Secrecy Outage Probability and Strictly Positive Secrecy Capacity of Cognitive Radio Networks with Adaptive Transmit Power

Abstract

In this article, we derive the Secrecy Outage Probability (SOP) and the Probability of Strictly Positive Secrecy Capacity (SPSC) for Cognitive Radio Networks with Adaptive Transmit Power. Our analysis takes into consideration the interference between primary and secondary nodes. The Secondary Source $S_S$ and relays $R_k$ adapt their power to generate low interference at Primary Receiver ($P_R$). We derive the SOP and SPSC at the secondary destination in the presence of an eavesdropper.

Appendices

Appendix A

We can write

$$\begin{aligned}{} & {} P\left( \frac{T|h_{SE}|^{2}}{|h_{SP_{{R}}}|^{2}N_{0}}<x||h_{SP_{{R}}}|^{2}>\frac{T}{E_{max}}\right) \nonumber \\{} & {} =P\left( |h_{SE}|^{2}<\frac{x|h_{SP_{{R}}}|^{2}N_{0} }{T}||h_{SP_{{R}}}|^{2}>\frac{T}{E_{max}}\right) \nonumber \\{} & {} =e^{\frac{T}{\beta _{SP_{{R}}}^{2}E_{max}}}\int _{\frac{T}{E_{max}} }^{+\infty }P\left( |h_{SE}|^{2}<\frac{xuN_{0}}{T}\right) \frac{e^{-\frac{u}{ \beta _{SP_{{R}}}^{2}}}}{\beta _{SP_{{R}}}^{2}}du \nonumber \\{} & {} =e^{\frac{T}{\beta _{SP_{{R}}}^{2}E_{max}}}\int _{\frac{T}{E_{max}} }^{+\infty }\left[ 1-e^{-\frac{xuN_{0}}{T\beta _{SE}^{2}}}\right] \frac{e^{- \frac{u}{\beta _{SP_{{R}}}^{2}}}}{\beta _{SP_{{R}}}^{2}}du \nonumber \\{} & {} =1-\frac{e^{-\frac{N_{0}x}{\beta _{SE}^{2}E_{max}}}}{1+\frac{\beta _{SP_{{R}}}^{2}xN_{0}}{T\beta _{SE}^{2}}} \end{aligned}$$

(22)

Appendix B

If $\frac{T}{|h_{SP_R}|^2}>E_{max}$, the SINR at E is equal to

$$\begin{aligned} \Gamma _{E}=\frac{E_{max}|h_{SE}|^{2}}{E_{P_{T}}|h_{P_TE}|^{2}+N_{0}} \end{aligned}$$

(23)

The CDF of the SINR is equal to

$$\begin{aligned} F_{\Gamma _{E}}(x)=P(E_{max}|h_{SE}|^{2}<x (E_{P_{T}}|h_{P_TE}|^{2}+N_{0})) \end{aligned}$$

(24)

We deduce

$$\begin{aligned} F_{\Gamma _{E}}(x )=\, & {} P(E_{max}|h_{SE}|^{2}<x(N_{0}+E_{P_{T}}|h_{P_TE}|^{2})) \nonumber \\= & {} \int _{N_{0}}^{+\infty }F_{E_{max}|h_{SE}|^{2}}(x u)f_{E_{P_{T}}|h_{P_TE}|^{2}}(u-N_{0})du \end{aligned}$$

(25)

We have

$$\begin{aligned} F_{\Gamma _{E}}(x )=\, & {} \int _{N_{0}}^{+\infty }\left[ 1-e^{-\frac{ x u}{E_{max}\beta _{SE}^{2}}}\right] e^{-\frac{(u-N_{0})}{E_{P_T}\beta _{P_{T}E}^{2}}}\frac{1}{E_{P_T}\beta _{P_TE}^{2}}du \nonumber \\= & {} 1-\frac{E_{max}\beta _{SE}^{2}}{E_{max}\beta _{SE}^{2}+x E_{P_T}\beta _{P_TE}^{2}} e^{-\frac{N_{0}x }{E_{max}\beta _{SE}^{2}}} \end{aligned}$$

(26)

Appendix C

If $\frac{T}{|h_{SP_R}|^2}<E_{max}$, the SINR at E is equal to

$$\begin{aligned} \Gamma _{E}=\frac{T|h_{SE}|^{2}}{|h_{SP_{R}}|^{2}(N_0+E_{P_{T}}|h_{P_TE}|^{2})} \end{aligned}$$

(27)

For Rayleigh channels, $|h_{SE}|^{2}$,$|h_{SP_{R}}|^{2}$ and $|h_{P_TE}|^{2}$ have an exponential distribution with mean $\beta _{i}=E(U_{i})$.

We have

$$\begin{aligned}{} & {} P\left( \Gamma _{E}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E_{max}\right) =P\left( \Gamma _{E}<x||h_{SP_{R}}|^{2}>\frac{T}{E_{max}}\right) \nonumber \\{} & {} =P\left( T|h_{SE}|^{2}<x|h_{SP_{R}}|^{2}(N_0+E_{P_{T}}|h_{P_TE}|^{2})||h_{SP_{R}}|^{2}>\frac{T}{E_{max}} \right) \end{aligned}$$

(28)

Let $U=N_0+E_{P_{T}}|h_{P_TE}|^{2}$. The CDF and PDF of U are equal to

$$\begin{aligned} F_{U}(w)= & {} F_{|h_{P_TE}|^{2}}\left( \frac{w-N_0}{E_{P_{T}}}\right) \end{aligned}$$

(29)

$$\begin{aligned} f_{U}(w)= & {} \frac{1}{E_{P_{T}}}f_{|h_{P_TE}|^{2}}\left( \frac{w-N_0}{E_{P_{T}}}\right) . \end{aligned}$$

(30)

Equation 28 can be expressed as

$$\begin{aligned}{} & {} P\left( T|h_{SE}|^{2}<x|h_{SP_{R}}|^{2}(N_0+E_{P_{T}}|h_{P_TE}|^{2})||h_{SP_{R}}|^{2}>\frac{T}{E_{max}}\right) \nonumber \\{} & {} =\int _{\frac{T}{E_{max}}}^{+\infty }\int _{N_0}^{+\infty }P(T|h_{SE}|^{2}<xvw)f_{|h_{SP_{R}}|^{2}||h_{SP_{R}}|^{2}>\frac{T}{E_{max}}}(v)f_{U}(w)dvdw\nonumber \\{} & {} =\int _{N_0}^{+\infty }\int _{\frac{T}{E_{max}}}^{+\infty }e^{\frac{T}{ E_{\max }\beta ^2_{SP_R}}}\left[ 1-e^{-\frac{xvw}{T\beta ^2_{SE}}}\right] \frac{e^{-\frac{v}{\beta ^2_{SP_R}}}}{\beta ^2_{SP_R}}dv\frac{e^{-\frac{(w-N_0)}{ E_{P_{T}}\beta ^2_{P_TE}}}}{E_{P_{T}}\beta ^2_{P_TE}}dw \end{aligned}$$

(31)

We have

$$\begin{aligned} \int _{\frac{T}{E_{max}}}^{+\infty }e^{\frac{T}{E_{\max }\beta ^2_{SP_R}}} \left[ 1-e^{-\frac{xvw}{T\beta ^2_{SE}}}\right] \frac{e^{-\frac{v}{ \beta ^2_{SP_R}}}}{\beta ^2_{SP_R}}dv=1-\frac{e^{-\frac{Txw}{E_{max}e_{1\beta ^2_{SE}}}}}{1+\frac{\beta ^2_{SP_R}xw}{\beta ^2_{SE}T}} \end{aligned}$$

(32)

Using (31) and (32), we deduce

$$\begin{aligned}{} & {} P\left( \Gamma _{E}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E_{max}\right) =\int _{N_0}^{+\infty }\left[ 1-\frac{e^{-\frac{Txw}{E_{max}e_{1\beta ^2_{SE}}}}}{1+\frac{\beta ^2_{SP_R}xw}{\beta ^2_{SE}T}}\right] \frac{e^{-\frac{ (w-N_0)}{E_{P_{T}}\beta ^2_{P_TE}}}}{E_{P_{T}}\beta ^2_{P_TE}}dw\nonumber \\{} & {} =1-\frac{e^{\frac{N_0}{E_{P_{T}}\beta ^2_{P_TE}}}}{E_{P_{T}}\beta ^2_{P_TE}} \int _{N_0}^{+\infty }\frac{e^{-w\left( \frac{1}{E_{P_{T}}\beta ^2_{P_TE}}+\frac{xT }{E_{max}T\beta ^2_{SE}}\right) }}{1+\frac{\beta ^2_{SP_R}xw}{\beta ^2_{SE}T}}dw \end{aligned}$$

(33)

Let

$$\begin{aligned} z=1+\frac{\beta ^2_{SP_R}xw}{\beta ^2_{SE}T} \end{aligned}$$

(34)

We deduce

$$\begin{aligned}{} & {} P\left( \Gamma _{E}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E_{max}\right) =1-\frac{ e^{\frac{N_0}{E_{P_{T}}\beta ^2_{P_TE}}}}{E_{P_{T}}\beta ^2_{P_TE}}\frac{\beta ^2_{SE}T}{\beta ^2_{SP_R}x}\nonumber \\{} & {} \qquad \times \int _{1+\frac{\beta ^2_{SP_R}xN_0}{\beta ^2_{SE}T} }^{+\infty }\frac{e^{-\left( z-1\right) \frac{\beta ^2_{SE}T}{\beta ^2_{SP_R}x}\left( \frac{1}{E_{P_{T}}\beta ^2_{P_TE}}+\frac{xT}{E_{max}T\beta ^2_{SE} }\right) }}{z}dz\nonumber \\{} & {} \quad =1-\frac{e^{\frac{N_0}{E_{P_{T}}\beta ^2_{P_TE}}}}{E_{P_{T}}\beta ^2_{P_TE}}\frac{ \beta ^2_{SE}T}{\beta ^2_{SP_R}x}e^\frac{\beta ^2_{SE}T}{E_{P_{T}} \beta ^2_{P_TE}\beta ^2_{SP_R}x}\nonumber \\{} & {} \qquad +\frac{T}{E_{max}\beta ^2_{SP_R}}\times E_{i}\left( \left( \frac{ \beta ^2_{SE}T}{\beta ^2_{SP_R}x}+N_0\right) \left( \frac{1}{E_{P_{T}}\beta ^2_{P_TE}}+\frac{xT}{E_{max}T\beta ^2_{SE}}\right) \right) \end{aligned}$$

(35)

where

$$\begin{aligned} E_{i}(x)=\int _{x}^{+\infty }\frac{e^{-t}}{t}dt. \end{aligned}$$

(36)