Secure Energy Harvesting Communications with Relay Selection over Nakagami-m Fading Channels

Abstract

In this paper, an energy harvesting relay network over Nakagami-m fading is investigated. In the considered system, the power beacon can provide wireless energy for the source and relays which deploy time-switching-based radio frequency energy harvesting technique. Two relay selection schemes, namely partial relay selection and optimal relay selection, are proposed in order to enhance the system performance. In the former, the source only has the channel state information of the first hop, while in the latter it has the full knowledge of the channel state information. The eavesdropper is able to wiretap to the signal transmitted from the source and the relays. The exact closed-form expressions of secrecy outage probability are derived. The results show that optimal relay selection performs better than partial relay selection. With increasing number of relays, the considered system shows better performance. In addition, the energy harvesting duration has a significant effect on the secrecy outage probability.

Appendices

Appendix A: Proof of lemma 1

The SNR of first hop and second hop in PRS scheme can be written as

$$\begin{array}{@{}rcl@{}} \gamma_{1\mathsf{PRS}} &=& \frac{1 + {\gamma}_{\mathsf{M}} \xi |h_{\mathsf{B} \mathsf{S}}|^2 |h_{\mathsf{S} \mathsf{R}_{k^*}}|^2}{1+ {\gamma}_{\mathsf{E}} \xi |h_{\mathsf{B} \mathsf{S}}|^2 |h_{\mathsf{S} \mathsf{E}}|^2}, \end{array} $$

(A.1)

$$\begin{array}{@{}rcl@{}} \gamma_{2\mathsf{PRS}} &=& \frac{1+ {\gamma}_{\mathsf{M}} \xi |h_{\mathsf{B} \mathsf{R}_{k^*}}|^2 |h_{\mathsf{R}_{k^*} \mathsf{D}}|^2}{1+{\gamma}_{\mathsf{E}} \xi |h_{\mathsf{B} \mathsf{R}_{k^*}}|^2 |h_{\mathsf{R}_{k^*} \mathsf{E}}|^2}. \end{array} $$

(A.2)

The CDF of γ _{1P R S} is expressed as

The CDF of γ _{2P R S} is expressed as

The SOP of the considered system in PRS scheme is formulated as follows

$$ F_{{\gamma_{\mathsf{PRS}}}}\left( {\beta}\right)=1-\left[(1-F_{{\gamma_{1\mathsf{PRS}}}}\left( {\beta}\right))(1-F_{{\gamma_{2\mathsf{PRS}}}}\left( {\beta}\right))\right] $$

(A.5)

After performing some mathematical manipulations, Eq. 22 can be achieved with the help of [30, Eq. (3.471.9)].

Appendix B: Proof of lemma 2

In ORS scheme, the SNR of the first and second hop can be derived as

$$\begin{array}{@{}rcl@{}} \gamma_{1\mathsf{ORS}} &= &\frac{1 + {\gamma}_{\mathsf{M}} \xi |h_{\mathsf{B} \mathsf{S}}|^2 |h_{\mathsf{S} \mathsf{R}_k}|^2}{1+ {\gamma}_{\mathsf{E}} \xi |h_{\mathsf{B} \mathsf{S}}|^2 |h_{\mathsf{S} \mathsf{E}}|^2}, \\ \gamma_{2\mathsf{ORS}} &=& \frac{1+ {\gamma}_{\mathsf{M}} \xi |h_{\mathsf{B} \mathsf{R}_{k}}|^2 |h_{\mathsf{R}_k \mathsf{D}}|^2}{1+{\gamma}_{\mathsf{E}} \xi |h_{\mathsf{B} \mathsf{R}_{k}}|^2 |h_{\mathsf{R}_k \mathsf{E}}|^2}. \end{array} $$

(B.1)

We denote that Y _o = |h _{B S} |², and Z _o = |h _{S E} |². The SOP of the considered system in the ORS scheme is calculated as follows

$$\begin{array}{@{}rcl@{}} F_{{\gamma_{\mathsf{ORS}}}}\left( {x}\right)\!&=&\! \int\limits_{0}^{\infty}\!\int\limits_{0}^{\infty}\left[1-(1-F_{{\gamma_{1\mathsf{ORS}} |Y_{o},Z_{o}}}\left( {x}\right))\right.\\&&\left.\!\times(1\,-\,F_{{\gamma_{2\mathsf{ORS}}}}\left( {x}\right))\right]^{K}\!f_{{Y_{o}}}\left( {y}\right)f_{{Z_{o}}}\left( {z}\right)\mathit{dy}\ \mathit{dz}. \end{array} $$

(B.2)

Where in the first hop

Where in the second hop

$$\begin{array}{@{}rcl@{}} &&F_{{\gamma_{2\mathsf{ORS}}}}\left( {x}\right) \\ &&=1-\sum\limits_{u=0}^{m_{\mathsf{R}\mathsf{D}}-1} \sum\limits_{v=0}^{u}\binom{u}{v}\frac{1}{u!}\frac{(x-1)^{u-v}({\gamma}_{\mathsf{E}} x )^{v}}{{\gamma}_{\mathsf{M}}^{u}\xi^{u-v}\theta_{\mathsf{R}\mathsf{D}}^{u}}\\&&\times\frac{1}{\Gamma(m_{\mathsf{R}\mathsf{E}})\theta_{\mathsf{R}\mathsf{E}}^{m_{\mathsf{R}\mathsf{E}}}} {\Gamma}(v+m_{\mathsf{R}\mathsf{E}}) \left( \frac{{\gamma}_{\mathsf{E}} x}{{\gamma}_{\mathsf{M}} \theta_{\mathsf{R}\mathsf{D}}}+\frac{1}{\theta_{\mathsf{R}\mathsf{E}}}\right)^{-(v+m_{\mathsf{R}\mathsf{E}})} \\ &&\times 2\times \left( \frac{[x-1]\theta_{\mathsf{B}\mathsf{R}}}{{\gamma}_{\mathsf{M}} \xi \theta_{\mathsf{R}\mathsf{D}}}\right)^{\frac{v-u+m_{\mathsf{B}\mathsf{R}}}{2}} \frac{1}{\Gamma(m_{\mathsf{B}\mathsf{R}}) \theta_{\mathsf{B}\mathsf{R}}^{m_{\mathsf{B}\mathsf{R}}}} \mathbf{K}_{{v-u+m_{\mathsf{B}\mathsf{R}}}}\\&&\times\left( {2\sqrt{\frac{x-1}{{\gamma}_{\mathsf{M}} \xi \theta_{\mathsf{R}\mathsf{D}} \theta_{\mathsf{B}\mathsf{R}}}}}\right). \end{array} $$

(B.4)

After performing some mathematical manipulations, Eq. 26 can be achieved with the help of [30, Eq. (3.471.9)].