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Evaluating the Performance of Cooperative NOMA with Energy Harvesting Under Physical Layer Security

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Abstract

In this paper, we consider a non-orthogonal multiple access system with imperfect successive interference cancellation and physical layer security where the source node communicates with Users via an energy harvesting relay node. This node uses a power-switching architecture to harvest energy from the sources signals and applies an amplify-and-forward protocol to forward signals. Moreover, the transmit jamming or artificial noise is generated by source node to enhance the security of system in the case an eavesdropper tries to overhear the confidential information from the source. To evaluate the secrecy performance of the proposed system, the asymptotic secrecy outage probability over the Rayleigh fading channel is studied. These results are compared with the secrecy performance of the orthogonal multiple access system and the system without the help of relay. Monte-Carlo results are presented to verify the theoretical results.

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Funding

This work was supported by the 2019 Research Fund of the University of Ulsan.

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Authors and Affiliations

  1. University of Ulsan, Ulsan, Republic of Korea

    Thi Anh Le & Hyung Yun Kong

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  1. Thi Anh Le
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  2. Hyung Yun Kong
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Correspondence to Thi Anh Le.

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Appendices

Appendix A Proof of Theorem 1 In (25)

This appendix derives \({P_{{SOP}_1}}\) in (25), at User 1 for NOMA-AF imperfect SIC with EH-relay and physical layer security.

According to (24), in the case that the SNR is very high, then \({m_5}<<{m_2}+{m_3}\); \(d_3^m\sigma _E^2< < P{\alpha _2} + P{\alpha _3}\) . Therefore, (22) can be rewritten as

$$\begin{aligned} {P_{SO{P_1}}}&= \Pr \left( {\tfrac{{{m_1}g_0^{}{g_1}}}{{{m_2}g_0^{}{g_1} + {m_3}{g_1} + {m_4}}}< {\beta _1} + \left( {{\beta _1} + 1} \right) \tfrac{{P{\alpha _1}}}{{P{\alpha _2} + P{\alpha _3}}}} \right) \nonumber \\&= \Pr \left( {\frac{{{m_1}g_0^{}{g_1}}}{{{m_2}g_0^{}{g_1} + {m_3}{g_1} + {m_4}}}< \beta _1^*} \right) \nonumber \\&= \Pr \left[ {\left( {{m_1} - {m_2}\beta _1^*} \right) g_0^{}{g_1} < \beta _1^*{m_3}{g_1} + \beta _1^*{m_4}} \right] \end{aligned}$$
(36)

where \(\beta _1^* = {\beta _1} + \left( {{\beta _1} + 1} \right) \frac{{{\alpha _1}}}{{{\alpha _2} + {\alpha _3}}}\).

In (36), the factor in the denominator, \({m_1} - {m_2}\beta _1^*\) , can be positive or negative; hence, we consider the following two cases.

+ In the case \(\beta _1^* > \frac{{{m_1}}}{{{m_2}}} = \frac{{{\alpha _1}}}{{{\alpha _2}}}\), from (36) we have the secrecy outage probability of U1 can be expressed as

$$\begin{aligned} {P_{SO{P_1}}} = \Pr \left[ {g_0^{}{g_1} > \underbrace{\frac{{\beta _1^*{m_3}{g_1} + \beta _1^*{m_4}}}{{{m_1} - {m_2}\beta _1^*}}}_{ < 0}} \right] = 1 \end{aligned}$$
(37)

+ In the case \(\beta _1^* \le \frac{{{m_1}}}{{{m_2}}} = \frac{{{\alpha _1}}}{{{\alpha _2}}}\), from (36) we have

$$\begin{aligned} {P_{SO{P_1}}} = \Pr \left[ {g_0^{} < {A_1} + \frac{{{A_2}}}{{{g_1}}}} \right] \end{aligned}$$
(38)

We can express (38) as

$$\begin{aligned} {P_{SO{P_1}}} = \int \limits _0^{ + \infty } {{f_Y}\left( y \right) } \Pr \left[ {X < \left( {{A_1} + {A_2}\frac{1}{y}} \right) } \right] dy \end{aligned}$$
(39)

Substituting the PDF of Y and the CDF of X given by (1) into (38), we have the following.

$$\begin{aligned} {P_{SO{P_1}}}&= \int \limits _0^{ + \infty } {{\lambda _1}{e^{ - {\lambda _1}y}}} \Pr \left[ {X < \left( {{A_1} + {A_2}\frac{1}{y}} \right) } \right] dy\nonumber \\&= \int \limits _0^{ + \infty } {{\lambda _1}{e^{ - {\lambda _1}y}}} {F_X}\left( {{A_1} + {A_2}\frac{1}{y}} \right) dy\nonumber \\&= \int \limits _0^{ + \infty } {{\lambda _1}{e^{ - {\lambda _1}y}}\left( {1 - {e^{ - {\lambda _0}\left( {{A_1} + {A_2}\frac{1}{y}} \right) }}} \right) } dy\nonumber \\&= {\lambda _1}\int \limits _0^{ + \infty } {{e^{ - {\lambda _1}y}}dy - } {\lambda _1}{e^{ - {\lambda _0}{A_1}}}\int \limits _0^{ + \infty } {{e^{ - \left( {{\lambda _1}y + {\lambda _0}{A_2}\frac{1}{y}} \right) }}} dy\nonumber \\&= 1 - {\lambda _1}{e^{ - {\lambda _0}{A_1}}}\int \limits _0^{ + \infty } {{e^{ - \left( {{\lambda _1}y + {\lambda _0}{A_2}\frac{1}{y}} \right) }}} dy \end{aligned}$$
(40)

Using [ [23], Eq. (3.324.1)], the SOP of User 1 can be expressed as (25). This ends the proof of Theorem 1.

Appendix B Proof of Theorem 2 In (2)

This appendix shows how to find the expression of the SOP of User 2 in (29). The following expression from (29) is given as

$$\begin{aligned} {P_{SO{P_2}}} =&\Pr \left[ \begin{array}{l} \frac{{{m_1}{g_0}{g_2}}}{{{m_2}{g_0}{g_2} + {m_3}{g_2} + {m_6}}}> {\beta _{1t}},\\ \frac{{1 + \frac{{{m_2}{g_0}{g_2}}}{{\xi {m_1}{g_0}{g_2} + {m_3}{g_2} + {m_6}}}}}{{1 + \frac{{{\alpha _2}}}{{\xi {\alpha _1} + {\alpha _3}}}}}< {\beta _2} \end{array} \right] \nonumber \\&+\Pr \left[ {\frac{{{m_1}{g_0}{g_2}}}{{{m_2}{g_0}{g_2} + {m_3}{g_2} + {m_6}}}< {\beta _{1t}}} \right] \nonumber \\ =&\Pr \left[ \begin{array}{l} \left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_0} > {m_3}{\beta _{1t}} + \frac{{{m_6}{\beta _{1t}}}}{{{g_2}}},\\ \left( {{m_2} - \xi {m_1}{k_1}} \right) {g_0}< {m_3}{k_1} + \frac{{{m_6}{k_1}}}{{{g_2}}} \end{array} \right] \nonumber \\&+ \Pr \left[ {\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_0} < {m_3}{\beta _{1t}} + \frac{{{m_6}{\beta _{1t}}}}{{{g_2}}}} \right] \end{aligned}$$
(41)

Because \({m_1} - {\beta _{1t}}{m_2}\) and \({m_2} - \xi {m_1}{k_1}\)can be positive or negative, (41) is given as follows:

+ In the case \({\beta _{1t}}< \frac{{{m_1}}}{{{m_2}}} < \frac{1}{{\xi {k_1}}}\), (41) can be rewritten as

$$\begin{aligned} {P_{sout2}} =&\Pr \left[ \begin{array}{l} \frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}< {g_0}\ {g_0}< \frac{{{m_3}{k_1}}}{{{m_2} - \xi {m_1}{k_1}}} + \frac{{{m_6}{k_1}}}{{\left( {{m_2} - \xi {m_1}{k_1}} \right) {g_2}}} \end{array} \right] \nonumber \\&+ \Pr \left[ {{g_0}< \frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}} \right] \nonumber \\ =&\Pr \left[ {{A_3} + \frac{{{A_4}}}{{{g_2}}}< {g_0}< {A_5} + \frac{{{A_6}}}{{{g_2}}}} \right] +\nonumber \\&\Pr \left[ {{g_0} < {A_3} + \frac{{{A_4}}}{{{g_2}}}} \right] \nonumber \\ =&\int \limits _0^{ + \infty } {{\lambda _2}{e^{ - {\lambda _2}y}}} \left[ {{F_X}\left( {{A_5} + \tfrac{{{A_6}}}{y}} \right) - {F_X}\left( {{A_3} + \tfrac{{{A_4}}}{y}} \right) } \right] dy\nonumber \\&+ \int \limits _0^{ + \infty } {{\lambda _2}{e^{ - {\lambda _2}y}}} {F_X}\left( {{A_3} + \tfrac{{{A_4}}}{y}} \right) dy\nonumber \\ =&\int \limits _0^{ + \infty } {{\lambda _2}{e^{ - {\lambda _2}y}}} {F_X}\left( {{A_5} + \tfrac{{{A_6}}}{y}} \right) dy\nonumber \\ =&1 - {\lambda _2}{e^{ - {\lambda _0}{A_5}}}\sqrt{\tfrac{{4{\lambda _0}{A_6}}}{{{\lambda _2}}}} {K_1}\left( {\sqrt{4{\lambda _0}{\lambda _2}{A_6}} } \right) \end{aligned}$$
(42)

+ In the case \(\frac{{{m_1}}}{{{m_2}}} > m\mathrm{{ax}}\left\{ {{\beta _{1t}};\frac{1}{{\xi {k_1}}}} \right\}\) (41) can be expressed as

$$\begin{aligned} {P_{sout2}} =&\Pr \left[ \begin{array}{l} {g_0}> \frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}},\\ {g_0} > \underbrace{\frac{{{m_3}{k_1}}}{{{m_2} - \xi {m_1}{k_1}}} + \frac{{{m_6}{k_1}}}{{\left( {{m_2} - \xi {m_1}{k_1}} \right) {g_2}}}}_{< 0} \end{array} \right] \nonumber \\&+ \Pr \left[ {{g_0}< \frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}} \right] \nonumber \\ =&\Pr \left[ {{A_3} + \frac{{{A_4}}}{{{g_2}}}< {g_0}} \right] + \Pr \left[ {{g_0} < {A_3} + \frac{{{A_4}}}{{{g_2}}}} \right] \nonumber \\ =&1 \end{aligned}$$
(43)

+ In the case \(\frac{{{m_1}}}{{{m_2}}} < \min \left\{ {{\beta _{1t}};\frac{1}{{\xi {k_1}}}} \right\}\) from (41) we have

$$\begin{aligned} {P_{sout2}} =&\Pr \left[ \begin{array}{l} {g_0}< \underbrace{\frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}}_{< 0},\\ {g_0}< \underbrace{\frac{{{m_3}{k_1}}}{{{m_2} - \xi {m_1}{k_1}}} + \frac{{{m_6}{k_1}}}{{\left( {{m_2} - \xi {m_1}{k_1}} \right) {g_2}}}}_{> 0} \end{array} \right] \nonumber \\&+ \Pr \left[ {{g_0} > \underbrace{\frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}}_{ < 0}} \right] \nonumber \\ =&0 + 1 = 1 \end{aligned}$$
(44)

+ In the case \(\frac{1}{{\xi {k_1}}}< \frac{{{m_1}}}{{{m_2}}} < {\beta _{1t}}\) we express (41) as

$$\begin{aligned} {P_{sout2}} =&\Pr \left[ \begin{array}{l} {g_0}< \underbrace{\frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}}_{< 0}, \\ {g_0}< \underbrace{\frac{{{m_3}{k_1}}}{{{m_2} - \xi {m_1}{k_1}}} + \frac{{{m_6}{k_1}}}{{\left( {{m_2} - \xi {m_1}{k_1}} \right) {g_2}}}}_{< 0} \end{array} \right] \nonumber \\&+ \Pr \left[ {{g_0} > \underbrace{\frac{{{m_3}{\beta _{1t}}}}{{{m_1} - {\beta _{1t}}{m_2}}} + \frac{{{m_6}{\beta _{1t}}}}{{\left( {{m_1} - {\beta _{1t}}{m_2}} \right) {g_2}}}}_{ < 0}} \right] \nonumber \\ =&0 + 1 = 1 \end{aligned}$$
(45)

This ends the proof of Theorem 2.

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Le, T.A., Kong, H.Y. Evaluating the Performance of Cooperative NOMA with Energy Harvesting Under Physical Layer Security. Wireless Pers Commun 108, 1037–1054 (2019). https://doi.org/10.1007/s11277-019-06452-5

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