Research on Throughout with SWIPT in Two-Way Relay System Under Power Splitting Receiver

Abstract

We consider the Simultaneous Wireless Information and Power Transfer strategy optimization in the two-way transmission relay system under Power Splitting (PS) receiver. In such case, we propose two operational strategies for two-way transmission under PS receiver structure, (1) Delay-Limited-Relay strategy (PS-LDR) and (2) Delay-Tolerated-Relay strategy (PS-TDR). PS-LDR strategy is that both sending node and receiving node need synchronization of information transmission, so communication systems have higher Signal Noise Ratio (SNR) threshold for channels, or communication may break down. In order to explore the performance of PS-LDR strategy, we use outage probability to calculate the analytical expression of the actual maximum throughput (AMT). PS-TDR strategy allows that the information received by the receiving node delays with the information sent by the sending node. Communication system in PS-TDR strategy has lower SNR threshold than that in PS-LDR strategy. The analytical expression of the AMT is calculated by the ergodic capacity in PS-TDR strategy. The influences of system parameters on the optimal throughput in two strategies are discussed.

Appendices

Appendix

Appendix 1: Proof of Theorem 1

In this part, we prove the outage probability $p_{out1}$ in the communication process in formula (14). Substituting formula (13) into formula (1) to calculate the outage probability $p_{out1}$, we obtain

$$\begin{aligned} p_{out1} & = p\left( {\frac{{\eta (1 - \rho )\rho P_{1}^{2} \left| h \right|^{4} \left| g \right|^{2} }}{{\eta \rho P_{1} \left| h \right|^{2} \left| g \right|^{2} \sigma_{R}^{2} + (1 - \rho )P_{1} \left| h \right|^{2} \sigma_{D}^{2} + (1 - \rho )\sigma_{R}^{2} \sigma_{D}^{2} }} < \gamma_{0} } \right) \\ & = p((a_{1} \left| h \right|^{4} - b_{1} \left| h \right|^{2} )\left| g \right|^{2} < c_{1} \left| h \right|^{2} + d_{1} ) \\ \end{aligned}$$

(23)

To solve this inequality, since the positive and negative values of $a_{1} \left| h \right|^{4} - b_{1} \left| h \right|^{2}$ is uncertain, we need to classify and discuss it:

$$p_{out1} = \left\{ {\begin{array}{*{20}l} {p\left( {\left| g \right|^{2} < \frac{{c_{1} \left| h \right|^{2} + d_{1} }}{{a_{1} \left| h \right|^{4} - b_{1} \left| h \right|^{2} }}} \right)} \hfill & {\left| h \right|^{2} > \frac{{b_{1} }}{{a_{1} }}} \hfill \\ {p\left( {\left| g \right|^{2} > \frac{{c_{1} \left| h \right|^{2} + d_{1} }}{{a_{1} \left| h \right|^{4} - b_{1} \left| h \right|^{2} }}} \right) = 1} \hfill & {\left| h \right|^{2} < \frac{{b_{1} }}{{a_{1} }}} \hfill \\ \end{array} } \right.$$

(24)

In (24), in the condition of $\left| h \right|^{2} < b_{1} /a_{1}$ and there is $c_{1} \left| h \right|^{2} + d_{1} > 0$, while channel gain $\left| g \right|^{2}$ is a constant positive value. Therefore, through Eq. (24), $p_{out1}$ can be expressed as (14). This ends the proof for Theorem 1.□

Appendix 2: Proof of Theorem 3

In order to explore the expression of ergodic capacity, we need the PDF function $f_{{\gamma_{{_{L1} }} }} (\gamma )$ of $\gamma_{L1}$, which can be obtained from

$$f_{{\gamma_{{_{L1} }} }} (\gamma ) = \frac{{\partial F_{{\gamma_{L1} }} (\gamma )}}{\partial \gamma }$$

(25)

According to Eqs. (1) and (14), we can calculate the probability distribution $F_{{\gamma_{{_{L1} }} }} (\gamma )$ of $\gamma_{L1}$which is given as

$$F_{{\gamma_{{L_{1} }} }} (\gamma ) = \int_{0}^{{\frac{{b_{3} \gamma }}{{a_{3} }}}} {f_{h} (z)dz + } \int_{{\frac{{b_{3} \gamma }}{{a_{3} }}}}^{\infty } {f_{h} (z) \cdot \left( {1 - e^{{ - \frac{{c_{3} \gamma z + d_{3} \gamma }}{{\lambda_{g} (a_{{_{3} }} z^{2} - b_{{_{3} }} \gamma z)}}}} } \right)} dz$$

(26)

The PDF function of $f_{{\gamma_{{_{L1} }} }} (\gamma )$ can be obtained by using formula (27):

$$f_{{\gamma_{{_{{L_{1} }} }} }} (\gamma ) = \int_{{\frac{{b_{3} \gamma }}{{a_{3} }}}}^{\infty } {\frac{{a_{3} c_{3} z^{3} + a_{3} d_{3} z^{2} }}{{\lambda_{h} (a_{3} z^{2} - b_{3} \gamma z)^{2} }}} \cdot e^{{ - \frac{z}{{\lambda_{h} }} - \frac{{c_{3} \gamma z + d_{{_{3} }} \gamma }}{{\lambda_{g} (a_{{_{3} }} z^{2} - b_{3} \gamma z)}}}} dz$$

(27)

According to formula (19) and formula (27), the ergodic capacity can be obtained in formula (20). This ends the proof for Theorem 3.□