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

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.

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).