# Outage Analysis and Optimization of SWIPT in Network-Coded Two-Way Relay Networks

Authors: Ruihong Jiang; Ke Xiong; Yu Zhang; Zhangdui Zhong
Journal: Mobile Information Systems
 (2017)

---

## Abstract

This paper investigates the outage performance of simultaneous wireless information and power transfer (SWIPT) in network-coded two-way relay systems, where a relay first harvests energy from the signals transmitted by two sources and then uses the harvested energy to forward the received information to the two sources. We consider two transmission protocols, power splitting two-way relay (PS-TWR) and time switching two-way relay (TS-TWR) protocols. We present two explicit expressions for the system outage probability of the two protocols and further derive approximate expressions for them in high and low SNR cases. To explore the system performance limits, two optimization problems are formulated to minimize the system outage probability. Since the problems are nonconvex and have no known solution methods, a genetic algorithm- (GA-) based algorithm is designed. Numerical and simulation results validate our theoretical analysis. It is shown that, by jointly optimizing the time assignment and SWIPT receiver parameters, a great performance gain can be achieved for both PS-TWR and TS-TWR. Moreover, the optimized PS-TWR always outperforms the optimized TS-TWR in terms of outage performance. Additionally, the effects of parameters including relay location and transmit powers are also discussed, which provide some insights for the SWIPT-enabled two-way relay networks.


---

## Body

## 1. Introduction

### 1.1. Background

One of the main challenges in energy-constrained wireless communication networks, for example, the battery-powered wireless sensor networks, is the limited lifetime due to the fixed energy-supplying devices with finite service life. In order to provide a perpetual energy supply to maintain reliable wireless communications, harvesting energy from the surrounding environment is considered as an effective solution [1–9]. Traditional energy harvesting (EH) harvests energy from the external energy sources including solar, wind, and vibration, which are greatly affected and limited by environment or climate, and cannot provide stable energy supply for wireless communication networks [10, 11], while harvesting energy from ambient radio-frequency (RF) signals [12], especially from dedicated radio-frequency radiation, is regarded as a stable and promising energy source for future energy-constrained wireless networks that cannot be offered by conventional battery or grid power-operated wireless communication networks. Because RF signals are independent with the external conditions of the environment and climate and can be easily designed and controlled. Since RF signals can carry energy as well as information at the same time, energy and information can be delivered simultaneously via RF signals, which is known as the simultaneous information and power transfer (SWIPT) advocated in [13–28].


### 1.2. Related Work

A number of works on SWIPT, thus far, can be found in the literature. In [14], the authors proposed two SWIPT receiver architectures, where energy harvesting and information detection are operated in either power splitting (PS) architecture or time switching (TS) architecture at the receiver, making it possible to realize SWIPT in practical wireless communication systems. In the past few years, the two receiver architectures have been widely adopted to various wireless communication systems. In [15], the PS and TS architectures were investigated in point-to-point communication systems with SWIPT technology. Later, they were studied in more SWIPT-enabled complex network systems such as nonrelaying two-way transmission systems (see, e.g., [16]), two-hop transmissions (see, e.g., [17]), MIMO communication systems (see, e.g., [18]), and nonregenerative MIMO-OFDM relaying systems (see, e.g., [19]). Besides, SWIPT is also be benefit to the relay networks, so that some existing works focused on the system design and performance analysis for SWIPT-enabled relay networks, see for example, [20–23], where one-way two-hop relay systems with amplify-and-forward (AF) or decode-and-forward (DF) relaying operation were studied.As is known, the two-way relay network, where two sources exchange their information with a helping relay node, is able to greatly improve system spectral efficiency and throughput by employing some advanced coding technologies such as network coding (NC) [29–31] and the two-way relay transmission has wide applications in various wireless networks, so some recent works began to investigate SWIPT in two-way relay networks; see, for example, [24–28]. Specifically, the work in [24] investigated SWIPT in an AF energy harvesting two-way relay system with TS receiver, where however, only the system bit error rate (BER) performance was analyzed. In [25], the authors studied the outage performance for an asymmetric two-way relaying system with PS receiver architecture, where however, only AF relaying protocol was considered. In [26], the authors analyzed and optimized the system outage performance for two-way relay system, where however, just traditional four-phase two-way EH relay system was considered and NC transmission was not considered. In [27], the authors studied the network-coded two-way relay network from an information theoretical viewpoint, where their goal was to explore the system throughput limit with data rate fairness and the system outage probability was not analyzed, although, in [28], the authors considered SWIPT for a network-coded two-way relay system by using DF relaying and derived the outage probability for the system, where only TS receiver architecture was considered and the time allocation for the two-way relay information transmission was not involved.


### 1.3. Motivations and Contributions

In this paper, we also study a network-coded two-way relay system with SWIPT and DF relaying operation, where two sources with sufficient energy supply desire to exchange their individual information through an EH relay node. Due to the shadowing of the obstacles like high buildings or the long distance, the direct link between two sources is unavailable.Compared with previous works, two main differences in our work are stressed as follows.Firstly, both TS and PS receiver architectures are took into account in our work, where the power splitting factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for PS-TWR, and the time switching factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for TS-TWR.Secondly, our goal is to explore the system outage performance limits for the PS-TWR and TS-TWR, where network coding is involved in our work.The main contributions of this paper are summarized as follows.First, we derive the closed-form expression of the outage probability for PS-TWR as well as the TS-TWR.Second, to explore the system performance limits, two optimization problems are formulated to minimize the outage probability for PS-TWR and TS-TWR, where the power splitting factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for PS-TWR, and the time switching factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for TS-TWR.Third, since two optimization problems are nonconvex and cannot be solved by directly using traditional solution methods, a genetic algorithm (GA)-based scheme is designed to solve them.Fourth, in order to achieve simplicity, two approximate expressions of the system outage probabilities of PS-TWR and TS-TWR are derived at the high and low SNR regions, respectively.Finally, we provide some numerical and simulation results demonstrate our theoretical analysis associated with the PS-TWR and TS-TWR systems. It is shown that by jointly optimizing the time assignment and the SWIPT receiver parameters, a great performance gain can be achieved for both PS-TWR and TS-TWR. It is also shown that the optimized PS-TWR always outperforms the optimized TS-TWR in terms of outage performance. In addition, the effects of various system parameters including the relay location, the transmit power of the two sources are also discussed, which provide some insights for the SWIPT-enabled two-way relay networks.


### 1.4. Organization

The rest of the paper is organized as follows. Section2 describes the system model and the PS-TWR and TS-TWR protocols. Sections 3 and 4 analyze the system outage probability for the two protocols, respectively, where the exact and approximate outage probability expressions of for PS-TWR and TS-TWR are presented. Section 6 formulates two optimization problems to minimize the outage probabilities of PS-TWR and TS-TWR and then designs a GA-based optimization algorithm to solve the problems. Section 7 provides some numerical and simulation results and, finally, Section 8 summarizes the paper with some conclusions.


## 1.1. Background

One of the main challenges in energy-constrained wireless communication networks, for example, the battery-powered wireless sensor networks, is the limited lifetime due to the fixed energy-supplying devices with finite service life. In order to provide a perpetual energy supply to maintain reliable wireless communications, harvesting energy from the surrounding environment is considered as an effective solution [1–9]. Traditional energy harvesting (EH) harvests energy from the external energy sources including solar, wind, and vibration, which are greatly affected and limited by environment or climate, and cannot provide stable energy supply for wireless communication networks [10, 11], while harvesting energy from ambient radio-frequency (RF) signals [12], especially from dedicated radio-frequency radiation, is regarded as a stable and promising energy source for future energy-constrained wireless networks that cannot be offered by conventional battery or grid power-operated wireless communication networks. Because RF signals are independent with the external conditions of the environment and climate and can be easily designed and controlled. Since RF signals can carry energy as well as information at the same time, energy and information can be delivered simultaneously via RF signals, which is known as the simultaneous information and power transfer (SWIPT) advocated in [13–28].


## 1.2. Related Work

A number of works on SWIPT, thus far, can be found in the literature. In [14], the authors proposed two SWIPT receiver architectures, where energy harvesting and information detection are operated in either power splitting (PS) architecture or time switching (TS) architecture at the receiver, making it possible to realize SWIPT in practical wireless communication systems. In the past few years, the two receiver architectures have been widely adopted to various wireless communication systems. In [15], the PS and TS architectures were investigated in point-to-point communication systems with SWIPT technology. Later, they were studied in more SWIPT-enabled complex network systems such as nonrelaying two-way transmission systems (see, e.g., [16]), two-hop transmissions (see, e.g., [17]), MIMO communication systems (see, e.g., [18]), and nonregenerative MIMO-OFDM relaying systems (see, e.g., [19]). Besides, SWIPT is also be benefit to the relay networks, so that some existing works focused on the system design and performance analysis for SWIPT-enabled relay networks, see for example, [20–23], where one-way two-hop relay systems with amplify-and-forward (AF) or decode-and-forward (DF) relaying operation were studied.As is known, the two-way relay network, where two sources exchange their information with a helping relay node, is able to greatly improve system spectral efficiency and throughput by employing some advanced coding technologies such as network coding (NC) [29–31] and the two-way relay transmission has wide applications in various wireless networks, so some recent works began to investigate SWIPT in two-way relay networks; see, for example, [24–28]. Specifically, the work in [24] investigated SWIPT in an AF energy harvesting two-way relay system with TS receiver, where however, only the system bit error rate (BER) performance was analyzed. In [25], the authors studied the outage performance for an asymmetric two-way relaying system with PS receiver architecture, where however, only AF relaying protocol was considered. In [26], the authors analyzed and optimized the system outage performance for two-way relay system, where however, just traditional four-phase two-way EH relay system was considered and NC transmission was not considered. In [27], the authors studied the network-coded two-way relay network from an information theoretical viewpoint, where their goal was to explore the system throughput limit with data rate fairness and the system outage probability was not analyzed, although, in [28], the authors considered SWIPT for a network-coded two-way relay system by using DF relaying and derived the outage probability for the system, where only TS receiver architecture was considered and the time allocation for the two-way relay information transmission was not involved.


## 1.3. Motivations and Contributions

In this paper, we also study a network-coded two-way relay system with SWIPT and DF relaying operation, where two sources with sufficient energy supply desire to exchange their individual information through an EH relay node. Due to the shadowing of the obstacles like high buildings or the long distance, the direct link between two sources is unavailable.Compared with previous works, two main differences in our work are stressed as follows.Firstly, both TS and PS receiver architectures are took into account in our work, where the power splitting factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for PS-TWR, and the time switching factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for TS-TWR.Secondly, our goal is to explore the system outage performance limits for the PS-TWR and TS-TWR, where network coding is involved in our work.The main contributions of this paper are summarized as follows.First, we derive the closed-form expression of the outage probability for PS-TWR as well as the TS-TWR.Second, to explore the system performance limits, two optimization problems are formulated to minimize the outage probability for PS-TWR and TS-TWR, where the power splitting factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for PS-TWR, and the time switching factor for the energy receiver and the time assignment factor for the transmission phases are jointly optimized for TS-TWR.Third, since two optimization problems are nonconvex and cannot be solved by directly using traditional solution methods, a genetic algorithm (GA)-based scheme is designed to solve them.Fourth, in order to achieve simplicity, two approximate expressions of the system outage probabilities of PS-TWR and TS-TWR are derived at the high and low SNR regions, respectively.Finally, we provide some numerical and simulation results demonstrate our theoretical analysis associated with the PS-TWR and TS-TWR systems. It is shown that by jointly optimizing the time assignment and the SWIPT receiver parameters, a great performance gain can be achieved for both PS-TWR and TS-TWR. It is also shown that the optimized PS-TWR always outperforms the optimized TS-TWR in terms of outage performance. In addition, the effects of various system parameters including the relay location, the transmit power of the two sources are also discussed, which provide some insights for the SWIPT-enabled two-way relay networks.


## 1.4. Organization

The rest of the paper is organized as follows. Section2 describes the system model and the PS-TWR and TS-TWR protocols. Sections 3 and 4 analyze the system outage probability for the two protocols, respectively, where the exact and approximate outage probability expressions of for PS-TWR and TS-TWR are presented. Section 6 formulates two optimization problems to minimize the outage probabilities of PS-TWR and TS-TWR and then designs a GA-based optimization algorithm to solve the problems. Section 7 provides some numerical and simulation results and, finally, Section 8 summarizes the paper with some conclusions.


## 2. System Model and Protocols

### 2.1. Network Model

Consider a SWIPT-enabled two-way relay system as shown in Figure1, where node 1 and node 2, namely, S1 and S2, are with sufficient energy supply and desire to exchange their individual information with each other. However, due to the shadowing of some obstacles like the high buildings or the long distance, the direct link between S1 and S2 is nonexistent, so that the information exchange between S1 and S2 has to be helped by an intermediate node, node 3 (also referred to as node R). It is assumed that node 3 is an energy-selfish/energy contained node, which is not willing to/has no power to help node 1 and node 2 to exchange their information. By employing RF-EH technique, node 3 is able to harvest energy from the RF signals transmitted by node 1 and node 2 and then use the harvested energy to help the information transmission. All three nodes are equipped with single antenna. It is worth noting that such an energy-constrained two-way relay model can be applied to many different wireless scenarios, for example, the wireless sensor network (WSN) in a toxic environments and the wireless body area network (WBAN).Figure 1
System model.Quasi-block fading channel is considered, which means that all channel coefficients associated with the links can be regarded to be constant in each fading block and then change independently from one fading block to the next following Rayleigh distribution. The channel gains of the links fromSi to R and from R to Si are denoted by hi and fi, respectively, and the distances from Si to R are denoted as dS,i, where i=1,2.Moreover,S1 and S2 have large amount of data to be exchanged and R is with a capacity-limited rechargeable battery buffer, which means that all harvested energy at R in a fading block must be used for the information transmission in the same fading block.Based on the system mode described above, we shall present the two protocols, PS-TWR and TS-TWR, for the SWIPT-enabled two-way relay system in the following two subsections.


### 2.2. PS-TWR Protocol

Figure2(a) illustrates the transmission framework of PS-TWR. Let T be the time period of each fading block. In PS-TWR, T is divided into three parts by the time assignment factors θi, that is, θ1T, θ2T and (1-θ1-θ2)T, where 0<θi<1 and 0<θ1+θ2<1. In the θiT parts, Si transmits information to R and, in the (1-θ1-θ2)T part, R forwards the network-coded information to the two sources.Figure 2
(a) The framework of PS-TWR. (b) Architecture of the relay (R) receiver for PS-TWR.
(a) (b)In first two transmission parts (i.e., source transmission phases),Si transmits its information to R, where (1-ρi)PR,i power is used for the information transmission to R, and the rest ρiPR,i power is for energy harvested by R. Here 0<ρi<1 denotes the power splitting factor of the energy harvesting receiver, and PR,i is the power of the received signal transmitted from Si at R, which can be expressed as(1)PR,i=Pihi2dS,im,where m is the path loss exponent and Pi is the Si transmit power.In the third phase (R forwarding phase), R first decodes the received information transmitted from Si in first two transmission phases and mixes the two flows of the received information with NC, and then broadcasts the mixed information to S1 and S2 using all the harvested energy ∑i=12‍ρiPR,i, simultaneously. Since S1 and S2 know their own prior knowledge, they can decode the desired information from the mixed information once they receive the broadcasted information from R.The receiver architecture ofR for PS-TWR is depicted in Figure 2(b), where the received signal yR is split into two parts for information processing and energy harvesting by a power splitter in ρi:1-ρi proportion. So, a part of the received signal ρiyR is allocated to the EH receiver, the other part 1-ρiyR is for the information receiver. And, the received signal yR at R is interfered by two noises, nR,a and nR,c, where nR,a~CN(0,σR,a2) is the receiving antenna noise at the relay node R and σR,a2 denotes the noise power. nR,c~CN(0,σR,c2) is the sampled AWGN introduced by the RF-band to baseband signal conversion at the information receiver as shown in Figure 2(b) and σR,c2 is the noise power.


### 2.3. TS-TWR Protocol

Figure3(a) shows the transmission framework of TS-TWR, where T is also divided into three phases by the time assignment factors θi as those defined in PS-TWR. However, in TS-TWR, the first two θiT time intervals are further divided into two subphases by the time switching ratio αi of the TS receiver architecture, where 0<αi<1 and i=1,2.Figure 3
(a) The framework of TS-TWR. (b) Architecture of the relay (R) receiver for TS-TWR.
(a) (b)In the first two phases (i.e., source transmission phases),αiθiT is assigned to R for harvesting energy from the received signals transmitted by Si and the remaining part (1-αi)θiT is used for R to receive information from Si during the first two subphases. In the third phase (R forwarding phase), R first decodes the received information from Si and mixes the two flows of information with NC, and then broadcasts the mixed information to S1 and S2 simultaneously by using all the harvested energy.The block diagram of TS-TWR the receiver architecture atR is shown in Figure 3(b), where the received RF signal, yR, at R is input into the energy harvesting receiver in the αiθiT subphase, and then input into to the information receiver for information decoding during the (1-αi)θiT subphase. nR,a and nR,c are AWGNs received at R, similar to those defined in PS-TWR.


## 2.1. Network Model

Consider a SWIPT-enabled two-way relay system as shown in Figure1, where node 1 and node 2, namely, S1 and S2, are with sufficient energy supply and desire to exchange their individual information with each other. However, due to the shadowing of some obstacles like the high buildings or the long distance, the direct link between S1 and S2 is nonexistent, so that the information exchange between S1 and S2 has to be helped by an intermediate node, node 3 (also referred to as node R). It is assumed that node 3 is an energy-selfish/energy contained node, which is not willing to/has no power to help node 1 and node 2 to exchange their information. By employing RF-EH technique, node 3 is able to harvest energy from the RF signals transmitted by node 1 and node 2 and then use the harvested energy to help the information transmission. All three nodes are equipped with single antenna. It is worth noting that such an energy-constrained two-way relay model can be applied to many different wireless scenarios, for example, the wireless sensor network (WSN) in a toxic environments and the wireless body area network (WBAN).Figure 1
System model.Quasi-block fading channel is considered, which means that all channel coefficients associated with the links can be regarded to be constant in each fading block and then change independently from one fading block to the next following Rayleigh distribution. The channel gains of the links fromSi to R and from R to Si are denoted by hi and fi, respectively, and the distances from Si to R are denoted as dS,i, where i=1,2.Moreover,S1 and S2 have large amount of data to be exchanged and R is with a capacity-limited rechargeable battery buffer, which means that all harvested energy at R in a fading block must be used for the information transmission in the same fading block.Based on the system mode described above, we shall present the two protocols, PS-TWR and TS-TWR, for the SWIPT-enabled two-way relay system in the following two subsections.


## 2.2. PS-TWR Protocol

Figure2(a) illustrates the transmission framework of PS-TWR. Let T be the time period of each fading block. In PS-TWR, T is divided into three parts by the time assignment factors θi, that is, θ1T, θ2T and (1-θ1-θ2)T, where 0<θi<1 and 0<θ1+θ2<1. In the θiT parts, Si transmits information to R and, in the (1-θ1-θ2)T part, R forwards the network-coded information to the two sources.Figure 2
(a) The framework of PS-TWR. (b) Architecture of the relay (R) receiver for PS-TWR.
(a) (b)In first two transmission parts (i.e., source transmission phases),Si transmits its information to R, where (1-ρi)PR,i power is used for the information transmission to R, and the rest ρiPR,i power is for energy harvested by R. Here 0<ρi<1 denotes the power splitting factor of the energy harvesting receiver, and PR,i is the power of the received signal transmitted from Si at R, which can be expressed as(1)PR,i=Pihi2dS,im,where m is the path loss exponent and Pi is the Si transmit power.In the third phase (R forwarding phase), R first decodes the received information transmitted from Si in first two transmission phases and mixes the two flows of the received information with NC, and then broadcasts the mixed information to S1 and S2 using all the harvested energy ∑i=12‍ρiPR,i, simultaneously. Since S1 and S2 know their own prior knowledge, they can decode the desired information from the mixed information once they receive the broadcasted information from R.The receiver architecture ofR for PS-TWR is depicted in Figure 2(b), where the received signal yR is split into two parts for information processing and energy harvesting by a power splitter in ρi:1-ρi proportion. So, a part of the received signal ρiyR is allocated to the EH receiver, the other part 1-ρiyR is for the information receiver. And, the received signal yR at R is interfered by two noises, nR,a and nR,c, where nR,a~CN(0,σR,a2) is the receiving antenna noise at the relay node R and σR,a2 denotes the noise power. nR,c~CN(0,σR,c2) is the sampled AWGN introduced by the RF-band to baseband signal conversion at the information receiver as shown in Figure 2(b) and σR,c2 is the noise power.


## 2.3. TS-TWR Protocol

Figure3(a) shows the transmission framework of TS-TWR, where T is also divided into three phases by the time assignment factors θi as those defined in PS-TWR. However, in TS-TWR, the first two θiT time intervals are further divided into two subphases by the time switching ratio αi of the TS receiver architecture, where 0<αi<1 and i=1,2.Figure 3
(a) The framework of TS-TWR. (b) Architecture of the relay (R) receiver for TS-TWR.
(a) (b)In the first two phases (i.e., source transmission phases),αiθiT is assigned to R for harvesting energy from the received signals transmitted by Si and the remaining part (1-αi)θiT is used for R to receive information from Si during the first two subphases. In the third phase (R forwarding phase), R first decodes the received information from Si and mixes the two flows of information with NC, and then broadcasts the mixed information to S1 and S2 simultaneously by using all the harvested energy.The block diagram of TS-TWR the receiver architecture atR is shown in Figure 3(b), where the received RF signal, yR, at R is input into the energy harvesting receiver in the αiθiT subphase, and then input into to the information receiver for information decoding during the (1-αi)θiT subphase. nR,a and nR,c are AWGNs received at R, similar to those defined in PS-TWR.


## 3. Outage Analysis for PS-TWR

### 3.1. Exact Expression of the Outage Probability for PS-TWR

As depicted in Figure2, the individual information is transmitted from Si to R in the ith source transmission phase, and after the processing of the relay information receiver, the sampled baseband signal yR,i(PS) at R can be given by(2)yR,iPS=1dS,im1-ρiPihixi+1-ρinR,ai+nR,ci,where xi is the sampled and normalized information signal from Si; that is, E{xi2}=1. E·=1 is the expectation operator and · is the absolute value operator. hi is the channel gain of the link Si to R. nR,ai and nR,ci are the AWGNs with the variances σR,ai2 and σR,ci2, respectively. Based on (1), the data rate RR,i(PS) at R from Si can be given by (3)RR,iPS=θilog⁡1+1-ρiPR,iσR,i2=θilog⁡1+1-ρiPihi2dS,imσR,i2,where nR,i≜(1-ρi)nR,ai+nR,ci with σR,i2≜(1-ρi)σR,ai2+σR,ci2 is the overall AWGNs at R in the first two transmission phases.Meanwhile, using (1), the RF-band energy QR,i(PS) harvested by R from Si at the relay energy harvesting receiver can be given by (4)QR,iPS=ζρiPR,iθiT=ζρiPihi2dS,imθiT,where 0<ζ≤1 is used to describe the energy conversion efficiency, and we assume ζ=1 in this paper.After the first two transmission phases, the total energy may be harvested byR from S1 and S2 can be expressed as QR,1(PS)+QR,2(PS), which is consumed by R to forward the received information in the third relay forwarding phase. As the fact that R communicates with Si for the time (1-θ1-θ2)T as shown in Figure 2(a) and using QR,i(PS) in (4), the transmit power of the relay, QR(PS), in the third phase can be given by (5)QRPS=QR,1PS+QR,2PS1-θ1-θ2T=11-θ1-θ2∑i=12ζρiPihi2dS,imθi.At the end ofR forwarding phase, the received signal yS,i(PS) at Si from R can be given by(6)yS,iPS=1dS,imQRPSfixj+nS,ai+nS,ci,where j=1,2 and j≠i. fi is the channel gain of the link R to Si. nS,ai is the baseband AWGN with variance σS,ai2 and nS,ci is the sampled AWGN with variance σS,ci2. The achievable date rate at Si from R can be given by(7)RS,iPS=1-θ1-θ2log⁡1+QRPSfi2dS,imσS,i2,where nS,i≜nS,ai+nS,ci with σS,i2≜σS,ai2+σS,ci2.Due to the impact of channelfi and the employed NC, the broadcast data rate of R is limited in the third phase. Therefore, the actual achievable broadcast rate, RSR(PS), at Si from R can be given by (8)RSRPS=minRS,1PS,RS,2PS=1-θ1-θ2log1+SNRSR,where SNRSR=minQR(PS)f12/dS,1mσS,12,QR(PS)f22/dS,2mσS,22.As is known, in traditional one-way relay networks, an outage occurs if either link’s rate of two hops falls below the given target rateRsr. Comparatively, the two-way relay networks have two communication tasks which refer to three transmission links: S1 to R link, S2 to R link and R forwarding link. Unfortunately, any link’s transmission failure can result in system outage. So, we define the system outage probability in terms of the analysis in [32, 33] as follows: (9)PoutPS=1-PoPS,where Po(PS)=Pr[RR,1(PS)≥Rsr]·Pr[RR,2(PS)≥Rsr]·Pr[RSR(PS)≥Rsr].Theorem 1.
For PS-TWR protocol, given a predetermined transmission rateRsr, the system outage probability can be calculated as (11) in terms of Pi, θi, ρi, ζ, and dS,i, where(10)a=θ11-θ1-θ2ζρ1P1dS,1-m,b=θ21-θ1-θ2ζρ2P2dS,2-m,c=u1dS,1mσR,121-ρ1P1,d=u2dS,2mσR,221-ρ2P2,u1=21/θ1Rsr-1,u2=21/θ2Rsr-1,u3=21/1-θ1-θ2Rsr-1,z0=dS,1mσS,12λf1+dS,2mσS,22λf2.K1· and K2· are the first-order and second-order modified Bessel functions of the second kind, respectively. λhi and λfi denote the parameter of the exponential random variables hi2 and fi2, respectively.(11)PoutPS=1-2λh1λh2aλh2-bλh1e-λh1c-λh2daz0u3λh1K12z0u3λh1a-bz0u3λh2K12z0u3λh2b,aλh2≠bλh1,1-2λh1z0u3ae-λh1c-λh2dK22z0u3λh2b,aλh2=bλh1.Proof.
See the Appendix.


### 3.2. Approximate Expressions of the Outage Probability for PS-TWR

Since the theoretical analysis expression of the system outage probabilityPout(PS) in (11) is complicated with the modified Bessel functions, it is difficult to obtain an analytic solution of Pout(PS) by using traditional methods. For the sake of simplicity, the approximate expressions of Pout(PS) at extremely high and low SNR regions are formulated in the following theorems.Theorem 2.
At high SNR region, the approximate expression of the system outage probabilityPout(PS) in (11) for PS-TWR protocol can be formulated as PPS(high) in(12)PPShigh=1-e-λh1c-λh2d,aλh2≠bλh1,1-bλh1aλh2e-λh1c-λh2d,aλh2=bλh1.Proof.
If0<x≤v+1, the modified Bessel function of the second kind follows from the approximation [34] given by(13)Kvx≈Γv22xv,v>0,where Γ(v) is the Gamma function. Therefore, we can obtain the approximate expressions of K1(·) and K2(·) as (14)K1x=1x,K2x=2x2.At high SNR region, the variables in K1(·) and K2(·) of the system outage probability Pout(PS) are very close to zero with the relatively high variables a and b, which are contained in the denominators of the variables in K1(·) and K2(·). Substituting (14) into (11), the approximate expression for Pout(PS) at high SNR region can be calculated as (12).Theorem 3.
At low SNR region, the approximate expression of the system outage probabilityPout(PS) in (11) for PS-TWR protocol can be formulated as PPS(low) in(15)PPSlow=1-2λh1λh2aλh2-bλh1e-λh1c-λh2daz0u3λh1π4az0u3λh1e-4z0u3λh1/a-bz0u3λh2π4bz0u3λh2e-4z0u3λh2/b,aλh2≠bλh1,1-2λh1z0u3ae-λh1c-λh2dπ4bz0u3λh2e-4z0u3λh2/b,aλh2=bλh1.Proof.
Ifx>v2-1/4, the modified Bessel function of the second kind follows from the approximation [34] as(16)Kvx≈π2xe-x,v>0.
At low SNR region, the denominators of the variables inK1(·) and K2(·) are either a or b, which is relatively low leading to the variables in K1(·) and K2(·) of the system outage probability Pout(PS) are very close to infinity. Thus, substituting (16) into (11), the approximate expression of Pout(PS) at low SNR region can be achieved as (15).


## 3.1. Exact Expression of the Outage Probability for PS-TWR

As depicted in Figure2, the individual information is transmitted from Si to R in the ith source transmission phase, and after the processing of the relay information receiver, the sampled baseband signal yR,i(PS) at R can be given by(2)yR,iPS=1dS,im1-ρiPihixi+1-ρinR,ai+nR,ci,where xi is the sampled and normalized information signal from Si; that is, E{xi2}=1. E·=1 is the expectation operator and · is the absolute value operator. hi is the channel gain of the link Si to R. nR,ai and nR,ci are the AWGNs with the variances σR,ai2 and σR,ci2, respectively. Based on (1), the data rate RR,i(PS) at R from Si can be given by (3)RR,iPS=θilog⁡1+1-ρiPR,iσR,i2=θilog⁡1+1-ρiPihi2dS,imσR,i2,where nR,i≜(1-ρi)nR,ai+nR,ci with σR,i2≜(1-ρi)σR,ai2+σR,ci2 is the overall AWGNs at R in the first two transmission phases.Meanwhile, using (1), the RF-band energy QR,i(PS) harvested by R from Si at the relay energy harvesting receiver can be given by (4)QR,iPS=ζρiPR,iθiT=ζρiPihi2dS,imθiT,where 0<ζ≤1 is used to describe the energy conversion efficiency, and we assume ζ=1 in this paper.After the first two transmission phases, the total energy may be harvested byR from S1 and S2 can be expressed as QR,1(PS)+QR,2(PS), which is consumed by R to forward the received information in the third relay forwarding phase. As the fact that R communicates with Si for the time (1-θ1-θ2)T as shown in Figure 2(a) and using QR,i(PS) in (4), the transmit power of the relay, QR(PS), in the third phase can be given by (5)QRPS=QR,1PS+QR,2PS1-θ1-θ2T=11-θ1-θ2∑i=12ζρiPihi2dS,imθi.At the end ofR forwarding phase, the received signal yS,i(PS) at Si from R can be given by(6)yS,iPS=1dS,imQRPSfixj+nS,ai+nS,ci,where j=1,2 and j≠i. fi is the channel gain of the link R to Si. nS,ai is the baseband AWGN with variance σS,ai2 and nS,ci is the sampled AWGN with variance σS,ci2. The achievable date rate at Si from R can be given by(7)RS,iPS=1-θ1-θ2log⁡1+QRPSfi2dS,imσS,i2,where nS,i≜nS,ai+nS,ci with σS,i2≜σS,ai2+σS,ci2.Due to the impact of channelfi and the employed NC, the broadcast data rate of R is limited in the third phase. Therefore, the actual achievable broadcast rate, RSR(PS), at Si from R can be given by (8)RSRPS=minRS,1PS,RS,2PS=1-θ1-θ2log1+SNRSR,where SNRSR=minQR(PS)f12/dS,1mσS,12,QR(PS)f22/dS,2mσS,22.As is known, in traditional one-way relay networks, an outage occurs if either link’s rate of two hops falls below the given target rateRsr. Comparatively, the two-way relay networks have two communication tasks which refer to three transmission links: S1 to R link, S2 to R link and R forwarding link. Unfortunately, any link’s transmission failure can result in system outage. So, we define the system outage probability in terms of the analysis in [32, 33] as follows: (9)PoutPS=1-PoPS,where Po(PS)=Pr[RR,1(PS)≥Rsr]·Pr[RR,2(PS)≥Rsr]·Pr[RSR(PS)≥Rsr].Theorem 1.
For PS-TWR protocol, given a predetermined transmission rateRsr, the system outage probability can be calculated as (11) in terms of Pi, θi, ρi, ζ, and dS,i, where(10)a=θ11-θ1-θ2ζρ1P1dS,1-m,b=θ21-θ1-θ2ζρ2P2dS,2-m,c=u1dS,1mσR,121-ρ1P1,d=u2dS,2mσR,221-ρ2P2,u1=21/θ1Rsr-1,u2=21/θ2Rsr-1,u3=21/1-θ1-θ2Rsr-1,z0=dS,1mσS,12λf1+dS,2mσS,22λf2.K1· and K2· are the first-order and second-order modified Bessel functions of the second kind, respectively. λhi and λfi denote the parameter of the exponential random variables hi2 and fi2, respectively.(11)PoutPS=1-2λh1λh2aλh2-bλh1e-λh1c-λh2daz0u3λh1K12z0u3λh1a-bz0u3λh2K12z0u3λh2b,aλh2≠bλh1,1-2λh1z0u3ae-λh1c-λh2dK22z0u3λh2b,aλh2=bλh1.Proof.
See the Appendix.


## 3.2. Approximate Expressions of the Outage Probability for PS-TWR

Since the theoretical analysis expression of the system outage probabilityPout(PS) in (11) is complicated with the modified Bessel functions, it is difficult to obtain an analytic solution of Pout(PS) by using traditional methods. For the sake of simplicity, the approximate expressions of Pout(PS) at extremely high and low SNR regions are formulated in the following theorems.Theorem 2.
At high SNR region, the approximate expression of the system outage probabilityPout(PS) in (11) for PS-TWR protocol can be formulated as PPS(high) in(12)PPShigh=1-e-λh1c-λh2d,aλh2≠bλh1,1-bλh1aλh2e-λh1c-λh2d,aλh2=bλh1.Proof.
If0<x≤v+1, the modified Bessel function of the second kind follows from the approximation [34] given by(13)Kvx≈Γv22xv,v>0,where Γ(v) is the Gamma function. Therefore, we can obtain the approximate expressions of K1(·) and K2(·) as (14)K1x=1x,K2x=2x2.At high SNR region, the variables in K1(·) and K2(·) of the system outage probability Pout(PS) are very close to zero with the relatively high variables a and b, which are contained in the denominators of the variables in K1(·) and K2(·). Substituting (14) into (11), the approximate expression for Pout(PS) at high SNR region can be calculated as (12).Theorem 3.
At low SNR region, the approximate expression of the system outage probabilityPout(PS) in (11) for PS-TWR protocol can be formulated as PPS(low) in(15)PPSlow=1-2λh1λh2aλh2-bλh1e-λh1c-λh2daz0u3λh1π4az0u3λh1e-4z0u3λh1/a-bz0u3λh2π4bz0u3λh2e-4z0u3λh2/b,aλh2≠bλh1,1-2λh1z0u3ae-λh1c-λh2dπ4bz0u3λh2e-4z0u3λh2/b,aλh2=bλh1.Proof.
Ifx>v2-1/4, the modified Bessel function of the second kind follows from the approximation [34] as(16)Kvx≈π2xe-x,v>0.
At low SNR region, the denominators of the variables inK1(·) and K2(·) are either a or b, which is relatively low leading to the variables in K1(·) and K2(·) of the system outage probability Pout(PS) are very close to infinity. Thus, substituting (16) into (11), the approximate expression of Pout(PS) at low SNR region can be achieved as (15).


## 4. Outage Analysis for TS-TWR

### 4.1. Exact Expression of the Outage Probability for TS-TWR

As shown in Figure3(b), the RF received signal yR is first sent to the energy harvesting receiver to collect energy by R over the αiθiT time in the Si transmission phase. Thus, the received RF signal yR,i(TS) at R for harvesting energy can be given by(17)yR,iTS=1dS,imPihixi+nR,ai+nR,ci,where xi, m, hi, nR,ai, and nR,ci are defined as (2).Using (1), the achieved data rate RR,i(TS) at R from Si in the (1-αi)θiT subphase can be given by (18)RR,iTS=1-αiθilog1+PR,iσR,i2=1-αiθilog1+Pihi2dS,imσR,i2,where nR,i≜nR,ai+nR,ci with σR,i2≜σR,ai2+σR,ci2.During the first two transmission phases, the harvested RF-band energyQR,i(TS) by R from Si in the αiθiT subphase can be given by(19)QR,iTS=ζPihi2dS,imαiθiT,where ζ is the energy conversion efficiency.At the end of the first twoSi transmission phases, the total energy may be harvested by R from S1 and S2 can be expressed as QR,1(TS)+QR,2(TS), which is used by R for forwarding the received information to S1 and S2 simultaneously in the third relay forwarding phase. Using QR,i(TS) in (19), the transmit power of R broadcasting in the third phase can be given by (20)QRTS=QR,1TS+QR,2TS1-θ1-θ2T=11-θ1-θ2∑i=12ζPihi2dS,imαiθi.At the end of the thirdR forwarding phase, the sampled received signal at Si from R can be given by(21)yS,iTS=1dS,imQRTSfixj+nS,ai+nS,ci,where j=1,2 and j≠i and fi, nS,ai, and nS,ci are defined as (6).Finally, the achievable data rate atSi from R is given by(22)RS,iTS=1-θ1-θ2log⁡1+QRTSfi2dS,imσS,i2,where nS,i≜nS,ai+nS,ci with σS,i2≜σS,ai2+σS,ci2.In the third relay forwarding phase, the broadcast data rate ofR is limited by the effect of the channel fi and the employed digital NC, so that the actual achievable broadcast rate at Si from R is (23)RSRTS=minRS,1TS,RS,2TS=1-θ1-θ2log1+SNRSR,where SNRSR=minQR(TS)f12/dS,1mσS,12,QR(TS)f22/dS,2mσS,22.As described above for PS-TWR, we also define the system outage probability based on the analysis in [32, 33] as follows: (24)PoutTS=1-PoTS,where Po(TS)=Pr[RR,1(TS)≥Rsr]·Pr[RR,2(TS)≥Rsr]·Pr[RSR(TS)≥Rsr].Theorem 4.
For TS-TWR protocol, given a predetermined transmission rateRsr, the system outage probability also can be calculated as (26) in terms of variants Pi, θi, αi, ζ, and dS,i, where(25)a0=α1θ11-θ1-θ2ζP1dS,1-m,b0=α2θ21-θ1-θ2ζP2dS,2-m,c0=w1dS,1mσR,12P1,d0=w2dS,2mσR,22P2,w1=21/1-α1θ1Rsr-1,w2=21/1-α2θ2Rsr-1,w3=21/1-θ1-θ2Rsr-1,e0=dS,1mσS,12λf1+dS,2mσS,22λf2.K1·, K2·, λf1, and λf2 are set to be similar to PS-TWR protocol.(26)PoutTS=1-2λh1λh2a0λh2-b0λh1e-λh1c0-λh2d0a0e0w3λh1K12e0w3λh1a0-b0e0w3λh2K12e0w3λh2b0,a0λh2≠b0λh1,1-2λh1e0w3a0e-λh1c0-λh2d0K22e0w3λh2b0,a0λh2=b0λh1.Proof.
The detailed derivation of the system outage probabilityPout(TS) for TS-TWR is skipped here, because it follows the same steps as given in the Appendix.


### 4.2. Approximating Expressions of the Outage Probability for TS-TWR

Due to the theoretical analysis expression of the system outage probabilityPout(TS) in (26) is complicated containing the modified Bessel functions, it is difficult to obtain an analytic solution of Pout(TS) by using traditional methods. For the sake of simplicity, the approximate expressions of Pout(TS) at extremely high and low SNR regions are formulated in the following theorems.Theorem 5.
For TS-TWR protocol, the approximate expression of the system outage probabilityPout(TS) in (26) at high SNR region can be formulated as PTS(high) in(27)PTShigh=1-e-λh1c0-λh2d0,a0λh2≠b0λh1,1-b0λh1a0λh2e-λh1c0-λh2d0,a0λh2=b0λh1.Proof.
The detailed derivation ofPTS(high) for TS-TWR protocol is omitted here because it follows the same steps as given in Theorem 2 for PS-TWR protocol.Theorem 6.
For TS-TWR protocol, the approximate expression of the system outage probabilityPout(TS) in (26) at low SNR region can be formulated as PTS(low) in(28)PTSlow=1-2λh1λh2a0λh2-b0λh1e-λh1c0-λh2d0a0e0w3λh1π4a0e0w3λh1e-4e0w3λh1/a0-b0e0w3λh2π4b0e0w3λh2e-4e0w3λh2/b0,a0λh2≠b0λh1,1-2λh1e0w3a0e-λh1c0-λh2d0π4b0e0w3λh2e-4e0w3λh2/b0,a0λh2=b0λh1.Proof.
The detailed derivation ofPTS(low) for TS-TWR protocol is also omitted here because it follows the same steps as given in Theorem 3 for PS-TWR protocol.


## 4.1. Exact Expression of the Outage Probability for TS-TWR

As shown in Figure3(b), the RF received signal yR is first sent to the energy harvesting receiver to collect energy by R over the αiθiT time in the Si transmission phase. Thus, the received RF signal yR,i(TS) at R for harvesting energy can be given by(17)yR,iTS=1dS,imPihixi+nR,ai+nR,ci,where xi, m, hi, nR,ai, and nR,ci are defined as (2).Using (1), the achieved data rate RR,i(TS) at R from Si in the (1-αi)θiT subphase can be given by (18)RR,iTS=1-αiθilog1+PR,iσR,i2=1-αiθilog1+Pihi2dS,imσR,i2,where nR,i≜nR,ai+nR,ci with σR,i2≜σR,ai2+σR,ci2.During the first two transmission phases, the harvested RF-band energyQR,i(TS) by R from Si in the αiθiT subphase can be given by(19)QR,iTS=ζPihi2dS,imαiθiT,where ζ is the energy conversion efficiency.At the end of the first twoSi transmission phases, the total energy may be harvested by R from S1 and S2 can be expressed as QR,1(TS)+QR,2(TS), which is used by R for forwarding the received information to S1 and S2 simultaneously in the third relay forwarding phase. Using QR,i(TS) in (19), the transmit power of R broadcasting in the third phase can be given by (20)QRTS=QR,1TS+QR,2TS1-θ1-θ2T=11-θ1-θ2∑i=12ζPihi2dS,imαiθi.At the end of the thirdR forwarding phase, the sampled received signal at Si from R can be given by(21)yS,iTS=1dS,imQRTSfixj+nS,ai+nS,ci,where j=1,2 and j≠i and fi, nS,ai, and nS,ci are defined as (6).Finally, the achievable data rate atSi from R is given by(22)RS,iTS=1-θ1-θ2log⁡1+QRTSfi2dS,imσS,i2,where nS,i≜nS,ai+nS,ci with σS,i2≜σS,ai2+σS,ci2.In the third relay forwarding phase, the broadcast data rate ofR is limited by the effect of the channel fi and the employed digital NC, so that the actual achievable broadcast rate at Si from R is (23)RSRTS=minRS,1TS,RS,2TS=1-θ1-θ2log1+SNRSR,where SNRSR=minQR(TS)f12/dS,1mσS,12,QR(TS)f22/dS,2mσS,22.As described above for PS-TWR, we also define the system outage probability based on the analysis in [32, 33] as follows: (24)PoutTS=1-PoTS,where Po(TS)=Pr[RR,1(TS)≥Rsr]·Pr[RR,2(TS)≥Rsr]·Pr[RSR(TS)≥Rsr].Theorem 4.
For TS-TWR protocol, given a predetermined transmission rateRsr, the system outage probability also can be calculated as (26) in terms of variants Pi, θi, αi, ζ, and dS,i, where(25)a0=α1θ11-θ1-θ2ζP1dS,1-m,b0=α2θ21-θ1-θ2ζP2dS,2-m,c0=w1dS,1mσR,12P1,d0=w2dS,2mσR,22P2,w1=21/1-α1θ1Rsr-1,w2=21/1-α2θ2Rsr-1,w3=21/1-θ1-θ2Rsr-1,e0=dS,1mσS,12λf1+dS,2mσS,22λf2.K1·, K2·, λf1, and λf2 are set to be similar to PS-TWR protocol.(26)PoutTS=1-2λh1λh2a0λh2-b0λh1e-λh1c0-λh2d0a0e0w3λh1K12e0w3λh1a0-b0e0w3λh2K12e0w3λh2b0,a0λh2≠b0λh1,1-2λh1e0w3a0e-λh1c0-λh2d0K22e0w3λh2b0,a0λh2=b0λh1.Proof.
The detailed derivation of the system outage probabilityPout(TS) for TS-TWR is skipped here, because it follows the same steps as given in the Appendix.


## 4.2. Approximating Expressions of the Outage Probability for TS-TWR

Due to the theoretical analysis expression of the system outage probabilityPout(TS) in (26) is complicated containing the modified Bessel functions, it is difficult to obtain an analytic solution of Pout(TS) by using traditional methods. For the sake of simplicity, the approximate expressions of Pout(TS) at extremely high and low SNR regions are formulated in the following theorems.Theorem 5.
For TS-TWR protocol, the approximate expression of the system outage probabilityPout(TS) in (26) at high SNR region can be formulated as PTS(high) in(27)PTShigh=1-e-λh1c0-λh2d0,a0λh2≠b0λh1,1-b0λh1a0λh2e-λh1c0-λh2d0,a0λh2=b0λh1.Proof.
The detailed derivation ofPTS(high) for TS-TWR protocol is omitted here because it follows the same steps as given in Theorem 2 for PS-TWR protocol.Theorem 6.
For TS-TWR protocol, the approximate expression of the system outage probabilityPout(TS) in (26) at low SNR region can be formulated as PTS(low) in(28)PTSlow=1-2λh1λh2a0λh2-b0λh1e-λh1c0-λh2d0a0e0w3λh1π4a0e0w3λh1e-4e0w3λh1/a0-b0e0w3λh2π4b0e0w3λh2e-4e0w3λh2/b0,a0λh2≠b0λh1,1-2λh1e0w3a0e-λh1c0-λh2d0π4b0e0w3λh2e-4e0w3λh2/b0,a0λh2=b0λh1.Proof.
The detailed derivation ofPTS(low) for TS-TWR protocol is also omitted here because it follows the same steps as given in Theorem 3 for PS-TWR protocol.


## 5. Theoretical Comparison of PS-TWR and TS-TWR

As mentioned previously, it can be obtained that the expressions of the harvested RF energyQR,i(PS/TS) by R in the first two phases and the corresponding transmit power of the relay broadcasting QR(PS/TS) in the third phase in PS-TWR and TS-TWR systems, respectively. In order to compare the two systems, we list the terms QR(PS/TS), that is, (5) and (20), in Table 1 with the only difference ρi and αi. For the two systems, PS-TWR and TS-TWR, the system outage probabilities can be given by (9) and (24), respectively. Defining Po1(PS/TS)=Pr[RR,1(PS/TS)≥Rsr], Po2(PS/TS)=Pr[RR,2(PS/TS)≥Rsr] and Po3(PS/TS)=Pr[RSR(PS/TS)≥Rsr], the system outage probabilities can be rewritten as Pout(PS)=1-Po1(PS)Po2(PS)Po3(PS) for PS-TWR, and for TS-TWR, it is Pout(TS)=1-Po1(TS)Po2(TS)Po3(TS), as shown in Table 1. Moreover, for PS-TWR, Po1(PS)=e-λh,1c, Po2(PS)=e-λh,2d, and for TS-TWR, Po1(TS)=e-λh,1c0, Po2(TS)=e-λh,2d0. And the analysis of Po3((PS/TS)) can be found in the Appendix in terms of variables a,b,u3,z0 or a0,b0,w3,e0, so that the whole system outage probabilities for the two systems can be obtained as shown in Table 1.Table 1
PS-TWR versus TS-TWR.
Items PS-TWR TS-TWR Q R ( P S / T S ) 1 1 - θ 1 - θ 2 ∑ i = 1 2 ζ P i h i 2 d S , i m ( ρ i θ i ) 1 1 - θ 1 - θ 2 ∑ i = 1 2 ζ P i h i 2 d S , i m ( α i θ i )  Pout(PS/TS) 1 - P o 1 P S P o 2 P S P o 3 P S = 1 - e - λ h 1 c - λ h 2 d P o 3 ( P S ) 1 - P o 1 T S P o 2 T S P o 3 T S = 1 - e - λ h 1 c 0 - λ h 2 d 0 P o 3 ( T S )  Variables a = ( ρ 1 θ 1 ) 1 - θ 1 - θ 2 ζ P 1 d S , 1 - m a 0 = ( α 1 θ 1 ) 1 - θ 1 - θ 2 ζ P 1 d S , 1 - m b=(ρ2θ2)1-θ1-θ2ζP2dS,2-m b0=(α2θ2)1-θ1-θ2ζP2dS,2-m c = u 1 d S , 1 m σ R , 1 2 ( 1 - ρ 1 ) P 1 c 0 = w 1 d S , 1 m σ R , 1 2 P 1 d = u 2 d S , 2 m σ R , 2 2 ( 1 - ρ 1 ) P 2 d0=w2dS,2mσR,22P2 u 1 = 2 1 / θ 1 R s r - 1 w 1 = 2 1 / ( 1 - α 1 ) θ 1 R s r - 1 u 2 = 2 1 / θ 2 R s r - 1 w 2 = 2 1 / ( 1 - α 2 ) θ 2 R s r - 1 u 3 = 2 1 / 1 - θ 1 - θ 2 R s r - 1 w 3 = 2 1 / 1 - θ 1 - θ 2 R s r - 1 z 0 = d S , 1 m σ S , 1 2 λ f 1 + d S , 2 m σ S , 2 2 λ f 2 e0=dS,1mσS,12λf1+dS,2mσS,22λf2From Table1, as 0<ρi,αi,θi<1, it can be obtained that c<c0, d<d0 with u1<w1, u2<w2. Thus, Po1(PS)>Po1(TS) and Po2(PS)>Po2(TS) with the properties of function f(x)=e-x, that is, Po1(PS)Po2(PS)>Po1(TS)Po2(TS).Further, we discuss the system outage probabilities for PS-TWR and TS-TWR by the following cases onρi, αi.Case 1 (ρi=αi).
In this case, it can be observed thata=a0,b=b0, u3=w3, and z0=e0 with the same scenarios for PS-TWR and TS-TWR, which results in the same outage probabilities in the third phase of the two systems; that is, Po3(PS)=Po3(TS). In other words, the whole system outage probabilities for PS-TWR and TS-TWR are determined by the outage probabilities of the first two phases, that is, Po1(PS),Po2(PS),Po1(TS),Po2(TS), in terms of c,d,c0,d0, as shown in Table 1. Because Po1(PS)Po2(PS)>Po1(TS)Po2(TS), we have that Pout(PS)=1-Po1(PS)Po2(PS)Po3(PS)<Pout(TS)=1-Po1(TS)Po2(TS)Po3(TS). Therefore, in Case 1, the system outage probability for PS-TWR outperforms that for TS-TWR.Case 2 (ρi<αi).
In this case,a<a0, b<b0, u3=w3, and z0=e0, so Po3(PS)>Po3(TS). Moreover, since Po1(PS)Po2(PS)>Po1(TS)Po2(TS), Pout(PS)=1-Po1(PS)Po2(PS)Po3(PS)<Pout(TS)=1-Po1(TS)Po2(TS)Po3(TS). Therefore, in Case 2, the system outage probability for PS-TWR is lower than that for TS-TWR.Case 3 (ρi>αi).
In this case,a>a0, b>b0, u3=w3, and z0=e0, so Po3(PS)<Po3(TS). However, Po1(PS)Po2(PS)>Po1(TS)Po2(TS), the system outage of the two systems are difficult to compare theoretically. Thus, we compare them via numerical results in Section 7.Based on the above analysis, it can be also achieved that the approximating outage probabilities of TS-TWR are higher than that of PS-TWR even at high and low SNR regions.


## 6. GA-Based Algorithm for Solving the Optimization Problems

### 6.1. Optimization Problem Formulation for PS-TWR

From the description in Section3, it can be seen that for PS-TWR, the transmit power Pi of Si and all channel parameters are known, while the rest parameters such as the conversion efficiency ζ for the energy harvesting, the power splitting factor ρi, the time assignment factor θi and the distance dS,i between S1 and S2 are all required to be determined and configured. Besides, all the parameters maybe effect on the system outage performance so that jointly designing them for achieving the minimal system outage probability is meaningful and necessary. In this subsection, we formulate an optimization problem to minimize the outage performance of PS-TWR by jointly optimizing ρi and θi, which can be expressed as (29)min⁡ρ=ρ1,ρ2;θ=θ1,θ2PoutPSs.t.0<ρi<1,0<θi<1,0<θ1+θ2<1.


### 6.2. Optimization Problem Formulation for TS-TWR

For TS-TWR protocol, we also formulate an optimization problem to explore its minimal outage probability by jointly optimizing the time switching factorαi of the energy receiver and the time assignment factor θi. The optimization problem can be expressed by(30)min⁡θ=θ1,θ2;α=α1,α2PoutTSs.t.0<αi<1,0<θi<1,0<θ1+θ2<1.


### 6.3. GA-Based Algorithm

Since both problems in (29) and (30) are nonconvex and not easy to derive closed-form solutions by using conventional solution methods, to solve the two problems, we present a GA-based algorithm as follows.GA-based algorithm is a metaheuristic, which iteratively refines an initial population of candidate solution until some termination criterions are satisfied; it starts with randomly creating an initial population of individuals, which is a group of chromosomes. Then each individual is evaluated using a fitness function and a fitness value is allocated to each individual indicating how to close a candidate solution is to an ideal solution. Based on the survival of the fittest, individuals with better fitness will be selected for evolution, or abandoned. The common genetic operations are selection, crossover and mutation. Selection is used to select parents to create new offsprings including tournament selection, roulette wheel selection and rank selection. Crossover varies the programming of an individual or individuals from one generation to the next including one-point crossover, two-point crossover, and “cut and splice.” Mutation causes a certain level of an organism’s genes to change at a certain probability so that the problem of local extreme value can be avoided with proper perturbation introduced by mutation including mut, mutate, and mutbga. Finally, once the termination conditions are satisfied, the process of evolution is stopped [26, 28, 35].In our work, the main steps of GA-based algorithm are simply described in Algorithm1 for PS-TWR and TS-TWR.Algorithm 1:GA-based algorithm for PS-TWR and TS-TWR.
Step  1. Setting the population size: N, selection rate: Ps, crossover rate: Pc, mutation rate: Pm; Step  2. Population initialization: coding to individuals (ρi,θi) for PS-TWR or (αi,θi) for TS-TWR randomly generatesN chromosomes to from the initial population: Chrom=crtbp(⋅), setting generations count:gen=0; Step  3. Fitness evaluation: evaluating individuals (ρi,θi) fitness by the objective function in (11) for PS-TWR or (26) for TS-TWR: FitnV=ranking(Pout(⋅)); Step  4. If the end condition is met, algorithm ends. Otherwise, go to Step  3. Step  5. Selection: select(r′ws′,Chrom,FitnV,Ps); Step  6. Crossover: recombin(x′ovsp′,Pc); Step  7. Mutation: Smut(SelCh,Pm); Step  8. Inserting offsprings to parent produces a new population: reins(⋅).  gen=gen+1, go to Step  2.


## 6.1. Optimization Problem Formulation for PS-TWR

From the description in Section3, it can be seen that for PS-TWR, the transmit power Pi of Si and all channel parameters are known, while the rest parameters such as the conversion efficiency ζ for the energy harvesting, the power splitting factor ρi, the time assignment factor θi and the distance dS,i between S1 and S2 are all required to be determined and configured. Besides, all the parameters maybe effect on the system outage performance so that jointly designing them for achieving the minimal system outage probability is meaningful and necessary. In this subsection, we formulate an optimization problem to minimize the outage performance of PS-TWR by jointly optimizing ρi and θi, which can be expressed as (29)min⁡ρ=ρ1,ρ2;θ=θ1,θ2PoutPSs.t.0<ρi<1,0<θi<1,0<θ1+θ2<1.


## 6.2. Optimization Problem Formulation for TS-TWR

For TS-TWR protocol, we also formulate an optimization problem to explore its minimal outage probability by jointly optimizing the time switching factorαi of the energy receiver and the time assignment factor θi. The optimization problem can be expressed by(30)min⁡θ=θ1,θ2;α=α1,α2PoutTSs.t.0<αi<1,0<θi<1,0<θ1+θ2<1.


## 6.3. GA-Based Algorithm

Since both problems in (29) and (30) are nonconvex and not easy to derive closed-form solutions by using conventional solution methods, to solve the two problems, we present a GA-based algorithm as follows.GA-based algorithm is a metaheuristic, which iteratively refines an initial population of candidate solution until some termination criterions are satisfied; it starts with randomly creating an initial population of individuals, which is a group of chromosomes. Then each individual is evaluated using a fitness function and a fitness value is allocated to each individual indicating how to close a candidate solution is to an ideal solution. Based on the survival of the fittest, individuals with better fitness will be selected for evolution, or abandoned. The common genetic operations are selection, crossover and mutation. Selection is used to select parents to create new offsprings including tournament selection, roulette wheel selection and rank selection. Crossover varies the programming of an individual or individuals from one generation to the next including one-point crossover, two-point crossover, and “cut and splice.” Mutation causes a certain level of an organism’s genes to change at a certain probability so that the problem of local extreme value can be avoided with proper perturbation introduced by mutation including mut, mutate, and mutbga. Finally, once the termination conditions are satisfied, the process of evolution is stopped [26, 28, 35].In our work, the main steps of GA-based algorithm are simply described in Algorithm1 for PS-TWR and TS-TWR.Algorithm 1:GA-based algorithm for PS-TWR and TS-TWR.
Step  1. Setting the population size: N, selection rate: Ps, crossover rate: Pc, mutation rate: Pm; Step  2. Population initialization: coding to individuals (ρi,θi) for PS-TWR or (αi,θi) for TS-TWR randomly generatesN chromosomes to from the initial population: Chrom=crtbp(⋅), setting generations count:gen=0; Step  3. Fitness evaluation: evaluating individuals (ρi,θi) fitness by the objective function in (11) for PS-TWR or (26) for TS-TWR: FitnV=ranking(Pout(⋅)); Step  4. If the end condition is met, algorithm ends. Otherwise, go to Step  3. Step  5. Selection: select(r′ws′,Chrom,FitnV,Ps); Step  6. Crossover: recombin(x′ovsp′,Pc); Step  7. Mutation: Smut(SelCh,Pm); Step  8. Inserting offsprings to parent produces a new population: reins(⋅).  gen=gen+1, go to Step  2.


## 7. Numerical Results

In this section, some simulation and numerical results are presented to validate our analysis and discuss the performance of the proposed PS-TWR and TS-TWR protocols. In our simulations, we consider a typical three-node relaying network as shown in Figure4, in which there is an obstacle between S1 and S2. The relay R is located on the top of the obstacle to help the information exchanging between S1 and S2. Note that the height of the obstacle is varied according to the different environment conditions.Figure 4
The graph of experimental scene for PS-TWR and TS-TWR.The distance betweenS1 and S1 is denoted as dS,S, which is regarded as a reference distance. The variable, 0<φ<1, is applied to describe the ratio of the distance from S1 to the obstacle. And h is the height of the obstacle, where we define it as the setting height of R. Therefore, the distance from Si to R can be given by (31)dS,1=φdS,S2+h2,dS,2=1-φdS,S2+h2.Note that, ifh=0, R is located on the direct line S1-to-S2, which is generally considered as a model to discuss that R is moving along the direct S1-to-S2 link. If h≠0, R is located on the top of the obstacle hindering S1 and S2 communication, which is common in our real life as shown in Figure 1. The power spectral density of the receiving noise at Si is set to be −100 dBm as well as at R. Besides, the path loss effect is also considered, where the path loss exponent is 3, and the distance of the direct S1-to-S2 link, dS,S, is set to be 100 m.Based on the simulation network model as presented in Figure4, we consider five different system configurations (or scenarios) for comparison, as shown in Table 2 for PS-TWR and Table 3 for TS-TWR, respectively. Besides, unless specifically stated, we set Rsr=1 bit/sec/Hz, ζ=1. For simplicity, all the mean values of the exponential random variables h12, h22, f12 and f22 are set to be 1.Table 2
Five scenarios for PS-TWR.
Number Scenarios 1 Fixedρ1=ρ2=1/2, θ1=θ2=1/3 2 Optimalρ, with θ1=θ2=1/3 and ρ1=ρ2=ρ 3 Optimalρ, θ1, θ2 with ρ1=ρ2=ρ 4 Optimalρ1, ρ2, with θ1=θ2=1/3 5 Optimalρ1, ρ2, θ1, θ2Table 3
Five scenarios for TS-TWR.
Number Scenarios 1 Fixedα1=α2=1/2, θ1=θ2=1/3 2 Optimalα, with θ1=θ2=1/3 and α1=α2=α 3 Optimalα, θ1, θ2 with α1=α2=α 4 Optimalα1, α2, with θ1=θ2=1/3 5 Optimalα1, α2, θ1, θ2


### 7.1. Verification of Analytical Results

In this subsection, some simulation results are presented to verify our analytical expressions of the system outage probability and the effectiveness of our designed GA-based algorithm, where the Monte Carlo simulation is adopted by using up to105 simples. In Figure 5, we present the system outage performances of both PS-TWR and TS-TWR versus the available S1 transmit power P1 for φ=0.2 with P2=P1 and h=0 based on the 5th scenario as shown in Tables 2 and 3, respectively.Figure 5
System outage probability: numerical versus simulation for PS-TWR and TS-TWR withP2=P1, h=0, φ=0.2, and dS,S=100m.From Figure5, firstly, it can be seen that the numerical and simulation results match very well for PS-TWR and TS-TWR, which demonstrates our theoretical analysis for the system outage probability and GA-based algorithm. Moreover, it shows that the two protocols can achieve lower system outage probability with the increment of P1. The reason is that more P1 will lead to higher SNR. It is also shown that the system outage performance of TS-TWR is higher than that of PS-TWR, because in the same channel conditions, the system outage performances of PS-TWR and TS-TWR rely on the information received and energy harvested by R. To well balance information transmission and energy harvesting, for TS-TWR, except for the system transmission time assignment and the power allocation, information decoding and energy harvesting processed by TS receiver at R are sequenced over the Si transmission phase with the different time switching factor. But for PS-TWR, except for the system transmission time assignment and the power allocation, information decoding and energy harvesting are indeed processed simultaneously by PS receiver at R over the Si transmission phase with the different power splitting factor. PS-TWR comparing with TS-TWR is more intelligent to allocate the system resources, so it may greatly improve the system outage performance compared with TS-TWR as shown in Figure 5.


### 7.2. Effect ofSi’s Transmit Power on System Performance

Figures6 and 7 depict the system outage performances of both PS-TWR and TS-TWR versus P1 for φ=0.5 and φ=0.2 with P2=P1 and h=0, respectively, which mean that in Figure 6, R is placed on the middle of the straight line S1-to-S2, and in Figure 7, R is closer to S1 than S2.Figure 6
System outage probability versusP1 for PS-TWR with P2=P1, h=0, φ=0.5, and dS,S=100m.Figure 7
System outage probability versusP1 for PS-TWR with P2=P1, h=0, φ=0.2, and dS,S=100m.From Figure6, it can be observed that the optimal system outage probabilities of the five scenarios as shown in Table 2 decrease as the increment of P1, and the system outage performance of the 1st scenario is much higher than that of the other four scenarios, which demonstrates that jointly optimizing the flexible power splitting factor and time assignment factor is superior to only optimizing the fixed power splitting factor and time assignment factor. The curves of the 2nd and the 4th scenarios are almost coincident with each other by only optimizing the power splitting factor ρi with fixed time assignment factor θi=1/3. And the curves of the 3rd and the 5th scenarios are also almost coincident with each other with jointly optimizing the power splitting factor ρi and the time assignment factor θi. But the system outage performances of the 3rd and the 5th scenarios outperform that of the 2nd and the 4th scenarios. In other words, the system outage performance obtained by jointly optimizing the power splitting factor ρi and the time assignment factor θi is better than by only optimizing the power splitting factor ρi with fixed time assignment factor θi=1/3, or the time assignment factor θi can greatly affect the system outage performance with jointly optimizing the power splitting factor ρi.The similar results can also be seen in Figure7, but the curves of the 3rd and the 5th scenarios are close to each other when P1 is within the interval of 0 dBm to 25 dBm and almost coincident when P1 goes to the rest of the value, because more harvested energy should be allocated to P1 in order to reduce the growing path loss firstly with φ=0.2, and then P1 is large enough to meet the system requirements. Besides, the system outage performance gain between the 3rd, 5th scenarios and the 2nd, 4th scenarios is larger than that in Figure 6, which mean that no matter whether R is placed in the middle position of S1 and S2 or not, jointly optimizing the power splitting factor ρi and the time assignment factor θi on the system outage performance outperforms only optimizing the power splitting factor ρi with the fixed time assignment factor θi=1/3. Thus, depending on those different influences, we can reasonably configure system parameters in practice.Figure8 shows the system outage probability versus P1 and φ for PS-TWR based on the 5th scenario as shown in Table 2, where it shows that the system outage probability achieves its minimum at R deployed closer to source nodes with better channel quality. Figure 9 plots the optimal region of P1 and φ for system outage probability, which may guide us to select a proper P1 and φ pair to obtain the optimal system outage probability.Figure 8
Three-dimensional graph of system outage probability for PS-TWR withP2=50-P1, h=0, and dS,S=100m.Figure 9
Optimal region ofP1 and φ for system outage probability with P2=50-P1, h=0, and dS,S=100m for PS-TWR.


### 7.3. Effect of Relay Location on System Outage Probability

In this subsection, we shall discuss the system performances of PS-TWR and TS-TWR by varying the location ofR with different h. As shown in Figure 4, the relay location is described by the ratio φ changing in (0,1), so that we set it varies from 0.1 to 0.9 with h=0 and h=50m in Figures 10 and 11, respectively.Figure 10
System outage probability versusφ with h=0, dS,S=100m, and P1=P2=50 dBm.Figure 11
System outage probability versusφ with h=50, dS,S=100m, and P1=P2=50 dBm.From Figure10, we can see that the optimal system outage probabilities of the five scenarios as shown in Table 2 first increase and then decrease with the increment of φ and achieve their maximums at φ=0.5, that is, when R is positioned on the middle of the direct link S1-to-S2. It is noteworthy that the system outage probability is differ from the traditional case where EH is not considered at the relay and the minimal outage probability is achieved when R is deployed in the middle of Si and S2.While, in Figure11, the system outage probabilities of the five scenarios first decrease and then increase with the increment of φ and achieve their minimums at φ=0.5, these results are opposite to that in Figure 10 but similar to the traditional case. The reason may be that when φ is small, R is closer to S1, a higher energy harvesting efficiency can be yielded at R from S1 and S2. When φ is 0.5, R is placed on the middle of the direct link Si-to-S2 and now lower energy harvesting efficiency can be obtained at R, which may have effect on the system outage problem. When φ is large, R is closer to S2, which leading to the result is opposite for the symmetric channel model. Besides, from Figures 10 and 11, we can also find that PS-TWR can greatly improve the system outage performance compared with TS-TWR.To further investigate the effect of the relay location on the system outage performance, we set the heighth to be 0 m and 100 m, and the distance ratio φ to vary from 0.1 to 0.9. Figure 12 plots the system outage probability when P1=P2=50 dBm and dS,S=100m. It can be observed that, as h increasing, the optimal outage probability is significantly different. The curve of the optimal system outage probability is convex when h varies from 0 m to 40 m. However, when h changes in (40,100), the curve is concave. As a result, the maximum outage probability is achieved when the relay is deployed in the middle of the two sources with h changing in (0,40) as shown in Figure 10, so that, in order to make the system outage performance better, the relay node should be deployed closer to the source node with better channel quality, whereas, when h changes in (40,100), the result is exactly the opposite when h changes in (0,40) as shown in Figure 11.Figure 12
Three-dimensional graph of system outage probability for PS-TWR withP1=P2=50 dBm and dS,S=100m.Besides, Figure13 plots the optimal region of the relay location versus P1 and φ for system outage probability corresponding to Figure 12, where the obtained results may guide us to select a suitable location for the relay to achieve the optimal outage probability.Figure 13
Optimal region ofP1 and φ for system outage probability with P1=P2=50 dBm and dS,S=100m for PS-TWR.


### 7.4. Convergence Behavior of GA-Based Algorithm

Figure14 plots the convergence behaviors of the system outage probability by using the GA-based algorithm during 25 generations from the 2nd to the 5th scenarios as shown in Table 2. It can been seen that since the number of optimized parameters is different, the curve of the 5th scenario converges slower than the other three scenarios. For the 2nd and 4th scenarios, the convergence point is found within 5 runs, and the predefined precision of the 3rd and the 5th scenarios is achieved after almost 15 generations.Figure 14
Convergence behavior of GA-based algorithm.


### 7.5. Comparison of the Exact and the Approximate Outage Probabilities

The approximate versus the theoretical analysis system outage probabilities based on the 5th scenario as shown in Table2 for PS-TWR and Table 3 for TS-TWR are plotted in Figures 15(a) and 15(b), respectively. It can be seen that all approximate outage probability results obtained by our analytical expressions in Theorems 2, 3, 5, and 6 are very close to those of the exact ones obtained in Theorems 1 and 4, respectively, which validate our analytical results for the approximate outage operability of PS-TWR and TS-TWR.Figure 15
(a) Approximate outage probability for PS-TW. (b) Approximate outage probability for TS-TWR.
(a) (b)


## 7.1. Verification of Analytical Results

In this subsection, some simulation results are presented to verify our analytical expressions of the system outage probability and the effectiveness of our designed GA-based algorithm, where the Monte Carlo simulation is adopted by using up to105 simples. In Figure 5, we present the system outage performances of both PS-TWR and TS-TWR versus the available S1 transmit power P1 for φ=0.2 with P2=P1 and h=0 based on the 5th scenario as shown in Tables 2 and 3, respectively.Figure 5
System outage probability: numerical versus simulation for PS-TWR and TS-TWR withP2=P1, h=0, φ=0.2, and dS,S=100m.From Figure5, firstly, it can be seen that the numerical and simulation results match very well for PS-TWR and TS-TWR, which demonstrates our theoretical analysis for the system outage probability and GA-based algorithm. Moreover, it shows that the two protocols can achieve lower system outage probability with the increment of P1. The reason is that more P1 will lead to higher SNR. It is also shown that the system outage performance of TS-TWR is higher than that of PS-TWR, because in the same channel conditions, the system outage performances of PS-TWR and TS-TWR rely on the information received and energy harvested by R. To well balance information transmission and energy harvesting, for TS-TWR, except for the system transmission time assignment and the power allocation, information decoding and energy harvesting processed by TS receiver at R are sequenced over the Si transmission phase with the different time switching factor. But for PS-TWR, except for the system transmission time assignment and the power allocation, information decoding and energy harvesting are indeed processed simultaneously by PS receiver at R over the Si transmission phase with the different power splitting factor. PS-TWR comparing with TS-TWR is more intelligent to allocate the system resources, so it may greatly improve the system outage performance compared with TS-TWR as shown in Figure 5.


## 7.2. Effect ofSi’s Transmit Power on System Performance

Figures6 and 7 depict the system outage performances of both PS-TWR and TS-TWR versus P1 for φ=0.5 and φ=0.2 with P2=P1 and h=0, respectively, which mean that in Figure 6, R is placed on the middle of the straight line S1-to-S2, and in Figure 7, R is closer to S1 than S2.Figure 6
System outage probability versusP1 for PS-TWR with P2=P1, h=0, φ=0.5, and dS,S=100m.Figure 7
System outage probability versusP1 for PS-TWR with P2=P1, h=0, φ=0.2, and dS,S=100m.From Figure6, it can be observed that the optimal system outage probabilities of the five scenarios as shown in Table 2 decrease as the increment of P1, and the system outage performance of the 1st scenario is much higher than that of the other four scenarios, which demonstrates that jointly optimizing the flexible power splitting factor and time assignment factor is superior to only optimizing the fixed power splitting factor and time assignment factor. The curves of the 2nd and the 4th scenarios are almost coincident with each other by only optimizing the power splitting factor ρi with fixed time assignment factor θi=1/3. And the curves of the 3rd and the 5th scenarios are also almost coincident with each other with jointly optimizing the power splitting factor ρi and the time assignment factor θi. But the system outage performances of the 3rd and the 5th scenarios outperform that of the 2nd and the 4th scenarios. In other words, the system outage performance obtained by jointly optimizing the power splitting factor ρi and the time assignment factor θi is better than by only optimizing the power splitting factor ρi with fixed time assignment factor θi=1/3, or the time assignment factor θi can greatly affect the system outage performance with jointly optimizing the power splitting factor ρi.The similar results can also be seen in Figure7, but the curves of the 3rd and the 5th scenarios are close to each other when P1 is within the interval of 0 dBm to 25 dBm and almost coincident when P1 goes to the rest of the value, because more harvested energy should be allocated to P1 in order to reduce the growing path loss firstly with φ=0.2, and then P1 is large enough to meet the system requirements. Besides, the system outage performance gain between the 3rd, 5th scenarios and the 2nd, 4th scenarios is larger than that in Figure 6, which mean that no matter whether R is placed in the middle position of S1 and S2 or not, jointly optimizing the power splitting factor ρi and the time assignment factor θi on the system outage performance outperforms only optimizing the power splitting factor ρi with the fixed time assignment factor θi=1/3. Thus, depending on those different influences, we can reasonably configure system parameters in practice.Figure8 shows the system outage probability versus P1 and φ for PS-TWR based on the 5th scenario as shown in Table 2, where it shows that the system outage probability achieves its minimum at R deployed closer to source nodes with better channel quality. Figure 9 plots the optimal region of P1 and φ for system outage probability, which may guide us to select a proper P1 and φ pair to obtain the optimal system outage probability.Figure 8
Three-dimensional graph of system outage probability for PS-TWR withP2=50-P1, h=0, and dS,S=100m.Figure 9
Optimal region ofP1 and φ for system outage probability with P2=50-P1, h=0, and dS,S=100m for PS-TWR.


## 7.3. Effect of Relay Location on System Outage Probability

In this subsection, we shall discuss the system performances of PS-TWR and TS-TWR by varying the location ofR with different h. As shown in Figure 4, the relay location is described by the ratio φ changing in (0,1), so that we set it varies from 0.1 to 0.9 with h=0 and h=50m in Figures 10 and 11, respectively.Figure 10
System outage probability versusφ with h=0, dS,S=100m, and P1=P2=50 dBm.Figure 11
System outage probability versusφ with h=50, dS,S=100m, and P1=P2=50 dBm.From Figure10, we can see that the optimal system outage probabilities of the five scenarios as shown in Table 2 first increase and then decrease with the increment of φ and achieve their maximums at φ=0.5, that is, when R is positioned on the middle of the direct link S1-to-S2. It is noteworthy that the system outage probability is differ from the traditional case where EH is not considered at the relay and the minimal outage probability is achieved when R is deployed in the middle of Si and S2.While, in Figure11, the system outage probabilities of the five scenarios first decrease and then increase with the increment of φ and achieve their minimums at φ=0.5, these results are opposite to that in Figure 10 but similar to the traditional case. The reason may be that when φ is small, R is closer to S1, a higher energy harvesting efficiency can be yielded at R from S1 and S2. When φ is 0.5, R is placed on the middle of the direct link Si-to-S2 and now lower energy harvesting efficiency can be obtained at R, which may have effect on the system outage problem. When φ is large, R is closer to S2, which leading to the result is opposite for the symmetric channel model. Besides, from Figures 10 and 11, we can also find that PS-TWR can greatly improve the system outage performance compared with TS-TWR.To further investigate the effect of the relay location on the system outage performance, we set the heighth to be 0 m and 100 m, and the distance ratio φ to vary from 0.1 to 0.9. Figure 12 plots the system outage probability when P1=P2=50 dBm and dS,S=100m. It can be observed that, as h increasing, the optimal outage probability is significantly different. The curve of the optimal system outage probability is convex when h varies from 0 m to 40 m. However, when h changes in (40,100), the curve is concave. As a result, the maximum outage probability is achieved when the relay is deployed in the middle of the two sources with h changing in (0,40) as shown in Figure 10, so that, in order to make the system outage performance better, the relay node should be deployed closer to the source node with better channel quality, whereas, when h changes in (40,100), the result is exactly the opposite when h changes in (0,40) as shown in Figure 11.Figure 12
Three-dimensional graph of system outage probability for PS-TWR withP1=P2=50 dBm and dS,S=100m.Besides, Figure13 plots the optimal region of the relay location versus P1 and φ for system outage probability corresponding to Figure 12, where the obtained results may guide us to select a suitable location for the relay to achieve the optimal outage probability.Figure 13
Optimal region ofP1 and φ for system outage probability with P1=P2=50 dBm and dS,S=100m for PS-TWR.


## 7.4. Convergence Behavior of GA-Based Algorithm

Figure14 plots the convergence behaviors of the system outage probability by using the GA-based algorithm during 25 generations from the 2nd to the 5th scenarios as shown in Table 2. It can been seen that since the number of optimized parameters is different, the curve of the 5th scenario converges slower than the other three scenarios. For the 2nd and 4th scenarios, the convergence point is found within 5 runs, and the predefined precision of the 3rd and the 5th scenarios is achieved after almost 15 generations.Figure 14
Convergence behavior of GA-based algorithm.


## 7.5. Comparison of the Exact and the Approximate Outage Probabilities

The approximate versus the theoretical analysis system outage probabilities based on the 5th scenario as shown in Table2 for PS-TWR and Table 3 for TS-TWR are plotted in Figures 15(a) and 15(b), respectively. It can be seen that all approximate outage probability results obtained by our analytical expressions in Theorems 2, 3, 5, and 6 are very close to those of the exact ones obtained in Theorems 1 and 4, respectively, which validate our analytical results for the approximate outage operability of PS-TWR and TS-TWR.Figure 15
(a) Approximate outage probability for PS-TW. (b) Approximate outage probability for TS-TWR.
(a) (b)


## 8. Conclusion

This paper investigated SWIPT in a network-coded two-way relay system with exploring the system outage performance. For such a system, we presented two protocols, PS-TWR and TS-TWR to analyze the system outage performance. Besides, we also derived the corresponding approximate expressions of the system outage probability at the high and low SNR regions, respectively. In order to explore effects of the relay position and the sources transmit power on the system outage performance, we formulated two optimization problems to minimize the system outage probability by jointly optimizing the power splitting factor of the energy harvesting receiver and the time assignment factor of transmission phase for PS-TWR, and by jointly optimizing the time splitting factor of the energy harvesting receiver and the time assignment factor of transmission phase for TS-TWR. To solve the problems, we designed GA-based algorithm for them. Numerical and simulation results validated our theoretical analysis. It is shown that by jointly optimizing the time assignment and the SWIPT receiver parameters, a great performance gain can be achieved for both PS-TWR and TS-TWR. It is also shown that the optimized PS-TWR always outperforms the optimized TS-TWR in terms of outage performance. In addition, the effects of various system parameters including the relay location, the transmit power of the two sources are also discussed, which provide some insights for the SWIPT-enabled two-way relay networks.



---

*Source: 2516035-2017-02-22.xml*