A New Protocol for Energy Harvesting Decode-and-Forward Relaying Networks

Abstract

This paper investigates radio frequency energy harvesting decode-and-forward (DF) multi-relay networks with hybrid power transfer architecture, i.e. time switching (TS) combines with power splitting (PS). Specifically, this system consists of one source, one destination and multiple energy constraint relay nodes which help to the source transfer information to the destination over Nakagami fading channels. In order to improve the performance and reduce the load of this considered network, an efficient protocol for this system is proposed based on adaptive power splitting ratio adjustment and best relay selection. We aim to evaluate the performance of this considered system by deriving the closed-form expressions for the outage probability (OP) based on the statistical characteristics of signal-to-noise ratio (SNR). In addition, the results show that this scheme outperforms the random relay selection scheme, including transmit SNR, number of relays, active energy and fading severity factor. Our analysis is also verified by Monte Carlo simulation.

Appendix A: Proof of Proposition 1

From (4), (5) and (8), the Eq. (10) can be rewritten as

$$\begin{aligned} {P_{out}}= & {} 1-\Pr [\eta (1-\varepsilon _i)\alpha T{P_0}X_{1i}> E_0), (a_i\gamma _0 X_{1i}-\beta )X_{2i} \ge (2^{\frac{R}{(1-\alpha )T}}-1)] \\= & {} 1-\Pr \left[ \phi _1 \bigg (1-\frac{\varOmega _1}{X_{1i}} \bigg ) X_{1i}> E_0, \bigg ( \phi _2 \bigg (1-\frac{\varOmega _1}{X_{1i}} \bigg ) X_{1i}-\beta \bigg ) X_{2i} \ge \varOmega _2 \right] \\= & {} 1-\Pr \left[ \phi _1 X_{1i}> E_0+\phi _1 \varOmega _1, \bigg ( \phi _2 X_{1i}-\beta -\phi _2 \varOmega _1 \bigg ) X_{2i} \ge \varOmega _2 \right] \\= & {} 1- \Pr \bigg [ X_{1i} > b_1, X_{2i} \ge \frac{\varOmega _2}{(\phi _2 X_{1i}-b_2)} \bigg ] \\= & {} 1- \int \limits _{b_1}^\infty {\left[ 1 - F_{X_{2i}} \left( \frac{\varOmega _2}{\phi _2 x-{b_2}} \right) \right] {f_{{X_{1i}}}}\left( {{x}} \right) dx} \end{aligned}$$

$$\begin{aligned} {P_{out}}= & {} 1 - \sum \limits _{k = 0}^{{m_2} - 1} \frac{{m_1^{m_1}m_2^k}}{{k!(m_1-1)!\lambda _1^{m_1} \lambda _2^k}} \int \limits _{b_1}^\infty {\left( \frac{\varOmega _2}{\phi _2 x-b_2} \right) ^k}x^{m_1-1}{e^{- \frac{{{m_2}}}{{{\lambda _2}}}\frac{\varOmega _2}{\phi _2 x-b_2}{-\textstyle {{{m_1}x} \over {{\lambda _1}}}}}}dx \nonumber \\= & {} 1 - \sum \limits _{k = 0}^{{m_2} - 1} \frac{{m_1^{m_1}m_2^k \varOmega _2^k e^{-\frac{m_1 b_2}{\lambda _1 \phi _2}}}}{{k!(m_1-1)!\lambda _1^{m_1} \lambda _2^k \phi ^{m_1}_2}} \int \limits _0^\infty {y^{-k}}{(y+b_2)}^{m_1-1}{e^{- \frac{{{m_2\varOmega _2}}}{{{\lambda _2 y}}}{-\textstyle {{{m_1}y} \over {{\lambda _1 \phi _2}}}}}}dy \nonumber \\= & {} 1 - \sum \limits _{k = 0}^{{m_2} - 1} \sum \limits _{j = 0}^{{m_1} - 1} \frac{C_{m_1-1}^j {m_1^{m_1}m_2^k \varOmega _2^k b^{m_1-j-1}_2 e^{-\frac{m_1 b_2}{\lambda _1 \phi _2}}}}{{k!(m_1-1)!\lambda _1^{m_1} \lambda _2^k \phi ^{m_1}_2}} \int \limits _0^\infty {y^{j-k}}{e^{- \frac{{{m_2\varOmega _2}}}{{{\lambda _2 y}}}{-\textstyle {{{m_1}y} \over {{\lambda _1 \phi _2}}}}}}dy \nonumber \\= & {} 1 - 2 \sum \limits _{k = 0}^{{m_2} - 1} \sum \limits _{j = 0}^{{m_1} - 1} \frac{{m_2^k \varOmega _2^k b^{m_1-j-1}_2 e^{-\frac{m_1 b_2}{\lambda _1 \phi _2}}}}{{k!j!(m_1-j-1)! \lambda _2^k }} \bigg (\frac{m_1}{\lambda _1 \phi _2} \bigg )^{m_1+k-j-1} u^{j-k+1} \mathcal {K}_{j-k+1} (2u), \end{aligned}$$

(18)

where $\varOmega _1 = \frac{2^{\frac{R}{\alpha T}}-1}{\gamma _0}$, $\varOmega _2 = 2^{\frac{R}{(1-\alpha )T}}-1$, $\phi _1=\eta \alpha T{P_0}$, $\phi _2=\frac{{\eta \alpha \gamma _0}}{{1 - \alpha }}$, $b_1=\varOmega _1+\frac{E_0}{\phi _1}$, $b_2=\beta +\phi _2 \varOmega _1$, $u=\sqrt{\frac{m_1 m_2 \varOmega _2}{\lambda _1 \lambda _2 \phi _2}}$, and $\mathcal {K}_{\nu }(.)$ is the modified Bessel function of the second kind and $\nu ^{th}$ order [13]. Substituting $\varOmega _2=\varOmega$, $b_2=b$ and $\phi _2=\phi$ into (18), the Proposition 1 is completely proved.