Exploiting cooperative relays to enhance the performance of energy-harvesting systems over Nakagami-m fading channels

Abstract

This paper investigates a time-switching energy-harvesting system in which a source communicates with a destination via energy-constrained amplify-and-forward relays. To exploit the benefit of using multiple relays, we propose a relay scheduling called consecutive relay selection (CRS), which allows all relays to assist the source-to-destination communication, to improve the transmission efficiency of the time-switching policy. The partial relay selection (PRS) is examined for performance comparison. The selected relay in the PRS protocol is considered in two cases: in one, it is selected based on the first-hop channel gains (PRS-1 protocol), and in the other, it is selected based on the second-hop channel gains (PRS-2 protocol). For performance evaluation, the analytical expressions of the outage probability and throughput for Nakagami-m fading channel are derived. Our results show that the CRS protocol outperforms the PRS protocol in terms of throughput, the PRS-1 protocol achieves better performance than the PRS-2 protocol. Moreover, we discuss the effects of various key system parameters on system performance, such as the energy-harvesting ratio, source transmission rate, and locations of relays, to provide insights into the various design choices.

Appendices

Proof of Proposition 1

According to [5, 7], the OP in (7) does not admit an exact closed-form expression due to the complex structure of $\gamma _{e2e,a}$ in (6). However, at sufficiently high values of $\rho _s$, the effect of ${\mathbb {c}}_a$ is negligible compared to the rest factors in the denominator; hence, the OP in (7) can be lower bounded as

$$\begin{aligned}&{\mathcal {P}_{\text {out,low}}}\left( \gamma \right) \nonumber \\&= \Pr \left\{ {\frac{{{{\mathbb {a}}_a}{|h_{1,a}|^2 |h_{2,a}|^2}}}{{{{\mathbb {b}}_a}{|h_{2,a}|^2} + 1}} < \gamma } \right\} \nonumber \\&= 1-\Pr \left\{ {{|h_{2,a}|^2} \left( \hat{{\mathbb {a}}}_a {|h_{1,a}|^2} - {\mathbb {b}}_a \right) > 1} \right\} \nonumber \\&= 1 - \int \limits _0^{ + \infty } \left( 1- F_{|h_{2,a}|^2}\left( \frac{1}{\hat{{\mathbb {a}}}_at}\right) \right) f_{|h_{1,a}|^2}\left( t + \frac{{\mathbb {b}}_a}{\hat{{\mathbb {a}}}_a} \right) dt, \end{aligned}$$

(22)

where $\hat{{\mathbb {a}}}_a = \frac{{\mathbb {a}}_a}{\gamma }$ and $t={|h_{1,a}|^2}-\frac{{\mathbb {b}}_a}{\hat{{\mathbb {a}}}_a}$.

It can be seen from (22) that it is possible to derive the closed-form expression for ${\mathcal {P}_{\text {out,low}}}$. Moreover, the simulation results in Sect. 4 show that ${\mathcal {P}_{\text {out,low}}}$ is very tight to the exact SOP on the entire SNR range.

1.1 The OP for the consecutive relay selection protocol

Setting $\alpha _T=\frac{K-2}{K}$ yields $\mathbb {a}_{\mathbb {k}}$ and $\mathbb {b}_{\mathbb {k}}$. Then, substituting $f_{\mathcal {X}}\left( t+\frac{\mathbb {b}_{\mathbb {k}}}{\hat{{\mathbb {a}}}_{\mathbb {k}}} ;m_1,\frac{\lambda _1}{m_1}\right)$ and $F_{\mathcal {X}} \left( \frac{1}{\hat{{\mathbb {a}}}_{\mathbb {k}} t} ;m_2,\frac{{\lambda }_2}{m_2} \right)$ into (22), $\mathcal {P}_{\text {out,low}}^{\text {CRS}}(\gamma )$ can be calculated as

$$\begin{aligned} \mathcal {P}_{\text {out,low}}^{\text {CRS}}(\gamma )&= 1 - \frac{m_1^{m_1}}{j!\varGamma (m_1){\lambda _1}^{m_1}} e^{ - \frac{m_1 \mathbb {b}_{\mathbb {k}}}{\lambda _1 \hat{{\mathbb {a}}}_{\mathbb {k}}}} \nonumber \\ {}&\quad \times \sum \limits _{i = 0}^{{m_1} - 1} {\sum \limits _{j = 0}^{{m_2} - 1} {\left( {\begin{array}{c}{m_1} - 1\\ i\end{array}}\right) } } \nonumber \\&\quad \times {\left( {\frac{{{m_2}}}{{{\lambda _2 \hat{{\mathbb {a}}}_{\mathbb {k}}}}}} \right) ^j}{\left( {\frac{{{\mathbb {b}_{\mathbb {k}}}}}{{{{\hat{{\mathbb {a}}}_{\mathbb {k}}}}}}} \right) ^{{m_1} - i - 1}} \nonumber \\ {}&\quad \times \int \limits _0^{ + \infty } {{t^{i - j}}{e^{ - \frac{{{m_2}}}{{{\lambda _2}{{\hat{{\mathbb {a}}}_{\mathbb {k}}}}t}} - \frac{{{m_1}}}{{{\lambda _1}}}t}}} dt. \end{aligned}$$

(23)

The integration in (23) can be solved using the identity in [12, Eq. (3.471.9)].

1.2 The OP for the partial relay selection protocol

To calculate the OP for the PRS protocol, we need to study the PDF and CDF of the maximum value of K gamma RVs. Letting $\mathcal {X}_1,\ldots ,\mathcal {X}_{K}$ be K i.i.d. gamma RVs with parameters $(m,\lambda )$. According to [17], the PDF and CDF of the best RV $\mathcal {X}_{b}= \underset{1\leqslant k \leqslant K}{\max } \{\mathcal {X}_k\}$ are given by

$$\begin{aligned} f_{\mathcal {X}_{b}} (x;m,\lambda )&= K{f_\mathcal {X} \left( x;m,\lambda \right) } F_\mathcal {X} \left( x;m,\lambda \right) ^{K - 1}, \end{aligned}$$

(24)

$$\begin{aligned} F_{\mathcal {X}_{b}} (x;m,\lambda )&= F_\mathcal {X} \left( x;m,\lambda \right) ^{K}. \end{aligned}$$

(25)

1.2.1 Calculation for $\mathcal {P}_{\text {out,low}}^{\text {PRS-1}}(\gamma )$

Substituting $f_{\mathcal {X}_b}\left( t+\frac{{\mathbb {b}}_b}{\hat{{\mathbb {a}}}_b} ;m_1,\frac{\lambda _1}{m_1}\right)$ and $F_{\mathcal {X}} \left( \frac{1}{\hat{{\mathbb {a}}}_b t} ;m_2,\frac{{\lambda }_2}{m_2} \right)$ into (22), respectively, and using [18, Eq. (26.4.9)], the close-form expression of $\mathcal {P}_{\text {out,low}}^{\text {PRS-1}}\left( \gamma \right)$ can be obtained as in (11) by following the same steps of the calculation for $\mathcal {P}_{\text {out,low}}^{\text {CRS}}(\gamma )$.

1.2.2 Calculation for $\mathcal {P}_{\text {out,low}}^{\text {PRS-2}}(\gamma )$

Substituting $f_{\mathcal {X}}\left( t+\frac{{\mathbb {b}}_b}{\hat{{\mathbb {a}}}_b} ;m_1,\frac{\lambda _1}{m_1}\right)$ and $F_{\mathcal {X}_b} \left( \frac{1}{\hat{{\mathbb {a}}}_b t} ;m_2,\frac{{\lambda }_2}{m_2} \right)$ into (22), and following the same steps of the calculation for $\mathcal {P}_{\text {out,low}}^{\text {PRS-1}}(\gamma )$, $\mathcal {P}_{\text {out,low}}^{\text {PRS-2}}(\gamma )$ can be expressed as in (12).

This ends the proof for Proposition 1.

Proof of Lemma 1

According to (15), the calculation of the ergodic capacity involves solving the derivative of $\varXi ( {\gamma ;{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }} )$ with respect to $\gamma$. By using [12, Eq. (8.486.14)] and some additional manipulations, we can obtain

$$\begin{aligned}&\tfrac{d}{d\gamma }\varXi ( {\gamma ;{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }} ) \nonumber \\&\quad = - {\beta _\epsilon }\varXi \left( {\gamma ;{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }} \right) + \tfrac{{\left( {{\alpha _\epsilon } - {\upsilon _\epsilon }} \right) {\mu _\epsilon }}}{2} \nonumber \\&\qquad \times \varXi \left( {\gamma ;{\alpha _\epsilon } - 2,{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }} \right) \nonumber \\&\qquad - {\mu _\epsilon }\varXi \left( {\gamma ;{\alpha _\epsilon } - 1,{\beta _\epsilon },{\upsilon _\epsilon } - 1,{\mu _\epsilon }} \right) . \end{aligned}$$

(26)

Substituting (26) into the expression of $\varPhi ({{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }})$ and expressing $\ln (1+\gamma )$ as in [19, Eq. (07.34.03.0456.01)], $e^{-\beta _\epsilon \gamma }$ as in [19, Eq. (07.34.03.0228.01)], and $K_{\upsilon _\epsilon }(2 \sqrt{\mu _\epsilon \gamma })$ as in [19, Eq. (07.34.03.0605.01)], $\varPhi ({{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }})$ can be expressed as

$$\begin{aligned}&\varPhi ({{\alpha _\epsilon },{\beta _\epsilon },{\upsilon _\epsilon },{\mu _\epsilon }}) \nonumber \\&\quad = \frac{\mu _\epsilon ^\frac{\alpha _\epsilon }{2}}{{\ln \left( 2 \right) }} \left( -\frac{\beta _\epsilon }{2} \int _{0}^{\infty } {\gamma ^\frac{\alpha _\epsilon }{2}}G_{2,2}^{1,2}\left( {\left. \gamma \right| _{1,0}^{1,1}} \right) G_{0,1}^{1,0}\left( {\left. {\beta _\epsilon \gamma } \right| _0^ - } \right) \right. \nonumber \\&\quad \quad \times G_{0,2}^{2,0}\left( {\left. {\mu _\epsilon \gamma } \right| _{{\frac{\upsilon _\epsilon }{2}},{\frac{-\upsilon _\epsilon }{2}}}^ - } \right) d\gamma + {\frac{\alpha _\epsilon - \upsilon _\epsilon }{4}} \int _{0}^{\infty } {\gamma ^{\frac{\alpha _\epsilon }{2} - 1}}G_{2,2}^{1,2}\left( {\left. \gamma \right| _{1,0}^{1,1}} \right) \nonumber \\&\quad \quad \times G_{0,1}^{1,0}\left( {\left. {\beta _\epsilon \gamma } \right| _0^ - } \right) G_{0,2}^{2,0}\left( {\left. {\mu _\epsilon \gamma } \right| _{\frac{\upsilon _\epsilon }{2},\frac{-\upsilon _\epsilon }{2}}^ - } \right) d\gamma - \frac{\sqrt{\mu _\epsilon }}{2} \int _{0}^{\infty } {\gamma ^\frac{\alpha _\epsilon - 1}{2}} \nonumber \\&\quad \quad \times \left. G_{2,2}^{1,2} \left( {\left. \gamma \right| _{1,0}^{1,1}} \right) G_{0,1}^{1,0}\left( {\left. {\beta _\epsilon \gamma } \right| _0^ - } \right) G_{0,2}^{2,0}\left( {\left. {\mu _\epsilon \gamma } \right| _{{\frac{\upsilon _\epsilon - 1}{2}},{\frac{1 - \upsilon _\epsilon }{2}}}^ - } \right) d\gamma \right) . \end{aligned}$$

(27)

The integrations in (27) can be solved with the aid of [19, Eq. (07.34.21.0081.01)]. Then, Lemma 1 is proved.