Secure Communication for RF Energy Harvesting NOMA Relaying Networks with Relay-User Selection Scheme and Optimization

Abstract

In this paper, we study the physical layer security (PLS) of a radio frequency energy harvesting (RF EH) non-orthogonal multiple access (NOMA) relaying network where relay-user selection scheme and secrecy performance optimization are investigated. Specifically, we consider a scenario, where the energy-constrained users from two different user clusters harvest RF energy from the selected relay to transmit their messages to a destination with the help of a relay cluster in the presence of a passive eavesdropper. To enhance the PLS performance of the considered system, we propose a relay and user selection scheme and an optimal algorithm to find the best system parameter set including time switching ratio and power allocation ratio, which can bring the best secrecy performance for the system. Accordingly, to evaluate the secrecy performance, the closed-form expressions of secrecy outage probability are derived. Moreover, the impacts of the network parameters on the secrecy performance are investigated to achieve more insight into the behavior of NOMA relaying network with RF EH. Finally, the accuracy of our analysis is verified by Monte-Carlo simulation results.

Appendices

Appendix A: Proof of Lemma 1

Here, from Eq. 34 we derive the closed-form expression of $P^{A}_{out}$ as follows.

$$\begin{array}{@{}rcl@{}} P^{A}_{out} &=& \Pr(C_{x_{A}} < C_{x_{A}}^{th})=\Pr \left( \frac{\rho_{0} \beta \gamma_{0} {g^{2}_{A}}}{(1-\rho_{0}) \beta \gamma_{0} {g^{2}_{B}}+1}<\gamma^{A}_{th}\right) \\ &=& {\int}_{0}^{\infty} F_{g_{A}}\left( \sqrt{a_{1} x^{2}+b_{1}}\right)f_{g_{B}}(x)dx \\ &=& \sum\limits_{i=0}^{M}\sum\limits_{j=1}^{N}\left( \begin{array}{c} M \\ i \end{array}\right)\left( \begin{array}{c} N \\ j \end{array}\right)\frac{(-1)^{i+j+1}j}{\lambda_{B}}\\ &&*{\int}_{0}^{\infty} e^{-\frac{i\sqrt{a_{1} x^{2}+b_{1}}}{\lambda_{A}}-\frac{jx}{\lambda_{B}}}dx \\ &&\overset{(a)}{=} \frac{\pi}{\lambda_{B} Q}\sum\limits_{i=0}^{M}\sum\limits_{j=1}^{N}\sum\limits_{q=1}^{Q}\left( \begin{array}{c} M \\ i \end{array}\right)\left( \begin{array}{c} N \\ j \end{array}\right)(-1)^{i+j+1}j\\ &&*e^{-\frac{i\sqrt{a_{1}{\theta_{q}^{2}}+b_{1}}}{\lambda_{A}}-\frac{j\theta_{q}}{\lambda_{B}}}\sqrt{\frac{1-\phi_{q}}{1+\phi_{q}}}. \end{array}$$

(A-1)

where $a_{1}=\frac {(1-\rho _{0})\gamma ^{A}_{th}}{\rho _{0}}$, $b_{1}=\frac {\gamma ^{A}_{th}}{\rho _{0} \beta \gamma _{0}}$, $\phi _{q}=\cos \limits \left (\frac {2q-1}{2Q}\pi \right )$, $\theta _{q}=\frac {\phi _{q}+1}{2}$. Note that step (a) is obtained by applying the Gaussian-Chebyshev quadrature method with Q are the complexity-vs-accuracy trade-off coefficients. This ends our proof.

Appendix B: Proof of Lemma 3

Here, we derive the closed-form expression of ${P^{A}_{s}}$ as follows.

$$\begin{array}{@{}rcl@{}} {P^{A}_{s}} &=& {\int}_{0}^{\infty}{\int}_{0}^{2^{{R^{A}_{s}}}(y+1)-1} f_{\gamma^{x_{A}}_{D}}\left( x\right)f_{\gamma^{x_{A}}_{E}}(y)dxdy \\ &=& {\int}_{0}^{\infty} F_{\gamma^{x_{A}}_{D}}\left( 2^{{R^{A}_{s}}}(y+1)-1\right)f_{\gamma^{x_{A}}_{E}}(y)dy \\ &=& \begin{cases} \sum\limits_{k=0}^{K}\left( \begin{array}{c} K \\ k \end{array}\right)\frac{(-1)^{k} \rho_{1}}{\lambda_{E}\gamma_{E}}\\ *{\int}_{0}^{c_{2}} \frac{e^{-\frac{k[2^{{R^{A}_{s}}}(y+1)-1]}{\lambda_{D} \gamma_{0} \{\rho_{1}-\rho_{2}[2^{{R^{A}_{s}}}(y+1)-1]\}}-\frac{y}{\lambda_{E} \gamma_{E}[\rho_{1}-(\rho_{2}+\rho_{3})y]}}}{[\rho_{1}-(\rho_{2}+\rho_{3})y]^{2}}dy\\ , c_{1} \geq c_{2}\\ \sum\limits_{k=0}^{K}\left( \begin{array}{c} K \\ k \end{array}\right)\frac{(-1)^{k} \rho_{1}}{\lambda_{E}\gamma_{E}}\\ *{\int}_{0}^{c_{1}} \frac{e^{-\frac{k[2^{{R^{A}_{s}}}(y+1)-1]}{\lambda_{D} \gamma_{0} \{\rho_{1}-\rho_{2}[2^{{R^{A}_{s}}}(y+1)-1]\}}-\frac{y}{\lambda_{E} \gamma_{E}[\rho_{1}-(\rho_{2}+\rho_{3})y]}}}{[\rho_{1}-(\rho_{2}+\rho_{3})y]^{2}}dy\\ +{\int}_{c_{1}}^{c_{2}}f_{\gamma^{x_{A}}_{E}}(y)dy\\ , c_{1} < c_{2} \end{cases} \\ &=& \begin{cases} I_{A}, & c_{1} \geq c_{2}\\ I_{A}+e^{-\frac{c_{1}}{\lambda_{E} \gamma_{E}[\rho_{1}-(\rho_{2}+\rho_{3})c_{1}]}}, & c_{1} < c_{2} \end{cases}, \end{array}$$

(B-2)

where

$$\begin{array}{@{}rcl@{}} I_{A} &&\overset{(b)}{=} \frac{\pi c}{2Q}\sum\limits_{k=0}^{K}\sum\limits_{q=1}^{Q}\left( \begin{array}{c} K \\ k \end{array}\right) * \sqrt{1-{\phi_{q}^{2}}}\\ &&\frac{(-1)^{k} \rho_{1}e^{-\frac{k[2^{{R^{A}_{s}}}(\theta_{q}+1)-1]}{\lambda_{D} \gamma_{0} \{\rho_{1}-\rho_{2}[2^{{R^{A}_{s}}}(\theta_{q}+1)-1]\}}-\frac{\theta_{q}}{\lambda_{E}\gamma_{E}[\rho_{1}-(\rho_{2}+\rho_{3})\theta_{q}]}}}{\lambda_{E}\gamma_{E}[\rho_{1}-(\rho_{2}+\rho_{3})\theta_{q}]^{2}}, \end{array}$$

$c_{1} = \frac {\rho _{1}+\rho _{2}(1-2^{{R^{A}_{s}}})}{\rho _{2} 2^{{R^{A}_{s}}}}$, $c_{2} = \frac {\rho _{1}}{\rho _{2}+\rho _{3}}$, $c=\min \limits \{c_{1},c_{2}\}$, $\phi _{q}=\cos \limits \left (\frac {2q-1}{2Q}\pi \right )$, $\theta _{q}=\frac {(\phi _{q}+1)c}{2}$. Note that step (b) is obtained by applying the Gaussian-Chebyshev quadrature method with Q are the complexity-vs-accuracy trade-off coefficients. This concludes our proof.