# Design of Artificial Noise for Physical Layer Security in Visible Light
Systems with Clipping

Thanh V. Pham1, Steve Hranilovic2, and Susumu Ishihara1 Emails:
1{pham.van.thanh<EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Mathematical and Systems Engineering, Shizuoka University,
Shizuoka, Japan.  2Department of Electrical and Computer Engineering, McMaster
University, ON, Canada.

###### Abstract

Though visible light communication (VLC) systems are contained in a given
room, ensuring their security amongst users in a room is essential. In this
paper, the design of artificial noise (AN) to enhance physical layer security
in VLC systems is studied in the context of input signals with no explicit
amplitude constraint (such as multicarrier systems). In such systems, clipping
is needed to constrain the input signals within the limited linear ranges of
the LEDs. However, this clipping process gives rise to non-linear clipping
distortion, which must be incorporated into the AN design. To facilitate the
solution of this problem, a sub-optimal design approach is presented using the
Charnes-Cooper transformation and the convex-concave procedure (CCP).
Numerical results show that the clipping distortion significantly reduces the
secrecy level, and using AN is advantageous over the no-AN scheme in improving
the secrecy performance.

###### Index Terms:

VLC, physical layer security, artificial noise, clipping distortion.

##  I Introduction

Visible light communications (VLC) is a promising wireless technology to
complement existing radio-based wireless communication systems [1]. Research
and development of VLC systems have been motivated due to the increasing need
for high-speed wireless connections, the scarcity of radio frequency (RF)
spectrum, and the popularity of using light-emitting diodes (LEDs) for
illumination.

As visible light is confined by opaque objects, VLC systems are expected to be
more secure than their RF counterparts. Nonetheless, there is still a security
risk within the illumination areas of VLC systems due to the broadcast nature
of the visible light signal. In this regard, over the past few years, there
has been an increasing interest in applying physical layer security (PLS) to
enhance message confidentiality in VLC systems [2]. Fundamental analyses on
the secrecy capacity of systems consisting of a single transmitter, single
legitimate user, and single eavesdropper were presented in [3], where lower
and upper secrecy capacity bounds were derived under an average optical
intensity constraint, and both average and peak optical intensity constraints.
In indoor scenarios, multiple LED luminaries are often deployed to provide
sufficient illumination. As a result, multi-transmitter VLC systems are more
relevant in practice [4].

The use of multiple transmitter luminaires offers spatial degrees of freedom
that can be exploited to improve the secrecy performance by means of precoding
[5, 6] and artificial noise (AN). In essence, AN is a jamming signal which is
purposely generated by the transmitters to degrade the quality of the
eavesdropper channel while introducing little or no interference to the
legitimate user channel. The use of AN in improving the secrecy performance of
VLC systems has been studied intensively in the literature [7, 8, 9, 10, 11,
12].

It is well-known that LEDs have specific input ranges over which the emitted
optical power is linearly related to the amplitude of the input drive current.
Due to this characteristic, previous work assumed that the AN and information-
bearing signals are both amplitude-constrained to ensure that the amplitude of
the LED input drive current can be properly constrained within the LED linear
range. However, the assumption of amplitude-constrained signals does not apply
to multicarrier signals such as orthogonal frequency division multiplexing
(OFDM), which are expected to be widely employed for high-speed VLC systems
[13]. Multicarrier signals can often have large peak-to-average power ratios.
Thus clipping is unavoidable in VLC OFDM systems and it is necessary to
constrain the signals within the LED linear range. For the design of AN for
VLC OFDM systems, the nonlinear clipping distortion introduced renders optimal
design extremely difficult. Hence, in this paper a sub-optimal design is
presented which uses the Charnes-Cooper transformation and the convex-concave
procedure (CCP). Simulation results show that the clipping distortion has a
significant impact on the secrecy level, and using AN can considerably improve
the secrecy performance.

The balance of the paper is organized as follows. Sec. II outlines the system
and channel models. The AN transmission scheme and design approach are then
described. Numerical results are discussed in Sec. III. Finally, Sec. IV
concludes the paper.

Notation: The following notations are used throughout the paper. Bold
uppercase and lowercase letters (e.g., $\mathbf{H}$ and $\mathbf{h}$)
represent matrices and vectors, respectively. $[\mathbf{H}]_{m,n}$ indicate
the element at the $m$-th row and $n$-th column of $\mathbf{H}$ while
$[\mathbf{h}]_{n}$ indicate the $n$-th element of $\mathbf{h}$. Also,
$\mathbb{E}[\cdot]$ is the expected value operation, $\lVert\cdot\rVert$ is
the Euclidean norm, and $\odot$ is the element-wise product operation.

##  II System and AN Transmission Models

Figure 1: System Configuration.

Consider a multi-transmitter VLC system that consists of a legitimate user
(Bob) and an eavesdropper (Eve) coexisting in a room of dimension
$d_{L}~{}\text{(m)}\times d_{W}~{}\text{(m)}\times d_{H}~{}\text{(m)}$ as
illustrated in Fig. 1. There are $N_{T}$ LED luminaries connected to a Central
Processing Unit (CPU) through wired connections. Assume that Bob and Eve are
both equipped with a receiver having a single photodiode (PD), and all
receivers are on the same receiver plane which is $d_{R}~{}\text{(m)}$ above
the floor. For the sake of simplicity, only the line-of-sight (LoS)
propagation channel is considered. Let
$\mathbf{h}_{\text{B}}=\begin{bmatrix}h_{1,\text{B}}&h_{2,\text{B}}&\ldots&h_{N_{T},\text{B}}\end{bmatrix}^{T}$
be the channel vector of Bob where $h_{k,\text{B}}$ is the channel gain from
the $k$-th LED luminary111The LoS channel gain model in the case of Lambertian
emission pattern can be found in [2] and references therein.. Similarly,
denote
$\mathbf{h}_{\text{E}}=\begin{bmatrix}h_{1,\text{E}}&h_{2,\text{E}}&\cdots&h_{N_{T},\text{E}}\end{bmatrix}^{T}$
as the channel vector of Eve.

###  II-A AN transmission scheme

The IEEE 802.11bb standard advocates for the use of DC-biased optical (DCO)
OFDM to support high data rates (up to 10 Gbps) transmission in VLC [13].
Unlike PAM and OOK signaling considered in previous studies, OFDM signals are
inherently not subject to amplitude constraint and can have multiple high
peaks. Hence, clipping is necessary to fit the signals into the LED linear
range.

Let $d$ and $z$ be the information-bearing and AN signals, respectively. In
previous studies, $d$ and $z$ were either assumed to follow the uniform
distribution [8, 12] or the truncated Gaussian distribution [14, 10, 15] over
the normalized amplitude range $[-1~{}1]$. In the case of OFDM, when a large
number of sub-carriers is used (e.g., 64 or more), the signals can be well
approximated to be Gaussian and have domain over the real numbers. Hence, for
the sake of analysis, let $d$ and $z$ be time-domain Gaussian information-
bearing and AN signals. Without loss of generality, assume that $d$ and $z$
are independent zero-mean and have unit variance. At the $n$-th luminaire,
denote $v_{n}$ and $w_{n}$ as the precoders of $d$ and $z$, respectively. The
combined information-bearing and AN current is given by

$\displaystyle x_{n}=v_{n}d+w_{n}z.$ (1)

Figure 2: AN transmission scheme for non-amplitude constrained signals.

Since $d$ and $z$ are not amplitude-constrained, $x_{n}$ must be clipped to
limit its amplitude within the LED operational range, which is assumed to be
in the range $[I_{\text{min}},~{}I_{\text{max}}]$. Since
$d\sim\mathcal{N}(0,1)$ and $z\sim\mathcal{N}(0,1)$, it follows that
$x_{n}\sim\mathcal{N}\left(0,v^{2}_{n}+w^{2}_{n}\right)$. As illustrated in
Fig. 2, a nonlinear clipping is applied to $x_{n}$ at a bottom level
$I_{\text{min}}-I_{\text{DC}}$ and at a top level
$I_{\text{max}}-I_{\text{DC}}$, where $I_{\text{DC}}$ is the DC-bias current
which is added to the clipped signal. Thus, the resulting distorted signal
$\tilde{x}_{n}$ can be modeled using the Bussgang theorem as

$\displaystyle\tilde{x}_{n}=R_{n}\left(v_{n}d+w_{n}z\right)+I_{\text{DC}}+\zeta_{n},$
(2)

where $R_{n}\in[0,~{}1]$ is the attenuation factor and $\zeta_{n}$ is the
clipping noise [16]. An increased severity of clipping distortion is
characterized by a small value of $R_{n}$ and a large variance of $\zeta_{n}$.
If we denote
$\alpha_{n}=\frac{I_{\text{min}}-I_{\text{DC}}}{\sqrt{v^{2}_{n}+w^{2}_{n}}}$
and
$\beta_{n}=\frac{I_{\text{max}}-I_{\text{DC}}}{\sqrt{v^{2}_{n}+w^{2}_{n}}}$
then $R_{n}=Q(\alpha_{n})-Q(\beta_{n})$. The clipping noise $\zeta_{n}$ can be
approximately modeled as a zero-mean Gaussian random variable whose variance
is given by

$\displaystyle\sigma^{2}_{\text{clip},n}$ $\displaystyle=$
$\displaystyle\Big{(}R_{n}+\alpha_{n}\phi(\alpha_{n})-\beta_{n}\phi(\beta_{n})+\alpha_{n}^{2}(1-Q(\alpha_{n}))+\beta_{n}^{2}Q(\beta_{n})\big{.}$
$\displaystyle\left.-\big{(}\phi(\alpha_{n})-\phi(\beta_{n})+(1-Q(\alpha_{n}))\alpha_{n}+Q(\beta_{n})\beta_{n}\big{)}^{2}\right.\\!\\!\\!-R_{n}^{2}\Big{)}$
$\displaystyle\times\left(v^{2}_{n}+w^{2}_{n}\right),$ (3)

where $\phi(t)=\frac{1}{2\pi}\exp\left(\frac{-t^{2}}{2}\right)$ and
$Q(t)=\frac{1}{2\pi}\int_{t}^{\infty}\exp\left(\frac{-u^{2}}{2}\right)\text{d}u$.

Suppose that there are $n_{c}$ closely packed LED chips in each luminaire. The
instantaneous emitted optical power from the $n$-th LED luminary is then

$\displaystyle
P_{n,\text{optical}}=n_{c}\eta\left(R_{n}\left(v_{n}d+w_{n}z\right)+I_{\text{DC}}+\zeta_{n}\right),$
(4)

where $\eta$ is the LED conversion factor. Since the LED chips in a given
luminaire are assumed to be closely packed, it is reasonable to assume that
the channel gains from them to a receiver are identical. The output electrical
signal for Bob from all luminaires in the room and after the removal of the DC
bias current is

$\displaystyle r_{\text{B}}=\gamma\eta
n_{c}\left(\left(\mathbf{h}_{\text{B}}\odot\mathbf{R}\right)^{T}\mathbf{v}d+\left(\mathbf{h}_{\text{B}}\odot\mathbf{R}\right)^{T}\mathbf{w}z+\mathbf{h}^{T}_{\text{B}}\boldsymbol{\zeta}\right)+n_{\text{B}},$
(5)

where
$\mathbf{R}=\begin{bmatrix}R_{1}&R_{2}&\cdots&R_{N_{T}}\end{bmatrix}^{T}$,
$\mathbf{v}=\begin{bmatrix}v_{1}&v_{2}&\cdots&v_{N_{T}}\end{bmatrix}^{T}$,
$\mathbf{w}=\begin{bmatrix}w_{1}&w_{2}&\cdots&w_{N_{T}}\end{bmatrix}^{T}$, and
$\boldsymbol{\zeta}=\begin{bmatrix}\zeta_{1}&\zeta_{2}&\cdots&\zeta_{N_{T}}\end{bmatrix}^{T}$.
The receiver noise $n_{\text{B}}$ can be well approximated being Gaussian
distributed with zero mean and variance $\sigma^{2}_{\text{B}}$ given by

$\displaystyle\sigma^{2}_{\text{B}}=2e\gamma\overline{P}_{r,\text{B}}B_{\text{mod}}+4\pi
e\gamma
A_{r}\chi_{\text{amb}}\left(1-\cos\Psi\right)B_{\text{mod}}+i^{2}_{\text{amp}}B_{\text{mod}},$
(6)

where $e$ is the elementary charge, $\gamma$ is the PD responsivity,
$B_{\text{mod}}$ is the signal bandwidth, $\chi_{\text{amb}}$ is the ambient
light photocurrent, and $i^{2}_{\text{amp}}$ is the pre-amplifier noise
current density. $A_{r}$ and $\Psi$ are the active area and the optical field
of view (FoV) of the PD, respectively.
$\overline{P}_{r,\text{B}}=n_{c}\eta\sum_{n=1}^{N_{T}}h_{n,\text{B}}\mathbb{E}[\tilde{x}_{n}]$
is the average received optical power at Bob’s receiver. The signal-to-
interference-plus-noise ratio (SINR) of Bob’s received signal is then given by

$\displaystyle\text{SINR}_{\text{B}}=\frac{\left(n_{c}\left(\mathbf{h}_{\text{B}}\odot\mathbf{R}\right)^{T}\mathbf{v}\right)^{2}}{\left(n_{c}\left(\mathbf{h}_{\text{B}}\odot\mathbf{R}\right)^{T}\mathbf{w}\right)^{2}+\left(n_{c}\mathbf{h}^{T}_{\text{B}}\boldsymbol{\sigma}_{\text{clip}}\right)^{2}+{\sigma}^{2}_{\text{B,
norm}}},$ (7)

where
$\boldsymbol{\sigma}_{\text{clip}}=\begin{bmatrix}\sigma_{\text{clip},1}&\sigma_{\text{clip},2},&\ldots&\sigma_{\text{clip},N_{T}}\end{bmatrix}^{T}$
and ${\sigma}^{2}_{\text{B,
norm}}=\frac{\sigma^{2}_{\text{B}}}{\left(\gamma\eta\right)^{2}}$.

It is often unrealistic to assume that $\mathbf{h}_{\text{E}}$ is known at the
transmitter due to the fact that Eve tends to hide itself from being detected.
Nonetheless, the design of AN to provide an upper limit on secrecy performance
when $\mathbf{h}_{\text{E}}$ is known is investigated here. Similar to Bob,
the SINR of Eve’s wire-taped signal is then given by

$\displaystyle\text{SINR}_{\text{E}}=\frac{\left(n_{c}\left(\mathbf{h}_{\text{E}}\odot\mathbf{R}\right)^{T}\mathbf{v}\right)^{2}}{\left(n_{c}\left(\mathbf{h}_{\text{E}}\odot\mathbf{R}\right)^{T}\mathbf{w}\right)^{2}+\left(n_{c}\mathbf{h}^{T}_{\text{E}}\boldsymbol{\sigma}_{\text{clip}}\right)^{2}+{\sigma}^{2}_{\text{E,
norm}}},$ (8)

where $\sigma^{2}_{\text{E,
norm}}=\frac{\sigma^{2}_{\text{E}}}{\left(\gamma\eta\right)^{2}}$.

###  II-B AN design

Optimization problems in physical layer security often utilize the secrecy
rate as the objective function. In addition to the secrecy rate, the secrecy
performance can also be evaluated indirectly through the pair of SINRs of
Bob’s and Eve’s received signal, which is often easier to handle. Moreover, by
taking the SINR of Eve’s received signal into the design, one can precisely
(or qualitatively) control the performance of Eve’ channel, such as the bit-
error rate. In this work, we, therefore, study AN design problems using
$\text{SINR}_{\text{B}}$ and $\text{SINR}_{\text{E}}$. Specifically,
$\text{SINR}_{\text{B}}$ is maximized while constraining
$\text{SINR}_{\text{E}}$ to a predefined maximum threshold, denoted as
$\lambda$. Additionally, the sum-power of the information-bearing and AN
signals at the $n$-th luminary is constrained to a maximum allowable value of
$P_{n}$. The AN design problem is thus given as follows

$\displaystyle\underset{\mathbf{v},\mathbf{w}}{\text{maximize}}$
$\displaystyle\text{SINR}_{\text{B}}$ (9a) subject to
$\displaystyle\text{SINR}_{\text{E}}\leq\lambda,$ (9b)
$\displaystyle\left[\mathbf{v}\right]^{2}_{n}+\left[\mathbf{w}\right]^{2}_{n}\leq
P_{n},~{}~{}\forall n=1,2,...,N_{T}.$ (9c)

Note that, instead of (9c), one can also apply a constraint on the sum-power
of the information-bearing and AN signals over all luminaries, which is

$\displaystyle\left\lVert\mathbf{v}\right\rVert^{2}+\left\lVert\mathbf{w}\right\rVert^{2}\leq
P,$ (10)

where $P=\sum_{n=1}^{N_{T}}P_{n}$. Compared with (10), the advantage of using
(9c) is two-fold. Firstly, it is clear that (9c) is a generalization of (10).
Secondly, using (9c) facilitates the handling of $\text{SINR}_{\text{B}}$ and
$\text{SINR}_{\text{E}}$ as it can be seen that $\mathbf{R}$,
$\boldsymbol{\sigma}_{\text{clip}}$, $\sigma^{2}_{\text{B, norm}}$, and
$\sigma^{2}_{\text{E, norm}}$ are functions of
$\left[\mathbf{v}\right]^{2}_{n}+\left[\mathbf{w}\right]^{2}_{n}$ at each
individual luminary. Nonetheless, due to the complex non-linearity of
$\mathbf{R}$, $\boldsymbol{\sigma}_{\text{clip}}$, ${\sigma}^{2}_{\text{B,
norm}}$, and ${{\sigma}}^{2}_{\text{E, norm}}$, it is extremely challenging
(if not impossible) to optimally solve the above problem. Therefore, a sub-
optimal design can be constructed by considering alternative simplified
expressions for $\text{SINR}_{\text{B}}$ and $\text{SINR}_{\text{E}}$ under
certain assumptions. Specifically, define $\widetilde{\mathbf{R}}$,
$\widetilde{\boldsymbol{\sigma}}_{\text{clip}}$,
$\widetilde{{\sigma}}^{2}_{\text{B, norm}}$, and
$\widetilde{{{\sigma}}}^{2}_{\text{E, norm}}$ as the values of $\mathbf{R}$,
$\boldsymbol{\sigma}_{\text{clip}}$, ${\sigma}^{2}_{\text{B, norm}}$, and
${{{\sigma}}}^{2}_{\text{E, norm}}$, respectively, under the assumption that
$\left[\mathbf{v}\right]^{2}_{n}+\left[\mathbf{w}\right]^{2}_{n}=P_{n}$
($\forall n=1,2,...,N_{T}$). In this case,
$\widetilde{\text{SINR}}_{\text{B}}$ and $\widetilde{\text{SINR}}_{\text{E}}$
represent $\text{SINR}_{\text{B}}$ and ${\text{SINR}}_{\text{E}}$ when
$\mathbf{R}=\widetilde{\mathbf{R}}$,
$\boldsymbol{\sigma}_{\text{clip}}=\widetilde{\boldsymbol{\sigma}}_{\text{clip}}$,
$\sigma^{2}_{\text{B, norm}}=\widetilde{\sigma}_{\text{B, norm}}$, and
$\widetilde{{\sigma}}^{2}_{\text{E, norm}}={\sigma}^{2}_{\text{E, norm}}$,
respectively. We are now focusing on designing $\mathbf{v}$ and $\mathbf{w}$
based on $\widetilde{\text{SINR}}_{\text{B}}$ and
$\widetilde{\text{SINR}}_{\text{E}}$. An intuitive reasoning for this sub-
optimal design approach is it has been observed that as long as (9b) is
satisfied, $\left[\mathbf{v}\right]^{2}_{n}+\left[\mathbf{w}\right]^{2}_{n}$
tends to achieve its maximum allowable value when (9a) is maximized. Indeed,
numerical results in Sec. III show that $\widetilde{\text{SINR}}_{\text{B}}$
provides a very good lower bound for ${\text{SINR}}_{\text{B}}$ (less than 5%
difference).

For the sake of analysis, define
$\widetilde{\mathbf{h}}_{\text{B}}=\mathbf{h}_{\text{B}}\odot\widetilde{\mathbf{R}}$
and
$\widetilde{\mathbf{h}}_{\text{E}}={\mathbf{h}}_{\text{E}}\odot\widetilde{\mathbf{R}}$.
The sub-optimal design problem is then given by

$\displaystyle\underset{\mathbf{v},\mathbf{w}}{\text{maximize}}$
$\displaystyle\widetilde{\text{SINR}}_{\text{B}}$ (11a) subject to
$\displaystyle\widetilde{\text{SINR}}_{\text{E}}\leq\lambda,$ (11b)
$\displaystyle\left[\mathbf{v}\right]^{2}_{n}+\left[\mathbf{w}\right]^{2}_{n}\leq
P_{n},~{}~{}\forall n=1,2,...,N_{T}.$ (11c)

It is easy to see that there exists $\mathbf{v}$ and $\mathbf{w}$ satisfying
both (11b) and (11c) regardless of the values of $\lambda$ and $P_{n}$’s
(e.g., $\mathbf{v}=\mathbf{w}=\mathbf{0}$). Hence, (11) is always feasible.
Also, it is seen that (11) is a fractional programming problem, which can
often be solved using the Charnes-Cooper transformation [17]. Specifically,
define

$\displaystyle\mathbf{v}$ $\displaystyle=\frac{\overline{\mathbf{v}}}{t},$
$\displaystyle\ltx@label{a}\tagform@$ and $\displaystyle\mathbf{w}$
$\displaystyle=\frac{\overline{\mathbf{w}}}{t},$ (12b)

where $t>0$. The problem in (11) can then be transformed into the following
non-fractional programming problem.

$\displaystyle\underset{\overline{\mathbf{v}},\overline{\mathbf{w}},t}{\text{maximize}}$
$\displaystyle\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{v}}\right)^{2}$
(12ma) subject to
$\displaystyle\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}\right)^{2}+t^{2}\left(\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}\right)=1$ (12mb)
$\displaystyle\frac{1}{\lambda}\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{v}}\right)^{2}\leq\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}\right)^{2}$
$\displaystyle\hskip
54.06023pt+t^{2}\left(\left(n_{c}{\mathbf{h}}^{T}_{\text{E}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\overline{\sigma}}^{2}_{\text{E,
norm}}\right),$ (12mc)
$\displaystyle\left[\overline{\mathbf{v}}\right]^{2}_{n}+\left[\overline{\mathbf{w}}\right]^{2}_{n}\leq
t^{2}P_{n},~{}~{}\forall n=1,2,\cdots,N_{T}.$ (12md)

From (12mb), we get
$t^{2}=\frac{1-\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}\right)^{2}}{\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}}.$ Substituting into (12mc) and (12md) gives

$\displaystyle\underset{\overline{\mathbf{v}},\overline{\mathbf{w}}}{\text{maximize}}\hskip
5.69054pt\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{v}}\right)^{2}$
(12na) subject to
$\displaystyle\frac{1}{\lambda}\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{v}}\right)^{2}\leq\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}\right)^{2}$
$\displaystyle+\frac{1-\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}\right)^{2}}{\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}}\left(\left(n_{c}{\mathbf{h}}^{T}_{\text{E}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\overline{\sigma}}^{2}_{\text{E,
norm}}\right),$ (12nb)
$\displaystyle\left[\overline{\mathbf{v}}\right]^{2}_{n}+\left[\overline{\mathbf{w}}\right]^{2}_{n}\leq\frac{1-\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}\right)^{2}}{\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}}P_{n},~{}~{}\forall n=1,\cdots,N_{T}.$ (12nc)

The problem (12n) is not convex due to the maximization of a convex function
in (12na) and the non-convexity of the constraint in (12nb). A well-known
technique to solve (12n) is CCP, which is based on an iterative procedure to
find a local optimum [18, 19]. Specifically, at each iteration, the first-
order Taylor approximation is utilized to approximately linearlize (12na) and
the first term in the right-hand side of (12nb) as follows.

$\displaystyle\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{v}}^{(i)}\right)^{2}\approx$
$\displaystyle\left(n_{c,1}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{v}}^{(i-1)}\right)^{2}$
$\displaystyle+2n^{2}_{c}\left[\overline{\mathbf{v}}^{(i-1)}\right]^{T}\widetilde{\mathbf{h}}_{\text{B}}\widetilde{\mathbf{h}}^{T}_{\text{B}}\left(\overline{\mathbf{v}}^{(i)}-\overline{\mathbf{v}}^{(i-1)}\right),$
(12o)

and

$\displaystyle\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}\right)^{2}\approx$
$\displaystyle
n_{c}^{2}\Big{(}\left(\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}^{(i-1)}\right)^{2}$
$\displaystyle+2\left[\widetilde{\mathbf{h}}_{\text{E}}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}^{(i-1)}\right]^{T}\left(\overline{\mathbf{w}}^{(i)}-\overline{\mathbf{w}}^{(i-1)}\right)\Big{)},$
(12p)

where $\overline{\mathbf{v}}^{(i)}$ and $\overline{\mathbf{w}}^{(i)}$ are
instances of $\overline{\mathbf{v}}$ and $\overline{\mathbf{w}}$ at the $i$-th
iteration. Using the above approximations, the CCP involves solving the
problem in (12q) (on top of the following page), which is convex.

$\displaystyle\underset{\overline{\mathbf{v}}^{(i)},\overline{\mathbf{w}}^{(i)}}{\text{maximize}}$
$\displaystyle\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{v}}^{(i-1)}\right)^{2}+2n^{2}_{c}\left[\overline{\mathbf{v}}^{(i-1)}\right]^{T}\widetilde{\mathbf{h}}_{\text{B}}\widetilde{\mathbf{h}}^{T}_{\text{B}}\left(\overline{\mathbf{v}}^{(i)}-\overline{\mathbf{v}}^{(i-1)}\right)$
(12qa) subject to
$\displaystyle\frac{1}{\lambda}\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{E}}\left[\overline{\mathbf{v}}^{(i)}\right]\right)^{2}\leq
n_{c}^{2}\left(\left(\widetilde{\mathbf{h}}^{T}_{\text{E}}\left[\overline{\mathbf{w}}^{(i-1)}\right]\right)^{2}+2\left[\widetilde{\mathbf{h}}_{\text{E}}\widetilde{\mathbf{h}}^{T}_{\text{E}}\overline{\mathbf{w}}^{(i-1)}\right]^{T}\left(\overline{\mathbf{w}}^{(i)}-\overline{\mathbf{w}}^{(i-1)}\right)\right)$
$\displaystyle\hskip
91.04881pt+\frac{1-\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}\right)^{2}}{\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}}\left(\left(n_{c}\mathbf{h}^{T}_{\text{E}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\overline{\sigma}}^{2}_{\text{E,
norm}}\right),$ (12qb)
$\displaystyle\left[\overline{\mathbf{v}}^{(i)}\right]_{n}^{2}+\left[\overline{\mathbf{w}}^{(i)}\right]_{n}^{2}\leq\frac{1-\left(n_{c}\widetilde{\mathbf{h}}^{T}_{\text{B}}\overline{\mathbf{w}}^{(i)}\right)^{2}}{\left(n_{c}\mathbf{h}^{T}_{\text{B}}\widetilde{\boldsymbol{\sigma}}_{\text{clip}}\right)^{2}+\widetilde{\sigma}^{2}_{\text{B,
norm}}}P_{n},~{}\forall n=1,\cdots,N_{T},$ (12qc)

The detailed procedure is outlined in Algorithm 1.

1 Choose the maximum number of iterations $L_{\text{max}}$ and the error
tolerance $\epsilon>0$ for the algorithm.

2 Choose a feasible initial point
$\left(\overline{\mathbf{v}}^{(0)},\overline{\mathbf{w}}^{(0)}\right)$ to
(12q).

3 Set $i\leftarrow 1$.

4 while convergence = False and $i\leq L_{\text{max}}$ do

5 Solve (12q) using
$\left(\overline{\mathbf{v}}^{(i-1)},\overline{\mathbf{w}}^{(i-1)}\right)$
obtained from the previous iteration.

6 if
$\frac{\left\lVert\overline{\mathbf{v}}^{(i)}-\overline{\mathbf{v}}^{(i-1)}\right\rVert}{\left\lVert\overline{\mathbf{v}}^{(i)}\right\rVert}\leq\epsilon$
and$\frac{\left\lVert\overline{\mathbf{w}}^{(i)}-\overline{\mathbf{w}}^{(i-1)}\right\rVert}{\left\lVert\overline{\mathbf{w}}^{(i)}\right\rVert}\leq\epsilon$
then

7 convergence = True.

8 $\overline{\mathbf{v}}^{*}\leftarrow\overline{\mathbf{v}}^{(i)}$.

9 $\overline{\mathbf{w}}^{*}\leftarrow\overline{\mathbf{w}}^{(i)}$.

10

11 else

12 convergence $\leftarrow$ False.

13 end if

14 $i\leftarrow i+1$.

15

16 end while

Return $\overline{\mathbf{v}}^{*}$ and $\overline{\mathbf{w}}^{*}$ then
calculate $\mathbf{v}^{*}$ and $\mathbf{w}^{*}$ from (LABEL:a) and (12b).

Algorithm 1 CCP-type algorithm to solve (12n)

The obtained sub-optimal precoders $\mathbf{v}^{*}$ and $\mathbf{w}^{*}$ are
used to calculate the actual values of $\text{SINR}_{\text{B}}$,
${\text{SINR}}_{\text{E}}$ given in (7) and (8), respectively. For comparison,
these values are denoted by $\text{SINR}^{*}_{\text{B}}$ and
${\text{SINR}}^{*}_{\text{E}}$ while the respective values obtained though the
approximation of Sec. II-B are denoted by
$\widetilde{\text{SINR}}^{*}_{\text{B}}$ and
$\widetilde{{\text{SINR}}}^{*}_{\text{E}}$. Similarly,
$\text{SINR}^{*}_{\text{B, NoAN}}$, ${\text{SINR}}^{*}_{\text{E, NoAN}}$,
$\widetilde{\text{SINR}}^{*}_{\text{B, NoAN}}$, and
$\widetilde{{\text{SINR}}}^{*}_{\text{E, NoAN}}$ are defined in the same
manner for the case of the no-AN scheme (i.e., $\mathbf{w}=0$).

##  III Numerical Results and Discussions

This section presents simulation results to demonstrate the secrecy
performance of the proposed AN designs. The system configuration and
parameters are given in Table I, and simulation results are obtained by
averaging over 5000 randomly located positions of Bob and Eve.

TABLE I: System Parameters Room dimension, | 5 $\times$ 5 $\times$ 3
---|---
Length (m) $\times$ Width (m) $\times$ Height (m) |
LED luminary positions | Luminary 1 : ($-\sqrt{2}$, $-\sqrt{2}$, 3)
| Luminary 2 : ($\sqrt{2}$, $-\sqrt{2}$, 3)
| Luminary 3 : ($\sqrt{2}$, $\sqrt{2}$, 3)
| Luminary 4 : ($-\sqrt{2}$, $\sqrt{2}$, 3)
LED bandwidth, $B_{\text{mod}}$ | 20 MHz
Number of chips in each luminary, $n_{c}$ | 24
LED beam angle, $\phi$ | $120^{\circ}$
(LED Lambertian order is 1) |
LED conversion factor, $\eta$ | 0.44 W/A
$[I_{\text{min}},~{}~{}I_{\text{max}}]$ | $[0\text{A},~{}~{}1\text{A}]$
Active area, $A_{r}$ | 1 $\text{cm}^{2}$
Responsivity, $\gamma$ | 0.54 A/W
Field of view (FoV), $\Psi$ | $70^{\circ}$
Ambient light photocurrent, $\chi_{\text{amp}}$ | 10.93 $\text{A}/(\text{m}^{2}\cdot\text{Sr}$)
Preamplifier noise current density, $i_{\text{amp}}$ | 5 $\text{pA}/\text{Hz}^{-1/2}$

Firstly, Fig. 3 depicts the convergence behavior of Algorithm 1 with respect
to the number of iterations $L$ when $\lambda=0$ and $-5$dB. It is shown that,
on average, the objective function in (12qa) approaches its optimal value when
$L\approx 3$, which corresponds to relative errors of $\overline{\mathbf{v}}$
and $\overline{\mathbf{w}}$ being greater than $10^{-2}$, respectively.
Accordingly, the error tolerance $\epsilon=10^{-2}$ and the maximum number of
iteration $L_{\text{max}}=10$ can be chosen to guarantee a satisfactory
convergence of (12qa).

Figure 3: Convergence behaviors of Algorithm 1.

((a)) $\lambda=0$ dB.

((b)) $\lambda=-5$ dB.

Figure 4: Secrecy performance versus $\sigma_{P}$ for different thresholds on
Eve’s SINR.

Assuming DCO-OFDM is employed to generate the information-bearing signal and
AN, a widely used setting for $I_{\text{DC}}$ is $I^{2}_{\text{DC}}=4P_{n}$ is
employed, which is equivalent to 7dB bias [20]. If we denote
$\sigma_{P_{n}}=\sqrt{P_{n}}$ (i.e., $\sigma_{P_{n}}$ represents the maximum
allowable standard deviation for the combined information-bearing and AN
signals), then $I_{\text{DC}}=2\sigma_{P_{n}}$. Consider that
$\sigma_{P_{n}}\text{'s}=\sigma_{P}$, Fig. 4(a) and Fig. 4(b) present the
secrecy performance as a function of $\sigma_{P}$ for $\lambda=0$ dB and $-5$
dB, respectively. Firstly, the left sub-figures compare the AN and no-
approaches in terms of Bob’s and Eve’ SINR performances. In the AN scheme,
notice that the difference between $\text{SINR}^{*}_{\text{B}}$ and
$\widetilde{\text{SINR}}^{*}_{\text{B}}$ are small, especially at high values
of $I_{\text{DC}}$. More importantly, the differences between
$\text{SINR}^{*}_{\text{E}}$ and $\widetilde{\text{SINR}}^{*}_{\text{E}}$ are
seen to be negligible. These two observations confirm the approximation of the
proposed sub-optimal approach presented in Sec. II-B. In the case of no-AN
scheme, the gap between $\text{SINR}^{*}_{\text{B, NoAN}}$ and
$\widetilde{\text{SINR}}^{*}_{\text{B, NoAN}}$ are more clear but still near 3
dB. Nonetheless, there are considerable gaps between
$\text{SINR}^{*}_{\text{E, NoAN}}$ and $\widetilde{\text{SINR}}^{*}_{\text{E,
NoAN}}$, where Eve’s SINRs are about 5 dB higher than the results obtained
from using the approximate expression.

For a better illustration of the secrecy performance, the right sub-figures
compare the AN and no-AN schemes in terms of achievable secrecy rate. Since
the information-bearing and AN signals are approximated to be Gaussian, the
achievable secrecy rates corresponding to the pairs
$\left(\text{SINR}^{*}_{\text{B}},~{}\text{SINR}^{*}_{\text{E}}\right)$ and
$\left(\widetilde{\text{SINR}}^{*}_{\text{B}},~{}\widetilde{\text{SINR}}^{*}_{\text{E}}\right)$
in the case of known $\mathbf{h}_{\text{E}}$ are given by
$C^{*}_{s}=\log_{2}\left(1+\text{SINR}^{*}_{\text{B}}\right)-\log_{2}\left(1+{\text{SINR}}^{*}_{\text{E}}\right)$
and
$\widetilde{C}^{*}_{s}=\log_{2}\left(1+\widetilde{\text{SINR}}^{*}_{\text{B}}\right)-\log_{2}\left(1+\widetilde{{\text{SINR}}}^{*}_{\text{E}}\right)$,
respectively. Note that $C^{*}_{s,\text{NoAN}}$ and
$\widetilde{C}^{*}_{s,\text{NoAN}}$ are similarly defined. Then, it is clear
to see the superiority of the AN scheme over the no-AN approach. For example,
at $\sigma_{P}=0.25A$, the use of AN improves the achievable secrecy rate by
about $1.2$ and $0.4$ bits/s/Hz when $\lambda=0$ and $-5$ dB, respectively.
Moreover, observe that at low $\sigma_{P}$ values (i.e., below $0.2$ A), the
achievable secrecy rate stays nearly unchanged. This arises because, for low
$\sigma_{P}$, the impact of increasing $\sigma_{P}$ is balanced by the
clipping distortion. However, when $\sigma_{P}>0.2$ A, the impact of the
clipping distortion becomes dominant, resulting in a dramatic degradation of
the achievable secrecy rate.

##  IV Conclusion

In this paper, we have studied an AN design for VLC systems considering the
impact of clipping distortion. Simulation results revealed that compared with
the no-AN scheme, the proposed design could effectively constrain Eve’s SINR
to the predefined threshold while did not affect considerably Bob’s SINR. As a
result, using AN was beneficial in improving the secrecy performance.
Furthermore, significant impacts of the clipping distortion on the secrecy
performance were also observed. Our future work will focus on novel signal
transmission schemes to alleviate the clipping distortion, AN design for the
case of multi-user, and examine the case that Eve’s CSI is unknown at the
transmitters.

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