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Relay Selection and Performance Analysis of Wireless Energy Harvesting Networks

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Abstract

We proposed an integrated information relay and wireless power supply assisted RF energy harvesting-based cooperative dual-hope decode-and-forward (DF) relaying communication model. The relay node not only aids the communication between energy constrained source and destination but also supply power to them using time switching (TS) protocol. We also proposed a relay selection protocol where the source is capable of selecting an appropriate relay link on the basis of channel gain condition. The performance of the system in terms of outage probability and achievable ergodic capacity over Rayleigh fading channels are thoroughly analyzed. Closed from analytical expression of outage probability of the considered system is derived and authenticated by the Monte-Carlo simulation result. The results show the impact of the number of relay nodes on outage probability and achievable ergodic capacity. Simulation results also demonstrated the optimum energy harvesting time for which system achieves maximum throughput and minimum outage probability.

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Correspondence to Dipen Bepari.

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Appendix 1

Appendix 1

1.1 Proof of Proposition 1 in (7)

Assume \({{P_{{t_i}}^{'}}|{h_{{R_i}S}}|^2} =Y\) and \(\max \{|h_{SR_i}|^2\}=Z\). CDF of Y is given by [34]

$$\begin{aligned} \begin{aligned} {F_Y}(y) = \frac{{\Gamma \left( {L,\frac{y}{{{P_{{t_i}}^{'}}{\lambda _y}}}} \right) }}{{{N_0}}} \end{aligned} \end{aligned}$$
(17)

And PDF of Y is given by [34]

$$\begin{aligned} \begin{aligned} {f_Y}(y) = \frac{{{y^{L - 1}}{e^{ - \frac{y}{{{P_{{t_i}}^{'}}{\lambda _y}}}}}}}{{\Gamma \left( L \right) {{\left( {{P_{{t_i}}^{'}}{\lambda _y}} \right) }^L}}} \end{aligned} \end{aligned}$$
(18)

CDF of Z is given by [21]

$$\begin{aligned} \begin{aligned} {F_Z}\left( z \right)&= {\left( {1 - {e^{ - {\textstyle {z \over {{\lambda _x}}}}}}} \right) ^L} \\&= \sum \limits _{i = 0}^L {{{\left( { - 1} \right) }^l}\left( {_l^L} \right) {e^{ - l\frac{z}{{{\lambda _x}}}}}} \end{aligned} \end{aligned}$$
(19)

The CDF of SNR at relay i.e. \(F_{\gamma _R}(\gamma _{th})\) using (6) is given by

$$\begin{aligned} \begin{aligned}&F_{\gamma _R}(\gamma _{th}) = {\mathcal{P}_r}\left\{ {\frac{{\left\{ {\eta \sum \limits _{i = 1}^L {{P_{{t_i}}^{'}}|{h_{{R_i}S}}{|^2}\frac{{2\alpha }}{{1 - \alpha }}} } \right\} \left\{ {\max \left\{ {|{h_{S{R_i}}}{|^2}} \right\} } \right\} }}{{{N_0}}} < {\gamma _{th}}} \right\} \\&\quad ={\mathcal{P}_r}\{ YZ \le w\} \\&\quad = {\mathcal{P}_r}\left\{ {Z \le {{\left. {\frac{w}{y}} \right| }_{Y = y}}} \right\} \\&\quad = \int _0^\infty {{F_Z}\left( {\frac{w}{y}} \right) {f_Y}\left( y \right) dy} \\&\quad =\int _0^\infty {\sum \limits _{l = 0}^L {{{\left( { - 1} \right) }^l}\left( {_l^L} \right) } {e^{ - l\frac{w}{{{\lambda _x}y}}}}\frac{{{y^{L - 1}}{e^{ - \frac{y}{{{P_{{t_i}}^{'}}{\lambda _y}}}}}}}{{\Gamma \left( L \right) {{\left( {{P_{{t_i}}^{'}}{\lambda _y}} \right) }^L}}}dy} \\&\quad = \frac{1}{{\Gamma \left( L \right) {{\left( {{P_{{t_i}}^{'}}{\lambda _y}} \right) }^L}}}\sum \limits _{l = 0}^L {{{\left( { - 1} \right) }^l}\left( {_l^L} \right) \int \limits _0^\infty {{e^{ - l\frac{w}{{{\lambda _x}y}}}}{{{y^{L - 1}}{e^{ - \frac{y}{{{P_{{t_i}}^{'}}{\lambda _y}}}}}}}dy} } \\&\quad = \frac{1}{{\Gamma \left( L \right) {{\left( {{P_{{t_i}}^{'}}{\lambda _y}} \right) }^L}}}\sum \limits _{l = 0}^L {{{\left( { - 1} \right) }^l}\left( {_l^L} \right) \int \limits _0^\infty {{y^{L - 1}}{e^{ - \left( {\frac{y}{{{P_{{t_i}}^{'}}{\lambda _y}}} + l\frac{w}{{{\lambda _x}y}}} \right) }}dy} }\\&\quad = \frac{1}{{\Gamma \left( L \right) {{\left( {{P_{{t_i}}^{'}}{\lambda _y}} \right) }^L}}}{\sum \limits _{l = 0}^L {{{\left( { - 1} \right) }^l}\left( {_l^L} \right) 2\left( {l\frac{w}{{{\lambda _x}}}{P_{{t_i}}^{'}}{\lambda _y}} \right) } ^L/2}{K_L}\left( {2\sqrt{l\frac{w}{{{\lambda _x}{\lambda _y}{P_{{t_i}}^{'}}}}} } \right) \\ \end{aligned} \end{aligned}$$
(20)

where\(\int \limits _0^\infty x^{\nu -1} e^{ - \frac{\beta }{x} - \gamma x}dx = 2\left( \frac{\beta }{\gamma }\right) ^\frac{\nu }{2}K_{\nu }\left( {2\sqrt{\beta \gamma } } \right)\) [35, §3.471.9] is used and \(K_{\nu }(.)\) is the \(\nu\)th order modified Bessel function of the second kind.

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Biswas, S., Bepari, D. & Mondal, S. Relay Selection and Performance Analysis of Wireless Energy Harvesting Networks. Wireless Pers Commun 114, 3157–3171 (2020). https://doi.org/10.1007/s11277-020-07522-9

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