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Outage Probability Analysis for Energy Harvesting Cooperative Relays in a Clustered Environment

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Abstract

This paper presents the outage probability analysis of the energy harvesting (EH) decode and forward (DF) cooperative relay network when more than one relays are available to assist the communication between source and destination in the presence of the direct connection. The relays use power splitting (PS) protocol with adaptive PS ratio for EH. As wireless EH can be more beneficial over smaller distances therefore a clustered environment is considered in which the source, destination and relays are located in a small area. First, we analyze the performance of selection cooperation (SC) which requires channel state information (CSI) at the source. High signal to noise ratio approximation of the outage probability is provided for this case. Secondly, we present the performance of all relays cooperation (ARC) scheme which requires no CSI at the source. Lower and upper bounds of the outage probability are presented for smaller number of relays in ARC scheme whereas high signal to noise ratio approximation is provided for higher number of relays. Simulation results validate the analytical results and show that SC scheme outperforms ARC scheme at the expense of CSI requirement.

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Acknowledgments

This research was supported by Institute for Information and Communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (B0126-16-1051) and National Research Foundation (NRF) Korea under grant number NRF-2015R1D1A1A09059026.

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Authors and Affiliations

  1. Department of Information and Communication Engineering, Sejong University, Seoul, 05006, Republic of Korea

    Mateen Ashraf & Kyung-Geun Lee

  2. Department of Electronic Engineering, Sogang University, Seoul, 04107, Republic of Korea

    Ju Wook Jang

Authors
  1. Mateen Ashraf
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Correspondence to Kyung-Geun Lee.

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Appendix

Appendix

Proof of Lemma 1

We know the cdf of individual g i ’s and since they are independent therefore we can write the outage probability for SC scheme as

$$\begin{array}{@{}rcl@{}} &&{{\int}_{0}^{C}}\left[1-2\sqrt{\frac{\lambda^{2}(C-|h_{s_{n}d}|^{2})}{\eta}}e^{-\lambda(C-|h_{s_{n}d}|^{2})}\right.\\ &&\qquad\left.\times K_{1}\left( 2\sqrt{\frac{\lambda^{2}(C-|h_{s_{n}d}|^{2})}{\eta}}\right)\right]^{M}\\ &&\times\lambda e^{-\lambda|h_{s_{n}d}|^{2}}d|h_{s_{n}d}|^{2}. \end{array} $$
(36)

After some mathematical manipulations, the above integral can be written as

$$\begin{array}{@{}rcl@{}} P_{out}&=&e^{-\lambda C}{{\int}_{0}^{C}} \left[1-e^{-\lambda x}2\sqrt{\frac{\lambda^{2}x}{\eta}}K_{1}\left( 2\sqrt{\frac{\lambda^{2}x}{\eta}}\right)\right]^{M}\\ &&\times\lambda e^{\lambda x}dx. \end{array} $$
(37)

For high SNR, we use following approximations

$$\begin{array}{@{}rcl@{}} xK_{1}(x)\simeq1+\frac{x^{2}}{2}\ln{\frac{x}{2}} \end{array} $$
(38)
$$\begin{array}{@{}rcl@{}} e^{kx}\simeq1+kx \end{array} $$
(39)
$$\begin{array}{@{}rcl@{}} (1+x)^{k}\simeq 1+kx \end{array} $$
(40)

Hence Eq. 37 can be simplified to

$$\begin{array}{@{}rcl@{}} P_{out}^{SC}&\simeq& \lambda^{M+1}e^{-\lambda C}{{\int}_{0}^{C}} x^{M}\left[1+\frac{\lambda}{\eta}(1-\lambda x)\ln{\left( \frac{\eta}{x \lambda^{2}}\right)}\right]^{M}\\ &&\times (1+\lambda x) dx, \end{array} $$
(41)

we can use binomial expansion to get

$$\begin{array}{@{}rcl@{}} P_{out}^{SC}&\simeq& \lambda^{M+1}e^{-\lambda C}{\sum}_{p=0}^{M} \binom{M}{p}{{\int}_{0}^{C}}\left( \frac{\lambda}{\eta}\right)^{M-p}\\ &&\times\left( 1-\left( M-p\right)\left( \lambda x\right)\right)x^{M}\ln^{M-p}\\ &&\times{\left( \frac{\eta}{x \lambda^{2}}\right)}\left( 1+\lambda x\right)dx, \end{array} $$
(42)

the result of the inner integration can be written as

$$\begin{array}{@{}rcl@{}} (\frac{\lambda}{\eta})^{M-p}\!\left[A\Theta(M,\!p,\!1)\,-\,\!B\Theta(M,\!p,\!2)\,-\,\!D\Theta(M,\!p,\!3)\right]. \end{array} $$
(43)

□

By putting Eq. 43 into Eq. 42 Lemma 2 is proved.

Proof of Lemma 2

The pdf of g can be found using Eq. 18

$$\begin{array}{@{}rcl@{}} p_{g}(x)\!\!&=&\!\!\frac{\partial Pr(g<x)}{\partial x}\\ \!\!&=&\!\!e^{-\lambda x}\left[\lambda K_{1}\left( \sqrt{\frac{4\lambda^{2} x}{\eta}}\right)\sqrt{\frac{4 \lambda^{2} x}{\eta}}\,-\,\sqrt{\frac{\lambda^{2}}{\eta x}}K_{1}\left( \sqrt{\frac{4\lambda^{2} x}{\eta}}\right)\right.\\ &&\quad\quad\quad\left.+\! K_{0}\left( \sqrt{\frac{4\lambda^{2} x}{\eta}}\right)\frac{\lambda^{2}}{\eta}\,+\,K_{2}\left( \sqrt{\frac{4\lambda^{2} x}{\eta}}\right)\frac{\lambda^{2}}{\eta}\right], \end{array} $$
(44)

and \(\overline {g}\) and \({\sigma _{g}^{2}}\) are given by

$$\begin{array}{@{}rcl@{}} \overline{g}={\int}_{0}^{\infty} xp_{g}(x) dx, \end{array} $$
(45)
$$\begin{array}{@{}rcl@{}} {\sigma_{g}^{2}}={\int}_{0}^{\infty} x^{2} p_{g}(x) dx -\overline{g}^{2}, \end{array} $$
(46)

Now we can use following identity from [26]

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{\infty} x^{\mu-\frac{1}{2}}e^{-\gamma x}K_{2\nu}(2\beta \sqrt{x})dx\\&=&\frac{\Gamma(\mu\,+\,\nu\,+\,\frac{1}{2}) \Gamma(\mu\,-\,\nu\,+\,\frac{1}{2})}{2\beta}e^{\frac{\beta^{2}}{2\gamma}}\gamma^{\,-\,\mu}W_{\,-\,\mu,\nu} \left( \frac{\beta^{2}}{\gamma}\right), \end{array} $$
(47)

where W α,β (x) is the Whittaker function. Putting Eq. 44 into Eqs. 45, 46 and using Eq. 47 we can get Eqs. 34 and 35. Now we can write the right hand side of Eq. 27 as follows

$$\begin{array}{@{}rcl@{}} P_{out}^{ARC}&\simeq&{\int}_{0}^{C_{1}} \frac{1}{2}\left[1+erf\left( \frac{C_{1}-|h_{s_{n}d}|^{2}-\omega}{\sigma\sqrt{2}}\right)\right]\\ &&\times\lambda e^{-\lambda |h_{s_{n}d}|^{2}} d|h_{s_{n}d}|^{2}. \end{array} $$
(48)

After simplification [27] we can get Eq. 33. □

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Ashraf, M., Jang, J.W. & Lee, KG. Outage Probability Analysis for Energy Harvesting Cooperative Relays in a Clustered Environment. Mobile Netw Appl 23, 1208–1219 (2018). https://doi.org/10.1007/s11036-017-0813-1

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