Optimal Throughput Under Time Power Switching Based Relaying Protocol in Energy Harvesting Cooperative Networks

Abstract

This paper propose novel protocol for energy harvesting enabled relaying networks. To evaluate performance, we investigate how the harvested power at relay node affects on signal to noise ratio, outage probability and optimal throughput. Specifically, we develop outage and throughput expression in terms of time and power factors in the proposed time power switching based relaying (TPSR) protocol. A highly accurate closed-form formula of outage probability and throughput are also derived. It is shown that the maximized throughput critically depends on optimal time switching and optimal power splitting coefficients of the proposed protocol. In addition, we compare performance of the energy harvesting protocol in optimal case together with balanced receiver at relay node. The impressive results in this work proved that proposed protocol outperforms power splitting based relaying protocol presented in the literature. The tightness of our proposed protocol is determined through Monte Carlo simulation results. Finally, our results provide useful guidelines for design of the energy harvesting relay node in cooperative networks.

Appendix

Proof

(Proof of Proposition 1): It should be noted that $h_S$ and $h_D$ are two independent random variables. In addition, it is observed and also can be verified that the factor in the denominator, $a|h_S|^2-c \ne 0$. Thus, $P_{out}^{\textit{TPSR}}$ is rewritten in two cases as

• If $|h_S|^2>c/a$ then

  $$\begin{aligned} P_{out}^{\textit{TPSR}}= \Pr \left( \left| h_D\right| ^2<\frac{b}{a\left| h_S\right| ^2-c}\right) \end{aligned}$$

  (26)

• If $|h_S|^2<c/a$ then

  $$\begin{aligned} P_{out}^{\textit{TPSR}} = Pr\left( {{{\left| {{h_D}} \right| }^2} > \frac{b }{{a {{\left| {{h_S}} \right| }^2} - c }}} \right) = 1. \end{aligned}$$

  (27)

We obtain the equality in (27) due to the fact that if the value of $|h_S|^2<c/a$ is the factor in denominator $a|h_S|^2-c$ will be a negative number and probability of $|h_D|^2$ being greater than some negative number is always equal to 1. Therefore, $P_{out}^{\textit{TPSR}}$ can be written as

$$\begin{aligned} P_{out}^{\textit{TPSR}}= & {} \int \limits _0^{c/a} \Pr \left( \left| h_D \right| ^2 > \frac{b}{a x - c } \right) f_{\left| {h_S}\right| ^2}(x)dx \\&+ \int \limits _{c/a}^\infty \Pr \left( \left| h_D\right| ^2<\frac{b}{ax-c}\right) f_{\left| {h_S}\right| ^2}(x)dx \end{aligned}$$

(28)

Substituting (27) into (28) yields

$$\begin{aligned} P_{out}^{\textit{TPSR}}= & {} \int \limits _0^{c/a} f_{\left| {h_S}\right| ^2}(x)dx \\&+ \int \limits _{c/a}^\infty \Pr \left( 1- \exp \left( -\frac{b}{\left( ax-c\right) \varOmega _D}\right) \right) f_{\left| {h_S}\right| ^2}(x)dx \end{aligned}$$

(29)

where is the integration variable, $f_{|{h_S}|^2}(x) \buildrel \varDelta \over = \frac{1}{\varOmega _S} e^{ - \frac{x}{\varOmega _S}}$ is the probability density function (PDF) of exponential distributed random variable $|h_S|^2$ while $F_{|h_D|^2}(x) \buildrel \varDelta \over =\Pr (|h_D|^2<x)=1-e^{-x/\varOmega _D}$ is the cumulative distribution function (CDF) of the exponential distributed random variable $|h_D|^2$. Thus, $P_{out}^{\textit{TPSR}}$ can be calculated by

$$\begin{aligned} P_{out}^{\textit{TPSR}} = 1 - \frac{1}{\varOmega _S}\int \limits _{c/a}^\infty \exp \left( - \frac{x}{\varOmega _S} - \frac{b }{\left( a x - c \right) \varOmega _D} \right) dx. \end{aligned}$$

(30)

Let us define a new integration variable $y=ax-c$. Thus, we obtain expression as

$$\begin{aligned} P_{out}^{\textit{TPSR}}=1- \frac{1}{a\varOmega _S}\exp \left( -\frac{c}{a\varOmega _S}\right) \int \limits _{0}^\infty \exp \left( -\frac{y}{a\varOmega _S}-\frac{b}{y\varOmega _D}\right) dy. \end{aligned}$$

(31)

Finally, to obtain (15), we use (3.324.1) given in [28]. This completes the proof.