#  Optimal Discrete Constellation Inputs for Aggregated LiFi-WiFi Networks

Shuai Ma, Fan Zhang, Songtao Lu, Hang Li, , Ruixin Yang, Sihua Shao, Jiaheng
Wang, , Shiyin Li  S. Ma, F. Zhang, R. Yang and S. Li are with the School of
Information and Control Engineering, China University of Mining and
Technology, Xuzhou 221116, China. (e-mail: {mashuai001;zhangfan; ray.young;
lishiyin}@cumt.edu.cn).S. Lu is with the IBM Thomas J. Watson Research Center,
Yorktown Heights, NY 10598 USA (e-mail: songtao@ibm.com).H. Li is with the
Shenzhen Research Institute of Big Data, Shenzhen 518172, Guangdong, China.
(email: hangdavidli@163.com).S. Shao is with the Department of Electrical
Engineering, New Mexico Tech, Socorro, NM 87801 USA. (email:
sihua.shao@nmt.edu).J. Wang is with the National Mobile Communications
Research Laboratory, Southeast University, Nanjing 210096, China, and also
with the Purple Mountain Laboratories, Nanjing 211111, China. (e-mail:
jhwang@seu.edu.cn).

###### Abstract

In this paper, we investigate the performance of a practical aggregated LiFi-
WiFi system with the discrete constellation inputs from a practical view. We
derive the achievable rate expressions of the aggregated LiFi-WiFi system for
the first time. Then, we study the rate maximization problem via optimizing
the constellation distribution and power allocation jointly. Specifically, a
multilevel mercy-filling power allocation scheme is proposed by exploiting the
relationship between the mutual information and minimum mean-squared error
(MMSE) of discrete inputs. Meanwhile, an inexact gradient descent method is
proposed for obtaining the optimal probability distributions. To strike a
balance between the computational complexity and the transmission performance,
we further develop a framework that maximizes the lower bound of the
achievable rate where the optimal power allocation can be obtained in closed
forms and the constellation distributions problem can be solved efficiently by
Frank-Wolfe method. Extensive numerical results show that the optimized
strategies are able to provide significant gains over the state-of-the-art
schemes in terms of the achievable rate.

###### Index Terms:

Aggregated LiFi-WiFi system, power allocation, probabilities allocation,
discrete constellation inputs.

##  I Introduction

###  I-A Motivation and Contributions

The increasing number of Internet of Things (IoT) devices exerts tremendous
bandwidth burden continuously on the wireless networks. According to Ericsson’
report [1], more than $80\%$ of the wireless data are generated in indoor
environments. Visible light communication (VLC) or light fidelity (LiFi), with
a vast license-free visible band in 400-790 THz, can support both high speed
data transmission and illumination simultaneously. LiFi exploits the off-the-
shelf light emitting diodes (LEDs) and photodiodes (PDs) as transceivers,
which can be integrated into IoT devices. Although LiFi serves as a
competitive candidate for the next generation wireless solution, its
vulnerability to the blockage and small its signal coverage still impose many
challenges on various indoor applications. Therefore, a more practical
solution is to aggregate the LiFi and WiFi systems and exploit their unique
advantages under certain conditions.

In this paper, we consider an aggregated LiFi-WiFi system from a practical
communication perspective. First of all, we derive the achievable rate
expression of the system with the discrete constellation input signals, rather
than the Gaussian inputs adopted in most of the existing works. Then, we
further investigate the optimal input distribution and power allocation for
the considered system. Our results provide a relatively practical design
framework for the aggregated LiFi-WiFi communication system. Specifically, the
main contributions of this work are given as follows:

  * •

Generally, the inputs of the practical communications systems follow an
finite-set discrete distribution rather than Gaussian distributions. To obtain
the performance description of the aggregated LiFi-WiFi system with an
arbitrary discrete distribution, we derive the achievable rate expressions of
LiFi links and WiFi links, respectively. Comparing with the existing rate
expressions with equiprobable discrete constellation points, the derived
results are more general and practical. Given that such rate expression is not
in closed-form, we further derive both the lower and the upper bounds. All
these results can be used as the performance metric for the considered system.

  * •

We jointly optimize the discrete constellation input distribution and the
power allocation to maximize the derived achievable rate. To handle this non-
convex problem, we propose a multi-level mercy-filling method to obtain the
suboptimal power allocation scheme, which exploits the relationship between
the mutual information and the minimum mean square error (MMSE). The optimal
probability distributions of the discrete constellation are calculated by the
inexact gradient descent method.

  * •

To reduce the computation complexity in the previous design problem, we
further adopt the derived lower bound as the performance metric. Specifically,
we jointly optimize the discrete constellation input distribution and the
power allocation to maximize the derived lower bound. To overcome the
difficulty of the nonconvex problem, we iteratively optimize the power
allocation and discrete probability distribution, where the optimal power
allocation scheme is derived with closed expression, and the discrete
probability distribution sub-problems is optimized via the proposed Frank-
Wolfe method.

###  I-B Related Works and Organization

There are two schemes for the LiFi-WiFi transmission: hard-switching (hybrid)
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and aggregating [10, 11, 12, 13, 14, 15].
The former realizes the transmission via LiFi link or WiFi link, while the
later uses both links simultaneously. In general, the former scheme usually
results in a lower spectral efficiency and may cause frequent link switches.
The LiFi-WiFi aggregated systems, on the contrary, can increase the system
data rate and provide reliable communication.

The LiFi-WiFi aggregated system receives great attentions in recent years. For
example, by leveraging the bonding technique in the Linux operating system,
the authors in [10] prove that the aggregated system outperforms the
conventional WiFi. Based on the cross-layer analysis of the physical and
datalink layers, link selection approaches were proposed in [11] to maximize
the average data arrival rate and minimize the non-asymptotic bounds on data
buffering delay. In [12], the coverage probability and rate were analyzed for
the aggregated RF/VLC networks. It has been shown that selecting the correct
intensity of OBSs (optical BSs) plays a crucial role, and the performance of
the aggregated scheme outperforms all other schemes (RF-only, VLC-only, and
opportunistic RF/VLC). In [13], the access point selection strategies were
optimized via the multi-armed bandit scheme. In [14], both subchannel
allocation and power control scheme were developed for improving the energy
efficiency of the aggregated VLC/RF network.

In most of existing works, e.g. [10, 7, 8, 9, 13, 14, 11, 12, 15], the
achievable rate of the considered system is derived based on the assumption
that the input signal follows the Gaussian distribution. However, Gaussian
signals may not be an accurate representation of the inputs in practical
communication systems. The inputs of the practical RF-only system are always
generated based on the discrete constellations, such as pulse amplitude
modulation (PAM), quadrature amplitude modulation (QAM), and phase shift
keying (PSK), etc., rather than the Gaussian codebook. Thus, those resource
allocation strategies based on the Gaussian assumption would lead to serious
performance loss. It has been shown that the mutual information maximization
strategies can improve performance of the practical communications systems
[16]. Meanwhile, it is proved that the optimal inputs of the VLC-only system
follow a finite-set discrete distribution [17]. Although some existing works
studied the achievable rate based on the discrete constellation points with
equal probability [18, 19, 20, 16, 21, 22], those results cannot be directly
extended to the LiFi-WiFi systems with arbitrary discrete inputs. Therefore,
it is necessary to investigate the optimal transmission scheme with the
discrete inputs for the aggregated LiFi-WiFi system.

The rest of this paper is organized as follows. We provide the model of the
aggregated LiFi-WiFi system in Section II. The optimal power allocation and
optimal probability distributions schemes of the aggregated LiFi-WiFi system
are presented in Section III. In Section IV, we provide the solutions for the
lower bound as the throughput metric. The simulation results are presented in
Section V. Finally, Section VI concludes the paper.

_Notations_ : ${\left(\cdot\right)^{\rm{T}}}$, $\left(\cdot\right)^{*}$,
$\left\|\cdot\right\|$, and ${\rm{Tr}}\left(\cdot\right)$ represent the
transpose, conjugate, Frobenius norm, and trace of a matrix respectively. The
Hadamard product of ${\bf{A}}$ and ${\bf{B}}$ is denoted as
${\bf{A}}\odot{\bf{B}}$. $\mathcal{M}\triangleq\left\\{{1,2,...,M}\right\\}$
and $\mathcal{N}\triangleq\left\\{{1,2,...,N}\right\\}$.

##  II System model

Figure 1:  Schematic diagram of an aggregated LiFi-WiFi system.

As illustrated in Fig. 1, we consider the downlink transmission of an
aggregated LiFi-WiFi system, where the transmitter is equipped with single LED
and single WiFi antenna, and the receiver is equipped with single PD and
single RF antenna. The transmitter simultaneously transmits information via
both the LiFi link and the WiFi link, where the bandwidths of LiFi link and
WiFi link are $B_{1}$Hz and $B_{2}$Hz, respectively. Let
${\bf{x}}\buildrel\Delta\over{=}\left[{{x_{1}},{x_{2}}}\right]^{T}$ denote the
transmitted signal vector, where ${x_{1}}\in{\mathbb{R}}$ and
${x_{2}}\in{\mathbb{C}}$ denote the independently transmitted signals of the
LiFi link and WiFi link, respectively.

In a practical LiFi-WiFi communication system, the transmitted signals are
distributed in discrete constellation. Suppose that the LiFi link signal is
sent via $M$-pulse-amplitude modulation ($M$-PAM) and the WiFi link signal is
sent via $N$-quadrature amplitude modulation ($N$-QAM). To be more specific,
the signal $x_{1}$ is taken from a non-negative real discrete constellation
set ${\Omega_{1}}$ with cardinality $M$, which is given as

$\displaystyle{\Omega_{1}}\buildrel\Delta\over{=}$
$\displaystyle\left\\{{{x_{1}}\left|\begin{array}[]{l}\Pr\left({{x_{1}}={x_{1,k}}}\right)={p_{1,k}},\\\
0\leq{x_{1,k}}\leq A,\\\ \sum\limits_{k=1}^{M}{{p_{1,k}}}=1,\\\
\sum\limits_{k=1}^{M}{{p_{1,k}}}{x_{1,k}}\leq\bar{\mu},\\\
\sum\limits_{k=1}^{M}{{p_{1,k}}}x_{1,k}^{2}\leq P_{{\rm{e}},1},\\\
x_{1,k}\in{\mathbb{R}},k=1,...,M,\end{array}\right.}\right\\},$ (7)

where ${x_{1,k}}$ denotes the constellation point, ${{p_{1,k}}}$ represents
probability that $x_{1}$ equals ${x_{1,k}}$, and parameters $A$, $\bar{\mu}$,
and $P_{{\rm{e}},1}$ denotes the peak optical power, maximum average optical
power, and maximum average electric power of $x_{1}$, respectively. On the
other hand, the WiFi signal ${x_{2}}$ is taken from a complex discrete
constellation set ${\Omega_{2}}$ with cardinality $N$, which is given as

$\displaystyle{\Omega_{2}}\buildrel\Delta\over{=}\left\\{x_{1}\left|\begin{array}[]{l}\Pr\left({{x_{2}}={x_{2,l}}}\right)={p_{2,l}},\\\
\sum\limits_{l=1}^{N}{{p_{2,l}}}=1,\\\
\sum\limits_{l=1}^{N}{{p_{2,l}}{{\left|{{x_{2,l}}}\right|}^{2}}}\leq{P_{{\rm{e}},2}},\\\
{x_{2,l}}\in{\mathbb{C}},l=1,...,N,\end{array}\right.\right\\}$ (12)

where ${x_{2,l}}$ denotes the constellation point, ${{p_{2,l}}}$ denotes the
probability that the constellation ${x_{2,l}}$ is chosen, $P_{{\rm{e}},2}$
denotes the maximum average electric power of $x_{2}$, respectively.

Let ${q}_{1}\in{\mathbb{R}}$ and ${{q}}_{2}\in{\mathbb{C}}$ denote the power
amplification factors for ${x_{1}}$ and ${x_{2}}$, respectively. Here,
${q}_{1}$ and ${{q}}_{2}$ need to satisfy the average electrical power
constraint, i.e.,

$\displaystyle{\eta_{1}}{\varepsilon_{\text{1}}}q_{1}^{2}+{\eta_{2}}{\varepsilon_{\text{2}}}{\left|{{q_{2}}}\right|^{2}}\leq{P_{\text{T}}},$
(13)

where ${\eta_{1}}$ and ${\eta_{2}}$ denote the efficiency of the power
amplifier of the LiFi link and WiFi link, respectively;
${\varepsilon_{\text{1}}}=\sum\limits_{k=1}^{M}{{p_{1,k}}}x_{1,k}^{2}$,
${\varepsilon_{\text{2}}}=\sum\limits_{l=1}^{N}{{p_{2,l}}}{{\left|{{x_{2,l}}}\right|}^{2}}$;
and ${P_{\text{T}}}$ denotes the total electrical power threshold. Moreover,
for human eye safety considerations, the power control over the LiFi signal
also needs to meet the average optical power and peak optical power
requirement:
${\mathbb{E}}\left\\{{{{q_{1}}}{x_{1}}}\right\\}=q_{1}\mu\leq{P_{\rm{o}}}$,
and $q_{1}A\leq{P_{\rm{ins}}}$, where
$\mu=\sum\limits_{k=1}^{M}{{p_{1,k}}}{x_{1,k}}$; ${P_{\rm{o}}}$ and
${P_{\rm{ins}}}$ denote the maximum average optical power threshold and the
instant optical power threshold, respectively.

Let
${{\bf{g}}}\buildrel\Delta\over{=}\left[{{{g}}_{1}},{{{g}}_{2}}\right]^{T}$
denotes the channel vector, where ${g}_{1}$ and ${g}_{2}$ are the channel
parameters of the LiFi link and WiFi link, respectively. Assume that the
channel parameters $g_{1}$ and $g_{2}$ are quasi-static in the paper.
Specifically, ${g}_{1}$ is given by [23]

$\displaystyle{g_{1}}=\frac{{\left({m+1}\right){A_{1}}}}{{2\pi
d_{1}^{2}}}{\cos^{m}}\left(\phi\right)\cos\varphi{g_{f}}{g_{c}}\left(\varphi\right),$
(14)

where $m$ is the order of the Lambertian emission; $A_{1}$ is the detector
area of PD receiver; $d_{1}$ is the distance between the LED and PD; ${\phi}$
and $\varphi$ are the radiance and incidence angle of the LiFi link; $g_{f}$
denotes the gain of the optical filter; ${g_{c}}\left(\varphi\right)$ denotes
the gain of optical concentrator, which is given as

$\displaystyle{g_{c}}\left(\varphi\right)=\left\\{{\begin{array}[]{*{20}{c}}{\frac{{{n^{2}}}}{{{{\sin}^{2}}\left({{\Psi_{c}}}\right)}},}&{0\leq\varphi\leq{\Psi_{c}},}\\\
{0,}&{{\rm{otherwise}},}\end{array}}\right.$ (17)

where $n$ is the refractive index, $\Psi_{c}$ represents the field-of-view
(FoV) of the LiFi receiver,

As to the WiFi link, ${g}_{2}$ is modeled by[8]

$\displaystyle{g_{2}}=\left\\{{\begin{array}[]{*{20}{l}}{{g_{r}}{{10}^{-\left({{{L_{\rm{F}}}\left({{d_{2}}}\right)+{L_{\sigma}}}}\right)/{{20}}}},{d_{2}}\leq{d_{\rm{B}}}}\\\
{{g_{r}}{{10}^{-\left({{L_{\rm{F}}\left({{d_{2}}}\right)+35{{\log}_{10}}\left({\frac{{{d_{2}}}}{{{d_{\rm{B}}}}}}\right)+{L_{\sigma}}}}\right)/{{20}}}},{d_{2}}>{d_{\rm{B}}},}\end{array}}\right.$
(20)

where
$g_{r}=\left({{\sqrt{\frac{K}{{K+1}}}{e^{j\psi}}+\sqrt{\frac{1}{{K+1}}}{a}}}\right)$
denotes the small-scale fading gain, $a\sim{\cal C}{\cal
N}\left({0,1}\right)$; $\psi$ is the angle of arrival/departure of the WiFi
link; $K$ denotes the Ricean $K$-factor; $d_{2}$ denotes the distance between
user and the RF antenna; $d_{\rm{B}}$ is the breakpoint distance;
${L_{\sigma}}\sim{\cal N}\left({0,\sigma^{2}}\right)$ denotes the shadow
fading, ${\sigma}=3$dB with ${d_{2}}\leq{d_{\rm{B}}}$, ${\sigma}=5$dB with
other conditions; ${L_{F}}\left(d_{2}\right)$ is the free space loss at the
central carrier frequency $f_{c}$ as follows:

$\displaystyle{L_{F}}\left(d_{2}\right)=20{\log_{10}}\left(d_{2}\right)+20{\log_{10}}\left({{f_{c}}}\right)-147.5.$
(21)

Let $y_{1}$ and $y_{2}$ denote the received signals from the LiFi link and
WiFi link, respectively, which can be written in the vector form as

$\displaystyle\left[\begin{array}[]{l}{y_{1}}\\\ {y_{2}}\\\
\end{array}\right]=\left[{\begin{array}[]{*{20}{c}}{{g_{1}}{q_{1}}{x_{1}}}\\\
{g_{2}^{*}{q_{2}}{x_{2}}}\\\
\end{array}}\right]+\left[{\begin{array}[]{*{20}{c}}{{z_{1}}}\\\ {{z_{2}}}\\\
\end{array}}\right],$ (28)

where $z_{1}\sim{\cal N}\left({0,\sigma^{2}_{1}}\right)$ is the received real
Gaussian noise from the LiFi link, and ${z_{2}}\sim{\cal C}{\cal
N}\left({0,\sigma^{2}_{2}}\right)$ denotes the received complex Gaussian noise
from the WiFi link. Then, the signal-to-noise ratio (SNR) of the LiFi link and
WiFi link can be defined as

$\displaystyle\mathrm{SNR}_{1}$
$\displaystyle\triangleq\frac{{g_{1}^{2}{q_{1}^{2}}{\varepsilon_{1}}}}{{{B_{1}}\sigma_{1}^{2}}},~{}\mathrm{SNR}_{2}\triangleq\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\left|q_{2}\right|^{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}.$
(29)

Note that, for the aggregated LiFi-WiFi system with the finite-alphabet
inputs, the achievable rate is still unknown. To address this issue, we define
the achievable rate ${{{R}}_{{\rm{LiFi-WiFi}}}}$ as

$\displaystyle{R_{{\rm{LiFi-
WiFi}}}}={{I}}\left({{x_{1}},{x_{2}};{y_{1}},{y_{2}}}\right)={R_{{\rm{LiFi}}}}+{R_{{\rm{WiFi}}}},$
(30)

where ${{{R}}_{{\rm{LiFi}}}}\triangleq{{{I}}\left({{x_{1}};{y_{1}}}\right)}$
and ${{{R}}_{{\rm{WiFi}}}}\triangleq{{{I}}\left({{x_{2}};{y_{2}}}\right)}$
denote the achievable rates of the LiFi link and WiFi link, respectively.

_Proof:_ According to the mutual information, the upper bound of
$R_{\mathrm{LiFi-WiFi}}$ is given as

$\displaystyle R_{\mathrm{LiFi-WiFi}}$
$\displaystyle=I\left(x_{1},x_{2};y_{1},y_{2}\right)$ (31a)
$\displaystyle=h\left(y_{1}\right)+h\left(y_{2}|y_{1}\right)-h\left(y_{1},y_{2}|x_{1},x_{2}\right)$
(31b) $\displaystyle\leq I\left(x_{1};y_{1}\right)+I\left(x_{2};y_{2}\right)$
(31c) $\displaystyle=R_{\mathrm{LiFi}}+R_{\mathrm{WiFi}},$ (31d)

where the equality (31c) holds if and only if $x_{1}$ and $x_{2}$ are
independent. Thus, when $x_{1}$ and $x_{2}$ are independent, the upper bound
(31d) is achievable, i.e., the upper bound (31d) is the channel capacity of
the aggregated LiFi-WiFi system. In other words, for the considered aggregated
LiFi-WiFi system, the optimal inputs of $x_{1}$ and $x_{2}$ are independent,
and the achievable rate is ${R_{{\rm{LiFi-
WiFi}}}}={{I}}\left({{x_{1}},{x_{2}};{y_{1}},{y_{2}}}\right)={R_{{\rm{LiFi}}}}+{R_{{\rm{WiFi}}}}$.
∎

Lemma 1: With the finite-alphabet inputs and the given bandwidths $B_{1}$ and
$B_{2}$, the achievable rates of the aggregated LiFi-WiFi system
${{{R}}_{{\rm{LiFi}}}}$ and ${{{R}}_{{\rm{WiFi}}}}$ are respectively given as

$\displaystyle{{{R}}_{{\rm{LiFi}}}}=-\frac{B_{1}}{{\ln
2}}-2B_{1}\sum\limits_{k=1}^{{M}}{{p_{1,k}}{\mathbb{E}_{{z_{1}}}}\left\\{{{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}\exp\left({\Lambda_{k,m}}\right)}}\right\\}},$
(32a) $\displaystyle{{{R}}_{{\rm{WiFi}}}}=-\frac{{B_{2}}}{{\ln
2}}-B_{2}\sum\limits_{l=1}^{{N}}{{p_{2,l}}{\mathbb{E}_{{z_{2}}}}\left\\{{{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}\exp\left(\Gamma_{l,n}\right)}}\right\\}},$
(32b)

where
$\Lambda_{k,m}\buildrel\Delta\over{=}{-\frac{{{{\left({{g_{1}}{q_{1}}\left({{x_{1,k}}-{x_{1,m}}}\right)+\sqrt{{B_{1}}}{z_{1}}}\right)}^{2}}}}{{2B_{1}\sigma_{1}^{2}}}}$,
$\Gamma_{l,n}\buildrel\Delta\over{=}{-\frac{{{{\left|{{g_{2}^{*}}{q_{2}}\left({{x_{2,l}}-{x_{2,n}}}\right)+\sqrt{{B_{2}}}{z_{2}}}\right|}^{2}}}}{{B_{2}\sigma_{2}^{2}}}}$.
111 The rate expression (10) is based on the unit bandwidth, while the Lemma 1
is for a given bandwidth. Thus, their units is different, i.e., bits/sec/Hz
and bits/sec, respectively.

_Proof:_ Please find the proof in Appendix A.∎

With Lemma 1, the achievable rate (30) of the considered system can be
obtained.

##  III Optimal Discrete Constellation Inputs of the Aggregated LiFi-WiFi
System

Based on the derived expression of ${R_{{\rm{LiFi-WiFi}}}}$, the following
question is to optimize the signal distributions and power allocation schemes
for the two links to obtain the maximal achievable rate of the aggregated
system. The optimization problem is formulated as

$\displaystyle\mathop{\max}\limits_{q_{1},q_{2},\left\\{{{p_{1,k}}}\right\\},\left\\{{{p_{2,l}}}\right\\}}\quad$
$\displaystyle{R_{{\rm{LiFi-WiFi}}}}$ (33a)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad\quad\quad$
$\displaystyle{\eta_{1}}P_{\mathrm{e},1}q_{1}^{2}+{\eta_{2}}P_{\mathrm{e},2}{\left|{{q_{2}}}\right|^{2}}\leq{P_{\rm{T}}},$
(33b) $\displaystyle
q_{1}\leq\min\left({{{{P_{o}}/\bar{\mu}}},{{P_{\rm{ins}}}/A}{}}\right),$ (33c)
$\displaystyle\sum\limits_{k=1}^{M}{{p_{1,k}}}{x_{1,k}}\leq\bar{\mu},\sum\limits_{k=1}^{M}{{p_{1,k}}}x_{1,k}^{2}\leq
P_{{\rm{e}},1},$ (33d)
$\displaystyle\sum\limits_{k=1}^{M}{{p_{1,k}}}=1,{p_{1,k}}\geq 0,\forall
k\in{\cal M},$ (33e)
$\displaystyle\sum\limits_{l=1}^{N}{{p_{2,l}}{{\left|{{x_{2,l}}}\right|}^{2}}}\leq
P_{{\rm{e}},2},$ (33f)
$\displaystyle\sum\limits_{l=1}^{N}{{p_{2,l}}}=1,{p_{2,l}}\geq 0,\forall
l\in{\cal N},$ (33g)

where (33b) denotes the total average electrical power of two links; (33c) and
(33d) denote the maximal optical power constraint, average optical power
constraint, and average electrical power constraint on the LiFi link,
respectively; (33f) denotes the average electrical power constraint on the
WiFi link; (33e) and (33g) denote the probability distribution constraint.

$\displaystyle{R_{{\rm{LiFi-
WiFi}}}}=-2{B_{1}}\sum\limits_{k=1}^{M}{{p_{1,k}}{\mathbb{E}_{{z_{1}}}}\left\\{{{{\log}_{2}}\sum\limits_{m=1}^{M}{{p_{1,m}}\exp\left({-\frac{{{{\left({{g_{1}}\sqrt{{{\hat{q}}_{1}}}\left({{x_{1,k}}-{x_{1,m}}}\right)+\sqrt{{B_{1}}}{z_{1}}}\right)}^{2}}}}{{2{B_{1}}\sigma_{1}^{2}}}}\right)}}\right\\}}$
$\displaystyle-{B_{2}}\sum\limits_{l=1}^{N}{{p_{2,l}}{\mathbb{E}_{{z_{2}}}}\left\\{{{{\log}_{2}}\sum\limits_{n=1}^{N}{{p_{2,n}}\exp\left({-\frac{{{{\left|{\left|{{g_{2}}}\right|\sqrt{{{\hat{q}}_{2}}}\left({{x_{2,l}}-{x_{2,n}}}\right)+\sqrt{{B_{2}}}{z_{2}}}\right|}^{2}}}}{{{B_{2}}\sigma_{2}^{2}}}}\right)}}\right\\}}-\frac{{{B_{1}}+{B_{2}}}}{{\ln
2}}.$ (15)

For problem (33), the optimal phase of $q_{2}$ is the same as that of $g_{2}$,
which can be proved by the counter-evidence method due to the property of the
complex multiplication. Thus, the optimal $q_{2}$ can be written as

$\displaystyle
q_{2}=\sqrt{{{\hat{q}}_{2}}}\frac{{{g_{2}}}}{{\left|{{g_{2}}}\right|}},$ (34)

where ${\hat{q}_{2}}\buildrel\Delta\over{=}{\left|{{q_{2}}}\right|^{\rm{2}}}$.
Furthermore, by substituting (34) into (30) and
${{{\hat{q}}_{1}}}\buildrel\Delta\over{=}q_{1}^{2}$, the achievable rate
${R_{{\rm{LiFi-WiFi}}}}$ can be rewrote as (III).

By defining
${\hat{\bf{x}}_{1}}\buildrel\Delta\over{=}{\left[{{x_{1,1}},\ldots,{x_{1,{M}}}}\right]^{\rm{T}}}$
and
${{\bf{p}}_{1}}\buildrel\Delta\over{=}{\left[{{p_{1,1}},\ldots,{p_{1,{M}}}}\right]^{\rm{T}}}$,
constraints (33f) and (33d) can be reformulated as

$\displaystyle\Upsilon_{1}\triangleq\left\\{\mathbf{p}\left|\mathbf{1}_{M}^{\mathrm{T}}\mathbf{p}=1,\mathbf{p}\succeq\mathbf{0},{\widehat{\mathbf{x}}_{1}^{\mathrm{T}}\mathbf{p}\leq\bar{\mu}},{\left(\widehat{\mathbf{x}}_{1}\odot\widehat{\mathbf{x}}_{1}\right)^{\mathrm{T}}\mathbf{p}\leq
P_{e,1}}\right.\right\\},$ $\displaystyle\mathbf{p}_{1}\in\Upsilon_{1}.$ (16)

Similarly, by defining
${\hat{\bf{x}}_{2}}\buildrel\Delta\over{=}{\left[{{x_{2,1}}\ldots{x_{2,{N}}}}\right]^{\rm{T}}}$
and
${{\bf{p}}_{2}}\buildrel\Delta\over{=}{\left[{{p_{2,1}}\ldots{p_{2,{N}}}}\right]^{\rm{T}}}$,
constraints (33e) and (33g) can be reformulated as

$\displaystyle\Upsilon_{2}$
$\displaystyle\triangleq\left\\{\mathbf{p}\left|\mathbf{1}_{N}^{\mathrm{T}}\mathbf{p}=1,\mathbf{p}\succeq\mathbf{0},{\left(\widehat{\mathbf{x}}_{2}\odot\widehat{\mathbf{x}}_{2}\right)^{\mathrm{T}}\mathbf{p}\leq
P_{e,2}}\right.\right\\},\mathbf{p}_{2}\in\Upsilon_{2}.$ (17)

Next, we introduce auxiliary variables, i.e.,

$\displaystyle{\bf{w}}\buildrel\Delta\over{=}{\left[{{{{\log}_{2}}{\bf{p}}_{1}^{\rm{T}}{{{\bf{\hat{w}}}}_{1}}},\ldots,{{{\log}_{2}}{\bf{p}}_{1}^{\rm{T}}{{{\bf{\hat{w}}}}_{M}}}}\right]^{\rm{T}}},$
(18a)
$\displaystyle{{{\bf{\hat{w}}}}_{k}}\buildrel\Delta\over{=}{\left[{{{\hat{w}}_{k,1}},\ldots,{{\hat{w}}_{k,M}}}\right]^{\rm{T}}},$
(18b)
$\displaystyle{{\hat{w}}_{k,m}}\buildrel\Delta\over{=}\exp\left({-\frac{{{{\left({{g_{1}}\sqrt{{{\hat{q}}_{1}}}\left({{x_{1,k}}-{x_{1,m}}}\right)+\sqrt{{B_{1}}}{z_{1}}}\right)}^{2}}}}{{2{B_{1}}\sigma_{1}^{2}}}}\right),$
(18c)
$\displaystyle{\bf{r}}\buildrel\Delta\over{=}{\left[{{{{\log}_{2}}{\bf{p}}_{2}^{\rm{T}}{{{\bf{\hat{r}}}}_{1}}},\ldots,{{{\log}_{2}}{\bf{p}}_{2}^{\rm{T}}{{{\bf{\hat{r}}}}_{N}}}}\right]^{\rm{T}}},$
(18d)
$\displaystyle{{{\bf{\hat{r}}}}_{l}}\buildrel\Delta\over{=}{\left[{{{\hat{r}}_{l,1}},\ldots,{{\hat{r}}_{l,N}}}\right]^{\rm{T}}},$
(18e)
$\displaystyle{{\hat{r}}_{l,n}}\buildrel\Delta\over{=}\exp\left({-\frac{{{{\left|{\left|{{g_{2}}}\right|\sqrt{{{\hat{q}}_{2}}}\left({{x_{2,l}}-{x_{2,n}}}\right)+\sqrt{{B_{2}}}{z_{2}}}\right|}^{2}}}}{{{B_{2}}\sigma_{2}^{2}}}}\right),$
(18f)

where $k$ and $m\in{\cal M}$, $l$ and $n\in{\cal N}$, Then, we rewrite the
achievable rate ${R_{{\rm{LiFi-WiFi}}}}$ as

$\displaystyle{R_{{\rm{LiFi-WiFi}}}}=$
$\displaystyle-\frac{{{B_{1}}+{B_{2}}}}{{\ln
2}}-2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{\bf{p}}_{1}^{\rm{T}}{\bf{w}}\right\\}-{B_{2}}{\mathbb{E}_{{z_{2}}}}\left\\{{\bf{p}}_{2}^{\rm{T}}{\bf{r}}\right\\}.$
(19)

Based on the above definitions, problem (33) can be equivalent to a compact
from as

$\displaystyle\mathop{\min}\limits_{\hat{q}_{1},\hat{q}_{2},{\bf{p}}_{1},{\bf{p}}_{2}}\quad$
$\displaystyle
2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{\bf{p}}_{1}^{\rm{T}}{\bf{w}}\right\\}+{B_{2}}{\mathbb{E}_{{z_{2}}}}\left\\{{\bf{p}}_{2}^{\rm{T}}{\bf{r}}\right\\}$
(20a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad\quad$
$\displaystyle{\eta_{1}}P_{\mathrm{e},1}\hat{q}_{1}+{\eta_{2}}P_{\mathrm{e},2}\hat{q}_{2}\leq{P_{\rm{T}}},$
(20b) $\displaystyle\hat{q}_{1}\leq\tau^{2},$ (20c)
$\displaystyle{{\bf{p}}_{1}\in{\Upsilon_{1}}},{{\bf{p}}_{2}\in{\Upsilon_{2}}},$
(20d)

where
$\tau\buildrel\Delta\over{=}\min\left({{{{P_{o}}/\bar{\mu}}},{{P_{\rm{ins}}}/A}{}}\right)$.
Note that in problem (20), the power allocation variables ${{\hat{q}_{1}}}$
and ${{\hat{q}_{2}}}$ are only contained in constraints (20b) and (20c), while
the distribution variables ${\bf{p}}_{1}$ and ${\bf{p}}_{2}$ are only
contained in the constraint (20d). Therefore, the optimization problem (18)
can be decomposed into two sub-problems solved alternately until the objective
function converges : _power allocation sub-problem 1_ : optimizing
${{\hat{q}_{1}}}$ and ${{\hat{q}_{2}}}$ with given ${\bf{p}}_{1}$ and
${\bf{p}}_{2}$, and _probability distribution sub-problem 2_ : optimizing
${\bf{p}}_{1}$ and ${\bf{p}}_{2}$ with given ${{\hat{q}_{1}}}$ and
${{\hat{q}_{2}}}$. Next, we will present the solutions to these two sub-
problems.

###  III-A Power Allocation Sub-Problem

With given ${\bf{p}}_{1}$, ${\bf{p}}_{2}$, problem (20) can be simplified to
an optimal power allocation problem as

$\displaystyle\mathop{\min}\limits_{\hat{q}_{1},\hat{q}_{2}}\quad$
$\displaystyle h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)$ (21a)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{\eta_{1}}P_{\mathrm{e},1}\hat{q}_{1}+{\eta_{2}}P_{\mathrm{e},2}{\hat{q}_{2}}\leq{P_{\rm{T}}},$
(21b) $\displaystyle{\hat{q}_{1}}\leq\tau^{2},$ (21c)

where
$h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)\buildrel\Delta\over{=}2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{\bf{p}}_{1}^{\rm{T}}{\bf{w}}\right\\}+{B_{2}}{\mathbb{E}_{{z_{2}}}}\left\\{{\bf{p}}_{2}^{\rm{T}}{\bf{r}}\right\\}$.

Note that
$-\frac{\left(g_{1}\sqrt{\hat{q}_{1}}\left(x_{1,k}-x_{1,m}\right)+\sqrt{B_{1}}z_{1}\right)^{2}}{2B_{1}\sigma_{1}^{2}}$
and
$-\frac{\left|\left|g_{2}\right|\sqrt{\hat{q}_{2}}\left(x_{2,\ell}-x_{2,n}\right)+\sqrt{B_{2}}z_{2}\right|^{2}}{B_{2}\sigma_{2}^{2}}$
are convex over $\hat{q}_{1}$ and $\hat{q}_{2}$, respectively. Furthermore,
since $\log\sum\limits_{i}{{e^{{f_{i}}\left(x\right)}}}$ is convex as long as
${{f_{i}}\left(x\right)}$ is convex[24], problem (21) is convex over
$\hat{q}_{1}$ and $\hat{q}_{2}$. To obtain the optimal power allocation
$\hat{q}_{1}$ and $\hat{q}_{2}$, we write the Lagrangian function of problem
(21) as

$\displaystyle{\cal L}\left({\hat{q}_{1}},{\hat{q}_{2}},\gamma,\nu\right)=$
$\displaystyle
h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)+\gamma\left({{\eta_{1}}{{P_{e,1}}}\hat{q}_{1}+{\eta_{2}}{{P_{e,2}}}{\hat{q}_{2}}-{P_{\rm{T}}}}\right)$
$\displaystyle~{}+\nu\left({{\hat{q}_{1}}-\tau^{2}}\right),$ (22)

where $\gamma\geq 0$ and $\nu\geq 0$ are the Lagrangian multipliers associated
with constraints (21b) and (21c), respectively. Then, the Karush-Kuhn-Tucker
(KKT) conditions of problem (21) are

$\displaystyle\frac{{\partial
h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)}}{{\partial{\hat{q}_{1}}}}+\gamma{\eta_{1}}P_{\mathrm{e},1}+\nu=0,$
(23a) $\displaystyle\frac{{\partial
h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)}}{{\partial{\hat{q}_{2}}}}+\gamma{\eta_{2}}P_{\mathrm{e},2}=0,$
(23b)
$\displaystyle\gamma\left({{\eta_{1}}P_{\mathrm{e},1}\hat{q}_{1}+{\eta_{2}}P_{\mathrm{e},2}{\hat{q}_{2}}-{P_{\rm{T}}}}\right)=0,~{}\gamma\geq
0,$ (23c) $\displaystyle\nu\left({{\hat{q}_{1}}-\tau^{2}}\right)=0,~{}\nu\geq
0.$ (23d)

Furthermore, based on the definition of
${h\left({{{\hat{q}}_{1}},{{\hat{q}}_{2}}}\right)}$, we have

$\displaystyle\frac{{\partial
h\left({{{\hat{q}}_{1}},{{\hat{q}}_{2}}}\right)}}{{\partial{{\hat{q}}_{1}}}}$
$\displaystyle=-\frac{{g_{1}^{2}{\varepsilon_{1}}}}{{{B_{1}}\sigma_{1}^{2}}}\frac{{\partial{I}\left({{x_{1}};{y_{1}}}\right)}}{{\partial{\rm{SN}}{{\rm{R}}_{\rm{1}}}}},$
$\displaystyle\frac{{\partial
h\left({{{\hat{q}}_{1}},{{\hat{q}}_{2}}}\right)}}{{\partial{{\hat{q}}_{2}}}}$
$\displaystyle=-\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}\frac{{\partial{\rm{I}}\left({{x_{2}};{y_{2}}}\right)}}{{\partial{\rm{SN}}{{\rm{R}}_{2}}}}.$
(24)

Moreover, according to relationship between the mutual information and the
MMSE [22], we have

$\displaystyle\begin{aligned}
\frac{{\partial{{I}}\left({{x_{1}};{y_{1}}}\right)}}{{\partial{\rm{SNR}_{1}}}}&=\frac{1}{2}{\rm{MMS}}{{\rm{E}}_{1}}\left({{\rm{SNR}}_{1}}\right),\\\
\frac{{\partial{{I}}\left({{x_{2}};{y_{2}}}\right)}}{{\partial{\rm{SN}}{{\rm{R}}_{2}}}}&={\rm{MMSE}}_{2}\left({{\rm{SN}}{{\rm{R}}_{2}}}\right),\end{aligned}$
(25)

where
$\mathrm{MMSE}_{i}\left(\mathrm{SNR}_{i}\right)\triangleq\mathbb{E}\left\\{\left|x_{i}-\hat{x}_{i}\right|\right\\}$
denotes the MMSE between $x_{i}$ and $\hat{x}_{i}$, $i=1,2$, and $\hat{x}_{i}$
is the conditional mean of the MMSE estimate of $x_{i}$, i.e.
$\hat{x}_{1}\triangleq\mathbb{E}\left\\{x_{1}|y_{1}=g_{1}q_{1}x_{1}+z_{1}\right\\}$
and
$\hat{x}_{2}\triangleq\mathbb{E}\left\\{x_{2}|y_{2}=g_{2}^{*}q_{2}x_{2}+z_{2}\right\\}$.
Note that the calculation of the above MMSE involves non-holonomic function,
i.e., it is non-trivial to compute the exact MMSE for an arbitrary input
distribution. Meanwhile, it is an opening problem to find the lower and upper
bounds of the MMSE for the discrete inputs [25]. Thus, most of existing works
adopt numerical methods such as Monte Carlo integral to obtain the results,
which may lead to sub-optimality and high complexity. To overcome this issue,
we approximate the MMSEs by a scaled linear MMSE upper bound [25], i.e.,

$\displaystyle\begin{aligned}
\rm{MMSE}_{1}\left(\rm{SNR_{1}}\right)&\approx\rm{LMMSE}_{\varepsilon_{1}}\left(\rm{SNR_{1}}\right),\\\
\rm{MMSE}_{2}\left(\rm{SNR_{2}}\right)&\approx\rm{LMMSE}_{\varepsilon_{2}}\left(\rm{SNR_{2}}\right),\end{aligned}$
(26)

where ${\rm{LMMSE}}_{t}\left(x\right)=\frac{t}{1+tx}$. Although the
approximation based on LMMSE, like a continuous Gaussian MMSE, would lead to
the optimality loss, it can provide a fast acceptable closed-form method.
Then, (25) can be reformulated as

$\displaystyle\begin{aligned}
\frac{{\partial{{I}}\left({{x_{1}};{y_{1}}}\right)}}{{\partial{\rm{SNR}_{1}}}}&=\frac{1}{2}{\rm{LMMSE}}_{\varepsilon_{1}}\left(\rm{SNR_{1}}\right),\\\
\frac{{\partial{{I}}\left({{x_{2}};{y_{2}}}\right)}}{{\partial{\rm{SN}}{{\rm{R}}_{2}}}}&={\rm{LMMSE}}_{\varepsilon_{2}}\left(\rm{SNR_{2}}\right).\end{aligned}$
(27)

With (29) and (27), we arrive at

$\displaystyle\begin{aligned} \frac{{\partial
h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)}}{{\partial{\hat{q}_{1}}}}&=-\frac{{g_{1}^{2}{\varepsilon_{1}}}}{{2{B_{1}}\sigma_{1}^{2}}}{\rm{LMMSE}}_{\varepsilon_{1}}\left({\frac{{g_{1}^{2}\hat{q}_{1}{\varepsilon_{1}}}}{{{B_{1}}\sigma_{1}^{2}}}}\right),\\\
\frac{{\partial
h\left({\hat{q}_{1}},{\hat{q}_{2}}\right)}}{{\partial{\hat{q}_{2}}}}&=-\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}{\rm{LMMSE}}_{\varepsilon_{2}}\left({\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\hat{q}_{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}}\right).\end{aligned}$
(28)

Substituting equation (28) into equation (23), we have

$\displaystyle\frac{{g_{1}^{2}{\varepsilon_{1}}}}{{2{B_{1}}\sigma_{1}^{2}}}{\rm{LMMSE}}_{\varepsilon_{1}}\left({\frac{{g_{1}^{2}\hat{q}_{1}{\varepsilon_{1}}}}{{{B_{1}}\sigma_{1}^{2}}}}\right)-\gamma{\eta_{1}}P_{\mathrm{e},1}-\nu=0,$
(29a)
$\displaystyle\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}{\rm{LMMSE}}_{\varepsilon_{2}}\left({\frac{{{{\left|{{g_{2}}}\right|}^{2}}{\hat{q}_{2}}{\varepsilon_{2}}}}{{{B_{2}}\sigma_{2}^{2}}}}\right)-\gamma{\eta_{2}}P_{\mathrm{e},2}=0.$
(29b)

Then, $\widehat{q}_{1}$ and $\widehat{q}_{2}$ can be derived through solving
the equation (29) as

$\displaystyle{{\hat{q}}_{1}}=\frac{{{B_{1}}\sigma_{1}^{2}}}{{g_{1}^{2}{\varepsilon_{1}}}}\rm{LMMSE}_{\varepsilon_{1}}^{{\rm{-1}}}\left({\frac{{2{B_{1}}\sigma_{1}^{2}\left({\gamma{\eta_{1}}P_{\mathrm{e},1}{\rm{+}}\nu}\right)}}{{g_{1}^{2}{\varepsilon_{1}}}}}\right),$
(30a)
$\displaystyle{{\hat{q}}_{2}}=\frac{{{B_{2}}\sigma_{2}^{2}}}{{{{\left|{{g_{2}}}\right|}^{2}}{\varepsilon_{2}}}}\rm{LMMSE}_{\varepsilon_{2}}^{{\rm{-1}}}\left({\frac{{{B_{2}}\sigma_{2}^{2}\gamma{\eta_{2}}P_{\mathrm{e},2}}}{{{{\left|{{g_{2}}}\right|}^{2}}{\varepsilon_{2}}}}}\right),$
(30b)

where ${\rm{LMMSE}}_{t}^{-1}\left(x\right)=\frac{1}{x}-\frac{1}{t}$. The
Lagrangian multipliers $\gamma$ and $\nu$ can be solved by the Water-filling
(WF) Method. The detailed algorithm is given in Algorithm 1.

Algorithm 1 Water-filling (WF) Method for Power Allocation Sub-Problem (21).

1:Given $\zeta\geq 0$, $\gamma\in\left[0,\hat{\gamma}\right]$ ;

2:Initialization: ${\gamma_{\min}}=0$, ${\gamma_{\max}}=\hat{\gamma}$;

3:repeat.

4: Set $\gamma\leftarrow\left({{\gamma_{\min}}+{\gamma_{\max}}}\right)/2$;

5: Obtain $\hat{q}_{2}$ by (30b);

6: Find the minimum $\nu$ satisfying the constraint
${\hat{q}_{1}}\leq\tau^{2}$;

7: Obtain $\hat{q}_{1}$ by (30a);

8: If
${\eta_{1}}P_{\mathrm{e},1}\hat{q}_{1}+{\eta_{2}}P_{\mathrm{e},2}{\hat{q}_{2}}\leq{P_{\rm{T}}}$,
set ${\gamma_{\max}}\leftarrow\gamma$; otherwise,
${\gamma_{\min}}\leftarrow\gamma$;

9:until $\left|{{\nu_{\max}}-{\nu_{\min}}}\right|\leq\zeta$.

10:return $\hat{q}_{1}$ and $\hat{q}_{2}$;

###  III-B Probability Distribution Sub-Problem

When $\hat{q}_{1}$ and $\hat{q}_{2}$ are given, problem (20) can be
reformulated as

$\displaystyle\mathop{\min}\limits_{{\bf{p}}_{1},{\bf{p}}_{2}}\quad$
$\displaystyle
2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{\bf{p}}_{1}^{\rm{T}}{\bf{w}}\right\\}+{B_{2}}{\mathbb{E}_{{z_{2}}}}\left\\{{\bf{p}}_{2}^{\rm{T}}{\bf{r}}\right\\}$
(31a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{{\bf{p}}_{1}\in{\Upsilon_{1}}},~{}{{\bf{p}}_{2}\in{\Upsilon_{2}}},$
(31b)

which is a convex optimization problem with two variables ${\bf{p}}_{1}$ and
${\bf{p}}_{2}$. However, there is no analytical expression of the objective
function, which prevents us from calculating the optimal probability
distributions.

To overcome this bottleneck, we adopt the inexact gradient descent method [17]
to calculate the optimal probability distribution. Besides, the Blahut-Arimoto
algorithm [26, 27, 28] can also solve this problem. Let
$\phi_{1}\left({{{\bf{p}}_{1}}}\right)\buildrel\Delta\over{=}2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{\bf{p}}_{1}^{\rm{T}}{\bf{w}}\right\\}$
and ${\nabla}\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$ denote the its gradient
as

$\displaystyle{\nabla}\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$
$\displaystyle=2{B_{1}}{\mathbb{E}_{{z_{1}}}}\left\\{{{\bf{w}}+{\bf{W}}{{\bf{p}}_{1}}}\right\\}$
$\displaystyle=2{B_{1}}\int_{-\infty}^{\infty}{{f_{{z_{1}}}}\left({{z_{1}}}\right)\left({{\bf{w}}+{\bf{W}}{{\bf{p}}_{1}}}\right)}dz_{1},$
(32)

where ${{\bf{W}}}\buildrel\Delta\over{=}{\left[{{{W}}_{i,j}}\right]}$,
${{{W}}_{i,j}}\buildrel\Delta\over{=}\frac{{{\bf{\hat{w}}}_{j}^{\rm{T}}{{\bf{e}}_{i}}}}{{{\bf{\hat{w}}}_{j}^{\rm{T}}{{\bf{p}}_{1}}\ln
2}}$, ${{\bf{e}}_{i}}$ denotes the unit vector in which the $i$th element is 1
and the other elements are zeros, and
${f_{{z_{1}}}}\left({{z_{1}}}\right)\buildrel\Delta\over{=}\frac{1}{{\sqrt{2\pi}{\sigma_{1}}}}\exp\left({-\frac{{z_{1}^{2}}}{{\sigma_{1}^{2}}}}\right)$
denotes the probability density function of $z_{2}$.

Since neither ${\phi_{j}}\left({{{{\bf{p}}}_{j}}}\right)$ nor
$\nabla{\phi_{j}}\left({{{\bf{\hat{p}}}_{j}}}\right)$ has a closed-form
expression, we perform the truncation of the integration from infinity to an
finite interval. More specifically, let
$\left[{-{\tau_{1}},{\tau_{1}}}\right]$ and
$\left[{-{\tau_{2}},{\tau_{2}}}\right]$ denote the integration intervals of
$\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$ and
${\nabla}\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$. Then, let
$\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)$ and
${\nabla}\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)$ denote the
approximation of $\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$ and
${\nabla}\phi_{1}\left({{{{\bf{p}}}_{1}}}\right)$ respectively, which are
given by

$\displaystyle\hat{\phi}_{1}\left({{{\bf{p}}_{1}}}\right)\buildrel\Delta\over{=}2{B_{1}}\int_{-{\tau_{1}}}^{\tau_{1}}{f_{{z_{1}}}}\left({{z_{1}}}\right){\bf{p}}_{1}^{\rm{T}}{\bf{w}}dz,$
(33a)
$\displaystyle{\nabla}\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)\buildrel\Delta\over{=}2{B_{1}}\int_{-{\tau_{2}}}^{\tau_{2}}{{f_{{z_{1}}}}\left({{z_{1}}}\right)\left({{\bf{w}}+{\bf{W}}{{\bf{p}}_{1}}}\right)}dz,$
(33b)

where ${\tau_{1}}>0$ and ${\tau_{2}}>0$.

With the approximated objective function and its gradient, i.e.,
$\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)$ and
${\nabla}\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)$, we adopt the gradient
projections method to solve problem (31). Specifically, let
${{{\bf{p}}^{\left[{i}\right]}_{1}}}$ denote the $n$th iteration feasible
point. With inexact gradient
${\nabla}\hat{\phi}_{1}\left({{{{\bf{p}}}_{1}}}\right)$, the gradient
projection iteration between ${{{\bf{p}}^{\left[{i}\right]}_{1}}}$ and
${{{\bf{p}}^{\left[{i+1}\right]}_{1}}}$ is given by

$\displaystyle{{{\bf{p}}^{\left[{i+1}\right]}_{1}}}={\rm{Pro}}{{\rm{j}}_{{\Upsilon_{1}}}}\left({{\bf{p}}_{1}^{\left[i\right]}-\alpha_{1}^{\left[i\right]}{\nabla}\hat{\phi}_{1}\left({\bf{p}}_{1}^{\left[i\right]}\right)}\right),$
(34)

where $\alpha_{1}^{\left[i\right]}$ denotes the $i$th-iteration step size,
${\rm{Pro}}{{\rm{j}}_{{\Upsilon_{1}}}}\left({\bf{\hat{x}}}\right)$ denotes the
projection of ${\bf{x}}$ onto ${\Upsilon_{1}}$:

$\displaystyle{{\rm{Pro}}{{\rm{j}}_{{\Upsilon_{1}}}}\left({{{{\bf{{x}}}}}}\right)=\left\\{{\begin{array}[]{*{20}{l}}{{{{\bf{{x}}}}},{\rm{if}}\;{{{\bf{{x}}}}}\in{\Upsilon_{1}},}\\\
{\mathop{\arg\min}\limits_{{{{{\bf{{\hat{x}}}}}}}\in{\Upsilon_{1}}}\left\|{{{{\bf{{x}}}}}-{{{{\bf{{\hat{x}}}}}}}}\right\|^{2},{\rm{otherwise}}.}\end{array}}\right.}$
(37)

The details of the inexact gradient descent method are listed in Algorithm 2,
and the distribution of LiFi link ${\bf{p}}_{1}$ can be obtained. The optimal
distribution of WiFi link ${\bf{p}}_{2}$ can be obtained analogously. Thus, we
omit the detailed derivations for brevity. The optimal probability
distribution $\mathbf{p}_{1}$, $\mathbf{p}_{2}$ can be achieved by the
distribution matching (DM) [29, 30, 31].

In summary, the achievable rate maximization problem (33) can be solved by
Algorithm 3 , therefore the maximal achievable rate $R_{{\rm{LiFi-WiFi}}}$ can
be achieved.

Algorithm 2 Inexact Gradient Descent Method for Probability Distribution Sub-
Problem(31)

1:Initialization: Given $\delta\geq 0$, set $i=1$, and choose
${{{{\bf{p}}}^{\left[{i}\right]}_{1}}}\in{\Upsilon_{1}}$.

2:repeat

3: $i\leftarrow i+1$.

4: Update $\hat{\phi}_{1}\left({{{{\bf{p}}}^{\left[{i-1}\right]}_{1}}}\right)$
and
${\nabla}\hat{\phi}_{1}\left({{{{\bf{p}}}^{\left[{i-1}\right]}_{1}}}\right)$
by (33).

5: Compute the step size $\alpha_{1}^{\left[i1-\right]}$ by Armijo rule[24].

6: Update
${{{\bf{p}}^{\left[{i}\right]}_{1}}}={\rm{Pro}}{{\rm{j}}_{{\Upsilon_{1}}}}\left({{\bf{p}}_{1}^{\left[i-1\right]}-\alpha_{1}^{\left[i-1\right]}{\nabla}\hat{\phi}_{1}\left({\bf{p}}_{1}^{\left[i-1\right]}\right)}\right)$.

7:until
$\left\|{{{\bf{p}}^{\left[{i}\right]}_{1}}}-{{{\bf{p}}^{\left[{i-1}\right]}_{1}}}\right\|\leq\delta$.

8:return ${{{\bf{p}}_{1}}}={{{\bf{p}}^{\left[{i}\right]}_{1}}}$.

Algorithm 3 Optimal discrete constellation inputs of problem (20)

1:Initialization: Given $\xi\geq 0$, set $k=1$, and choose
${{{{\bf{p}}}_{1}^{\left[1\right]}}}\in{\Upsilon_{1}}$,
${{{{\bf{p}}}_{2}^{\left[1\right]}}}\in{\Upsilon_{2}}$.

2:Update ${{\hat{q}}_{1}^{\left[1\right]}}$ and
${{\hat{q}}_{2}^{\left[1\right]}}$ by Algorithm 1.

3:Obtain $R_{\mathrm{LiFi-WiFi}}^{\left[1\right]}$ by substituting
$\hat{q}_{1}^{\left[1\right]}$, $\hat{q}_{2}^{\left[1\right]}$,
$\mathbf{p}_{1}^{\left[1\right]}$ and $\mathbf{p}_{2}^{\left[1\right]}$ into
formulation (19).

4:repeat

5: $k\leftarrow k+1$.

6: Update ${{\bf{p}}_{1}^{\left[k\right]}}$ and
${{\bf{p}}_{2}^{\left[k\right]}}$ based on Algorithm 2 with given
${\hat{q}}_{1}^{\left[k-1\right]}$ and ${\hat{q}}_{2}^{\left[k-1\right]}$.

7: Update ${\hat{q}}_{1}^{\left[k\right]}$ and
${\hat{q}}_{2}^{\left[k\right]}$ by Algorithm 1 with given
${{\bf{p}}_{1}^{\left[k\right]}}$ and ${{\bf{p}}_{2}^{\left[k\right]}}$.

8: Obtain $R^{\left[k\right]}_{{\rm{LiFi-WiFi}}}$ by substituting
${\hat{q}}^{\left[k\right]}_{1}$ , ${\hat{q}}^{\left[k\right]}_{2}$,
${{\bf{p}}^{\left[k\right]}_{1}}$ and ${{\bf{p}}^{\left[k\right]}_{2}}$ into
formulation (19).

9:until $\left|{R^{\left[k\right]}_{{\rm{LiFi-
WiFi}}}-R^{\left[k-1\right]}_{{\rm{LiFi-WiFi}}}}\right|\leq\xi$.

10:return ${{{\bf{p}}_{1}}}={{{\bf{p}}^{\left[k\right]}_{1}}}$,
${{{\bf{p}}_{2}}}={{{\bf{p}}^{\left[k\right]}_{2}}}$,
${\hat{q}}_{1}={\hat{q}}^{\left[k\right]}_{1}$ and
${\hat{q}}_{2}={\hat{q}}^{\left[k\right]}_{2}$,$R_{{\rm{LiFi-
WiFi}}}=R^{\left[k-1\right]}_{{\rm{LiFi-WiFi}}}$.

##  IV Optimal Discrete Constellation Input Distributions based on Lower
Bounds and Upper Bounds

Recall that, neither the achievable rate (32a) nor (32b) is in a closed-form
expression, and thus the calculation of (30) is computationally inefficient.
To reduce the computational complexity, we may replace the objective function
with an explicit expression. Thus, we turn to the capacity bound of LiFi link
and WiFi link.

Lemma 2: With the finite-alphabet inputs, the closed-form upper bound
${R^{\rm{U}}_{{\rm{LiFi}}}}$ and lower bound ${R^{\rm{L}}_{{\rm{LiFi}}}}$ of
the LiFi link achievable rate are respectively given as

$\displaystyle{R^{\rm{U}}_{{\rm{LiFi}}}}=-2B_{1}\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}\exp\left(2\hat{\Lambda}_{k,m}\right)},$
(38a) $\displaystyle{R^{\rm{L}}_{{\rm{LiFi}}}}=B_{1}-\frac{B_{1}}{{\ln
2}}-2B_{1}\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}\exp\left(\hat{\Lambda}_{k,m}\right)},$
(38b)

where
$\hat{\Lambda}_{k,m}\buildrel\Delta\over{=}{-\frac{{{{\left({{x_{1,k}}-{x_{1,m}}}\right)}^{2}}{{\left({{g_{1}}{q_{1}}}\right)}^{2}}}}{{4B_{1}\sigma_{1}^{2}}}}$.

_Proof:_ Please find the proof in Appendix B.∎

Furthermore, we develop the closed-form upper bound and lower bound of the
achievable rate for WiFi link ${{{R}}_{{\rm{WiFi}}}}$.

Lemma 3: The upper bound ${R^{\rm{U}}_{{\rm{WiFi}}}}$ and lower bound
${R^{\rm{L}}_{{\rm{WiFi}}}}$ of the achievable rate for the WiFi link are
respectively given as

$\displaystyle{R^{\rm{U}}_{{\rm{WiFi}}}}=-B_{2}\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}}\exp\left(2\hat{\Gamma}_{l,n}\right)},$
(39a) $\displaystyle{R^{\rm{L}}_{{\rm{WiFi}}}}=B_{2}-\frac{B_{2}}{{\ln
2}}-B_{2}\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}}\exp\left(\hat{\Gamma}_{l,n}\right)},$
(39b)

where
$\hat{\Gamma}_{l,n}\buildrel\Delta\over{=}{-\frac{{{{\left|{{x_{2,l}}-{x_{2,n}}}\right|}^{2}}{{\left|{g_{2}^{*}{q_{2}}}\right|}^{2}}}}{{2B_{2}\sigma_{2}^{2}}}}$.

_Proof:_ Please see the proof in the Appendix C.∎

Then, let $R_{{\rm{LiFi-WiFi}}}^{\rm{L}}$ and $R_{{\rm{LiFi-WiFi}}}^{\rm{U}}$
respectively denote the lower bound and upper bound of ${R_{{\rm{LiFi-
WiFi}}}}$, which are given by

$\displaystyle R_{{\rm{LiFi-WiFi}}}^{\rm{L}}\triangleq$
$\displaystyle{R^{\rm{L}}_{{\rm{LiFi}}}}+{R^{\rm{L}}_{{\rm{WiFi}}}}.$ (40a)
$\displaystyle R_{{\rm{LiFi-WiFi}}}^{\rm{U}}\triangleq$
$\displaystyle{R^{\rm{U}}_{{\rm{LiFi}}}}+{R^{\rm{U}}_{{\rm{WiFi}}}}.$ (40b)

The lower bound $R_{\mathrm{LiFi}}^{\mathrm{L}}$,
$R_{\mathrm{WiFi}}^{\mathrm{L}}$ $R_{\mathrm{LiFi-WiFi}}^{\mathrm{L}}$ are not
tight, and might be negative [32]. When $P_{\mathrm{T}}\to 0$ and
$P_{\mathrm{T}}\to+\infty$, there is a constant gap, i.e.,
$B_{1}\left(\frac{1}{\ln 2}-1\right)$, between the $R_{\mathrm{LiFi}}$ (32a)
and $R_{\mathrm{LiFi}}^{\mathrm{L}}$ (38b). There also is a constant gap,
i.e., $B_{2}\left(\frac{1}{\ln 2}-1\right)$, between the $R_{\mathrm{WiFi}}$
(32b) and $R_{\mathrm{WiFi}}^{\mathrm{L}}$ (39b). Thus, when
$P_{\mathrm{T}}\to 0$ and $P_{\mathrm{T}}\to+\infty$, there is a constant gap,
i.e., $\left(\frac{1}{\ln 2}-1\right)\left(B_{1}+B_{2}\right)$, between the
$R_{\mathrm{LiFi-WiFi}}$ (30) and $R_{\mathrm{LiFi-WiFi}}^{\mathrm{L}}$ (40a).
If the constant gap is directly added to $R_{\mathrm{LiFi-
WiFi}}^{\mathrm{L}}$, there might be intersections between $R_{\mathrm{LiFi-
WiFi}}$ and $R_{\mathrm{LiFi-WiFi}}^{\mathrm{L}}$ as similar as Fig. 1-3 in
[32].

Based on the lower bound of the achievable rate $R_{{\rm{LiFi-
WiFi}}}^{\rm{L}}$, we aim to obtain the optimal distribution of the signal
magnitudes and power allocation jointly, which can be formulated as

$\displaystyle\mathop{\max}\limits_{q_{1},q_{2},\left\\{{{p_{1,k}}}\right\\},\left\\{{{p_{2,l}}}\right\\}}\quad$
$\displaystyle R_{{\rm{LiFi-WiFi}}}^{\rm{L}}$ (41)
$\displaystyle{\rm{s.t.}}\quad\quad\quad$
$\displaystyle\eqref{max_rate_accuracya},\eqref{max_rate_accuracyb},\eqref{max_rate_accuracyc},\eqref{max_rate_accuracyd},\eqref{max_rate_accuracye},\eqref{max_rate_accuracyf}.$

Furthermore, by defining ${{{\hat{q}}_{1}}}\buildrel\Delta\over{=}q_{1}^{2}$
and ${{{\hat{q}}_{2}}}\buildrel\Delta\over{=}{\left|{{q_{2}}}\right|^{2}}$,
the lower bound of the achievable rate $R_{{\rm{LiFi-WiFi}}}^{\rm{L}}$ can be
rewritten as

$\displaystyle R_{{\rm{LiFi-WiFi}}}^{\rm{L}}=$
$\displaystyle\left({1-\frac{1}{{\ln
2}}}\right)\left({{{B_{1}}+{B_{2}}}}\right)$
$\displaystyle~{}~{}~{}~{}~{}-2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({{{\hat{q}}_{1}}}\right)-{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({{{\hat{q}}_{2}}}\right),$
(42)

where

$\displaystyle{\bf{u}}\left({{\hat{q}_{1}}}\right)\buildrel\Delta\over{=}{\left[{{\log_{2}}{\bf{a}}_{1}^{\rm{T}}\left({{\hat{q}_{1}}}\right){{\bf{p}}_{1}},\ldots,{\log_{2}}{\bf{a}}_{M}^{\rm{T}}\left({{\hat{q}_{1}}}\right){{\bf{p}}_{1}}}\right]^{\rm{T}}},$
(43a)
$\displaystyle{{\bf{a}}_{k}}\left({{\hat{q}_{1}}}\right)\buildrel\Delta\over{=}{\left[{{a_{k,1}}\left({{\hat{q}_{1}}}\right),\ldots,{a_{k,M}}\left({{\hat{q}_{1}}}\right)}\right]^{\rm{T}}},$
(43b)
$\displaystyle{{a_{k,m}}\left({{{\hat{q}}_{1}}}\right)}\buildrel\Delta\over{=}\exp\left({-\frac{{{{\left({{x_{1,k}}-{x_{1,m}}}\right)}^{2}}{g_{1}^{2}}{{{\hat{q}}_{1}}}}}{{4B_{1}\sigma_{1}^{2}}}}\right),\forall{k,m}\in{\cal
M},$ (43c)
$\displaystyle{\bf{v}}\left({{\hat{q}_{2}}}\right)\buildrel\Delta\over{=}{\left[{{\log_{2}}{\bf{b}}_{1}^{\rm{T}}\left({{\hat{q}_{2}}}\right){{\bf{p}}_{2}},\ldots,{\log_{2}}{\bf{b}}_{N}^{\rm{T}}\left({{\hat{q}_{2}}}\right){{\bf{p}}_{2}}}\right]^{\rm{T}}},$
(43d)
$\displaystyle{{\bf{b}}_{l}}\left({{\hat{q}_{2}}}\right)\buildrel\Delta\over{=}{\left[{{b_{l,1}}\left({{\hat{q}_{2}}}\right),\ldots,{b_{l,N}}\left({{\hat{q}_{2}}}\right)}\right]^{\rm{T}}},$
(43e)
$\displaystyle{{b_{l,n}}\left({{{\hat{q}}_{2}}}\right)}\buildrel\Delta\over{=}\exp\left({-\frac{{{{\left|{{x_{2,l}}-{x_{2,n}}}\right|}^{2}}{{\left|{{g_{2}}}\right|}^{2}}{{{\hat{q}}_{2}}}}}{{2B_{2}\sigma_{2}^{2}}}}\right),\forall{l,n}\in{\cal
N}.$ (43f)

Then, problem (41) can be formulated as

$\displaystyle\mathop{\min}\limits_{{{{\hat{q}}_{1}}},{{{\hat{q}}_{2}}},{{\bf{p}}_{1}},{{\bf{p}}_{2}}}\quad$
$\displaystyle
2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({{{\hat{q}}_{1}}}\right)+{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({{{\hat{q}}_{2}}}\right)$
(44) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad\quad$
$\displaystyle{\eqref{max_rate_accuracy_2b},\eqref{max_rate_accuracy_2c},\eqref{max_rate_accuracy_2d}.}$

It can be easily checked that the objection function (44) is convex with
respect to ${{{\hat{q}}_{1}}}$ and ${{{\hat{q}}_{2}}}$, and constraints (20b),
(20c), and (20d) are linear in terms of ${{{\hat{q}}_{1}}}$, and
${{{\hat{q}}_{2}}}$. Therefore, problem (44) is convex with respect to
(w.r.t.) ${{{\hat{q}}_{1}}}$ and ${{{\hat{q}}_{2}}}$. While, the problem (44)
is nonconvex w.r.t. ${\bf{p}}_{1}$ and ${\bf{p}}_{2}$. To solve problem (44),
we by solving the following two sub-problems iteratively, and until the
objective function converges.

###  IV-A Power Allocation Sub-problem

Firstly, we solve the power allocation sub-problem: optimizing the LiFi and
the WiFi links power allocation, i.e., ${{\hat{q}_{1}}}$ and
${{\hat{q}_{2}}}$, with fixed ${\bf{p}}_{1}$ and ${\bf{p}}_{2}$. Then, problem
(44) is reformulated as

$\displaystyle\mathop{\min}\limits_{{{{\hat{q}}_{1}}},{{{\hat{q}}_{2}}}}\quad$
$\displaystyle
2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({{{\hat{q}}_{1}}}\right)+{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({{{\hat{q}}_{2}}}\right)$
(45a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{\eta_{1}}P_{\mathrm{e},1}{{{\hat{q}}_{1}}}+{\eta_{2}}P_{\mathrm{e},2}{{{\hat{q}}_{2}}}\leq{P_{\rm{T}}},$
(45b) $\displaystyle\hat{q}_{1}\leq\tau^{2},$ (45c)
$\displaystyle{{{\hat{q}}_{1}}}\geq 0,{{{\hat{q}}_{2}}}\geq 0.$ (45d)

Since ${\log_{2}}x$ is a concave function, at least one of constraint (45b)
and constraint (45c) is active for the optimal power allocation, i.e.,
${\eta_{1}}P_{\mathrm{e},1}{{{\hat{q}}_{1}}}+{\eta_{2}}P_{\mathrm{e},2}{{{\hat{q}}_{2}}}={P_{\rm{T}}}$
or $\hat{q}_{1}=\tau^{2}$. When $\hat{q}_{1}=\tau^{2}$, constraint (45b) is
also active. Therefore, constraint (45b) is always active, i.e.,
${\eta_{1}}P_{\mathrm{e},1}{{{\hat{q}}_{1}}}+{\eta_{2}}P_{\mathrm{e},2}{{{\hat{q}}_{2}}}={P_{\rm{T}}}$.
In the following, we will discuss the optimal power allocation of problem (45)
based on whether constraint (45c) is active or not.

####  IV-A1 ${\eta_{1}}{\varepsilon_{1}}\tau^{2}>{P_{\rm{T}}}$

In this case, constraint (45c) is inactive, i.e., $\hat{q}_{1}<\tau^{2}$.
Problem (45) can be reformulated as

$\displaystyle\mathop{\min}\limits_{{{{\hat{q}}_{1}}},{{{\hat{q}}_{2}}}}\quad$
$\displaystyle
2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({{{\hat{q}}_{1}}}\right)+{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({{{\hat{q}}_{2}}}\right)$
(46a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{\eta_{1}}P_{\mathrm{e},1}{{{\hat{q}}_{1}}}+{\eta_{2}}P_{\mathrm{e},1}{{{\hat{q}}_{2}}}={P_{\rm{T}}},$
(46b) $\displaystyle{{{\hat{q}}_{1}}}\geq 0,{{{\hat{q}}_{2}}}\geq 0.$ (46c)

Substituting
${{{\hat{q}}_{2}}}=\frac{{{P_{\rm{T}}}-{\eta_{1}}{\varepsilon_{1}}{{{\hat{q}}_{1}}}}}{{{\eta_{2}}{\varepsilon_{2}}}}$
into problem (46), the objective function (46a) is reformulated as

$\displaystyle\Phi\left({{\hat{q}}_{1}}\right)\buildrel\Delta\over{=}2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({{{\hat{q}}_{1}}}\right)+{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{\hat{v}}}\left({{{\hat{q}}_{1}}}\right),$
(47)

where
${\bf{\hat{v}}}\left({{\hat{q}_{1}}}\right)\buildrel\Delta\over{=}{\left[{{\log_{2}}{\bf{\hat{b}}}_{1}^{\rm{T}}\left({{\hat{q}_{1}}}\right){{\bf{p}}_{1}},\ldots,{\log_{2}}{\bf{\hat{b}}}_{N}^{\rm{T}}\left({{\hat{q}_{1}}}\right){{\bf{p}}_{1}}}\right]^{\rm{T}}}$,
${{\bf{\hat{b}}}_{l}}\left({{\hat{q}_{1}}}\right)\buildrel\Delta\over{=}{\left[{{\hat{b}_{l,1}}\left({{\hat{q}_{1}}}\right),\ldots,{\hat{b}_{l,N}}\left({{\hat{q}_{1}}}\right)}\right]^{\rm{T}}},\forall{l}\in{\cal
N}$, and
${{\hat{b}}_{l,n}}\left({{{\hat{q}}_{1}}}\right)\buildrel\Delta\over{=}\exp\left({-\frac{{{{\left|{{x_{2,l}}-{x_{2,n}}}\right|}^{2}}{{\left|{{g_{2}}}\right|}^{2}}\left({{P_{\rm{T}}}-{\eta_{1}}{\varepsilon_{1}}{{\hat{q}}_{1}}}\right)}}{{2{B_{2}}\sigma_{2}^{2}{\eta_{2}}{\varepsilon_{2}}}}}\right),\forall{l,n}\in{\cal
N}$.

Thus, problem (45) can be reformulated as

$\displaystyle\mathop{\min}\limits_{{{{\hat{q}}_{1}}}}$
$\displaystyle\quad\Phi\left({{\hat{q}}_{1}}\right)$ (48a)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$ $\displaystyle\quad
0\leq{{{\hat{q}}_{1}}}\leq\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}.$
(48b)

Let ${{\hat{q}}_{1}}^{{\rm{opt}}}$ denote the optimal solution of problem
(48), and $\Phi^{{\rm{opt}}}$ denote the maximum rate of the aggregated LiFi -
WiFi system. Then, we can obtain a stationary point
${{\hat{q}}_{1}}^{\rm{sta}}$ by setting
$\frac{{\partial\Phi\left({{\hat{q}}_{1}}\right)}}{{\partial{{{\hat{q}}_{1}}}}}=0$
so that the optimal solution ${{\hat{q}}_{1}}^{\rm{sta}}$ satisfies the
following equation

$\displaystyle
2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{\tilde{a}}}\left({{{\hat{q}}_{1}}}\right)={B_{2}}{\bf{p}}_{2}^{\rm{T}}{{{\bf{\tilde{b}}}}}\left({{{\hat{q}}_{1}}}\right),$
(49)

where

$\displaystyle{\bf{\tilde{a}}}\left({{{\hat{q}}_{1}}}\right)={\left[{{{\tilde{a}}_{1}}\left({{{\hat{q}}_{1}}}\right),\ldots,{{\tilde{a}}_{M}}\left({{{\hat{q}}_{1}}}\right)}\right]^{\rm{T}}},$
(50a)
$\displaystyle{{\tilde{a}}_{k}}\left({{{\hat{q}}_{1}}}\right)=\frac{{{{{{\bf{c}}^{\rm{T}}_{k}\odot{{\bf{a}}^{\rm{T}}_{k}}\left({{{\hat{q}}_{1}}}\right)}}}{{\bf{p}}_{1}}}}{{{{\bf{a}}^{\rm{T}}_{k}}\left({{{\hat{q}}_{1}}}\right){{\bf{p}}_{1}}}},$
(50b)
$\displaystyle{{\bf{c}}_{k}}\buildrel\Delta\over{=}{\frac{{g_{1}^{2}}}{{4{B_{1}}\sigma_{1}^{2}}}}{\left[{{{\left({{x_{1,k}}-{x_{1,1}}}\right)}^{2}},\ldots,{{\left({{x_{1,k}}-{x_{1,M}}}\right)}^{2}}}\right]^{\rm{T}}},\forall{k}\in{\cal
M},$ (50c)
$\displaystyle{\bf{\tilde{b}}}\left({{{\hat{q}}_{1}}}\right)={\left[{{{\tilde{b}}_{1}}\left({{{\hat{q}}_{1}}}\right),\ldots,{{\tilde{b}}_{N}}\left({{{\hat{q}}_{1}}}\right)}\right]^{\rm{T}}},$
(50d)
$\displaystyle{{\tilde{b}}_{l}}\left({{{\hat{q}}_{1}}}\right)=\frac{{{{{{\bf{d}}^{\rm{T}}_{l}\odot{{{\bf{\hat{b}}}}^{\rm{T}}_{l}}\left({{{\hat{q}}_{1}}}\right)}}}{{\bf{p}}_{2}}}}{{{{{\bf{\hat{b}}}}^{\rm{T}}_{l}}\left({{{\hat{q}}_{1}}}\right){{\bf{p}}_{2}}}},$
(50e)
$\displaystyle{{\bf{d}}_{l}}\buildrel\Delta\over{=}{\frac{{{\eta_{1}}{\varepsilon_{1}}{{\left|{{g_{2}}}\right|}^{2}}}}{{2{B_{2}}\sigma_{2}^{2}{\eta_{2}}{\varepsilon_{2}}}}}{\left[{{{\left|{{x_{2,l}}-{x_{2,1}}}\right|}^{2}},\ldots,{{\left|{{x_{2,l}}-{x_{2,N}}}\right|}^{2}}}\right]^{\rm{T}}},\forall{l}\in{\cal
N}.$ (50f)

Let $\Phi^{{\rm{opt},1}}$ and ${{\hat{q}}_{1}}^{{\rm{opt},1}}$ respectively
denote the minimal objective function and the optimal power allocation
${{\hat{q}}_{1}}$ of problem (45). Thus, if
$0\leq{{\hat{q}}_{1}}\leq\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}$,
we have

$\displaystyle\Phi^{{\rm{opt},1}}=\min\left\\{{\Phi\left({{{{\hat{q}}_{1}}^{\rm{sta}}}}\right),\Phi\left(0\right),\Phi\left(\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}\right)}\right\\},$
(51a)
$\displaystyle{{\hat{q}}_{1}}^{{\rm{opt},1}}=\mathop{\arg\min}\limits_{{{\hat{q}}_{1}}}\left\\{{\Phi\left({{{{\hat{q}}_{1}}^{\rm{sta}}}}\right),\Phi\left(0\right),\Phi\left(\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}\right)}\right\\}.$
(51b)

Otherwise, we have

$\displaystyle\Phi^{{\rm{opt},1}}=\min\left\\{{\Phi\left(0\right),\Phi\left(\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}\right)}\right\\},$
(52a)
$\displaystyle{{\hat{q}}_{1}}^{{\rm{opt},1}}=\mathop{\arg\min}\limits_{{{\hat{q}}_{1}}}\left\\{{\Phi\left(0\right),\Phi\left(\frac{P_{\rm{T}}}{{{{\eta_{1}}{\varepsilon_{1}}}}}\right)}\right\\}.$
(52b)

####  IV-A2 ${\eta_{1}}{{P_{e,1}}}\tau^{2}\leq{P_{\rm{T}}}$

When ${\eta_{1}}{{P_{e,1}}}\tau^{2}\leq{P_{\rm{T}}}$, constraint (45c) could
be either active or inactive, i.e., $\hat{q}_{1}=\tau^{2}$ or
$\hat{q}_{1}<\tau^{2}$. When constraint (45c) is inactive, the minimal
objective function of problem (45) is the same as the case
${\eta_{1}}{{P_{e,1}}}\tau^{2}<{P_{\rm{T}}}$. Otherwise, the optimal allocated
power of problem (45) are $\hat{q}_{1}=\tau^{2}$, and
${{{\hat{q}}_{2}}}=\frac{{{P_{\rm{T}}}-{\eta_{1}}{{P_{e,1}}}\tau^{2}}}{{{\eta_{2}}{{P_{e,2}}}}}$.
Here, the corresponding minimal objective function of problem (45) is
$\Phi\left(\tau^{2}\right)$. Let $\Phi^{{\rm{opt},2}}$ and
${{\hat{q}}_{1}}^{{\rm{opt},2}}$ respectively denote the minimal objective
function and the optimal power allocation ${{\hat{q}}_{1}}$ of problem (45),
which are given by

$\displaystyle\Phi^{{\rm{opt},2}}=\min\left\\{{\Phi^{{\rm{opt},1}},\Phi\left(\tau^{2}\right)}\right\\},$
(53a)
$\displaystyle{{\hat{q}}_{1}}^{{\rm{opt},2}}=\mathop{\arg\min}\limits_{{{\hat{q}}_{1}}}\left\\{{\Phi^{{\rm{opt},1}},\Phi\left(\tau^{2}\right)}\right\\}.$
(53b)

By solving the power allocation sub-problem, we can obtain the optimal power
of LiFi and WiFi links $q_{1}$ and $q_{2}$.

###  IV-B Probability Distribution Sub-problem

Next, we solve the probability distribution sub-problem: optimizing the LiFi
and the WiFi links probability distribution, i.e., ${\bf{p}}_{1}$ and
${\bf{p}}_{2}$, with given ${{\hat{q}_{1}}}$ and ${{\hat{q}_{2}}}$. Then, the
maximization problem of the aggregated LiFi-WiFi system (45) can be formulated
as

$\displaystyle\mathop{\min}\limits_{{{\bf{p}}_{1}},{{\bf{p}}_{2}}}\quad$
$\displaystyle
2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({\bf{p}}_{1}\right)+{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({\bf{p}}_{2}\right)$
(54a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{{\bf{p}}_{1}\in{\Upsilon_{1}}},~{}{{\bf{p}}_{2}\in{\Upsilon_{2}}},$
(54b)

where
${\bf{u}}\left({{{\bf{p}}_{1}}}\right)={\left[{{\log_{2}}{{\bf{a}}^{\rm{T}}_{1}}\left({{{\hat{q}}_{1}}}\right){{\bf{p}}_{1}},\ldots,{\log_{2}}{{\bf{a}}^{\rm{T}}_{M}}\left({{{\hat{q}}_{1}}}\right){{\bf{p}}_{1}}}\right]^{\rm{T}}}$,
and
${\bf{v}}\left({{{\bf{p}}_{2}}}\right)={\left[{{\log_{2}}{{\bf{b}}^{\rm{T}}_{1}}\left({{{\hat{q}}_{2}}}\right){{\bf{p}}_{2}},\ldots,{\log_{2}}{{\bf{b}}^{\rm{T}}_{N}}\left({{{\hat{q}}_{2}}}\right){{\bf{p}}_{2}}}\right]^{\rm{T}}}$.
Then, problem (54) can be divided into two independent subproblems as

$\displaystyle\mathop{\min}\limits_{{{\bf{p}}_{1}}}\quad$ $\displaystyle
f_{1}\left({{{{\bf{p}}}_{1}}}\right)$ (55a)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{{\bf{p}}_{1}\in{\Upsilon_{1}}},$ (55b)

and

$\displaystyle\mathop{\min}\limits_{{{\bf{p}}_{2}}}\quad$ $\displaystyle
f_{2}\left({{{{\bf{p}}}_{2}}}\right)$ (56a)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{{\bf{p}}_{2}\in{\Upsilon_{2}}}.$ (56b)

where
$f_{1}\left({{{{\bf{p}}}_{1}}}\right)\buildrel\Delta\over{=}2{B_{1}}{\bf{p}}_{1}^{\rm{T}}{\bf{u}}\left({\bf{p}}_{1}\right)$,
and
$f_{2}\left({{{{\bf{p}}}_{2}}}\right)\buildrel\Delta\over{=}{B_{2}}{\bf{p}}_{2}^{\rm{T}}{\bf{v}}\left({\bf{p}}_{2}\right)$.
We can adopt the Frank-Wolfe method [33] to solve problem (55) and (56).
Specifically, let ${{\bf{p}}^{\left[{i}\right]}_{1}}$ denote a feasible point
at the $i$th iteration. The first-order Taylor expansion of
$f_{1}\left({{{{\bf{p}}}_{1}}}\right)$ is given by

$\displaystyle{f_{1}}\left({{{\bf{p}}_{1}}}\right)\approx{f_{1}}\left({{{\bf{p}}^{\left[{i}\right]}_{1}}}\right)+{\nabla_{{{\bf{p}}_{1}}}}f_{\rm{1}}^{\rm{T}}\left({{{\bf{p}}^{\left[{i}\right]}_{1}}}\right)\left({{{\bf{p}}_{1}}-{{\bf{p}}^{\left[{i}\right]}_{1}}}\right),$
(57)

where
${\nabla_{{{\bf{p}}_{1}}}}f_{\rm{1}}^{\rm{T}}\left({{{\bf{p}}^{\left[{i}\right]}_{1}}}\right)$
denotes the gradient of the objective function
$f_{1}\left({{{{\bf{p}}}_{1}}}\right)$. Since
${{\bf{p}}^{\left[{i}\right]}_{1}}$ is obtained, problem (55) is equivalent to

$\displaystyle\mathop{\min}\limits_{{{\bf{p}}_{1}}}\quad$
$\displaystyle{\nabla_{{{\bf{p}}_{1}}}}f_{\rm{1}}^{\rm{T}}\left({{{\bf{p}}^{\left[{i}\right]}_{1}}}\right){{\bf{p}}_{1}}$
(58a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle{{\bf{p}}_{1}\in{\Upsilon_{1}}}.$ (58b)

Then, the $i+1$th iteration point ${{{\bf{p}}^{\left[{i+1}\right]}_{1}}}$ is
updated by

$\displaystyle{{{\bf{p}}^{\left[{i+1}\right]}_{1}}}={{{\bf{p}}^{\left[{i}\right]}_{1}}+{\lambda^{\left[{i}\right]}_{1}}{{\bf{d}}^{\left[{i}\right]}_{1}}},$
(59)

where ${\lambda^{\left[{i}\right]}_{1}}$ denotes the stepsize of the $i$th
iteration, and ${{\bf{d}}^{\left[{i}\right]}_{1}}$ denotes the feasible
descending direction of the $i$th iteration. The details of the Frank-Wolfe
method are summarized in Algorithm 4, which outputs the probability
distribution of the LiFi link, i.e., ${\bf{p}}_{1}$. Similar as problem (55),
we can obtain the optimal probability distribution of the WiFi link, i.e.,
${\bf{p}}_{2}$, by solving problem (56) through Frank-Wolfe method. We omit
the detailed derivations for brevity. The optimal probability distribution
$\mathbf{p}_{1}$, $\mathbf{p}_{2}$ can be achieved by the distribution
matching (DM) [29, 30, 31].

In summary, the achievable rate maximization of the aggregated LiFi-WiFi
system (44) can be solved by Algorithm 5. The optimal power allocation of LiFi
and WiFi links ${{\hat{q}_{1}}}$ and ${{\hat{q}_{2}}}$, the probability
distribution of LiFi and WiFi links ${\bf{p}}_{1}$ and ${\bf{p}}_{2}$, and the
maximal lower bound of the achievable rate ${\hat{R}}_{\rm{LiFi-
WiFi}}^{\rm{L}}$ can be obtained by Algorithm 3. Note that, based on the upper
bound of the achievable rate $R_{{\rm{LiFi-WiFi}}}^{\rm{U}}$, we can also find
the optimal distribution of the signal magnitudes and power allocation to
maximize the achievable rate of the aggregated LiFi-WiFi system, which is
similar to the lower bound case.

Algorithm 4 Frank-Wolfe Method for the Probability Distribution Sub-Problem
(54).

1:Initialization: Given $\varsigma\geq 0$, set $i=1$, and choose
${{{{\bf{p}}}^{\left[{1}\right]}_{1}}}\in{\Upsilon_{1}}$.

2:repeat

3: $i\leftarrow i+1$.

4: Obtain ${{{\bf{\hat{p}}}}^{\left[{i-1}\right]}_{1}}$ by solving problem
(58).

5: Construct feasible descending direction
${{\bf{d}}^{\left[{i-1}\right]}_{1}}={{{\bf{\hat{p}}}}^{\left[{i-1}\right]}_{1}}-{{{{\bf{p}}}^{\left[{i-1}\right]}_{1}}}$.

6: Use the bisection search to obtain the optimal
${\lambda^{\left[{i-1}\right]}_{1}}=\mathop{\arg\min}\limits_{{\lambda_{1}}}{f_{1}}\left({{\bf{p}}_{1}^{\left[{i-1}\right]}+{\lambda_{1}}{\bf{d}}_{1}^{\left[{i-1}\right]}}\right)$.

7: Update
${{{\bf{p}}^{\left[{i}\right]}_{1}}}={{{\bf{p}}^{\left[{i-1}\right]}_{1}}+{\lambda^{\left[{i-1}\right]}_{1}}{{\bf{d}}^{\left[{i-1}\right]}_{1}}}$.

8:until
$\left|{\nabla_{{{\bf{p}}_{1}}}}f_{\rm{1}}^{\rm{T}}\left({{{\bf{p}}^{\left[{i-1}\right]}_{1}}}\right){{\bf{d}}^{\left[{i-1}\right]}_{1}}\right|\leq\varsigma$.

9:return ${{{\bf{p}}_{1}}}={{{\bf{p}}^{\left[{i+1}\right]}_{1}}}$.

Algorithm 5 Optimal discrete constellation input distributions based on lower
bounds

1:Initialization: Given $\xi\geq 0$, set $k=1$, and choose
${{{{\bf{p}}}_{1}^{\left[1\right]}}}\in{\Upsilon_{1}}$,
${{{{\bf{p}}}_{2}^{\left[1\right]}}}\in{\Upsilon_{2}}$.

2:Update ${{\hat{q}}_{1}^{\left[1\right]}}$ and
${{\hat{q}}_{2}^{\left[1\right]}}$ by solving problem (45).

3:Obtain ${R}^{{\rm{L}}\left[1\right]}_{\rm{LiFi-WiFi}}$ by substituting
${\hat{q}}_{1}^{\left[1\right]}$, ${\hat{q}}_{2}^{\left[1\right]}$,
${{\bf{p}}_{1}^{\left[1\right]}}$ and ${{\bf{p}}_{2}^{\left[1\right]}}$ into
formulation (IV).

4:repeat.

5: $k\leftarrow k+1$.

6: Update ${\hat{q}}_{1}^{\left[k\right]}$ and
${\hat{q}}_{2}^{\left[k\right]}$ by solving problem (45).

7: Update ${{\bf{p}}_{1}^{\left[k\right]}}$ and
${{\bf{p}}_{2}^{\left[k\right]}}$ by Frank-Wolfe method (in Algorithm 4).

8: Obtain ${{R}_{\rm{LiFi-WiFi}}^{{\rm{L}}{\left[k\right]}}}$ by substituting
${\hat{q}}_{1}^{\left[k\right]}$, ${\hat{q}}_{2}^{\left[k\right]}$,
${{\bf{p}}_{1}^{\left[k\right]}}$ and ${{\bf{p}}_{2}^{\left[k\right]}}$ into
formulation (IV).

9: until $\left|{R}^{{\rm{L}}\left[k\right]}_{\rm{LiFi-
WiFi}}-{R}^{{\rm{L}}\left[k-1\right]}_{\rm{LiFi-WiFi}}\right|\leq\xi$.

10:return ${{{\bf{p}}_{1}}}={{{\bf{p}}^{\left[k\right]}_{1}}}$,
${{{\bf{p}}_{2}}}={{{\bf{p}}^{\left[k\right]}_{2}}}$,
${\hat{q}}_{1}={\hat{q}}^{\left[k\right]}_{1}$ and
${\hat{q}}_{2}={\hat{q}}^{\left[k\right]}_{2}$,${R}_{\rm{LiFi-
WiFi}}^{\rm{L}}={R}_{\rm{LiFi-WiFi}}^{{\rm{L}}\left[k\right]}$ .

##  V Simulation Results and Discussions

In this section, we illustrate the performance of the proposed schemes for the
considered aggregated LiFi-WiFi system. Moreover, the simulation results also
demonstrate the impact of key parameters on both the achievable rate and
energy efficiency of the aggregated LiFi-WiFi system, such as total power
threshold, bandwidths, etc. In the simulations, we consider the locations of
the LED and PD as $\left(0,0,5.7\right)$m and $\left(0,0,1.7\right)$m,
respectively. More detailed system parameters of the LiFi and WiFi links are
given in Table I.

TABLE I: Basic parameters  LiFi Link | WiFi Link
---|---
Parameters | Value | Parameters | Value
${\sigma_{1}^{2}}$ | $10^{-21}\rm{A^{2}/Hz}$ | ${\sigma_{2}^{2}}$ | -57dBm/MHz
FoV ${\Psi_{c}}$ | ${90^{\circ}}$ | $d_{B}$ | 5 m
$A_{\rm{PD}}$ | $1{\rm{c}}{{\rm{m}}^{2}}$ | $d_{2}$ | 4 m
$I_{\rm{H}}$ | 8 A | $f_{c}$ | 2.4 GHz
${\theta_{1/2}}$ | ${60^{\circ}}$ | $\psi$ | ${45^{\circ}}$
$B_{1}$ |  $40$MHz | $B_{2}$ |  $20$MHz

###  V-A Performance of the Aggregated LiFi-WiFi System based on
${{{R}}_{{\rm{LiFi-WiFi}}}}$ (30)

In the following, we evaluate the performance of the proposed transmission
schemes in Fig. 2 and Fig. 3.

(a)

(b)

(c)

(d)

Figure 2: (a) The optimal probability distribution of input
$\left\\{{{x_{1,k}}},{{p_{1,k}}}\right\\}$ versus ${\rm{SNR}}_{1}$ of LiFi
link;(b) The optimal probability distribution of input
$\left\\{{{x_{2,l}}},{{p_{2,l}}}\right\\}$ with ${\rm{SNR}}_{2}$=4dB of WiFi
link. (c) Achievable rate of LiFi link ${{{R}}_{{\rm{LiFi}}}}$ versus
${\rm{SNR}}_{1}$; (d) Achievable rate of WiFi link ${{{R}}_{{\rm{WiFi}}}}$
versus ${\rm{SNR}}_{2}$.

Fig. 2 (a) illustrates the optimal probability distribution of input
$\left\\{{{x_{1,k}}},{{p_{1,k}}}\right\\}$ versus ${\rm{SNR}}_{1}$ (SNR) of
the LiFi link, where $K_{1}=8$. It shows that in the low SNR region, the
optimal input positions include two discrete points with an equal probability.
While in the high SNR region, the optimal input positions have more than two
discrete points. As the increase of SNR, the optimal probability distribution
is closer to the equiprobable distribution. Fig. 2 (c) illustrates the
achievable rate of LiFi link ${{{R}}_{{\rm{LiFi}}}}$ versus SNR. We can
observe that the achievable rate of LiFi link ${{{R}}_{{\rm{LiFi}}}}$ with the
proposed method is higher than that equiprobable distribution. Moreover, with
the increase of the SNR, the gap between the proposed method and the
equiprobable distribution shrinks.

Fig. 2 (b) shows the optimal probability distribution of input
$\left\\{{{x_{2,l}}},{{p_{2,l}}}\right\\}$ with ${\rm{SNR}}_{2}=$ 4dB of the
WiFi link, where $K_{2}=16$. It shows that for $\mathrm{SNR}_{2}$=4dB, the
optimal input positions include sixteen discrete points, and the optimal
probability distribution is equiprobable. Fig. 2(d) illustrates the achievable
rate of WiFi link ${{{R}}_{{\rm{WiFi}}}}$ versus $\mathrm{SNR}_{2}$. We can
observe that the achievable rate of WiFi link ${{{R}}_{{\rm{WiFi}}}}$ with the
proposed method is higher than that obtained by the equiprobable distribution.
Moreover, with the increase of the $\mathrm{SNR}_{2}$, the gap between the
proposed method and the equiprobable distribution decreases.

(a)

(b)

Figure 3: Achievable rate of the aggregated LiFi-WiFi system
${{{R}}_{{\rm{LiFi-WiFi}}}}$: (a) versus the total power threshold
$P_{\rm{T}}$; (b) versus the instant optical power threshold $P_{\rm{ins}}$.

Fig. 3 (a) shows the achievable rate ${{{R}}_{{\rm{LiFi-WiFi}}}}$ and link
transmit power of the aggregated LiFi-WiFi system versus total power threshold
$P_{\rm{T}}$, respectively. We observe that the achievable rates
${{{R}}_{{\rm{LiFi-WiFi}}}}$ with both the proposed method and equiprobable
distribution increase as total power threshold $P_{\rm{T}}$ increases.

Fig. 3 (b) illustrates the achievable rate of aggregated LiFi-WiFi system
${{{R}}_{{\rm{LiFi-WiFi}}}}$ versus instant optical power threshold
$P_{\rm{ins}}$. We observe that the achievable rates ${{{R}}_{{\rm{LiFi-
WiFi}}}}$ with the proposed method increase as the instant optical power
threshold $P_{\rm{ins}}$ first increases, and remains constant. Moreover, as
instant optical power threshold $P_{\rm{ins}}$ increases, the transmit power
of LiFi link $\hat{q}_{1}$ first increases, and then keeps as a constant. The
transmit power of WiFi link $\hat{q}_{2}$ first decreases, and then keeps as a
constant. This is due to
$\tau\buildrel\Delta\over{=}\min\left({{{{P_{o}}/\bar{\mu}}},{{P_{\rm{ins}}}/A}{}}\right)$,
and we assume that ${P_{o}}=0.8{P_{\rm{ins}}}$, and $\bar{\mu}=0.5A$. For a
lower $P_{\rm{ins}}$, $\hat{q}_{1}$ is constrained by $\tau$, while for a high
$P_{\rm{ins}}$, $\hat{q}_{1}$ is constrained by the total electrical power
threshold $P_{\rm{T}}$.

###  V-B Optimal Discrete Constellation Input Distributions Based on
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ (40a)

In the following, the performance of the proposed aggregated LiFi-WiFi system
based on the lower bound of achievable rate is investigated in Fig. 4 (a) and
(b).

(a)

(b)

Figure 4: Lower bound of the achievable rates of the aggregated LiFi-WiFi
system ${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$: (a) versus the total electrical
power threshold $P_{\rm{T}}$; (b) versus the instant optical power threshold
$P_{\rm{ins}}$.

Fig. 4 (a) demonstrates the lower bound of achievable rate
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ and link transmit power of aggregated
LiFi-WiFi system versus total power threshold $P_{\rm{T}}$, respectively. We
observe that the lower bound of achievable rates ${{{R}}^{\rm{L}}_{{\rm{LiFi-
WiFi}}}}$ with both the proposed method and equiprobable distribution increase
as total power threshold $P_{\rm{T}}$ increases. Moreover, as total power
threshold ${P_{T}}$ increases, the transmit power of the LiFi link, i.e.,
$\hat{q}_{1}$, first increases, and remains as a constant.and the transmit
power of the WiFi link, i.e., $\hat{q}_{2}$, increases.The reason is that
$\hat{q}_{1}$ is also limited by the optical power constraint.222Due to adding
the constant gap, $R_{\mathrm{LiFi-WiFi}}$ might be less than
$R_{\mathrm{LiFi-WiFi}}^{\mathrm{L}}$ in Fig. 4-6 as similar as Fig. 1-3 of
[32].

Fig. 4 (b) describes the lower bound of achievable rate of aggregated LiFi-
WiFi system ${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ versus instant optical power
threshold $P_{\rm{ins}}$, and we assume that ${P_{o}}=0.8{P_{\rm{ins}}}$, and
$\bar{\mu}=0.5A$. We observe that the lower bound of achievable rates
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ with the proposed method increases as
instant optical power threshold $P_{\rm{ins}}$ first increases, and remains as
a constant. Moreover, as instant optical power threshold $P_{\rm{ins}}$
increases, the transmit power of the LiFi link, i.e., $\hat{q}_{1}$, first
increases, and then keeps as a constant. The transmit power of WiFi link
$\hat{q}_{2}$ first decreases, and then keeps as a constant. This is due to
$\tau\buildrel\Delta\over{=}\min\left({{{{P_{o}}/\bar{\mu}}},{{P_{\rm{ins}}}/A}{}}\right)$.
For a lower $P_{\rm{ins}}$, $\hat{q}_{1}$ is constrained by $\tau$, while for
high $P_{\rm{ins}}$, $\hat{q}_{1}$ is constrained by the total electrical
power threshold $P_{\rm{T}}$.

###  V-C The Rate Performance

In the following, we investigate the comparison between the achievable rate
${{{R}}_{{\rm{LiFi-WiFi}}}}$ (30), it’s lower bound
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ (40a) and the upper bound
${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ (40b) of the proposed aggregated LiFi-
WiFi system in Fig. 5 (a) and (b), Fig. 6 (a) and (b).

(a)

(b)

Figure 5: ${{{R}}_{{\rm{LiFi-WiFi}}}}$, ${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$
and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ versus the total power threshold
$P_{\rm{T}}$: (a) with the optimal solutions of problem (33); (b) with the
optimal solutions of problem (41).

Fig. 5 (a) shows the achievable rate ${{{R}}_{{\rm{LiFi-WiFi}}}}$ (30), it’s
lower bound ${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ (40a) and the upper bound
${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ (40b) of the aggregated LiFi-WiFi system
versus total power threshold $P_{\rm{T}}$ with the optimal solutions of
problem (33), respectively. And Fig. 5 (b) illustrates the achievable rate
${{{R}}_{{\rm{LiFi-WiFi}}}}$ (30), it’s lower bound
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ (40a) and the upper bound
${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ (40b) of the aggregated LiFi-WiFi system
versus total power threshold $P_{\rm{T}}$ with the optimal solutions of
problem (41), respectively. Both Fig. 5 (a) and (b) show that for a low
${P_{T}}$, the gap between ${{{R}}_{{\rm{LiFi-WiFi}}}}$ and
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ is larger than the one of between
${{{R}}_{{\rm{LiFi-WiFi}}}}$ and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$, and
for a high ${P_{T}}$, the gap between ${{{R}}_{{\rm{LiFi-WiFi}}}}$ and
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ is lower than the one of between
${{{R}}_{{\rm{LiFi-WiFi}}}}$ and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$.
Moreover, the ${{{R}}_{{\rm{LiFi-WiFi}}}}$ , ${{{R}}^{\rm{L}}_{{\rm{LiFi-
WiFi}}}}$ and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ in Fig. 5 (a) are lower
than those in Fig. 5 (b). This is because that solutions of Algorithm 3 are
suboptimal.

(a)

(b)

Figure 6: ${{{R}}_{{\rm{LiFi-WiFi}}}}$, ${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$
and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$ versus the LiFi link bandwidth
$B_{1}$ with the WiFi link bandwidth $B_{2}=20\mathrm{MHz}$: (a) with the
optimal solutions of problem (33); (b) with the optimal solutions of problem
(41).

Fig. 6 (a) and (b) shows that the achievable rate versus the LiFi link
bandwidth $B_{1}$ with the WiFi link bandwidth $B_{2}=20\mathrm{MHz}$ and the
two solutions. Fig. 5 (a) based on the solutions of problem (33), and Fig. 5
(b) is with the solutions of problem (41). Both Fig. 6 (a) and (b) show that
for a low ${B_{1}}$, the gap between ${{{R}}_{{\rm{LiFi-WiFi}}}}$ and
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ is larger than the one of between
${{{R}}_{{\rm{LiFi-WiFi}}}}$ and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$, and
for a high ${B_{1}}$, the gap between ${{{R}}_{{\rm{LiFi-WiFi}}}}$ and
${{{R}}^{\rm{L}}_{{\rm{LiFi-WiFi}}}}$ is lower than the one of between
${{{R}}_{{\rm{LiFi-WiFi}}}}$ and ${{{R}}^{\rm{U}}_{{\rm{LiFi-WiFi}}}}$.

###  V-D The Average Executing Time

To evaluate the computational complexity, Table II shows the average executing
times versus modulation order ($M=N$) of Algorithm 3 and Algorithm 5,
respectively.

TABLE II: The average executing time of different scheme versus modulation order($M=N$). $M=N$ Time(s) Scheme | Algorithm 3 | Algorithm 5
---|---|---
$4$ | $5.47$ | $0.2466$
$8$ | $30.16$ | $4.60$
$16$ | $79.01$ | $7.68$
$32$ | $311.44$ | $29.75$
$64$ | $4217.26$ | $264.95$

For the given modulation order, the average executing time is calculated over
$200$ independent trials, which are performed by MATLAB (2021a) with Intel(R)
Core(TM) i5-9500 3.00 GHz CPU and 8 GB RAM. As showed in Table II, the CPU
time of the two algorithms increases as the modulation order increases, and
the CPU time of Algorithm 5 is significantly less than that of Algorithm 3.

##  VI Conclusion

In this paper, we investigated the aggregated LiFi-WiFi system, which
simultaneously used both the LiFi link and WiFi link to transmit information.
First of all, with both LiFi link and WiFi link bandwidths consideration, we
derived the achievable rate expression of the system with the discrete
constellation input signals. Then, we further investigated the optimal input
distribution and power allocation for the considered system. Moreover, we
derived the upper and lower bounds of the achievable rate expression of the
system with the discrete constellation input signals, and we investigated the
optimal input distribution and power allocation for the considered system. At
last, the effects of critical parameters of the aggregated LiFi-WiFi system on
the maximum achievable rate, such as total threshold, optical power, and
bandwidth, etc., are numerically analyzed. In the future, we would further
investigate both imperfect CSI and interference management for multi-user
LiFi-WiFi networks.

##  Appendix A Proof of the formulation Lemma 1

The achievable rates of the LiFi link ${{{R}}_{{\rm{LiFi}}}}$ are given as
follows

$\displaystyle{R_{{\rm{LiFi}}}}$
$\displaystyle={\rm{h}}\left({{y_{1}}}\right)-{\rm{h}}\left({{y_{1}}\left|{{x_{1}}}\right.}\right)$
(60a)
$\displaystyle=-\int_{-\infty}^{\infty}{\frac{{\sum\limits_{k=1}^{M}{{p_{1,k}}\exp\left({-\frac{{z_{\rm{1}}^{\rm{2}}}}{{2\sigma_{1}^{2}}}}\right)}}}{{\sqrt{2\pi}{\sigma_{1}}}}{{\log}_{2}}}\frac{{\sum\limits_{m=1}^{M}{{p_{1,m}}\exp\left({{{\overline{\Lambda}}_{k,m}}}\right)}}}{{\sqrt{2\pi}{\sigma_{1}}}}\,\mathrm{d}{z_{1}}$
$\displaystyle~{}~{}~{}-\frac{1}{2}{\log_{2}}2\pi e\sigma_{1}^{2}$ (60b)
$\displaystyle=-\frac{1}{{2\ln
2}}-\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\mathbb{E}}_{z_{1}}}\left\\{{{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}\exp\left({{\overline{\Lambda}_{k,m}}}\right)}}\right\\}},$
(60c)

where
${\overline{\Lambda}_{k,m}}\buildrel\Delta\over{=}-\frac{{{{\left({{g_{1}}{q_{1}}\left({{x_{1,k}}-{x_{1,m}}}\right)+{z_{1}}}\right)}^{\rm{2}}}}}{{2\sigma_{1}^{2}}}$.

Suppose that the bandwidth of the LiFi link is $B_{1}$ (Hz). Then, both the
input and output can be represented by samples taken by $\frac{1}{{2B_{1}}}$
seconds apart [34]. The power spectral density of the noise is
$\frac{\sigma_{1}^{2}}{2}$ watts/Hz, which leads to the noise power
${{\sigma_{1}^{2}}}B_{1}$. For the time interval $\left[{0,T}\right]$, there
are $2BT$ noise samples, and the variance of each sample is
$\frac{{{\sigma_{1}^{2}}B_{1}T}}{{{2}B_{1}T}}=\frac{\sigma_{1}^{2}}{2}$.
Moreover, if the power of signal is $P_{1}$, the energy of signal per sample
is $\frac{{{P_{1}}T}}{{{2}B_{1}T}}=\frac{P_{1}}{{{2}B_{1}}}$. Therefore, we
obtain the achievable rate expression of the LiFi link with the bandwidth of
$B_{1}$ as given in (32a).

Then, the achievable rates of the WiFi link and ${{{R}}_{{\rm{WiFi}}}}$ are
given as follows

$\displaystyle{R_{{\rm{WiFi}}}}$
$\displaystyle=-\int_{-\infty}^{\infty}{\frac{{\sum\limits_{l=1}^{N}{{p_{2,l}}\exp\left({-\frac{{{{\left|{{z_{2}}}\right|}^{\rm{2}}}}}{{\sigma_{\rm{2}}^{\rm{2}}}}}\right)}}}{{\pi\sigma_{\rm{2}}^{\rm{2}}}}}{\log_{2}}\frac{{\sum\limits_{n=1}^{N}{{p_{2,n}}\exp\left({\overline{\Gamma}_{l,n}}\right)}}}{{\pi\sigma_{\rm{2}}^{\rm{2}}}}\,\mathrm{d}{z_{2}}$
$\displaystyle~{}~{}~{}-{\log_{2}}\pi e\sigma_{2}^{2}$ (61a)
$\displaystyle=-\frac{{1}}{{\ln
2}}-\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\mathbb{E}}_{{z_{2}}}}\left\\{{{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}\exp\left({\overline{\Gamma}_{l,n}}\right)}}\right\\}},$
(61b)

where
${\overline{\Gamma}_{l,n}}\buildrel\Delta\over{=}-\frac{{{{\left|{g_{\rm{2}}^{\rm{*}}{q_{\rm{2}}}\left({{x_{2,l}}-{x_{2,n}}}\right)+{z_{2}}}\right|}^{\rm{2}}}}}{{\sigma_{\rm{2}}^{\rm{2}}}}$.

Suppose that the bandwidth of the WiFi link is $B_{2}$ (Hz). Then, both the
input and output can be represented by complex samples taken by
$\frac{1}{{B_{2}}}$ seconds apart. Note that since the noise is independent in
the I and Q components, each use of the complex channel can be thought of as
two independent uses of a real AWGN channel. If the power of signal is
$P_{2}$, the noise variance and the power constraint per real symbol are
$\frac{\sigma_{2}^{2}}{2}$ and $\frac{{{P_{2}}}}{{2{B_{2}}}}$, respectively.
Thus, we have the achievable rate expression of the WiFi link with the
bandwidth of $B_{1}$ as given in (32b).

Hence, the achievable rates of the aggregated LiFi-WiFi system ${R_{{\rm{LiFi-
WiFi}}}}$ is given in (14).

##  Appendix B Derivation of the formulation Lemma 2

Since $\log_{2}\left({\sum{\exp\left(x\right)}}\right)$ is a convex function
with respect to $x$, we can obtain the upper bound of ${R_{{\rm{LiFi}}}}$ by
Jensen’s inequality.

$\displaystyle{R_{{\rm{LiFi}}}}$ $\displaystyle\leq-\frac{1}{{2\ln
2}}-\sum\limits_{k=1}^{M}{{p_{1,k}}}{\log_{2}}\sum\limits_{m=1}^{M}{{p_{1,m}}\exp\left({{{\mathbb{E}}_{z_{1}}}\left\\{{{{\overline{\Lambda}}_{k,m}}}\right\\}}\right)}$
(62a)
$\displaystyle=-\sum\limits_{k=1}^{M}{{p_{1,k}}}{\log_{2}}\sum\limits_{m=1}^{M}{{p_{1,m}}\exp\left({2B_{1}\hat{\Lambda}_{k,m}-\frac{{{{\mathbb{E}}_{{z_{\rm{1}}}}}\left\\{{z_{1}^{2}}\right\\}}}{{2\sigma_{1}^{2}}}}\right)}$
$\displaystyle~{}~{}~{}-\frac{1}{{2\ln 2}}$ (62b)
$\displaystyle=-\sum\limits_{k=1}^{M}{{p_{1,k}}}{\log_{2}}\sum\limits_{m=1}^{M}{{p_{1,m}}\exp\left(2B_{1}\hat{\Lambda}_{k,m}\right)}.$
(62c)

The upper bound of the LiFi link ${R_{{\rm{LiFi}}}}$ with the bandwidth of
$B_{1}$ is given by

$\displaystyle{R_{{\rm{LiFi}}}}\leq-2B_{1}\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}\exp\left({2B_{1}\hat{\Lambda}_{k,m}}\right)}.$
(63)

Since $\log_{2}\left(x\right)$ is a concave function, we can obtain the lower
bound of ${R_{{\rm{LiFi}}}}$ by Jensen’s inequality as follows

$\displaystyle{R_{{\rm{LiFi}}}}$ $\displaystyle\geq-\frac{1}{{2\ln
2}}-\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}{{\mathbb{E}}_{z_{1}}}\left\\{{\exp\left({{{{\overline{\Lambda}}_{k,m}}}}\right)}\right\\}}$
(64a) $\displaystyle=-\frac{1}{{2\ln
2}}+\frac{1}{{2}}-\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}\exp\left({B_{1}\hat{\Lambda}_{k,m}}\right)}.$
(64b)

The lower bound of the LiFi link ${R_{{\rm{LiFi}}}}$ with the bandwidth of
$B_{1}$ is given by

$\displaystyle{R_{{\rm{LiFi}}}}\geq B_{1}-\frac{B_{1}}{{\ln
2}}-2B_{1}\sum\limits_{k=1}^{{M}}{{p_{1,k}}{{\log}_{2}}\sum\limits_{m=1}^{{M}}{{p_{1,m}}}\exp\left({\hat{\Lambda}_{k,m}}\right)}.$
(65)

##  Appendix C Derivation of the formulation Lemma 3

The upper bound of ${R_{{\rm{WiFi}}}}$ is given as

$\displaystyle{R_{{\rm{WiFi}}}}\leq-\frac{1}{{\ln
2}}-\sum\limits_{l=1}^{N}{{p_{2,l}}}{\log_{2}}\sum\limits_{n=1}^{N}{{p_{2,n}}\exp\left({{{\mathbb{E}}_{{z_{\rm{2}}}}}\left\\{{{\overline{\Gamma}_{l,n}}}\right\\}}\right)}$
(66a)
$\displaystyle=-\sum\limits_{l=1}^{N}{{p_{2,l}}}{\log_{2}}\sum\limits_{n=1}^{N}{p_{2,n}}\exp\left(2{B_{2}}{{\hat{\Gamma}}_{l,n}}-{{\mathbb{E}}_{{z_{\rm{2}}}}}\left\\{{\frac{{{{\left|{{z_{2}}}\right|}^{\rm{2}}}}}{{\sigma_{\rm{2}}^{\rm{2}}}}}\right\\}\right)$
(66b)
$\displaystyle=-\sum\limits_{l=1}^{N}{{p_{2,l}}}{\log_{2}}\sum\limits_{n=1}^{N}{{p_{2,n}}\exp\left({2B_{2}{\widehat{\Gamma}_{l,n}}}\right)}.$
(66c)

Then, the upper bound of the WiFi link ${R_{{\rm{WiFi}}}}$ with the bandwidth
of $B_{2}$ is given by

$\displaystyle{R_{{\rm{WiFi}}}}\leq-
B_{2}\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}}\exp\left({2{\widehat{\Gamma}_{l,n}}}\right)}.$
(67)

Since $\log_{2}\left(x\right)$ is a concave function, the lower bound of
${R_{{\rm{WiFi}}}}$ can be obtained by Jensen’s inequality as well

$\displaystyle{R_{{\rm{WiFi}}}}$ $\displaystyle\geq-\frac{1}{{\ln
2}}-\sum\limits_{l=1}^{N}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{N}{{p_{1,n}}}\frac{1}{{\sigma_{2}^{2}\pi}}\int\limits_{-\infty}^{\infty}{\int\limits_{-\infty}^{\infty}{\exp\left({}\right.}}}$
$\displaystyle~{}~{}\left.{-\frac{{{{\left({{c_{\rm{R}}}+{z_{2,{\rm{R}}}}}\right)}^{2}}+z_{2,{\rm{R}}}^{2}}}{{{\sigma^{2}}}}-\frac{{{{\left({{c_{\rm{I}}}+{z_{2,{\rm{I}}}}}\right)}^{2}}+z_{2,{\rm{I}}}^{2}}}{{{\sigma^{2}}}}}\right)d{z_{2,{\rm{R}}}}d{z_{2,{\rm{I}}}}$
(68a) $\displaystyle=$ $\displaystyle-\frac{1}{{\ln
2}}+1-\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{{2},n}}}\exp\left({B_{2}{\widehat{\Gamma}_{l,n}}}\right)},$
(68b)

where ${z_{2,{\rm{R}}}}\buildrel\Delta\over{=}{{\mathop{\rm
Re}\nolimits}}\left({z_{2}}\right)$,${z_{2,{\rm{I}}}}\buildrel\Delta\over{=}{{\mathop{\rm
Im}\nolimits}}\left({z_{2}}\right)$,
${c_{\rm{R}}}\buildrel\Delta\over{=}{{\mathop{\rm
Re}\nolimits}}\left({{{g}}^{*}_{2}{q}_{2}\left({{x_{2,l}}-{x_{2,n}}}\right)}\right)$
and ${c_{\rm{R}}}\buildrel\Delta\over{=}{{\mathop{\rm
Im}\nolimits}}\left({{{g}}^{*}_{2}{q}_{2}\left({{x_{2,l}}-{x_{2,n}}}\right)}\right)$;
(68a) is true due to
${{\mathbb{E}}_{z_{2}}}\left\\{{f\left({z_{2}}\right)}\right\\}=\int\limits_{z_{2}}{{p_{z_{2}}}}f\left({z_{2}}\right){d_{z_{2}}}$,
and $z_{2}$ follows the complex Gaussian distribution.

Finally, the lower bound of the WiFi link ${R_{{\rm{WiFi}}}}$ with the
bandwidth of $B_{2}$ is given by

$\displaystyle{R_{{\rm{WiFi}}}}\geq B_{2}-\frac{B_{2}}{{\ln
2}}-B_{2}\sum\limits_{l=1}^{{N}}{{p_{2,l}}{{\log}_{2}}\sum\limits_{n=1}^{{N}}{{p_{2,n}}}\exp\left({{\widehat{\Gamma}_{l,n}}}\right)}.$
(69)

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