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

Abstract

In this paper, a secure energy harvesting relay communication system with partial relay selection over Nakagami-m fading channels is proposed. A power beacon can provide wireless energy for the source and relay. A time-switching-based (TS) radio frequency energy harvesting technique is deployed at the power beacon. An eavesdropper is able to wiretap to the signal transmitted from the source and the relays. The exact closed-form expression of secrecy outage probability is derived. The results show that with increasing number of relays the system performs better in terms of secrecy outage probability (SOP). In addition, the energy harvesting duration has a significant effect on the secrecy outage probability. There exist an optimal energy harvesting duration that can achieve the lowest SOP and therefore this parameter should be carefully designed.

A Appendices

1.1 A.1 Proof of Lemma 1

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

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

(A.1)

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

(A.2)

The CDF of $\gamma _{1\mathsf {PRS}}$ is expressed as

$$\begin{aligned}&P{(\gamma _{1\mathsf {PRS}}<x)}= 1+\sum _{k=1}^{K}\sum _{v=0}^{l}\sum _{l=0}^{k(m_{\mathsf {S}\mathsf {R}}-1)}\left( {\begin{array}{c}K\\ k\end{array}}\right) \left( {\begin{array}{c}v\\ l\end{array}}\right) (-1)^{k}(x-1)^{l-v}(\xi )^{v-l}\left( {\gamma }_{\mathsf {E}}x\right) ^{v}\nonumber \\&\quad \times \frac{\varGamma (v+m_{\mathsf {S}\mathsf {E}})}{\varGamma (m_{\mathsf {S}\mathsf {E}})\theta _{\mathsf {S}\mathsf {E}}^{m_{\mathsf {S}\mathsf {E}}}}\frac{w(l,k,m_{\mathsf {S}\mathsf {R}})}{\theta _{\mathsf {S}\mathsf {R}}^{l} {\gamma }_{\mathsf {M}}^{l}} \frac{1}{\varGamma (m_{\mathsf {B}\mathsf {S}})\theta _{\mathsf {B}\mathsf {S}}^{m_{\mathsf {B}\mathsf {S}}}} \left( \frac{\theta _{\mathsf {S}\mathsf {R}}{\gamma }_{\mathsf {M}}\theta _{\mathsf {S}\mathsf {E}}}{k {\gamma }_{\mathsf {E}}x \theta _{\mathsf {S}\mathsf {E}}+ \theta _{\mathsf {S}\mathsf {R}}{\gamma }_{\mathsf {M}}}\right) ^{v+ m_{\mathsf {S}\mathsf {E}}}\nonumber \\&\quad \times \mathbf {K}_{{m_{\mathsf {B}\mathsf {S}}-l+v}}\left( {2\sqrt{\frac{k(x-1)}{\theta _{\mathsf {S}\mathsf {R}}\theta _{\mathsf {B}\mathsf {S}}{\gamma }_{\mathsf {M}}\xi }}}\right) \times 2\times \left( \frac{k(x-1)\theta _{\mathsf {B}\mathsf {S}}}{\theta _{\mathsf {S}\mathsf {R}}{\gamma }_{\mathsf {M}}\xi }\right) ^{\frac{m_{\mathsf {B}\mathsf {S}}-l+v}{2}} \end{aligned}$$

(A.3)

The CDF of $\gamma _{2\mathsf {PRS}}$ is expressed as

$$\begin{aligned}&P{(\gamma _{2\mathsf {PRS}}<x)} =1-\sum _{i=0}^{m_{\mathsf {R}\mathsf {D}}-1}\sum _{j=0}^{i}\left( {\begin{array}{c}i\\ j\end{array}}\right) \frac{1}{i!}\frac{(x-1)^{i-j}({\gamma }_{\mathsf {E}}x)^{j}}{{\gamma }_{\mathsf {M}}^{i} \xi ^{i-j}\theta _{\mathsf {R}\mathsf {D}}^{i}}\frac{\varGamma (v+m_{\mathsf {R}\mathsf {E}})}{\varGamma (m_{\mathsf {R}\mathsf {E}})\theta _{\mathsf {R}\mathsf {E}}^{m_{\mathsf {R}\mathsf {E}}}\varGamma (m_{\mathsf {B}\mathsf {R}})\theta _{\mathsf {B}\mathsf {R}}^{m_{\mathsf {B}\mathsf {R}}}}\nonumber \\&\quad \times \left( \frac{x {\gamma }_{\mathsf {E}}}{{\gamma }_{\mathsf {M}}\theta _{\mathsf {R}\mathsf {D}}}+\frac{1}{\theta _{\mathsf {R}\mathsf {E}}}\right) ^{-(v+m_{\mathsf {R}\mathsf {E}})} \mathbf {K}_{{m_{\mathsf {B}\mathsf {R}}-i+j}}\left( {2\sqrt{\frac{x-1}{\theta _{\mathsf {R}\mathsf {D}}\theta _{\mathsf {B}\mathsf {R}}{\gamma }_{\mathsf {M}}\xi }}}\right) \nonumber \\&\quad \times 2 \times \left( \frac{\theta _{\mathsf {B}\mathsf {R}}(x-1)}{\theta _{\mathsf {R}\mathsf {D}}{\gamma }_{\mathsf {M}}\xi }\right) ^{\frac{m_{\mathsf {B}\mathsf {R}}-i+j}{2}} \end{aligned}$$

(A.4)

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

$$\begin{aligned} F_{{\gamma _{\mathsf {PRS}}}}\left( {\beta }\right) =1-\left[ (1-F_{{\gamma _{1\mathsf {PRS}}}}\left( {\beta }\right) ) (1-F_{{\gamma _{2\mathsf {PRS}}}}\left( {\beta }\right) )\right] \end{aligned}$$

(A.5)

After performing some mathematical manipulations, (20) can be achieved with the help of [25, Eq. (3.471.9)].