Performance Analysis on Wireless Power Transfer Wireless Sensor Network with Best AF Relay Selection over Nakagami-m Fading

Abstract

In this paper, we present the performance analysis of energy harvesting amplify-and-forward (AF) relaying wireless sensor network with best relay selection scheme over Nakagami-m fading. Specifically, this considered network consists of one sink, multiple energy-constrained relays, and one destination sensor node. The best relay is chosen to amplify and forward the message to the destination after powered by the sink. In order to evaluate the performance, the closed-form expression of outage probability and throughput are derived by applying the discrete optimal power splitting ratio. Based on this expression, we investigate the behavior of this network according to the key parameters such as transmit power, number of relays, time switching ratio and the distance.

Appendix

Here, we derive the expression of $P^*_{out}$ as (12) on the top of next page. Substituting (12) into (8), we obtain the closed-form expression of outage probability for this system.

$$\begin{aligned}&P^*_{out} = \Pr (\gamma _{e2e}^{*}<2^{\frac{2R}{1-\alpha }}-1) \nonumber \\&= 1-\overset{L-1}{\underset{l=1}{\sum }}\Pr \left( \dfrac{c\gamma _{0}\gamma _{1}\gamma _{2}(1-\rho _{l})\rho _{l}}{c(1-\rho _{l})\gamma _{2}+\rho _{l}}>\gamma _{th},b_{l}\le \gamma _{2}<b_{l+1}\right) \nonumber \\&= 1-\overset{L-1}{\underset{l=1}{\sum }}\overset{b_{l+1}}{\underset{b_l}{\int }}\left[ 1-F_{\gamma _{1}}\left( \dfrac{c(1-\rho _{l})\gamma _{th}x+\rho _{l}\gamma _{th}}{c(1-\rho _{l})\rho _{l}\gamma _{0}x}\right) \right] f_{\gamma _{2}}(x)dx\nonumber \\&= 1-\overset{L-1}{\underset{l=1}{\sum }}\overset{m_{1}-1}{\underset{j=0}{\sum }}\overset{j}{\underset{i=0}{\sum }}\dfrac{e^{-\tfrac{m_{1}\gamma _{th}}{\lambda _{1}\rho _{l}\gamma _{0}}}}{i!(j-i)!(m_{2}-1)!c^{i}(1-\rho _{l})^{i}\rho _{l}^{j-i}}\left( \dfrac{m_{1}\gamma _{th}}{\lambda _{1}\gamma _{0}}\right) ^{j}\left( \dfrac{m_{2}}{\lambda _{2}}\right) ^{m_{2}}\nonumber \\&\times \overset{b_{l+1}}{\underset{b_l}{\int }}x^{m_{2}-i-1}e^{-\tfrac{m_{1}\gamma _{th}}{\lambda _{1}c(1-\rho _{l})\gamma _{0}x}-\tfrac{m_{2}x}{\lambda _{2}}}dx \end{aligned}$$

$$\begin{aligned}= & {} 1-\overset{L-1}{\underset{l=1}{\sum }}\overset{m_{1}-1}{\underset{j=0}{\sum }}\overset{j}{\underset{i=0}{\sum }}\dfrac{e^{-\tfrac{m_{1}\gamma _{th}}{\lambda _{1}\rho _{l}\gamma _{0}}}}{i!(j-i)!(m_{2}-1)!c^{i}(1-\rho _{l})^{i}\rho _{l}^{j-i}}\left( \dfrac{m_{1}\gamma _{th}}{\lambda _{1}\gamma _{0}}\right) ^{j}\left( \dfrac{m_{2}}{\lambda _{2}}\right) ^{m_{2}}\nonumber \\\times & {} \left[ \overset{\infty }{\underset{b_l}{\int }}x^{m_{2}-i-1}e^{-\tfrac{m_{1}\gamma _{th}}{\lambda _{1}c(1-\rho _{l})\gamma _{0}x}-\tfrac{m_{2}x}{\lambda _{2}}}dx-\overset{\infty }{\underset{b_{l+1}}{\int }}x^{m_{2}-i-1}e^{-\tfrac{m_{1}\gamma _{th}}{\lambda _{1}c(1-\rho _{l})\gamma _{0}x}-\tfrac{m_{2}x}{\lambda _{2}}}dx\right] \nonumber \end{aligned}$$

(12)

where $\gamma _{th}=2^{\tfrac{2R}{(1-\alpha )}}-1$. Note that step (b) and (c) are obtained by the help of (1.211-1) and (3.381-3), respectively, in [22].

This concludes our proof.