Abstract
This paper mainly focus on the performance of secrecy capacity in the physical layer security (PLS) for the eavesdropping channel in visible light communication (VLC) system. In this system, due to the effects of thermal and shoot noises, the main interference of the channel is not only from additive white Gaussian noise (AWGN), but also dependent on the input signal. Considering a practical scenery, based on the input-dependent Gaussian noise, the closed-form expression of the upper and lower bounds of secrecy capacity are derived under the constraints of non-negative and average optical intensity. Specifically, since the entropy of the output signal is always greater than the input signal, on this basis, the derivation of lower bound is using the variational method to obtain a better input distribution. The upper bound is derived by the dual expression of channel capacity. We verified the performance of secrecy capacity through numerical results. The results show that the upper and lower bounds are relatively tight when optical intensity is high, which proves validity of the expression. In the low signal-to-noise ratio (SNR) scheme, the result of bounds with more input-dependent noise is better than less noise. And in the high SNR scheme, the result of bounds with less input-dependent noise outperforms that noise is more.
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6 Appendix
6 Appendix
1.1 6.1 Appendix A
For expression (7), according to [10], we have
According to Theorem 17.2.3 in [12], an upper bound of \(\mathcal {H}\left( {{Y_\mathrm{{E}}}} \right) \) is given by
Substituting (17) and (18) into (7), \({C_s}\) can be written as
where \({f_{{\mathrm {low}}}}\left( {\xi P} \right) \), \(\mathcal {H}\left( {{Y_{\mathrm {B}}}\left| X \right. } \right) \) and \(\mathcal {H}\left( {{Y_{\mathrm {E}}}\left| X \right. } \right) \) are given by
Then \({C_s}\) in (19) can be written as
where the \({E_{{f_X}}}\left\{ {\ln \left[ {{{\left( {1 + \varsigma _2^2X} \right) } / {\left( {X + \varsigma _1^2{X^2}} \right) }}} \right] } \right\} \) is tends to zero when X is infinite.
We select an input distribution \({f_X}\left( x \right) \) to maximizes the under the input constrains (3) and (4). Such an optimization problem as following can be solved to find the better input PDF.
Then an optimal distribution problem can be transformed as
This problem can be solved by variational method. Assuming that optimal result in (24) is \({f_X}\left( x \right) \), then define a perturbation function as
where \(\varepsilon \) is a variable, and \(\eta \left( x \right) \) is a function, where x is the independent variable of the function. And the perturbation function in (26) should also satisfy the constraints in (24). So we have
Then define a function \(\rho \left( \varepsilon \right) \), where \(\varepsilon \) is the independent variable of the function.
The extremum value is obtained when \(\varepsilon = 0\), the first variation can be expressed as
So we have
where b and c are the free parameters. Submitting (30) into the constrains in (25), we have
And \({f_X}\left( x \right) \) in (30) can be written as
The unknowns \(\mathcal {H}\left( X \right) \), \(\frac{1}{2}{E_{{f_X}}}\left\{ {\ln \left( X \right) } \right\} \) and \(\frac{1}{2}{E_{{f_X}}}\big \{ {\ln } \) \( {\left[ {{{\left( {1 + \varsigma _2^2X} \right) } / {\left( {X + \varsigma _1^2{X^2}} \right) }}} \right] } \big \}\) in (23) can be solved as following
As for \({\mathop {\mathrm {var}}} \left( {{Y_{\mathrm {E}}}} \right) \), we have
The \({\mathrm {var}}\left( X \right) \) and \({\mathop {\mathrm {var}}} \left( {\sqrt{X} {Z_{2}}} \right) \) can be written as
Submitting (36) and (37) into (35), we have
Then substituting (33), (34) and (38) into (23), (8) can be derived.
1.2 6.2 Appendix B
Expression (14) can be written as
\({I_1}\) can be written as
Obtained by (21) and (22), \(\mathcal {H}\left( {\left. {{Y_\mathrm{{B}}}} \right| {X^*}} \right) \) and \(\mathcal {H}\left( {\left. {{Y_\mathrm{{E}}}} \right| {X^*}} \right) \) can be written as
As for \(H\left( {\left. {{Y_\mathrm{{E}}}} \right| {X^*},{Y_\mathrm{{B}}}} \right) \), the conditional PDF \({f_{\left. {{Y_\mathrm{{E}}}} \right| {Y_\mathrm{{B}}}}}\left( {\left. {{y_\mathrm{{E}}}} \right| {y_\mathrm{{B}}}} \right) \) can be expressed as
\(\mathcal {H}\left( {\left. {{Y_\mathrm{{E}}}} \right| {X^*},{Y_\mathrm{{B}}}} \right) \) can be expressed as
Substituting (43) and (41) into (40), we have
One of the difficulties in solving \({I_2}\) is that the input signal X has no peak intensity constraints so it is difficult to find the bounds of upper bound because the signal range can be arbitrarily large. In order to get \({I_2}\), \({g_{\left. {{Y_\mathrm{{B}}}} \right| {Y_\mathrm{{E}}}}}\left( {\left. {{y_\mathrm{{B}}}} \right| {y_\mathrm{{E}}}} \right) \) can be chosen as [10]
where s and \(\mu \) are free parameters.
Let \(p \!=\! \left[ {{{H_\mathrm{{E}}^2}/{H_\mathrm{{B}}^2}} + \left( {{{H_\mathrm{{E}}^2}/ {{H_\mathrm{{B}}}}}} \right) x\varsigma _1^2} \right] \sigma _\mathrm{{B}}^2\mathrm{{ + }} \left( {1 + {H_\mathrm{{E}}}x\varsigma _2^2} \right) \sigma _\mathrm{{E}}^2\) and \(q = \left( {1 + {H_\mathrm{{B}}}x\varsigma _1^2} \right) \sigma _\mathrm{{B}}^2\). Therefore, \({f_{\left. {{Y_\mathrm{{B}}}{Y_\mathrm{{E}}}} \right| X}}\left( {\left. {{y_\mathrm{{B}}},{y_\mathrm{{E}}}} \right| X} \right) \) can be written as
Substituting (46) into (39), \({I_2}\) can be written as
Then \({I_2}\) can be upper-bounded by
Case1: when \(\mu < 0\), \({I_3}\) is given by
Case2: when , \({I_3}\) is given by
So \({I_3}\) can be lower-bounded by
Case3: when , \({I_3}\) is given by
From three cases, \({I_3}\) can be expressed as
Substituting (53) into (48) \({I_2}\) can be written as
Then substituting (44) and (54) into (39), the secrecy capacity (15) can be derived.
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Huang, B., Dai, J. (2019). Secrecy Capacity Analysis for Indoor Visible Light Communications with Input-Dependent Gaussian Noise. In: Gui, G., Yun, L. (eds) Advanced Hybrid Information Processing. ADHIP 2019. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 301. Springer, Cham. https://doi.org/10.1007/978-3-030-36402-1_4
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