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On the average secrecy capacity for indoor visible light communication systems

Abstract

For visible light communication (VLC), the light signals are transmitted without optical fibers or any sort of wave-guiding. Due to the inherent broadcast nature, physical-layer security emerges as a promising method to protect information delivery from eavesdropping. As for the secrecy capacity of VLC channel, there exist two features. In one way, the limited optical power makes the common capacity expressions in radio-frequency (RF) communication unapplicable for VLC. In another way, several correlated geometrical parameters directly alters the Lambertian model of indoor VLC channel, which gives the secrecy capacity more meanings. However, the issue considering both aspects has not been studied recently. In this paper, from the practical scenarios, we extract a typical geometrical model to reveal the mobility principles of the legitimate receiver and the eavesdroppers. Then, we character two typical distributions of the geometrical parameter. Correspondingly, we derive the upper and lower bounds on the average secrecy capacity, which have the closed forms. Finally, simulation results show that our upper and lower bounds are tight at high optical signal-to-noise rates (OSNRs). Moreover, the geometrical features of VLC systems and distribution parameters of the receiver mobility are effectively reveal by the bounds.

Introduction

In recent years, because of the light-emitting diodes (LEDs) widely deployed for energy conservation in practical illumination, visible light communication (VLC) is currently considered as one of the most indispensable green communication technologies and has attracted considerable attention [13]. For VLC, despite LOS propagation and better signal confinement, the light signals are transmitted without optical fibers or any sort of wave-guiding. The inherent broadcast nature makes VLC links inherently susceptible to eavesdropping by unintended or unauthorized users having access to the physical area illuminated by the data transmitters. Typical scenarios include public areas such as classrooms, offices, libraries, cafes and so on. Therefore, physical-layer security emerged as a promising method [4] to complement conventional encryption techniques and provide a first line of defense against eavesdropping attacks.

By contrast to radio-frequency (RF) communication, there exist two main features in the secrecy capacity. Intensity modulation with direct detection (IM/DD), which modulates the input data into the light beams emitted from LEDs [57], is always employed in indoor VLC system.

Feature 1: The optical intensity signals are limited to be a nonnegative real variable and ought to be less than the eye-safety standard[5, 6]. So, Gaussian distribution is not necessarily followed by the capacity-achieving input distribution [8], thus making the secrecy capacity expressions in RF communication not suitable for VLC.

Feature 2: Compared with the log-normal or Gaussian distribution, the Lambertian is often assumed to be a model of indoor VLC channel, which directly changes with several geometrical parameters, relating to the transmitters and receivers spatial locations [57]. Obviously, the geometrical feature give the secrecy performance of VLC channels more meanings.

In [9], Shannon pioneered the framework of information-theoretic security. Wyner in [10] proposed a fundamental infromation-theoretic security measure, termed as the secrecy capacity. Motivated by Wyner’s work, many studies [1116] on the secrecy performance of Gaussian wiretap channels were considered. However, despite that there exist many studies on the secrecy performance for RF communication, the corresponding study for VLC channel is rare.

As for IM/DD channels, Shannon referred to the challenge in deriving an analytical capacity expression for peak-limited channels [17]. In [8], the upper and lower capacity bounds were derived with the additive Gaussian channel, while a discrete Gaussian distribution is utilized in [18] to research the problem [1922] studied the capacity bounds of optical wireless channels. Furthermore, as for the secrecy performance, [23] derived the lower and upper bounds on the secrecy capacity of the scalar Gaussian wiretap channel subject to amplitude constraints. Mukherjee [24] analyzed the secret-key transmission strategy of multiple-input single-output (MISO) VLC systems after deriving the upper and lower bounds. Shen et al. [25] employed transmit beamforming and jamming techniques to enhance the secrecy for MISO-VLC systems. However, they all didn’t consider the geometrical property of indoor VLC systems. Particularly, although [26] derive the closed-form analytical expressions for secrecy outage probability and the average secrecy capacity, the study was based on the secrecy capacity expression of Gaussian channel. From the above, the issue considering both features has not been studied currently.

Because of the above-mentioned factors, in this paper, we research the relation between the secrecy capacity with the geometrical property, based on the power-limited channel capacity. Particularly, we extract a typical geometrical model from the practical scenarios, which reveals the mobility principles of the legitimate receiver and the eavesdroppers. Then, we character two typical distributions of the geometrical parameter. Accordingly, the upper and lower bounds of the average secrecy capacity are derived, which have the closed forms and are related to the geometrical parameters of VLC systems and distribution parameters of the receiver mobility. Finally, simulation results show that our upper and lower bounds are tight at high OSNRs and the geometrical feature are effectually revealed.

For conciseness, the key notations throughout the paper are summarized in the following Table 1.

Table 1 Key notations used in this paper

System model

In this paper, a typical indoor VLC system is considered, which with a LED as the transmitter and a photo-detector as a legitimate receiver. Moreover, there exist some eavesdroppers, which locate in the same communication region and try to eavesdrop the information between the LED and the legitimate receiver. As shown in Fig. 1, the scenario is common, such as the office, the classroom, the airport lounge and so on.

Fig. 1
figure1

The sketch map of a typical indoor VLC system and the Lambertian model parameters expression

The signals received by the legitimate receiver and the ith (i≥1) eavesdroppers are given by

$$ \begin{aligned} & Y = h_{0}X + w_{0}, \\ & Z_{i} = h_{i}X + w_{i}, \end{aligned} $$
(1)

where \( w_{0},w_{i} \!\! \sim \!\! \mathcal {N}_{\mathbb {R}}\left (0,\sigma ^{2}\right)\) denote the additive Gaussian noise. For simplicity, let us denote the legitimate receiver as the 0th receiver.

  1. 1.

    Optical Power Constraint. For illumination and eye-safety consideration, the optical average power ε, the input signal X, and peak power A are constrained as

    $$ \begin{aligned} \text{Pr}(0 \leq X \leq A) =1, \end{aligned} $$
    (2)
    $$ \mathbb{E}[\!X] \leq \varepsilon, $$
    (3)

    \(X \in \mathbb {R}_{0}^{+}\). The optical average-to-peak power ratio (APPR) is defined as

    $$ \alpha \triangleq \frac{\varepsilon}{A}, $$
    (4)

    where 0<α≤1.

  2. 2.

    Geometrical Property. Although RF channel often considers multipath effect, line-of-sight (LOS) path is often assumed to exist in the typical indoor point-to-point channel. The channel gain hi, (i≥0) is derived by the Lambertian model [8, 9] as

    $$ h_{i} =\left\{ \begin{aligned} & \frac{{{S}_{r}}\left(m\text{+}1 \right)}{2\pi D_{ij}^{2}}{{\cos }^{m}}\left(\phi \right)\cos \left(\psi \right),0\le \psi \le {{\Psi }_{c}},\\ & 0,\psi >{{\Psi }_{c}}, \\ \end{aligned} \right. $$
    (5)

    where ϕ is the emergence angle of transmitter and ψ being the incidence angle of receiver. Sr is the detector acreage, Dij denotes the distance between the transmitter and receiver, and Ψc is the detector field-of-view (FOV) semiangle. \(\Phi _{\frac {1}{2}}\) is the transmitter half-power semiangle. The Lamberts mode order is derived by

    $$ m = -\frac{\ln(2)}{\ln\left(\cos\left(\Phi_{\frac{1}{2}}\right)\right)}. $$
    (6)

In this paper, let ϕ=ψ and meanwhile \(\cos (\phi)\! =\! \cos (\psi)\! =\! \frac {l}{D}\). Additional, (5) can be rewritten as

$$ h_{i} = \frac{S_{r}(m+1)l^{(m+1)}}{2 \pi \left(l^{2} + d_{i}^{2}\right)^{\frac{m+3}{2}}}, (i \geq 0). $$
(7)

When scenario is given, the Sr, m being fixed, h is related to the movement of the receiver, calculated by l and di. \(M =l\cdot \tan {\Phi _{\frac {1}{2}}}\) represents the maximum value of d, which is the radius of illumination region \(\mathcal {R}\). Here, The distribution of di under the given l mainly attract our attention.

$$ d_{i}^{2}=x_{i}^{2}+y_{i}^{2}, (i \geq 0), $$
(8)

where xi, yi are the coordinate values.

According to the distinct features of VLC channel, our problem can be summarized as follows:

Problem 1:

Considering the geometrical property and the optical power constraint, how to derive the closed-form expression for average secrecy capacity. The mathematical model involved is given as ([27])

$$ \begin{aligned} & \bar{\mathcal{C}}_{s} = \mathbb{E}_{h_{0}, h_{i}} [\mathcal{C}_{s} ] \\ & ~~~ = \mathbb{E}_{h_{0}, h_{i}} \left[ \max_{p_{X}} (\mathbb{I}(X;Y) - \mathbb{I}(X;Z))\right] \\ & \hspace{1mm} s.t. \hspace{9mm} \left\{ \begin{aligned} &\text{Pr}(0 \leq X \leq A) =1 \\ &\mathbb{E}[\!X] \leq \varepsilon. \\ \end{aligned} \right. \end{aligned} $$
(9)

Combining (7) and (9), the average secrecy capacity \(\bar {\mathcal {C}}_{s}\) of the legitimate receiver with the ith (i≥1) eavesdropper can be given as

$$ \begin{aligned} \bar{\mathcal{C}}_{s}(h_{0}, h_{i}) & = \mathbb{E}_{h_{0}, h_{i}} [\mathcal{C}_{s}(h_{0}, h_{i})] \\ & = \mathbb{E}_{d_{0}, d_{i}} [\mathcal{C}_{s}(d_{0}, d_{i})] \\ & = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{s}(d_{0}, d_{i}) f(d_{0}, d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & \underset{=}{(c_{0})} \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{s}(d_{0}, d_{i}) f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i}, \end{aligned} $$
(10)

where \( \mathcal {C}_{s}(d_{0}, d_{i})\) is analyzed in (11), (12) and (13). The term f(d0,di) is the joint probability density function (PDF) and the terms of f(d0),f(di) represent the indoor human mobility, respectively. Without loss of generality, we assume that the mobilities of the eavesdroppers and the legitimate receiver are independent with each other. Thus, we have f(d0,di)=f(d0)f(di), making the transformation of (c0).

Average secrecy capacity

Based on (10), we firstly analyze the bounds on the secrecy capacity \(\mathcal {C}_{s}\). Then, by characterising the geometrical property of the practical scenarios, we extract two typical distribution model of di,(i≥0), which is related to hi,(i≥0). Further, we correspondingly derive the lower bounds on the average secrecy capacity.

Bounds on secrecy capacity \(\mathcal {C}_{s}\)

Based on the definition of capacity [28], due to the constraints (2) and (3) for VLC, the capacity-achieving input distribution does not always follow Gaussian distribution [8]. Thus, the common Shannon formula in RF communication can not be applied in VLC.

According to [23], the secrecy capacity of the legitimate receiver with the ith (i≥1) eavesdropper is lower-bounded by each of the following two bounds

$$ {}\begin{aligned} \mathcal{C}_{s} \geq & \frac{1}{2} \log\left(1 + \frac{2 h_{0}^{2} A^{2}}{\pi e \sigma^{2}} \right) \\ & - \left(1- 2\mathcal{Q} \left(\frac{\delta + h_{i} A}{\sigma} \right) \right) \log \frac{2(h_{i}A + \delta)}{\sqrt{2\pi \sigma^{2}} \left(1-2\mathcal{Q}\left(\frac{\delta}{\sigma}\right) \right)} \\ & - \mathcal{Q}\left(\frac{\delta}{\sigma} \right) - \frac{\delta}{\sqrt{2\pi \sigma^{2}}} e^{-\frac{\delta^{2}}{2\sigma^{2}}} + \frac{1}{2}, \end{aligned} $$
(11)
$$ \mathcal{C}_{s} \geq \frac{1}{2} \log \frac{6 h_{0}^{2} A^{2} + 3 \pi e \sigma^{2}}{\pi e h_{i}^{2} A^{2} + 3 \pi e \sigma^{2}}, \hspace{35mm} $$
(12)

and is upper-bounded by

$$ \mathcal{C}_{s} \leq \frac{1}{2} \log \frac{h_{0}^{2} A^{2} + \sigma^{2} }{h_{i}^{2} A^{2} + \sigma^{2}}. $$
(13)

Note that as analyzed in [23], the lower bounds and upper bounds on the secrecy capacity are tight at asymptotically high SNR. Where δ is a free parameter such that δ>0 and σ2 is variance of the additive Gaussian noise. Other parameter is metioned above. Due to that (12) and (13) have the simpler expression, we will utilize them to analyze the average secrecy capacity in the following.

Mobility model of di

Here, we firstly model the geometrical distribution of typical scenarios. For the practical scenarios such as the classroom, the airport lounge and so on, we tend to move within a relatively region. Without loss of generality, let us assume that each receiver locates at a relatively-fixed region.

Figure 2 shows the sketch of our mobility model. The ith (i≥0) receiver moves within the corresponding circle with the radius ui. ti is the distance between the center of the circle Oi to the center of illumination region. We use a 3-D cartesian coordinate system at the receivers height to identify their locations. The origin (0,0) corresponds to the room center. Then, we have

$$ \begin{aligned} & 0 \leq (x_{i}-t_{i1})^{2} + (y_{i} - t_{i2})^{2} \leq u_{i}^{2}, \\ & x_{i}^{2} + y_{i}^{2} = d_{i}^{2}, \\ & t_{i1}^{2} + t_{i2}^{2} = t_{i}^{2}. \end{aligned} $$
(14)
Fig. 2
figure2

The sketch map of a typical indoor VLC system and the parameters in Lambertian model

where ti1 and ti2 are coordinate of Oi. Although our human mobility is various from person to person, there exist some basic principles. Here two typical mobility types are mainly proposed as follows:

  • Type of main location. This means that the possibility of sitting in a main location is often much larger than the surrounding area. In the practical restaurant, cafes and so on, we tend to sit at the desks or the other relatively-fixed location. Typically, as for the ith receiver, we model f(di) in (10) as the 2-dimension Gaussian distribution at the receiver height, i.e.,

    $$ \begin{aligned} & f(d_{i}) = \frac{1}{2\pi \zeta_{i1} \zeta_{i2} }e^{- \frac{(x_{i} - t_{i1})^{2}}{2 \zeta_{i1}^{2}} - \frac{(y_{i} -t_{i2})^{2}}{2 \zeta_{i2}^{2}}}, (i \geq 0). \end{aligned} $$
    (15)

    Note that it is simple to extend (15) to the specific human mobility, by adjusting ζi1,ζi2,ti1,ti2. Here, the other scenarios are based on a scenario of one LED and one main location. Furthermore, let us assumeFootnote 1ζi1=ζi2=ζi≠0. Then, (15) can be transformed as

    $$ \begin{aligned} & f(d_{i}) = \frac{1}{2\pi \zeta_{i}^{2} }e^{- \frac{(x_{i} - t_{i1})^{2}+(y_{i} -t_{i2})^{2}}{2 \zeta_{i}^{2}} }, (i \geq 0). \end{aligned} $$
    (16)
  • Type of random walk. The meaning is that the possibilities of the receiver at any point within the circle are same. For example, we often randomly and steadyly walk in the relatively-fixed region. Typically, f(di) is modeled as the uniform distribution, i.e.,

    $$ \begin{aligned} f(d_{i}) = \frac{1}{\pi u_{i}^{2}}, (i \geq 0), \end{aligned} $$
    (17)

    and the other random-walk scenarioswhich is based on it. For the practical random walkFootnote 2, we can readily deduce the corresponding average secrecy capacity with the same manner as the case of (17).

For simplicity, let us make some notations.

$$ T_{1} \triangleq \frac{S_{r} (m+1)l^{m+1}}{2\pi}, ~~ T_{2} \triangleq \frac{m+3}{2}. $$
(18)

For the legitimate receiver and the ith (i≥1) eavesdropper,

$$ \begin{aligned} & h_{0L} \triangleq T_{1} (l^{2} + t_{0}^{2})^{-T_{2}}, \\ & h_{0R} \triangleq T_{1} (l^{2} + (t_{0}+u_{0})^{2})^{-T_{2}}, \\ & h_{iL} \triangleq T_{1} (l^{2} + (t_{i}-u_{i})^{2})^{-T_{2}}, \\ & h_{iR} \triangleq T_{1} (l^{2} + t_{i}^{2})^{-T_{2}}. \end{aligned} $$
(19)

And then, the following Theorem 1 illuminates the results for the average secrecy capacity, Appendix Appendix illuminates the proof of which.

For indoor VLC system, the upper bounds on the average secrecy capacity under different geometrical model are given as follows:

  • 1) When \(f(d_{i}) = \frac {1}{2\pi \zeta _{i}^{2} }e^{- \frac {(x_{i} - t_{i1})^{2}+(y_{i} -t_{i2})^{2}}{2 \zeta _{i}^{2}} }\). \(\bar {\mathcal {C}}_{s}\) is upper-bounded by

    $$ \begin{aligned} & \bar{\mathcal{C}}_{s} \leq \frac{\left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right)}{4} \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}), \end{aligned} $$
    (20)

    \(\bar {\mathcal {C}}_{s}\) is lower-bounded by one of the following bounds

    $$ \begin{aligned} & \bar{\mathcal{C}}_{s} \geq \frac{\left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right)}{4} \mathcal{G}_{2}(u_{0},t_{0},u_{i},t_{i}),~~~ \end{aligned} $$
    (21)
    $$ \begin{aligned} & \bar{\mathcal{C}}_{s} \geq \left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right) \left[{\vphantom{\frac{ p_{1}(h_{0L}) +p_{1}(h_{0R}) }{4}}} - p_{3} +\right.\\ & \hspace{10mm} \left.\frac{ p_{1}(h_{0L}) +p_{1}(h_{0R}) }{4} -\frac{p_{2}(h_{iL}) + p_{2}(h_{iR})}{2} \right]. \end{aligned} $$
    (22)
  • 2)When \(f(d_{i}) = \frac {1}{\pi u_{i}^{2}}\). \(\bar {\mathcal {C}}_{s}\) is upper-bounded as

    $$ \begin{aligned} & \bar{\mathcal{C}}_{s} \leq \frac{1}{4}\mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}), \end{aligned} $$
    (23)

\(\bar {\mathcal {C}}_{s}\) is lower-bounded by one of the following bounds

$$ \begin{aligned} & \bar{\mathcal{C}}_{s} \geq \frac{1}{4}\mathcal{G}_{2}(u_{0},t_{0},u_{i},t_{i}), \end{aligned} $$
(24)
$$ \bar{\mathcal{C}}_{s} \geq \frac{ p_{1}(h_{0L}) +p_{1}(h_{0R}) }{4} - \frac{p_{2}(h_{iL}) + p_{2}(h_{iR})}{2}-p_{3}. $$
(25)

Among them, p1(·),p2(·),p3 are defined in (54), (55) and (56). The terms of \(\mathcal {G}_{1}(u_{0},t_{0},u_{i},t_{i}),\mathcal {G}_{2}(u_{0},t_{0},u_{i},t_{i})\) are given in (26) and (27).

In Theorem 1,

$$ \begin{aligned} \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}) \,=\, & \log \left[\sigma^{2}+A^{2} T_{1}^{2} \left[ l^{2} + (t_{0} - u_{0})^{2}\right]^{-2 T_{2}}\right]\\ &+ \log \left[\sigma^{2}+A^{2} T_{1}^{2} \left(l^{2} + t_{0}^{2}\right)^{-2 T_{2}}\right]\\ & - \log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+t_{i}^{2}\right)^{-2 T_{2}} \right]\\ &- \log \left[\sigma^{2} \,+\, A^{2} T_{1}^{2} \left(l^{2}+(t_{i}+u_{i})^{2}\right)^{-2 T_{2}} \right], \end{aligned} $$
(26)
$$ \begin{aligned} \mathcal{G}_{2}(u_{0},t_{0},u_{i},t_{i}) = & \log\left [3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} \left(l^{2}+t_{0}^{2}\right)^{-2 T_{2}}\right]\\ & + \log [3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} (l^{2}+(t_{0}+u_{0})^{2})^{-2 T_{2}} ]\\ & - \log [3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} (l^{2}+t_{i}^{2})^{-2 T_{2}}]\\ &- \log [3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} (l^{2}+(t_{i}-u_{i})^{2})^{-2 T_{2}}].\overleftarrow{} \end{aligned} $$
(27)

Simulation results

In this section, the tightness and the geometrical feature of our upper and lower bounds on the average secrecy capacity are analyzed in Theorem 1. We consider a scenario with several users under a LED as our simulation environment. \(\frac {A}{\sigma }\) represents the OSNR. High OSNRs are the practical application zone, because of typical indoor illumination levels can offer at least 40dB OSNR at the receiver [5]. Note that we use “Ratio” to express \(\frac {u_{i}}{\zeta _{i}}\). Notation "U-B" denotes the simulation for the upper bound and Notation "L-B" for the lower bound.

Figure 3 shows the average secrecy capacity bounds under l = 1.5m, 2.5m, 3.5m, with di yielding to the Gaussian distribution. Figure 4 plots the similar curves under different \(\frac {u_{i}}{\zeta _{i}}\) = 1, 2, 3. Figure 5 shows that under different ti = 2.2m, 5.2m, 7.2m. As shown in figures, when l increases, the average secrecy capacity degrades. When the distribution parameters of \(\frac {u_{i}}{\zeta _{i}}\), ti increase, the average secrecy capacity can be improved. As shown in Theorem 1, the bounds are effective to reveal the geometrical features of VLC systems and distribution parameters of the receiver mobility.

Fig. 3
figure3

Average secrecy capacity under different l, for Gaussian distribution

Fig. 4
figure4

Average secrecy capacity under various \(\frac {u_{i}}{\zeta _{i}}\), for Gaussian distribution

Fig. 5
figure5

Average secrecy capacity under different ti, for Gaussian distribution

Figure 6 shows the average secrecy capacity bounds under different l, with di yielding to the uniform distribution. Figure 7 plots the similar curves under different ti.In one way, we can observe that the same principle that when l increases, the average secrecy capacity degrades. When the distribution parameters of ti increase, the average secrecy capacity can be improved. In another way, by combing Fig. 3 with Figs. 5, 6 with Fig. 7, different PDF of the receiver mobility has different secrecy capacity even if under same parameters. This also reveals the influence of the geometrical property on the secrecy capacity, as shown in Theorem 1.

Fig. 6
figure6

Average secrecy capacity under different l, for uniform distribution

Fig. 7
figure7

Average secrecy capacity under different ti, for Gaussian distribution

Moreover, according to Figs. 2, 3, 4 and 5, the lower bound in 22 keeps the relatively-stable gap with the upper bound in (21), while the lower bound in (23) has a relatively-varying gap. At the lower OSNRs, the former gap is smaller than the latter gap. At the higher OSNRs, the former gap is larger than the latter gap. Due to that typical indoor illumination levels can offer high OSNR at the receiver [5], high OSNRs are the practical application zone. From the results, under different conditions, our bounds just perform the better tightness at high OSNRs.

Conclusion

Due to the inherent broadcast nature of VLC channels, physical-layer security emerged as a promising method to protect information delivery from eavesdropping. There exist two features within the secrecy capacity. The limited optical power makes the common capacity expressions in radio-frequency (RF) communication not suitable for VLC. In another way, the Lambertian model of indoor VLC channel directly changes with several related geometrical parameters, which makes the secrecy capacity more meanings. However, the study considering both aspects does not been carried out currently. This paper extracted a typical geometrical model from the practical scenario to reveal the mobility principles of the legitimate receiver and the eavesdroppers. Then, by characterizing two typical distributions of the geometrical parameter, we derived the corresponding upper- and lower- bounds on the average secrecy capacity, which have the closed forms. Simulation results reveal that our bounds are tight at high optical signal-to-noise rates (OSNRs). Moreover, the geometrical features of VLC systems and distribution parameters of the receiver mobility can be effectively revealed by the bounds.

Appendix

For simplicity, let

$$ h_{i} = T_{1} \left(l^{2} + d_{i}^{2}\right)^{-T_{2}}, $$
(28)

where \(T_{1} = \frac {S_{r} (m+1)l^{m+1}}{2\pi }\), \(T_{2} = \frac {m+3}{2}\). For indoor VLC system, \(m = -\frac {\ln (2)}{\ln \left (\cos \left (\Phi _{\frac {1}{2}}\right)\right)}>0\). Thus

$$ T_{1} > 0,~ T_{2} > 0. $$
(29)
Fig. 8
figure8

The sketch map of a typical indoor VLC system and the parameters in Lambertian model

Note that in (28) and (33), the subscript of i=0 is for the legitimate receiver, while the subscripts of i≥1 are for the eavesdroppers.

Here, we divide the circle \(\mathcal {S}_{i}\), as shown in Fig. 8. For i[0,N], we have

$$ \begin{aligned} &~~~~ \mathcal{S}_{i} = \mathcal{S}_{il} + \mathcal{S}_{ir}, \\ & \int\!\!\!\!\int\limits_{\mathcal{S}_{il}} f(d_{i}) ~\mathrm{d} \tau_{i} =\int\!\!\!\!\int\limits_{\mathcal{S}_{ir}} f(d_{i}) ~\mathrm{d} \tau_{i} = \frac{1}{2}\int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i}, \end{aligned} $$
(30)

which means that the possibilities of the receiver di(xi,yi) locating at \(\mathcal {S}_{il}\), \(\mathcal {S}_{ir}\) are equal. When di(xi,yi) lies in the half-circle \(\mathcal {S}_{il}\), we have

$$ t_{i} - u_{i} \leq d_{i} \leq t_{i}. $$
(31)

When di(xi,yi) lies in the half-circle \(\mathcal {S}_{ir}\), we have

$$ t_{i} \leq d_{i} \leq t_{i} + u_{i}. $$
(32)

By using the polar coordinate system, let

$$ \begin{aligned} & x_{i} = t_{i1} + \rho_{i} \cos\theta_{i}, \\ & y_{i} = t_{i2} + \rho_{i} \sin\theta_{i}. \end{aligned} $$
(33)

Proof of upper bounds (20) and (23)

We derive the upper bounds based on (13) and (10). Then, we have

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{s}(d_{0}, d_{i}) f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & \leq \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \frac{1}{2} \log \frac{h_{B}^{2} A^{2} + \sigma^{2} }{h_{E}^{2} A^{2} + \sigma^{2}} f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & = \frac{1}{2} \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \log (h_{B}^{2} A^{2} + \sigma^{2}) f(d_{0}) ~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i} }_{w_{1}} \\ & ~~ - \frac{1}{2} \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} f(d_{0}) ~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \log (h_{E}^{2} A^{2} + \sigma^{2}) f(d_{i}) ~\mathrm{d} \tau_{i} }_{w_{2}} \end{aligned} $$
(34)

Based on (28), the terms of w1,w2 are given by

$$ \begin{aligned} w_{1} & = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+d_{0}^{2}\right)^{-2 T_{2}}\right] \\ & \hspace{5mm} \cdot f(d_{0}) ~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i}, \end{aligned} $$
(35)
$$ \begin{aligned} w_{2} & = \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+d_{i}^{2}\right)^{-2 T_{2}}\right] \\ & \hspace{5mm} \cdot f(d_{i}) ~\mathrm{d} \tau_{i} \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} f(d_{0}) ~\mathrm{d} \tau_{0}. \end{aligned} $$
(36)

Next, we derive the upper bounds under two typical distribution.

1) When Gaussian distribution.

For (35) and (36), let us denote

$$ w_{1} = \frac{1}{4\pi^{2} \zeta_{0}^{2} \zeta_{i}^{2}} ~ a_{1}(d_{0}) \cdot a_{3}, $$
(37)
$$ w_{2} = \frac{1}{4\pi^{2} \zeta_{0}^{2} \zeta_{i}^{2}} ~ a_{1}(d_{i})\cdot a_{2}, (i\geq 1) $$
(38)

where the terms of a1(d0),a1(d1),a2,a3 are given in the following,

$$ \begin{aligned} a_{1}(d_{i}) = & \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \left(\log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+d_{i}^{2}\right)^{-2 T_{2}} \right] \right) \\ & ~~~~ \cdot e^{-\frac{(x_{i} - t_{i1})^{2} + (y_{i} - t_{i2})^{2} }{2\zeta_{0}^{2}}} ~\mathrm{d} \tau_{0}, (i \geq 0). \end{aligned} $$
(39)
$$ \begin{aligned} a_{2} & = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} e^{-\frac{(x_{0} - t_{01})^{2} + (y_{0} - t_{02})^{2} }{2\zeta_{0}^{2}}} ~\mathrm{d} \tau_{0} \\ & = \int_{0}^{2\pi}\!\!\!\! \int_{0}^{u_{0}} e^{-\frac{\rho_{0}^{2}}{2\zeta_{0}^{2}}} \rho_{0} ~\mathrm{d} \rho_{0} \mathrm{d} \theta_{0} = 2\pi \zeta_{0}^{2} \left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right). \end{aligned} $$
(40)

Similarly,

$$ \begin{aligned} a_{3} = \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} e^{-\frac{(x_{i} - t_{i1})^{2} + (y_{i} - t_{i2})^{2} }{2\zeta_{i}^{2}}} ~\mathrm{d} \tau_{i} = 2\pi \zeta_{i}^{2} \left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right). \end{aligned} $$
(41)

With T2>0 in (29), a2(d0) can be upper-bounded by

$$ \begin{aligned} & a_{1}(d_{0}) \underset{=}{(c_{1})} \int\!\!\!\!\int\limits_{\mathcal{S}_{0l}} + \int\!\!\!\!\int\limits_{\mathcal{S}_{0r}} \\ & \underset{\leq}{(c_{2})} \log \left[\sigma^{2}+ A^{2} T_{1}^{2} \left[ l^{2} + (t_{0} - u_{0})^{2}\right]^{-2 T_{2}} \right] \\ & ~~~ \cdot \int\!\!\!\!\int\limits_{\mathcal{S}_{0l}} \!\! e^{-\frac{(x_{0} - t_{01})^{2} + (y_{0} - t_{02})^{2} }{2\zeta_{0}^{2}}} \!\! \mathrm{d} \tau_{0} \!\! + \!\! \int\!\!\!\!\int\limits_{\mathcal{S}_{0r}} \!\! e^{-\frac{(x_{0} - t_{01})^{2} + (y_{0} - t_{02})^{2} }{2\zeta_{0}^{2}}} \!\! \mathrm{d} \tau_{0} \\ & ~~~ \cdot \log \left[\sigma^{2}+ A^{2} T_{1}^{2} \left[ l^{2} +t_{0}^{2}\right]^{-2 T_{2}} \right] \\ & = \frac{a_{2}}{2} \left(\log \left[\sigma^{2}+A^{2} T_{1}^{2} [ l^{2} + (t_{0} - u_{0})^{2}]^{-2 T_{2}}\right] +\right.\\ & ~~~~\left.\log \left[\sigma^{2}+A^{2} T_{1}^{2} (l^{2} + t_{0}^{2})^{-2 T_{2}}\right] \right), \end{aligned} $$
(42)

where the transformation of (c1) is due to (30) and the approximation of (c2) is based on (31), (32). Note that a2 is calculated in (40).

The term of a1(di) can be bounded by

$$ \begin{aligned} & a_{1}(d_{i}) \underset{=}{(c_{3})} \int\!\!\!\!\int\limits_{\mathcal{S}_{il}} + \int\!\!\!\!\int\limits_{\mathcal{S}_{ir}} \\ &\underset{\geq}{(c_{4})}\log \left[\sigma^{2}+ A^{2} T_{1}^{2}\left(l^{2}+t_{i}^{2}\right)^{-2 T_{2}} \right] \\ & ~~ \cdot \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \!\! e^{-\frac{(x_{i} - t_{i1})^{2} + (y_{i} - t_{i2})^{2} }{2\zeta_{i}^{2}}} \!\! \mathrm{d} \tau_{i} \!\! + \!\! \int\!\!\!\!\int\limits_{\mathcal{S}_{i}}\!\! e^{-\frac{(x_{i} - t_{i1})^{2} + (y_{i} - t_{i2})^{2} }{2\zeta_{i}^{2}}} \!\! \mathrm{d} \tau_{i} \\ & ~~ \cdot \log \left[\sigma^{2}+ A^{2} T_{1}^{2} \left(l^{2}+(t_{i}+u_{i})^{2}\right)^{-2 T_{2}} \right] \\ & = \frac{a_{3}}{2}\left(\log \left[\sigma^{2} + A^{2} T_{1}^{2} (l^{2}+t_{i}^{2})^{-2 T_{2}} \right] +\right. \\ & ~~~~ \left.\log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+(t_{i}+u_{i})^{2}\right)^{-2 T_{2}} \right] \right), \end{aligned} $$
(43)

where the transformation of (c3) is due to (30) and the approximation of (c4) is based on (31), (32). Note that a3 is calculated in (41).

Finally, combining (34) with (42), (43), we can obtain

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) \leq \frac{1}{2}(w_{1} - w_{2}) \\ & = \frac{1}{8\pi^{2} \zeta_{0}^{2} \zeta_{i}^{2}} ~ [ a_{1}(d_{0}) \cdot a_{3} - a_{1}(d_{i}) \cdot a_{2} ] \\ & \leq \frac{a_{2} * a_{3}}{16\pi^{2} \zeta_{0}^{2} \zeta_{i}^{2}} \cdot \left(\log \left[\sigma^{2}+A^{2} T_{1}^{2} [ l^{2} + (t_{0} - u_{0})^{2}]^{-2 T_{2}}\right] \right.\\ & \hspace{7mm} + \log \left[\sigma^{2}+A^{2} T_{1}^{2} \left(l^{2} + t_{0}^{2}\right)^{-2 T_{2}}\right] - \\ & \hspace{10mm} \log \left[\sigma^{2} + A^{2} T_{1}^{2} (l^{2}+t_{i}^{2})^{-2 T_{2}} \right] - \\ & \hspace{10mm} \left.\log \left[\sigma^{2} + A^{2} T_{1}^{2} (l^{2}+(t_{i}+u_{i})^{2})^{-2 T_{2}} \right] \right) \\ & = \frac{\left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right)}{4} \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}). \end{aligned} $$
(44)

This completes the proof of (20).

2) When uniform distribution.

For the ith (i≥0) receiver,

$$ \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i} = \frac{1}{\pi u_{i}^{2}} \int_{0}^{2\pi}\!\!\! \int_{0}^{u_{i}} \rho_{i}~ \mathrm{d} \rho_{i} \mathrm{d} \theta_{i} = 1. $$
(45)

Then, using the similar method of proofing (44), we have

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) \leq \frac{1}{2}(w_{1} - w_{2}) \\ & \leq \frac{ \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \!\! f(d_{0}) \mathrm{d} \tau_{0}}{4} \cdot \left(\log\left[\sigma^{2} \!+ \! A^{2} T_{1}^{2} \left[ l^{2} \! +\! (t_{0} \,-\, u_{0})^{2}\right]^{-2 T_{2}}\right] \right.\\ & \hspace{6mm} + \left.\log \left[\sigma^{2}+A^{2} T_{1}^{2} \left(l^{2} + t_{0}^{2}\right)^{-2 T_{2}}\right] \right) \\ & \hspace{3mm}- \frac{ \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \!\! f(d_{i}) \mathrm{d} \tau_{i}}{4} \left(\log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+t_{i}^{2}\right)^{-2 T_{2}} \right] \right.\\ & \hspace{6mm} +\left. \log \left[\sigma^{2} + A^{2} T_{1}^{2} \left(l^{2}+(t_{i}+u_{i})^{2}\right)^{-2 T_{2}} \right] \right)\\ & = \frac{1}{4} \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}). \end{aligned} $$
(46)

This completes the proof of (23).

Proof of lower bound (21) and (24)

We derive the upper bounds based on (12) and (10). Then, we have

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{s}(d_{0}, d_{i}) f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & \geq \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \frac{1}{2} \log \frac{6 h_{B}^{2} A^{2} + 3 \pi e \sigma^{2}}{\pi e h_{E}^{2} A^{2} + 3 \pi e \sigma^{2}} f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & = \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \frac{ \log \left(6 h_{B}^{2} A^{2} + 3 \pi e \sigma^{2}\right)}{2} f(d_{0}) \mathrm{d} \tau_{0} }_{b_{1}(d_{0})} \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \! f(d_{i}) \mathrm{d} \tau_{i}}_{b_{3}} \\ & ~ - \!\! \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} f(d_{0}) \mathrm{d} \tau_{0} }_{b_{2}} \underbrace{ \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \frac{ \log \left(\pi e h_{E}^{2} A^{2} + 3 \pi e \sigma^{2}\right)}{2} f(d_{i}) \mathrm{d} \tau_{i}}_{b_{1}(d_{i})} \\ & = b_{1}(d_{0}) \cdot b_{3} - b_{2} \cdot b_{1}(d_{i}). \end{aligned} $$
(47)

1) When Gaussian Distribution. It is easy to know that

$$ b_{2} = a_{2}/\left(2\pi \zeta_{0}^{2}\right), ~ b_{3} = a_{3}/\left(2\pi \zeta_{i}^{2}\right), $$
(48)

where a2,a3 are in (40) and (41). Applying to the similar method of proofing (44), we have

$$ {}\begin{aligned} b_{1}(d_{0}) \geq & \frac{b_{2}}{4}\left(\log \left[3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} \left(l^{2}+t_{0}^{2}\right)^{-2 T_{2}}\right] + \right.\\ & \left.\log \left[3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} \left(l^{2}+(t_{0}+u_{0})^{2}\right)^{-2 T_{2}} \right] \right), \end{aligned} $$
(49)
$$ {}\begin{aligned} b_{1}(d_{i}) \leq & \frac{b_{3}}{4}\left(\log \left[3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} \left(l^{2}+t_{i}^{2}\right)^{-2 T_{2}}\right] + \right.\\ & \left.\log \left[3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} \left(l^{2}+(t_{i}-u_{i})^{2}\right)^{-2 T_{2}} \right] \right), \end{aligned} $$
(50)

Then, the average secrecy capacity is lower-bounded by

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{s}(d_{0}, d_{i}) f(d_{0}) f(d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & \geq b_{1}(d_{0}) \cdot b_{3} - b_{2} \cdot b_{1}(d_{i}) \\ & \geq \frac{\left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right)}{4} \\ & \hspace{4mm} \cdot \left(\log \left[3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} \left(l^{2}+t_{0}^{2}\right)^{-2 T_{2}}\right]\right. \\ & \hspace{7mm} + \log \left[3\pi e \sigma^{2} +6 A^{2} T_{1}^{2} \left(l^{2}+(t_{0}+u_{0})^{2}\right)^{-2 T_{2}} \right] \\ & \hspace{7mm} - \log \left[3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} \left(l^{2}+t_{i}^{2}\right)^{-2 T_{2}}\right] \\ & \hspace{7mm} -\left. \log \left[3\pi e \sigma^{2} + \pi e A^{2} T_{1}^{2} \left(l^{2}+(t_{i}-u_{i})^{2}\right)^{-2 T_{2}} \right] \right)\\ & \triangleq \frac{\left(1- e^{-\frac{u_{0}^{2}}{2\zeta_{0}^{2}}}\right)\left(1- e^{-\frac{u_{i}^{2}}{2\zeta_{i}^{2}}}\right)}{4} \cdot \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}). \end{aligned} $$
(51)

2) When Uniform Distribution. It is easy to get

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) \geq \frac{1}{4} \mathcal{G}_{1}(u_{0},t_{0},u_{i},t_{i}). \end{aligned} $$
(52)

This completes the proofs of (21) and (24

Proof of lower bounds (22) and (25)

For simplicity, let us denote the lower bound in (11) as

$$ \begin{aligned} \mathcal{C}_{L1}(h_{B}, h_{E}) \triangleq \frac{p_{1}(h_{B})}{2}- p_{2}(h_{E})- p_{3}, \end{aligned} $$
(53)

where the functions of p1(x),p2(x),p3 are

$$ p_{1}(x) = \log\left(1 + \frac{2 x^{2} A^{2}}{\pi e \sigma^{2}} \right), $$
(54)
$$ {}p_{2}(x) = \left(1- 2\mathcal{Q}\left(\frac{\delta + x A}{\sigma} \right) \right) \log \frac{2(xA + \delta)}{\sqrt{2\pi \sigma^{2}} \left(1-2\mathcal{Q}\left(\frac{\delta}{\sigma}\right) \right)}, $$
(55)
$$ p_{3} = \left(\mathcal{Q}\left(\frac{\delta}{\sigma}\right) + \frac{\delta}{\sqrt{2\pi \sigma^{2}}} e^{-\frac{\delta^{2}}{2\sigma^{2}}} - \frac{1}{2} \right). $$
(56)

We can observe that p1(hB) has no relation with hE, while p2(hE) has no relation with hB. Moreover, p3 has no relation with hB,hE. Then, the corresponding \(\bar {\mathcal {C}}_{s}(h_{B}, h_{E})\) can be lower-bounded as

$$ \begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) \geq \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \mathcal{C}_{L1}(h_{B}, h_{E}) f(d_{0},d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} \left(\frac{p_{1}(h_{B})}{2} - p_{2}(h_{E}) - p_{3} \right) f(d_{0},d_{i}) ~\mathrm{d} \tau_{0} \mathrm{d} \tau_{i} \\ & = \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} \frac{ p_{1}(h_{B})}{2} \cdot f(d_{0}) ~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i} \\ & \hspace{5mm} - \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} p_{2}(h_{E}) \cdot f(d_{i}) ~\mathrm{d} \tau_{i} \int\!\!\!\!\int\limits_{\mathcal{S}_{0}} f(d_{0}) ~\mathrm{d} \tau_{0} \\ & \hspace{5mm} - p_{3} \int\!\!\!\!\int\limits_{\mathcal{S}_{0}}f(d_{0})~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i}. \end{aligned} $$
(57)

Applying to the similar method of proofing (44), (57) can be further approximated as

$$ {}\begin{aligned} & \bar{\mathcal{C}}_{s}(h_{B}, h_{E}) \geq \int\!\!\!\!\int\limits_{\mathcal{S}_{0}}f(d_{0})~\mathrm{d} \tau_{0} \int\!\!\!\!\int\limits_{\mathcal{S}_{i}} f(d_{i}) ~\mathrm{d} \tau_{i} \\ & \cdot \left[ \frac{1}{4} p_{1}(h_{0L}) + \frac{1}{4} p_{1}(h_{0R})- \frac{1}{2} p_{2}(h_{iL}) -\frac{1}{2} p_{2}(h_{iR}) - p_{3} \right], \end{aligned} $$
(58)

where

$$ h_{0L} = T_{1} \left(l^{2} + t_{0}^{2}\right)^{-T_{2}}, ~ h_{0R} = T_{1} \left(l^{2} + \left(t_{0}+u_{0}\right)^{2}\right)^{-T_{2}}, $$
(59)
$$ h_{iL} = T_{1} \left(l^{2} + (t_{i}-u_{i})^{2}\right)^{-T_{2}}, ~ h_{iR} = T_{1} \left(l^{2} + t_{i}^{2}\right)^{-T_{2}}. $$
(60)

When Uniform Distribution, \(\int \!\!\!\!\int \limits _{\mathcal {S}_{0}}f(d_{0})~\mathrm {d} \tau _{0} \int \!\!\!\!\int \limits _{\mathcal {S}_{i}} f(d_{i}) ~\mathrm {d} \tau _{i} = 1\). When Gaussian Distribution, \(\int \!\!\!\!\int \limits _{\mathcal {S}_{0}}f(d_{0})~\mathrm {d} \tau _{0} \int \!\!\!\!\int \limits _{\mathcal {S}_{i}} f(d_{i}) ~\mathrm {d} \tau _{i} = \left (1- e^{-\frac {u_{0}^{2}}{2\zeta _{0}^{2}}}\right)\left (1- e^{-\frac {u_{i}^{2}}{2\zeta _{i}^{2}}}\right) \).

This completes the proofs of (22) and (25).

Availability of data and materials

Availability of data and materials.

Notes

  1. 1.

    Here, we consider the simple case of ζi1=ζi2, which can be simply transformed to the other practical scenarios.

  2. 2.

    Although our mobility may not be the whole region \(\mathcal {R}\), there only exists a change of the integration region in (10).

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Acknowledgements

Authors sincerely thank all teachers and graduate students of National Digital Switching System Engineering Technological Research Center for their constant support to the research activity. Especially for Yijun Zhu, Chao Wang.

Funding

The work of this paper was supported in part by the National Key Research and Development Project under Grant 2018YFB1801903, in part by the National Natural Science Foundation of China (NSFC) under Grant 61671477, 61901524 and in part by the China Postdoctoral Science Foundation under Grant 2019M663477.

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Project administration and Writing–original draft, Jun Zheng Zhang and Ke Ke. All authors read and approved the final manuscript.

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Correspondence to Ke Ke.

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Zhang, J.Z., Ke, K. On the average secrecy capacity for indoor visible light communication systems. J. Eur. Opt. Soc.-Rapid Publ. 16, 11 (2020). https://doi.org/10.1186/s41476-020-00132-9

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Keywords

  • Visible light communication
  • Geometrical property
  • Average secrecy capacity
  • Upper and lower bounds