# Outage Analysis and Power Allocation Optimization for Multiple Energy-Harvesting Relay System Using SWIPT

Authors: Kang Liu; Qi Zhu; Ying Wang
Journal: Mobile Information Systems
 (2018)

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## Abstract

Energy harvesting (EH) combined with cooperative relying plays a promising role in future wireless communication systems. We consider a wireless multiple EH relay system. All relays are assumed to be EH nodes with simultaneous wireless and information transfer (SWIPT) capabilities, which means the relays are wirelessly powered by harvesting energy from the received signal. Each EH node separates the input RF signal into two parts which are, respectively, for EH and information transmission using the power splitting (PS) protocol. In this paper, a closed-form outage probability expression is derived for the cooperative relaying system based on the characteristic function of the system’s probability density function (PDF) with only one relay. With the approximation of the outage probability expression, three optimization problems are built to minimize the outage probability under different constraints. We use the Lagrange method and Karush–Kuhn–Tucker (KKT) condition to solve the optimization problems to jointly optimize the relay’s PS factors and the transmit power. Numerical results show that our derived expression of the outage probability is accuracy and gives insights into the effect of various system parameters on the performance of protocols. Meanwhile, compared with the no optimal condition, our proposed optimization algorithms can all offer superior performance under different system constraints.


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## Body

## 1. Introduction

Recently, with the development of wireless communication networks, energy consumption has become a serious increasing problem. As a result, green communications have attracted many researchers’ attention. Energy harvesting (EH), which can gain energy from the environment, appears to be an effective technique for green communications and has great potential to extend the lifetime of communication devices [1, 2]. On the contrary, cooperative relaying is an effective technique to improve the coverage of wireless networks, avoid the decrease of communication quantity caused by channel fading, and increase the channel gain. So, how to use EH in such networks with multiple relays is an interesting topic.Plenty of studies have already been carried out about wireless systems which harvest energy from the environment. An optimal packet-scheduling strategy is proposed in [3] in order to minimize the transmission time in the EH wireless communication system using additive white Gaussian noise (AWGN) channels. The traffic load and available energy are considered in order to adaptively change the transmission rate. Under the same AWGN channel scenario, an optimal energy allocation scheme which maximizes the throughput is obtained in [4] with the use of dynamic programming and convex optimization techniques. The AWGN channel capacity under energy constrains is studied in [5]. It is proved that given sufficient energy supply, the AWGN channel capacity of the EH model equals the capacity of the continuous energy supply model. A peer-to-peer communication system is analyzed in [3–5] without considering relays. In the literature [6], a communication system where the source and relays are EH nodes is considered. Its main work is to maximize the peer-to-peer system throughput using joint relay selection and power allocation schemes. For a slow-fading channel, Li et al. [7] analyze an EH cooperative relaying network and derive the closed-form outage probability expression for the proposed protocol. Different from others, multiple source-destination pairs with an EH relay in a wireless cooperative network is considered in [8], and a power allocation scheme of the harvested energy is proposed. Meanwhile, the water-flooding method is used to achieve a better balance between the system performance and complexity.Apart from the strategies that harvest energy from the environment, a new way is to use the radiofrequency (RF) signal [9]. The idea is first studied in [10], and a new energy transfer technology is proposed, that is, simultaneous wireless and information transfer (SWIPT). Two schemes are further proposed in [11] coordinating the information processing (IP) and EH, which are time switching (TS) and power splitting (PS). In [12–16], a relaying network where an energy-constrained relay node using SWIPT and TS or PS schemes is considered. In particular, the outage probability expressions and the ergodic capacity are derived in [12, 16], and effects of various system parameters are illustrated. The work in [12] is extended in [13] by assuming that the source also harvests energy from the relay. The outage performances of two schemes where one uses the EH relay and the other conveys data via direct links are analyzed in [14], and an optimization problem which minimizes the transmit power is also formulated. In two-way full-duplex (FD) relay networks, the capability is first investigated in [15], and two relay selection schemes are proposed in order to minimize the outage probability and maximize the sum capacity, respectively. Considering cooperative EH communications, a multiple-relay selection scheme using geometric programming (GP) is proposed in [17] in order to maximize data rate-based utilities over multiple coherent time slots. The work in [18] is extended to a cognitive two-way EH relay network. Specifically, closed-form expressions of the throughput are derived to maximize interference temperature apportioning parameter (ITAP) and the PS parameter. It is shown that they can be optimized separately.Recent research work mainly focuses on the system performance of wireless networks with a single EH relay. However, few studies have been presented to analyze the system performance of multiple-relay EH systems using the PS protocol. In this paper, we consider an amplify-and-forward (AF) wireless network with multiple EH relays. We assume that all the relays use SWIPT to harvest energy from the RF signal and adopt the PS protocol for EH and IP. We analyze the system performance and derive closed-form expressions for the outage probability of the multiple-relay EH system. We also jointly optimize relays’ PS factors and their transmit power levels to minimize the outage probability. The main contributions of this paper are summarized as follows:(1)
We use fitting algorithms to simplify the outage probability of the system with only one relay, and the result is approximated as a combination of several exponential functions. The closed-form expression for the outage probability of the multiple-relay EH system is obtained using the characteristic function of the probability density function (PDF).(2)
Based on the approximation of the outage probability expression, we propose three optimization algorithms to obtain power allocation policies which minimize the outage probability under different constraints. In particular, Algorithm1 is to optimize relays’ PS factors without considering the power constrains. Algorithm 2 is to optimize relay’s transmit power levels with their total transmit power limited. Algorithm 3 is to jointly optimize the relay’s PS factors and the transmit power of the source under the total power constrains of the whole system.(3)
For each algorithm, we use the Lagrange method and Karush–Kuhn–Tucker condition to solve the optimization problems. Our derived expression of the outage probability is consistent with simulation results and gives insights into the effect of various system parameters on the system performance. Meanwhile, our proposed optimization algorithms can offer superior performance, and the improvement is the same for different system conditions.Algorithm 1:The optimal power splitting factor design under each relay’s power constraint.
(1)
Using the transmit powerPs and the energy conversion efficiency ηi in problem (37) to obtain power splitting factor ρi for each relay.
(2)
Usingρi in Pri=ηiρiPsγ¯ri to obtain each relay’s transmit power Pri_temp.
(3)
if Pri_temp≥Prmaxi,
(4)
Pri=Prmaxi;
(5)
else Pri=Pri_temp
(6)
end
(7)
UsingPri to achieve the final outage probability.Algorithm 2:The optimal power splitting factor design under all relays’ total power constraint.
(1)
Initialization: Select a feasibleμi1. Choose the step size k and the error tolerance σ, and let n=1
(2)
while (1)
(3)
Usingμin in dLμ/dρi=0 to solve the power splitting factor ρin for each relay.
(4)
Usingρin and ρin−1 in logPout to obtain logPoutn and logPoutn−1, respectively.
(5)
if logPoutn−logPoutn−1<σ
(6)
ρi∗=ρin;
(7)
break;
(8)
else
(9)
Usingρi in formula (43) to update μin+1.
(10)
n=n+1;
(11)
end
(12)
ifn>100
(13)
break;
(14)
end
(15)
end
(17)
UsingPri to achieve the final outage probability.Algorithm 3:The optimal power splitting factor design under the system’s power constraint.
(1)
Initialization: Letφ=−N/Pmax and select a feasible Ps1. Choose the step size k and the error tolerance σ, and let n=1
(2)
while (1)
(3)
UsingPsn in dLφ/dρi=0 to solve the power splitting factor ρin for each relay.
(4)
Usingρin and ρin−1 in logPout to obtain logPoutn and logPoutn−1, respectively.
(5)
if logPoutn−logPoutn−1<σ
(6)
Ps∗=Psn
(7)
ρi∗=ρin;
(8)
break;
(9)
else
(10)
Usingρi in formula (50) to update power Psn+1
(11)
n=n+1;
(12)
end
(13)
if n>100
(14)
break;
(15)
end
(16)
end
(17)
UsingPs∗,ρi∗ to achieve the final outage probability.


## 2. System Model

As shown in Figure1, we consider a wireless communication system containing a source node, S, a destination node, D, and N relay nodes, Ri, i=1,2,3,…,N. The information is transferred from the source node to the destination node through relay nodes. Only relays can harvest energy from the RF signal. We assume that the channel state information (CSI) is only available at the source node, which means all relays are independent from each other. The source node broadcasts the information to all the relays, and the relays adopt the AF scheme to transfer the information to the destination node. Also, there is no direct link between the source node and the destination node.Figure 1
System diagram.At the relays, the received signals are split according to the PS protocol. As Figure2 shows, the information transformation is conducted in two time slots. In the first slot, the relays receive signals from the source nodes. In the other slot, the relays send the signals to the destination node. In Figure 2, yrt is the signal received at the relay. P is the power of the received signal. The relay splits the signal into two parts according to a power ratio of 1−ρ:ρ. One part is for IP and the other is for EH. The harvested energy is used to power the transmission circuits.Figure 2
Relay node diagram [9].We set the channel gain from the source nodeS to the relay node Ri as hi and from the relay node Ri to the destination node D as gi. The distances between the source node S and relay node Ri and between the relay node and the destination D are d1i and d2i, respectively. m is the path loss exponent. So that we define the equivalent channel gains as γsri=hi2/d1im and γsri=gi2/d2im.Assume that the information signal from the source issk. In the first time slot, the signal at the relay used for information transmission can be given by(1)yrik=1−ρiPsγsrisk+1−ρinarik+ncrik,where Ps is the transmit power at the source, narik is the AWGN incurred at the radiofrequency, and ncrik is the AWGN incurred in the RF-to-baseband conversion. Both of the noise signals are considered as a Gaussian random variable with mean zero and variance σnari2 and σncri2, respectively. ρi is the power factor that the received signal splits for EH. The energy that the relay harvests is given by Ehri=ηiρiPsγsriT/2, where 0<ηi<1 is the energy conversion efficiency and T/2 is the duration of the time slot.In the second time slot, the relay transfers the signal to the destination node. So the received signal at the destination node is given by(2)ydik=γrdiPribpiyrik+nadik+ncdik,where Pri=Ehri/T/2=ηiρiPsγsri is the transmit power at the relay and bpi is the normalized transmit factor which is bpi=1/1−ρPsγsri+1−ρσnari2+σncri2. nadik and ncdik are the RF-front noise and the conversion noise with mean zero and variance σnadi2 and σncdi2, respectively.In order to simplify the formula, we usenrik=1−ρinarik+ncrik and ndrik=nadik+ncdik to represent the total AWGNs at the relay and destination node, respectively. Substituting (1) into (2),(3)ydik=bpi1−ρiPriPsγsriγrdisk+bpiPrinrik+ndrik.Assume that the variances for the noise signal satisfyσnri2=1−ρiσnari2+σncri2, σndri2=σnadi2+σncdi2 and σnari2=σncri2=σ2, σnadi2=σncdi2=σ2. Substituting Pri and bpi in (3), the signal-to-noise ratio (SNR) at the destination is given by(4)SNRi=Ps21−ρiηiρiγsri2γrdiPs1−ρiγsriσndri2+Psηiρiγsriγrdiσnri2+σndri2σnri2.Let(5)Ai=Ps21−ρiηiρi,Bi=Ps1−ρiσndri2,Ci=Psηiρiσnri2.The final expression is given by(6)SNRi=Aiγsri2γrdiBiγsriγrdi+Ciγsri+σndri2σnri2.


### 2.1. Outage Probability Analysis

Assume that all the relays take part in the transmission. According to the maximal-ratio combining (MRC), the final SNR at the destination node is given by(7)SNR=∑i=1NSNRi.Then, based on the Shannon capacity formula, the overall throughout is derived as(8)C=12log21+SNR.LetR be the minimized rate demand for users. So the outage probability is defined as Pout=PrC<R. Using (6)–(8), it becomes(9)Pout=Pr∑i=1NSNRi<22R−1=Pr∑i=1NAiγsri2γrdiBiγsriγrdi+Ciγsri+σndri2σnri2<22R−1.Define the random variablesωi=Aiγsri2γrdi/Biγsriγrdi+Ciγsri+σndri2σnri2, and the cumulative distribution function (CDF) of ωi is given in [3] as(10)Poutx=Prωi<x=1−e−Cix/Aiγ¯riuiK1ui,where ui=4Bix/Aiγ¯riγ¯di, γ¯ri=Ehi2/d1im, and γ¯ri=Egi2/d2im. K1⋅ is the first-order modified Vessel function of the second kind [20].Defining the random variableW=∑i=1N1ωi, our goal is to obtain the CDF of the W, PrW≤x. According to [13], we will use the characteristic function to achieve the final expression.In formula (10), the CDF of ωi can be split into two parts. One part e−Cix/Aiγ¯ri is an exponential random variables, the other uiK1ui is too difficult to obtain the CDF directly. However, it is achievable if we approximate this part. Notice that uiK1ui is a function of an independent variable ui, and ui=4Bix/Aiγ¯riγ¯ri=4σndri2x/Psηiρiγ¯riγ¯ri, which means the value of ui is mainly dependent on σnri2/Psγ¯riγ¯ri. Due to the fact that γ¯ri and γ¯ri represent the equivalent channel gain for the source node to the relay and the relay to the destination node, respectively, Psγ¯riγ¯ri can be recognized as the received power Pr. According to the basic demand of the system, the SNR at the receiver is always larger than 0dB, which means σnri2/Psγ¯riγ¯ri<1. So the independent variable ui can be approximated as ui<1.Letfui=uiK1ui. Considering the character of Vessel function, if we let ui=0, then fui=1 and if let ui⟶∞, then fui=0. It is the same for the exponential random variables. Also, noticing that the former part of Poutx is already an exponential random variable, it is not hard to consider using a combination of several exponential functions to approximate fui. fui can be approximated as(11)fui=a1⋅e−b1ui2+a2⋅e−b2ui2+⋯+aL⋅e−bLui2,where a1+a2+⋯+aL=1.We use fitting tools in MATLAB to obtain the coefficients of the approximate function offui and getwhenL=1, a1=1, b1=0.6whenL=2, a1=0.12, a2=0.88, b1=6.2, b2=0.38whenL=3, a1=0.9, a2=−2.3, a3=2.4, b1=0.4, b2=7.8, b3=7.8Figure3 shows the approximate result of fui. For L=1, the performance of fitting approach is not good, and the estimation errors between two functions are too large. But for L=2 and L=3, the results reflect the trend of the original function perfectly, and the degree of those two fitting function are nearly 0.99. Also, the complexity for L=2 is lower than that for L=3, so the final function for Poutx can be approximated as(12)Poutx=1−e−Cix/Aiγ¯ri⋅0.12e−6.2ui2+0.88e−0.38ui2.Figure 3
Different fitting function forL=1,2,3.Usingui in formula (12), the CDF of ωi is given by(13)Prωi<x≈1−e−λi1x−e−λi2x,where λi1=Ci/Aiγ′ri+0.62u′i2, λi2=Ci/Aiγ′ri+0.38u′i2, and u′i=4Bi/Aiγ¯riγ¯ri. Using the characteristic function, the CDF of W is(14)PrW≤x=∑A∈U,B∈U∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi11−e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi21−e−λi2w.Proof. The probability density function (PDF) of ωi is(15)fx=λi1e−λi1x+λi2e−λi2x.The characteristic function ofωi is(16)gwit=1−jtλi1−1+1−jtλi2−1.According to the character of the characteristic function, the characteristic function ofW is(17)gWt=∏i=1N11−jtλi1−1+1−jtλi2−1.It can be rewritten as(18)gWt=∑A∈U,B∈U∏i∈A1−jtλi1−1∏j∈B1−jtλj2−1,where U=1,2,3,…,N1 and A, B denote all the subsets of U that satisfy A+B=U. From formula (18), gWt is a combination of 2N1 product terms. We only need to consider the PDF of one particular term and can obtain the others in the same way.ForA=U, B=∅, the characteristic function is(19)gA=U,B=∅t=∏i=1N11−jtλi1−1.Based on the connection between the PDF and the characteristic function,(20)fWw=12π∫−∞∞e−jtwgWtdt.Using the method of partial fraction,gA=U,B=∅t can be rewritten as(21)gA=U,B=∅t=∏i=1N11−jtλi1−1=∑i=1N1Niλi1−jt,where Ni=gA=U,B=∅t⋅λi1−jtλi1=jt=λi1⋅∏k=1,k≠iN1λi1/λk1−λi1.Then,(22)fA=U,B=∅ω=∑i=1N1∏k=1,k≠iN1λk1λk1−λi1λi1e−λi1w.Formula (22) is the PDF of this particular case. For random A and B, we can obtain(23)fA,Bω=∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi1e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi2e−λi2w,because(24)fWw=∑A∈U,B∈UfA,Bω.The CDF ofW is(25)PrW≤x=∑A∈U,B∈U∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi11−e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi21−e−λi2w.


### 2.2. Power Allocation and Scheduling Design

In this subsection, we propose different algorithms based on different power constrains by optimizing the transmit power of the source node and the power splitting factor of the relays, in order to minimize the outage probability. However, formula (14) is too complex for optimization. So, we choose a new approximate function for the outage probability which will give up some accuracy but is easy for further analysis.The outage probability is now approximated as(26)Poutx=1−e−Cix/Aiγ¯ri⋅e−0.6ui2+1.4ui−1.3ui2.Proof. According to the formula in [20], ∫0∞e−β/4x−γxdx=β/γK1βγ; let ui=βγ, we can get(27)uiK1ui=γ⋅βγK1βγ=γ⋅∫0∞e−β/4x−γxdx.Lett=γx, the integration can be rewritten as(28)uiK1ui=γ⋅∫0∞e−β/4x−γxdx=∫0∞e−β/4t−tdt=∫0∞e−ui2/4t⋅e−tdt.By using the mean value theorem of integrals, there must exit anε between 0,+∞ and(29)uiK1ui=e−ui2/4ε∫0∞e−tdt=e−ui2/4ε.The value ofε changes with ui. Assuming that uiK1ui=ef′ui⋅ui2 and using the fitting tools in MATLAB, the analytical expression is(30)f′ui=−0.6ui2+1.4ui−1.3.For optimization problems, it is not necessary to estimate the outage probability accurately. So we letfui=0.5 for further simplification.The CDF ofωi is(31)Prωi<x≈1−e−λix,where λi=Ci/Aiγ¯ri+0.5⋅u′i2 and u′i=4Bi/Aiγ¯riγ¯ri. Using the characteristic function, the CDF of W is(32)PrW≤x≃∑i=1N1∏k=1,k≠iN1λkλk−λi1−e−λix.According to the conclusion in [19], the CDF of W can be written as(33)PrW≤x≤1N1+1!⋅∏i=1N1λi⋅22+1x−1N1+1.Based on the formula (33), we propose three algorithms to obtain power allocation policies which minimize the outage probability under different constraints. In particular,Algorithm1: with the transmit power of the source node fixed, we consider each relay’s transmit power is limited. The maximal power for each relay is Prmaxi.Algorithm2: with the transmit power of the source node fixed, we consider the total transmit power of all the relays is limited. The maximal power is Prmax.Algorithm3: the transmit power of the source node is not fixed and the total transmit power of the system is limited. The maximal power is Pmax.


#### 2.2.1. Algorithm1: The Optimization Algorithm under Each Relay’s Power Constraint

The optimal power allocation scheme is based on minimizing the outage probability bound in (33). Considering Pri=ηiρiPsγ¯ri, we can transfer the problem into finding the optimal power splitting factor ρi∗ to minimizing the outage probability bound. Assuming that the power of the transmission signal from the source node is fixed, the optimization can be written as(34)P1:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅22+1x−1N1+1s.t.Pri≤Prmaxi0<ρi<1,where Pri=ηiρiPsγ¯ri. Using the monotonicity of the log function, our optimization problem can be written as(35)P1′:argminρilogPout=∑i=1N1logλi′s.t.ηiρiPsγri¯≤Prmaxi0<ρi<1,where the fixed terms in the optimization bound is removed. It is easy to know that cost function in (35) is convex in ρi, which means the minimal value exists in the range 0,1. Using λ′i and Pri in the cost function, we can obtain the derivative of the function as(36)∂logPout∂ρi=22+1x−1N1+1N1+1!⋅ηiγdi¯−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri.Let∂logPout/∂ρi=0, we can get(37)ρi=−4+16+8ηiγ¯ri−22ηiγ¯ri−2negative  reject.We summerize it in Algorithm1.


#### 2.2.2. Algorithm2: The Optimization Algorithm under All Relays’ Total Power Constraint

According to [19, 21], it is necessary to consider the total power consumed by the system. In our condition, we first consider the total power of all the relays. Based on minimizing the outage probability bound in (33), we consider that the total power of all the relays are limited and formulate an optimization problem as(38)P2:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.∑iN1Pri≤Prmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(39)P2′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλi′s.t.∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is a convex optimization problem inρi. In order to solve the problem, we can use the Lagrangian method. We split the question into two steps. The first step is to use the Lagrangian of this optimization problem to solve the power splitting factor for each relay. It can be written as(40)Lμ=C⋅∑i=1N1logλ′i+μPrmax−∑iN1Pri,where C=2N1+1x−1N1+1/N1+1! and μ represents the Lagrange multiplier associated with the power constraint in (39). The derivative of the function is(41)dLμdρi=C⋅ηiγ¯di−2⋅ρi2+4ρi−2ηiρiγ¯di+21−ρiηiPs1−ρiρiγ¯riγ¯di−μηiPsγ¯ri.Applying the KKT conditions, we letdLμ/dρi=0 and obtain the following necessary and sufficient conditions for a fixed μ as(42)ρi=0,dLμdρi=0has no answer within0,1,ρi∗,dLμdρi=0has answerρi∗within0,1.Step two is to update the Lagrange multiplier by the power constraint in problem (39) as follows at each iteration n.(43)μin+1=μin−kPrmax−∑iN1Pri,where k is the step size at the nth iteration.We summerize it in Algorithm2.


#### 2.2.3. Algorithm3: The Optimization Algorithm under the System’s Power Constraint

Now, we consider the total power of the system. Based on minimizing the outage probability bound in (33), we consider that the total power of the whole system is limited and formulate an optimization problem as(44)P3:argminPs,ρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.Ps+∑iN1Pri≤Pmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(45)P3′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλj′s.t.Ps+∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is also a convex optimization problem in bothPs and ρi and can be solved by the Lagrangian method. We can obtain the Lagrangian of this optimization problem:(46)Lφ=C⋅∑i=1N1logλ′i+φPmax−Ps−∑iN1Pri,where φ represents the Lagrange multiplier. The derivative for Ps of the function is(47)dLφdPs=−1Ps−φ1+∑i=1Nρiηi.LetdLφ/dPs=0 and we can obtain φ=−N/Pmax.Also the derivative forρi of the function is(48)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri−φηiPsγ¯ri.Useφ=−N/Pmax in formula (48) and obtain(49)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri+NPmaxηiPsγ¯ri,In order to solve the optimization problem, we split it into two steps.(1)
Optimal power splitting factor for each relay.For a givenPsn, let dLφ/dρi=0, we can obtain the optimal power splitting factor ρin for each relay.(2)
Update the transmit power of the source node.By the power constraint in problem (45), we can update Psn using the following formula as(50)Psn+1=Pmax1+∑i=1Nρin.We summerize it in Algorithm3.


### 2.3. Numerical Results

We use MATLAB to conduct some simulations to prove the theoretical analysis presented in the former part. We assumed that the system contains a source node, a destination node, and multirelays. We set the distance between the source node and the destination noded=2.4. All the relays are located in a circular manner whose the center is at the midpoint between the source node and the destination with radius r=1.2. We adopt Ps=5w and set σ2=0.01 [11]. The large scale fading exponent of the channel m is 2.7, and the small scale fading coefficients are generated as independent and identical (i.i.d.) Rayleigh random variables with mean zero and variance 1.Figure4 shows the outage probabilities for the theoretical analysis and the simulation versus normalized user transmission rate R bits/sec/Hz. From Figure 4, it is clear that the theoretical analysis is basically the same with the simulation results for different relay numbers. Also the outage probability increases as R increases. Meanwhile, comparing the outage probability for different relay numbers, we know that the outage probability decreases as relay number increases, which proves multiple relays can achieve the diversity gain of the channel.Figure 4
Outage probabilities for the theoretical analysis and the real scenario with different relay numbers.Figure5 shows the effect of the power splitting factor ρ. From the figure, we know that the outage probability is increasing when ρ is smaller but later, it starts decreasing, which means there is an optimal value for ρ. When the value of ρ is smaller than the optimal ρ, there is less power available for EH. Therefore, the transmit power of the relay node is not enough, which leads to the increase of the outage probability. On the contrary, when ρ is large than the optimal value, although the power for EH is enough, the power left for information transmission is low. Because of that, the signal strength at the destination node is not enough, and the outage probability increases too. Meanwhile, comparing different relay numbers N=4 and N=5 and different relay distribution r=1.2 and r=1, the trend of the outage probability is similar, but the value is much lower for N=5 and r=1. That is because when the relays are located near the midpoint between the source node and the destination, both the channel gain hi and gi are better.Figure 5
Outage probabilities vs. power splitting factorρ for different relay numbers N=4 and N=5 and different source to destination distances r=1.2 and r=1.Figure6 plots the effect of the source transmit power Ps. As shown in Figure 6, the outage probability of the system decreases as Ps increases. This is because the large value of Ps leads to more received power at the relay nodes, which means more power for both EH and information transmission. Consequently, the outage probability for the system is small. However, the outage probability does not change much when Ps is large enough, which means there is no need to continually increase transmit power in order to decrease the outage probability. Also, changing the relay number and the source to destination distance, the trend is the same, and the value for the large relay number and short source to destination distance is lower. Meanwhile, we notice that the variation of the tendency for the relay number N=5 is not the obvious comparing with that for N=4.Figure 6
Outage probabilities vs. the source transmit powerPs for different relay numbers N=4 and N=5 and different source to destination distances d=1.2 and d=1.4.Figure7 depicts the effect of the energy conversion efficiency. From this, we can observe that for smaller values of η, the outage probability is higher, and for larger values, the probability is lower, because the value of η decides the power from the relay node as Pri∝ηi. Also, changing the relay number and the source to destination distance, the trend is the same and the value for large relay number and short source to destination distance is lower. But from the figure, when the relay number N=5, the value of the outage probability is almost unchanged. That means we should also limit the number of relays because the large number of relays may cause the restriction of the system performance.Figure 7
Outage probabilities vs. the effect of the energy conversion efficiencyη for different relay numbers N=4 and N=5 and different source to destination distances d=1.2 and d=1.4.Figures8–10 show a comparison between no optimal power allocation and optimal power allocation schemes in Algorithm 2 and Algorithm 3, respectively. From Figure 8, we observe that the outage probability is lower by using the scheme in Algorithm 1. The improvement is the same for schemes in Algorithm 2 and Algorithm 3. Here, in order to make the numerical result more reliable, we set the power consumption of the two schemes equal. In particular, for Algorithm 2, we set the power splitting factor ρ=0.5 for no optimal scheme, and the power constraint for all relays is Prmax=ρ⋅Ps⋅N. For Algorithm 3, the power splitting factor is also ρ=0.5, and the power constraint for the total system is Prmax=ρ⋅Ps⋅N. Based on this condition, we can observe that the outage probability for optimal schemes is lower than that for no optimal situation. And it also has the improvement for different source power Ps and relay number N. From the figures, we can also see that the outage probability is growing with the increasing of the transmission rate R, and the variation between no optimal power allocation and optimal power allocation schemes is almost the same for a different source power Ps. But for the relay number N=5, the growing rate and the variation between two algorithms are much higher than that for N=3. This also proves that when we want to use multiple relays, the number should be under control to prevent the rapid deterioration of the system performance.Figure 8
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm1.Figure 9
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm2.Figure 10
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm3.


## 2.1. Outage Probability Analysis

Assume that all the relays take part in the transmission. According to the maximal-ratio combining (MRC), the final SNR at the destination node is given by(7)SNR=∑i=1NSNRi.Then, based on the Shannon capacity formula, the overall throughout is derived as(8)C=12log21+SNR.LetR be the minimized rate demand for users. So the outage probability is defined as Pout=PrC<R. Using (6)–(8), it becomes(9)Pout=Pr∑i=1NSNRi<22R−1=Pr∑i=1NAiγsri2γrdiBiγsriγrdi+Ciγsri+σndri2σnri2<22R−1.Define the random variablesωi=Aiγsri2γrdi/Biγsriγrdi+Ciγsri+σndri2σnri2, and the cumulative distribution function (CDF) of ωi is given in [3] as(10)Poutx=Prωi<x=1−e−Cix/Aiγ¯riuiK1ui,where ui=4Bix/Aiγ¯riγ¯di, γ¯ri=Ehi2/d1im, and γ¯ri=Egi2/d2im. K1⋅ is the first-order modified Vessel function of the second kind [20].Defining the random variableW=∑i=1N1ωi, our goal is to obtain the CDF of the W, PrW≤x. According to [13], we will use the characteristic function to achieve the final expression.In formula (10), the CDF of ωi can be split into two parts. One part e−Cix/Aiγ¯ri is an exponential random variables, the other uiK1ui is too difficult to obtain the CDF directly. However, it is achievable if we approximate this part. Notice that uiK1ui is a function of an independent variable ui, and ui=4Bix/Aiγ¯riγ¯ri=4σndri2x/Psηiρiγ¯riγ¯ri, which means the value of ui is mainly dependent on σnri2/Psγ¯riγ¯ri. Due to the fact that γ¯ri and γ¯ri represent the equivalent channel gain for the source node to the relay and the relay to the destination node, respectively, Psγ¯riγ¯ri can be recognized as the received power Pr. According to the basic demand of the system, the SNR at the receiver is always larger than 0dB, which means σnri2/Psγ¯riγ¯ri<1. So the independent variable ui can be approximated as ui<1.Letfui=uiK1ui. Considering the character of Vessel function, if we let ui=0, then fui=1 and if let ui⟶∞, then fui=0. It is the same for the exponential random variables. Also, noticing that the former part of Poutx is already an exponential random variable, it is not hard to consider using a combination of several exponential functions to approximate fui. fui can be approximated as(11)fui=a1⋅e−b1ui2+a2⋅e−b2ui2+⋯+aL⋅e−bLui2,where a1+a2+⋯+aL=1.We use fitting tools in MATLAB to obtain the coefficients of the approximate function offui and getwhenL=1, a1=1, b1=0.6whenL=2, a1=0.12, a2=0.88, b1=6.2, b2=0.38whenL=3, a1=0.9, a2=−2.3, a3=2.4, b1=0.4, b2=7.8, b3=7.8Figure3 shows the approximate result of fui. For L=1, the performance of fitting approach is not good, and the estimation errors between two functions are too large. But for L=2 and L=3, the results reflect the trend of the original function perfectly, and the degree of those two fitting function are nearly 0.99. Also, the complexity for L=2 is lower than that for L=3, so the final function for Poutx can be approximated as(12)Poutx=1−e−Cix/Aiγ¯ri⋅0.12e−6.2ui2+0.88e−0.38ui2.Figure 3
Different fitting function forL=1,2,3.Usingui in formula (12), the CDF of ωi is given by(13)Prωi<x≈1−e−λi1x−e−λi2x,where λi1=Ci/Aiγ′ri+0.62u′i2, λi2=Ci/Aiγ′ri+0.38u′i2, and u′i=4Bi/Aiγ¯riγ¯ri. Using the characteristic function, the CDF of W is(14)PrW≤x=∑A∈U,B∈U∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi11−e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi21−e−λi2w.Proof. The probability density function (PDF) of ωi is(15)fx=λi1e−λi1x+λi2e−λi2x.The characteristic function ofωi is(16)gwit=1−jtλi1−1+1−jtλi2−1.According to the character of the characteristic function, the characteristic function ofW is(17)gWt=∏i=1N11−jtλi1−1+1−jtλi2−1.It can be rewritten as(18)gWt=∑A∈U,B∈U∏i∈A1−jtλi1−1∏j∈B1−jtλj2−1,where U=1,2,3,…,N1 and A, B denote all the subsets of U that satisfy A+B=U. From formula (18), gWt is a combination of 2N1 product terms. We only need to consider the PDF of one particular term and can obtain the others in the same way.ForA=U, B=∅, the characteristic function is(19)gA=U,B=∅t=∏i=1N11−jtλi1−1.Based on the connection between the PDF and the characteristic function,(20)fWw=12π∫−∞∞e−jtwgWtdt.Using the method of partial fraction,gA=U,B=∅t can be rewritten as(21)gA=U,B=∅t=∏i=1N11−jtλi1−1=∑i=1N1Niλi1−jt,where Ni=gA=U,B=∅t⋅λi1−jtλi1=jt=λi1⋅∏k=1,k≠iN1λi1/λk1−λi1.Then,(22)fA=U,B=∅ω=∑i=1N1∏k=1,k≠iN1λk1λk1−λi1λi1e−λi1w.Formula (22) is the PDF of this particular case. For random A and B, we can obtain(23)fA,Bω=∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi1e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi2e−λi2w,because(24)fWw=∑A∈U,B∈UfA,Bω.The CDF ofW is(25)PrW≤x=∑A∈U,B∈U∑i∈A∏k∈A,k≠iλk1λk1−λi1∏k∈B,k≠iλk2λk2−λi1λi11−e−λi1w+∑i∈B∏k∈A,k≠iλk1λk1−λi2∏k∈B,k≠iλk2λk2−λi2λi21−e−λi2w.


## 2.2. Power Allocation and Scheduling Design

In this subsection, we propose different algorithms based on different power constrains by optimizing the transmit power of the source node and the power splitting factor of the relays, in order to minimize the outage probability. However, formula (14) is too complex for optimization. So, we choose a new approximate function for the outage probability which will give up some accuracy but is easy for further analysis.The outage probability is now approximated as(26)Poutx=1−e−Cix/Aiγ¯ri⋅e−0.6ui2+1.4ui−1.3ui2.Proof. According to the formula in [20], ∫0∞e−β/4x−γxdx=β/γK1βγ; let ui=βγ, we can get(27)uiK1ui=γ⋅βγK1βγ=γ⋅∫0∞e−β/4x−γxdx.Lett=γx, the integration can be rewritten as(28)uiK1ui=γ⋅∫0∞e−β/4x−γxdx=∫0∞e−β/4t−tdt=∫0∞e−ui2/4t⋅e−tdt.By using the mean value theorem of integrals, there must exit anε between 0,+∞ and(29)uiK1ui=e−ui2/4ε∫0∞e−tdt=e−ui2/4ε.The value ofε changes with ui. Assuming that uiK1ui=ef′ui⋅ui2 and using the fitting tools in MATLAB, the analytical expression is(30)f′ui=−0.6ui2+1.4ui−1.3.For optimization problems, it is not necessary to estimate the outage probability accurately. So we letfui=0.5 for further simplification.The CDF ofωi is(31)Prωi<x≈1−e−λix,where λi=Ci/Aiγ¯ri+0.5⋅u′i2 and u′i=4Bi/Aiγ¯riγ¯ri. Using the characteristic function, the CDF of W is(32)PrW≤x≃∑i=1N1∏k=1,k≠iN1λkλk−λi1−e−λix.According to the conclusion in [19], the CDF of W can be written as(33)PrW≤x≤1N1+1!⋅∏i=1N1λi⋅22+1x−1N1+1.Based on the formula (33), we propose three algorithms to obtain power allocation policies which minimize the outage probability under different constraints. In particular,Algorithm1: with the transmit power of the source node fixed, we consider each relay’s transmit power is limited. The maximal power for each relay is Prmaxi.Algorithm2: with the transmit power of the source node fixed, we consider the total transmit power of all the relays is limited. The maximal power is Prmax.Algorithm3: the transmit power of the source node is not fixed and the total transmit power of the system is limited. The maximal power is Pmax.


### 2.2.1. Algorithm1: The Optimization Algorithm under Each Relay’s Power Constraint

The optimal power allocation scheme is based on minimizing the outage probability bound in (33). Considering Pri=ηiρiPsγ¯ri, we can transfer the problem into finding the optimal power splitting factor ρi∗ to minimizing the outage probability bound. Assuming that the power of the transmission signal from the source node is fixed, the optimization can be written as(34)P1:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅22+1x−1N1+1s.t.Pri≤Prmaxi0<ρi<1,where Pri=ηiρiPsγ¯ri. Using the monotonicity of the log function, our optimization problem can be written as(35)P1′:argminρilogPout=∑i=1N1logλi′s.t.ηiρiPsγri¯≤Prmaxi0<ρi<1,where the fixed terms in the optimization bound is removed. It is easy to know that cost function in (35) is convex in ρi, which means the minimal value exists in the range 0,1. Using λ′i and Pri in the cost function, we can obtain the derivative of the function as(36)∂logPout∂ρi=22+1x−1N1+1N1+1!⋅ηiγdi¯−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri.Let∂logPout/∂ρi=0, we can get(37)ρi=−4+16+8ηiγ¯ri−22ηiγ¯ri−2negative  reject.We summerize it in Algorithm1.


### 2.2.2. Algorithm2: The Optimization Algorithm under All Relays’ Total Power Constraint

According to [19, 21], it is necessary to consider the total power consumed by the system. In our condition, we first consider the total power of all the relays. Based on minimizing the outage probability bound in (33), we consider that the total power of all the relays are limited and formulate an optimization problem as(38)P2:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.∑iN1Pri≤Prmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(39)P2′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλi′s.t.∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is a convex optimization problem inρi. In order to solve the problem, we can use the Lagrangian method. We split the question into two steps. The first step is to use the Lagrangian of this optimization problem to solve the power splitting factor for each relay. It can be written as(40)Lμ=C⋅∑i=1N1logλ′i+μPrmax−∑iN1Pri,where C=2N1+1x−1N1+1/N1+1! and μ represents the Lagrange multiplier associated with the power constraint in (39). The derivative of the function is(41)dLμdρi=C⋅ηiγ¯di−2⋅ρi2+4ρi−2ηiρiγ¯di+21−ρiηiPs1−ρiρiγ¯riγ¯di−μηiPsγ¯ri.Applying the KKT conditions, we letdLμ/dρi=0 and obtain the following necessary and sufficient conditions for a fixed μ as(42)ρi=0,dLμdρi=0has no answer within0,1,ρi∗,dLμdρi=0has answerρi∗within0,1.Step two is to update the Lagrange multiplier by the power constraint in problem (39) as follows at each iteration n.(43)μin+1=μin−kPrmax−∑iN1Pri,where k is the step size at the nth iteration.We summerize it in Algorithm2.


### 2.2.3. Algorithm3: The Optimization Algorithm under the System’s Power Constraint

Now, we consider the total power of the system. Based on minimizing the outage probability bound in (33), we consider that the total power of the whole system is limited and formulate an optimization problem as(44)P3:argminPs,ρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.Ps+∑iN1Pri≤Pmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(45)P3′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλj′s.t.Ps+∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is also a convex optimization problem in bothPs and ρi and can be solved by the Lagrangian method. We can obtain the Lagrangian of this optimization problem:(46)Lφ=C⋅∑i=1N1logλ′i+φPmax−Ps−∑iN1Pri,where φ represents the Lagrange multiplier. The derivative for Ps of the function is(47)dLφdPs=−1Ps−φ1+∑i=1Nρiηi.LetdLφ/dPs=0 and we can obtain φ=−N/Pmax.Also the derivative forρi of the function is(48)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri−φηiPsγ¯ri.Useφ=−N/Pmax in formula (48) and obtain(49)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri+NPmaxηiPsγ¯ri,In order to solve the optimization problem, we split it into two steps.(1)
Optimal power splitting factor for each relay.For a givenPsn, let dLφ/dρi=0, we can obtain the optimal power splitting factor ρin for each relay.(2)
Update the transmit power of the source node.By the power constraint in problem (45), we can update Psn using the following formula as(50)Psn+1=Pmax1+∑i=1Nρin.We summerize it in Algorithm3.


## 2.2.1. Algorithm1: The Optimization Algorithm under Each Relay’s Power Constraint

The optimal power allocation scheme is based on minimizing the outage probability bound in (33). Considering Pri=ηiρiPsγ¯ri, we can transfer the problem into finding the optimal power splitting factor ρi∗ to minimizing the outage probability bound. Assuming that the power of the transmission signal from the source node is fixed, the optimization can be written as(34)P1:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅22+1x−1N1+1s.t.Pri≤Prmaxi0<ρi<1,where Pri=ηiρiPsγ¯ri. Using the monotonicity of the log function, our optimization problem can be written as(35)P1′:argminρilogPout=∑i=1N1logλi′s.t.ηiρiPsγri¯≤Prmaxi0<ρi<1,where the fixed terms in the optimization bound is removed. It is easy to know that cost function in (35) is convex in ρi, which means the minimal value exists in the range 0,1. Using λ′i and Pri in the cost function, we can obtain the derivative of the function as(36)∂logPout∂ρi=22+1x−1N1+1N1+1!⋅ηiγdi¯−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri.Let∂logPout/∂ρi=0, we can get(37)ρi=−4+16+8ηiγ¯ri−22ηiγ¯ri−2negative  reject.We summerize it in Algorithm1.


## 2.2.2. Algorithm2: The Optimization Algorithm under All Relays’ Total Power Constraint

According to [19, 21], it is necessary to consider the total power consumed by the system. In our condition, we first consider the total power of all the relays. Based on minimizing the outage probability bound in (33), we consider that the total power of all the relays are limited and formulate an optimization problem as(38)P2:argminρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.∑iN1Pri≤Prmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(39)P2′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλi′s.t.∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is a convex optimization problem inρi. In order to solve the problem, we can use the Lagrangian method. We split the question into two steps. The first step is to use the Lagrangian of this optimization problem to solve the power splitting factor for each relay. It can be written as(40)Lμ=C⋅∑i=1N1logλ′i+μPrmax−∑iN1Pri,where C=2N1+1x−1N1+1/N1+1! and μ represents the Lagrange multiplier associated with the power constraint in (39). The derivative of the function is(41)dLμdρi=C⋅ηiγ¯di−2⋅ρi2+4ρi−2ηiρiγ¯di+21−ρiηiPs1−ρiρiγ¯riγ¯di−μηiPsγ¯ri.Applying the KKT conditions, we letdLμ/dρi=0 and obtain the following necessary and sufficient conditions for a fixed μ as(42)ρi=0,dLμdρi=0has no answer within0,1,ρi∗,dLμdρi=0has answerρi∗within0,1.Step two is to update the Lagrange multiplier by the power constraint in problem (39) as follows at each iteration n.(43)μin+1=μin−kPrmax−∑iN1Pri,where k is the step size at the nth iteration.We summerize it in Algorithm2.


## 2.2.3. Algorithm3: The Optimization Algorithm under the System’s Power Constraint

Now, we consider the total power of the system. Based on minimizing the outage probability bound in (33), we consider that the total power of the whole system is limited and formulate an optimization problem as(44)P3:argminPs,ρiPout=1N1+1!⋅∏i=1N1λi′⋅2N1+1x−1N1+1s.t.Ps+∑iN1Pri≤Pmax0<ρi<1,where Pri=ηiρiPsγsri. Similar to Algorithm 1, we can rewrite the optimization problem as(45)P3′:argminρilogPout=2N1+1x−1N1+1N1+1!⋅∑i=1N1logλj′s.t.Ps+∑i=1N1ηiρiPsγri¯≤Prmaxi0<ρi<1.This is also a convex optimization problem in bothPs and ρi and can be solved by the Lagrangian method. We can obtain the Lagrangian of this optimization problem:(46)Lφ=C⋅∑i=1N1logλ′i+φPmax−Ps−∑iN1Pri,where φ represents the Lagrange multiplier. The derivative for Ps of the function is(47)dLφdPs=−1Ps−φ1+∑i=1Nρiηi.LetdLφ/dPs=0 and we can obtain φ=−N/Pmax.Also the derivative forρi of the function is(48)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri−φηiPsγ¯ri.Useφ=−N/Pmax in formula (48) and obtain(49)dLφdρi=C⋅ηiγ¯ri−2⋅ρi2+4ρi−2ηiρiγ¯ri+21−ρiηiPs1−ρiρiγ¯riγ¯ri+NPmaxηiPsγ¯ri,In order to solve the optimization problem, we split it into two steps.(1)
Optimal power splitting factor for each relay.For a givenPsn, let dLφ/dρi=0, we can obtain the optimal power splitting factor ρin for each relay.(2)
Update the transmit power of the source node.By the power constraint in problem (45), we can update Psn using the following formula as(50)Psn+1=Pmax1+∑i=1Nρin.We summerize it in Algorithm3.


## 2.3. Numerical Results

We use MATLAB to conduct some simulations to prove the theoretical analysis presented in the former part. We assumed that the system contains a source node, a destination node, and multirelays. We set the distance between the source node and the destination noded=2.4. All the relays are located in a circular manner whose the center is at the midpoint between the source node and the destination with radius r=1.2. We adopt Ps=5w and set σ2=0.01 [11]. The large scale fading exponent of the channel m is 2.7, and the small scale fading coefficients are generated as independent and identical (i.i.d.) Rayleigh random variables with mean zero and variance 1.Figure4 shows the outage probabilities for the theoretical analysis and the simulation versus normalized user transmission rate R bits/sec/Hz. From Figure 4, it is clear that the theoretical analysis is basically the same with the simulation results for different relay numbers. Also the outage probability increases as R increases. Meanwhile, comparing the outage probability for different relay numbers, we know that the outage probability decreases as relay number increases, which proves multiple relays can achieve the diversity gain of the channel.Figure 4
Outage probabilities for the theoretical analysis and the real scenario with different relay numbers.Figure5 shows the effect of the power splitting factor ρ. From the figure, we know that the outage probability is increasing when ρ is smaller but later, it starts decreasing, which means there is an optimal value for ρ. When the value of ρ is smaller than the optimal ρ, there is less power available for EH. Therefore, the transmit power of the relay node is not enough, which leads to the increase of the outage probability. On the contrary, when ρ is large than the optimal value, although the power for EH is enough, the power left for information transmission is low. Because of that, the signal strength at the destination node is not enough, and the outage probability increases too. Meanwhile, comparing different relay numbers N=4 and N=5 and different relay distribution r=1.2 and r=1, the trend of the outage probability is similar, but the value is much lower for N=5 and r=1. That is because when the relays are located near the midpoint between the source node and the destination, both the channel gain hi and gi are better.Figure 5
Outage probabilities vs. power splitting factorρ for different relay numbers N=4 and N=5 and different source to destination distances r=1.2 and r=1.Figure6 plots the effect of the source transmit power Ps. As shown in Figure 6, the outage probability of the system decreases as Ps increases. This is because the large value of Ps leads to more received power at the relay nodes, which means more power for both EH and information transmission. Consequently, the outage probability for the system is small. However, the outage probability does not change much when Ps is large enough, which means there is no need to continually increase transmit power in order to decrease the outage probability. Also, changing the relay number and the source to destination distance, the trend is the same, and the value for the large relay number and short source to destination distance is lower. Meanwhile, we notice that the variation of the tendency for the relay number N=5 is not the obvious comparing with that for N=4.Figure 6
Outage probabilities vs. the source transmit powerPs for different relay numbers N=4 and N=5 and different source to destination distances d=1.2 and d=1.4.Figure7 depicts the effect of the energy conversion efficiency. From this, we can observe that for smaller values of η, the outage probability is higher, and for larger values, the probability is lower, because the value of η decides the power from the relay node as Pri∝ηi. Also, changing the relay number and the source to destination distance, the trend is the same and the value for large relay number and short source to destination distance is lower. But from the figure, when the relay number N=5, the value of the outage probability is almost unchanged. That means we should also limit the number of relays because the large number of relays may cause the restriction of the system performance.Figure 7
Outage probabilities vs. the effect of the energy conversion efficiencyη for different relay numbers N=4 and N=5 and different source to destination distances d=1.2 and d=1.4.Figures8–10 show a comparison between no optimal power allocation and optimal power allocation schemes in Algorithm 2 and Algorithm 3, respectively. From Figure 8, we observe that the outage probability is lower by using the scheme in Algorithm 1. The improvement is the same for schemes in Algorithm 2 and Algorithm 3. Here, in order to make the numerical result more reliable, we set the power consumption of the two schemes equal. In particular, for Algorithm 2, we set the power splitting factor ρ=0.5 for no optimal scheme, and the power constraint for all relays is Prmax=ρ⋅Ps⋅N. For Algorithm 3, the power splitting factor is also ρ=0.5, and the power constraint for the total system is Prmax=ρ⋅Ps⋅N. Based on this condition, we can observe that the outage probability for optimal schemes is lower than that for no optimal situation. And it also has the improvement for different source power Ps and relay number N. From the figures, we can also see that the outage probability is growing with the increasing of the transmission rate R, and the variation between no optimal power allocation and optimal power allocation schemes is almost the same for a different source power Ps. But for the relay number N=5, the growing rate and the variation between two algorithms are much higher than that for N=3. This also proves that when we want to use multiple relays, the number should be under control to prevent the rapid deterioration of the system performance.Figure 8
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm1.Figure 9
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm2.Figure 10
Comparison between no optimal power allocation and optimal power allocation scheme in Algorithm3.


## 3. Conclusion

In this paper, we consider a wireless cooperative network with multiple relay nodes. All relays are considered as EH nodes with SWIPT capabilities. We derive a closed-form expression for the outage probability of the whole system. Based on the approximation of the final expression, we also propose three optimization algorithms to obtain power allocation policies which minimize the outage probability under different constraints. In the end, we show that the analytical expression of the outage probability is suitable for most communication scenarios and illustrate the effect of various system parameters on the performance of the protocols. We also show that optimal schemes in different algorithms can all result in a performance improvement compared to no optimal power allocation scheme.



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