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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.

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Acknowledgements

This work was supported by the 2018 Research Fund of the University of Ulsan.

Author information

Correspondence to Van Phu Tuan.

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.

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)].

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)

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 \(\mathcal {P}_{\text {out,low}}^{\text {CRS}}(\gamma )\).

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.

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Tuan, V.P., Kong, H.Y. Exploiting cooperative relays to enhance the performance of energy-harvesting systems over Nakagami-m fading channels. Telecommun Syst 69, 477–487 (2018). https://doi.org/10.1007/s11235-018-0448-y

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Keywords

  • Wireless energy harvesting
  • Time switching
  • Cooperative communication
  • Relay selection
  • Amplify-and-forward
  • Nakagami-m fading