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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022 11179
Physical-Layer Security for Indoor VLC Wiretap
Systems Under Multipath Reflections
Fan Yang ,Student Member, IEEE, Jingjing Wang ,Senior Member, IEEE,
and Yuhan Dong ,Senior Member, IEEE
Abstract— In this paper, we consider the physical-layer security
for single-input single-output (SISO) indoor visible light commu-
nication (VLC) wiretap systems in the presence of multipath
reflections. We derive both the lower and upper bounds on
the secrecy capacity in the context of both the symbol- and
block-based transmission policies. To enhance the secrecy per-
formance relying on block transmission policy, we propose a
low-complexity amplitude scaling (AS) scheme by scaling the
amplitudes of different symbols in each block to maximize the
achievable secrecy rate. We further provide the upper bound
on the optimal secrecy rate achieved by precoding for the
sake of validating the effectiveness of the proposed AS scheme.
Numerical results suggest that the secrecy performance is severely
degraded by inter-symbol interference (ISI) imposed by multi-
path reflections, which yet can be alleviated by long-block based
transmission. Moreover, the proposed AS scheme can enhance
the secrecy performance significantly and achieve a secrecy rate
very close to the optimal solution.
Index Terms—Visible light communications, physical-layer
security, secrecy capacity, multipath reflection.
I. INTRODUCTION
VISIBLE light communication (VLC) relying on visi-
ble light beams emitted and captured by light-emitting
diodes (LEDs) and photo-diodes (PDs) respectively, has been
regarded as one of the most promising wireless technologies
considering its high data rate and wide unlicensed band-
width [1]. Moreover, benefiting from combining the data
transmission and illumination, VLC plays a critical role in the
future Internet of everything for indoor applications, depend-
ingonbothhighdatarateand communication security.
In 1975, Wyner proposed a wiretap channel model and
conceived the performance metric of secrecy capacity defined
as the maximum information transmission rate under secrecy
Manuscript received 23 September 2021; revised 26 March 2022
and 21 June 2022; accepted 1 July 2022. Date of publication 20 July 2022;
date of current version 12 December 2022. This work was supported in part
by the Guangdong Basic and Applied Basic Research Foundation under Grant
2022A1515010209 and in part by the Shenzhen Natural Science Foundation
under Grant JCYJ20200109143016563. The associate editor coordinating the
review of this article and approving it for publication was C. Masouros.
(Corresponding author: Yuhan Dong.)
Fan Yang and Yuhan Dong are with the Shenzhen International Graduate
School, Tsinghua University, Shenzhen 518055, China, and also with the
Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China (e-mail: yangf19@tsinghua.org.cn; dongyuhan@sz.tsinghua.edu.cn).
Jingjing Wang is with the School of Cyber Science and Technology, Beihang
University, Beijing 100191, China (e-mail: drwangjj@buaa.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TWC.2022.3190362.
Digital Object Identifier 10.1109/TWC.2022.3190362
constraints [2]. One important solution to the secure commu-
nications is to realize physical-layer security by utilizing the
dissimilarity between the legitimate and eavesdropping chan-
nels. In recent years, many physical-layer security approaches
for radio frequency (RF) systems has been proposed, such
as artificial noise (AN) [3], physical-layer authentication [4],
physical-layer key generation technology [5], etc. Moreover,
physical-layer security has also been investigated in various
complex RF scenarios, such as multiple-antenna channels
[6], [7], energy- and cost-efficient systems [8], anonymous
communications systems [9], etc.
As a promising member of the next-generation commu-
nication family, VLC is also susceptible to eavesdropping
due to the open and broadcast nature, especially when users
are deployed in public areas. However, VLC systems cannot
directly adopt the RF physical-layer security approaches since
they transmit real-valued and unipolar signals with the aid
of intensity modulation and direct detection (IM/DD) scheme
rather than complex signals in RF systems [10]. Besides, VLC
systems generally impose amplitude constraint on the channel
input due to the limited dynamic range of LEDs. Therefore,
it is necessary to investigate the physical-layer security in VLC
systems in terms of both secrecy performance evaluation and
enhancement. As for secrecy performance evaluation, Mostafa
and Lampe [11] first of all derived the upper and lower bounds
on secrecy capacity for single-input single-output (SISO)
indoor VLC wiretap channels under a peak optical intensity
constraint. Then, Wang et al. considered both average and
peak optical intensity constraints, and derived the closed-form
expression of the upper and lower bounds on secrecy capac-
ity [12]. Moreover, the secrecy performance evaluations for
a range of complex VLC scenarios were carried out, such
as multiple-input single-output (MISO) VLC wiretap chan-
nels [13], multiple-input multiple-output (MIMO) VLC wire-
tap channels [14], multi-user VLC networks with randomly
located eavesdroppers [15], spatial modulation based indoor
VLC systems with multiple transmitters [16], random VLC
networks with imperfect channel state information (CSI) [17],
etc. By contrast, as for secrecy performance enhancement,
Mostafa and Lampe [18] utilized null-steering beamforming
and AN strategies to achieve positive secrecy rates. Then, they
further proposed a robust beamforming scheme for the sake of
adapting the uncertainty of both the legitimate and eavesdrop-
ping channels [19]. For the multi-eavesdropper VLC channels,
the zero-forcing (ZF) beamforming scheme was highlighted by
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11180 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
Cho et al. in [20]. Additionally, the AN scheme was further
investigated in the context of, e.g., MISO generalized space-
shift-keying (GSSK) VLC systems [21], MISO VLC wiretap
channels with randomly located eavesdropper [22], energy effi-
cient VLC networks with multiple eavesdroppers [23], SISO
DC-biased optical orthogonal frequency division multiplexing
(DCO-OFDM) systems [24], etc.
For indoor VLC systems, multipath reflections typically
exist and will introduce both the line-of-sight (LoS) com-
ponent and non-line-of-sight (NLoS) components. The NLoS
components will impose inter-symbol interference (ISI) on
the received signals of both legitimate receivers and eaves-
droppers and then affect the overall secrecy performance.
However, most of the existing works about secure VLC
systems only considered the LoS component for channel
modeling and impractically ignored the NLoS components,
due to the absence of closed-form expressions of NLoS
components. However, NLoS components cannot be ignored
especially in high-reflectivity environments where the optical
power of NLoS components is comparable to that of LoS
component [25]. To address this issue, the impact of multipath
reflections imposed on security performance of VLC systems
was first studied in [26] based on low data rate and non-ISI
assumptions. In [27], Chen et al. derived the upper and lower
bounds on secrecy capacity to evaluate the impact of both
specular and diffusive reflections imposed, while impractically
assumed that the ISI was only correlated with the multipath
channel and uncorrelated with the amplitude of transmitted
signals.
Motivated by these prior works, in this paper, we investigate
the impact of multipath reflections imposed on the secrecy per-
formance of indoor VLC SISO wiretap systems each includes
a transmitter, a legitimate receiver and an eavesdropper. The
transmitted signals are subject to amplitude constraint, and
captured by both the legitimate receiver and the eavesdropper
with the same strategy. The main contributions are summarized
as follows:
1) We thoroughly analyze the characteristics of SISO
indoor VLC wiretap systems in the presence of mul-
tipath reflection, and theoretically derive both the lower
and upper bounds on the secrecy capacity in the con-
text of both the symbol- and block-based transmission
policies.
2) To enhance the secrecy performance, we propose a
low-complexity amplitude scaling (AS) scheme by scal-
ing the amplitudes of different symbols in each block
with CSI of the eavesdropping channel known or
unknown at the transmitter. We further present an upper
bound on the optimal secrecy rate achieved by precod-
ing for the sake of validating the effectiveness of the
proposed AS scheme.
3) Extensive simulations are conducted to validate the
derived bounds and the proposed AS scheme. Numer-
ical results demonstrate that the secrecy performance
is severely degraded by the ISI due to the multipath
reflections, and our proposed AS scheme can achieve
a closely near-optimal secrecy performance against the
optimal solution.
Fig. 1. The indoor SISO VLC wiretap system with multipath reflections.
The remainder of the paper is organized as follows.
In Section II, we introduce the model of indoor VLC wiretap
systems in the context of multipath reflections. In Section III,
we analyze the secrecy performance by evaluating the lower
bound on the secrecy capacity. We propose the AS scheme to
enhance the secrecy performance in Section IV and derive the
upper bound on the secrecy capacity in Section V. Section VI
presents the numerical results, followed by our conclusions in
Section VII.
Notations: Throughout this paper, the following notations
are adopted. R,R+,Z,andZ+denote the sets of real numbers,
non-negative real numbers, integers, and non-negative integers,
respectively. Rn×mis the set of n×mreal-valued matrices.
Dn×nrepresents the set of n×nreal-valued diagonal matrices.
E{·},h{·},Tr{·},andVar{·} denote mathematical expec-
tation, differential entropy, trace, and variance, respectively.
[·]Tis the transpose operator, while · denotes the ceiling
function. |·| represents the determinant of a matrix or the
absolute value of a real number, and ·ldenotes the l-norm
of vector or matrix. [a]idenotes the i-th element of vector a.
[A]i,j denotes the element at the i-th row and j-th column
of matrix A, while [A]i,:and [A]:,j represent the i-th row
vector and the j-th column vector of matrix A, respectively.
[x]+denotes max{x, 0},andI(x;y)is the mutual information
between random variables xand y.⊗represents the convolu-
tion operator. Diag(x1,x
2,...,x
n)denotes the n×ndiagonal
matrix with diagonal elements of x1,x
2,...,x
n.
II. SYSTEM MODEL
We consider an indoor SISO VLC wiretap system as shown
in Fig. 1. The transmitter (Alice) with one LED intends to send
light signals to the legitimate receiver (Bob) in the presence
of the eavesdropper (Eve). Alice is attached to the center of
the ceiling, and the two receivers are located on the work
plane above from the floor. Due to the diffusive reflections
of walls, two receivers capture multipath optical signals from
Alice over the channel. We assume that Bob and Eve utilize
the same receiving strategy.
A. Indoor VLC Channel With Multipath Reflections
Without available closed-form expression for the VLC
channel impulse response (CIR) under multipath reflections,
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11181
Fig. 2. Block diagram of data transmission model.
we choose the iterative algorithm in [25] and its improved
version in [28] to simulate the CIR. First we decompose the
CIR into LoS component and NLoS components as follows
h(t;S,R)=
∞
l=0
h(l)(t;S,R),(1)
where Sand Rdenote the source and the receiver, respec-
tively. t∈R+denotes the elapsed time. h(l)(t;S,R)is the
CIR of the light undergoing l∈Z+reflections. If l=0,
h(0)(t;S,R)corresponds to the LoS component of CIR, which
can be calculated by
h(0)(t;S,R)
=⎧
⎨
⎩
m+1
2πcosm(φ)APD
d2cos(θ)δt−d
c,|θ|≤θFoV,
0,|θ|>θ
FoV ,
(2)
where mis the order of Lambertian emission, APD is the PD
geometrical area, dis the distance between Sand R,φand θ
are the angles of emission and incidence, cis the light speed
and δ(·)is the Dirac delta function giving the time delay, θFoV
is the receiver semi-angle field-of-view (FoV). On the other
hand if l>0,h(l)(t;S,R)∞
l=1 correspond to the NLoS
components of CIR, which can be approximated by breaking
the reflecting surface into numerous small diffusive reflecting
elements and summing up the contributions from all these
elements to the impulse response as follows
h(l)(t;S,R)≈
n
i=1
ρih(l−1)(t;S,ε
i)⊗h(0)(t;εi,R),(3)
where nis the total number of reflecting elements, εisignifies
the i-th element with reflectivity of ρi.
As lincreases, the contribution of h(l)(t;S,R)decreases
to zero due to the reflection attenuation. Hence we can take
finite bounces to calculate the CIR in (1). According to [28],
more than five bounces should be sufficient for accuracy.
B. Signal Transmission Model
We consider an equivalent signal transmission model with
direct current (DC) biased IM/DD as shown in Fig. 2. For
the sake of simplicity, Bob and Eve are both represented by
the receiver in Fig. 2 due to the same receiving strategy at
two receivers. We therefore do not distinguish them in the
following of this sub-section.
At the transmitter side, the discrete-time signals {xk}∈
Rwith zero mean are passed through a transmit filter with
impulse response ζ(t)at a sampling period of T, to generate
the continuous-time transmitted signal ˜x(t)as follows
˜x(t)=
∞
k=−∞
xk·ζ(t−kT).(4)
Then ˜x(t)is superimposed on a fixed bias current IDC ∈R+
to modulate the instantaneous optical power emitted from the
LED. In order to maintain linear current-to-light conversion,
the amplitude of ˜x(t)should be constrained such that |˜x(t)|≤
αIDC,whereα∈[0,1] denotes the modulation index [29].
At the receiver side, the collected optical power is trans-
formed into current signal by a PD. After the removal of the
DC bias, the received signal is passed through a receive filter
with impulse response ξ(t). We assume that ζ(t)and ξ(t)
are both Nyquist pulses with normalized amplitude [30], i.e.,
ζ(0) = ξ(0) = 1 and ζ(kT)=ξ(kT)=0for k=0. Hence
the continuous-time received signal ˜y(t)can be expressed as
˜y(t)= ηκ2RT
sin2(θFoV )·˜x(t)⊗h(t;S,R)⊗ξ(t)+˜n(t),(5)
where η(W/A) is the current-to-light conversion efficiency of
the LED, κis the refractive index of the optical concentrator,
R(A/W) is the PD responsibility, T(V/A) is the transim-
pedance amplifier gain at the receiver. ˜n(t)∈Rdenotes the
additive white Gaussian noise (AWGN) at the receiver.
After sampling ˜y(t)at the rate of 1/T, we can obtain the
k-th discrete-time received signal yk=˜y(t)|t=kT.Using(4)
and (5), we can rewrite ykas
yk=xk⊗hk+nAWG N
k,(6)
where nAWG N
k=˜n(t)|t=kT,andhk∈Rrepresents the overall
discrete-time channel filter tap given by
hk=ηκ2RT
sin2(θFoV )·ζ(t)⊗h(t;S,R)⊗ξ(t)|t=kT.(7)
Note that the convolutions in (5) and (7) are over the con-
tinuous parameter twhile the convolution in (6) is over the
discrete parameter k.
Since ζ(t)is Nyquist pulse with normalized amplitude as
discussed earlier, the amplitude constraint |˜x(t)|≤αIDC is
equivalent to
−A
2≤xk≤A
2,∀k∈Z,(8)
where A=2αIDC represents the amplitude range constraint
on the discrete-time transmitted signal.
III. LOWER BOUND ON SECRECY CAPACITY
For the wiretap channel with amplitude-constrained input,
the closed-form expressions of secrecy capacity have not yet
been addressed [11]. Hence we choose to derive the lower
bound (in this section) and the upper bound (in Section V)
on secrecy capacity for the indoor SISO VLC wiretap system
discussed in Section II. In this system, the signals can be trans-
mitted by symbol [27] or block (i.e., multiple symbols) [31],
and then received and decoded by two receivers without
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11182 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
Fig. 3. Diagram of symbol- and block-based transmission polices with the
impact of ISI.
reserving previous received symbols/blocks.1The diagram of
symbol- and block-based transmission policies with the impact
of ISI is depicted in Fig. 3, where the highlighted rectangles
represent the current transmitted and received symbols/blocks.
The main differences between two transmission policies are
that: 1) the transmission block can be tailored to the multipath
channel using some inter-symbol precoding schemes; 2) the
ISI impact within the same block (corresponds to the blue
dotted lines in Fig. 3(b)) can be alleviated using some post-
processing techniques.2We now calculate the lower bounds
on the secrecy capacity of the two transmission policies
respectively, both given that the transmitted data symbols {sk}
are independent and identically distributed (i.i.d.) with zero
mean.
A. Symbol Transmission
In this case as shown in Fig. 3(a), the data symbols are
directly transmitted without inter-symbol precoding. We hence
assume that xk=sk(k∈Z) without loss of generality.
Since the received signals are affected by ISI due to the
multipath reflections, the k-th received signals at Bob and Eve
are formulated as
yB,k =hB,0sk+
nISI
B,k
∞
i=1
hB,isk−i+nAWG N
B,k
nB,k
,(9)
yE,k =hE,0sk+
nISI
E,k
∞
i=1
hE,isk−i+nAWG N
E,k
nE,k
,(10)
where hB,i and hE,i denote the discrete-time channel filter
taps derived from (7) by replacing Swith Alice and replacing
Rwith Bob and Eve. nAWG N
B,k ∼N0,σ
2
Band nAWG N
E,k ∼
N0,σ
2
Edenote the AWGN at Bob and Eve and both
are independent of the transmitted symbols. Because of the
memoryless receiver, the ISI can only be regarded as noise.
1According to the information theory, reserving previous received sym-
bols/blocks can improve the mutual information between the transmitter and
receiver. Whereas it is not included in this paper and will be our future work.
2Any post-processing techniques working on the received signals do not
improve the mutual information, hence they are not included in this work
which focuses on the information-theoretic security.
We denote the ISI noises as nISI
B,k ∞
i=1 hB,isk−iand
nISI
E,k ∞
i=1 hE,isk−i, which are also independent of the
transmitted symbols. Hence we can define the sum noises as
nB,k nISI
B,k +nAWG N
B,k and nE,k nISI
E,k +nAWG N
E,k . Accord-
ing to the central limit theorem (CLT), the ISI noise nISI
B,k
and nISI
E,k can be approximated by Gaussian variables.3This
approximation will be validated in Section VI-A.
In this wiretap system, the secrecy capacity, i.e., the maxi-
mum achievable secrecy rate, is given by
Cs=max
p(sk)[I(sk;yB,k)−I(sk;yE,k)]+,(11)
where p(sk)is the probability density function (PDF) of sk.
Recall that sk=xkas assumed earlier, hence skis also subject
to the amplitude constraint in (8). It was shown in [11] that
the optimal distribution for the amplitude-constrained wiretap
channel is discrete with a finite number of mass points, and
there is no closed-form expression for the optimal distribution.
Nevertheless, we can use some optimization procedures to
obtain a sub-optimal distribution. First we define the number
of constellation points as Mand let Prob {sk=ai}=pi
for i=1,...,M,where{ai}M
i=1 ∈Rand {pi}M
i=1 ∈R+
denote the constellation points and corresponding probabili-
ties, respectively. Then the mutual information I(sk;yB,k)in
(11) can be calculated as
I(sk;yB,k)=h(yB,k)−h(yB,k|sk)
=−−∞
∞
p(yB,k)log
2p(yB,k)dyB,k
−1
2log2[2πeVar {nB,k}],(12)
where p(yB,k)is the PDF of yB,k,givenby
p(yB,k)
=1
2πVar {nB,k}
M
i=1
piexp −(yB,k −hB,0ai)2
2Var {nB,k},
(13)
and the variance Var {nB,k}is given by
Var {nB,k}=σ2
B+⎡
⎣
M
i=1
pia2
i−M
i=1
piai2⎤
⎦·
∞
i=1
h2
B,i.
(14)
Likewise, we can also calculate I(sk;yE,k). Therefore, for a
fixed M, the maximization problem in (11) can be formulated
as follows
max
{pi}M
i=1,{ai}M
i=1
I(sk;yB,k)−I(sk;yE,k)(15a)
s.t.
M
i=1
pi=1 (15b)
0≤pi≤1,i=1,2,...,M,(15c)
−A
2≤ai≤A
2,i=1,2,...,M.(15d)
3The CLT normally requires i.i.d. summands, which is not satisfied due to
the coefficients
hB,i
. We will present in Section VI-A that the values of
hB,i
approximate to each other within a certain range of i.Andsoare
hE,i
. Hence the CLT is still applicable in general.
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11183
Clearly, (15) is a non-convex optimization problem. One
can use the interior-point method to obtain a local optimum
(depends on the selection of initial point) as a sub-optimal
solution to (11). This locally optimized value can be regarded
as a lower bound on the exact secrecy capacity. However, this
kind of lower bound has no closed-form expression and may
not be practical for system design purposes. Thus, we then
provide a closed-form lower bound on the secrecy capacity in
the following.
According to [32], under a very high signal-to-noise
ratio (SNR) with the amplitude constraint, the input with
uniform distribution can achieve the capacity. So we take the
uniform distribution as input to derive the lower bound on Cs,
such that
p(sk)=⎧
⎨
⎩
1
A,−A
2≤sk≤A
2,
0,otherwise.
(16)
Then the variances of ISI noises are given by VarnISI
B,k=
A2
12 ∞
i=1 h2
B,i and Var nISI
E,k=A2
12 ∞
i=1 h2
E,i, respectively.
Hence the variances of the sum noises nB,k and nE,k can be
expressed as
Var {nB,k}=σ2
B+A2
12
∞
i=1
h2
B,i,(17)
Var {nE,k}=σ2
E+A2
12
∞
i=1
h2
E,i.(18)
For the amplitude-constrained Gaussian wiretap channel with
noise power of Var {nB,k}and Var {nE,k}, the secrecy capac-
ity is lower bounded by [12, Eq. (20)] as follows
Cs≥1
2log2⎡
⎣
6Var {nE,k}A2h2
B,0+2πeVar {nB,k}
πeVar {nB,k}A2h2
E,0+12Var {nE,k}⎤
⎦
=1
2log2⎡
⎢
⎢
⎣
12σ2
E+A2∞
i=1
h2
E,i
12σ2
B+A2∞
i=1
h2
B,i
×
12σ2
B+6
πeA2h2
B,0+A2∞
i=1
h2
B,i
12σ2
E+A2∞
i=0
h2
E,i
⎤
⎥
⎥
⎦
Rs,(19)
where [Rs]+can be regarded as an achievable secrecy rate.
B. Block Transmission
In this case as shown in Fig. 3(b), each transmission
block consists of Ntransmitted symbols, and is generated by
one of the inter-symbol precoding strategies. Without loss of
generality, we consider a linear precoding scheme such that
x=Fs,(20)
where s=[sk,s
k+1,...,s
k+N−1]Tand x=
[xk,x
k+1,...,x
k+N−1]Tare the current transmitted data
symbol vector and signal vector, respectively. F∈RN×N
is the precoding matrix. In order to satisfy the amplitude
constraint in (8), we impose the constraints on Fand ssuch
that
F∞≤1,(21)
s∞≤A
2.(22)
Since the precoding is adopted to each transmission block,
the previous transmitted signals can be expressed as
ˇ
x=ˇ
Fˇ
s,(23)
where ˇ
s=[...,s
k−2,s
k−1]Tand ˇ
x=[...,x
k−2,x
k−1]Tare
the previous transmitted data symbol vector and signal vector,
respectively. Note that ˇ
sand ˇ
xboth have infinite number of
elements. ˇ
Fis an infinite matrix defined as
ˇ
F⎡
⎢
⎣
...
F
F
⎤
⎥
⎦.(24)
At each receiver side, the received signals are decoded by
block each of which consists of Nsignal symbols. With the
impact of ISI, the received blocks at Bob and Eve can be
expressed as
yB=HBFs +
nISI
B
GBˇ
Fˇ
s+nAWG N
B
nB
,(25)
yE=HEFs +
nISI
E
GEˇ
Fˇ
s+nAWG N
E
nE
,(26)
where yB=[yB,k,y
B,k+1,...,y
B,k+N−1]Tand yE=
[yE,k,y
E,k+1,...,y
E,k+N−1]Tare the received signal vectors
at Bob and Eve respectively. nAWGN
B∼N
0,σ
2
BINand
nAWG N
E∼N0,σ
2
EINare the AWGN vectors. HB∈RN×N
is a Toeplitz matrix and denotes the legitimate channel matrix
of current transmitted signals, which is given by
HB=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
hB,000··· 0
.
.
.hB,00··· 0
hB,i ··· ...··· .
.
.
.
.
....··· ...0
hB,N−1··· hB,i ··· hB,0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(27)
GB∈RN×∞ is an infinite matrix and corresponds to the
legitimate channel matrix of previous transmitted signals,
which is given by
GB=⎡
⎢
⎢
⎢
⎢
⎣
··· hB,N+1 hB,N ··· hB,2hB,1
··· ··· ......··· hB,2
··· ··· ··· .......
.
.
··· ··· ··· ··· hB,N+1 hB,N
⎤
⎥
⎥
⎥
⎥
⎦
.
(28)
And so are the eavesdropping channel matrices HEand GE
by replacing {hB,i}∞
i=0 with {hE,i}∞
i=0. Although GBand
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11184 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
GEtheoretically have infinite number of columns, the effect
of previous signals from a long time ahead can be ignored
in practice due to the reflection attenuation. In other words,
when iis large enough, the channel filter taps hB,i and hE,i
approximate to zero. So we could just take the last Lcolumns
of GBand GE,andthelastLrows of ˇ
xfor calculation, where
L∈Z+denotes the maximum discrete delay spread satisfying
hB,i,h
E,i ≈0when i>L.ThevalueofLdepends on the
sampling rate of Alice, since higher sampling rate leads to
more transmitted signals per unit time and hence requires a
larger L.
In this wiretap system, the secrecy capacity per symbol is
given by
Cs=1
Nmax
p(s),F[I(s;yB)−I(s;yE)]+.(29)
Similarly to Section III-A, we assume that {sk}are uni-
formly distributed as defined in (16), then the lower bound
on (29) can be derived accordingly. Just as discussed in
Section III-A, the ISI noises nISI
BGBˇ
Fˇ
sand nISI
EGEˇ
Fˇ
s
can be approximated by Gaussian random vectors according
to the CLT. This approximation will also be validated in
Section VI-A. Then the covariance matrices of the ISI noises
are given by Var nISI
B=A2
12 GBˇ
Fˇ
FTGT
Band Var nISI
E=
A2
12 GEˇ
Fˇ
FTGT
E, respectively. We define the sum noises as
nBnISI
B+nAWG N
Band nEnISI
E+nAWG N
E. Hence the
covariance matrices of nBand nEare given by
Var {nB}=σ2
BIN+A2
12 GBˇ
Fˇ
FTGT
B,(30)
Var {nE}=σ2
EIN+A2
12 GEˇ
Fˇ
FTGT
E.(31)
Then the secrecy capacity is lower bounded by (32), shown
at the bottom of the page, where [Rs(F)]+can be regarded
as an achievable secrecy rate.
Proof: See Appendix.
Specially, for the case of N=1, the matrix Fdegenerates
into a scalar, denoted by [F]1,1∈[−1,1]. It can be calculated
that when [F]1,1=±1, the secrecy rate in (32) is equivalent
to the case of symbol transmission in (19). Hence the symbol
transmission can be regarded as a special case of block
transmission.
By appropriately designing the precoding matrix F,the
secrecy performance of the indoor SISO VLC wiretap system
can be enhanced accordingly. The achievable secrecy rate
Rs(F)in (32) provides a reasonable optimization objective
function. Considering the constraint in (21), the optimization
problem can be formulated as
max
F∈RN×NRs(F)(33a)
s.t. F∞≤1.(33b)
Note that the problem (33) is based on the assumption that the
CSI of the legitimate channel (referred to as Bob’s CSI) and
eavesdropping channel (referred to as Eve’s CSI) are perfectly
known at Alice. Clearly, (33) is a non-convex optimization
problem and hard to be solved explicitly. One can use brute-
force (BF) search method to solve (33) but suffer from the time
complexity increasing exponentially as the square of block
length, i.e., 2O(N2). Therefore, we propose a sub-optimal but
low-complexity scheme to solve (33) with details given in
Section IV.
IV. PROPOSED AMPLITUDE SCALING SCHEME
To acquire better communication performance in the pres-
ence of multipath propagation, a widely adopted scheme in RF
systems is adding guard interval (GI) between two continuous
transmission blocks to mitigate the impact of ISI. Among
several GI schemes, zero padding (ZP) is commonly used to
reduce the signal amplitudes to zero during the process [33].
Based on ZP insertion scheme, we are motivated to extend
it to a more general amplitude scaling scheme to enhance
the secrecy performance of the indoor SISO VLC wiretap
systems discussed in Section III. In the proposed AS scheme,
the amplitudes of different elements in each transmission block
are reduced to varying degrees instead of zero only. Hence we
can express the precoding matrix corresponding to AS scheme
as
FAS =Diag(√m1,√m2,...,√mN),(34)
where √mi∈[0,1] for ∀i∈{1,2,...,N}denotes the
amplitude scaling factor of the i-th element in each block.
Clearly FAS satisfies the constraint in (21). Note that the
square root in (34) is used for offsetting the quadratic form of
Fin the secrecy rate expression (32). We define a diagonal
matrix √MFAS, then (32) is transformed into (35), shown
at the bottom of the page, where the definition of ˇ
Mis similar
to that of ˇ
F, just by replacing Fwith Min (24).
In fact, the proposed AS scheme is equivalent to limit-
ing the feasible set of the original problem (33) to the set
Cs≥1
2log2⎡
⎢
⎢
⎣12σ2
EIN+A2GEˇ
Fˇ
FTGT
E
1
N6
πeA2h2
B,0|F|2
N+12σ2
BIN+A2GBˇ
Fˇ
FTGT
B
1
N
12σ2
BIN+A2GBˇ
Fˇ
FTGT
B
1
N12σ2
EIN+A2HEFFTHT
E+A2GEˇ
Fˇ
FTGT
E
1
N
⎤
⎥
⎥
⎦
Rs(F)(32)
Rs√M=1
2log2⎡
⎢
⎢
⎣12σ2
EIN+A2GEˇ
MGT
E
1
N6
πeA2h2
B,0|M|1
N+12σ2
BIN+A2GBˇ
MGT
B
1
N
12σ2
BIN+A2GBˇ
MGT
B
1
N12σ2
EIN+A2HEMHT
E+A2GEˇ
MGT
E
1
N
⎤
⎥
⎥
⎦(35)
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11185
of non-negative diagonal matrices with elements not larger
than 1. Thus the AS scheme is just a sub-optimal solution
to the original optimization problem (33). However, by this
means the number of optimization scalar variables is reduced
from N2(corresponds to F)toN(corresponds to M). We now
provide the optimal Mdesigning procedures in accordance
with two scenarios: Eve’s CSI is known or unknown at Alice,
both given that Bob’s CSI is perfectly known at Alice.
A. Known Eve’s CSI
In some scenarios, Eve is active that transmits signals while
eavesdropping, then the signals can be captured by Alice
and used for estimating Eve’s CSI. Even if Eve works in
a passive manner, its CSI can still be obtained by Alice
using the modeling method mentioned in Section II since
Eve’s location is fixed and known by Alice. In this case, the
secrecy rate Rs(√M)in (35) can be precisely calculated by
Alice. Hence the optimization of secrecy performance is to
directly maximize Rs(√M)over the matrix M, which can be
formulated as
max
M∈DN×NRs√M(36a)
s.t. 0≤[M]i,i ≤1,i=1,2,...,N. (36b)
Recall that [M]i,i is equal to mi. As can be seen, the
objective function (36a) is non-concave in M, which implies
that the optimization problem (36) is non-convex. In addition,
GB,GEand ˇ
Min (36a) are all infinite matrices, then the
problem (36) requires some transformations for solving.
For the infinite matrix GB, we divide it into several
sub-matrices as GB=[..., GB,2,GB,1],whereGB,k ∈
RN×Nfor k∈{1,2,...}contains the last (kN −N+1)-
th through the (kN)-th columns of GB. As mentioned in
Section III-B, the channel filter tap hB,i approximates to zero
when i>L. Accordingly, we have
GBˇ
MGT
B≈
γ
k=1
GB,kMGT
B,k,(37)
where γ L
N!. Similarly, we let GE=[..., GE,2,GE,1]
and also have
GEˇ
MGT
E≈
γ
k=1
GE,kMGT
E,k.(38)
Substituting (37) and (38) into (35), we can approximate
the secrecy rate as
Rs√M≈1
N[f1(M)−f2(M)] ,(39)
where f1(M)and f2(M)are respectively given by
f1(M)= 1
2log2
σ2
EIN+A2
12
γ
k=1
GE,kMGT
E,k
+N
2log22πe
σ2
BIN+A2
12
γ
k=1
GB,kMGT
B,k
1
N
+A2h2
B,0|M|1
N,(40)
f2(M)= 1
2log2
σ2
BIN+A2
12
γ
k=1
GB,kMGT
B,k
+1
2log2
σ2
EIN+A2
12 HEMHT
E
+A2
12
γ
k=1
GE,kMGT
E,k
+N
2log2(2πe).(41)
Note that although (39) gives an approximated secrecy rate, the
approximation deviation can be neglected as long as γis large
enough. We therefore regard (39) as the actual secrecy rate
with negligible deviation and then transform the optimization
problem (36) into
max
M∈DN×Nf1(M)−f2(M)(42a)
s.t. 0≤[M]i,i ≤1,i=1,2,...,N. (42b)
According to the scalar composition theorem in
[34, Sec. 3.2.4], f1(M)in (40) and f2(M)in (41) are
both concave functions of M. In addition, the feasible set
of (42) is convex, hence (42) is a difference of convex
programming problem and can be effectively solved using
the convex-concave procedure (CCP) algorithm which is a
heuristic method used to find local solutions [35].
The basic idea of CCP algorithm is to apply the first-order
Taylor series approximation to transform the difference of
convex programming problem into a sequence of convex
problems, and then using iterative method to find the solution.
Specifically, at the j-th iteration of the problem (42), the
first-order Taylor series approximation to f2(M)around the
linearization point Mjcan be expressed as
ˆ
f2(M,Mj)=f2(Mj)+Tr ∇f2(Mj)T(M−Mj),
(43)
where ∇f2(M)denotes the gradient of the function f2(M)
and is given by
∇f2(M)
=log2e
2
×"γ
k=1
GT
B,k 12σ2
B
A2IN+
γ
k=1
GB,kMGT
B,k−1
GB,k
+HT
E12σ2
E
A2IN+HEMHT
E+
γ
k=1
GE,kMGT
E,k−1
HE
+
γ
k=1
GT
E,k12σ2
E
A2IN+HEMHT
E
+
γ
k=1
GE,kMGT
E,k−1
GE,k#.(44)
It can be seen that ˆ
f2(M,Mj)is a affine function of
M. By replacing the concave function f2(M)in (42) with
ˆ
f2(M,Mj), we can obtain the optimization problem at the
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11186 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
j-th iteration as follows,
max
M∈DN×Nf1(M)−f2(Mj)−Tr ∇f2(Mj)T(M−Mj)
(45a)
s.t. 0≤[M]i,i ≤1,i=1,2,...,N. (45b)
Algorithm 1 CCP Algorithm for Solving Problem (42)
Given an initial feasible point M0∈DN×N.
set j=0.
repeat
1) Solve the convex optimization problem (45).
2) Assign the solution to Mj+1.
3) Update iteration j←j+1.
until one of following stopping conditions is satisfied:
1) Rs√Mj−RsMj−1<δ,whereδis a
predefined convergence threshold.
2) j=J,whereJis a predefined maximum number of
iterations.
Clearly, (45) is a convex optimization problem. The opti-
mized solution is assigned to Mj+1 and taken as the lineariza-
tion point at the next iteration. Then the iteration procedure
is repeated until stopping criterion is satisfied. The detailed
CCP algorithm is given in Algorithm 1. Note that the CCP
algorithm guarantees a convergence to a local solution, which
may depend on the initial point M0[35]. Therefore we can
execute Algorithm 1 with several different initial points, and
choose the solution which maximizes the objective function
in (42a) as the final solution, denoted as M∗
K.
To solve the convex optimization problem (45) invoked
in Algorithm 1, we adopt the interior-point method whose
worst-case time complexity grows with Nlog N,whereN
is the number of inequality constraints [34, Sec. 11.5.3].
Hence the time complexity of solving (45) in each iteration
is O(Nlog N). Besides, the number of iterations in Algo-
rithm 1 is upper-bounded by a constant J. Therefore, the time
complexity of the overall CCP algorithm is also O(Nlog N),
which is significantly lower than that of the BF search method,
i.e., 2O(N2).
B. Unknown Eve’s CSI
In some practical scenarios, Eve is passive and randomly
located in the room, which leads Eve’s CSI unavailable at
Alice. Then, the secrecy rate in (35) cannot be calculated
by Alice and therefore cannot be optimized directly. Without
loss of generality, we assume that Eve is uniformly randomly
located on the work plane in the room. Alternatively, we can
choose the average secrecy rate as the optimization objective,
which can be formulated as
max
M∈DN×N
EHE,GERs√M (46a)
s.t. 0≤[M]i,i ≤1,i=1,2,...,N. (46b)
However, it is hard to calculate and optimize the expectation
in (46a) mathematically. Therefore, we utilize a sub-optimal
but tractable optimization approach similar to [22], by substi-
tuting (46a) with the secrecy rate under the average channel
gain of Eve. Then the optimization problem (46) is trans-
formed into (47), shown at the bottom of the page, where
¯
HEE{HE}and ¯
GEE{GE}denote the average
channel matrices of Eve, which can be obtained by averaging
the channel matrices of numerous sampling locations on the
work plane. The sampling locations are uniformly selected
since Eve is uniformly randomly located as assumed earlier.
Similar to the approximating and solving approach of the
problem (36) in Section IV-A, the problem (47) can also be
solved using Algorithm 1 in which it suffices to replace HE
and GEwith ¯
HEand ¯
GE, respectively. We denote the final
solution as M∗
UK.
C. Relaxation for Upper-Bounding Achievable Secrecy Rate
As mentioned before, the proposed AS scheme is
sub-optimal to the original optimization problem (33). To eval-
uate the gap in the secrecy rate between the proposed
sub-optimal scheme and the theoretical optimal value, we now
provide a relaxed solution to (33) for upper-bounding the
optimal secrecy rate achieved by precoding.
The precoding matrix constraint in (33b) is equivalent to
F∞≤1⇐⇒ $
$
$%FT&:,i$
$
$1≤1,∀i=1,2,...,N. (48)
Then using the norm equivalence theorem [36], we have
$
$
$%FT&:,i$
$
$1≤1=⇒$
$
$%FT&:,i$
$
$2='(FFT)i,i ≤1.(49)
Hence the constraint F∞≤1can be relaxed as
*[FFT]i,i ≤1. We define a symmetric semidefinite matrix
W∈RN×Nas WFFTand substitute Winto the original
problem (33) which then can be relaxed as
max
W∈RN×NRs√W(50a)
s.t. W0(50b)
0≤[W]i,i ≤1,i=1,2,...,N, (50c)
Clearly, (50) has a convex feasible set and therefore can also
be solved using the approximating and solving procedure of
max
M∈DN×N
1
2log2⎡
⎢
⎢
⎣12σ2
EIN+A2¯
GEˇ
M¯
GT
E
1
N6
πeA2h2
B,0|M|1
N+12σ2
BIN+A2GBˇ
MGT
B
1
N
12σ2
BIN+A2GBˇ
MGT
B
1
N12σ2
EIN+A2¯
HEM¯
HT
E+A2¯
GEˇ
M¯
GT
E
1
N
⎤
⎥
⎥
⎦(47a)
s.t. 0≤[M]i,i ≤1,i=1,2,...,N, (47b)
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11187
the problem (36) in Section IV-A. We denote the final solution
as W∗
R.
Due to the constraint relaxation, the feasible set of
the relaxed problem (50) includes that of the original
problem (33). Therefore, the optimal value of (50), i.e.,
Rs(W∗
R), can be regarded as an upper bound on the secrecy
rate achieved by precoding, i.e., Rs(F).
V. U PPER BOUND ON SECRECY CAPACITY
Recall that the achievable secrecy rate optimized above is
just a lower bound on the secrecy capacity. In this section,
we further provide an upper bound on the secrecy capacity.
The amplitude constraint in (8) can be relaxed as the average
power constraint as follows
|xk|≤A
2=⇒E{x2
k}≤A2
4.(51)
As discussed earlier, the noises at Bob and Eve are
both approximated by Gaussian variables. For such a
power-constrained Gaussian wiretap channel, the secrecy
capacity is the difference between the capacities of legitimate
and eavesdropping channels, where the Gaussian input is
optimal [37]. We denote the covariance matrix of xas K
E{xxT}, then the secrecy capacity under block transmission
in (29) can be upper-bounded by
Cs≤max
K
1
2Nlog2|Va r {yB}|
|Var {nB}| −log2|Va r {yE}|
|Var {nE}|
=max
K
1
2Nlog2σ2
BIN+HBKHT
B+GBˇ
KGT
B
σ2
BIN+GBˇ
KGT
B
−log2σ2
EIN+HEKHT
E+GEˇ
KGT
E
σ2
EIN+GEˇ
KGT
E
≈max
K
1
N[g1(K)−g2(K)] ,(52)
where the definition of ˇ
Kis similar to that of ˇ
Fby replacing
Fwith Kin (24). g1(K)and g2(K)are given by
g1(K)=1
2log2
σ2
BIN+HBKHT
B+
γ
k=1
GB,kKGT
B,k
+1
2log2
σ2
EIN+
γ
k=1
GE,kKGT
E,k
,(53)
g2(K)=1
2log2
σ2
EIN+HEKHT
E+
γ
k=1
GE,kKGT
E,k
+1
2log2
σ2
BIN+
γ
k=1
GB,kKGT
B,k
,(54)
both of which are concave functions of K. Similar to (39),
the approximation deviation in (52) can be neglected as long
as γis large enough. Hence we can regard (52) as the actual
secrecy capacity upper bound with negligible deviation.
Since the average power constraint E{x2
k}≤A2/4is
equivalent to [K]i,i ≤A2/4for i=1,2,...,N, the maximal
value of (52) can be obtained by an optimization problem as
follows
max
K∈RN×Ng1(K)−g2(K)(55a)
TAB L E I
SIMULATION PARAMETERS
s.t. K0(55b)
0≤[K]i,i ≤A2
4,i=1,2,...,N, (55c)
which is a difference of convex programming problem and can
be solved using CCP algorithm as mentioned in Section IV.
We denote the optimal value as Cup
scorresponding to an
upper bound on the secrecy capacity under block transmission.
Specially when block length N=1, the matrix Kdegenerates
into a scalar and Cup
scan also be regarded as an upper bound
on the secrecy capacity under symbol transmission.
VI. NUMERICAL RESULTS
In this section, we present some numerical results to illus-
trate how the ISI introduced by multipath reflections affects the
secrecy performance of the indoor VLC wiretap systems. The
room configuration and simulation parameters are summarized
in Table I. We assume that the wall reflectivity is identical
and set it to ρ=0.75. A Cartesian coordinate system is used
as shown in Fig. 1 to identify the locations with the units of
meters. Since both Bob and Eve are located on the work plane
with the same height of 0.85 m, we omit the z-coordinate
of the locations of two receivers for brevity hereafter of this
section. We assume that two receivers have the same AWGN
power σ2
B=σ2
E=σ2due to the same receiving strategy, and
σis set to 10 log10 σ2=−98.82 dBm.
A. Indoor VLC Channels Under Multipath Reflections
Fig. 4(a) and Fig. 4(c) plot the continuous-time optical CIRs
h(t)(Sand Rin h(t;S,R)are omitted) at different receiving
locations on the work plane. The proportions of LoS and NLoS
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11188 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
Fig. 4. Optical continuous-time CIR h(t)and discrete-time channel filter
taps {hk}at different receiving locations on the work plane.
Fig. 5. PDFs of the ISI noise nISI
Bwith uniformly i.i.d. data symbols {si}
with block length N=32and amplitude constraint 20 l og10 A=20dB.
components are also drawn in these two sub-figures. Then
by sampling the continuous function in (7), the discrete-time
channel filter taps {hk}are derived and depicted as circle lines
as shown in Fig. 4(b) and Fig. 4(d). It can be observed that
the proportion of LoS component is higher when the receiver
is closer to the transmitter. Moreover, the channel filter taps
{hk}keep decreasing as kincreases, and approximate to zero
when k>8. Hence we can set the maximum discrete delay
spread as L=8, and have γ 8
N!which is large enough
for negligible approximation deviations in (39) and (52).
Fig. 5 shows the PDFs of ISI noise nISI
Bin (25) with
uniformly i.i.d. data symbols {si}. The block length is set to
N=32and amplitude constraint is set to 20 log10 A=20dB.
Bob is located at (2.5, 2.5), and Eve is located at (0.5, 0.5).
The PDFs are approximated using Monte Carlo method by
averaging 106samples of signals. Fig. 5(a) corresponds to
the signals without precoding, i.e., F=IN, while Fig. 5(b)
Fig. 6. Secrecy capacity bounds under symbol transmission versus amplitude
constraint Awith NLoS components.
Fig. 7. Secrecy capacity bounds under symbol transmission versus amplitude
constraint Awith and without NLoS components.
corresponds to the signals with precoding in the case of
known Eve’s CSI, i.e., F=M∗
K. Note that %nISI
B&1without
precoding in Fig. 5(a) is equivalent to the ISI noise under
symbol transmission in (9). For comparison, the Gaussian
distribution function with the same variance as %nISI
B&iis also
drawn in each sub-figure. We observe that the two functions
agree very well with each other, validating that the ISI noise
can indeed be approximately regarded as Gaussian random
vectors, just as discussed in Section III.4
B. Secrecy Performance
Fig. 6 and Fig. 7 show the secrecy capacity bounds under
symbol transmission versus amplitude constraint A.Bobis
located at (2.5, 2.5) and Eve is located at (0.5 0.5). The
4We also simulate the PDFs of ISI noise of Bob and Eve at other locations
and obtain the simulation results all supporting this conclusion. For brevity,
they are not presented in this paper.
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11189
Fig. 8. Achievable secrecy rates of the AS scheme with known Eve’s CSI
versus amplitude constraint Afor various block lengths Nand Eve’s locations.
unclosed-form and closed-form lower bounds are obtained
from (15) (the number of constellation points is set to M=
32) and (19) respectively. From Fig. 6 we can see that the
unclosed-form lower bound is tighter than the closed-form
one and can reach the upper bound under a high transmit
amplitude constraint. Fig. 7 depicts that with the impact of
multipath reflections, the secrecy performance is misestimated
with a large deviation under the non-NLoS assumption. For
comparison, we also simulate the secrecy rates in (11), i.e.,
[I(sk;yB,k)−I(sk;yE,k)]+, by Monte Carlo method using
106samples of uniformly distributed symbols, denoted by
the dashed lines in Fig. 7. It is shown that the uniformly
distributed transmitted signals cannot achieve the upper bound
under a high transmit amplitude constraint. The reason is that
in the presence of multipath reflections, the received SNR
does not keep increasing as the signal power increases due to
the simultaneous increment of the ISI power. Only for a high
enough SNR, the uniformly distributed signal can achieve the
capacity of amplitude-constrained channel.
Fig. 8 shows the achievable secrecy rates obtained by the
AS scheme with known Eve’s CSI, i.e., [Rs(M∗
K)]+,versus
amplitude constraint Afor different block lengths Nand
Eve’s locations. Bob is located at (2.5, 2.5). The secrecy
rates are derived from (32). Note that block transmission with
N=1is equivalent to symbol transmission in (19), just as
discussed in Section III-B. For comparison, we also plot the
resulting curves without impact of ISI denoted by the dashed
lines in Fig. 8, which is derived by setting the ISI noise
in (19) to zero.5It is shown that the secrecy performance is
severely degraded due to the ISI. As Nincreases, the secrecy
performance is enhanced and tends towards the case without
ISI. It is due to the facts that block transmission can alleviate
the impact of ISI within the same block as shown in Fig. 3(b)
and longer block leads to a better alleviation.
5Under non-ISI assumption or non-NLoS assumption, the communication
performance is uncorrelated with {hk}∞
k=1 and is only correlated with h0.
However, these two assumptions are not exactly equivalent since h0depends
on both LoS and NLoS components.
Fig. 9. Achievable secrecy rates obtained from the AS scheme with known
Eve’s CSI, the AS scheme with unknown Eve’s CSI and the relaxed solution
versus amplitude constraint Afor block length N=4and various Eve’s
locations.
Fig. 10. Achievable secrecy rates obtained from the AS scheme with known
Eve’s CSI, the AS scheme with unknown Eve’s CSI and the relaxed solution
versus block length Nfor amplitude constraint 20 log10 A=20dB and
various Eve’s locations.
Fig. 9 and Fig. 10 show the achievable secrecy rates
obtained by the AS scheme with known Eve’s CSI, i.e.,
[Rs(M∗
K)]+, the AS scheme with unknown Eve’s CSI, i.e.,
[Rs(M∗
UK)]+, and the relaxed solution, i.e., [Rs(W∗
R)]+,
versus amplitude constraint Aand block length N, respec-
tively. In Fig. 9, the block length is set to N=4,andin
Fig. 10 the amplitude constraint is set to 20 log10 A=20dB.
In both two figures, Bob is located at the same position as
(2.5, 2.5). We also plot the secrecy rates without precoding,
i.e., [Rs(IN)]+, in Fig. 10, and the upper bounds on secrecy
capacity, i.e., Cup
s, in both Fig. 9 and Fig. 10. It can be
observed that the AS scheme can efficiently enhance the
secrecy performance under the impact of ISI. The case with
known Eve’s CSI exhibits a slightly better performance than
that with unknown Eve’s CSI. Furthermore, the difference
between [Rs(M∗
K)]+and [Rs(W∗
R)]+is small, which
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11190 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 21, NO. 12, DECEMBER 2022
Fig. 11. Amplitude scaling factor
√mi
N
i=1 of the AS scheme with known
and unknown Eve’s CSI versus time slot index ifor various block lengths N
with amplitude constraint of 20 log 10 A=20 dB.
Fig. 12. Achievable secrecy rates obtained from the AS scheme with known
Eve’s CSI versus Eve’s locations for different block lengths Nwith amplitude
constraint of 20 log10 A=20 dB.
demonstrates that the proposed AS scheme can achieve a
secrecy performance very close to the optimal solution. As N
increases, [Rs(M∗
K)]+,[Rs(M∗
UK)]+,and[Rs(W∗
R)]+
tend to be almost identical since the ISI impact is significantly
alleviated as mentioned before. Hence less performance gain
can be achieved by the precoding schemes.
Fig. 11 shows the amplitude scaling factor √miN
i=1
obtained from the AS scheme with known and unknown Eve’s
CSI versus time slot index ifor different block lengths Nwith
amplitude constraint of 20 log10 A=20dB. Bob is located
at (2.5, 2.5) and Eve is located at (0.5, 0.5). It is shown that
the AS scheme tends to only reduce the amplitudes of the
last min(L, N )symbols, since the ISI impact mainly comes
from the previous transmitted Lsymbols. By this means the
ISI impact can be alleviated and the secrecy performance is
enhanced.
Fig. 12 and Fig. 13 show the achievable secrecy rates
obtained by the AS scheme with known Eve’s CSI versus
Eve’s and Bob’s locations, respectively. Bob is located at
(2.0, 2.0) in Fig. 12 and Eve is located at (0.5, 0.5) in
Fig. 13. In both two figures, the amplitude constraint is set
Fig. 13. Achievable secrecy rates of the AS scheme with known Eve’s CSI
versus Bob’s locations for various block lengths Nwith amplitude constraint
of 20 log10 A=20dB.
to 20 log10 A=20dB. It is observed that the secrecy rates
decrease as Eve gets closer to Alice while increase as Bob
gets closer to Alice. When Eve is too close or Bob is too far
w.r.t. Alice, the eavesdropping channel is much better than the
legitimate channel, which decreases the secrecy rates to zero
and cannot guarantee the secure transmission.
VII. CONCLUSION
In this paper, we investigated the secrecy performance of
SISO indoor VLC wiretap systems in the presence of multipath
reflections. We derived the lower and upper bounds on the
secrecy capacity under amplitude constraint for both policies
of symbol- and block-based transmission. A low-complexity
(O(Nlog N)) AS scheme was proposed to enhance the
secrecy performance relying on block transmission policy for
known and unknown eavesdropper’s CSI. Numerical results
suggested that the secrecy performance is severely degraded
by the ISI imposed by the multipath reflections and this
degradation can be alleviated with the aid of long-block based
transmission. In addition, the proposed AS scheme can achieve
a near-optimal secrecy performance very close to the optimal
solution especially for a very long block length. In this case,
the ISI impact is significantly alleviated and less precoding
gain can be achieved.
It would be interesting to extend this work to multiple-
eavesdropper systems, where the secrecy performance can be
evaluated by calculating the secrecy rate of each eavesdropping
channel using the calculation methods in this paper, and then
combining into the overall secrecy performance measures.6
However, it is hard to optimize the multiple-eavesdropper
secrecy performance measures, which deserves more in-depth
research in our future works.
APPENDIX
PROOF OF THE ACHIEVABLE SECRECY RAT E I N (32)
By taking the uniform distribution of the transmitted signal
as input, the secrecy capacity per symbol can be lower-
6Typical multiple-eavesdropper secrecy performance measures include the
max-min fairness, harmonic mean, proportional fairness, weighted fairness,
and etc. However, the optimization problems subject to these measures
are more complicated than that subject to single-eavesdropper secrecy rate
according to [10].
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YANG et al.: PHYSICAL-LAYER SECURITY FOR INDOOR VLC WIRETAP SYSTEMS UNDER MULTIPATH REFLECTIONS 11191
bounded by
Cs≥1
N[I(s;yB)−I(s;yE)]
=1
N[h(yB)−h(yB|s)−h(yE)+h(yE|s)]
=1
N[h(HBFx +nB)−h(nB)
−h(HEFx +nE)+h(nE)] .(56)
For the noises, we define that ΦBVa r {nB}and ΦE
Var {nE}. Hence we have
h(nB)=N
2log2(2πe)+ 1
2log2|ΦB|,(57)
h(nE)=N
2log2(2πe)+ 1
2log2|ΦE|.(58)
Then h(HBFs +nB)can be lower-bounded using the
entropy-power inequality as follows
h(HBFs +nB)
≥N
2log2(22
Nh(HBFs)+22
Nh(nB))
=N
2log2(22
N[log2|HBF|+h(s)] +22
N[N
2log2(2πe)+ 1
2log2|ΦB|])
=N
2log2(A2h2
B,0|F|2
N+2πe |ΦB|1
N).(59)
h(HEFs +nE)can be upper-bounded by the differential
entropy of a Gaussian random vector with covariance matrix
Var {HEFs +nE}as follows
h(HEFs +nE)
≤N
2log2(2πe)+1
2log2|Va r {HEFs +nE}|
=N
2log2(2πe)+1
2log2
A2
12 HEFFTHT
E+ΦE.(60)
Then, by substituting (57), (58), (59) and (60) into (56),
we have
Cs≥1
2log2⎡
⎣
6|ΦE|1
NA2h2
B,0|F|2
N+2πe |ΦB|1
N
πe |ΦB|1
NA2HEFFTHT
E+12ΦE
1
N⎤
⎦.
(61)
Finally, using the expressions of Var {nB}and Var {nE}in
(30) and (31), we can obtain the lower bound in (32).
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Fan Yang (Student Member, IEEE) received the
B.Eng. degree in telecommunication engineering
from Sun Yat-sen University, Guangzhou, China,
in 2019. He is currently pursuing the M.Eng.
degree in electronics and communication engineer-
ing with the Department of Electronic Engineering,
Tsinghua University, Beijing, China. He is also a
Research Assistant with the Modern Communica-
tion Laboratory, Shenzhen International Graduate
School, Tsinghua University. His research inter-
ests include physical-layer security and visible light
communications.
Jingjing Wang (Senior Member, IEEE) received
the B.S. degree (Hons.) in electronic information
engineering from the Dalian University of Technol-
ogy, Liaoning, China, in 2014, and the Ph.D. degree
(Hons.) in information and communication engi-
neering from Tsinghua University, Beijing, China,
in 2019. From 2017 to 2018, he visited the Next
Generation Wireless Group chaired by Prof. Lajos
Hanzo, University of Southampton, U.K. He is
currently an Associate Professor at the School of
Cyber Science and Technology, Beihang University.
He has published over 100 IEEE journals/conference papers. His research
interests include AI enhanced next-generation wireless networks and swarm
intelligence and confrontation. He was a recipient of the Best Journal Paper
Award of IEEE ComSoc Technical Committee on Green Communications &
Computing in 2018 and the Best Paper Award of IEEE ICC and IWCMC
in 2019.
Yuhan Dong (Senior Member, IEEE) received the
B.S. and M.S. degrees in electronic engineering from
Tsinghua University, Beijing, China, in 2002 and
2005, respectively, and the Ph.D. degree in elec-
trical engineering from North Carolina State Uni-
versity, Raleigh, NC, USA, in 2009. Since January
2010, he has been with the Shenzhen International
Graduate School, Tsinghua University, where he
is currently an Associate Professor and a mem-
ber of the Modern Communication Laboratory. His
research interests include wireless communications
and networking, machine learning and optimization, and optical wireless
communications. He was a recipient of the 2008 IEEE GLOBECOM Best
Paper Award.
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