{g}$ values

Interestingly, the multipliers $\lambda_{g}$ and $\xi_{g}$ must also be
updated to guarantee high quality of service for the $K_{g}$ users of each
group $g$, where at their optimal values, the power allocated to each private
message determined from equation (38) at the optimal values of
$\beta^{*[k]}_{g}$ and $\nu^{*}_{g}$ must be modified to satisfy the first and
second constraints in equation (35). First, at a given value of $\xi_{g}$, the
optimal value of $\lambda_{g}$ can be determined by applying the KKT
conditions. That is, $\frac{\partial\mathcal{F}}{\partial\Gamma}=0$, where
$\Gamma=\min\limits_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}$.
From equation (36), $\sum_{g\in G}\lambda_{g}=1$. Therefore, the optimum power
value $P^{*[k,g]}_{p}$ allocated to the private message intended to each user
$k\in K_{g}$ belonging to group $g$ can be modified according to

$\frac{\partial\mathcal{F}_{g}}{\partial
P^{[k,g]}_{p}}=({\lambda}_{g}+\widetilde{\xi}_{g})\frac{\partial{R^{[k,g]}_{p}}(P^{[k,g]}_{p})}{\partial
P^{[k,g]}_{p}}-(\delta\lambda_{g}+\widetilde{\nu}^{*}_{g})-\beta^{*[k]}_{g}+\widetilde{\xi}_{g}\frac{\partial{R^{[g]}_{c}}(P^{\dagger}_{c},P^{[k,g]}_{p})}{\partial
P^{[k,g]}_{p}}=0,$ (42)

where $\widetilde{\xi}_{g}={\xi}_{g}\sum_{g\in G}\lambda_{g}$,
$\widetilde{\nu}_{g}={\nu}_{g}\sum_{g\in G}\lambda_{g}$ and
$\widetilde{\xi}_{g}={\xi}_{g}\sum_{g\in G}\lambda_{g}$. Similar to equations
(40) and (41), the gradient projection method can be applied to obtain an
updated value for the Lagrangian multiplier $\lambda_{g}$ as follows

$\lambda_{g}(\kappa)=\bigg{[}\lambda_{g}(\kappa-1)-\epsilon_{3}(\kappa-1)\bigg{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})(\kappa-1)-\rho-\min\limits_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}\bigg{)}\bigg{]}^{+},$
(43)

where $\epsilon_{3}(\kappa-1)$ is a sufficiently small step size at a given
iteration $(\kappa-1)$. Note that the optimal value $\lambda^{*}_{g}$
determines the optimal power value allocated to the private message intended
to each user for a given value $\xi_{g}\geq 0$. To find the optimal value
$\lambda^{*}_{g}$, according to the first constraint in (35),
$\Big{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{)}\geq\Gamma^{*}$,
where
$\Gamma^{*}=\min\limits_{g}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}$.
Let $\widetilde{G}$ give the total number of the groups with
$\Big{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{)}>\Gamma^{*}$,
and then, using the KKT conditions, the optimal value $\lambda^{*}_{g}=0$
given that

$\lambda^{*}_{g}\bigg{\\{}\Big{(}{R^{*[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\delta\Big{(}P^{\dagger}_{c}+\sum\limits_{k\in
K_{g}}P^{*[k,g]}_{p}\Big{)}\Big{)}-\Gamma^{*}\bigg{\\}}=0.$ (44)

Therefore, for a given value of $\xi_{g}\geq 0$, if
$\Gamma^{*}>\Big{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{)}$,
${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho$ cannot be less
than
$\min\limits_{g}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}$.
As a consequence, the optimal value for any group $g^{\prime}$,
$g^{\prime}\neq g$ not within the group set $\widetilde{G}$ is obtained when
$\lambda^{*}_{g}>0$ and
$\Gamma^{*}=\Big{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{)}$.
Otherwise,
$\Big{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{)}>\Gamma^{*}$
for any group with $\lambda^{*}_{g}=0$.

The optimal value of $\xi^{*}_{g}$ must also be determined according to

$\xi_{g}(\kappa)=\bigg{[}\xi_{g}(\kappa-1)-\epsilon_{4}(\kappa-1)\bigg{(}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})(\kappa-1)-R^{[g]}_{min}\bigg{)}\bigg{]}^{+},$
(45)

where $\epsilon_{4}$ is a sufficiently small step size. It is worth pointing
out that the overall algorithm iterates over the power allocated to the
private message intended to each user, until the optimal value
$P^{*[k,g]}_{p}$ is determined, which can significantly maximize the sum rate
of the $K_{g}$ users belonging to each group with respect to power
consumption. For better understanding, Algorithm 1 presents a summary of the
overall algorithm, where $\epsilon$ is a small tolerance value.

TABLE I: Simulation Parameters Parameter | Value
---|---
Transmitter Bandwidth | 5 GHz
Laser Wavelength | 850 nm
Laser beam waist |  $5\mu$m
Physical area of the photodiode | 15 $\text{mm}^{2}$
Receiver FOV | 60 deg
Detector responsivity | 0.9 A/W
Gain of optical filter | 1.0
Laser noise |  $-155~{}dB/H$z

##  VI performance evaluation

In an indoor environment with 8m$\times$ 8m$\times$ 3m dimensions, $L=4\times
4$ APs are deployed on the ceiling, each AP with $L_{v}\times L_{v}$ VCSELs,
to serve $K=20$ users randomly distributed on the receiving plane located at
2m distance from the ceiling. Each user is equipped with an optical detector
consisting of $M$ photodiodes that has the ability to provide a set of
linearly independent channel responses and guarantee connectivity to almost
all the APs in the room. All other simulation parameters are listed in Table
1.

Figure 5: User grouping versus a set of iterations for the proposed dynamic
algorithm.

In Fig. 5, the sum rates of the user grouping methods are depicted against the
iterations considering different numbers of groups $G=\\{4,6,8\\}$. It is
shown that the dynamic algorithm provides sub-optimal solutions significantly
close to the optimal solutions obtained from solving the optimization problem
in (14) through exhaustive search. It is worth noticing that the number of
groups affects the sum rate of the proposed limited CSI-RS scheme with fixed
power allocation. For instance, the proposed scheme provides higher sum rate
26.6 [bits/s/Hz] with $G=4$, i.e., $\mathcal{K}_{m}=4$ and
$\mathcal{K}_{ed}=16$, compared to the scenarios of $6$ and $8$ groups. This
is because of the trade-offs between the sum rate and the number of groups. In
other words, increasing the number of groups results in a larger transmission
block and noise enhancement (see equations (29), (30) and (31)), which limit
the performance of BIA used as an outer precoder to align inter-group
interference. Note that, having a high number of groups reduces the complexity
of MUI management belonging to each group using RS. However, it is at the cost
of enhancing the noise among the formed groups.

Figure 6: The sum rate of the proposed limited CSI-RS scheme after power
allocation under different constraints. $K=20$, $\mathcal{K}_{m}=4$ and
$\mathcal{K}_{ed}=16$.

In Fig. 6, the performance of the proposed scheme is shown against a set of
iterations after performing the optimization of power allocation. It can be
seen that the sum rate of the proposed scheme can be enhanced considerably
after maximizing the minimum sum rate of the users belonging to each group
where the power budget can be utilized efficiently compared to simply dividing
the power among the common and private messages intended to the users
regardless of their requirements. The figure further shows that the
reformulation of the optimization problem using multiple multipliers relaxes
the complexity while providing solutions close to the optimal. Specifically,
the optimization problem is solved using the multipliers $\beta^{[k]}_{g}$ and
$\nu_{g}$ in the red curve, and therefore, it can be seen that the sum rate is
30.2 [bits/s/Hz] compared to 38 [bits/s/Hz] achieved from solving the main
power allocation problem, which is due to the fact that the power is allocated
to maximize the sum rate of each group independently and without guaranteeing
the quality of service of each user, i.e., without the constraints
$\min\limits_{g\in
G}\Big{\\{}{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho\Big{\\}}\leq{R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})-\rho$
and ${R^{[g]}_{\text{sum}}}(P^{\dagger}_{c},P^{[k,g]}_{p})\geq R^{[g]}_{min}$.
On the other hand, when the overall algorithm iterates over the four
multipliers $\beta^{[k]}_{g}$, $\nu_{g}$, $\lambda_{g}$ and $\xi_{g}$ in the
green curve, a significant solution is provided close to the optimal one. This
solution is considered for the rest of the results due to its practicality in
terms of complexity.

Figure 7: The sum rate of the limited CSI-RS scheme versus SNR compared with
benchmark schemes. $K=20$, $\mathcal{K}_{m}=4$ and $\mathcal{K}_{ed}=16$.

The sum rate of the proposed scheme is shown in Fig. 7 against a range of SNR
values to demonstrate its superiority compared with BIA and RS schemes used to
derive its principles. It is shown that the proposed scheme provides higher
sum rates than BIA and RS at different SNR values. It is worth mentioning that
the proposed scheme divides the users into multiple groups, and then, by
following the transmission block of BIA determined by the number of groups,
the outer precoding matrices of the groups are designed with no CSI to
transmit the common and private messages intended to each group. In BIA, the
number of users determines the length of the transmission block, and each user
must subtract the information transmitted to $K-1$ users. Therefore, the sum
rate of the users slightly increases with increase in the SNR. Furthermore, RS
suffers performance degradation in a high density network consisting of a high
number of transmitters serving multiple users due to the cost of providing CSI
at transmitters in such scenarios, and the fact that each user performs SIC to
cancel the interference caused by the transmission to all other users, which
has a negative impact on the achievable sum rate of the network.

Figure 8: The sum rate of the limited CSI-RS scheme versus $K$ users compared
with benchmark schemes.

In Fig. 8, the performance of the proposed scheme is depicted against
different numbers of users compared to BIA and RS. It can been seen that the
sum rate of the network increases with the number of users regardless of the
transmission scheme considered for multi-user interference management since it
represents the aggregate data rate determined from the sum of the user rates.
The proposed scheme is more suitable for the OWC network as the number of
users increases, providing higher sum rate in all the scenarios due to its
unique way in handling interference while minimizing the noise resulting from
interference cancellation. Note that, the application of the proposed power
allocation technique further enhances the performance of the limited CSI-RS
scheme compared to the fixed power allocation method. Moreover, the sum rates
of RS and BIA schemes increase slightly with the number of users as each user
receives a low data rate subject to high noise and errors caused by the
limitations of these schemes in serving such a high number of users.

Figure 9: BER for 2-PAM Modulation. $\mathcal{K}_{m}=4$ and
$\mathcal{K}_{ed}=16$.

To evaluate the BER of the limited CSI-RS scheme, multilevel N-ary pulse
amplitude modulation (N-PAM) is considered where the mean of the transmitted
signal is equal to the current, which ensures the linear response of the
optical transmitter. In Fig. 9, BER for $K=20$ users is plotted against a
range of SNR values, (5,15) dB, considering different transmission schemes. It
can be seen that the limited CSI-RS is superior compared to RS and BIA schemes
at different SNR values. For instance, the proposed scheme achieves a BER
value less than $10^{-3}$ at 14 dB SNR with and without the power allocation
optimization problem, while RS and BIA schemes at the same value of SNR
achieve BER values higher than $10^{-3}$ and slightly less than $10^{-2}$,
respectively. This is owing to the fact that each user $k$ experiences high
noise resulting from the interference cancellation of the information
transmitted to all other users, $k^{\prime}\neq k$, in the case of RS and BIA
schemes.

Figure 10: Energy efficiency of the limited CSI-RS scheme versus $K$ users.

To further test the effectiveness of the proposed scheme, its energy
efficiency, which is one of the most important metrics in wireless
communications, is depicted in Fig. 10 against a higher number of users
compared with the conventional RS scheme. It can be seen that the energy
efficiency of the network decreases as the number of users increases up to
$40$ users due to the fact that more power is consumed at a high number of
users. However, the limited CSI-RS scheme is more energy efficient than the RS
scheme as the number of users increases from 20 to 40 users. This behavior is
expected due to the implementation of the user grouping algorithm and the use
of the BIA-based outer precoder to eliminate inter-group interference.
Furthermore, it can be seen that the overall power allocation algorithm
distributes the power among the messages of the users more efficiently based
on the sum rate maximization of the network resulting in higher energy
efficiency compared with the fixed power allocation scheme that has low
complexity at the cost of serving users regardless of the quality of service.

##  VII Conclusions

In this work, a novel RS-based transmission scheme is proposed in an OWC
network to serve multiple users simultaneously maximizing the spectral
efficiency of the network. We first formulate an optimization problem where
users are divided into multiple groups under fixed power allocation. Then, a
dynamic algorithm is designed to relax the complexity, while guaranteeing the
formation of each group with a unique set of users. After that, a limited CSI-
RS scheme is derived to manage inter-group interference using an outer
precoder designed with limited CSI to the channel coherence time and the
distribution of users, while the users belonging to each group receive their
desired information following the methodology of RS. Finally, an optimization
problem is formulated to maximize the minimum sum rate within each group
through finding the optimum power allocated to the private message intended to
each user. The power allocation is reformulated via multiple multipliers with
the aim of reducing complexity, while providing sub-optimal solutions. The
results show that the proposed limited CSI-RS scheme achieves high performance
in terms of sum rate, BER and energy efficiency compared to the benchmark
schemes considered.

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