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Self-Powered Wireless Two-Way Relaying Networks: Model and Throughput Performance with Three Practical Schemes

Abstract

In this paper, we analyse the throughput performance for two, three and four time slot transmission schemes, (2TS, 3TS and 4TS) for two-way amplify-and-forward relaying networks, in which we use RF signal for the energy harvesting (EH) enabled relay node to assist the exchange of information. Most importantly, we derive expression for delay-limited throughput and the approximate expressions for outage probability, and we also compare these results in case of EH and non-EH. Additionally, the trade-off between the distance allocation between source to relay, and relay to destination is comprehensively investigated, in which the large scale path loss is considered to obtain the optimal throughput. Thanks to the numerical results, we consider a scenario in each scheme, where the throughput of 2TS is higher regardless of values of power splitting coefficients compared to other two schemes. Numerical results provide an interesting trade-off between the considered EH parameters in the system design, and reveal the improvement of bandwidth and power efficiency. The proposed schemes confirm that the appropriate placement of nodes can help achieve low outage probability and optimal throughput.

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Acknowledgements

This research is funded by Foundation for Science and Technology Development of Ton Duc Thang University (FOSTECT), website: http://fostect.tdt.edu.vn, under Grant FOSTECT.2016.BR.21.

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Correspondence to Dinh-Thuan Do.

Appendices

Appendix

A. Proof of Proposition 1

First, the coming up proposition 1 is established to derive the outage probability and then the output SNRs at A and B for 2TS, 3TS (in case of EH and Ideal EH) is rewritten by

$$\begin{aligned} Z = \frac{{{Q_1}XY}}{{{Q_2}X + {Q_3}}}, \end{aligned}$$
(32)

where variables X and Y are \({\left| {{h_A}} \right| ^2}\) and \({\left| {{h_B}} \right| ^2}\). Constant \(Q_1^{Non,uTS} \ge 0\), \(Q_2^{Non,uTS} \ge 0\) and \(Q_3^{Non,uTS} \ge 0\) are distinguished factors from X and Y. We have

$$\begin{aligned} {P_{out}}= & {} {P_r}\left( {\frac{{{Q_1}XY}}{{{Q_2}X + {Q_3}}}< {\gamma _0}} \right) = {P_r}\left( {X< \frac{{{\gamma _0}{Q_3}}}{{{Q_1}Y - {\gamma _0}{Q_2}}}} \right) {1_{Y> \frac{{{\gamma _0}{Q_2}}}{{{Q_1}}}}}\nonumber \\&+ {P_r}\left( {X > \frac{{{\gamma _0}{Q_3}}}{{{Q_1}Y - {\gamma _0}{Q_2}}}} \right) {1_{Y < \frac{{{\gamma _0}{Q_2}}}{{{Q_1}}}}}, \end{aligned}$$
(33)

where \({\gamma _0} = R_{0,i}^{uTS}\) and \({1_{Y < \frac{{{\gamma _0}{Q_2}}}{{{Q_1}}}}}\):\({P_r}\left( {X > \frac{{{\gamma _0}{Q_3}}}{{{Q_1}Y - {\gamma _0}{Q_2}}}} \right) = 1\) because \({Q_1}Y - {\gamma _0}{Q_2}\) is a negative number and the probability of X is always 1, since X is a positive number. \({P_{out}}\) is given by

$$\begin{aligned} {P_{out}}= & {} \int \limits _0^{{\gamma _0}{Q_2}/{Q_1}} {{f_{{{\left| {{h_B}} \right| }^2}}}\left( z \right) } dz \qquad + \int \limits _{{\gamma _0}{Q_2}/{Q_1}}^\infty {{f_{{{\left| {{h_B}} \right| }^2}}}\left( z \right) } Pr\left( {X < \frac{{{\gamma _0}{Q_3}}}{{{Q_1}z - {\gamma _0}{Q_2}}}} \right) dz \nonumber \\= & {} 1 - \frac{1}{{{\varOmega _B}}}\int \limits _{{\gamma _0}{Q_2}/{Q_1}}^\infty {{e^{ - \left( {\frac{z}{{{\varOmega _B}}} + \frac{{{\gamma _0}{Q_3}}}{{\left( {{Q_1}z - {\gamma _0}{Q_2}} \right) {\varOmega _A}}}} \right) }}} dz . \end{aligned}$$
(34)

The cumulative distribution function (CDF) of the exponential random variable is denoted by \({\left| {{h_A}} \right| ^2}\) and we have \(P(X < z) = 1 - {e^{ - z/{\varOmega _A}}}\) and the average of the exponential random variable \({\left| {{h_B}} \right| ^2}\) is designated by the probability density function (PDF) of exponential random variable \({f_{{{\left| {{h_B}} \right| }^2}}}\left( z \right) \buildrel \varDelta \over = \frac{1}{{{\varOmega _B}}}{e^{ - \frac{z}{{{\varOmega _B}}}}}\). The new integration is established: \(x \buildrel \varDelta \over = {Q_1}z - {\gamma _0}{Q_2}\)

$$\begin{aligned} \begin{array}{c} {P_{out}} = 1 - \frac{1}{{{\varOmega _B}}}\int \limits _0^\infty {{e^{ - \left( {\frac{{x + {\gamma _0}{Q_2}}}{{{Q_1}{\varOmega _B}}} + \frac{{{\gamma _0}{Q_3}}}{{x{\varOmega _A}}}} \right) }}} dx = 1 - \frac{{{e^{ - \left( {\frac{{{\gamma _0}{Q_2}}}{{{Q_1}{\varOmega _B}}}} \right) }}}}{{{\varOmega _B}}}\int \limits _0^\infty {{e^{ - \left( {\frac{x}{{{Q_1}{\varOmega _B}}} + \frac{{{\gamma _0}{Q_3}}}{{x{\varOmega _A}}}} \right) }}} dx, \end{array} \end{aligned}$$
(35)

To find closed-form expression of this outage performance it need be solve the intergral function by applying \(\int \limits _0^\infty {{e^{ - \frac{\delta }{{4x}} - \lambda x}}} dx = \sqrt{\frac{\delta }{\lambda }} {K_1}\left( {\sqrt{\delta \lambda } } \right)\) thanks to formula ([20], 3.324.1), in which \({K_1}(.)\) is the modified Bessel function of the second kind. Hence, the cumulative density function (CDF) of Z is computed by

$$\begin{aligned} \begin{array}{c} {F_{{\gamma _A}}}\left( {{\gamma _0}} \right) \approx 1 - \exp \left( { - \frac{{{Q_2}{\gamma _0}}}{{{Q_1}{\varOmega _B}}}} \right) \sqrt{\frac{{\left( {4{Q_3}} \right) {\gamma _0}}}{{{Q_1}\left( {{\varOmega _A}{\varOmega _B}} \right) }}} {K_1}\left( {\sqrt{\frac{{\left( {4{Q_3}} \right) {\gamma _0}}}{{{Q_1}\left( {{\varOmega _A}{\varOmega _B}} \right) }}} } \right) , \end{array} \end{aligned}$$
(36)

where the average of the exponential random variable \({\left| {{h_A}} \right| ^2}\) is characterized by \({\varOmega _A}\) and \({\left| {{h_B}} \right| ^2}\) is characterized by \({\varOmega _B}\). This ends the proof for Proposition 1.

B. Proof of Proposition 2

In terms of non-EH, a general form for the received SNR gain is given by

$$\begin{aligned} Z = \frac{{{Q_1}XY}}{{{Q_2}X + {Q_3}Y}}. \end{aligned}$$
(37)

Based on the proof of Proposition 1, we have

$$\begin{aligned} {P_{out}}= \,& {P_r}\left( {\frac{{{Q_1}XY}}{{{Q_2}X + {Q_3}Y}}< {\gamma _0}} \right) = {P_r}\left( {X< \frac{{{\gamma _0}{Q_3}Y}}{{{Q_1}Y - {\gamma _0}{Q_2}}}} \right) {1_{Y> \frac{{{\gamma _0}{Q_2}}}{{{Q_1}}}}}\nonumber \\&+ {P_r}\left( {X > \frac{{{\gamma _0}{Q_3}Y}}{{{Q_1}Y - {\gamma _0}{Q_2}}}} \right) {1_{Y < \frac{{{\gamma _0}{Q_2}}}{{{Q_1}}}}}. \end{aligned}$$
(38)

The \(P_{out}\) is given by

$$\begin{aligned} {P_{out}}= & {} \int \limits _0^{{\gamma _0}{Q_2}/{Q_1}} {{f_{{{\left| {{h_B}} \right| }^2}}}\left( z \right) } dz + \int \limits _{{\gamma _0}{Q_2}/{Q_1}}^\infty {{f_{{{\left| {{h_B}} \right| }^2}}}\left( z \right) } p\left( {X < \frac{{{\gamma _0}{Q_3}z}}{{{Q_1}z - {\gamma _0}{Q_2}}}} \right) dz\nonumber \\= & {} 1 - \frac{1}{{{\varOmega _B}}}\int \limits _{{\gamma _0}{Q_2}/{Q_1}}^\infty {{e^{ - \left( {\frac{z}{{{\varOmega _B}}} + \frac{{{\gamma _0}{Q_3}z}}{{\left( {{Q_1}z - {\gamma _0}{Q_2}} \right) {\varOmega _A}}}} \right) }}} dz. \end{aligned}$$
(39)

Then, the CDF of Z is computed by

$$\begin{aligned} {F_{A,{\gamma _0}}}\left( {{\gamma _0}} \right) = 1 - \exp \left( { - \frac{{{\gamma _0}{Q_2}}}{{{\varOmega _B}{Q_1}}} - \frac{{{\gamma _0}{Q_3}}}{{{\varOmega _A}{Q_1}}}} \right) \sqrt{\frac{{4{{\left( {{\gamma _0}} \right) }^2}{Q_2}{Q_3}}}{{{Q_1}^2{\varOmega _B}{\varOmega _A}}}} {K_1}\left( {\sqrt{\frac{{4{{\left( {{\gamma _0}} \right) }^2}{Q_2}{Q_3}}}{{{Q_1}^2{\varOmega _B}{\varOmega _A}}}} } \right) \end{aligned}$$
(40)

This ends the proof for Proposition 2.

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Nguyen, HS., Do, DT., Bui, AH. et al. Self-Powered Wireless Two-Way Relaying Networks: Model and Throughput Performance with Three Practical Schemes. Wireless Pers Commun 97, 613–631 (2017). https://doi.org/10.1007/s11277-017-4526-3

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Keywords

  • Two-way
  • Amplify-and-forward
  • Energy harvesting
  • Throughput