11institutetext: Department of Mathematics
Faculty of Sciences, University of Cádiz
Campus de Puerto Real, 11510, Puerto Real, Cádiz, Spain.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
A.J. Calderón, F.J. Navarro, J.M. Sánchez

# Modules over linear spaces admitting a multiplicative basis

Antonio J. Calderón Martín The first and the third authors are supported by
the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI
with project numbers FQM298, FQM7156 and by the project of the Spanish
Ministerio de Educación y Ciencia MTM2010-15223. Third author acknowledges the
University of Cadiz for the contract research. Francisco J. Navarro Izquierdo
José M. Sánchez Delgado

###### Abstract

We study the structure of certain modules $V$ over linear spaces $W$ with
restrictions neither on the dimensions nor on the base field $\mathbb{F}$. A
basis $\mathfrak{B}=\\{v_{i}\\}_{i\in I}$ of $V$ is called multiplicative
respect to the basis $\mathfrak{B}^{\prime}=\\{w_{j}\\}_{j\in J}$ of $W$ if
for any $i\in I,j\in J$ we have either $v_{i}w_{j}=0$ or $0\neq
v_{i}w_{j}\in\mathbb{F}v_{k}$ for some $k\in I$. We show that if $V$ admits a
multiplicative basis then it decomposes as the direct sum
$V=\bigoplus_{k}V_{k}$ of well-described submodules admitting each one a
multiplicative basis. Also the minimality of $V$ is characterized in terms of
the multiplicative basis and it is shown that the above direct sum is by means
of the family of its minimal submodules, admitting each one a multiplicative
basis.

Keywords: Multiplicative basis, infinite dimensional linear space, module over
an algebra, representation theory, structure theory.

##  1 Introduction and previous definitions

We begin by noting that throughout this paper linear spaces $V$ and $W$ are
considered of arbitrary dimensions and over an arbitrary base field
$\mathbb{F}$, and also the increasing interest in the study of modules over
different classes of algebras, and so over linear spaces, specially motivated
by their relation with mathematical physics (see [6], [7], [8], [9], [11],
[12], [13]).

###### Definition 1

Let $V$ be a vector space over an arbitrary base field $\mathbb{F}$. It is
said that $V$ is moduled by a linear space $W$ (over same base field
$\mathbb{F}$), or just that $V$ is a module over $W$ if it is endowed with a
bilinear map $V\times W\to V$, $\hskip 5.69046pt(v,w)\mapsto vw$.

Any kind of algebra is an example of a module over itself. Since the even part
$L^{0}$ of the standard embedding of a Lie triple system $T$ is a Lie algebra,
the natural action of $L^{0}$ over $T$ makes of $T$ a (Lie) module over
$L^{0}$. Hence the present paper extend the results in [5].

###### Definition 2

Let $V$ be a module over the linear space $W$. Given a basis
$\mathfrak{B}^{\prime}=\\{w_{j}\\}_{j\in J}$ of $W$ we say that a basis
$\mathfrak{B}=\\{v_{i}\\}_{i\in I}$ of $V$ is multiplicative respect
$\mathfrak{B}^{\prime}$ if for any $i\in I$ and $j\in J$ we have either
$v_{i}w_{j}=0$ or $0\neq v_{i}w_{j}\in\mathbb{F}v_{k}$ for some (unique) $k\in
I$.

To construct examples of modules over linear spaces admitting a multiplicative
basis, we just have to fix two non-empty sets $I$, $J$ and two arbitrary
mappings $\alpha:I\times J\to I$ and $\beta:I\times J\to\mathbb{F}.$ Then the
${\mathbb{F}}$-linear space $V$ with basis $\mathfrak{B}=\\{v_{i}\\}_{i\in I}$
is a module respect to the ${\mathbb{F}}$-linear space $W$ with basis
$\mathfrak{B}^{\prime}=\\{w_{j}\\}_{j\in J}$, under the action induced by
$v_{i}w_{j}:=\beta(i,j)v_{\alpha(i,j)}$, admitting $\mathfrak{B}$ as
multiplicative basis respect to $\mathfrak{B}^{\prime}$.

###### Remark 1

Definition 2 agree with the one for arbitrary algebras given in [5], and it is
a little bit more general than the usual one in the literature ([1, 2, 3, 4,
10]).

##  2 Connections in the set of indexes. Decompositions

From now on and throughout the paper, $V$ denotes a module over the linear
space $W$, both endowed with respective basis $\mathfrak{B}=\\{v_{i}\\}_{i\in
I},\mathfrak{B}^{\prime}=\\{w_{j}\\}_{j\in J}$, and being $\mathfrak{B}$
multiplicative respect to $\mathfrak{B}^{\prime}$. We denote by
$\mathcal{P}(I)$ the power set of $I$.

We begin this section by developing connection techniques among the elements
in the set of indexes $I$ as the main tool in our study. For each $j\in J$, a
new variable $\overline{j}\notin J$ is introduced and we denote by
$\overline{J}:=\\{\overline{j}:j\in J\\}$ the set of all these new symbols. We
will also write $\overline{(\overline{j})}:=j\in J$.

We consider the operation
$\star:I\times(J\;\dot{\cup}\;\overline{J})\to\mathcal{P}(I)$ given by:

  * •

If $i\in I$ and $j\in J$,

$\displaystyle i\star
j:=\left\\{\begin{array}[]{ccl}\emptyset&\text{if}&0=v_{i}w_{j}\\\
\\{k\\}&\text{if}&0\neq v_{i}w_{j}\in\mathbb{F}v_{k}\end{array}\right.$

  * •

If $i\in I$ and $\overline{j}\in\overline{J}$,

$i\star\overline{j}:=\\{k:0\neq v_{k}w_{j}\in\mathbb{F}v_{i}\\}$

Now, we also consider the mapping
$\phi:\mathcal{P}(I)\times(J\;\dot{\cup}\;\overline{J})\to\mathcal{P}(I)$
defined as $\phi(U,j):=\bigcup_{i\in U}(i\star j)$.

###### Lemma 1

Let $a,b\in I$ be. Given $j\in J\;\dot{\cup}\;\overline{J}$ we have that $a\in
b\star j$ if and only if $b\in a\star\overline{j}$.

###### Proof

Let us suppose that $a\in b\star j$. If $j\in J$ then
$v_{b}w_{j}\in{\mathbb{F}}v_{a}$, and if $j\in\overline{I}$ we have
$v_{a}w_{\overline{j}}\in{\mathbb{F}}v_{b}$. In any case $b\in
a\star\overline{j}$. To prove the converse we can argue in a similar way.

###### Lemma 2

Given $j\in J\;\dot{\cup}\;\overline{J}$ and $U\subset\mathcal{P}(I)$ then
$i\in\phi(U,j)$ if and only if $\phi(\\{i\\},\overline{j})\cap
U\neq\emptyset$.

###### Proof

Let us suppose that $i\in\phi(U,j)$. Then there exists $k\in U$ such that
$i\in k\star j$. By Lemma 1 we have $k\in
i\star\overline{j}=\phi(\\{i\\},\overline{j})$. So
$k\in\phi(\\{i\\},\overline{j})\cap U\neq\emptyset.$ By arguing in a similar
way the converse can be proven.

###### Definition 3

Let $i,k\in I$ be with $i\neq k$. We say that $i$ is connected to $k$ if there
exists a subset $\\{j_{1},\dots,j_{n}\\}\subset J\;\dot{\cup}\;\overline{J}$,
such that the following conditions hold:

  1. 1.

$\phi(\\{i\\},j_{1})\neq\emptyset$,
$\phi(\phi(\\{i\\},j_{1}),j_{2})\neq\emptyset,\dots,\phi(\phi(\dots\phi(\\{i\\},j_{2})\dots),j_{n-1})\neq\emptyset$.

  2. 2.

$k\in\phi(\phi(\dots\phi(\\{i\\},j_{1})\dots),j_{n}).$

We say that $\\{j_{1},\dots,j_{n}\\}$ is a connection from $i$ to $k$ and we
accept $i$ is connected to itself.

###### Lemma 3

Let $\\{j_{1},j_{2},\dots,j_{n-1},j_{n}\\}$ be any connection from some $i$ to
some $k$ where $i,k\in I$ with $i\neq k$. Then the set
$\\{\overline{j}_{n},\overline{j}_{n-1},\dots,\overline{j}_{2},\overline{j}_{1}\\}$
is a connection from $k$ to $i$.

###### Proof

Let us prove it by induction on $n$. For $n=1$ we have that
$k\in\phi(\\{i\\},j_{1})$. It means that $k\in i\star j_{1}$ and so, by Lemma
1, $i\in k\star\overline{j}_{1}=\phi(\\{k\\},\overline{j}_{1})$. Hence
$\\{\overline{j}_{1}\\}$ is a connection from $k$ to $i$.

Let us suppose that the assertion holds for any connection with $n\geq 1$,
elements and let us show this assertion also holds for any connection
$\\{j_{1},j_{2},\dots,j_{n},j_{n+1}\\}.$

By denoting the set $U:=\phi(\phi(\dots\phi(\\{i\\},j_{1})\dots),j_{n})$ and
taking into the account the second condition of Definition 3 we have that
$k\in\phi(U,j_{n+1}).$ Then, by Lemma 2, $\phi(\\{k\\},\overline{j}_{n+1})\cap
U\neq\emptyset$ and so we can take $h\in U$ such that

$h\in\phi(\\{k\\},\overline{j}_{n+1}).$ (1)

Since $h\in U$ we have that $\\{j_{1},j_{2},\dots,j_{n-1},j_{n}\\}$ is a
connection from $i$ to $h$. Hence
$\\{\overline{j}_{n},\overline{j}_{n-1},\dots,\overline{j}_{2},\overline{j}_{1}\\}$
connects $h$ with $i$. From here and by Equation (1) we obtain
$i\in\phi(\phi(\dots\phi(\phi(\\{k\\},\overline{j}_{n+1}),\overline{j}_{n})\dots),\overline{j}_{1}).$
So $\\{\overline{j}_{n+1},\dots,\overline{j}_{2},\overline{j}_{1}\\}$ connects
$k$ with $i$.

###### Proposition 1

The relation $\sim$ in $I$, defined by $i\sim k$ if and only if $i$ is
connected to $k$, is an equivalence relation.

###### Proof

The reflexive and symmetric character is given by Definition 3 an Lemma 3.

If we consider the connections $\\{a_{1},\dots,a_{m}\\}$ and
$\\{b_{1},\dots,b_{n}\\}$ from $a$ to $b$ and from $b$ to $c$ respectively,
then is easy to prove that $\\{a_{1},\dots,a_{m},b_{1},b_{2},\dots,b_{n}\\}$
is a connection from $a$ to $c$. So $\sim$ is transitive and consequently an
equivalence relation.

By the above Proposition we can introduce the quotient set
$I/\sim:=\\{[i]:i\in I\\},$ becoming $[i]$ the set of elements in $I$ which
are connected to $i$.

Recall that a submodule $Y$ of a module $V$ (respect to the linear space $W$)
is a linear subspace of $V$ such that $YW\subset Y$. Our next aim is to
associate an (adequate) submodule to each $[i]\in I/\sim$. We define the
linear subspace $V_{[i]}:=\bigoplus_{j\in[i]}\mathbb{F}v_{j}.$

###### Proposition 2

For any $i\in I/\sim$ we have that $V_{[i]}$ is a submodule of $V$.

###### Proof

We need to check $V_{[i]}W\subset V_{[i]}$. Suppose there exist
$i_{1}\in[i],j_{1}\in[j]$ such that $0\neq v_{i_{1}}w_{j_{1}}\in v_{n}$, for
some $n\in I$. Therefore $n\in\phi(\\{i_{1}\\},j_{1})$. Considering the
connection $\\{j_{1}\\}$ we get $i_{1}\sim n$, and by transitivity $n\in[i]$.

###### Definition 4

We say that a submodule $Y\subset V$ admits a multiplicative basis
$\mathfrak{B}_{Y}$ inherited from $\mathfrak{B}$ if
$\mathfrak{B}_{Y}\subset\mathfrak{B}$.

Observe that any submodule $V_{[i]}\subset V$ admits an inherited basis
$\mathfrak{B}_{[i]}:=\\{v_{j}:j\in[i]\\}$. So we can assert

###### Theorem 2.1

Let $V$ be a module admitting a multiplicative basis $\mathfrak{B}$ respect to
a fixed basis of $W$. Then

$V=\bigoplus_{[i]\in I/\sim}V_{[i]},$

being any $V_{[i]}\subset V$ a submodule admitting a multiplicative basis
$\mathfrak{B}_{[i]}$ inherited from $\mathfrak{B}$.

We recall that a module $V$ is simple if its only submodules are $\\{0\\}$ and
$V$.

###### Corollary 1

If $V$ is simple then any couple of elements of $I$ are connected.

###### Proof

The simplicity of $V$ applies to get that $V_{[i]}=V$ for some $[i]\in
I/\sim$. Hence $[i]=I$ and so any couple of elements in $I$ are connected.

##  3 The minimal components

In this section we show that, under mild conditions, the decomposition of $V$
of Theorem 2.1 can be given by means of the family of its minimal submodules.
We begin by introducing a concept of minimality for $V$ that agree with the
one for algebras in [5].

###### Definition 5

A module $V$, (over a linear space $W$), admitting a multiplicative basis
$\mathfrak{B}$ respect to fixed basis of $W$, is said to be minimal if its
unique nonzero submodule admitting a multiplicative basis inherited from
$\mathfrak{B}$ is $V$.

Let us also introduce the concept of $\star$-multiplicativity in the framework
of modules over linear spaces in a similar way to the analogous one for
arbitrary algebras (see [5] for these notions and examples).

###### Definition 6

We say that a module $V$ respect $W$ admits a $\star$-multiplicative basis
$\mathfrak{B}=\\{v_{i}\\}_{i\in I}$ respect to a fixed basis
$\mathfrak{B}^{\prime}=\\{l_{j}\\}_{j\in J}$ of $W$, if it is multiplicative
and given $a,b\in I$ such that $b\in a\star j$ for some $j\in
J\;\dot{\cup}\;\overline{J}$ then $v_{b}\in v_{a}W$.

###### Theorem 3.1

Let $V$ be a module respect $W$ admitting a $\star$-multiplicative basis
$\mathfrak{B}=\\{v_{i}\\}_{i\in I}$ respect to the basis
$\mathfrak{B}^{\prime}=\\{w_{j}\\}_{j\in J}$ of $W$. Then $V$ is minimal if
and only if the set of indexes $I$ has all of its elements connected.

###### Proof

The first implication is similar to Corollary 1. To prove the converse,
consider a nonzero submodule $Y\subset V$ admitting a multiplicative basis
inherited from $\mathfrak{B}$. Then, for a certain $\emptyset\neq I_{Y}\subset
I$, we can write $Y=\displaystyle\bigoplus_{i\in I_{Y}}\mathbb{F}v_{i}$. Fix
some $i_{0}\in I_{Y}$ being then

$0\neq v_{i_{0}}\in Y.$ (2)

Let us show by induction on $n$ that if $\\{j_{1},j_{2},\dots,j_{n}\\}$ is any
connection from $i_{0}$ to some $k\in I$ then for any
$h\in\phi(\phi(\cdots\phi(\\{i_{0}\\},j_{1})\dots),j_{n})$ we have that $0\neq
v_{h}\in Y$.

In case $n=1$, we get $h\in\phi(\\{i_{0}\\},j_{1})$. Hence $h\in i_{0}\star
j_{1}$, then, taking into account that $Y$ is a submodule of $V$, by
$\star$-multiplicativity of $\mathfrak{B}$ and Equation (2) we obtain
$v_{h}\in v_{i_{0}}W\subset Y$.

Suppose now the assertion holds for any connection
$\\{j_{1},j_{2},\dots,j_{n}\\}$ from $i_{0}$ to any $r\in I$ and consider some
arbitrary connection $\\{j_{1},j_{2},\dots,j_{n},j_{n+1}\\}$ from $i_{0}$ to
any $k\in I$. We know that for $x\in U$, where
$U:=\phi(\phi(\cdots\phi(\\{i_{0}\\},j_{1})\cdots),j_{n})$, the element

$0\neq v_{x}\in Y.$ (3)

Taking into account that the fact
$h\in\phi(\phi(\cdots\phi(\\{i_{0}\\},j_{1})\dots),j_{n+1})$ means
$h\in\phi(U,j_{n+1}),$ we have that $h\in x\star j_{n+1}$ for some $x\in U$.
From here, the $\star$-multiplicativity of $\mathfrak{B}$ and Equation (3)
allow us to get $v_{h}\in v_{x}W\subset Y$ as desired.

Since given any $k\in I$ we know that $i_{0}$ is connected to $k$, we can
assert by the above observation that $\mathbb{F}v_{k}\subset Y$. We have shown
$V=\bigoplus_{k\in I}\mathbb{F}v_{k}\subset Y$ and so $Y=V$.

###### Theorem 3.2

Let $V$ be a module, over the linear space $W$, admitting a
$\star$-multiplicative basis $\mathfrak{B}$ respect to a fixed basis of $W$.
Then $V=\bigoplus_{k}V_{k}$ is the direct sum of the family of its minimal
submodules, each one admitting a $\star$-multiplicative basis inherited from
$\mathfrak{B}$.

###### Proof

By Theorem 2.1 we have $V=\displaystyle\bigoplus_{[i]\in I/\sim}V_{[i]}$ is
the direct sum of the submodules $V_{[i]}$. Now for any $V_{[i]}$ we have that
$\mathfrak{B}_{[i]}$ is a $\star$-multiplicative basis where all of the
elements of $[i]$ are connected. Applying Theorem 3.1 to any $V_{[i]}$ we have
that the decomposition $V=\displaystyle\bigoplus_{[i]\in I/\sim}V_{[i]}$
satisfies the assertions of the theorem.

## References

  * [1] Bautista, R., Gabriel, P., Roiter, A.V. and Salmeron, L.: Representation-finite algebras and multiplicative basis. Invent. math. 81 (1985), 217–285.
  * [2] Bovdi, V.: On a filtered multiplicative bases of group algebras. II. Algebr. Represent. Theory 6 (2003), no. 3, 353–368.
  * [3] Bovdi, V., Grishkov, A. and Siciliano, S.: Filtered multiplicative bases of restricted enveloping algebras. Algebr. Represent. Theory 14 (2011), no. 4, 601–608.
  * [4] Bovdi, V., Grishkov, A. and Siciliano, S.: On filtered multiplicative bases of some associative algebras. Algebr. Represent. Theory. DOI 10.1007/s10468-014-9494-7.
  * [5] Calderón, A.J. and Navarro, F.J.: Arbitrary algebras with a multiplicative basis. Linear Algebra and its Applications. In press.
  * [6] Chu, Y.J., Huang, F., Zheng, Z.J.: A commutant of $\beta\gamma$-system associated to the highest weight module $V_{4}$ of $sl(2,\mathbb{C})$ . J. Math. Phys. 51(9), 092301, 32 pp, (2010).
  * [7] Dimitrov, I., Futorny, V., Penkov, I.: A reduction theorem for highest weight modules over toroidal Lie algebras. Comm. Math. Phys. 250(1), 47–63, (2004).
  * [8] Grantcharov, D., Jung, J.H., Kang, S.J., Kim, M.: Highest weight modules over quantum queer superalgebra $U_{q}({{\mathfrak{q}}(n)})$. Comm. Math. Phys. 296(3), 827–860, (2010).
  * [9] Iohara, K.: Unitarizable highest weight modules of the $N=2$ super Virasoro algebras: untwisted sectors. Lett. Math. Phys. 91(3), 289–305, (2010).
  * [10] Kupisch, H. and Waschbusch, J.: On multiplicative basis in quasi-Frobenius algebras. Math. Z. 186, (1984), 401-405.
  * [11] Liu, D., Gao, S., Zhu, L.: Classification of irreducible weight modules over $W$-algebra $W(2,2)$. J. Math. Phys. 49(1), 113503, 6 pp, (2008).
  * [12] Takemura, K.: The decomposition of level-$1$ irreducible highest-weight modules with respect to the level-$0$ actions of the quantum affine algebra. J. Phys. A 31, n. 5., 1467–1485, (1998).
  * [13] Zapletal, A.: Difference equations and highest-weight modules of $U_{q}[{\rm sl}(n)]$. J. Phys. A 31, n. 47, 9593–9600, (1998).


# Optimizing Energy-Harvesting Hybrid VLC/RF Networks with Random Receiver
Orientation

Amir Hossein Fahim Raouf, Chethan Kumar Anjinappa, and Ismail Guvenc A. H. F.
Raouf and I. Guvenc are with the Department of Electrical and Computer
Engineering, North Carolina State University, Raleigh, NC. E-mail:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>C. K. Anjinappa is with the
Ericsson Research, Santa Clara, CA 95054, USA. E-mail:
<EMAIL_ADDRESS>paper is presented in part at IEEE
Globecom Workshops (GC Wkshps), December 2022 [1]. This work is supported in
part by NSF under the grant CNS-1910153.

###### Abstract

In this paper, we consider an indoor hybrid visible light communication (VLC)
and radio frequency (RF) communication scenario with two-hop downlink
transmission. The LED carries both data and energy in the first phase, VLC, to
an energy harvester relay node, which then uses the harvested energy to re-
transmit the decoded information to the RF user in the second phase, RF
communication. The direct current (DC) bias and the assigned time duration for
VLC transmission are taken into account as design parameters. The optimization
problem is formulated to maximize the data rate with the assumption of decode-
and-forward relaying for fixed receiver orientation. The non-convex
optimization is split into two sub-problems and solved cyclically. It
optimizes the data rate by solving two sub-problems: fixing time duration for
VLC link to solve DC bias and fixing DC bias to solve time duration. The
effect of random receiver orientation on the data rate is also studied, and
closed-form expressions for both VLC and RF data rates are derived. The
optimization is solved through an exhaustive search, and the results show that
a higher data rate can be achieved by solving the joint problem of DC bias and
time duration compared to solely optimizing the DC bias.

###### Index Terms:

Hybrid VLC-RF, DC bias, Energy harvesting, Information rate.

##  1 Introduction

Wireless communication services and the emergence of new technologies have
created a large demand for radio frequency spectrum [2]. Visible light
communications (VLC) is introduced as a promising complementary technology to
radio-frequency (RF) based wireless systems, as it can be used to offload
users from RF bands for releasing spectrum resources for other users while
also simultaneously providing illumination [3]. Recently, hybrid VLC-RF
systems have been receiving significant attention in the literature, which can
be designed to take advantage of both technologies, e.g., high-speed data
transmission by VLC links while achieving seamless coverage through RF links
[4]. In particular, a hybrid VLC-RF system is desirable for indoor
applications such as the Internet of things (IoT) or wireless sensor networks
[5, 6]. Given that power constraint is a key performance bottleneck in such
networks, a potential approach is to scavenge energy from the surrounding
environment through energy harvesting (EH), as illustrated in Fig. 1.

Figure 1: The system model for the considered VLC-RF transmission scenario.
The VLC link carries both data and energy to the relay node. The harvested
energy is then used at the relay node to forward the data to the far RF user.

The existing literature on hybrid VLC-RF systems that also use EH is mainly
limited to finding the optimal value of DC bias to either maximize the data
rate or minimize the outage probability [7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19]. To our best knowledge, optimization of VLC and RF resources for
hybrid RF/VLC links for a multi-hop scenario as shown in Fig. 1 has not been
studied in any works in the literature. In this paper, we investigate the
performance of EH for an indoor hybrid VLC-RF scenario as shown in Fig. 1. In
particular, we allocate a portion of each transmission block to VLC and the
rest to RF transmission in an adaptive manner. The LED transmits both data and
energy to a relay node with energy harvesting capability in the first phase as
illustrated in Fig. 2 (i.e., VLC transmission). During the second phase (RF
communication), the relay transmits the decoded information to the distant RF
user using the harvested energy. Also, during this phase, the LED continues to
transmit power (no information) to the relay node, aiming to harvest energy
that can be utilized by the RF relay in the next transmission block. The key
contributions of this work are summarized as follows.

  * •

In a related study [20], a comparable policy was introduced for a single
indoor link that can be based either on VLC or infrared communications (IRC)
with the aim of maximizing the harvested energy; however, no RF links or
relays are considered, nor is the goal to maximize the data rate. In addition,
due to existence of a relay in the considered system model, its relative
distance to the RF user, and its random orientation, we dynamically allocate a
portion of each transmission block to VLC and the rest to RF transmission.

  * •

For this specific scenario, we formulate an optimization problem for
maximizing the data rate at the far user. In particular, different than any
existing work in the literature (see e.g., [7]), we incorporate the assigned
time duration to VLC link as the design parameters, in addition to the DC
bias. We split the joint non-convex optimization problem over these two
parameters into two sub-problems and solve them cyclically. First, we fix the
assigned time duration for VLC transmission and solve the non-convex problem
for DC bias by employing the majorization-minimization (MM) procedure [21] and
[22]. The second step involves fixing the DC bias obtained from the previous
step and solving an optimization problem for the assigned time duration of the
VLC link.

  * •

To the best of our knowledge, this is the first paper that attempts to
investigate the effect of random receiver orientation for the relay on the
achievable data rate for a hybrid VLC-RF network. Unlike the conventional RF
wireless networks, the orientation of devices has a significant impact on VLC
channel gain, especially for mobile users. Determining the exact information
rate is a formidable task and may not offer valuable insights for optimizing
the system’s information rate. As an alternative approach, we formulate the
average information rate for the VLC link and the harvested energy based on
the orientation distribution. To gain a better understanding of the influence
of system and channel parameters, we assume that receiver orientation follows
a uniform distribution. From this assumption, we derive a closed-form
expression for the lower bound on the average information rate of both VLC and
RF. To verify our analysis, we present the results for the VLC and RF
information rate using three methods; i.e., exact integral expressions,
simulations and the derived closed-form expressions. Based on the obtained
closed-form expressions, we find the optimal values of DC bias and time
allocation for the system model under consideration. Due to the complexity of
the problem, an exhaustive search is conducted to solve it.

Figure 2: The transmission block under consideration with consecutive time
periods dedicated for VLC (relay) and RF (access) links. The VLC link is used
both as a backhaul to relay the data and for energy harvesting.

The remainder of this paper is organized as follows. Section 2 presents the
literature review. In Section 3, we describe our system model. The
optimization framework is introduced in Section 4, while the optimization
problem and our approach to solve it are provided in Section 5. In Section 6,
we derive the closed-form expressions for both VLC and RF data rate by
considering random orientation, numerical results are presented in Section 7,
and finally, the conclusion and future works are suggested in Section 8.

##  2 Literature Review

There have been some recent studies on enabling EH for a dual-hop hybrid VLC-
RF communication system where the relay can harness energy from a VLC link
(first hop), for re-transmitting the data to the end user over the RF link
(second hop). For example, an optimal design in the sense of maximizing the
data rate with respect to the direct current (DC) bias by allocating an equal
time portion for VLC and RF transmission is introduced in [7]. In another
work, both energy and spectral efficiency are investigated in [8] by taking
into account the power consumption of the light-emitting diodes (LEDs), which
clearly shows the trade-off between energy and spectral efficiency and the
importance of optimizing the DC bias.

TABLE I: Summary of the most relevant works and their contributions.

Ref. |  | Wireless
---
technology
Scheme |  | Relay
---
power
| EH during
---
VLC trans.
| EH during
---
RF trans.
Performance metric | Optimization parameters |  | Random
---
orientation
[7] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✓ | Data rate | DC bias | ✗
[8] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ |  | Energy efficiency
---
spectral efficiency
No parameter optimization | ✗
[9] | Hybrid VLC-RF | Direct link | Harvested energy | ✓ | ✗ | Secrecy outage probability | No parameter optimization | ✗
[10] | Hybrid VLC-RF | Cooperative | External | ✗ | ✗ |  | Outage probability
---
symbol error rate
No parameter optimization | ✗
[11] | Hybrid VLC-RF | Direct link | Harvested energy | ✓ | ✗ | Outage probability | No parameter optimization | ✗
[12] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ | Outage probability | No parameter optimization | ✗
[13] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ | Outage probability |  | DC bias
---
peak amplitude
✗
[14] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ | Outage probability | No parameter optimization | ✗
[15] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ | Data rate |  | LED transmit power
---
UL/DL transmission time
✗
[16] | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✗ | Outage probability | DC bias | ✗
[17] | Hybrid VLC-RF | Cooperative | External | ✗ | ✗ | Data rate |  | Time duration
---
power allocation
✗
[18] | Hybrid VLC-RF | Collaborative | N/A | ✓ | ✓ | SNR |  | DC bias
---
RF beamformers
✗
[19] | Hybrid VLC-RF | Cooperative | External | ✓ | ✗ | Data rate |  | Access mode selection
---
DC bias
power allocation
✗
[23] | VLC only | Direct link | N/A | ✗ | ✗ | Bit error rate | No parameter optimization | ✓
[24] | VLC only | Direct link | N/A | ✗ | ✗ | Outage probability | No parameter optimization | ✓
[25] | VLC only | Direct link | N/A | ✗ | ✗ | Bit error rate | No parameter optimization | ✓
[26] | VLC only | Direct link | N/A | ✗ | ✗ | SNR | No parameter optimization | ✓
[27] | VLC only | Direct link | N/A | ✗ | ✗ | SNR | No parameter optimization | ✓
This Work | Hybrid VLC-RF | Dual-hop | Harvested energy | ✓ | ✓ | Data rate |  | DC bias
---
time allocation
✓

Using stochastic geometry, in [9] secrecy outage probability and the
statistical characteristics of the received signal-to-noise ratio (SNR) are
derived in the presence of an eavesdropper for a hybrid VLC-RF system. Outage
probability and symbol error rate are studied in [10] under the assumption
that the relay and destination locations are random. They consider both
decode-and-forward (DF) and amplify-and-forward (AF) schemes and derive the
approximated analytical and asymptotic expressions for the outage probability.
In [11], the outage performance of an IoT hybrid RF/VLC system is investigated
where the VLC is considered as the downlink from the LED to the IoT devices,
while RF utilizes a non-orthogonal multiple access (NOMA) scheme for the
uplink. Specifically, they report the approximated analytical expressions for
the outage probability by utilizing a stochastic geometry approach to model
the location and number of terminals in a 3-D room.

In [12], Peng et al. consider a mobile relay to facilitate the communications
between the source and destination. The end-to-end outage probability of the
system is analytically obtained and compared with the simulation results. They
further extend this work and study the minimization problem of end-to-end
outage probability under the constraints of both the average and the peak
powers of the LED source in [13]. In [14], the relay is chosen among multiple
IoT devices which are randomly distributed within the coverage of the source.
Based on the channel state information, an analytical approach is used to
determine the end-to-end outage probability for two different transmission
schemes. In [15], the problem of maximizing the sum throughput of multiple
users in a hybrid VLC-RF communication system were studied where the users
utilize the harvested energy during downlink (DL) for transmitting in uplink
(UL).

In [16], a cooperative hybrid VLC-RF relaying network has been considered and
the outage probability of the VLC and RF users are calculated. Further, they
derive a sub-optimal DC bias that effectively minimizes the outage probability
for the RF user. More recently in [17], Rallis et al. propose a hybrid VLC-RF
network where a VLC access point serves two user equipments, which also
function as RF relays to expand network coverage to a third user beyond the
VLC cell. Inspired by rate-splitting multiple access, the proposed protocol
aims at maximizing the weighted minimum achievable rate in the system. In
[18], a hybrid VLC-RF ultra-small network is introduced where the optical
transmitters deliver both the lightwave information and energy signals whereas
a multiple-antenna RF access point (AP) is employed to transfer wireless power
over RF signals. In [19], Guo et al. consider two types of users; i.e.,
information users and energy-harvesting users. The information users receive
information from the LED AP through time-division-multiple-access manner by
either the single-hop VLC-only mode or the relay-assisted dual-hop VLC-RF mode
where the relay has access to an external power source.

The existing literature on hybrid VLC-RF communication systems is mainly
limited to assuming the receiver is vertically upward and fixed, and the
effect of random receiver orientation on such systems has not yet been
reported. The receiver orientation affects the availability of line-of-sight
(LoS) links in VLC networks. In [23], Eroğlu et al. present the statistical
distribution of the VLC channel gain in the presence of random orientation for
mobile users. In another study [24], Fu et al. derive the average channel
capacity and the outage probability based on the statistical characteristics
of the channel when the VLC receivers have random locations and orientations.
In [28], Rodoplu et al. study the behavior of human users and the LoS
availability in an indoor environment. They further derive the outage
probability and analyze the random orientation effect on inter-symbol
interference. Using the Laplace distribution, Soltani et al. derive the
probability density function of SNR and bit error rate for an indoor scenario
[25]. There have been also some recent efforts on experimental measurements to
model the receiver orientation [26, 27]. In Table I, recent studies discussing
the hybrid VLC-RF communication systems and VLC receiver orientation are
summarized and compared with our current work.

##  3 System Model

Fig. 1 illustrates the hybrid VLC-RF system under consideration. We assume a
relay equipped with a single photo-detector (PD), energy-harvesting circuity,
and a transmit antenna for RF communications. Let $T_{\textrm{tot}}^{(i)}$
denote the $i^{\textrm{th}}$ block transmission time which is measured in
seconds. Also, $\tau_{i}$ (unitless value) is the portion of time that is
allocated to transmit information and energy to the relay node in the
$i^{\textrm{th}}$ time block. Thus, the duration of this phase in seconds is
$T_{\textrm{VLC},{i}}=\tau_{i}T_{\textrm{tot}}^{{(i)}}$. We assume that the
block transmission time is constant. Hereafter, we drop the superscript of
$T_{\textrm{tot}}^{(i)}$ in the sequel to simplify notation. Fig. 2 depicts
the transmission block under consideration. Without loss of generality, we
assume that $T_{\textrm{tot}}=1$ second.

###  3.1 VLC Link

In the first hop, the LED transmits both energy and information to the relay
node through the VLC link. The non-negativity of the transmitted optical
signal can be assured by adding DC bias (i.e., ${I_{\textrm{b},{i}}}$) to the
modulated signal, i.e.,
${{x_{\textrm{t},{i}}}\left(t\right)=P_{\textrm{LED}}\big{(}{{x_{\textrm{s},{i}}}\left(t\right)+{I_{{\textrm{b}},{i}}}}\big{)}}$
where $P_{\textrm{LED}}$ denotes the LED power per unit (in W/A) and
$x_{\textrm{s},{i}}$ is the modulated electrical signal. We assume that the
information-bearing signal is zero-mean, and satisfies the peak-intensity
constraint of the optical channel such that [20]

$A_{{i}}\leq\min\left({{I_{\textrm{b},{i}}}-{I_{\textrm{min}}},{I_{\textrm{max}}}-{I_{\textrm{b},{i}}}}\right),$
(1)

where $A_{{i}}$ denotes the peak amplitude of the input electrical signal
(i.e., $\max(\left|{{x_{\textrm{s},{i}}}(t)}\right|)=A_{{i}}$), and
${I_{\textrm{b},{i}}}\in\left[{{I_{\textrm{min}}},{I_{\textrm{max}}}}\right]$
with ${I_{\textrm{max}}}$ and ${I_{\textrm{min}}}$ being the maximum and
minimum input currents of the DC offset, respectively. Let
${B_{{\textrm{VLC}}}}$ denote the double-sided signal bandwidth.

Then, the information rate associated with the optical link between the AP and
relay node within a block with $T_{\textrm{tot}}=1$ second, is given as [20]

${R_{{\textrm{VLC}},{i}}}=T_{\textrm{VLC},{i}}{B_{{\textrm{VLC}}}}{\log_{2}}\left({1+\frac{e}{{2\pi}}\frac{{{{\left({\eta
P_{\textrm{LED}}A_{{i}}{H_{\textrm{VLC}}}}\right)}^{2}}}}{{\sigma_{{\textrm{VLC}}}^{2}}}}\right),$
(2)

where $\eta$ is the photo-detector responsivity in A/W and
${H_{{\textrm{VLC}}}}$ is the optical DC channel gain. In (2),
$\sigma_{{\textrm{VLC}}}^{2}$ is the power of shot noise at the PD which is
given as
$\sigma_{{\textrm{VLC}}}^{2}={q_{\textrm{e}}}{I_{i}}{B_{{\textrm{VLC}}}}$
where ${q_{\textrm{e}}}$ is the charge of an electron and ${I_{i}}$ is the
induced current due to the ambient light. One should note that the shot noise
is the dominant noise source in the VLC channel and we ignore the thermal
noise in our paper [29].The optical DC channel gain of the VLC link can be
written as

${H_{{\textrm{VLC}}}}=\frac{{\left({m+1}\right){A_{\textrm{p}}}}}{{2\pi\left({h_{\Delta}^{2}+d_{\textrm{r}}^{2}}\right)}}{\cos^{m}}\left({{\phi_{\textrm{r}}}}\right)\cos\left({{\theta_{\textrm{r}}}}\right)\Pi\left({\left|{{\theta_{\textrm{r}}}}\right|,\Phi}\right),$
(3)

where ${h_{\Delta}}$ and ${d_{\textrm{r}}}$ are the vertical and horizontal
distances, respectively, between AP and the relay node, and ${\phi_{r}}$ and
${\theta_{r}}$ are the respective angle of irradiance and incidence,
respectively. The Lambertian order is
$m={{-1}\mathord{\left/{\vphantom{{-1}{{{\log}_{2}}\left({\cos\left(\Theta\right)}\right)}}}\right.\kern-1.2pt}{{{\log}_{2}}\left({\cos\left(\Theta\right)}\right)}}$
where $\Theta$ is the half-power beamwidth of each LED, and ${A_{\textrm{p}}}$
and $\Phi$ are the detection area and half field-of-view (FoV) of the PD,
respectively. The function $\Pi\left({x,y}\right)$ is 1 whenever $x\leq y$,
and is 0 otherwise. We also assume that the relay node distance follows a
Uniform distribution with
$\mathcal{U}\left[{d_{\textrm{r},{\min}},d_{\textrm{r},{\max}}}\right]$, and
that relay PD is looking directly upward.

The harvested energy at this phase can be computed as [20]

${E_{\textrm{1},{i}}}=0.75T_{\textrm{VLC},{i}}{I_{{\textrm{DC}},{i}}}{V_{\textrm{t}}}\ln\left({1+\frac{I_{{\textrm{DC}},{i}}}{{{I_{0}}}}}\right),$
(4)

where ${V_{\textrm{t}}}$ is the thermal voltage, ${I_{0}}$ is the dark
saturation current, and ${I_{{\textrm{DC}},{i}}}$ is the DC part of the output
current given as
$I_{{\textrm{DC}},{i}}=\eta{H_{{\textrm{VLC}}}}P_{\textrm{LED}}{I_{\textrm{b},{i}}}$.
In the time period $T_{\textrm{RF},{i}}=1-T_{\textrm{VLC},{i}}$, the aim is to
maximize the harvested energy while the relay transmits the information to the
far user over the RF link. Thus, during the second phase, the LED eliminates
the alternating current (AC) part and maximizes the DC bias, i.e., $A_{{i}}=0$
and $I_{\textrm{b},{i}}=I_{\textrm{max}}$. Mathematically speaking, the
harvested energy during the second phase can be expressed as

${E_{2,{i}}}=0.75T_{\textrm{RF},{i}}{I_{\textrm{DC,
max}}}{V_{\textrm{t}}}\ln\\!\left(\\!{1+\frac{I_{\textrm{DC,
max}}}{I_{0}}}\\!\right),$ (5)

where $I_{\textrm{DC,
max}}=\eta{H_{{\textrm{VLC}}}}P_{\textrm{LED}}{I_{\textrm{max}}}$.

The total harvested energy at the relay that can be utilized for transmitting
the decoded symbol to the far user through an RF link can be calculated as

$\begin{split}E_{\textrm{h},{i}}=&{E_{1,{i}}}+{E_{2,{i-1}}}\\\
=&z\bigg{(}T_{\textrm{VLC},{i}}I_{\textrm{b},{i}}\ln{\Big{(}1+\frac{\eta
H_{\textrm{VLC}}P_{\textrm{LED}}I_{\textrm{b},{i}}}{I_{0}}\Big{)}}\\\
&+T_{\textrm{RF},{i-1}}I_{\textrm{max}}\ln{\Big{(}1+\frac{\eta
H_{\textrm{VLC}}P_{\textrm{LED}}I_{\textrm{max}}}{I_{0}}\Big{)}}\bigg{)},\end{split}$
(6)

where $z=0.75\eta H_{\textrm{VLC}}P_{\textrm{LED}}V_{\textrm{t}}$,
${E_{2,{i}}}$ represents the harvested energy during the RF transmission in
the previous transmission block. In this paper, we assume that the initial
harvested energy is 0 (i.e., ${E_{2,0}}=0$).

As it can be readily checked from (1), increasing $I_{\textrm{b},i}$ leads to
a decrease in $A_{i}$ and, consequently, it decreases the information rate
associated with the VLC link. On the other hand, decreasing $I_{\textrm{b},i}$
limits the harvested energy that can be obtained during VLC transmission
(i.e., $E_{\textrm{1},{i}}$).

###  3.2 RF Link

In the second hop, the relay re-transmits the information to the far user
through the RF link by utilizing the harvested energy. We assume that the
energy used for data reception at the relay is practically negligible and the
harvested energy is primarily employed for data transmission [7, 16]. The
relaying operation is of DF type. We assumed that the user is away from the AP
horizontally by a distance ${d_{\textrm{u}}}$, which follows a Uniform
distribution with $\mathcal{U}\left[{d_{\textrm{u, min}},d_{\textrm{u,
max}}}\right]$. Let ${B_{{\textrm{RF}}}}$ denote the bandwidth for the RF
system and ${N_{0}}$ denote the noise power which can be defined as
${{N_{0}}={P_{0}}+10{\log_{10}}\left({{B_{{\textrm{RF}}}}}\right)+{N_{\textrm{F}}}}$
where ${P_{0}}$ is the thermal noise power, and ${N_{\textrm{F}}}$ is the
noise figure. Further, assume that the relay re-transmits the electrical
signal with normalized power. The respective information rate is given as

${R_{\textrm{RF},{i}}}=T_{\textrm{RF},{i}}{B_{{\textrm{RF}}}}{\log_{2}}\left({1+\frac{{{{P_{\textrm{h},{i}}}}{{\left|{{h_{{\textrm{RF}}}}}\right|}^{2}}}}{{{G_{{\textrm{RF}}}}{N_{0}}}}}\right),$
(7)

where ${h_{{\textrm{RF}}}}$ denotes the Rayleigh channel coefficients,
${{P_{\textrm{h},{i}}}={E_{\textrm{h},{i}}}/T_{\textrm{RF},{i}}}$ is the
transmit power and ${G_{{\textrm{RF}}}}$ is the path loss model for RF link
and can be expressed as

${G_{{\textrm{RF}}}}={\left({\frac{{4\pi{d_{0}}}}{\lambda}}\right)^{2}}{\left({\frac{d_{\textrm{u}}}{{{d_{0}}}}}\right)^{\beta}},$
(8)

where $\lambda$ is the used RF carrier wavelength, ${d_{0}}=1$ m is the
reference distance, and $\beta$ is the path loss exponent, which generally
takes a value between [1.6, 1.8] [30].

The achievable information rate is limited by the smaller information rate
between the VLC link and the RF link and can be expressed as [19]

$R_{\textrm{VLC-RF},{i}}=\min{(R_{\textrm{VLC},{i}},R_{\textrm{RF},{i}})}.$
(9)

Fig. 3 illustrates the effect of time allocation and DC bias on information
data rate. Unless otherwise stated, the system and channel parameters can be
found in Table II. We assume the relay is located at $d_{\textrm{r}}=0$ m, the
user is at $d_{\textrm{u}}=4$ m, and the RF frequency sets as
$f_{\textrm{c}}=2.4$ GHz. Fig. 3(a) illustrates the information data rate for
the VLC and RF links versus the time allocation of the VLC link. In this
figure, we assume DC bias as $I_{\textrm{b}}\in\\{0.6,0.8\\}$ A. Since we
assume that the block transmission time is constant
($T_{\textrm{VLC},i}+T_{\textrm{RF},i}=1$), as the VLC time portion increases
the RF time portion decreases. As it can be observed from Fig. 3(a),
increasing the time allocation for the VLC link (i.e., $T_{\textrm{VLC}}$)
results in increasing the VLC data rate while it decreases the harvested
energy during the second phase (see (5)) and consequently decreases the RF
data rate. Fig. 3(b) depicts the information rate for VLC and RF link versus
DC bias. Here, we assume an equal time portion for VLC and RF transmission
(i.e., $T_{\textrm{VLC}}=T_{\textrm{RF}}=0.5$) as well as
$T_{\textrm{VLC}}=0.8$ (consequently $T_{\textrm{RF}}=0.2$). Recalling (1), we
can observe that increasing DC bias leads to a reduction in peak amplitude of
the input electrical signal (i.e., $A_{i}$) and subsequently VLC data rate.
However, as DC bias increases the harvested energy in the first phase (see
(4)) increases which eventually results in a higher RF data rate.

(a) Fixed $I_{\textrm{b}}$.

(b) Fixed $T_{\textrm{VLC}}$.

Figure 3: The VLC and RF information rate when $d_{\textrm{r}}=0$ m,
$d_{\textrm{u}}=4$ m, and $f_{\textrm{c}}=2.4$ GHz for (a) fixed DC bias and
(b) equal time allocation. Figure 4: Summary of the optimization problem.
Here, we consider $I_{\textrm{b},i}$, $T_{\textrm{VLC},i}$, and
$T_{\textrm{RF},i}$ as the optimization variables to maximize the system data
rate.

##  4 Optimization Framework

Our aim is to optimize the achievable information rate (i.e., (9)) over
$I_{\textrm{b},i}$ and $T_{\textrm{VLC},i}$. Fig. 4 summarizes the
optimization problem of maximizing the system data rate, with the optimization
variables represented in red. Recalling the information rate in VLC link
(i.e., (2)) and RF link (i.e., (7)), the optimization problem can be written
as

$\displaystyle\max_{I_{\textrm{b},{i}},T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}}}$
$\displaystyle R_{\textrm{VLC-RF},{i}}$ (10) s.t. $\displaystyle
c_{1}:I_{\textrm{min}}\leq I_{\textrm{b},{i}}\leq I_{\textrm{max}},$
$\displaystyle
c_{2}:T_{\textrm{VLC},{i}}\\!+\\!T_{\textrm{RF},{i}}\\!=\\!1,T_{\textrm{VLC},{i}}>0,T_{\textrm{RF},{i}}>0$
$\displaystyle c_{3}:R_{\textrm{th}}\leq R_{\textrm{RF}}^{i}.$

where $R_{\textrm{th}}$ is a predefined threshold value, and constraint
$c_{1}$ is imposed to avoid any clipping and guarantee that the LED operates
in its linear region. Since the relay re-transmits the information and the RF
far user is unable to receive data from the LED, $c_{3}$ is added to satisfy
the minimum required data rate.

The joint-optimization problem in (10) is non-smooth (due to the $\min$
operator) and non-convex (due to the objective function and constraint
$c_{3}$). We reformulate the above optimization problem in the epigraph form
to remove the non-smoothness in the objective function. Referring to [31,
Chapter 4], the epigraph form of (10) can be written as

$\displaystyle\max_{\phi,I_{\textrm{b},{i}},T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}}}\quad\phi$
(11) s.t. $\displaystyle c_{1},c_{2},c_{3},$ $\displaystyle c_{4}:\phi\leq
R_{\textrm{VLC},{i}}$ $\displaystyle c_{5}:\phi\leq R_{\textrm{RF},{i}}.$

The above equivalent optimization problem to (10) solves the non-smoothness,
while it is still non-convex. Let ${\alpha={e\left({\eta
P_{\textrm{LED}}{H_{\textrm{VLC}}}}\right)}^{2}/(2\pi\sigma_{{\textrm{VLC}}}^{2})}$,
${\beta=\eta{H_{{\textrm{VLC}}}}P_{\textrm{LED}}}$, and
${\zeta=|h_{\textrm{RF}}|^{2}/(G_{\textrm{RF}}N_{0})}$. Substituting (2) and
(7) in (11), we have

$\displaystyle\max_{\phi,I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}},T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}}}\
\phi$ (12) s.t. $\displaystyle c_{1}^{\prime},c_{2},$ $\displaystyle
c_{3}:T_{\textrm{RF},{i}}B_{\textrm{RF}}\\!\log_{2}\\!\\!\left(\\!\\!1\\!+\\!\frac{\zeta
E_{\textrm{h},{i}}}{T_{\textrm{RF},{i}}}\\!\right)\\!\\!\geq\\!\\!R_{\textrm{th}},$
$\displaystyle c_{4}:T_{\textrm{VLC},{i}}B_{\textrm{VLC}}\log_{2}(1+\alpha
A_{i}^{2})\\!\\!\geq\\!\phi,$ $\displaystyle
c_{5}:T_{\textrm{RF},{i}}B_{\textrm{RF}}\log_{2}\\!\left(\\!1\\!+\\!\frac{\zeta
E_{\textrm{h},{i}}}{T_{\textrm{RF},{i}}}\right)\\!\geq\\!\phi,$ $\displaystyle
c_{6}:\min\big{(}I_{\textrm{b},{i}}-I_{\textrm{min}},I_{\textrm{max}}-I_{\textrm{b},{i}}\big{)}\\!\geq\\!A_{i},$
$\displaystyle
c_{7}:z\Big{(}T_{\textrm{VLC},{i}}I_{\textrm{b},{i}}\ln{\big{(}1+\frac{\beta
I_{\textrm{b},{i}}}{I_{0}}\big{)}}$
$\displaystyle\quad+\\!T_{\textrm{RF},{i-1}}I_{\textrm{max}}\\!\ln\\!{\big{(}1\\!+\\!\frac{\beta
I_{\textrm{max}}}{I_{0}}\big{)}}\\!\Big{)}\\!\\!\geq\\!E_{\textrm{h},{i}}.$

In the optimization problem of (12), $A_{i}^{2}$ is used in $c_{4}$ and
$c_{6}$ is still non-smooth. Here, we relax $c_{6}$ by using Proposition 1
from [19]. Intuitively, as $I_{\textrm{b,i}}$ increases the harvested energy
increases; however, it has a negative effect on the rate beyond
$(I_{\textrm{min}}+I_{\textrm{max}})/2$. Thus, the optimal value of the term
$I_{\textrm{b,i}}$ would be within $(I_{\textrm{min}}+I_{\textrm{max}})/2$ and
$I_{\textrm{max}}$ (and not the other regime ${0\leq I_{\textrm{b, i}}\leq
I_{\textrm{max}})}$. The above restriction enforces $0\leq A_{i}\leq
I_{\textrm{max}}-I_{\textrm{b},{i}}$ benefiting in getting rid of the non-
smooth $\min$ operator (see (12)) as well. This leads to
${c_{1}^{\prime}:(I_{\textrm{min}}+I_{\textrm{max}})/{2}\leq
I_{\textrm{b},i}\leq I_{\textrm{max}}}$ and $c_{6}^{\prime}:0\leq A_{i}\leq
I_{\textrm{max}}-I_{\textrm{b},{i}}$. The constraints $c_{3},c_{4},c_{5}$, and
$c_{7}$ are jointly non-convex. In this regard, we split the joint
optimization problem into two sub-problems and solve them in a cyclic fashion
which will be elaborated in the next section.

##  5 Solution Approach

In this section, we consider two sub-problems for solving (12). In sub-problem
1, we solve the maximization problem for $\phi$ over
$I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}}$ by fixing the time allocation
$T_{\textrm{VLC},{i}}$. In the second sub-problem, we solve the maximization
problem for $\phi$ over
$T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}},E_{\textrm{h},{i}}$ by using
$I_{\textrm{b},{i}}$ obtained from sub-problem 1.

###  5.1 Sub-problem 1

First, we fix $T_{\textrm{VLC},{i}}$ (and hence
$T_{\textrm{RF},{i}}=1-T_{\textrm{VLC},{i}}$) and solve the maximization
problem for $\phi$ over $I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}}$. Sub-
problem 1 can be written as

$\displaystyle\max_{\phi,I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}}}$
$\displaystyle\phi$ (13) s.t. $\displaystyle
c_{1}^{\prime},c_{3},c_{4},c_{5},c_{7}$ $\displaystyle c_{6}^{\prime}:0\leq
A_{i}\leq I_{\textrm{max}}-I_{\textrm{b},{i}},$

where the constraints $c_{3},c_{4},c_{5}$ are conditionally convex.

Assumption: The typical illumination requirement in an indoor VLC environment
results in a high transmit optical intensity, which can provide a high SNR at
the receiver [32, 33]. In this paper, we assume that SNR for the VLC link is
much greater than 1 (in linear scale); i.e., $\alpha(A_{i})^{2}\gg 1$. In this
condition, we further utilize ${\log(1+x)\approx\log(x)}$ in the constraints
$c_{4}$. Thus, the optimization problem can be written as

$\displaystyle\max_{\phi,I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}}}$
$\displaystyle\phi$ (14) s.t. $\displaystyle
c_{1}^{\prime},c_{3},c_{5},c_{6}^{\prime}$ $\displaystyle
c_{4}^{\prime}:T_{\textrm{VLC},{i}}B_{\textrm{VLC}}\log_{2}(\alpha
A_{i}^{2})\geq\phi,$ $\displaystyle
c_{7}^{\prime}:z\bigg{(}T_{\textrm{VLC},{i}}I_{\textrm{b},{i}}\ln{\Big{(}\frac{\beta
I_{\textrm{b},{i}}}{I_{0}}\Big{)}}$
$\displaystyle\quad+\\!T_{\textrm{RF},{i-1}}I_{\textrm{max}}\ln\\!{\Big{(}\\!\frac{\beta
I_{\textrm{max}}}{I_{0}}\Big{)}}\bigg{)}\\!\geq\\!E_{\textrm{h},{i}}.$

In (14), $c_{4}^{\prime}$ is a convex constraint while $c_{7}^{\prime}$ is
still not convex. We further utilize the first-order Taylor series and MM
approach to relax this constraint [21, 22]. As a result, $c_{7}^{\prime}$ can
be replaced with

$c_{7}^{\star}:\quad
g(I_{\textrm{b},{i}})=g_{0}\big{(}I_{\textrm{b},{i}}(t)\big{)}+\frac{\partial
g\big{(}I_{\textrm{b},{i}}(t)\big{)}}{\partial
I_{\textrm{b},{i}}}\big{(}I_{\textrm{b},{i}}-I_{\textrm{b},{i}}(t)\big{)},$
(15)

where

$\displaystyle g_{0}(I_{\textrm{b},{i}}(t))=$ $\displaystyle
z\bigg{(}T_{\textrm{VLC},{i}}I_{\textrm{b},{i}}(t)\ln{\Big{(}\frac{\beta
I_{\textrm{b},{i}}(t)}{I_{0}}\Big{)}}$ (16)
$\displaystyle+T_{\textrm{RF},{i}-1}I_{\textrm{max}}\ln{\Big{(}\frac{\beta
I_{\textrm{max}}}{I_{0}}\Big{)}}\bigg{)},$

and

$\displaystyle\frac{\partial g(I_{\textrm{b},{i}}(t))}{\partial
I_{\textrm{b},{i}}}$
$\displaystyle=zT_{\textrm{VLC},{i}}\bigg{(}\ln{\Big{(}\frac{\beta
I_{\textrm{b},{i}}(t)}{I_{0}}\Big{)}}+\frac{\beta
I_{\textrm{b},{i}}(t)}{I_{0}+\beta I_{\textrm{b},{i}}(t)}\bigg{)}.$ (17)

In (15), the term $t$ is an index-term and denotes the iteration index for the
MM approach. The MM procedure on (15) operates iteratively. We first solve the
problem for some initial values of $I_{\textrm{b},{i}}(t)$. Then, we update
the value of $I_{\textrm{b},{i}}(t)$ at each iteration until it remains the
same for two consecutive iterations, or the change between two consecutive
iterations is not appreciable.

Overall, the optimization sub-problem 1 is as follows:

$\displaystyle\max_{\phi,I_{\textrm{b},{i}},A_{i},E_{\textrm{h},{i}}}\quad\phi$
(18) s.t. $\displaystyle
c_{1},c_{3},c_{4}^{\prime},c_{5},c_{6}^{\prime},c_{7}^{\star}.$

We iteratively solve the above sub-problem 1 until its convergence. Once the
above sub-problem converges, we continue with sub-problem 2 which is
elaborated in the following.

###  5.2 Sub-problem 2

In here, we fix $I_{\textrm{b},{i}}$ obtained from sub-problem 1 and solve the
problem for maximizing $\phi$ over the variables
$T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}},E_{\textrm{h},{i}}$. The
optimization problem can be expressed as

$\displaystyle\max_{\phi,T_{\textrm{VLC},{i}},T_{\textrm{RF},{i}},E_{\textrm{h},{i}}}\quad\phi$
(19) s.t. $\displaystyle c_{2},c_{3},c_{4}^{\prime},c_{5},c_{7}^{\prime}.$

In (19), the objective function and constraints
$c_{2},c_{4}^{\prime},c_{7}^{\prime}$ are linear, whereas the constraints
$c_{3}$ and $c_{5}$ are convex constraints which result in a convex
optimization problem.

Figure 5: The performance of our optimization framework when $d_{r}=0$ m and
$d_{u}=4$ m.

###  5.3 Convergence

Here, we study the convergence of our proposed optimization algorithm. We
assume that the relay location is at $d_{r}=0$ m while the far user distance
is $d_{u}=4$ m. As it can be observed from Fig. 5, the achievable information
rate obtained for sub-problem 1 is higher than the one obtained for sub-
problem 2.However, after five iterations, rates obtained in the two sub-
problems converge and the difference between the information rate of the sub-
problems becomes negligible.

##  6 Relay Random Orientation

In this section, we investigate the effect of relay random orientation on the
achievable data rate for the considered scenario in Figs. 1, 2, and 4. Random
orientation can significantly influence channel quality. This effect not only
degrades the VLC data rate but also influences the RF data rate since the
relay is empowered by the harvested energy. Here, we assume wide FOV where the
incidence angle $\theta_{\textrm{r}}$ in (3) is always smaller than $\Theta$
which implies $\Pi\left({\left|\theta_{\textrm{r}}\right|,\Phi}\right)=1$;
therefore the LED is always within the FOV. To separate the deterministic and
random parts, we can rearrange (3) as follows:

$H_{{\textrm{VLC}}}=\frac{{\left({m+1}\right){A_{\textrm{p}}}{h_{\Delta}^{m}}}}{{2\pi}}{\left({{h_{\Delta}^{2}}+{d_{\textrm{r}}^{2}}}\right)^{-\frac{{m+2}}{2}}}\cos\left(\theta_{\textrm{r}}\right),$
(20)

where we employ the geometrical relation

$\cos\left({{\phi}_{\textrm{r}}}\right)=\frac{h_{\Delta}}{\sqrt{d_{\textrm{r}}^{2}+{h_{\Delta}^{2}}}}.$

Let $H_{\textrm{VLC}}=h_{\textrm{c}}h_{\theta}$ where

${h_{\textrm{c}}=\frac{{\left({m+1}\right){A_{\textrm{p}}}{h_{\Delta}^{m}}}}{{2\pi}}{\left({{h_{\Delta}^{2}}+{d_{\textrm{r}}^{2}}}\right)^{-\frac{{m+2}}{2}}}}$

is the deterministic part of (20) and $h_{\theta}=\cos(\theta_{\textrm{r}})$.
The distribution of the square channel can be derived by considering the
probability density function (PDF) of
$h_{\theta}^{2}=\cos^{2}(\theta_{\textrm{r}})$ given as [23]

${f_{h_{\theta}^{2}}}\left(x\right)=\frac{{{c_{\theta}}}}{{\sqrt{4x\left({1-x}\right)}}}{f_{\theta}}\left({\frac{1}{2}{{\cos}^{-1}}\left({2x-1}\right)}\right),$
(21)

for $0\leq x\leq 1$, and 0 otherwise. In (21), $c_{\theta}$ is the
normalization constant and $f_{\theta}(\cdot)$ is the PDF of the random angle
$\theta$. As a result, the PDF of the square channel is readily given as [23]

$f_{h^{2}}(x)=\frac{1}{h_{c}^{2}}{f_{h_{\theta}^{2}}}\Big{(}\frac{x}{h_{c}^{2}}\Big{)}.$
(22)

###  6.1 Average VLC data rate

In this condition, the average data rate for the VLC link can be calculated as

$\begin{split}{R_{{\textrm{VLC}},{i}}}^{\textrm{avg}}=&\int
T_{\textrm{VLC},{i}}{B_{{\textrm{VLC}}}}\\\
&\times{\log_{2}}\left({1+\frac{e}{{2\pi}}\frac{{{{\left({\eta
P_{\textrm{LED}}A_{{i}}}\right)}^{2}}}}{{\sigma_{{\textrm{VLC}}}^{2}}}}x\right)f_{h^{2}}(x)\text{{d}}x,\end{split}$
(23)

It is worth mentioning that $f_{\textrm{h}^{2}}$ is 0 for $x<0$ and
$x>h_{\textrm{c}}^{2}$. In this paper, we assume $\theta$ follows $\theta\sim
U[\theta_{1},\theta_{2}]$. As a result, We can rewrite (22) as

$f_{h^{2}}(x)=\frac{{c_{\theta}}}{(\theta_{2}-\theta_{1}){\sqrt{4x\left({1-x}\right)}}}.$
(24)

Inserting (24) into (23), and substituting $t\coloneqq x/h_{\textrm{c}}^{2}$
we have

${R_{{\textrm{VLC}},{i}}}^{\textrm{avg}}=L_{1}\int\frac{{\log_{2}}{\left({1+L_{2}}t\right)}}{{\sqrt{4t\left({1-t}\right)}}}\text{{d}}t,$
(25)

where
$L_{1}=\frac{{c_{\theta}}T_{\textrm{VLC},{i}}{B_{{\textrm{VLC}}}}}{\theta_{2}-\theta_{1}}$
and $L_{2}=\frac{e}{{2\pi}}\frac{{{{\left({\eta
P_{\textrm{LED}}A_{{i}}}{h_{\textrm{c}}}\right)}^{2}}}}{{\sigma_{{\textrm{VLC}}}^{2}}}$.
Utilizing the first two terms of Puiseux series[34] $x=1$ for
${1\mathord{\left/{\vphantom{1{\sqrt{4x(1-x)}}}}\right.\kern-1.2pt}{\sqrt{4x(1-x)}}}$,
a lower bound on (25) can be written as

$\begin{split}{R_{{\textrm{VLC}},{i}}}^{\textrm{avg}}\\!\geq\\!L_{1}\\!\Bigg{(}\\!\\!\int\\!\\!\frac{{\log_{2}}{\left({1\\!+\\!L_{2}}t\right)}}{2{\sqrt{\left({1-t}\right)}}}\text{{d}}t\\!\\!+\\!\\!\int\\!\\!\frac{\sqrt{1-t}{\log_{2}}\left({1\\!+\\!L_{2}}t\right)}{4}\text{{d}}t\\!\\!\Bigg{)}.\end{split}$
(26)

Using 2.727.5 of [35], the first integral term of (26) can be written as

$\begin{split}f_{1}(x,L_{2})&=\int\frac{{\log_{2}}\left({1+L_{2}}t\right)}{2{\sqrt{\left({1-t}\right)}}}\text{{d}}t\\\
&=\frac{-1}{\sqrt{L_{2}}}\bigg{(}\left(\ln(x)-2\right)\sqrt{L_{2}+1-x}\\\
&-2\sqrt{L_{2}+1}\ln{\frac{\sqrt{L_{2}+1-x}-\sqrt{L_{2}+1}}{\sqrt{x}}}\bigg{)}.\end{split}$
(27)

Using an integral solver [36], the second integral in (26) can be written as

$\begin{split}&f_{2}(x,L_{2})=\int\frac{\sqrt{1-t}{\log_{2}}\left({1+L_{2}}t\right)}{4}\text{{d}}t\\\
&=\frac{1}{4}\\!\Bigg{(}\\!\frac{2\sqrt{1\\!-\\!x}(-2L_{2}(x\\!-\\!4)\\!\\!+\\!\\!3L_{2}(x\\!-\\!1)\ln{(L_{2}x\\!+\\!1\\!)}\\!\\!+\\!\\!6)}{9L_{2}}\\\
&-\frac{4}{3}\left(\frac{L_{2}+1}{L_{2}}\right)^{(3/2)}{\tanh}^{-1}{\bigg{(}\sqrt{\frac{L_{2}(1-x)}{L_{2}+1}}\bigg{)}}\Bigg{)}.\end{split}$
(28)

The final result for (26) can then be expressed as

$\begin{split}{R_{{\textrm{VLC}},{i}}}^{\textrm{avg}}&\geq L_{1}\\\
&\times\bigg{(}f_{1}({\frac{\cos{(2\theta_{1})}+1}{2}},L_{2})+f_{2}(\\!{\frac{\cos{(2\theta_{1})}+1}{2}},L_{2}\\!)\\\
&-f_{1}({\frac{\cos{(2\theta_{2})}+1}{2}},L_{2})-f_{2}({\frac{\cos{(2\theta_{2})}+1}{2}},L_{2})\bigg{)}.\end{split}$
(29)

###  6.2 Average Energy Harvesting

Recalling (6) and utilizing $\ln{(1+x)}\approx\ln{(x)}$, the total harvesting
energy can be written as

$\begin{split}\tilde{E}_{\textrm{h},{i}}\approx
M_{1}H_{\textrm{VLC}}\ln{(M_{2}H_{\textrm{VLC}})}+M_{3}H_{\textrm{VLC}}\ln{(M_{4}H_{\textrm{VLC}})}\end{split}$
(30)

where

$M_{1}=0.75T_{\textrm{VLC}}\eta P_{\textrm{LED}}I_{\textrm{b,
i}}V_{\textrm{t}},$ (31) $M_{2}=\eta P_{\textrm{LED}}I_{\textrm{b, i}}/I_{0},$
(32) $M_{3}=0.75T_{\textrm{RF}}\eta
P_{\textrm{LED}}I_{\textrm{max}}V_{\textrm{t}},$ (33)

and

$M_{4}=\eta P_{\textrm{LED}}I_{\textrm{max}}/I_{0}.$ (34)

The PDF of $h_{\theta}$ can be expressed as

$f_{h_{\theta}}(x)=1/(\theta_{2}-\theta_{1})\sqrt{(1-\cos^{-1}(x))}$ (35)

for ${\cos{\theta_{1}}\leq x\leq\cos{\theta_{2}}}$. As a result, the PDF of
VLC channel can be calculated as
$f_{h}(x)=\frac{1}{h_{c}}f_{h_{\theta}}(\frac{x}{h_{c}})$. The average energy
harvesting can be calculated as

$\begin{split}\bar{E}_{\textrm{h},{i}}=&\int_{0}^{\infty}(M_{1}x\ln{(M_{2}x})+M_{3}x\ln{(M_{4}x)})f_{h}(x)\text{{d}}x\end{split}$
(36)

Utilizing $\ln{(ab)}=\ln{(a)}+\ln{(b)}$ and defining $f_{3}(x)$ and $f_{4}(x)$
as

$\begin{split}f_{3}(x)=&\int_{h_{c}\cos{\theta_{2}}}^{h_{c}\cos{\theta_{1}}}\frac{x}{\sqrt{1-\frac{x}{h_{c}}}}\text{{d}}x\\\
=&-h_{c}\sqrt{h_{c}^{2}-x^{2}}\end{split}$ (37)

and

$\begin{split}&f_{4}(x)=\int_{h_{c}\cos{\theta_{2}}}^{h_{c}\cos{\theta_{1}}}\frac{x\ln{x}}{\sqrt{1-\frac{x}{h_{c}}}}\text{{d}}x\\\
=&\\!-\\!h_{c}\Big{(}h_{c}\tanh^{-1}\big{(}\\!\frac{\sqrt{h_{c}^{2}\\!-\\!x^{2}}}{h_{c}}\\!\big{)}\\!\\!+\\!\\!\sqrt{h_{c}^{2}-x^{2}}\big{(}\ln{(x)}\\!-\\!1\big{)}\Big{)}.\end{split}$
(38)

Thus, the final expression for (36) can be written as

$\begin{split}\bar{E}_{\textrm{h},{i}}=&\frac{M_{1}}{h_{c}(\theta_{2}-\theta_{1})}\bigg{(}\ln{(M_{2})}f_{3}(h_{c}\cos{\theta_{1}})+f_{4}(h_{c}\cos{\theta_{1}})\\\
&-\ln{(M_{2})}f_{3}(h_{c}\cos{\theta_{2}})-f_{4}(h_{c}\cos{\theta_{2}})\bigg{)}\\\
+&\frac{M_{3}}{h_{c}(\theta_{2}-\theta_{1})}\bigg{(}\ln{(M_{4})}f_{3}(h_{c}\cos{\theta_{1}})+f_{4}(h_{c}\cos{\theta_{1}})\\\
&-\ln{(M_{4})}f_{3}(h_{c}\cos{\theta_{2}})-f_{4}(h_{c}\cos{\theta_{2}})\bigg{)}.\end{split}$
(39)

Therefore, a lower bound on the average data rate for RF link can be
calculated as

${R_{\textrm{RF},{i}}}^{\textrm{avg}}\geq
T_{\textrm{RF},{i}}{B_{{\textrm{RF}}}}{\log_{2}}\left({1+\frac{\bar{E}_{\textrm{h},i}{{{\left|{{h_{{\textrm{RF}}}}}\right|}^{2}}}}{T_{\textrm{RF},i}{{G_{{\textrm{RF}}}}{N_{0}}}}}\right).$
(40)

TABLE II: System and channel parameters that are used to generate the
numerical results.

Parameter | Numerical Value
---|---
User distance ($d_{\textrm{u},{\textrm{min}}}$,$d_{\textrm{u},{\textrm{max}}}$) | [4,8] m
Relay distance ($d_{\textrm{r},{\textrm{min}}}$,$d_{\textrm{r},{\textrm{max}}}$) | [0,2] m
LED power ($P_{\textrm{LED}}$) | 1.5 W/A
Noise figure ($N_{\textrm{F}}$) | 9 dB [8]
RF signal bandwidth $B_{\textrm{RF}}$ | 10 MHz [19]
VLC signal bandwidth $B_{\textrm{VLC}}$ | 10 MHz [19]
Thermal noise ($P_{0}$) | -174 dBm/Hz [8]
RF frequency ($f_{\textrm{c}}$) | {2.4, 5} GHz [8]
Minimum DC bias ($I_{\textrm{min}}$) | 100 mA [8]
Maximum DC bias ($I_{\textrm{max}}$) | 1 A [8]
Photo-detector responsivity ($\eta$ ) | 0.4 A/W [20]
Thermal voltage ($V_{\textrm{t}}$) | 25 mV [8]
Dark saturation current ($I_{0}$) |  $10^{-10}$ A [8]
Half FoV ($\Phi$) |  $60^{\circ}$ [8]
Half-power beamwidth ($\Theta$) |  $60^{\circ}$ [8]
Electron charge ($q_{\textrm{e}}$) | $1.6\times 10^{-19}$
Induced current ($I_{i}$) |  $5840\times 10^{-6}$ [8]
PD detection area ($A_{\textrm{p}}$) |  $10^{-4}$ $\textrm{m}^{2}$ [8]
AP relative height ($h_{\Delta}$) | 2 m [8]
Data rate threshold ($R_{\textrm{th}}$) |  $10^{6}$ b/s

##  7 Numerical Results

In this section, we demonstrate the performance of hybrid VLC-RF scheme under
consideration as in Fig. 1 using computer simulations. For the convenience of
the reader, unless otherwise stated, the channel and system parameters are
summarized in Table II.

###  7.1 Approximation and Closed-form Expressions

Figure 6: The performance of VLC data rate through the exact expression (i.e.,
(23)), simulation as illustrated with a black plus sign marker (+) and the
closed-form (i.e., (29)) as illustrated with a green circle marker (o) versus
the horizontal distance between the relay and the VLC AP.

(a) $\Phi=60^{\circ}$.

(b) $\Phi=90^{\circ}$.

Figure 7: The performance of RF data rate obtained using computer simulations,
using (36) as illustrated with circle marker and using the closed-form
expression (i.e., (39)) as illustrated with plus sign marker.

In this subsection, we first compare the performance of VLC with random
receiver orientation data rate using the exact expression in (23), simulation,
and the closed-form approximation in (29). We assume the relay location varies
as $d_{\textrm{r}}\in\\{0,2\\}$ and the half-power beamwidth as
$\Phi\in\\{60^{\circ},90^{\circ}\\}$. To verify our expressions, we consider
two case; $\theta\sim U[0^{\circ},10^{\circ}]$ and $\theta\sim
U[10^{\circ},40^{\circ}]$. We assume $A_{i}=0.2$, $T_{\textrm{VLC, i}}=0.8$.
Fig. 6 depicts the effect of random orientation on the average of VLC data
rate. As it can be observed from Fig. 6, the simulation results and the exact
expression are completely matched and there is a practically negligible
mismatch for the closed-form expression. As can be readily checked from (3),
as the half-power beamwidth (i.e., $\Phi$) increases, and consequently, the
Lambertian order decreases, the optical DC channel gain of the VLC decreases.

In addition to the effect of random orientation on the VLC data rate, random
orientation also influences the amount of harvested energy which consequently
affects the data rate in the RF link. In Fig. 7, we investigate the
performance of average RF data rate for two distinct relay locations, i.e.,
$d_{\textrm{r}}\in\\{0,2\\}$ versus the RF user distance by considering the
half-power beamwidth as $\Phi=60^{\circ}$ and $90^{\circ}$. The RF user
location follows ${d_{\textrm{u}}}\sim\mathcal{U}\left[{4,8}\right]$. It can
be readily seen that the utilized approximation is valid for the system under
consideration and the closed-form expression shows a close match with the
simulation results.

###  7.2 Achievable Data Rate without Relay Random Orientation

In this subsection, we analyze the achievable data rate for the system under
consideration while the relay orientation is fixed. We consider four distinct
cases:

Figure 8: The performance of optimal data rate for different user distances
when the relay is located at $d_{\textrm{r}}=0$ m.

(a) Optimal data rate

(b) Optimal DC bias

(c) Optimal duration of VLC transmission

Figure 9: The performance of system under consideration with
$f_{\textrm{c}}=2.4$ GHz when the user node distance follows
${d_{\textrm{u}}}\sim\mathcal{U}\left[{4,8}\right]$.

  * •

Case 1: Joint optimization (JO) of $I_{\textrm{b}}$ and $T_{\textrm{VLC}}$
with utilizing the harvested energy from previous transmission block;

  * •

Case 2: JO of $I_{\textrm{b}}$ and $T_{\textrm{VLC}}$ without utilizing the
harvested energy from previous transmission block (i.e., $E_{2,{i-1}}=0$);

  * •

Case 3: Optimization of $I_{\textrm{b}}$ when
$T_{\textrm{VLC}}=T_{\textrm{RF}}=0.5$, i.e., fixed time allocation (FTA),
with utilizing the harvested energy from previous transmission block (similar
to [7]);

  * •

Case 4: Optimization of $I_{\textrm{b}}$ when
$T_{\textrm{VLC}}=T_{\textrm{RF}}=0.5$ (FTA) without utilizing the harvested
energy from previous transmission block (i.e., $E_{2,{i-1}}=0$). Similar
condition was considered in [13].

(a) Optimal data rate

(b) Optimal DC bias

(c) Optimal duration of VLC transmission

Figure 10: The performance of system under consideration with
$f_{\textrm{c}}=5$ GHz when the user node distance follows
${d_{\textrm{u}}}\sim\mathcal{U}\left[{4,8}\right]$.

Fig. 8 depicts the performance of optimal data rate when the relay is located
at $d_{\textrm{r}}=0$ m and $d_{\textrm{r}}=2$ m. The user distance varies as
$d_{\textrm{u}}\in\\{4,5,6,7,8\\}$ m. We assume RF frequency as
${f_{\textrm{c}}=2.4}$ GHz. It can be observed that Case 1 and Case 3, where
the relay can harvest energy during RF transmission, outperform the other
cases. Note the optimal data rate for all cases decreases as the user distance
increases. Since the relay node is fixed and the achievable information rate
is limited by the smaller information rate between the VLC link and the RF
link, one can conclude that the restriction in the system under consideration
comes from the RF link. In such energy-deprived regimes, harvesting the energy
during the second-hop RF transmission (as proposed in our work) benefits
significantly in increasing the bottle-necked data rate.

Fig. 9 illustrates the performance of system under consideration assuming the
relay location varies as $d_{\textrm{r}}\in\\{0,2\\}$ m and the user node
distance follow Uniform distribution with
${{d_{\textrm{u}}}\sim\mathcal{U}\left[{4,8}\right]}$. Here, the RF frequency
is assumed as ${f_{\textrm{c}}=2.4}$ GHz. As it can be observed from Fig.
9(a), solving the joint optimization over $I_{\textrm{b}}$ and
$T_{\textrm{VLC}}$ leads to a higher data rate compared with the other
considered cases. In addition, Fig. 9(a) clearly shows that harvesting energy
during RF transmission (Case 1 and Case 3) results in an increase in optimal
data rate. To investigate the reason behind this, we further analyze the
optimal DC bias and the time duration assigned for the VLC link in Fig. 9(b)
and Fig. 9(c), respectively. The optimal value of DC bias for the joint
optimization (regardless of harvesting energy during RF transmission) is
higher than the ones obtained when $T_{\textrm{VLC}}=T_{\textrm{RF}}=0.5$ as
shown in Fig. 9(b). Recalling $c_{6}^{\prime}$ in (13), this leads to a
smaller amount of peak amplitude of the input electrical signal which
consequently limits the VLC data rate. In addition, it can be observed that
the optimal value of $T_{\textrm{VLC}}$ for Case 1 is less than 0.5 which
accordingly restrains the data rate of the VLC link (cf. (2)). Although the
difference between the optimal DC bias for Case 1 and Case 2 is practically
negligible, optimal $T_{\textrm{VLC}}$ for Case 2 is much higher than Case 1.
For Case 2, the power of the relay only relies on the harvested energy during
VLC transmission (according to (4)) which can be increased by allocating more
time for this phase. This behavior is because Case 2 ignores harvesting energy
during RF transmission and allocating more time for VLC transmission aims to
compensate for this effect.

To study the effect of RF frequency, Fig. 10 depicts the performance of system
under consideration with $f_{\textrm{c}}=5$ GHz when the user node distance
follows ${d_{\textrm{u}}}\sim\mathcal{U}\left[{4,8}\right]$ and the relay
location varies as $d_{\textrm{r}}\in\\{0,2\\}$ m. Comparing Fig. 10(a) and
Fig. 9(a) shows that the optimal data rate decreases as the RF frequency
increases. According to (8), as RF frequency increases the effect of path loss
becomes more pronounced. To further investigate the behavior of the
optimization problems, the optimal DC bias and the time duration assigned for
the VLC link are plotted in Fig. 10(b) and Fig. 10(c), respectively. For Case
1 and Case 2, the optimal DC bias decreases as the relay distance (i.e.,
$d_{\textrm{r}}$) increases which leads to a decrease in the VLC data rate.
Comparing Fig. 10(c) and Fig. 9(c) indicates that the optimal time duration
for VLC transmission increases as the RF frequency increases.

###  7.3 Achievable Data Rate with Relay Random Orientation

(a) Relay location: $d_{\textrm{r}}=0$ m.

(b) Relay location: $d_{\textrm{r}}=4$ m.

Figure 11: The effect of random orientation on the optimal data rate of the RF
user location follows a Uniform distribution with
$\mathcal{U}\left[4,8\right]$ when the half-power beamwidth is
$\Phi=60^{\circ}$.

(a) Relay location: $d_{\textrm{r}}=0$ m.

(b) Relay location: $d_{\textrm{r}}=4$ m.

Figure 12: The effect of random orientation on the optimal data rate of the RF
user location follows a Uniform distribution with
$\mathcal{U}\left[4,8\right]$ when the half-power beamwidth is
$\Phi=90^{\circ}$.

In this subsection, we investigate the effect of relay random orientation on
the maximum achievable data rate. Similar to the previous subsection, we
consider the same transmission policies. Due to the complexity of the
expressions for the average data rate of VLC and RF, the optimal values of
$T_{\textrm{VLC}}$ and $I_{\textrm{b}}$ are obtained through exhaustive
search. We assume the half-power beamwidth as $\Phi=60^{\circ}$, the RF user
location follows $d_{u}\sim U[4,8]$ and the RF frequency is considered as
${f_{\textrm{c}}=2.4}$ GHz. In Fig. 11(a), we assume $d_{\textrm{r}}=0$ m
whereas the relay distance is set to $d_{\textrm{r}}=4$ m in Fig. 11(b). We
assume that $\theta_{\textrm{r}}$ follows $\theta_{\textrm{r}}\sim
U[0,\theta_{2}]$ where $\theta_{2}$ varies within the range of
$(10^{\circ},50^{\circ})$. As can be seen, the optimal data rate decreases as
the range of random orientation increases. It can be readily checked that
increasing $\theta_{\textrm{r}}$ leads to a decrease in the VLC channel gain
(i.e., (20)). This also affects the harvested energy in the relay node which
is further utilized to empower the RF link. The results also reveal the
benefit of harvesting energy during RF transmission (Case 1 and Case 3) in
increasing the optimal data rate. Our proposed scheme outperforms others since
it is able to adjust the VLC link time duration (i.e., $T_{\textrm{VLC}}$) in
addition to DC bias (i.e., $I_{\textrm{b}}$). This effect becomes more
pronounced when the VLC channel gain decreases (i.e., the relay is farther
from the LED). In this case, the system fails to satisfy the data rate
threshold (i.e., $10^{6}$ b/s) if it ignores the harvesting energy during RF
transmission and/or adjusts the VLC link time duration.

In Fig. 12, we investigate the effect of Lambertian order on the achievable
data rate. Specifically, we consider the half-power beamwidth as
$\Phi=90^{\circ}$. As it can be seen from Fig. 12(a), the maximum achievable
data rate decreases as the half-power beamwidth increases when
$d_{\textrm{r}}=0$ m. As the relay distance increases, the effect of random
orientation becomes more pronounced. Comparing the results in Fig. 11(b) and
Fig. 12(b), it can be observed that the achievable data rate increases as the
half-power beamwidth increases.

##  8 Conclusion

In this paper, we proposed an EH hybrid VLC-RF scheme where the relay harvests
energy both during VLC transmission and RF communication. With DF relaying
assumed, we have formulated the optimization problem to maximize the
achievable data rate. In addition to the DC bias, we considered the assigned
time duration for VLC transmission as a design parameter. Specifically, the
joint optimization problem was divided into two sub-problems and solved
cyclically. Our first sub-problem involved fixing the time duration for VLC
transmission and solving the non-convex DC bias problem using MM approach. In
the second sub-problem, we solved the optimization problem for the assigned
VLC link time duration using the DC bias obtained in the previous step.
Furthermore, we investigated the effect of random receiver orientation for the
relay on the achievable data rate. Assuming a uniform random distribution, we
derived the closed-form expressions for both VLC and RF data rates. Due to the
complexity of the problem, the optimization was solved through an exhaustive
search. The results have shown that our proposed joint optimization approach
achieves a higher data rate compared to solely optimizing the DC bias. We have
also observed that the effect of random orientation becomes more pronounced as
the relay distance increases. Moreover, while the larger half-power beamwidth
leads to a reduction in the achievable data rate when the relay distance is
small, it can be beneficial to maintain the achievable data rate as the relay
distance increases. Potential future research directions include developing
hardware design, considering imperfect channel impulse response, and
investigating practical scenarios. For example, investigating the utilization
of cooperative communication becomes pertinent, particularly when the end user
can opportunistically leverage information from both VLC and RF sources.
Furthermore, taking into account the probability of blockages affecting the
VLC link while the end user maintains access to both VLC and RF communication
channels, such as WiFi, presents a compelling avenue for further exploration.

## References

  * Raouf et al. [2022] A. H. F. Raouf, C. K. Anjinappa, and I. Guvenc, “Optimal design of energy-harvesting hybrid VLC/RF networks,” in _IEEE Globecom Workshops (GC Wkshps)_ , 2022, pp. 705–710.
  * Matheus et al. [2019] L. E. M. Matheus, A. B. Vieira, L. F. Vieira, M. A. Vieira, and O. Gnawali, “Visible light communication: Concepts, applications and challenges,” _IEEE Commun. Surveys Tuts._ , vol. 21, no. 4, pp. 3204–3237, 2019.
  * Burchardt et al. [2014] H. Burchardt, N. Serafimovski, D. Tsonev, S. Videv, and H. Haas, “VLC: Beyond point-to-point communication,” _IEEE Commun. Mag._ , vol. 52, no. 7, pp. 98–105, 2014.
  * Basnayaka and Haas [2015] D. A. Basnayaka and H. Haas, “Hybrid RF and VLC systems: Improving user data rate performance of VLC systems,” in _Proc. IEEE Vehicular Technology Conference (VTC), Glasgow, U.K._, May 2015, pp. 1–5.
  * Delgado-Rajo et al. [2020] F. Delgado-Rajo, A. Melian-Segura, V. Guerra, R. Perez-Jimenez, and D. Sanchez-Rodriguez, “Hybrid RF/VLC network architecture for the Internet of Things,” _Sensors_ , vol. 20, no. 2, p. 478, 2020.
  * Pan et al. [2019a] G. Pan, P. D. Diamantoulakis, Z. Ma, Z. Ding, and G. K. Karagiannidis, “Simultaneous lightwave information and power transfer: Policies, techniques, and future directions,” _IEEE Access_ , vol. 7, pp. 28 250–28 257, 2019.
  * Rakia et al. [2016] T. Rakia, H.-C. Yang, F. Gebali, and M.-S. Alouini, “Optimal design of dual-hop VLC/RF communication system with energy harvesting,” _IEEE Commun. Lett._ , vol. 20, no. 10, pp. 1979–1982, 2016.
  * Yapıcı and Güvenç [2020] Y. Yapıcı and İ. Güvenç, “Energy-vs spectral-efficiency for energy-harvesting hybrid RF/VLC networks,” in _Proc. Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA_, Nov. 2020, pp. 1152–1156.
  * Pan et al. [2017] G. Pan, J. Ye, and Z. Ding, “Secure hybrid VLC-RF systems with light energy harvesting,” _IEEE Trans. Commun._ , vol. 65, no. 10, pp. 4348–4359, 2017\.
  * Zhang et al. [2018] C. Zhang, J. Ye, G. Pan, and Z. Ding, “Cooperative hybrid VLC-RF systems with spatially random terminals,” _IEEE Trans. Commun._ , vol. 66, no. 12, pp. 6396–6408, 2018.
  * Pan et al. [2019b] G. Pan, H. Lei, Z. Ding, and Q. Ni, “3-D hybrid VLC-RF indoor IoT systems with light energy harvesting,” _IEEE Trans. Green Commun. Netw._ , vol. 3, no. 3, pp. 853–865, 2019.
  * Peng et al. [2020] H. Peng, Q. Li, A. Pandharipande, X. Ge, and J. Zhang, “Performance analysis of a SLIPT-based hybrid VLC/RF system,” in _Proc. IEEE/CIC Int. Conf. Commun. China (ICCC), Chongqing, China_, Aug. 2020, pp. 360–365.
  * Peng et al. [2021] ——, “End-to-end performance optimization of a dual-hop hybrid VLC/RF IoT system based on SLIPT,” _IEEE Internet Things J._ , vol. 8, no. 24, pp. 17 356–17 371, 2021.
  * Zhang et al. [2021] Z. Zhang, Q. Li, H. Peng, A. Pandharipande, X. Ge, and J. Zhang, “A SLIPT-based hybrid VLC/RF cooperative communication system with relay selection,” in _Proc. IEEE/CIC Int. Conf. Commun. China (ICCC), Chongqing, China_, 2021, pp. 277–282.
  * Zargari et al. [2021] S. Zargari, M. Kolivand, S. A. Nezamalhosseini, B. Abolhassani, L. R. Chen, and M. H. Kahaei, “Resource allocation of hybrid VLC/RF systems with light energy harvesting,” _IEEE Trans. Green Commun. Netw._ , 2021.
  * Xiao et al. [2021] Y. Xiao, P. D. Diamantoulakis, Z. Fang, L. Hao, Z. Ma, and G. K. Karagiannidis, “Cooperative hybrid VLC/RF systems with SLIPT,” _IEEE Trans. Commun._ , vol. 69, no. 4, pp. 2532–2545, 2021.
  * Rallis et al. [2023] K. G. Rallis, V. K. Papanikolaou, S. A. Tegos, A. A. Dowhuszko, P. D. Diamantoulakis, M.-A. Khalighi, and G. K. Karagiannidis, “RSMA inspired user cooperation in hybrid VLC/RF networks for coverage extension,” in _IEEE Wireless Communications and Networking Conference (WCNC)_ , 2023, pp. 1–6.
  * Tran et al. [2019] H.-V. Tran, G. Kaddoum, P. D. Diamantoulakis, C. Abou-Rjeily, and G. K. Karagiannidis, “Ultra-small cell networks with collaborative RF and lightwave power transfer,” _IEEE Trans. Commun._ , vol. 67, no. 9, pp. 6243–6255, 2019.
  * Guo et al. [2021] Y. Guo, K. Xiong, Y. Lu, D. Wang, P. Fan, and K. B. Letaief, “Achievable information rate in hybrid VLC-RF networks with lighting energy harvesting,” _IEEE Trans. Commun._ , vol. 69, no. 10, pp. 6852–6864, 2021\.
  * Diamantoulakis et al. [2018] P. D. Diamantoulakis, G. K. Karagiannidis, and Z. Ding, “Simultaneous lightwave information and power transfer (SLIPT),” _IEEE Trans. Green Commun. Netw._ , vol. 2, no. 3, pp. 764–773, 2018.
  * Eroglu et al. [2019] Y. S. Eroglu, C. K. Anjinappa, I. Guvenc, and N. Pala, “Slow beam steering and NOMA for indoor multi-user visible light communications,” _IEEE Trans. Mobile Comput._ , vol. 20, no. 4, pp. 1627–1641, 2019.
  * Sun et al. [2016] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” _IEEE Trans. Signal Process._ , vol. 65, no. 3, pp. 794–816, 2016.
  * Eroğlu et al. [2018] Y. S. Eroğlu, Y. Yapıcı, and I. Güvenç, “Impact of random receiver orientation on visible light communications channel,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1313–1325, 2018.
  * Fu et al. [2021] X.-T. Fu, R.-R. Lu, and J.-Y. Wang, “Realistic performance analysis for visible light communication with random receivers,” _JOSA A_ , vol. 38, no. 5, pp. 654–662, 2021.
  * Soltani et al. [2019] M. D. Soltani, A. A. Purwita, I. Tavakkolnia, H. Haas, and M. Safari, “Impact of device orientation on error performance of LiFi systems,” _IEEE Access_ , vol. 7, pp. 41 690–41 701, 2019.
  * Soltani et al. [2018] M. D. Soltani, A. A. Purwita, Z. Zeng, H. Haas, and M. Safari, “Modeling the random orientation of mobile devices: Measurement, analysis and LiFi use case,” _IEEE Trans. Commun._ , vol. 67, no. 3, pp. 2157–2172, 2018.
  * Purwita et al. [2018] A. A. Purwita, M. D. Soltani, M. Safari, and H. Haas, “Impact of terminal orientation on performance in LiFi systems,” in _IEEE Wireless Communications and Networking Conference (WCNC)_ , 2018, pp. 1–6.
  * Rodoplu et al. [2020] V. Rodoplu, K. Hocaoğlu, A. Adar, R. O. Çikmazel, and A. Saylam, “Characterization of line-of-sight link availability in indoor visible light communication networks based on the behavior of human users,” _IEEE Access_ , vol. 8, pp. 39 336–39 348, 2020.
  * Komine and Nakagawa [2003] T. Komine and M. Nakagawa, “Integrated system of white LED visible-light communication and power-line communication,” _IEEE trans. Consum. Electron._ , vol. 49, no. 1, pp. 71–79, 2003.
  * Rappaport et al. [1996] T. S. Rappaport _et al._ , _Wireless communications: Principles and practice_. prentice hall PTR New Jersey, 1996, vol. 2.
  * Boyd and Vandenberghe [2004] S. P. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * Hanzo et al. [2012] L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: From 3G/4G to optical and quantum wireless,” _Proceedings of the IEEE_ , vol. 100, no. Special Centennial Issue, pp. 1853–1888, 2012.
  * Wang et al. [2018] J.-Y. Wang, C. Liu, J.-B. Wang, Y. Wu, M. Lin, and J. Cheng, “Physical-layer security for indoor visible light communications: Secrecy capacity analysis,” _IEEE Trans. Commun._ , vol. 66, no. 12, pp. 6423–6436, 2018\.
  * Winkler [2016] F. Winkler, “Commutative algebra and algebraic geometry,” _Lecture Notes_ , p. 2005, 2016.
  * Gradshteyn and Ryzhik [2014] I. S. Gradshteyn and I. M. Ryzhik, _Table of integrals, series, and products_. Academic press, 2014.
  * [36] WolframAlpha, https://www.wolframalpha.com.