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Power Splitting and Source-Relay Selection in Energy Harvesting Wireless Network

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Abstract

This paper investigates the performance of an energy-harvesting (EH) relay network, where multiple sources communicate with a destination via multiple EH decode-and-forward (DF) relays. The EH relays all equip with a power splitter to divide the received signal power into two parts, which are used for signal processing and information forwarding, respectively. The power splitting ratio depicts the trade-off between the relaying energy and decoding energy. We propose an optimal power splitting and joint source-relay selection (OPS-JSRS) scheme where the optimal power-splitting ratio is obtained and the best source-relay pair is selected to transmit the message. For the purpose of comparison, we present the optimal power splitting and round-robin (OPS-RR) and the traditional power splitting and joint source-relay selection (TPS-JSRS) schemes. The exact and asymptotic closed-form expressions of outage probability for OPS-RR, TPS-JSRS and OPS-JSRS schemes are derived over Rayleigh fading channels . Numerical results show that the outage probability of OPS-JSRS scheme is lower than that of OPS-RR and TPS-JSRS schemes, explaining that the proposed OPS-JSRS scheme outperforms TPS-JSRS and OPS-RR schemes. Additionally, the outage probability performance of OPS-JSRS scheme can be improved by increasing the number of sources and/or relays.

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The data that support the findings of this study are available from the corresponding author upon request.

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Funding

The subject is sponsored by the National Natural Science Foundation of P. R. China (No. 61872196, No. 61872194, No. 61902196 , No. 62102194 and No. 62102196), Scientific and Technological Support Project of Jiangsu Province (No. BE2019740, No. BK20200753 and No. 20KJB520001), Major Natural Science Research Projects in Colleges and Universities of Jiangsu Province (No. 18KJA520008), Six Talent Peaks Project of Jiangsu Province (No. RJFW-111), Natural Science Foundation of Jiangsu Province (No.BK20200753), Jiangsu Postdoctoral Science Foundation Funded Project (No. 2021K096A), Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX19_0909, No. KYCX19_0911, No. KYCX20_0759, No. KYCX21_0787, No. KYCX21_0788 and No. KYCX21_0799).

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Authors and Affiliations

  1. School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Xiao Jiang, Peng Li & Ruchuan Wang

  2. Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks, Nanjing, 210023, Jiangsu Province, China

    Xiao Jiang, Peng Li & Ruchuan Wang

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  1. Xiao Jiang
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  2. Peng Li
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Correspondence to Peng Li.

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Appendix

Appendix

Derivation of (14)

Notice that \({{\left| {\mathop h\nolimits _{SmRn} } \right| ^2}}\) follows exponential distributions with the mean of \({{\mathop \sigma \nolimits _{SmRn}^2 }}\). Thus, the cumulative density function (CDF) of \({{\left| {\mathrm{{ }}{h_{S\mathop m\nolimits ^* \mathop R\nolimits _n }}} \right| ^2}}\) can be expressed as

$$\begin{aligned} {\mathop F\nolimits _{{{\left| {\mathrm{{ }}{h_{S\mathop m\nolimits ^* \mathop R\nolimits _n }}} \right| ^2}}} (x) = \prod \limits _{m = 1}^M {\left[1 - \exp \left( - \frac{x}{{\mathop \sigma \nolimits _{SmRn}^2 }}\right)\right]} }. \end{aligned}$$
(38)

Therefore, the PDF of \({{\left| {\mathop h\nolimits _{Sm\mathrm{{*}}Rn} } \right| ^2}}\) can be obtained as

$$\begin{aligned} {\begin{array}{l} \mathrm{{ }}{f_{{{\left| {{h_{S\mathrm{{ }}{m^*}\mathrm{{ }}{R_n}}}} \right| }^2}}}(x) = \sum \limits _{i = 1}^M {\frac{1}{{\sigma _{SiRn}^2}}} \exp ( - \frac{x}{{\sigma _{SiRn}^2}})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \prod \limits _{m = 1,m \ne i}^M {[1 - \exp ( - \frac{x}{{\sigma _{SmRn}^2}})]}. \end{array}} \end{aligned}$$
(39)

Denoting \({X={\left| {\mathrm{{ }}{h_{S\mathop m\nolimits ^* \mathop R\nolimits _n }}} \right| ^2}}\),the \({\mathop P\nolimits _{out,1}^{\mathrm{{TPS - JSRS}}} }\) can be rewritten as

$$\begin{aligned} {\begin{array}{*{20}{l}} {\mathrm{{ }}P_{out,1}^{\mathrm{{TPS - JSRS}}} = \Pr \left( {X> a,{{\left| {\mathrm{{ }}{h_{RnD}}} \right| }^2} > \frac{\beta }{{{\rho _n}\eta X}}} \right) }\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int _a^\infty {\exp ( - \frac{\beta }{{\sigma _{RnD}^2{\rho _n}\eta x}})\sum \limits _{i = 1}^M {\frac{1}{{\sigma _{SiRn}^2}}} \exp ( - \frac{x}{{\sigma _{SiRn}^2}})} }\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \prod \limits _{m = 1,m \ne i}^M {[1 - \exp ( - \frac{x}{{\sigma _{SmRn}^2}})]} dx} \end{array}}, \end{aligned}$$
(40)

where \({a = \frac{{\mathop 2\nolimits ^{2R} - 1}}{{(1 - \mathop \rho \nolimits _n )\gamma }}}\). The term \({\prod \limits _{m = 1,m \ne i}^M {[1 - \exp ( - \frac{x}{{\mathop \sigma \nolimits _{SmRn}^2 }})]}}\) can be expanded as

$$\begin{aligned} {\begin{array}{l} \prod \limits _{m = 1,m \ne i}^M {[1 - \exp ( - \frac{x}{{\mathop \sigma \nolimits _{SmRn}^2 }})]} \\ \mathrm{{ = }}1\mathrm{{ + }}\sum \limits _{k = 1}^{\mathop 2\nolimits ^{M - 1} - 1} {\mathop {( - 1)}\nolimits ^{\left| {\mathop D\nolimits _k } \right| } \exp \left( - \sum \limits _{Sm \in \mathop D\nolimits _k } {\frac{x}{{\mathop \sigma \nolimits _{SmRn}^2 }}}\right )} \end{array}}, \end{aligned}$$
(41)

where \({\mathop D\nolimits _k }\) represents the k-th nonempty subset of M sources, \({\left| {\mathop D\nolimits _k } \right| }\) represents the number of elements in set \({\mathop D\nolimits _k }\). Thus, the \({\mathop P\nolimits _{out,1}^{\mathrm{{TPS - JSRS}}} }\) can be rewritten as

$$\begin{aligned} {\mathop P\nolimits _{out,1}^{\mathrm{{TPS - JSRS}}} = \mathop \Phi \nolimits _1 + \mathop \Phi \nolimits _2 } \end{aligned}$$
(42)

where

$$\begin{aligned} {\mathop \Phi \nolimits _1 = \sum \limits _{i = 1}^M {\int _a^\infty {\frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }}} } \exp ( - \frac{x}{{\mathop \sigma \nolimits _{SiRn}^2 }})\exp ( - \frac{\beta }{{\mathop {\mathop \sigma \nolimits _{RnD}^2 \rho }\nolimits _n \eta x}})dx }, \end{aligned}$$
(43)

and

$$\begin{aligned} {\begin{array}{l} \mathop \Phi \nolimits _2 = \sum \limits _{i = 1}^M {\frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }}\int _a^\infty {\sum \limits _{k = 1}^{\mathop 2\nolimits ^{M - 1} - 1} {\mathop {( - 1)}\nolimits ^{\left| {\mathop D\nolimits _k } \right| } \exp ( - \sum \limits _{Sm \in \mathop D\nolimits _k } {\frac{x}{{\mathop \sigma \nolimits _{SmRn}^2 }}} )} } } \\ \;\;\;\;\;\;\;\;\;\;\;\; \times \exp ( - \frac{x}{{\mathop \sigma \nolimits _{SiRn}^2 }})\exp ( - \frac{\beta }{{\mathop {\mathop \sigma \nolimits _{RnD}^2 \rho }\nolimits _n \eta x}})dx \end{array}}. \end{aligned}$$
(44)

Using the Maclaurin series expansion, we have

$$\begin{aligned} {\exp \left( - \frac{\beta }{{\mathop {\mathop \sigma \nolimits _{RnD}^2 \rho }\nolimits _n \eta x}}\right) = \sum \limits _{u = 0}^\infty {\frac{{\mathop {( - 1)}\nolimits ^u \mathop \beta \nolimits ^u }}{{u!\mathop \sigma \nolimits _{RnD}^{2u} \mathop \rho \nolimits _n^u \mathop \eta \nolimits ^u \mathop x\nolimits ^u }}} }. \end{aligned}$$
(45)

Substituting (45) into (43),we can obtain the \({\mathop \Phi \nolimits _1 }\) as (46) at the top of following page, where \({Ei(x) = \int _x^\infty {\frac{{\mathop e\nolimits ^t }}{t}} dt}\)

$$\begin{aligned} {\begin{array}{l} \mathop \Phi \nolimits _1 = \sum \limits _{i = 1}^M {\frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }}\int _a^\infty {\exp ( - \frac{x}{{\mathop \sigma \nolimits _{SiRn}^2 }})} } dx - \sum \limits _{i = 1}^M {\frac{\beta }{{\mathop \sigma \nolimits _{SiRn}^2 \mathop \sigma \nolimits _{RnD}^2 \mathop \rho \nolimits _n \eta }}\int _a^\infty {\frac{1}{x}\exp ( - \frac{x}{{\mathop \sigma \nolimits _{SiRn}^2 }})} } dx\\ \;\;\;\;\;\;\;\;\;\;+ \sum \limits _{i = 1}^M {\frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }}\sum \limits _{u = 2}^\infty {\frac{{\mathop {( - 1)}\nolimits ^u \mathop \beta \nolimits ^u }}{{u!\mathop \sigma \nolimits _{RnD}^{2u} \mathop \rho \nolimits _n^u \mathop \eta \nolimits ^u }}} } \mathop \Phi \nolimits _{1,u} \\ \;\;\;\;\; = \sum \limits _{i = 1}^M {\exp ( - \frac{a}{{\mathop \sigma \nolimits _{SiRn}^2 }})} - \sum \limits _{i = 1}^M {\frac{\beta }{{\mathop \sigma \nolimits _{SiRn}^2 \mathop \sigma \nolimits _{RnD}^2 \mathop \rho \nolimits _n \eta }}} Ei(\frac{a}{{\mathop \sigma \nolimits _{SiRn}^2 }}) + \sum \limits _{i = 1}^M {\frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }}\sum \limits _{u = 2}^\infty {\frac{{\mathop {( - 1)}\nolimits ^u \mathop \beta \nolimits ^u }}{{u!\mathop \sigma \nolimits _{RnD}^{2u} \mathop \rho \nolimits _n^u \mathop \eta \nolimits ^u }}} } \mathop \Phi \nolimits _{1,u} \end{array}} \end{aligned}$$
(46)

and

$$\begin{aligned} \begin{array}{*{20}{l}} {{\Phi _{1,u}} = \int _a^\infty {\frac{1}{{\mathrm{{ }}{x^u}}}} \exp ( - \frac{x}{{\sigma _{SiRn}^2}})dx}\\ {\;\;\;\;\;\;\;\;\mathrm{{ = }}\frac{1}{{\mathrm{{ }}{a^{u - 1}}}}\sum \limits _{k = 0}^{u - 2} {\frac{{{{(\mathrm{{ }} - \mathrm{{ }}1)}^k}\mathrm{{ }}{a^k}}}{{\sigma _{SiRn}^{2k}(u - 1)(u - 2) \cdots (u - k + 1)}}} \exp ( - \frac{a}{{\sigma _{SiRn}^2}})}\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {{(\mathrm{{ }} - \mathrm{{ }}1)}^u}\frac{1}{{(u - 1)!\sigma _{SiRn}^{2(u - 1)}}}Ei( - \frac{a}{{\sigma _{SiRn}^2}})} \end{array}. \end{aligned}$$
(47)

Similarly, substituting (45) into (44), we can obtain the \({\mathop \Phi \nolimits _2 }\) as

$$\begin{aligned} {\begin{array}{l} \mathop \Phi \nolimits _2 = \sum \limits _{i = 1}^M {\sum \limits _{k = 1}^{\mathop 2\nolimits ^{M - 1} - 1} {\frac{{\mathop {( - 1)}\nolimits ^{\left| {\mathop D\nolimits _k } \right| } }}{{\mathop {b\sigma }\nolimits _{SiRn}^2 }}\exp ( - ab)} } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;- \sum \limits _{i = 1}^M {\sum \limits _{k = 1}^{\mathop 2\nolimits ^{M - 1} - 1} {\frac{{\mathop {( - 1)}\nolimits ^{\left| {\mathop D\nolimits _k } \right| } \beta Ei(ab)}}{{\mathop {\mathop \rho \nolimits _n \eta \mathop \sigma \nolimits _{RnD}^2 \sigma }\nolimits _{SiRn}^2 }}} } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \sum \limits _{i = 1}^M {\sum \limits _{k = 1}^{\mathop 2\nolimits ^{M - 1} - 1} {\sum \limits _{u = 2}^\infty {\frac{{\mathop {( - 1)}\nolimits ^{\left| {\mathop D\nolimits _k } \right| + u} \mathop \beta \nolimits ^u }}{{\mathop \sigma \nolimits _{SiRn}^2 u!\mathop \sigma \nolimits _{RnD}^{2u} \mathop {\mathop \rho \nolimits _n }\nolimits ^u \mathop \eta \nolimits ^u }}} } } \mathop \Phi \nolimits _{2,u} \end{array} }. \end{aligned}$$
(48)

where \({b = \frac{1}{{\mathop \sigma \nolimits _{SiRn}^2 }} + \sum \limits _{Sm \in \mathop D\nolimits _k } {\frac{1}{{\mathop \sigma \nolimits _{SmRn}^2 }}} }\) and

$$\begin{aligned} {\begin{array}{*{20}{l}} {{\Phi _{2,u}} = \int _a^\infty {\frac{1}{{\mathrm{{ }}{x^u}}}} \exp ( - bx)dx}\\ {\;\;\;\;\;\;\;\;\mathrm{{ = }}\frac{1}{{\mathrm{{ }}{a^{u - 1}}}}\sum \limits _{k = 0}^{u - 2} {\frac{{{{(\mathrm{{ }} - \mathrm{{ }}1)}^k}\mathrm{{ }}{a^k}\mathrm{{ }}{b^k}}}{{(u - 1)(u - 2) \cdots (u - k + 1)}}} \exp ( - ab)}\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {{(\mathrm{{ }} - \mathrm{{ }}1)}^u}\frac{{\mathrm{{ }}{b^{u - 1}}}}{{(u - 1)!}}Ei( - ab)} \end{array}}. \end{aligned}$$
(49)

Substituting (46) and (48) into (42), \({\mathop P\nolimits _{out,1}^{\mathrm{{TPS - JSRS}}} }\) can be obtained as (14).

Derivation of (30)

Denoting \({Y = {\left| {\mathrm{{ }}{h_{RnD}}} \right| ^2}}\), the \({\mathop P\nolimits _{out}^{\mathrm{{OPS - JSRS}}} }\) can be rewritten as

$$\begin{aligned} {\begin{array}{l} \mathop P\nolimits _{out}^{\mathrm{{OPS - JSRS}}} = \prod \limits _{n = 1}^N {\int _0^\infty {\mathop f\nolimits _Y (y)\mathop F\nolimits _{{{\left| {\mathrm{{ }}{h_{S\mathop m\nolimits ^* \mathop R\nolimits _n }}} \right| }^2}} } } (\frac{\alpha }{y} + \alpha \eta )dy\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \prod \limits _{n = 1}^N {\int _0^\infty {\frac{1}{{\sigma _{RnD}^2}}\exp ( - \frac{y}{{\sigma _{RnD}^2}})} } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \prod \limits _{m = 1}^M {[1 - \exp ( - \frac{\alpha }{{\sigma _{SmRn}^2y}} - \frac{{\alpha \eta }}{{\sigma _{SmRn}^2}})]} dy \end{array}}. \end{aligned}$$
(50)

The term \({\prod \limits _{m = 1}^M {[1 - \exp ( - \frac{\alpha }{{\sigma _{SmRn}^2y}} - \frac{{\alpha \eta }}{{\sigma _{SmRn}^2}})]}}\) can be expanded as

$$\begin{aligned} {\begin{array}{l} \prod \limits _{m = 1}^M {[1 - \exp ( - \frac{\alpha }{{\sigma _{SmRn}^2y}} - \frac{{\alpha \eta }}{{\sigma _{SmRn}^2}})]} \\ = 1 + \sum \limits _{j = 1}^{\mathop 2\nolimits ^M - 1} {\mathop {( - 1)}\nolimits ^{\left| {\mathop A\nolimits _j } \right| } } \exp ( - \sum \limits _{Sm \in \mathop A\nolimits _j } {(\frac{\alpha }{{\sigma _{SmRn}^2y}} + \frac{{\alpha \eta }}{{\sigma _{SmRn}^2}})} ) \end{array}}. \end{aligned}$$
(51)

where \({\mathop A\nolimits _j }\) represents the j-th nonempty subset of M sources, \({\left| {\mathop A\nolimits _j } \right| }\) represents the number of elements in set \({\mathop A\nolimits _j }\). Substituting (51) into (50), we can obtain the outage probability of OPS-JSRS scheme \({\mathop P\nolimits _{out}^{\mathrm{{OPS - JSRS}}} }\) as (30).

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Jiang, X., Li, P. & Wang, R. Power Splitting and Source-Relay Selection in Energy Harvesting Wireless Network. Wireless Pers Commun 124, 2141–2160 (2022). https://doi.org/10.1007/s11277-021-09449-1

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