Artificial-noise aided secure transmission over visible light communication system under coexistent passive and active eavesdroppers

Ge Shi; Yong Li; Yong Li; Wei Cheng; Wei Cheng; Xiang Gao; Limeng Dong

doi:10.1364/OE.448860

1. Introduction

In the next generation of wireless systems, visible light communication (VLC) has been exploited as a promising technology to achieve broadband connectivity. High-rate connectivity can be guaranteed in sensitive environments by using VLC, such as hospitals, chemical plants, and airports, without causing any electromagnetic interference. Recently, VLC has been widely investigated in various indoor and outdoor scenarios [1,2]. Since the light cannot penetrate through walls, VLC was claimed to be more secure than radio frequency (RF) communications [3]. However, due to its broadcasting characteristic, a malicious eavesdropper (Eve) may exist inside public areas or beside large windows in the coverage area. The Eve intends to gather other users’ (UEs) confidential data and decode their private information. With the implementation of broadband connectivity (such as the internet of things and device-to-device), traditional cryptographic techniques may be inefficient or insufficient since the consumption of resources and signaling overhead for private key exchanges can be significant. Thus, physical layer security (PLS) has emerged as a promising approach to enhance the security by exploiting the characteristics of wireless channels. The advantages of employing PLS techniques are on two-folds. First, PLS techniques do not rely on computational complexity. As a result, even if the Eves are equipped with powerful computational devices, reliable communication can still be achieved. Second, PLS techniques can be used to safeguard secure transmission directly or generate the distribution of private keys for cryptographic techniques [4].

1.1 Related works

Many existing PLS techniques for VLC systems, such as beamforming [5–7], artificial noise (AN) [8–10], light emit diode (LED) selection [11], and relay-assisted transmission [12], were often designed to degrade either a passive Eve or an active Eve in one scenario. However, the broadcasting characteristic of VLC systems provides the possibility of being overheard from both passive and active Eves, especially in large public places. The difference between active and passive Eves lies in whether their channel state information (CSI) is known at a central control unit’s (CCU) side. Differing from active Eves, passive Eves rarely send any feedback through uplinks to the CCU, and thus their CSI is often imperfect, i.e., partially or completely unknown. If the CCU uses the imperfect CSI directly to design a transmit strategy, UEs may experience severe secrecy outages or not reach the required quality of service (QoS) level. Thus, in this case, robust transmit strategies were designed based on CSI error models, and the robustness of the strategies was proved by better secrecy performance of VLC systems compared to non-robust strategies [5,9,10,13].

In [5], a suboptimal solution based on zero-forcing (ZF) beamforming was utilized to suppress legitimate messages to the nullspace of active Eve’s CSI in order to eliminate its information leakage rate. A passive Eve with its CSI partially known was also considered, and the CSI error of the passive Eve originated from its uncertain position. A robust beamforming approach was devised to improve the worst-case secrecy rate under a bounded position of the Eve. In [6], the expression of the secrecy rate was investigated under the assumption of the input signaling following truncated generalized normal distribution, and an optimal beamforming was selected based on the eigenvector associated with the largest eigenvalue of the difference between the Gram matrices of legitimate and Eve’s channel vectors. In [7], several typical secrecy metrics were considered in a multi-user multi-input multi-output (MISO) broadcast channel, and a suboptimal precoding matrix was designed by using convex-concave procedure (CCP).

When the secrecy requirement is high or when Eve’s reception quality is better than UE’s, AN and cooperative jamming become more attractive for protecting VLC systems. In practical systems, Eve can be equipped with better devices than UEs, such as more photodiodes (PDs), higher gain optical lens, and even reception beamforming achitectures. However, with powerful devices, the Eve still cannot distinguish between random confidential signals and AN signals [14]. In our previous work [8], we proposed an optimal transmit strategy for a secrecy rate maximization (SRM) by emitting AN into the directions of active Eves with multi-PDs, and the optimality of beamforming was validated for the confidential transmission by using Karush–Kuhn–Tucker conditions. For passive Eves without CSI known, isotropic AN-aided method was often conducted to interfere with all directions except for UEs [15,16]. Assuming that the location of a passive Eve is known within a bounded area in [9], a discretization approach was applied to count all possible places of the Eve into an SRM problem, and a robust beamforming was designed by nulling out these positions. A non-orthogonal multiple access (NOMA)-enabled VLC system was investigated in [17] under the bounded CSI of both UEs and Eve, and a robust AN-aided beamforming was introduced to minimize the total transmit power while considering the QoS requirements of the UEs and the maximum data leakage tolerance of the Eve. In [10], a suboptimal AN-aided beamforming strategy was proposed to enhance the average secrecy rate of VLC systems with the presence of a randomly located Eve. The upper and lower bounds of the average secrecy rate were derived to replace the un-tractable expression of the average secrecy rate in the optimization problem. A LED selection strategy was investigated in [11] to reduce the computation complexity of a joint jamming and beamforming transmission when a large number of LED arrays was considered, which provided a practical and near-optimal solution.

Unfortunately, if both active and passive Eves coexist, these PLS schemes above are unable to safeguard VLC systems successfully or efficiently. A relaxed ZF-based beamforming strategy was designed in [18] to minimize the secrecy outages caused by passive Eves with Poisson point process distribution while simultaneously satisfying the target constraints on the signal to noise ratios (SNRs) of UE and active Eves. The outage area (where Eve’s SNR is higher than UE’s SNR) was discretized by Lloyd’s algorithm and minimized by applying CCP. However, when the reception quality of any Eve is better than that of UE or any Eve is nearby UE, only enhancing the SNR of UE by the beamforming strategy is not an efficient way to increase the secrecy performance anymore. Thus, in [19], the same author proposed a cooperative beamforming and jamming transmission strategy to maximize the signal to interference and noise ratio (SINR) of the legitimate UE, while suppressing the SINRs of the active and passive Eves.

1.2 Motivation and contributions

In this coexistent scenario, passive and active Eves are different in the availability of the CSI at the CCU. Compared to active Eves, it would be challenging to know the accurate CSI of passive Eves. The above research works introduced the CSI error of the passive Eves by assuming that their random positions follow a Poisson Point Process [18,19]. In this paper, we directly consider the Gaussian CSI error of the passive Eves, which is a popular model in RF systems [13,20]. The extra thermal noise is approximated as Gaussian distribution [21], which has been adopted as the CSI uncertainty in VLC systems [22–24]. In [22], a joint optimization for the precoder and equalizer was designed to minimize the robust minimum mean square error in a multi-input multi-output VLC system with the Gaussian channel uncertainty. In [23], the impact of the Gaussian CSI error on the bit error rate (BER) performance was quantified in NOMA-based VLC systems. Thus, It is natural to formulate the CSI uncertainty as a Gaussian model since it may originate from various sources such as the extra thermal noise, position derivation, and so on. However, passive Eves with the Gaussian CSI uncertainty has not been addressed in the security of VLC systems. With this uncertainty model, a secrecy-outage constraint is necessary to be considered to guarantee a low probability that any passive Eve drives the secrecy performance lower than the worst secrecy rate. The CSI of the active Eves is assumed to be perfect since it can calibrate or compensate their CSI through frequent feedback. In this case, a SRM problem becomes a secrecy-outage-constrained SRM problem. Unfortunately, the probabilistic constraint generally has no closed-form expression and is unlikely to be easily handled in an exact way. Thus, in this paper, safe approximations are resorted to making the problem solvable.

The pioneering research works [18,19] on the co-existent scenario focused on enhancing security with the perspective of SNR. In this paper, we consider a worse-case co-existent scenario containing Eves with multi-PDs, and an AN-aided transmission strategy is proposed to maximize the worst-case secrecy rate. The main contributions of this article are summarized as follows:

• An AN-aided transmission scheme is designed for securing a MISO VLC system, taking into account the secrecy outages of the passive Eves, total power budget, and the peak amplitude limit of LED arrays. The uncertainty of the passive Eves’ CSI is derived from the extra thermal noise and follows a Gaussian distribution. Joint optimization of confidential information covariance and AN covariance is considered without any restriction for their spatial patterns.
• Since the probabilistic constraint of the passive Eves do not possess any tractable form, a conservative approximation is presented to replace them with the worst-case secrecy constraints with spherically-bounded CSI errors. However, the resultant robust SRM problem is still non-convex and challenging to solve. Thus, we first apply semidefinite relaxation (SDR) to transform the problem into a convex one and then use $\cal S$-Procedure to convert the infinite matrix inequalities into finite ones. A Golden search-based algorithm is proposed to solve the transformed problem with fast convergence and low complexity.
• For different indoor scenarios, such as the only-active Eve scenario, the only-passive Eve scenario, and the coexistent scenario, secrecy rate distributions varying with the positions of the UE are investigated in the simulations. The simulation results show that the minimum secrecy rate depends on the positions of the Eves. Compared to the active Eves, the average secrecy rate is degraded by the passive Eves more.
• Furthermore, in order to validate the robustness of the proposed AN-aided method, we compare its performance with three other methods, including a non-robust AN-aided method, the isotropic AN-aided method, and a non-AN-aided method, i.e., the relaxed ZF beamforming method, by randomly selecting the positions of the Eves and the UE. The simulation results show that the proposed robust AN-aided method outperforms the above mentioned methods, especially when the power budget is high.

The rest of this paper is organized as follows. The system model description and problem formulation are given in Section II. In Section III, a conservative reformulation is provided to transform the probabilistic-outage constraint. Section IV proposes an optimization approach to handle the resultant robust SRM problem. In Section V, simulation results show the secrecy performance for different scenarios and compare the proposed method with other benchmark methods. Finally, Section VI concludes the paper.

Notation: The following notation is adopted throughout the paper. The set of $n$-dimensional, real-valued numbers is denoted by $\mathbb {R}^{n}$; $\{\cdot \}^{T}$, $\text {Tr}\{\cdot \}$, $\text {rank}\{\cdot \}$, and $\text {det}\{\cdot \}$ represent the transpose, trace, rank, and determinant of a matrix, respectively. $\mathbf {I}_n$ represents an identity matrix of size $n$. $||\cdot ||_{\text {max}}$ denotes the maximum norm for a matrix. $A\succeq 0$ means that $A$ is a positive semidefinite matrix.

2. System model

We consider a MISO-VLC downlink scenario in Fig. 1, wherein $M$ LED arrays send confidential messages to a single-PD UE in the presence of $K$ Eves. Each Eve is equipped with $L_{k}$ PDs to perform the optical-to-electricity conversion. LED arrays are connected to a CCU through backhaul networks. The coefficients of the transmit covariance matrix are designed by the CCU and allocated to the LED arrays to form a cooperative transmission strategy. Assuming that $K_p$ passive Eves and $K_a$ active Eves exist in the same illumination area with the legitimate UE. The active Eves are registered at the CCU and intend to overhear the UE. The parameters of the active Eves (such as position, type, and optoelectronic characteristics) are known by the CCU, and thus the CSI of active Eves is obtained by substituting above parameters to (3). However, the CSI may become out-of-date if the Eve moves to a new position. In order to update the positions of the active Eves, a received-signal-strength-based localization technique is utilized to measure the distances between the active Eves and the LED arrays. An uplink medium, such as VLC [25], and RF [26], is applied to transfer the distance information to the CCU. In CCU, the position of the Eve is calculated by using least-squares method [27] and updated to estimate the newest CSI. The CSI of the passive Eves is assumed to be partially known since their parameters are not registered in the CCU. Besides, the passive Eves rarely send feedback to the CCU and update their positions. The received signals of the UE and the $k$-th Eves are expressed as

(1)$$\begin{aligned}y(t)&=\mathbf{h}^{T} \mathbf{s}(t)+n(t), \end{aligned}$$

(2)$$\begin{aligned}\mathbf{y}_k(t)&=\mathbf{G}^{T}_k \mathbf{s}(t)+\mathbf{n}_{e,k}(t), k\in\cal K, \end{aligned}$$

where ${\cal K} \triangleq {\cal K}_p \cup {\cal K}_a=\{1,\dots,K\}$, wherein ${\cal K}_p$ and ${\cal K}_a$ denote the indices for the active Eves and the passive ones, respectively. $\mathbf {h} = \{h_{1}, h_{m}, \dots, h_{M}\}^{T} \in \mathbb {R}^{M}$ represents the channel gain response vector from the LED arrays to the UE. $\mathbf {G}_k = \{\mathbf {g}_1, \mathbf {g}_m, \dots, \mathbf {g}_M\}^{T}\in \mathbb {R}^{M\times L_k}$ is the channel gain response matrix from the LED arrays to the $k$-th Eve, where $\mathbf {g}_m = \{g_{m,1}, g_{m,l_k} ,\ldots, g_{m,L_k}\}^{T}$ denotes the channel gain vector from the $m$-th LED array to the $k$-th Eve. $n_b$ and $\mathbf {n}_{e,k}$ are independent and identically distributed circularly symmetric Gaussian noise with variances whose samples are ${\cal N}(0,\delta _b^{2})$ and ${\cal N}(0,\delta _{e,k}^{2}\mathbf {I}_{L_k})$, respectively. In this paper, only line-of-sight (LoS) link is considered since the signal strength of LoS component is far higher than that of non-line-of-sight component [28]. The LoS channel gain from the $m$-th LED array to the UE is expressed as [29]

(3)$$\begin{aligned} h_{m}= \begin{cases} \frac{(\alpha+1) A_{R}q^{2}}{2\pi d_{m}^{2}\sin^{2}(\varphi_{m})}\cos(\phi_{m})^{\alpha}\cos(\varphi_{m} )g_{of}, & 0\leq \varphi_{m} \leq \Theta,\\ 0, & \varphi_{m} \geq \Theta, \end{cases} \end{aligned}$$

where $\alpha =\frac {-1}{\log _{2}\cos {\theta _{1/2}}}$ denotes the Lambertian emission order, $\theta _{1/2}$ and $A_{R}$ denote the half-intensity radiation angle of the LED arrays and the area of the UE’s PD, respectively. $d_{m}$ represents the Euclidean distance from the $m$-th LED array to the UE. $\phi _{m}$ and $\varphi _{m}$ are the irradiance angle and incidence angle, respectively. $g_{of}$ is the gain of the optical filter. $q$ and $\Theta$ represent the refractive index and the field-of-view (FOV) of the PD. The channel gain $g_{m,l_k}$ from the $m$-th LED array to the $l_k$-th PD of the $k$-th Eve is similar to (3).

Fig. 1. System model.

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In order to enhance the secrecy performance, we adopt an AN-based scheme to degrade the reception of the multiple Eves. In such a scheme, the transmitted signal $\mathbf {s}(t)$ is characterized by the summation of the confidential information and AN, i.e.,

(4)$$\mathbf{s}(t)=\mathbf{x}(t)+\mathbf{z}(t)\in \mathbb{R}^{M},$$

where $\mathbf {x}(t)$ and $\mathbf {z}(t)$ denote the confidential signal and the AN signal, respectively. The confidential signal vector is assumed to follow a Gaussian distribution $\mathbf {x}(t)\sim {\cal N}(0,\mathbf {W})$, where $\mathbf {W}$ is the transmit covariance matrix. We assume $\mathbf {z}(t)\sim {\cal N}(0, \mathbf\Sigma)$, where $ {\mathbf\Sigma }$ is the AN covariance matrix. Although the exact secrecy capacity of the VLC channel remains unknown due to the peak amplitude constraint, a tight achievable lower bound on the secrecy capacity can be obtained by the minimum gap between the VLC channel capacity of the legitimate link and those of $K$ Eves’ links as follows:

(5)$$R_{s}=\min_{k\in {\cal K}}[R_b-R_{e,k}]^{+},$$

where $[x]^{+}=\max (0,x)$. $R_b$ and $R_{e,k}$ are the channel capacity of the UE’s link and the $k$-th Eve’s link, respectively. The achievable rates of the $\text {UE}$ and the $k$-th Eve are given by

(6)$$\begin{aligned} R_b&=\log_2(1+\frac{\mathbf{h}^{T}\mathbf{Wh}}{(\delta_{b}^{2}+\mathbf{h}^{T}\mathbf{\Sigma h})}), \end{aligned}$$

(7)$$\begin{aligned}R_{e,k}&=\log_2\det(\mathbf{I}_{L_k}+\frac{{\mathbf{G}_k}^{T}\mathbf{WG}_k}{(\delta_{e,k}^{2}\mathbf{I}_{L_k}+{\mathbf{G}_k}^{T}\mathbf{\Sigma G}_k)}). \end{aligned}$$

To formulate a SRM problem under the imperfect CSI, it is necessary to describe the CSI error model. In this paper, we assume that the CSI of the passive Eves is contaminated by the extra thermal noise. The addition of the thermal noise is generated by the low noise amplifier and mixers at receivers as well as other components, which appears as an increase of its variance [30]. The total effect of it to the CSI of the passive Eves is modeled as Gaussian distributed with zero-mean and variance $\sigma ^{2}$, which is a commonly adopted model for the channel uncertainty of passive Eves [20]. In this case, the estimated CSI of the $k$-th passive Eve is represented by

(8)$$\mathbf{G}_{k}=\hat{\mathbf{G}}_{k}+\mathbf \triangle \mathbf{G}_{k}, k \in {\cal K}_p,$$

where $\hat {\mathbf {G}}_{k}$ denotes the presumed CSI of $\mathbf {G}_{k}$. $\triangle \mathbf {G}_{k}\in \mathbb {R}^{M\times L_k}$ represents the CSI error matrix and is assumed to be independent of $\triangle \mathbf {G}_{j}$ for any $j$-th Eve’s link, $j\neq k$. The CSI error $\mathbf \triangle \mathbf {G}_{k}$ follows i.i.d Gaussian distribution,

(9)$$[ \triangle \mathbf{G}_{k}]_{m,l_k}\sim {\cal C }(0,\sigma^{2}), ~~\forall m,l_k, k \in {\cal K}_p,$$

where $[\triangle \mathbf {G}_{k}]_{m,l_k}$ denotes the $(m,l_k)$-th entry of the CSI error matrix. $\sigma$ is the error variance which denotes the expectation of the deviation between the presumed CSI and the estimated CSI. The CSI error of the passive Eves is unbounded, it may not be possible to obtain a perfectly safe design. Thus, an outage-probability of $(1-\gamma )\%$ is constrained to guarantee the robustness in the following SRM problem:

(10a)\begin{align} &\max_{\mathbf{W}\geq 0,\mathbf{\Sigma}\geq 0}~R_s\end{align}

(10b)\begin{align} \text{s.t.}~ &\Pr\{R_b-\max_{ k\in{\cal K}_p}R_{e,k}\geq R_s\}\geq 1-\gamma,\end{align}

(10c)\begin{align} &||\mathbf{W}+\mathbf{\Sigma}||_{\text{max}}\leq A^{2},\end{align}

(10d)\begin{align} &\text{Tr}(\mathbf{W+\Sigma})\leq P_{\text{max}},\end{align}

where $\gamma \in [0,1]$ is a given parameter specifying the probability that any passive Eve leads the worst-case secrecy rate to falling below $R_s$. $A$ and $P_{\text {max}}$ are the peak amplitude threshold and the total power budget for the LED arrays, respectively. The optical signals should be restricted by the peak amplitude constraint since the linear range of LEDs is limited.

3. Conservative formulation

Since $\mathbf {G}_{k}$ is independent to $\mathbf {G}_{j}$, $k\neq j$, the left-hand side of (10b) is equivalent to a multiplication of outage-probability constraints (11a) of all the passive Eves as follows:

(11a)$$\begin{aligned} &\prod_{k=1}^{K_p}\Pr\{R_b-R_{e,k}\geq R_s\}\geq 1-\gamma,\end{aligned}$$

(11b)$$\begin{aligned} \Leftarrow &\Pr\{R_b-R_{e,k}\geq R_s\}\geq (1-\gamma)^{\frac{1}{K_p}},k\in{\cal K}_p,\end{aligned}$$

The sufficient condition (11b) is derived to seperate the outage-probability constraints of each passive Eve in (11a). Any feasible points of the problem (10) satisfies (11b), and thus (11b) provides a conservative approximation for the constraint (10b). The main challenge of this problem lies in the objective function and the constraints (11b), which is impossible to have a tractable closed-form expression in this case. As a compromise, we consider an approximation of (11b) based on the Lemma as follows:

Lemma 1 Let $\triangle \mathbf {G}\in \mathbb {R}^{M\times L_k}$ be a continuous random matrix following certain statistical distribution and let $Q(\triangle \mathbf {G}): \mathbb {R}^{M\times L_k} \rightarrow \mathbb {R}$ be a function of $\triangle \mathbf {G}$. Let $r\geq 0$ be the radius of the ball $\{\triangle \mathbf {G}\arrowvert ~\Vert \mathbf \triangle \mathbf {G}\Vert ^{2}_F\leq r^{2}\}$ such that $\Pr (\Vert \triangle \mathbf {G}\Vert ^{2}_F\leq r^{2})\ge 1-\eta$ where $\eta \in (0,1]$. Then

(12)$$Q(\triangle \mathbf{G})\geq 0,~\forall \Vert \triangle \mathbf{G}\Vert^{2}_F\leq r^{2},$$

implies $\Pr (Q(\triangle \mathbf {G})\geq 0)\ge 1-\eta$.

The proof of the Lemma 1 is similar to the Lemma 1 in [31]. By applying Lemma 1 to (11b), we can see that the outage-probability constraints (11b) are satisfied whenever,

(13)$$R_b-R_{e,k}\geq R_s, \forall \Vert \triangle \mathbf{G}_{k}\Vert^{2}_F\leq r^{2}, k\in {\cal K}_p,$$

where the sphere boundary $r$ is chosen as follows:

(14)$$r=\sqrt{\frac{\sigma^{2}}{2}\Phi^{{-}1}_{\chi^{2}_{ML_{k}}} (1-\gamma)^{\frac{1}{K_p}}},$$

where $\Phi ^{-1}_{\chi ^{2}_{ML_{k}}}(\cdot )$ denotes the inverse cumulative distrubution function of a Chi-square random variable with $ML_{k}$ degrees of freedom. In (13), $(1-\gamma )^{\frac {1}{K_p}}\%$ realizations of $\triangle \mathbf {G}_{k}$ which satisfy (11b) lie in the sphere $r$. By applying the implications (11) and (13) to the constraint (10b), we obtain a safe approximation to the problem (10):

(15a)\begin{align} &\max_{\mathbf{W}\geq 0,\mathbf{\Sigma}\geq 0}~(R_b-\max_{ k\in{\cal K},\mathbf{G}_{k} \in {\cal B}_k}R_{e,k}) \end{align}

(15b)\begin{align}\text{s.t.}~&(10\mathrm{c}),\,(10\mathrm{d})\end{align}

where

(16)$$\begin{aligned}{\cal B}_k= \begin{cases} \{\mathbf{G}_{k}|\mathbf{G}_{k}=\hat{\mathbf{G}}_{k}+\mathbf ~\triangle \mathbf{G}_{k},\Vert \triangle \mathbf{G}_{k}\Vert^{2}_F\leq r^{2} \}, & \forall k \in {\cal K}_p,\\ ~\mathbf{G}_{k}, & \forall k \in {\cal K}_a, \end{cases} \end{aligned}$$

denotes the set of the admissible CSI for both active and passive Eves. Interestingly, the probabilistic constraint (10b) has been replaced by an infinite number of deterministic secrecy rate formulations with spherically bounded uncertainty matrix. Since the constraints (13) have the same form as the objective function, we rearrange them into the objective function by restricting the CSI based on the type of Eves. Note that if the Eves are all active, the problem (15) degrades to a previously investigated problem in [8].

4. SDR-based method for the problem

In this section, we proposed an SDR approach for handling the above problem. To start with, we assume that the optimal confidential transmission can be attained by using single-stream beamforming and define $\mathbf {W}=\mathbf {w}\mathbf {w}^{T}$. Meanwhile, we pose an equivalent transformation of the problem (15) by introducing a slack variable $\beta$

(17a)\begin{align} &\max_{\mathbf{W}\geq 0,\mathbf\Sigma\geq 0,\beta\geq 1}\log(1+\frac{\mathbf{h}^{T}\mathbf{Wh}}{\delta_{b}^{2}+\mathbf{h}^{T}\mathbf{\Sigma h}})-\log\beta\end{align}

(17b)\begin{align} \text{s.t.}~&\log\det(\mathbf{I}_{L_k}+\frac{{\mathbf{G}_k}^{T}\mathbf{WG}_k}{\delta_{e,k}^{2}\mathbf{I}_{L_k}+{\mathbf{G}_k}^{T}\mathbf{\Sigma G}_k})\leq \log\beta, ~\forall \mathbf{G}_{k} \in {\cal B}_k, k\in \cal K,\end{align}

(17c)\begin{align}&\text{rank} (\mathbf{W})= 1, (10\mathrm{c}),\,(10\mathrm{d}).\end{align}

where $\log \beta$ can be interpreted as the largest information leakage rate among all the Eves, and we can adjust the value of it by using $\beta$. In this problem, the challenging parts lie in the non-convex objective function (17a) and infinitely many constraints (17b). To overcome these difficulties, we first reformulate the fractional objective function (17a) by employing Charnes-Cooper transformation [32], specifically, we define

(18)$$\mathbf{W}=\frac{\bar{\mathbf{W}}}{\xi}, \mathbf\Sigma=\frac{\bar{\mathbf\Sigma}}{\xi}.$$

Now, the problem (17) is reformulated as follows:

(19a)\begin{align} &\Omega(\beta)=\max_{\bar{\mathbf{W}}\geq 0,\bar{\mathbf\Sigma}\geq 0,\beta\geq 1,\xi\geq 0}~~\xi\delta^{2}_{b}+\mathbf{h}^{T}(\bar{\mathbf{W}}+\bar{\mathbf{\Sigma}})\mathbf{h}\end{align}

(19b)\begin{align} \text{s.t.}~&\mathbf{h}^{T}\bar{\mathbf{\Sigma}}\mathbf{h}+\xi\delta_{b}^{2}=\beta^{{-}1}, \end{align}

(19c)\begin{align} &\log\det(\mathbf{I}_{L_k}+\frac{{\mathbf{G}_k}^{T}\bar{\mathbf{W}}\mathbf{G}_k}{\xi\delta_{e,k}^{2}\mathbf{I}_{L_k}+{\mathbf{G}_k}^{T}\mathbf{\bar\Sigma G}_k})\leq \log\beta, ~\forall \mathbf{G}_{k} \in {\cal B}_k, k\in \cal K,\end{align}

(19d)\begin{align} &\text{rank} (\mathbf{W})=1, \text{Tr}(\bar{\mathbf{W}}+\bar{\boldsymbol\Sigma})\leq \xi P_{\text{max}},~||\bar{\mathbf{W}}+\bar{\boldsymbol\Sigma}||_{\text{max}}\leq \xi A^{2},\end{align}

Note that $\xi > 0$ is replaced by $\xi \geq 0$, which does not lose optimality since it can be validated via the contradiction that any feasible solution $(\mathbf {W},\mathbf \Sigma )$ of the problem (19) cannot be zero. The goal of this step is to turn (17a) to a convex formulation by fixing the denominator, and that leads to a new convex constraint (19b). However, this problem remains non-convex because of (19c), and thus we introduce the following relaxation

Proposition 1 [13]: if $\text {rank}(\bar {\mathbf {W}})\leq 1$, (19c) can be replaced by a convex constraint:

(20)$$(\beta-1)(\xi\delta_{e,k}^{2}\mathbf{I}_{L_k}+{\mathbf{G}_k}^{T}\mathbf{\bar\Sigma G}_k)\succeq {\mathbf{G}_k}^{T}\mathbf{\bar W G}_k, \forall \mathbf{G}_{k} \in {\cal B}_k, k \in \cal K.$$

It indicates that any $(\bar {\mathbf {W}},\bar {\mathbf {\Sigma }},\beta )$ satisfying (19c) also satisfies (20) if $\text {rank}(\bar {\mathbf {W}})\leq 1$. The rank of the transmit covariance matrix can be derived by calculating the KKT condition of problem (19). While verifying rank-one beamforming as the optimal solution is not our main focus in this paper, we find via simulations that our problem usually yields rank-one solutions (more than 98% of the tested cases). Thus, we remove the non-convex one-rank constraint from (19), and the obtained rank-one solution can be used directly as a safe approximation solution.

For the active Eves, (20) is convex and easy to solve, but for the passive Eves, it corresponds to an infinite number of quadratic matrix inequalities. In order to make them computable, we further apply an extension of $\cal S$-procedure as follows:

Lemma 1 [33]: Let $\mathbf {A},\mathbf {B},\mathbf {C}\in {\mathbb {R}}^{M\times M}$, the following quadratic matrix inequality:

(21)$$\mathbf{C}+\mathbf{X}^{T}\mathbf{B}+\mathbf{B}^{T}\mathbf{X}+\mathbf{X}^{T}\mathbf{A}\mathbf{X}\succeq 0, \forall \mathbf{X}\in \{\mathbf{X}|\mathbf{I}-\mathbf{X}^{T}\mathbf{D}\mathbf{X}\succeq 0\},$$

is equivalent to a linear matrix inequality as follows:

(22)$$\begin{aligned} \begin{bmatrix}\mathbf{C} & \mathbf{B}^{T} \\ \mathbf{B} & \mathbf{A} \end{bmatrix}\in \{\mathbf{Z}|\mathbf{Z}-t \begin{bmatrix} \mathbf{I} & 0 \\ 0 & \mathbf -\mathbf{D} \end{bmatrix}\succeq 0,t\geq 0\}. \end{aligned}$$

By substituting (8) and (22) into (20), we obtain an equivalent and single (rather infinite) inequality instead of (20) as follows:

(23)\begin{align}&{\cal P}(\hat{\mathbf{G}},\hat{\mathbf{W}},\beta,\xi,t_k)=\nonumber\\&\begin{bmatrix}(\beta\xi\delta_{e,k}^2-\xi\delta_{e,k}^2-t_k)\mathbf{I}_{L_k}+\hat{\mathbf{G}}_k^T((\beta-1)\bar{\mathbf\Sigma} -\bar{\mathbf W})\hat{\mathbf{G}}_k&\hat{\mathbf{G}}_k^T((\beta-1)\bar{\mathbf\Sigma} -\bar{\mathbf W}) \\((\beta-1)\bar{\mathbf\Sigma} -\bar{\mathbf W})\hat{\mathbf{G}}_k& (\beta-1)\bar{\mathbf\Sigma} -\bar{\mathbf W}+\frac{t_k}{r^2}\mathbf{I}_{M}\end{bmatrix}{\underline \succ} 0,\end{align}

where $k \in {\cal K}_p$ and $\exists t_k \geq 0$, $\mathbf {X}=\Delta \mathbf {G}_k, \mathbf {A}= (\beta -1)\bar {\mathbf \Sigma } -\bar {\mathbf {W}}, \mathbf {B}=((\beta -1)\bar {\mathbf \Sigma } -\bar {\mathbf {W}})\hat {\mathbf {G}}_k , \mathbf {C}= (\beta -1)\xi \delta _{e,k}^{2}\mathbf {I}_{L_k}+\hat {\mathbf {G}}_k^{T}((\beta -1)\bar {\mathbf \Sigma } -\bar {\mathbf {W}})\hat {\mathbf {G}}_k$, and $\mathbf {D}=\mathbf {I}_{M}{r}^{-2}$. Therefore, we replace (19c) by (23) to obtain a relaxtion of the problem (19), which is given as follows:

(24a)\begin{align} &\Omega(\beta)=\max_{\bar{\mathbf{W}}\geq 0,\bar{\mathbf\Sigma}\geq 0,\beta\geq 1,\xi\geq 0}~~\xi\delta^{2}_{b}+\mathbf{h}^{T}(\bar{\mathbf{W}}+\bar{\mathbf{\Sigma}})\mathbf{h}\end{align}

(24b)\begin{align} \text{s.t.}~&\mathbf{h}^{T}\bar{\mathbf{\Sigma}}\mathbf{h}+\xi\delta_{b}^{2}=\beta^{{-}1}, \end{align}

(24c)\begin{align}&(\beta-1)(\xi\delta_{e,k}^{2}\mathbf{I}_{L_k}+{\mathbf{G}_k}^{T}\mathbf{\bar\Sigma G}_k)\succeq {\mathbf{G}_k}^{T}\mathbf{\bar W G}_k, \forall k \in {\cal K}_a, \end{align}

(24d)\begin{align}&{\cal P}(\bar{\mathbf{W}},\bar{\mathbf\Sigma},\beta,\xi,t_k) \succeq 0, \forall k \in {\cal K}_p, \end{align}

(24e)\begin{align}&(19\mathrm{d}), \end{align}

where the above problem (24) is a convex semidefinite program (SDP) problem for a fixed $\beta$. By applying a Golden search on $\beta$ over the feasible interval ${\beta }^{-1}\in [ \delta _{b}^{2}/(\text {min}(P_{\text {max}},MA^{2})||\mathbf {h}||^{2}+\delta _{b}^{2}), 1]$, the optimal solution of (24) can be obtained by choosing the best $\beta ^{*}$ that leads to a maximum worst secrecy rate as below

(25)$$R_s^{*}=\log\{\max_{\beta^{*}}\Omega(\beta^{*})\}.$$

Let $\theta =\text {min}(P_{\text {max}},MA^{2})/ \delta _{b}^{2}$, we proposed the Golden search-based algorithm in Algorithm 1. In Fig. 2, the convergence of the proposed algorithm outperforms the other one-dimensional search-based algorithms, such as bisearch [34] and uniform sampling search. Binary search begins by comparing boundary elements of the interval. If the result of the upper-boundary element is greater than that of the lower-boundary element, the search continues in the upper half of the interval. By doing this, the bisearch-based algorithm eliminates the half in which the maxima may not lie in each iteration. In Fig. 2, the bisearch-based algorithm has the lowest secrecy performance, since it falls into one of local optima nearby the global optimum.

The complexity of our proposed algorithm can be estimated through the complexity of solving the problem (24) times the number of Golden searches involved. For each certain $\beta$, the complexity lies in the standard SDP problem solved by the primal-dual interior-point method, which depends on the number of constraints $(M^{2}+6+K)$ and the dimension of the positive semidefinite core $M$. Thus, a worst-case complexity of solving this SDP problem is $U={\cal O}((M^{2}+6+K)^{4}M^{0.5})$. The complexity of both Golden search and bisearch are $T ={\cal O}(\log (N))$, where $N=\text {min}(P_{\text {max}},MA^{2})||\mathbf {h}||^{2}/ (\delta _{b}^{2}L)$ is the number of feasible solutions (searching values) and $L$ is the tolerance. In comparison, the complexity of the uniform sampling search is $T ={\cal O}(N)$. The total computation cost of Algorithm 1 is $UT$.

Fig. 2. Convergence performance of the proposed Golden search-based algorithm, uniform sampling search-based algorithm, and bisearch-based algorithm.

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5. Simulation results

In this section, we first investigate the secrecy performance of the one-Eve system and multi-Eves system with different positions of the UE in an office, respectively. Then, we apply Monte Carlo simulations to demonstrate the performance of the proposed robust AN-aided method and compare it with that of other AN-aided methods and a non-AN-aided method. The LED arrays are hung face-down on the ceiling, and the legitimate UEs and Eves are placed face-up at the same level of height. The parameters for simulations are illustrated in Table 1.The typical power of a LED chip often ranges from 0.2 W to 2 W, such as LUXEON 2083E ($3 \text {V}, 120 \text {mA}$) and LUXEON 5050HE ($6 \text {V}, 510 \text {mA}$). The total power budget is calculated by the total number of LED chips in 5 arrays times their typical power.

Algorithm 1. Proposed Golden Search-based Approach to Solve the Robust SRM Problem

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Table 1. Simulation Parameters.

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5.1 Secrecy rate of VLC system with one Eve

In the one-Eve scenario, the secrecy performance of the indoor VLC system varies with the location of the UE. To further explain it, the geometric distribution of the secrecy rate is investigated by choosing two different types of the Eves (i.e., active or passive) and two different positions of the Eve, i.e., at the corner or at the center. The above-mentioned two different positions of the Eve are shown in Fig. 3, where the dot and the asterisk indicate the location of the LED arrays and the Eve, respectively.The receiving plane of the room is evenly divided into $13\times 13$ small areas. The secrecy performance of each position is calculated when the UE occupies it. Figure 4 shows the distribution of the secrecy performance in the presence of different types and positions of the Eve. Table 2 describes the average, maximum, and minimum secrecy rates.

Fig. 3. Geometric distribution of the LED arrays and the Eve.

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Fig. 4. Secrecy rate with respect to the position of the UE.

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Table 2. Secrecy Performance Versus the Position of the UE in One-Eve Systems.

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Figure 4 shows that the maximum always exists below one of the LED arrays. In contrast, the minimum almost coincides with the location of the Eve, when the UE locates in the central area. When applying the non-AN-aided method as the transmission strategy, the Eve drives the nearby UE’s secrecy performance falling to zero, as depicted in Fig. 4. An expose of higher-quality confidential signals occurs if continue increasing its power portion. Thus, the AN-aided method is proposed to degrade the reception quality of the Eve. For the Eve at the corner, the minimum secrecy rate does not exist at the same position of the Eve but close to the corner because of the path loss. The type of Eve also impacts the average secrecy rate. The passive Eve reduces the average secrecy rate more than the active Eve. Due to the CSI error, AN is not align with the position of the passive Eve. However, in order to confuse the passive Eve, sending AN to all of its possible positions is necessary, which leads to a waste of power.

5.2 Secrecy rate of VLC system with multiple Eves

In this section, the secrecy performance of three different scenarios, i.e., two active Eves, two passive Eves, and a coexistance. The geometric distributions of the first two scenarios are shown in Fig. 5. The asterisk and plus present the coordinates of the Eves. The geometric distribution of the third scenario is similar to the previous ones, except that the plus and asterisk represent the coordinates of the active Eve and the passive Eve, respectively.

Figure 5 shows the distribution of the secrecy performance in three scenarios. Table 3 depicts the average, maximization, minimization secrecy rates in each scenario. As we can see that there is not much difference in the tendency of the secrecy performance between different scenarios, and the secrecy rate decreases rapidly and reaches the minimum at the position of any Eve. The apparent differences between the three scenarios lie in the average, minimum, and maximum secrecy rates. The average secrecy rate of the two-passive-Eve scenario is the lowest. The performance of the two-active-Eve scenario is the highest, and that of the coexistence scenario is moderate. As explained in the previous subsection, due to the imperfect CSI of the passive Eves, AN consumes more power to confuse all possible (but not the precise) positions of the Eves. Thus, every passive Eve in the room results in a secrecy degradation to the nearby area. Interestingly, the most fraction of the total power is distributed to enhance the SNR of the UE when it locates at the central regions (such as beneath the LED arrays), as shown in Fig. 6. In comparison, AN consumes more power to degrade the SINRs of the Eves when the UE appears at the corner or nearby any Eve, as denoted in Fig. 6(b).

Fig. 5. Secrecy rate with respect to the position of the UE.

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Fig. 6. Percentage of the optimal power allocation for both the confidential signal and AN signal at each position of the UE.

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Table 3. Secrecy Performance Versus the Position of the UE in Multi-Eves VLC Systems.

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Figure 7 displays the information leakage rate of Eves with different total power budgets. The information leakage rates of the passive Eves climb with the power budget, while those of the active Eves slightly decline after reaching the peak at the total power of 45 dBm. With the increasing power budget, not only the power of the confidential message increase but also that of the AN. Thus, when the power budget reaches over 45 dBm, a larger amount of the power is allocated to the AN, which leads to a degradation of the active Eves’ information leakage rates. In contrast, the information leakage rates of the passive Eves keep rising since AN cannot aim at the exact positions of the passive Eves but spread to numbers of possible positions.

Fig. 7. The information leakage rate of the Eves, $P_\text {max}=50$ dBm, $A_k=10~\text {Amp}$.

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5.3 Secrecy rate with respect to different parameters

In the simulation results, we compare the coexistence scenario with two other scenarios, including the only active-Eves scenario and only passive-Eves scenario. We also compare the robust AN-aided method with three other methods, i.e., non-robust AN-aided method [8], isotropic AN-aided method [16], and relaxed ZF beamforming method [18]. In the coexistence scenario, the simulation settings are $K_a=1$, $K_p=2$, $L_k=2$. In the only-active or only-passive Eves scenarios, the simulation settings are $K_a=4, K_p=0$ and $K_p=4,K_a=0$. The positions of the Eves and UE are randomly selected for each instance of the Monte Carlo-based simulation, and the value of the secrecy rate is averaged over various instances.

The secrecy performance of both the only active-Eves scenario and non-robust AN-aided method is calculated by solving the SRM problem from our previous work [8]. For the only active-Eves scenario, there is no CSI error, while, for the non-robust AN-aided method, the CSI contains a Gaussian error. The difference between these two benchmarks lies in whether the CSI contains the error or not when obtaining the optimal transmit and AN covariances in the SRM problem [8]. The final results of these two benchmarks are calculated by substituting the presumed (error-free) CSI of the passive Eves and the optimized covariances into (5). In the isotropic AN-aided design, AN is isotropically emitted in all directions except for the legitimate UE’s channel [16]. The solution of this method is obtained by the problem (24), but $\mathbf {\Sigma }$ is designed as the orthogonal complement projector of $\mathbf {h}$. In [18], the relaxed ZF beamforming method is designed to reduce secrecy outage in the co-existence scenario. In this method, the ZF constraints that beamforming vector has to lie in the null-space of the active Eves’ channel matrix are relaxed to lower-bounded constraints of the active Eves’ SNRs. For a fair comparison with the proposed method, the SNRs of the active Eves are not considered as constraints but the objective function in problem (24) to maximize the worst-case secrecy rate. Thus, the result of this method is obtained by solving the problem (24) but setting $\mathbf {\Sigma }$ as 0.

In Fig. 8, we can see that the performance of the coexistence scenario is always lower than that of the only-active scenario and higher than that of the only-passive scenario, which is consistent with the conclusion of the previous subsection. Interestingly, the non-robust AN-aided method yields a slightly better secrecy rate than the robust AN-aided method when $P<40 \text {dBm}$. In the proposed AN-aided method, the Euclidean norm of the actual CSI error $\Delta {\mathbf {G}}_k$ may be much smaller than the radii $r_k$ of the approximated sphere boundary, which leads to an over-conservative estimation. The over-estimated boundary for the CSI uncertainty gives rise to an extra power consumption in unnecessary positions. The proposed AN-aided method yields better performance when the power budget becomes sufficient for emitting AN. In contrast, the non-robust AN-aided method is sensitive to the CSI uncertainty and experiences a performance loss as $P$ increases. The isotropic AN-aided method allows AN to inject to every direction except for that of UE. Thus, the power of AN emitted to the directions without the existence of the Eves is wasted. In the relaxed ZF beamforming method, the total power can only be used to increase the SNR of the UE, but there is no way to prevent the SNR of the Eves from rising, especially when the reception quality of the Eve is better than that of the UE or the Eve is close to the UE.

Fig. 8. Secrecy rate with respect to various parameters.

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Figure 8(b) explains the slowly increasing tendency of the secrecy performance from the power range of 45 dBm to 50 dBm in Fig. 8. The peak amplitude constraint limits the increasing secrecy rate, which indicates that the LED arrays with a higher value of the peak-amplitude parameter are beneficial to the secrecy performance. Figure 8(c) depicts that the performance of all the schemes improves with the increasing number of LED arrays. More of the LED arrays light up the dark corners and provide a higher array gain for the SNR of the UE. The secrecy performance with the error variance of the passive Eves’ CSI is shown in Fig. 8(d). The performance of all the schemes decreases gradually with the increase of the error variance, except for the only active-Eves scenario. This is because a larger error variance leads to a larger derivation between the estimated CSI and the presumed CSI, which expands the boundary of the region where the passive Eves may exist. Thus, less AN is allocated to degrade the SINR of the passive Eves, and more AN is emitted to the places without the Eves. Compared to the robust AN-aided method, the non-robust AN-aided method is more sensitive to the CSI uncertainty, and its secrecy performance decreases rapidly as the error variance increases.

6. Conclusion

In this paper, a robust AN-aided transmission approach has been proposed to secure a VLC system. Unlike traditional scenarios, which can cope with either active Eves or passive Eves, but not both, the proposed robust AN-aided transmission strategy can safeguard the MISO VLC system against active and passive Eves simultaneously. A conservative approximation approach was applied to transform the probabilistic outage-secrecy constraint into the maximum secrecy constraint with the bounded CSI error. Semidefinite relaxation and $\cal S$-Procedure were introduced to reformulate the original problem into the form of SDP sequences. A Golden search-based algorithm was proposed to solve the series of SDP problems with the polynomial complexity. The simulation results show that every existence of passive Eve causes a worse degradation to the secrecy performance than active Eve does. When the position of the UE is nearby corners and Eves, it is more efficient to emit AN to degrade the Eves. In contrast, it is not necessary to allocate any power to emit AN if the position is beneath any LED array. Furthermore, the proposed robust AN-aided method performs better than the relaxed ZF beamforming method and isotropic AN-aided method. It outperforms the non-robust AN-aided method when total power is over 40 dBm.

As an extention of this work, we would like to discuss some challenges for doing field experiments. First of all, the maximum secrecy rate is difficult to verify in experiments. Although the maximum secrecy rate is not as tight as secrecy capacity, it may be the upper bound of any experiment results. Despite the transmit covariance, other underlying settings also need to be designed, such as modulation mode and coding method. All of them affect the experiment results of the proposed system. Thus, rather than verifying the values of the maximum secrecy rate, comparing the performance of the proposed method with the other methods would be a better idea. Second, the bit rate measured by oscilloscope cannot guarantee a low BER at the receiver’s side. Thus, fixing BER at a certain level is necessary to judge the value of the bit rate. Moreover, the transmit covariance may be realized by steering beams (optical signals) and shutting down parts of LED chips.

Funding

Fundamental Research Funds for the Central Universities (3102017zy026); National Natural Science Foundation of China (61401360).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter	Value	Parameter	Value
Office size ( $W \times L \times H$ )	$(6 \times 6 \times 3) m^{3}$	Refractive index ( $q$ )	1.5
Average electrical ambient noise ( $ξ$ )	-85 dBm	Typical power of a LED chip	$0.2 - 2$ W
Lambertian emission order $(β)$	1	PD surface area $(A_{R})$	1 ${cm}^{2}$
Half-intensity radiation angle $(θ_{1 / 2})$	$70^{\circ}$	PD FOV $(Θ)$	$80^{\circ}$
Secrecy-outage-probability requirement $(γ)$	0.01	Number of Eve’s PD $(L_{k})$	2
Numbers of LED chips per array	$1 - 10$	CSI error variance $(σ)$	0.05

Type of Scenarios	Average (bit/s)	Maximum Secrecy Rate		Minimum Secrecy Rate
Type of Scenarios	Average (bit/s)	Value (bit/s)	Coordinates	Value (bit/s)	Coordinates
One active Eve at the corner	8.084	10.734	(2.5,2.5,3)	0.654	(5.5,5.5,3)
One passive Eve at the corner	7.172	9.742	(2.5,2.5,3)	0.364	(6,5,3)
One active Eve at the center	8.034	9.691	(1.5,1.5,3)	0.997	(4,3,3)
One passive Eve at the center	7.162	8.814	(1.5,4.5,3)	0.997	(4,3,3)

Type of Scenarios	Average (bit/s)	Maximum Secrecy Rate		Minimum Secrecy Rate
Type of Scenarios	Average (bit/s)	Value (bit/s)	Coordinates	Value (bit/s)	Coordinates
Two active Eves	7.147	9.663	(1.5,4.5,3)	0.658	(5.5,5.5,3)
Two passive Eves	6.389	8.814	(1.5,4.5,3)	0.364	(6,5,3)
One active Eve and one passive Eve	6.550	8.851	(1.5,4.5,3)	0.654	(5.5,5.5,3)

Parameter	Value	Parameter	Value
Office size ( $W \times L \times H$ )	$(6 \times 6 \times 3) m^{3}$	Refractive index ( $q$ )	1.5
Average electrical ambient noise ( $ξ$ )	-85 dBm	Typical power of a LED chip	$0.2 - 2$ W
Lambertian emission order $(β)$	1	PD surface area $(A_{R})$	1 ${cm}^{2}$
Half-intensity radiation angle $(θ_{1 / 2})$	$70^{\circ}$	PD FOV $(Θ)$	$80^{\circ}$
Secrecy-outage-probability requirement $(γ)$	0.01	Number of Eve’s PD $(L_{k})$	2
Numbers of LED chips per array	$1 - 10$	CSI error variance $(σ)$	0.05

Type of Scenarios	Average (bit/s)	Maximum Secrecy Rate		Minimum Secrecy Rate
Type of Scenarios	Average (bit/s)	Value (bit/s)	Coordinates	Value (bit/s)	Coordinates
One active Eve at the corner	8.084	10.734	(2.5,2.5,3)	0.654	(5.5,5.5,3)
One passive Eve at the corner	7.172	9.742	(2.5,2.5,3)	0.364	(6,5,3)
One active Eve at the center	8.034	9.691	(1.5,1.5,3)	0.997	(4,3,3)
One passive Eve at the center	7.162	8.814	(1.5,4.5,3)	0.997	(4,3,3)

Artificial-noise aided secure transmission over visible light communication system under coexistent passive and active eavesdroppers

Abstract

1. Introduction

1.1 Related works

1.2 Motivation and contributions

2. System model

3. Conservative formulation

4. SDR-based method for the problem

5. Simulation results

5.1 Secrecy rate of VLC system with one Eve

5.2 Secrecy rate of VLC system with multiple Eves

5.3 Secrecy rate with respect to different parameters

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (4)

Equations (39)

Optics Express