Opportunistic Multiple Relay Selection Schemes in both Full-Duplex and Half-Duplex Operation for Decode-and-Forward Cooperative Networks

Abstract

An efficient technique used to prolong the lifetime of energy-constrained networks is energy harvesting (EH). In this paper, we investigate and develop the energy allocation methods for the relaying networks, in which opportunistic multiple relay selection schemes with both the full-duplex (FD) and half-duplex (HD) scheme for both EH and non-EH in decode-and-forward (DF) relaying mode. In addition, there are two policies proposed in this paper: (1) Max-Min with Self Interference Relay Selection (MMSI); (2) Max-Min Relay Selection (MMSR) are depicted for both EH and Non-EH relaying modes. Particularly, we derive closed-form expressions of outage probability to analyze the performance of systems. In addition, we propose the impact of self-interference on both policies to provide a practical insight. The results in numerical analysis reveal that the proposed MMSI scheme outperforms the MMSR mode in terms of outage probability.

Appendices

Appendix A

Proof of EH MMSI:

PDF and CDF of related variables as PDF of the SNR for self-interference

$$\begin{aligned} f_{\gamma _{{Ri}{Ri}}}({\gamma _0}) = \frac{1}{{{{\bar{\gamma }}_{{R}{R}}}}}{e^{\left( { - \frac{{{\gamma _0}}}{{{{\bar{\gamma }}_{{R}{R}}}}}} \right) }}, \end{aligned}$$

(A.1)

where the average residual self-interference power is denoted by ${{\bar{\gamma }}_{{RR}}}$. Thus, the CDF of the SNR at relay

$$\begin{aligned} F_{\gamma _{{Ri}}}({\gamma _0}) = 1 - \left( {1 - {e^{\left( { - \frac{{\left( {1 - \beta } \right) }}{{{\gamma _0}\rho {{\bar{\gamma }}_{RR}}}}} \right) }}} \right) . \end{aligned}$$

(A.2)

The outage probability is given by

$$\begin{aligned} \begin{array}{*{20}{ll}} {F_{{\gamma _D}}}({\gamma _0}) &{}= Pr\left( {\rho {\gamma _{SRi}}{\gamma _D}< {\gamma _0}} \right) = Pr\left( {{\gamma _D} < \frac{{{\gamma _0}}}{{\rho {\gamma _{SRi}}}}} \right) \\ &{}= \int _{x = 0}^\infty {{f_{{\gamma _{SRi}}}}(x)\left( {1 - {e^{( - \frac{{{\gamma _0}}}{{x\rho {{\bar{\gamma }}_{SR}}}})}}} \right) dx} \\ &{} = 1 - \sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{SR}}{{\bar{\gamma }}_D}}}} {K_1}\left( {\sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{SR}}{{\bar{\gamma }}_D}}}} } \right) \end{array}, \end{aligned}$$

(A.3)

where ${f_{{\gamma _{SRi}}}}(x) = \frac{1}{{{{\bar{\gamma }}_{SR}}}}{exp{\left( - \frac{x}{{{{\bar{\gamma }}_{SR}}}}\right) }}$, with $x \ge 0$ and $K_1(.)$ is the first-order modified Bessel function of the second kind. Additionally, the end-to-end SNR is $z = \min \left\{ {{\gamma _{Ri}},{\gamma _D}} \right\}$. Then, the outage probability of overall system is

$$\begin{aligned} {F_z}({\gamma _0}) = 1 - \left[ {1 - {F_{{\gamma _{Ri}}}}({\gamma _0})} \right] \times \left[ {1 - {F_{{\gamma _D}}}({\gamma _0})} \right] \end{aligned}$$

(A.4)

We have the overall outage probability

$$\begin{aligned} P_{out}^{SPC\_FD}({\gamma _0}) = 1 - \left( {1 - {e^{\left( { - \frac{{\left( {1 - \beta } \right) }}{{{\gamma _0}\rho {{\bar{\gamma }}_{RR}}}}} \right) }}} \right) \sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{SR}}{{\bar{\gamma }}_D}}}} {K_1}\left( {\sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{SR}}{{\bar{\gamma }}_D}}}} } \right) . \end{aligned}$$

(A.5)

This ends the proof Appendix A.

Appendix B

Proof of EH MMSR:

The CDF of SNR in link S-R as

$$\begin{aligned} {F_{{\gamma _{Ri}}}}({\gamma _0}) =1-\left( 1 - {e^{ - \left( {\frac{{{\gamma _0}}}{{{{\bar{\gamma }}_{SR}}{P_S}(1 - \beta )}}} \right) }}\right) . \end{aligned}$$

(B.1)

The PDF of random variances ${\gamma _D}$ is depicted in (A.3) as

$$\begin{aligned} {F_{{\gamma _D}}}({\gamma _0}) = 1 - \sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{RD}}{{\bar{\gamma }}_{SR}}}}} {K_1}\left( {\sqrt{\frac{{4{\gamma _0}}}{{\rho {{\bar{\gamma }}_{RD}}{{\bar{\gamma }}_{SR}}}}} } \right) \end{aligned}$$

(B.2)

In DF scheme, the outage probability is given by

$$\begin{aligned} \begin{array}{*{20}{ll}} {F_z}({\gamma _0}) &{} = 1 - \left[ {1 - {F_{{\gamma _{Ri}}}}({\gamma _0})} \right] \times \left[ {1 - {F_{{\gamma _D}}}({\gamma _0})} \right] \\ &{} = 1 - \left( {1 - {e^{ - \left( {\frac{{{\gamma _0}}}{{{{\bar{\gamma }}_{SR}}{P_S}(1 - \beta )}}} \right) }}} \right) \,.\,\sqrt{\frac{{4{\gamma _0}}}{{\rho {{\overline{\gamma }}_{RD}}{{\overline{\gamma }}_{SR}}}}} {K_1}\left( {\sqrt{\frac{{4{\gamma _0}}}{{\rho {{\overline{\gamma }}_{RD}}{{\overline{\gamma }}_{SR}}}}} } \right) \end{array} \end{aligned}$$

(B.3)

This ends the proof.