# Design of Energy-Efficient Artificial Noise for Physical Layer Security in
Visible Light Communications

Thanh V. Pham, Anh T. Pham, and Susumu Ishihara Thanh V. Pham and Susumu
Ishihara are with the Department of Mathematical and Systems Engineering,
Shizuoka University, Shizuoka, Japan (e-mail<EMAIL_ADDRESS>susumu.ishihara@shizuoka.ac.jp).Anh T. Pham is with the Department of Computer
Science and Engineering, The University of Aizu, Fukushima, Japan (e-mail:
pham@u-aizu.ac.jp).Part of this paper has been presented at the 2021 IEEE
International Conference on Communications (ICC 2021), Workshop on Green
Solutions for Smart Environment, Montreal, Canada.

###### Abstract

This paper studies the design of energy-efficient artificial noise (AN)
schemes in the context of physical layer security in visible light
communications (VLC). Two different transmission schemes termed selective AN-
aided single-input single-output (SISO) and AN-aided multiple-input single-
output (MISO) are examined and compared in terms of secrecy energy efficiency
(SEE). In the former, the closest LED luminaire to the legitimate user (Bob)
is the information-bearing signal’s transmitter. At the same time, the rest of
the luminaries act as jammers transmitting AN to degrade the channels of
eavesdroppers (Eves). In the latter, the information-bearing signal and AN are
combined and transmitted by all luminaries. When Eves’ CSI is unknown, an
indirect design to improve the SEE is formulated by maximizing Bob’s channel’s
energy efficiency. A low-complexity design based on the zero-forcing criterion
is also proposed. In the case of known Eves’ CSI, we study the design that
maximizes the minimum SEE among those corresponding to all eavesdroppers. At
their respective optimal SEEs, simulation results reveal that when Eves’ CSI
is unknown, the selective AN-aided SISO transmission can archive twice better
SEE as the AN-aided MISO does. In contrast, when Eves’ CSI is known, the AN-
aided MISO outperforms by 30%.

###### Index Terms:

VLC, energy efficiency, physical layer security, jamming, precoding.

##  I Introduction

###  I-A Background

Indoor wireless activities generate roughly 80% of the total wireless Internet
traffic [1]. In the case of indoor scenarios, a standard means of Internet
access is Wireless Fidelity (Wi-Fi), which is currently imposed by
increasingly stringent requirements for high data rates due to the tremendous
growth in the number of mobile devices and data-intensive applications. A
great deal of research effort has been devoted to satisfying this increasing
demand by exploring new communications technologies, among which visible light
communications (VLC) is a promising candidate. Compared to existing wireless
systems, VLC offers the unique advantage of simultaneously providing
illumination and communication, both essential for indoor activities.
Furthermore, the promised high data rate comes at no license-fee spectrum and
little infrastructural installation.

Over the years, numerous theoretical and experimental studies on VLC have
focused on improving its practicality and data-rate performance (see [2, 3, 4]
and references therein) for both indoor and outdoor scenarios. To realize
practical VLC systems, one must also incorporate security and privacy in the
design. Traditional approaches for acquiring secure transmissions over public
communication media are based on well-established key-based cryptographic
algorithms. The security of conventional cryptography relies on the extreme
computational complexity of solving certain mathematical problems (e.g., the
factoring problem). Given the current computing power of the classical
computer, it is likely that breaking the secret keys is not feasible within a
meaningful amount of time. However, quantum computers, which are expected to
be fully functional in the foreseeable future, can threaten the security of
current encryption schemes as they are exponentially faster than their
classical counterparts [5]. The vulnerability of traditional cryptography to
quantum computers has been the primary motivation for researching alternative
security measures not based on computational difficulty. In this context,
physical layer security (PLS) is a promising approach to complement and/or
replace cryptographic techniques as the information confidentiality can be
kept perfectly secure regardless of the computational power of the
eavesdropper[6].

###  I-B Related Works

While the literature on PLS in VLC systems is considerably extensive (see [7,
8] and references therein), the existing studies mainly focused on two
directions. The first focuses on analyzing different lower and upper bounds on
the secrecy capacity of the single-input single-output (SISO) wiretap VLC
channels [9, 10, 11]. Compared with the SISO, the multiple-input single-output
(MISO) configuration is more relevant in practice, as deploying multiple LED
luminaires is a norm in illumination. In the second direction, several
studies, therefore, exploited the spatial degree of freedom at the
transmitting side to investigate the application of precoding and artificial
noise (AN) to improve secrecy performance. Specifically, in the case of
precoding, Mostafa $et\leavevmode\nobreak\ al.$ studied the designs of zero-
forcing (ZF) and general precoders to maximize the achievable secrecy rate for
systems with a single legitimate user and a single eavesdropper considering
both perfect and uncertain estimation of the channel state information (CSI)
[12, 13]. Subsequent works then examined precoding designs for more general
system configurations with different objectives, such as multi-eavesdropper
with minimizing the total transmitted power [14], multi-legitimate users with
maximizing the achievable secrecy sum-rate [15], and multi-user systems, in
which each user treats others as eavesdroppers (hence, transmitted messages
for users must be kept mutually confidential)[16]. ZF precoding strategies to
deal with both active and passive eavesdroppers were also proposed in [17].

In addition to precoding, AN is another approach to enhancing PLS performance
[18, 19, 20]. In essence, AN is a jamming signal which is purposely generated
and transmitted by the transmitter so that it causes minimal or no
interference to the legitimate user while potentially degrading the quality of
the eavesdroppers’ channel. Note that AN should be deployed with precoding to
fully exploit the available spatial degree of freedom. Hence, precoding can be
considered a special case of AN-aided transmission where no power is allocated
to AN generation. Initial research on the topic considered systems where a
fixed LED luminaire transmits the information-bearing signal while the rest
broadcast jamming signals [21, 22]. We term this scheme as _AN-aided SISO
transmission_. Subsequent works then focused on the jamming scheme where the
information-bearing signal and AN are combined and transmitted by all
luminaires. This scheme is referred to as _AN-aided MISO transmission_. Joint
precoding and AN designs were investigated under different performance
criteria. For example, the authors in [23] studied the design that maximizes
the signal-to-interference-plus-noise ratio (SINR) of the legitimate user’s
channel while constraining that of the eavesdroppers’ channel to a predefined
threshold. The study in [24] examined the same objective yet under the
constraint on the average SINR of the eavesdroppers’ channel (assuming that
eavesdroppers’ CSI is unknown at the transmitters). Considering a multi-Bob
VLC system, AN designs to maximize the minimum SINR among Bobs’ channels were
investigated in [25]. Given specific constraints on the SINRs of Bob’s and
Eves’ channels, our study in [26] AN designs to minimize the total transmitted
power considering both perfect and imperfect CSI estimations.

Aside from improving the secrecy rate, there is also a need to optimize energy
efficiency (EE), which measures how much energy is consumed per a transmitted
information bit [27]. This has been of particular interest in recent years due
to the increased effort in reducing energy consumption to combat climate
change [28]. Although several studies have been conducted on EE in VLC and
hybrid VLC/RF systems [29, 30, 31, 32, 33, 34], little attention has been paid
to that in the context of PLS. In fact, to the best of the authors’ knowledge,
our previous studies in [35, 36] are the first works concerning the PLS in VLC
from the EE perspective. Defining secrecy energy efficiency (SEE) as the
consumed energy per a transmitted confidential bit, we designed precoding
schemes to maximize the sum SEE for multi-user VLC systems where messages
among users are kept mutually confidential.

###  I-C Contributions

Against the above background, this paper aims to study energy-efficient AN
designs for PLS in VLC systems. We consider the systems with a legitimate user
(i.e., Bob) and multiple eavesdroppers (i.e., Eves) whose CSI can be known or
unknown on the transmitting side. The AN-aided MISO and a refined AN-aided
SISO transmission scheme, which we term as _selective AN-aided SISO_ , are
investigated and compared in terms of the achievable SEE. Contrary to the
original AN-aided SISO, where the information-bearing signal is transmitted by
a fixed LED luminaire, in the proposed scheme, that is done by the luminaire,
which has the highest channel gain to Bob. The rest of the luminaires act as
jammers transmitting AN. By doing so, Bob can enjoy the strongest information-
bearing signal while suffering less from the AN. Hence, given the same amount
of consumed energy, the proposed scheme should achieve a better achievable
secrecy rate, resulting in a higher SEE compared with the original
counterpart. In the conference version of this paper [37], the design of the
selective AN-aided SISO scheme considering the unavailability of Eves’ CSI was
studied. Extending from this, the contributions of this paper are summarized
as follows.

  * •

In addition to the presented design in [37], which was solved using the
Dinkelbach algorithm and the convex-concave procedure (CCP), a simple and low-
complexity ZF AN-aided SISO design is investigated. Simulation results show
that this design approach achieves comperable SEE that of the design in [37]
with significantly reduced computational time.

  * •

We study an AN design when Eves’ CSI is available at the transmitting side.
The presence of multiple Eves results in multiple sub-wiretap channels, each
characterized by a different achievable secrecy rate. To ensure that the
transmitted information to Bob is secure against all Eves, the system’s
achievable secrecy rate is defined as the minimum among all sub-wiretap
channels. Then, an AN design is cast as a max-min SEE problem, which requires
a different approach to solve. Specifically, we transform the original problem
into a feasibility problem, which can then be solved using the bisection
method.

  * •

Comprehensive simulations are conducted to compare the complexity and
performance of the original AN-aided SISO, the proposed selective AN-aided
SISO, and the AN-aided MISO transmissions under various parameter settings.
Our main finding is that the selective AN-aided SISO scheme achieves a better
SEE than the AN-aided MISO does in the case of unknown Eves’ CSI, while it is
the opposite in the case of known Eves’ CSI as the AN-aided MISO scheme is
superior.

###  I-D Organization

The rest of the paper is organized as follows. The system models, including
the channel model, signal model, and energy consumption, are described in
Section II. Section III presents different AN-aided designs from the
perspective of SEE maximization. Numerical results with related discussions
are given in Section IV. Finally, Section V concludes the paper.

_Notation_ : The following notations are used in the paper. $\mathbb{R}$ and
$\mathbb{R}_{\geq 0}$ denote the set of real and nonnegative real numbers,
respectively. Uppercase and lowercase bold letters (e.g., $\mathbf{A}$ and
$\mathbf{a}$) represent matrices and column vectors, respectively.
$\mathbf{A}^{T}$ is the transpose of $\mathbf{A}$. In addition, $|\cdot|$,
$\lVert\cdot\rVert_{2}$, $\lVert\cdot\rVert_{\infty}$, and $\text{tr}(\cdot)$
are the absolute value, Euclidean norm, maximum norm, and trace operators,
respectively. Finally, $\mathbb{E}[x]$ is the expected value of $x$,
$[\mathbf{a}]_{n}$ denotes the $n$-th element of $\mathbf{a}$ and
$\mathbf{1}_{N}$ is the all-one column vector of size $N$.

##  II System Models

We consider a typical room as depicted in Figs. 1(a) and 1(b), where $N_{T}$
LED luminaries are deployed for both illumination and communications. These
LEDs are connected via wired connections to a central processing unit (CPU),
which is responsible for signal processing and coordination among LED
transmitters. There are one legitimate user (Bob) and multiple eavesdroppers
(Eves), who, assuming that, are non-colluding111In practice, Eves can
cooperate to maximize their eavesdropped information. This, however, may
complicate the analysis of the AN design. Hence, we leave this scenario for
future research.. It is assumed Bob feeds its estimated CSI back to the CPU
via a wireless uplink (e.g., infrared or Wi-Fi). By comparing the entries of
the received CSI vector, the CPU can determine the closest luminaire to Bob.
In the selective AN-aided SISO transmission scheme illustrated in Fig. 1(a),
the closest luminaire to Bob acts as Alice, who transmits the information-
bearing signal. The rest of the luminaries act as jammers transmitting AN. In
the AN-aided MISO scheme described in Fig. 1(b), all luminaries act as Alice
and jammer to transmit the combined information-bearing signal and AN. We also
consider that Eves’s CSI can be either known or unknown on the transmitting
side. Accordingly, different design approaches are presented considering the
availability of Eves’ CSI.

(a) Selective AN-aided SISO transmission.

(b) AN-aided MISO transmission.

Figure 1: AN-aided transmission schemes.

###  II-A Channel Model

In indoor environments, the transmitted signal propagates to a user via a
direct light-of-sight (LoS) and multiple non-light-of-sight paths (NLoS) (due
to reflections of walls and ceiling). Nevertheless, it was experimentally
verified that the total received optical power at the receiver is
predominantly contributed by the LoS signal. On the other hand, the NLoS
channel gain, which accounts for less than 5% of the total received optical
power, Therefore, only the LoS propagation path is considered in this work to
simplify the analysis.

The LoS channel coefficient between an LED luminaire and a user denoted as $h$
is given by [38]

$\displaystyle h=$ $\displaystyle\frac{A_{r}(m+1)}{2\pi
l^{2}}T_{s}(\psi)g(\psi)\cos^{m}(\phi)\cos(\psi)\times\mathds{1}_{[0,\Psi]}(\psi),$
(1)

where $\mathds{1}_{[x,y]}(\cdot)$ denotes the indicator function, $A_{r}$ is
the active area of the photodiode (PD), $l$ is the link length, $\phi$ is the
angle of irradiance, $\Psi$ is the optical field of view (FOV) of the PD, and
$\psi$ is the angle of incidence. Also, $m=-\frac{\ln(2)}{\ln(\Theta_{0.5})}$
is the order of the Lambertian emission where $\Theta_{0.5}$ is the LED’s
semi-angle for half illuminance. $T_{s}(\psi)$ is the gain of the optical
filter, and
$g(\psi)=\frac{\kappa^{2}}{\sin^{2}(\Psi)}\mathds{1}_{[0,\Psi]}(\psi)$ is the
gain of the optical concentrator, with $\kappa$ being the refractive index of
the optical concentrator.

###  II-B Signal Model

In this section, only the signal model of the selective AN-aided transmission
scheme is presented for the sake of conciseness. Without otherwise noted,
analysis in the case of the AN-aided MISO transmission scheme follows the same
manner. In the following, signal models for the case of unknown and known
Eves’s CSI are respectively described.

####  II-B1 Unknown Eves’ CSI

Let $d\in\mathbb{R}$ be a $M$-ary pulse amplitude modulation (M-PAM)-modulated
information signal intended for Bob. Also, let $\mathbf{z}\in\mathbb{R}$ be a
random signal representing the AN. Without loss of generality, it is assumed
that $d$ and $z$ are both zero-mean and normalized and uniformly distributed
over $[-1,\leavevmode\nobreak\ 1]$, which is notated as
$d,z\sim\mathcal{U}[-1,1]$. In our proposed scheme, $d$ is sent by Alice while
$\mathbf{z}$ is transmitted by the jammers. For Alice, the information-bearing
signal $d$ is first scaled by a weighting factor (i.e., precoder) $v$ and then
added by a DC-bias $I_{\text{DC}}$, which specifies the desired illumination
level. The resulting LED’s modulating signal is then represented by

$\displaystyle x_{d}=vd+I_{\text{DC}}.$ (2)

It is known that there exists a linear range (denoted as, for example,
$[I_{\text{min}},\leavevmode\nobreak\ I_{\text{max}}]$) for each particular
LED in which the output optical signal is linearly proportional to the
amplitude of the modulating signal $x_{d}$. Therefore, to ensure efficient use
of the LEDs, $x_{d}$ should be constrained within the linear range as

$\displaystyle I_{\text{min}}\leq x_{d}\leq I_{\text{max}}.$ (3)

As $d\sim\mathcal{U}[-1,\leavevmode\nobreak\ 1]$, the following constraint on
$v$ is needed to satisfy (3)

$\displaystyle|v|\leq\Delta_{\text{DC}},$ (4)

where
$\Delta_{\text{DC}}=\text{min}\left(I_{\text{DC}}-I_{\text{min}},I_{\text{max}}-I_{\text{DC}}\right)$.
When information regarding Eves (e.g., the number of Eves and their positions)
is unknown at the transmitters, the use of a single precoder
$\mathbf{w}\in\mathbb{R}^{N_{T}-1}$ for the AN is appropriate to simplify the
design. Similar to (2), it follows that

$\displaystyle\lVert\mathbf{w}\rVert_{\infty}\leq\Delta_{\text{DC}}.$ (5)

Let $h_{\text{R}}\in\mathbb{R}$ and
$\overline{\mathbf{h}}_{\text{R}}\in\mathbb{R}^{N_{T}-1}$ be the channel gain
and channel gain vector from Alice and jammers to Bob, respectively222The
subscript ‘R’ is used to denote Bob and Eves as receivers in general. When
necessary, the subscript ‘B’ is used to refer to Bob while ‘E’ is used to
refer to Eve.. If we denote $p_{d}=\eta\left(vd+I_{\text{DC}}\right)$ and
$\mathbf{p}_{z}=\eta\left(\mathbf{w}z+I_{\text{DC}}\right)$ with $\eta$ being
the LED conversion factor as the emitted optical power at Alice and jammers,
respectively, the received current signals at Bob and Eves can be written as

$\displaystyle
y_{\text{R}}=\gamma\mathbf{h}^{T}_{\text{R}}\mathbf{p}_{\text{t}}+n_{\text{R}}=\gamma\eta\left(h_{\text{R}}vd+\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}z+\mathbf{h}^{T}_{\text{R}}\textbf{I}_{\text{DC}}\right)+n_{\text{R}},$
(6)

where $\gamma$ is the photo-diode responsivity,
$\mathbf{h}_{\text{R}}=\begin{bmatrix}h_{\text{R}}&\overline{\mathbf{h}}^{T}_{\text{R}}\end{bmatrix}^{T}$,
$\mathbf{p}_{\text{t}}=\begin{bmatrix}p_{\text{d}}&\mathbf{p}_{z}^{T}\end{bmatrix}^{T}$,
$\mathbf{I}_{\text{DC}}=I_{\text{DC}}\mathbf{1}_{N_{T}}$ is the DC-bias
vector, and $n_{\text{R}}$ is the receiver noise. It is reasonable to model
$n_{\text{R}}$ as a zero-mean additive white Gaussian noise with the variance
being given by

$\displaystyle\sigma^{2}_{\text{R}}=2\gamma
e\overline{p}^{r}_{\text{R}}B_{\text{mod}}+4\pi
eA_{\text{d}}\gamma\chi_{\text{amb}}(1-\cos(\Psi))B+i^{2}_{\text{amp}}B_{\text{mod}},$
(7)

where $e$ is the elementary charge,
$\overline{p}^{r}_{\text{R}}=\mathbb{E}\left[p^{r}_{\text{R}}\right]=\eta\mathbf{h}^{T}_{\text{R}}\textbf{I}_{\text{DC}}$
is the average received optical power, $B_{\text{mod}}$ is the modulation
bandwidth of the LED, $\chi_{\text{amb}}$ is the ambient light photo-current,
and $i_{\text{amp}}$ is the pre-amplifier noise current density.

At the receiver, the DC current
$\mathbf{h}^{T}_{\text{R}}\mathbf{I}_{\text{DC}}$ is filtered out, leaving the
AC term for signal demodulation

$\displaystyle\overline{y}_{\text{R}}=\gamma\eta\left(h_{\text{R}}vd+\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}z\right)+n_{\text{R}}.$
(8)

The signal-to-interference-plus-noise ratios (SINRs) of Bob’s and Eves’
received signals are then calculated as

$\displaystyle\text{SINR}_{\text{R}}=\frac{\left(\gamma\eta\mathbb{E}[d^{2}]h_{\text{R}}v\right)^{2}}{\left(\gamma\eta\mathbb{E}[z^{2}]\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}\right)^{2}+{\sigma}^{2}_{\text{R}}}=\frac{\left(h_{\text{R}}v\right)^{2}}{\left(\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}\right)^{2}+\overline{\sigma}^{2}_{\text{R}}},$
(9)

where
$\overline{\sigma}^{2}_{\text{R}}=\frac{\sigma^{2}_{\text{R}}}{\frac{1}{3}\left(\gamma\eta\right)^{2}}$
(since $d,z\sim\mathcal{U}[-1,1]$, then
$\mathbb{E}[d^{2}]=\mathbb{E}[z^{2}]=\frac{1}{3}$).

####  II-B2 Known Eves’ CSI

In certain scenarios, the information intended for Bob must be kept
confidential; thus, any other active users should be treated as Eves. As a
result, Eves’ CSI can be assumed to be known on the transmitting side. To
better exploit the available spatial information, a separate AN signal should
be generated for each Eve. Thus, denote $z_{k}\sim\mathcal{U}[-1,1]$ as the AN
symbol intended for the $k$-th Eve and $\mathbf{w}_{k}\in\mathbb{R}^{N_{T}-1}$
as the corresponding precoder. The LED’s modulating current vector at the
jammers is then written as

$\displaystyle\mathbf{x}_{z}=\sum_{k=1}^{K}\mathbf{w}_{k}z_{k}+I_{\text{DC}},$
(10)

where $K$ is the number of Eves. Bounding $\mathbf{x}_{z}$ between
$\left[I_{\text{min}},\leavevmode\nobreak\ I_{\text{max}}\right]$ results in
the following constraint on $\mathbf{w}_{k}$’s

$\displaystyle\sum_{k=1}^{K}\left|[\mathbf{w}_{k}]_{n}\right|\leq\Delta_{\text{DC}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall n=1,2,\ldots,N_{T}-1.$ (11)

The received AC current signals at Bob and Eves are then given by

$\displaystyle
y_{\text{R}}=\gamma\mathbf{h}^{T}_{\text{R}}\mathbf{p}_{t}+n_{\text{R}}=\gamma\eta\left(h_{\text{R}}vd+\overline{\mathbf{h}}^{T}_{\text{R}}\sum_{k=1}^{K}\mathbf{w}_{k}z_{k}\right)+n_{\text{R}},$
(12)

where in this case
$\mathbf{p}_{\text{t}}=\begin{bmatrix}p_{d}&\mathbf{p}^{T}_{z}\end{bmatrix}^{T}$
with
$\mathbf{p}_{z}=\eta\left(\sum_{k=1}^{K}\mathbf{w}_{k}z_{k}+I_{\text{DC}}\right)$.

####  II-B3 Signal Model for the AN-aided MISO

In the case of AN-aided MISO, since each luminaire acts as Alice and jammer
simultaneously, notations of the channel and the precoder differ slightly from
those of the selective AN-aided SISO. These differences are summarized in
Table I.

Table I: Notational differences between Selective AN-aided SISO and AN-aided
MISO.

| Selective AN-aided SISO | AN-aided MISO
---|---|---
Alice-Receiver’s channel | $h_{\text{R}}\in\mathbb{R}_{\geq 0}$ | $\mathbf{h}_{\text{R}}\in\mathbb{R}_{\geq 0}^{N_{T}}$
Jammer-Receiver’s channel | $\overline{\textbf{h}}_{\text{R}}\in\mathbb{R}_{\geq 0}^{N_{T}-1}$
Precoder of information symbol | $v\in\mathbb{R}$ | $\mathbf{v}\in\mathbb{R}^{N_{T}}$
Precoder of AN | $\mathbf{w}\in\mathbb{R}^{N_{T}-1}$ | $\mathbf{w}\in\mathbb{R}^{N_{T}}$

With this, the LED’s drive current vector in the case of unknown Eves’ CSI and
known Eves’ CSI are given by
$\mathbf{x}=\mathbf{v}d+\mathbf{w}z+I_{\text{DC}}$ and
$\mathbf{x}=\mathbf{v}d+\sum_{k=1}^{K}\mathbf{w}_{k}z_{k}+I_{\text{DC}}$,
respectively. Then, constraints on $\mathbf{v}$, $\mathbf{w}$, and the
received signals at the receiver follow the same analyses presented for the
selective AN-aided SISO scheme.

###  II-C Energy Consumption and Energy Efficiency

####  II-C1 Unknown Eves’ CSI

Since Alice and jammers have no knowledge about Eves, we consider a worst-case
design based on an achievable rate of Bob’s channel. It is observed that the
channel in (8) is subject to a Gaussian channel with amplitude-constrained
input and interference. For such a channel, a lower bound on its channel
capacity is given by [39, Eq. (4)],

$\displaystyle
C_{\text{R},l}(v,\mathbf{w})=\frac{1}{2}\log_{2}\left(\frac{2\left(\left(h_{\text{R}}v\right)^{2}+\left(\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}\right)^{2}\right)+\pi
e\overline{\sigma}^{2}_{\text{R}}}{\pi
e\left(\frac{1}{3}{\left(\overline{\mathbf{h}}^{T}_{\text{R}}\mathbf{w}\right)^{2}}+\overline{\sigma}^{2}_{\text{R}}\right)}\right).$
(13)

One can also derive an upper bound on the channel capacity based on [39, Eq.
(7)]. However, the resulting design problem is not particularly different. For
the sake of conciseness, we thus consider designs with respect to the lower
bound capacity (or lower bound secrecy capacity as in the case of known Eves’
CSI).

In the considered system, energy is consumed for illumination, generation of
the information-bearing signal and AN, and circuitry operations. In general,
while the powers for illumination ($P_{\text{LEDs}}$) and circuitry operations
($P_{\text{circuit}}$) can be assumed to be fixed (i.e., the dimming level is
unchanged), the power level by the information-bearing signal
($P_{\text{data}}$) and AN ($P_{\text{AN}}$) needs to be optimized with
respect to the achievable secrecy rate. The total power can then be written as

$\displaystyle P_{\text{total}}$
$\displaystyle=P_{\text{circuit}}+P_{\text{LEDs}}+P_{\text{data}}+P_{\text{AN}}=P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\lVert\mathbf{w}\rVert^{2}\right),$
(14)

where $U_{\text{LEDs}}$ is the LEDs’ forward voltage and
$\zeta=\frac{1}{3}R_{\text{AC}}$ with $R_{\text{AC}}$ being the resistance of
the AC circuit. Given the achievable rate in (13), the EE of the system with
respect to Bob’s channel is expressed by

$\displaystyle\Phi_{\text{B}}(v,\mathbf{w})=\frac{C_{\text{B},l}(v,\mathbf{w})}{P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\lVert\mathbf{w}\rVert^{2}\right)}.$
(15)

####  II-C2 Known Eves’ CSI

In this scenario, due to the availability of Eves’ CSI, achievable secrecy
rates can be derived to calculate the SEE. In particular, a lower bound on the
secrecy capacity of the wiretap channel comprising Bob and the $k$th Eve is
written by

$\displaystyle
C^{k}_{s}\left(v,\mathbf{W}\right)=C_{\text{B},l}\left(v,\mathbf{W}\right)-C^{k}_{\text{E},u}\left(v,\mathbf{W}\right),$
(16)

where
$\mathbf{W}=\begin{bmatrix}\mathbf{w}_{1}&\mathbf{w}_{2}&\cdots&\mathbf{w}_{K}\end{bmatrix}$
and $C^{k}_{\text{E},u}\left(v,\mathbf{W}\right)$ is an upper bound on the
capacity of the $k$-th Eve’s channel, which can be derived from [39, Eq. (7)].
Using (13) gives the following expression for $C^{k}_{s}(v,\mathbf{W})$

$\displaystyle C^{k}_{\text{s}}(v,\mathbf{W})$
$\displaystyle=\frac{1}{2}\log_{2}\left(\frac{2\left(\left(h_{\text{B}}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2}\right)+\pi
e\overline{\sigma}^{2}_{\text{B}}}{\pi
e\left(\frac{1}{3}\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2}+\overline{\sigma}^{2}_{\text{B}}\right)}\right)$
$\displaystyle-\frac{1}{2}\log_{2}\left(\frac{\pi
e\left(\frac{1}{3}\left(\left(h_{\text{E},k}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}\right)+\overline{\sigma}^{2}_{\text{E},k}\right)}{2\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}+\pi
e\overline{\sigma}^{2}_{\text{E},k}}\right),$ (17)

with $h_{\text{E},k}$ and $\overline{\mathbf{h}}_{\text{E},k}$ being channel
coefficients between Alice, jammer and the $k$-th Eve, respectively. Moreover,
the constraint in (11) can rewritten in terms of $\mathbf{W}$ as

$\displaystyle\left\lVert\mathbf{W}\right\rVert_{\infty}\leq\Delta_{\text{DC}}.$
(18)

Using the same assumptions in the case of unknown Eves’ CSI, the total power
consumption is given by

$\displaystyle P_{\text{total}}$
$\displaystyle=P_{\text{circuit}}+P_{\text{LEDs}}+P_{\text{data}}+P_{\text{AN}}=P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\text{tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right).$
(19)

Note that to ensure the confidentiality of the transmitted signal against all
eavesdroppers, the information rate must be no higher than the lowest
achievable secrecy rate among $K$ values of $C^{k}_{s}(v,\mathbf{W})$.
Therefore, we are interested in an AN design that maximizes the SEE
corresponding to the minimum achievable secrecy rate, which is defined by

$\displaystyle\widetilde{\Phi}^{\text{min}}_{s}(v,\mathbf{W})=\frac{\underset{k}{\text{min}}\hskip
5.69054ptC^{k}_{s}(v,\mathbf{W})}{P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\text{tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right)}.$
(20)

##  III Secrecy Energy Efficiency Maximization

The availability of Eves’s CSI at Alice and the jammers is critical to our
design approaches. In the following, we, hence, separately consider two
scenarios: unknown and known Eves’ CSI. The designs are again presented for
the selective SISO transmission scheme, mentioning the MISO transmission
scheme when needed.

###  III-A Unknown Eves’ CSI

In most practical cases, Eves are passive eavesdroppers who do not feed their
CSI back to the transmitters, leading to the unavailability of Eves’ CSI on
the transmitting side. Without the knowledge of Eves’ CSI, direct optimization
of the SEE may not be possible. Instead, an indirect design approach is to
maximize the EE of Bob’s channel. The obtained solution $(v,\mathbf{w})$ is
then used to calculate the SEE. Due to the proportionality between the
capacity of Bob’s channel and the secrecy capacity, this design approach
possibly results in good SEE performance as well.

With constraints on the precoders $v$ and $\mathbf{w}$ given in (4) and (5), a
naive way to formulate an EE maximization problem for Bob’s channel is as
follows

$\displaystyle\bf{\mathcal{P}1}\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{w}}{\text{maximize}}$
$\displaystyle\Phi_{\text{B}}(v,\mathbf{w})$ (21a) subject to
$\displaystyle|v|\leq\Delta_{\text{DC}},$ (21b)
$\displaystyle\lVert\mathbf{w}\rVert_{\infty}\leq\Delta_{\text{DC}}.$ (21c)

######  Proposition 1.

The optimal AN solution to $\mathcal{P}\mathbf{{1}}$ is
$\mathbf{w}^{*}=\mathbf{0}$.

###### Proof.

An intuitive proof is that using AN does not improve the achievable rate of
Bob’s channel while increasing energy consumption. Thus, an AN-aided
transmission reduces the EE, which shows that the no-AN transmission (i.e.,
$\mathbf{w}=\mathbf{0}$) is optimal. A rigorous proof is given in Appendix A.
∎

Proposition 1 shows that in order to achieve the maximal
$\Phi_{\text{B}}(v,\mathbf{w})$, jammers should be inactive. However, this may
not lead to an improved SEE because without being interfered with by the AN,
Eves can enjoy high-quality communication channels (thus, lowering the
achievable secrecy rate). To enable the use of AN, one can set a constraint to
ensure that a certain amount of power is allocated to generating the AN
signal, for example, $\lVert\mathbf{w}\rVert^{2}_{2}\geq P_{\text{th}}$ where
$P_{\text{th}}>0$ is the minimal allocated power threshold. The choice of
$P_{\text{th}}$ is critical as due to (21c), this power allocation renders the
design problem infeasible if $P_{\text{th}}>(N_{T}-1)\Delta_{\text{DC}}^{2}$.
Moreover, although increasing $P_{\text{th}}$ decreases Eves’ SINRs, it
potentially degrades the quality of Bob’s channel as well and increases the
total consumption power. Consequently, the overall SEE might be reduced. To
prevent too much power from being allocated to the AN and guarantee that the
quality of Bob’s channel is above a desirable limit, a constraint on Bob’s
SINR is taken into consideration. Specifically, we study a modification for
$\bf{\mathcal{P}1}$ as given below

$\displaystyle\bf{\mathcal{P}2}\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{w}}{\text{maximize}}$
$\displaystyle\Phi(v,\mathbf{w})$ (22a) subject to
$\displaystyle\frac{\left(h_{\text{B}}v\right)^{2}}{\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}+\overline{\sigma}^{2}_{\text{B}}}\geq\delta_{\text{B}},$
(22b) $\displaystyle\lVert\mathbf{w}\rVert_{2}^{2}\geq P_{\text{th}},$ (22c)
$\displaystyle\eqref{P1:constraint1},\leavevmode\nobreak\
\eqref{P1:constraint2}.$

Here, $\delta_{\text{B}}$ is the minimum SINR required for Bob’s channel.

One should emphasize that the introduction of constraints in (22b) and (22c)
may render $\bf{\mathcal{P}2}$ infeasible. Indeed, it can be easily verified
that the problem is infeasible when
$\delta_{\text{B}}>\frac{\left(h_{\text{B}}\Delta_{\text{DC}}\right)^{2}}{\overline{\sigma}^{2}_{\text{B}}}$.
Therefore, we assume that $\delta_{\text{B}}$ and $P_{\text{th}}$ are
explicitly chosen so that $\bf{\mathcal{P}2}$ is feasible. It is seen that
$\bf{\mathcal{P}2}$ is a fractional non-convex optimization problem. Hence, we
employ the Dinkelbach algorithm, which is efficient in solving fractional
programming to tackle the problem. The principle of the Dinkelbach algorithm
is to get rid of the fractional objective function by introducing an
equivalent non-fractional parametric objective function, which can be handled
more conveniently. In particular to our problem, let
$D(v,\mathbf{w})=P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\lVert\mathbf{w}\rVert_{2}^{2}\right)$
and $\lambda\geq 0$ be the value of the objective function. The algorithm then
iteratively solves the following parametric problem

$\displaystyle\bf{\mathcal{P}3}(\lambda)\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{w}}{\text{maximize}}$ $\displaystyle
C_{\text{B}}(v,\mathbf{w})-\lambda D(v,\mathbf{w})$ (23a) subject to
$\displaystyle\eqref{P2:constraint1},\leavevmode\nobreak\
\eqref{P2:constraint2},\leavevmode\nobreak\
\eqref{P1:constraint1},\leavevmode\nobreak\ \eqref{P1:constraint2}.$

In each iteration, say the $i$th iteration, a new
$\lambda^{(i)}=\frac{C_{\text{B}}\left(v^{{(i)}^{*}},\mathbf{w}^{{(i)}^{*}}\right)}{D\left(v^{{(i)}^{*}},\mathbf{w}^{{(i)}^{*}}\right)}$
is updated, where $v^{{(i)^{*}}}$ and $\mathbf{w}^{{(i)^{*}}}$ are the
iteration’s optimal solutions to $v$ and $\mathbf{w}$, respectively. The
algorithm terminates when
$C_{\text{B}}\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)-\lambda^{(i)}D\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)$
converges to a predefined sufficiently low value. The algorithm is summarized
as follows.

1 Choose the maximum number of iterations $L_{\text{Din}}$ and the error
tolerance $\epsilon_{1}>0$.

2 Set $i\leftarrow 1$, $\lambda^{(0)}\leftarrow 0$.

3 while  _convergence = False and $i\leq L_{\text{Din}}$_ do

4 Given $\lambda^{(i-1)}$ from the previous iteration, solve
${\bf{\mathcal{P}3}}\left(\lambda^{(i-1)}\right)$ for $v^{{(i)^{*}}}$ and
$\mathbf{w}^{{(i)^{*}}}$.

5 Update
$\lambda^{(i)}\leftarrow\frac{C_{\text{B}}\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)}{D\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)}$.

6 if
_$C_{\text{B}}\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)-\lambda^{(i)}D\left(v^{{(i)^{*}}},\mathbf{w}^{{(i)^{*}}}\right)\leq\epsilon_{1}$_
then

7 convergence $\leftarrow$ True

8 $v^{*}\leftarrow v^{{(i)^{*}}}$,
$\mathbf{w}^{*}\leftarrow\mathbf{w}^{{(i)^{*}}}$

9

10 else

11 convergence $\leftarrow\textbf{False}$

12 end if

13 $i\leftarrow i+1$

14 end while

Algorithm 1 Dinkelbach-type algorithm

####  III-A1 General AN-aided SISO

In each iteration of Algorithm 1, it is required to solve
$\bf{\mathcal{P}3}(\lambda)$, which is not a convex optimization problem due
to the non-concavity of the objective function and non-convexity of (22b) and
(22c). Notice that (22b) can be convexified via an observation that there
exists an optimal solution $v^{*}$ satisfying that $h_{\text{B}}v^{*}\geq 0$.
Indeed, it is obvious that if there is any optimal solution $v^{\dagger}$ to
$\bf{\mathcal{P}3}(\lambda)$ that $h_{\text{B}}v^{\dagger}\leq 0$, then
$v^{*}=-v^{\dagger}$ is also feasible and offers the same objective value.
Hence, $v^{*}$ is also optimal. As a result, (22b) is equivalently rewritten
as

$\displaystyle\frac{1}{\sqrt{\delta_{\text{B}}}}h_{\text{B}}v\geq\sqrt{\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}+\overline{\sigma}^{2}_{\text{B}}},$
(24)

which is convex. The non-concavity of the objective function is due to the
fact that $C_{\text{B}}(v,\mathbf{w})$ is not concave. To overcome this
problem, one can make use of the following variable transformations

$\displaystyle
c_{\text{B},1}=\log_{2}\left(2\left(\left(h_{\text{B}}v\right)^{2}+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}\right)+\pi
e\overline{\sigma}^{2}_{\text{B}}\right),$ (25) $\displaystyle
c_{\text{B},2}=\log_{2}\left(\pi
e\left(\frac{1}{3}{\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}}+\overline{\sigma}^{2}_{\text{B}}\right)\right),$
(26) $\displaystyle
p_{\text{B},1}=\left(h_{\text{B}}v\right)^{2}+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2},$
(27) $\displaystyle
p_{\text{B},2}=\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}.$
(28)

Accordingly, we reformulate $\bf{\mathcal{P}3}(\lambda)$ as

$\displaystyle{\bf{\mathcal{P}4}}(\lambda)\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{w},c_{\text{B},1},c_{\text{B},2},p_{\text{B},1},p_{\text{B},2}}{\text{maximize}}$
$\displaystyle\frac{1}{2}\left(c_{\text{B},1}-c_{\text{B},2}\right)-\lambda
D(v,\mathbf{w})$ (29a) subject to $\displaystyle
c_{\text{B},1}\leq\log_{2}\left(2p_{\text{B},1}+\pi
e\overline{\sigma}^{2}_{\text{B}}\right),$ (29b) $\displaystyle
p_{\text{B},1}\leq\left(h_{\text{B}}v\right)^{2}+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2},$
(29c) $\displaystyle c_{\text{B},2}\geq\log_{2}\left(\pi
e\left(\frac{1}{3}{p_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\right),$
(29d) $\displaystyle
p_{\text{B},2}\geq\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2},$
(29e)
$\displaystyle\frac{1}{\sqrt{\delta_{\text{B}}}}h_{\text{B}}v\geq\sqrt{\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}+\overline{\sigma}^{2}_{\text{B}}},$
(29f)
$\displaystyle\eqref{P2:constraint2},\eqref{P1:constraint1},\eqref{P1:constraint2}.$

Note that without lost of optimality, the equalities in (25)-(28) are replaced
by their corresponding inequalities in (29b)-(29e) of
$\mathbf{\mathcal{P}4}(\lambda)$. Indeed, since the objective in (29a) is
monotically increasing with $c_{\text{B},1}$, at the opitmal solution,
$c_{\text{B},1}$ must attain its maximum value, which implies that (29b) and
(29c) hold at equality. The same argument can be used to validate (29d) and
(29e). Now, all constraints except (29c), (29d), and (22c) are convex. Observe
that one side in each of these inequality constraints is either constant or
linear. This enables the use of Taylor expansion to approximately linearize
the constraints. Then, the CCP is used to successively solve the problem.
Specifically, at the $j$th iteration of the CCP, the following first-order
Taylor approximations are used

$\displaystyle\left(h_{\text{B}}v^{(j)}\right)^{2}+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}\geq$
$\displaystyle\left(h_{\text{B}}v^{(j-1)}\right)^{2}+2h^{2}_{\text{B}}v^{(j-1)}\left(v^{(j)}-v^{(j-1)}\right)+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}^{(j-1)}\right)^{2}$
$\displaystyle+2\left[\mathbf{w}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{B}}\overline{\mathbf{h}}^{T}_{\text{B}}\left(\mathbf{w}^{(j)}-\mathbf{w}^{(j-1)}\right),$
(30) $\displaystyle\log_{2}\left(\pi
e\left(\frac{1}{3}{p_{\text{B,2}}}+\overline{\sigma}^{2}_{\text{B}}\right)\right)\leq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}{p^{(j-1)}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)+\frac{p^{(j)}_{\text{B},2}-p^{(j-1)}_{\text{B},2}}{\ln(2)\left(\\!\pi
e\left(\frac{1}{3}{p^{(j-1)}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)},$
(31)
$\displaystyle\left\lVert\mathbf{w}^{(j)}\right\rVert_{2}^{2}\geq\left\lVert\mathbf{w}^{(j-1)}\right\rVert_{2}^{2}+2\left[\mathbf{w}^{(j-1)}\right]^{T}\left(\mathbf{w}^{(j)}-\mathbf{w}^{(j-1)}\right),$
(32)

where $v^{(j-1)}$, $\mathbf{w}^{(j-1)}$, and $p^{(j-1)}_{\text{B, 2}}$ are the
solutions to $v$, $\mathbf{w}$, and $p_{\text{B, 2}}$ in the $(j-1)$th
iteration, respectively. Replacing (29c), (29d), and (22c) by their
corresponding constraints using (30), (31), and (32) results in

$\displaystyle{\bf{\mathcal{P}5}}^{(j)}(\lambda)\hskip 5.69054pt$
$\displaystyle\underset{\begin{subarray}{c}v^{(j)},\mathbf{w}^{(j)},c_{\text{B},1},\\\
c_{\text{B},2},p_{\text{B},1},p^{(j)}_{\text{B},2}\end{subarray}}{\text{maximize}}$
$\displaystyle\frac{1}{2}\left(c_{\text{B},1}-c_{\text{B},2}\right)-\lambda
D\left(v^{(j)},\mathbf{w}^{(j)}\right)$ (33a) subject to
$\displaystyle\left(h_{\text{B}}v^{(j-1)}\right)^{2}+2h_{\text{B}}\left(v^{(j)}-v^{(j-1)}\right)$
$\displaystyle+\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}^{(j-1)}\right)+2\left[\mathbf{w}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{B}}\overline{\mathbf{h}}^{T}_{\text{B}}\left(\mathbf{w}^{(j)}-\mathbf{w}^{(j-1)}\right)\geq
p_{\text{B, 1}},$ (33b) $\displaystyle c_{\text{B, 2}}\geq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}{p^{(j-1)}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)+\frac{p^{(j)}_{\text{B},2}-p^{(j-1)}_{\text{B},2}}{\ln(2)\left(\\!\pi
e\left(\frac{1}{3}{p^{(j-1)}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)},$
(33c)
$\displaystyle\left\lVert\mathbf{w}^{(j-1)}\right\rVert_{2}^{2}+2\left[\mathbf{w}^{(j-1)}\right]^{T}\\!\\!\left(\mathbf{w}^{(j)}-\mathbf{w}^{(j-1)}\right)\geq
P_{\text{th}},$ (33d)
$\displaystyle\eqref{P4:constraint1},\leavevmode\nobreak\
\eqref{P4:constraint4},\leavevmode\nobreak\
\eqref{P4:constraint5},\leavevmode\nobreak\
\eqref{P1:constraint1},\leavevmode\nobreak\ \eqref{P1:constraint2},$

which is a convex optimization problem and thus can be solved efficiently
using available software packages, e.g., CVX [40]. Then a sub-optimal solution
to $\bf{\mathcal{P}4}(\lambda)$ can be found via solving a sequence of
${\bf{\mathcal{P}5}}^{(j)}$ until a convergence criterion is satisfied. Note
that due to the tighter constraints used in (33b)-(33c), it is guaranteed that
a solution to ${\bf{\mathcal{P}5}}^{(j)}(\lambda)$ is feasible to
${\bf{\mathcal{P}5}}^{(j+1)}(\lambda)$. In summary, an CCP algorithm to solve
${\bf{\mathcal{P}4}}(\lambda)$ is described in Algorithm 2.

1Choose the maximum number of iterations $L_{\text{CCP}}$ and the error
tolerance $\epsilon_{2}$.

2 Choose the starting points $v^{(0)}$, $\mathbf{w}^{(0)}$, and
$p^{(0)}_{\text{B, 2}}$ satisfying that ${\bf{\mathcal{P}5}}^{(0)}(\lambda)$
is feasible.

3 Set $j\leftarrow 1$.

4

5while  _convergence = True and $j<L_{\text{CCP}}$_ do

6 Given $v^{(j-1)}$, $\mathbf{w}^{(j-1)}$, and $p^{(j-1)}_{\text{B, 2}}$
obtained from the previous iteration, solve
${\bf{\mathcal{P}5}}^{(j)}(\lambda)$.

7 if  _$\frac{\left|v^{(j)}-v^{(j-1)}\right|}{v^{(j)}}\leq\epsilon_{2}$ and
$\frac{\left\lVert\mathbf{w}^{(i)}-\mathbf{w}^{(i-1)}\right\rVert_{2}}{\left\lVert\mathbf{w}^{(i)}\right\rVert_{2}}\leq\epsilon_{2}$
and $\frac{\left|p^{(j)}_{\text{B, 2}}-p^{(j-1)}_{\text{B,
2}}\right|}{p^{(j)}_{\text{B},2}}\leq\epsilon_{2}$ _ then

8 convergence $\leftarrow$ True.

9 $v^{*}\leftarrow v^{(j)}$, $\mathbf{w}^{*}\leftarrow\mathbf{w}^{(j)}$,
$p^{*}_{\text{B},2}\leftarrow p^{(j)}_{\text{B},2}$.

10

11 else

12 convergence $\leftarrow$ False.

13 end if

14 $j\leftarrow j+1$.

15

16 end while

Algorithm 2 CCP-type algorithm for solving ${\bf{\mathcal{P}4}}(\lambda)$

####  III-A2 ZF AN-aided SISO

The presented general AN design requires two iterative procedures, which might
incur a high computational complexity. In order to simplify the design, we
present in this section a ZF design approach where the AN is constructed to
lie on the null-space of $\overline{\mathbf{h}}_{\text{B}}^{T}$. This ensures
that the generated AN does cause any interference on Bob’s channel. This,
however, also leads to a smaller search space of $\mathbf{w}$, possibly
resulting in a reduced SEE performance. To this end, let
$\widetilde{\mathbf{w}}$ be an orthogonal basic for the null-space of
$\overline{\mathbf{h}}_{\text{B}}^{T}$. The AN precoder is then given by
$\mathbf{w}=\sqrt{\phi}\widetilde{\mathbf{w}}$, where $\phi>0$ is a scaling
factor, which controls the AN power. Under this null-space strategy, the EE of
Bob’s channel is simplified to

$\displaystyle\Phi_{\text{B}}(v,\phi)=\frac{\log_{2}\left(1+\frac{2\left(h_{\text{B}}v\right)^{2}}{\pi
e\overline{\sigma}^{2}_{\text{B}}}\right)}{2\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\phi\right)\right)},$
(34)

and the design problem becomes

$\displaystyle\bf{\mathcal{P}6}\hskip 14.22636pt$
$\displaystyle\underset{v,\phi}{\text{maximize}}$
$\displaystyle\Phi_{\text{B}}(v,\phi)$ (35a) subject to
$\displaystyle\frac{\left(h_{\text{B}}v\right)^{2}}{\overline{\sigma}^{2}_{\text{B}}}\geq\delta_{\text{B}},$
(35b) $\displaystyle\phi\geq P_{\text{th}},$ (35c)
$\displaystyle|v|\leq\Delta_{\text{DC}},$ (35d)
$\displaystyle\sqrt{\phi}\leq\frac{\Delta_{\text{DC}}}{\lVert\widetilde{\mathbf{w}}\rVert_{\infty}}.$
(35e)

Note that (35c) holds because
$\left\lVert\widetilde{\mathbf{w}}\right\rVert_{2}=1$. Firstly, we can see
that $\bf{\mathcal{P}6}$ is feasible if and only if the two conditions
$\frac{\Delta_{\text{DC}}^{2}}{\left\lVert\widetilde{\mathbf{w}}\right\rVert^{2}_{\infty}}\geq{P_{\text{th}}}$
and
$\Delta_{\text{DC}}^{2}\geq\frac{\delta_{\text{B}}\overline{\sigma}^{2}_{\text{B}}}{h^{2}_{\text{B}}}$
are simultaneously satisfied. Assume that these conditions hold, because (34)
is monotonically decreasing with $\phi$, the optimal solution to $\phi$ is
obviously $\phi^{*}=P_{\text{th}}$. With this and by making change of variable
$V=v^{2}$, $\bf{\mathcal{P}6}$ reduces to

$\displaystyle\bf{\mathcal{P}7}(\phi^{*})\hskip 14.22636pt$
$\displaystyle\underset{V\geq 0}{\text{maximize}}$
$\displaystyle\Phi_{\text{B}}(V,\phi^{*})$ (36a) subject to
$\displaystyle\frac{h^{2}_{\text{B}}V}{\overline{\sigma}^{2}_{\text{B}}}\geq\delta_{\text{B}},$
(36b) $\displaystyle V\leq\Delta_{\text{DC}}^{2},$ (36c)

where
$\Phi_{\text{B}}(V,\phi^{*})=\frac{\log_{2}\left(1+\frac{2h^{2}_{\text{B}}V}{\pi
e\overline{\sigma}^{2}_{\text{B}}}\right)}{2\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(V+\phi^{*}\right)\right)}$.

######  Proposition 2.

Assume that $\bf{\mathcal{P}7}(\phi^{*})$ is feasible, a global solution
$V^{*}$ exists and is unique.

###### Proof.

The proof is given in Appendix B. ∎

The proof of Proposition 2 shows that there is no need to utilize the
Dinkelbach algorithm and the CCP to solve $\mathcal{P}\mathbf{6}$. In the
worst-case scenario, solving $\mathcal{P}\mathbf{6}$ only requires a bisection
procedure, which generally results in a considerably reduced computational
complexity. Note that the same solving approach presented in the proof of the
proposition may not be possible for the ZF AN-aided MISO scheme because of the
vector form of the precoder $\mathbf{v}$. Instead, a combination of the
Dinkelbach algorithm and the CCP is required to solve the design problem.
Nonetheless, the complexity can still be reduced due to the smaller search
space of $\mathbf{w}$ when the ZF constraint is applied.

###  III-B Known Eves’ CSI

Recall that when Eves’ CSI is known at the transmitter, we consider a direct
design to maximize the minimum SEE. Following the expression in (20), the AN
design problem, in this case, is formulated as

$\displaystyle{\bf{\mathcal{P}8}}\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{W}}{\text{maximize}}$
$\displaystyle\widetilde{\Phi}^{\text{min}}_{s}(v,\mathbf{W})$ (37a) subject
to $\displaystyle|v|\leq\Delta_{\text{DC}},$ (37b)
$\displaystyle\left\lVert\mathbf{W}\right\rVert_{\infty}\leq\Delta_{\text{DC}}.$
(37c)

It is seen that $\mathcal{P}\textbf{8}$ is a nonconvex maximin optimization
problem. A standard technique to handle the problem is to introduce a slack
variable $t=\widetilde{\Phi}^{\text{min}}_{s}(v,\mathbf{W})$ and then
transform $\mathcal{P}\textbf{8}$ to

$\displaystyle{\bf{\mathcal{P}9}}\hskip 14.22636pt$
$\displaystyle\underset{v,\mathbf{W},t}{\text{maximize}}\hskip 5.69054ptt$
(38a) subject to $\displaystyle C^{k}_{s}(v,\mathbf{W})\geq
t\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\text{tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right)\\!\\!\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall k=1,2,...,K,$ (38b)
$\displaystyle\eqref{P8:constraint1},\leavevmode\nobreak\
\eqref{P8:constraint2}.$

The use of the slack variable $t$ results in additional constraints
$\widetilde{\Phi}_{s}(v,\mathbf{W})\geq t,\hskip 2.0pt\forall k=1,2,...,K$,
which are equivalently represented in (38b), which is non-convex and difficult
to handle due to the product between optimizing variables on the right-hand
side. It is, therefore, challenging to directly solve ${\bf{\mathcal{P}9}}$.
However, one can transform ${\bf{\mathcal{P}9}}$ into a feasibility problem by
fixing the value of $t$. Then, a solution to the original problem can be found
using the bisection method. Specifically, given a fixed value of $t$,

$\displaystyle{{\mathcal{P}\bf{10}}(t)}\hskip 8.53581pt$
$\displaystyle\text{find}\hskip 5.69054ptv,\mathbf{W}$ (39a) subject to
$\displaystyle C^{k}_{s}(v,\mathbf{W})\geq
t\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\text{tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right)\\!\\!\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall k=1,2,...,K,$ (39b)
$\displaystyle\eqref{P8:constraint1},\leavevmode\nobreak\
\eqref{P8:constraint2}.$

Similar to (25)-(28), we make use of the following variable transformations

$\displaystyle
c_{\text{B},1}=\log_{2}\left(2\left(\left(h_{\text{B}}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2}\right)+\pi
e\overline{\sigma}^{2}_{\text{B}}\right),$ (40) $\displaystyle
c_{\text{B},2}=\log_{2}\left(\pi
e\left(\frac{1}{3}{\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2}}+\overline{\sigma}^{2}_{\text{B}}\right)\right),$
(41) $\displaystyle
p_{\text{B},1}=\left(h_{\text{B}}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2},$
(42) $\displaystyle
p_{\text{B},2}=\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2},$
(43) $\displaystyle c^{k}_{\text{E},1}=\log_{2}\left(\pi
e\left(\frac{1}{3}\left(\left(h_{\text{E},k}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}\right)+\overline{\sigma}^{2}_{\text{E},k}\right)\right),$
(44) $\displaystyle
c^{k}_{\text{E},2}=\log_{2}\left(2{\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}}+\pi
e\overline{\sigma}^{2}_{\text{E},k}\right),$ (45) $\displaystyle
p^{k}_{\text{E},1}=\left(h_{\text{E},k}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2},$
(46) $\displaystyle
p^{k}_{\text{E},2}=\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}.$
(47)

and then transform $\mathcal{P}{\bf{10}}(t)$ into the following

$\displaystyle{\bf{\mathcal{P}11}}(t)\hskip 14.22636pt$
$\displaystyle\text{find}\hskip
5.69054ptv,\mathbf{W},c_{\text{B},1},c_{\text{B},2},p_{\text{B},1},p_{\text{B},2},c^{k}_{\text{E},1},c^{k}_{\text{E},2},p^{k}_{\text{E},1},p^{k}_{\text{E},2}$
(48a) subject to
$\displaystyle\frac{1}{2}\left(c_{\text{B},1}-c_{\text{B},2}-c^{k}_{\text{E},1}+c^{k}_{\text{E},2}\right)\geq
t\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\text{tr}\left(\mathbf{W}\mathbf{W}^{T}\right)\right)\\!\\!\right),$
(48b) $\displaystyle c_{\text{B},1}\leq\log_{2}\left(2p_{\text{B},1}+\pi
e\overline{\sigma}^{2}_{\text{B}}\right),$ (48c) $\displaystyle
p_{\text{B},1}\leq\left(h_{\text{B}}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2},$
(48d) $\displaystyle c_{\text{B},2}\geq\log_{2}\left(\pi
e\left(\frac{1}{3}p_{\text{B},2}+\overline{\sigma}^{2}_{\text{B}}\right)\right),$
(48e) $\displaystyle
p_{\text{B},2}\geq\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2},$
(48f) $\displaystyle c^{k}_{\text{E},1}\geq\log_{2}\left(\pi
e\left(\frac{1}{3}p^{k}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\right)\right),$
(48g) $\displaystyle
p^{k}_{\text{E},1}\geq\left(h_{\text{E},k}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2},$
(48h) $\displaystyle
c^{k}_{\text{E},2}\leq\log_{2}\left(2p^{k}_{\text{E},2}+\pi
e\overline{\sigma}^{2}_{\text{E},k}\right),$ (48i) $\displaystyle
p^{k}_{\text{E},2}\leq\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2},$
(48j) $\displaystyle\eqref{P8:constraint1},\leavevmode\nobreak\
\eqref{P8:constraint2}.$

It is seen that constraint (48c), (48f), (48h), (48i), (37b), and (37c) are
convex while the rest are not. Using the first-order Taylor expansion to
approximately convexify those non-convex constraints results in

$\displaystyle\left(h_{\text{B}}v\right)^{2}+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}\right\rVert^{2}\geq$
$\displaystyle\left(h_{\text{B}}v^{(j-1)}\right)^{2}+2h^{2}_{\text{B}}v^{(j-1)}\left(v^{(j)}-v^{(j-1)}\right)+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}^{(j-1)}\right\rVert^{2}$
$\displaystyle+2\overline{\mathbf{h}}^{T}_{\text{B}}\left(\mathbf{W}^{(j)}-\mathbf{W}^{(j-1)}\right)\left[\mathbf{W}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{B}},$
(49) $\displaystyle\log_{2}\left(\pi
e\left(\frac{1}{3}{p_{\text{B,2}}}+\overline{\sigma}^{2}_{\text{B}}\right)\right)\leq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}{p^{{(j-1)}}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)+\frac{p^{{(j)}}_{\text{B},2}-p^{{(j-1)}}_{\text{B},2}}{\ln(2)\left(\\!\pi
e\left(\frac{1}{3}{p^{{(j-1)}}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)},$
(50) $\displaystyle\log_{2}\left(\pi
e\left(\frac{1}{3}p^{k}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\right)\right)\leq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}p^{k^{(j-1)}}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\\!\right)\right)+\frac{p^{k^{(j)}}_{\text{E},1}-p^{k^{(j-1)}}_{\text{E},1}}{\ln(2)\left(\pi
e\left(\frac{1}{3}p^{k^{(j-1)}}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\right)\right)},$
(51)
$\displaystyle\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}\right\rVert^{2}\geq\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}^{(j-1)}\right\rVert^{2}+2\overline{\mathbf{h}}^{T}_{\text{E},k}\left(\mathbf{W}^{(j)}-\mathbf{W}^{(j-1)}\right)\left[\mathbf{W}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{E},k},$
(52)

where $v^{(j)}$, $\mathbf{W}^{(j)}$, $p^{{(j)}}_{\text{B},2}$, and
$p^{k^{(j)}}_{\text{E},1}$ are the solutions to $v$, $\mathbf{W}$,
$p_{\text{B},2}$, and $p_{\text{E},1}$ at the $j$-th iteration of the CCP
procedure. Replacing the above approximating terms to (48d), (48e), (48g), and
(48j), respectively, resulting in

$\displaystyle{\bf{\mathcal{P}12}}(t)\hskip 14.22636pt$
$\displaystyle\text{find}\hskip
5.69054ptv^{(j)},\mathbf{W}^{(j)},c_{\text{B},1},c_{\text{B},2},p_{\text{B},1},p^{{(j)}}_{\text{B},2},c^{k}_{\text{E},1},c^{k}_{\text{E},2},p^{k^{(j)}}_{\text{E},1},p^{k}_{\text{E},2}$
(53a) subject to $\displaystyle
p_{\text{B},1}\leq\left(h_{\text{B}}v^{(j-1)}\right)^{2}+2h^{2}_{\text{B}}v^{(j-1)}\left(v^{(j)}-v^{(j-1)}\right)+\left\lVert\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{W}^{(j-1)}\right\rVert^{2}$
(53b) $\displaystyle\hskip
31.29802pt+2\overline{\mathbf{h}}^{T}_{\text{B}}\left(\mathbf{W}^{(j)}-\mathbf{W}^{(j-1)}\right)\left[\mathbf{W}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{B}},$
$\displaystyle c_{\text{B},2}\geq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}{p^{{(j-1)}}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)+\frac{p^{{(j)}}_{\text{B},2}-p^{{(j-1)}}_{\text{B},2}}{\ln(2)\left(\\!\pi
e\left(\frac{1}{3}{p^{{(j-1)}}_{\text{B},2}}+\overline{\sigma}^{2}_{\text{B}}\right)\\!\right)},$
(53c) $\displaystyle c^{k}_{\text{E},1}\geq\log_{2}\left(\\!\pi
e\left(\frac{1}{3}p^{k^{(j-1)}}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\\!\right)\right)+\frac{p^{k^{(j)}}_{\text{E},1}-p^{k^{(j-1)}}_{\text{E},1}}{\ln(2)\left(\pi
e\left(\frac{1}{3}p^{k^{(j-1)}}_{\text{E},1}+\overline{\sigma}^{2}_{\text{E},k}\right)\right)},$
(53d) $\displaystyle
p^{k}_{\text{E},2}\leq\left\lVert\overline{\mathbf{h}}^{T}_{\text{E},k}\mathbf{W}^{(j-1)}\right\rVert^{2}+2\overline{\mathbf{h}}^{T}_{\text{E},k}\left(\mathbf{W}^{(j)}-\mathbf{W}^{(j-1)}\right)\left[\mathbf{W}^{(j-1)}\right]^{T}\overline{\mathbf{h}}_{\text{E},k},$
(53e)
$\displaystyle\eqref{P11:constraint1},\eqref{P11:constraint3},\leavevmode\nobreak\
\eqref{P11:constraint6},\leavevmode\nobreak\
\eqref{P11:constraint8},\leavevmode\nobreak\
\eqref{P11:constraint9},\leavevmode\nobreak\
\eqref{P8:constraint1},\leavevmode\nobreak\ \eqref{P8:constraint2},$

which is a convex feasibility problem, hence can be efficiently handled. The
feasibility of $\mathcal{P}\textbf{11}(t)$ can then be determined if there
exists a solution to $\mathcal{P}\textbf{12}(t)$. Now, it is observed that the
solution to $\mathcal{P}\textbf{9}$ is the maximum value of $t$ at which
$\mathcal{P}\textbf{11}(t)$ is feasible. Therefore, the bisection method, as
described in the Algorithm 3 on top of the next page, can be utilized to solve
$\mathcal{P}\bf{9}$.

1 Choose $t_{1}$ and $t_{2}$ ($t_{1}<t_{2}$) so that
$\mathcal{P}\textbf{12}(t)$ is feasible at $t=t_{1}$ and is infeasible at
$t=t_{2}$.

2 Choose the error tolerance $\epsilon_{3}>0$.

3 while  _$t_{2}-t_{1} >\epsilon_{3}$_ do

4 $t\leftarrow\frac{t_{1}+t_{2}}{2}$.

5 Solve the feasibility problem $\mathcal{P}\textbf{12}(t)$.

6 if  _there exists a solution to $\mathcal{P}{\bf{12}}(t)$_ then

7 $t_{1}\leftarrow t$.

8 else

9 $t_{2}\leftarrow t$.

10 end if

11

12 end while

Algorithm 3 Bisection method for solving $\mathcal{P}\textbf{9}$.

##  IV Numerical Results and Discussions

In this section, simulation results are presented to demonstrate the SEE of
the AN designs for the two transmission schemes. A typical room of the size of
5 m (Length) $\times$ 5 m (Width) $\times$ 3 m (Height) equipped with 4 LED
luminaries as shown in Figs. 1(a) and 1(b) is considered. Bob’s and Eves’
receivers are placed 0.5 m above the floor. Locations of LED luminaires, Bob,
and Eves are determined through a Cartesian coordinate system whose origin is
the center of the floor. We also assume that $I_{\text{min}}=0$ and
$I_{\text{max}}\geq 2I_{\text{DC}}$, resulting in
$\Delta_{\text{DC}}=I_{\text{DC}}$. Without otherwise noted,
$\delta_{\text{B}}=0$ dB, $P_{\text{circuitry}}=8$ Watts, $U_{\text{DC}}=3.3$
Volts, $\zeta=2$, and error tolerances
$\epsilon_{1}=\epsilon{{}_{2}}=\epsilon_{3}=10^{-3}$ are used [36]. Other
system parameters are given in Table II. Furthermore, simulation results are
averaged over 2000 randomly generated channel realizations for Bob and Eves.
Without loss of generality, for the fixed AN-aided SISO scheme, luminaire 1 is
chosen to be Alice, while luminaires 2, 3, and 4 act as jammers. Since the
information-bearing and AN symbols are both assumed to be zero-mean, the
average emitted optical power of each luminaire is dependent solely on
$I_{\text{DC}}$ and given as $\overline{p}_{t}=\eta I_{\text{DC}}$.

###  IV-A Unknown Eves’ CSI

Firstly, it is necessary to specify the minimum allocated AN power
$P_{\text{th}}$ as described in (22c). According to the constraint in (21c),
the maximum allowable AN power in the selective AN-aided SISO scheme is
$(N_{T}-1)I^{2}_{\text{DC}}$ while it is $N_{T}I^{2}_{\text{DC}}$ in the case
of the AN-aided MISO transmission. Thus, to avoid the infeasibility of (21c)
in both AN transmission schemes, $P_{\text{th}}$ can be chosen to be
$P_{\text{th}}=(N_{T}-1)(\rho I_{\text{DC}})^{2}$, where
$\rho\in[0,\leavevmode\nobreak\ 1]$ is a constant that controls the maximum
amount of the AN power given a fixed value of $I_{\text{DC}}$.

Table II: System Parameters Room and LED configurations |
---|---
LED luminaire positions | luminaire 1 : ($-\sqrt{2}$, $-\sqrt{2}$, 3) | luminaire 2 : ($\sqrt{2}$, $-\sqrt{2}$, 3)
| luminaire 3 : ($\sqrt{2}$, $\sqrt{2}$, 3) | luminaire 4 : ($-\sqrt{2}$, $\sqrt{2}$, 3)
LED bandwidth, $B_{\text{mod}}$ | 20 MHz |
LED beam angle, $\phi$ | $120^{\circ}$ |
(LED Lambertian order is 1) |  |
LED conversion factor, $\eta$ | 0.44 W/A |
Receiver photodetectors |
Active area, $A_{r}$ | 1 $\text{cm}^{2}$ |
Responsivity, $\gamma$ | 0.54 A/W |
Field of view (FoV), $\Psi$ | $60^{\circ}$ |
Optical filter gain, $T_{s}(\psi)$ | 1 |
Refractive index of the concentrator, $\kappa$ | 1.5 |
Other parameters |
Ambient light photocurrent, $\chi_{\text{amp}}$ | 10.93 $\text{A}/(\text{m}^{2}\cdot\text{Sr}$) |
Preamplifier noise current density, $i_{\text{amp}}$ | 5 $\text{pA}/\text{Hz}^{-1/2}$ |

(a) $\overline{p}_{t}$ = 30 dBm.

(b) $\overline{p}_{t}$ = 35 dBm.

(c) $\overline{p}_{t}$ = 40 dBm.

Figure 2: Feasibility probability of AN transmission schemes.

Recall that depending on the choice of $\delta_{\text{B}}$ and $\rho$,
constraints (22b) and (22c) may not be simultaneously satisfied, which results
in an infeasibility of $\mathcal{P}\textbf{2}$ (similar arguments for the
fixed AN-aided SISO and AN-aided MISO schemes). Since $\delta_{\text{B}}=0$ dB
is assumed, we show in Figs. 2(a), 2(b), and 2(c) the impact of $\rho$ on the
feasibility of $\mathcal{P}\textbf{2}$ for different values of
$\overline{p}_{t}$. When $\rho$ is less than 0.5, except for the fixed SISO
scheme, all AN transmission schemes are always feasible. Their feasibility
probabilities, however, start decreasing as $\rho$ exceeds 0.5, especially for
the ZF AN designs (due to their reduced feasible regions imposed by the ZF
constraint). Also, the low feasibility probability of the fixed SISO scheme is
due to the fact that Bob can be out of the service zone as a fixed luminaire
is chosen to transmit the information-bearing signal. To ensure sufficiently
high feasibility probabilities of all transmission schemes, in the following,
$\rho=0.2$ is chosen for simulations.

To qualitatively compare the SEE performance of the examined AN-aided
transmission schemes, it suffices to consider the case that the number of Eves
is 1. Figs. 3(a) and 3(b) depict the EE of Bob’s channel (i.e., the solution
to $\mathcal{P}\textbf{1}$) and the resultant SEE with respect to the average
luminaire optical power $\overline{p}_{t}$. It is clearly shown that when only
Bob’s channel is considered, the MISO transmission performs better than the
selective and fixed SISO. This verifies the benefit of spatial precoding in
improving EE. Interestingly, the resultant SEE shows a contrary observation
where the proposed selective SISO scheme significantly outperforms the MISO
and the fixed SISO transmissions. For instance, the optimal SEE of the
selective SISO is more than twice that of the MISO scheme and about 80% higher
than that of the fixed SISO scheme.

(a) EE of Bob’s channel versus $\overline{p}_{t}$.

(b) SEE versus $\overline{p}_{t}$.

(c) SINR gap versus $\overline{p}_{t}$.

Figure 3: Performance of AN transmission schemes in the case of unknown Eves’
CSI.

This can be explained in Fig. 3(c), which plots the gap between the signal-to-
interference-plus-noise ratios (SINR) of Bob’s and Eve’s channels. It is seen
that the SINR gap in the case of the MISO scheme is smaller than that of the
selective SISO, especially in the low optical power region. Recall that the AN
design, in this case, tries to maximize the EE of Bob’s channel without
consideration of Eve due to its unknown CSI (i.e., indirect AN design).
Consequently, using more luminaires for transmitting the information-bearing
signal enables Eve to receive more signal power, which results in a lower
secrecy rate compared to that of the selective SISO scheme. As the transmit
optical power $\overline{p}_{t}$ increases, an increase in the SINR gap is
observed. Nevertheless, this improvement (due to increased $\overline{p}_{t}$)
does not result in a better SEE because of the dominance of the power
consumption. It is also evident that AN-aided transmission schemes offer
better SEE performance than their no-AN (i.e., without AN) counterparts.
Regarding the selective SISO transmissions, the ZF AN-aided design, despite
its low complexity, archives a comparable SEE with that of the general design.
Hence, it may be preferable for the system design when reducing the signal
processing time is of importance.

We now discuss the complexity of the general AN designs for the selective SISO
and MISO transmissions. As presented in Sec. III.A, the solving procedure
involves two iterative algorithms (i.e., Dinkelback-type and CCP-type
algorithms). Thus, Figs. 4(a) and 4(b) show the convergence error of the EE
with respect to the number of iterations of the Dikelbach-type (Algorithm 1)
and CCP-type (Algorithm 2) algorithms for the selective AN-aided SISO and the
AN-aided MISO schemes, respectively.

(a) Selective AN-aided SISO.

(b) AN-aided MISO.

Figure 4: Convergence behaviors of the two AN schemes.

For the former, there requires, on average, three iterations for both
algorithms to achieve the convergence error of $10^{-3}$. For the latter to
attain the same convergence error, at least six and four iterations are needed
for the CCP-type and the Dinkelbach-type algorithms, respectively. The higher
complexity of the AN-aided MISO is due to the larger sizes of its optimizing
variables (i.e., $N_{T}$ vs. 1 for the precoder of the information symbol and
$N_{T}$ vs. $N_{T}-1$ for the precoder of the AN). In practical scenarios,
when more LED luminaries are deployed (i.e., $N_{T}$ increases), the
complexity may increase substantially due to the bigger size of the
optimization variables.

###  IV-B Known Eves’ CSI

Figure 5: SEE versus $\overline{p}_{t}$.

Figure 6: SEE versus the number of Eves.

In Fig. 6, comparisons between AN-aided and no-AN transmission schemes in
terms of the SEE are illustrated when Eves’ CSI is known at the transmitters
when the number of Eves is set to 1. In sharp contrast to the case of unknown
Eves’ CSI, it is shown in this case that at their respective optimal SEE, the
MISO scheme outperforms the selective and fixed SISO counterparts by 30% and
167%, respectively. In the case of the indirect design when Eve’s CSI is not
available, due to the absence of Eve’s CSI in the AN design, only one
luminaire (that provides the highest channel gain) should be used to transmit
the information-bearing signal so that as little as possible the information
is eavesdropped on by Eves. This strategy is, however, not optimal for the
direct design when Eves’ CSI is considered. Since AN can be specifically
constructed to attenuate Eves’ channels, more luminaires should be employed to
transmit the information-bearing signal so that the SEE can be improved.

As we study VLC systems where multiple non-colluding Eves can coexist, Fig. 6
illustrates the SEE with respect to the number of Eves when
$\overline{p}_{t}=$ 26 dBm. It is highlighted that while the SEE in the case
of AN-aided transmission considerably deteriorates as the number of Eves
increases, it stays unchanged when AN is not used. Specifically, when the
number of Eves increases from 1 to 5, the SEE drops 22%, 28%, and 23% for the
selective AN-aided SISO, AN-aided MISO, and fixed SISO transmission schemes,
respectively. This SEE degradation is due to the proportionality between the
number of Eves and the number of AN streams. Hence, when more Eves are
present, more power should be allocated to the AN, leading to a reduction in
the SEE. Furthermore, it is seen that as the number of Eves increases, the SEE
of the AN-aided transmission asymptotes that of the no-AN scheme.

##  V Conclusion

In this paper, we have examined the selective AN-aided SISO and AN-aided MSIO
transmission schemes and studied their SEE performances. Simulation results
revealed that while the selective AN-aided SISO achieves better SEE than the
AN-aided MISO scheme when Eves’ CSI is unavailable at the transmitting side,
it is the opposite when Eves’ CSI is available. It was also observed that
using AN is advantageous over the no-AN transmission regardless of the
availability of Eves’ CSI. This work considered the case of a single
legitimate user, which is a relatively special assumption. It should be noted
that the proposed selective AN-aided SISO scheme was not designed for a more
general scenario with multiple Bobs. Therefore, our future research attempts a
modified transmission scheme to support such a setting.

##  Appendix A Proof of Proposition 1

We first prove that $C_{\text{B}}(v,\mathbf{w})$ is strictly monotonically
decreasing with
$\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}$. Indeed, the
first derivative of $C_{\text{B}}(v,\mathbf{w})$ with respect to
$\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}$ is given by

$\displaystyle C^{\prime}_{\text{B}}(v,\mathbf{w})=\frac{\left(2-\frac{\pi
e}{3}\right)\pi e\overline{\sigma}^{2}_{\text{B}}-\frac{2\pi
e\left(h_{\text{B}}v\right)^{2}}{3}}{2^{2C_{\text{B}}(v,\mathbf{w})+1}\log_{2}e\left(\pi
e\right)^{2}\left(\frac{\left(\overline{\mathbf{h}}^{T}_{\text{B}}\mathbf{w}\right)^{2}}{3}+\overline{\sigma}^{2}_{\text{B}}\right)^{2}},$
(54)

which is obviously negative. Thus, the optimal solution $\mathbf{w}^{*}$ to
maximize $C_{\text{B}}(v,\mathbf{w})$ satisfies
$\overline{\mathbf{h}}_{\text{B}}\mathbf{w}^{*}=0$, which corresponds to
$\mathbf{w}^{*}$ laying on the null-space of
$\overline{\mathbf{h}}^{T}_{\text{B}}$. It is then easily verified that
$\mathbf{w}^{*}=\mathbf{0}$ simultaneously maximizes
$C_{\text{B}}(v,\mathbf{w})$ and minimizes
$P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\left(v^{2}+\lVert\mathbf{w}\rVert^{2}\right)$
while satisfying the constraint in (21c). This proves that
$\mathbf{w}^{*}=\mathbf{0}$ is optimal to $\mathcal{P}\mathbf{1}$.

##  Appendix B Proof of Proposition 2

For expressional simplicity, we denote $a=\frac{2h^{2}_{\text{B}}}{\pi
e\overline{\sigma}^{2}_{\text{B}}}$ and
$b(\phi*)=2\left(P_{\text{circuit}}+\sum_{N_{T}}U_{\text{LEDs}}I_{\text{DC}}+\zeta\phi^{*}\right)$.
The derivative of $\Phi_{\text{B}}(V,\phi^{*})$ with respective to $V$ is then
given by

$\displaystyle\Phi^{\prime}_{\text{B}}(V,\phi*)=\frac{\frac{a(b(\phi^{*})+2\zeta
V)}{\ln(2)(1+aV)}-2\zeta\log_{2}(1+aV)}{\left(b(\phi^{*})+2\zeta
V\right)^{2}}.$ (55)

Let $f(V,\phi^{*})=\frac{a(b(\phi^{*})+2\zeta
V)}{\ln(2)(1+aV)}-2\zeta\log_{2}(1+aV)$. We then have the derivative of
$f(V,\phi*)$ with respective to $V$ is
$f^{\prime}(V,\phi^{*})=\frac{-a^{2}}{\ln(2)}\frac{b(\phi^{*})+2V}{(1+aV)^{2}}$.
Since $f^{\prime}(V,\phi^{*})<0$, $f(V,\phi^{*})$ is strictly decreasing with
$V$. As it is deduced from (36b) and (36c) that
$\frac{\delta_{\text{B}}\overline{\sigma}^{2}_{\text{B}}}{h^{2}_{\text{B}}}\leq
V\leq\Delta^{2}_{\text{DC}}$, let us consider the following three scenarios.

  * •

If
$f\left(\frac{\delta_{\text{B}}\overline{\sigma}^{2}_{\text{B}}}{h^{2}_{\text{B}}},\phi^{*}\right)\geq
0$ and $f(\Delta_{\text{DC}}^{2},\phi^{*})\leq 0$, a critical point (also the
optimal solution) $V^{*}$ to $\Phi_{\text{B}}(V,\phi^{*})$ exists and is
unique. Furthermore, $V^{*}$ can be found using the bisection method [41].

  * •

If $f(\Delta_{\text{DC}}^{2},\phi^{*})>0$ then $\Phi_{\text{B}}(V,\phi^{*})$
is strictly increasing with $V$. Therefore, the optimal solution
$V^{*}=\Delta_{\text{DC}}^{2}$.

  * •

If
$f\left(\frac{\delta_{\text{B}}\overline{\sigma}^{2}_{\text{B}}}{h^{2}_{\text{B}}},\phi^{*}\right)<0$
then $\Phi_{\text{B}}(V,\phi^{*})$ is strictly decreasing with $V$. Hence, the
optimal solution $V^{*}$ =
$\frac{\delta_{\text{B}}\overline{\sigma}^{2}_{\text{B}}}{h^{2}_{\text{B}}}$.

The proof is completed.

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