# Adaptive Energy Harvesting Relaying Protocol for Two-Way Half-Duplex System Network over Rician Fading Channels

Authors: Tan N. Nguyen; Tran Hoang Quang Minh; Phuong T. Tran; Miroslav Voznak
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

We investigate the system performance of a two-way amplify-and-forward (AF) energy harvesting relay network over the Rician fading environment. For details, the delay-limited (DL) and delay-tolerant (DT) transmission modes are proposed and investigated when both energy and information are transferred between the source node and the destination node via a relay node. In the first stage, the analytical expressions of the achievable throughput, ergodic capacity, the outage probability, and symbol error ratio (SER) were proposed, analyzed, and demonstrated. After that, the closed-form expressions for the system performance are studied in connection with all system parameters. Moreover, the analytical results are also demonstrated by Monte Carlo simulation in comparison with the closed-form expressions. Finally, the research results show that the analytical and the simulation results agree well with each other in all system parameters.


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## Body

## 1. Introduction

In the last decades, the Internet of Things (IoT) is a too hot research area over the world. With its potential to significantly influence all aspects of our daily lives, IoT is expected to have a significant impact on businesses by automating some processes and improving the control of many environment variables [1–6]. However, the large-scale realization of IoT is not easy, and numerous challenges need to be overcome before this concept is widely accepted. Energy limitation is the most significant problem in IoT technology revolution for long-term operation of wireless networks. In the beginning, this problem is solved by prolonging the lifetime or by battery recharging/replacement, which may not always be a perspective and practical solution for the wireless networks. In the last few years, energy harvesting has been considered as a prospective solution for prolonging the lifetime of wireless devices and a promising technique for IoT shortly. The sources for energy harvesting are an extensive range such as solar, wind, radio frequency (RF), and vibration. However, RF energy harvesting provides critical advantages over other energy collection methods due to its predictable and stable nature, small form factor, and low cost. The wireless powered communication network (WPCN), where network devices harvest energy from the signals transmitted by RF energy sources in the first step and then utilize this harvested energy for their communication needs, is the primary direction in the RF energy harvesting. In the last decade, many research papers focused on the WPCN and how to improve its efficiency. This concept of a tradeoff between EH and information transmission in WPCN was proposed and investigated in [7] and extended in [8]. Moreover, the concept of partial network level cooperation for EH networks was presented in detail in [9], and in [10] wireless EH and information transfer in cognitive relay networks was intensely analyzed. From this point of view, a wireless powered communication network (WPCN) with RF energy harvesting is the hot research area in tour decade.In WPCN, the two traditional time switching (TSP) and power splitting (PSP) protocols have been intensively studied in the literature, and many from these studies have compared the system performance of the two protocols under different scenarios [11–15]. From these studies, the TSP and PSP protocols have some drawbacks; for instance, TSP has to lose some information while it switches to the harvesting mode and PSP has a low coverage area. Furthermore, PSP requires a complicated hardware structure to make sure that a proper portion of energy from source signal is extracted for energy harvesting. In contrast, TSP can simplify the hardware at the expense of the throughput or achievable rate of the system. Based on the fact that both TSP and PSP protocols have their drawbacks, the prospective idea is to combine these two protocols to get the best out of them. This solution can be obtained in this paper by using an adaptive relaying protocol. Also, the adaptive relaying network is investigated for the two-way half-duplex system over the Rician fading channel. This model is more used in practice in comparison with the widespread Raleigh fading channel [16–20]In this paper, the system performance (in terms of the achievable throughput, ergodic capacity, the outage probability, and symbol errors ratio (SER) of a two-way amplify-and-forward (AF) adaptive relaying network over Rician fading channel) is investigated in detail. In this model, the energy and information are transferred between the destination node and source node via relay node by DL and DT transmission modes. In the first step, the analytical mathematical expressions of the achievable throughput, outage probability, ergodic capacity, and SER were proposed and demonstrated for both DL and DT transmission modes. In the first stage, the analytical expressions of the achievable throughput, ergodic capacity, the outage probability, and symbol error ratio (SER) were proposed, analyzed, and demonstrated. After that, the closed-form expressions for the system performance are studied in connection with all system parameters. Moreover, the analytical results are also demonstrated by Monte Carlo simulation in comparison with the closed-form expressions. Finally, the research results show that the analytical and the simulation results agree well with each other in all system parameters.The research results provided the practical solution for the progress of the WPCN in the near future. The main contributions are summarized as follows:( 1 ) The system model of the adaptive relaying network for the two-way half-duplex system over the Rician fading channel is proposed in both DT and DL transmission modes.( 2 ) The closed-form expressions of the outage probability, achievable throughput, ergodic capacity, and especially SER of the system model for the DL and DT transmission modes are proposed and derived.( 3 ) Use Monte Carlo simulation to clear the analysis results and compare the system performance of all TS, PS, and adaptive protocols.The structure of the remainder of this paper is proposed as follows. Section2 shows the system model of two-way AR adaptive relaying network over Rician fading channel in both DT and DL transmission modes. Section 3 proposed the system performance of the model system in detail. Section 4 presents the numerical results and some discussions. Finally, conclusions of the paper are proposed in Section 5.


## 2. System Model

In this section, amplify-and-forward (AF) adaptive relaying protocol for two-way half-duplex system network over Rician fading channel is presented in Figure1. In Figure 1, the information and energy are transferred between the sources (S1,S2) and the relay (R) in two-way with the half-duplex system. The energy harvesting and information processing of the adaptive relaying network are proposed in Figure 2. The all-time interval for the energy harvesting and information processing is denoted by T, which is divided into three intervals. The first portion of time αT is used for energy harvesting at R from the sourcesS1 andS2. In the second interval, whose length is (1-α)T/2, the source signal is divided into two streams. During this interval, a fraction of the power ρPS is used for energy harvesting from the sourcesS1 andS2 signal by the relay node, and a fraction 1-ρPS is used for decoding the information signal sent from the source nodes. The remaining interval of the length (1-α)T/2 is used for information forwarding from the relay to the destination nodesS1 andS2. Obviously, 0≤α≤1 and 0≤ρ≤1. If α=0, this scheme becomes PSP. If ρ=0, then it becomes the TSP protocol.Figure 1
System model.Figure 2
The energy harvesting and information processing by the adaptive relaying protocol.


### 2.1. Energy Harvesting

In the energy harvesting phase at the relay node, the received signal at the relay node can be calculated as(1)yr=g1xs1+g2xs2+nr.In this equation,xs1,2 is the energy-transmitted signal from sourcesS1 andS2 with Exs12=Exs22=Ps, and nr is the zero-mean additive white Gaussian noise (AWGN) with variance N0. Here E{·} denotes the expectation operation, and g1g2 are the Rician distribution factors.The energy harvested at the relay node is the combination of two components: the first one is the received energy during the first intervalαT (from TSR protocol), while the second one comes from the PSR interval (1-α)T/2. The amount of harvested energy can be calculated as(2)Er=ηαPsg12+ηαPsg22αT+ηρρPsg12+ηρρPsg221-αT2=ηααT+ηρρ1-αT2Psg12+Psg22,where ηα, ηρ represent the EH efficiency of the two stages, TS and PS, respectively, 0<ηα<1, and 0<ηρ<1.Finally, the received power at the relay node can be computed as(3)Pr=Er1-αT/2=ηααT+ηρρ1-αT/2Psg12+Psg221-αT/2=2ηαα1-α+ηρρPsg12+Psg22,Pr=κPsg12+Psg22,where we denote κ=2ηαα/(1-α)+ηρρ.


### 2.2. Information Transmission

In this subsection, the information transmission phase is considered. This interval time is divided into two subintervals. In the first interval, the received signal at the relay can be expressed as(4)yr=1-ρg1xs1+1-ρg2xs2+nr,where nr denotes the AWGN noise at the relay node.In research model, the amplify-and-forward protocol is used. Moreover, the received signal at relay node (which is amplified by a factorβ) is proposed by(5)β=xryr=Pr1-ρPsg12+g22+N0.During the second interval, the sourcesS1 andS2 are receiving the forwarded signal from the relay. Now, because of the symmetry ofS1 andS2, we first provide the expression of y1 for the sourceS1. The received signal at the sourceS1 is given by(6)y1=g1xr+n1.Heren1 is the additive white Gaussian noise (AWGN) at S1 with zero mean and variance N0. Substituting (4) in (6), (6) could be changed to (7)y1=g1xr+n1=g1βyr+n1=g1β1-ρg1xs1+1-ρg2xs2+nr+n1.In (7)S1 wants to extractS2 from y1. Since it knows its own transmitted symbolS1, it can perfectly eliminate the corresponding self-interference term β1-ρg12xs1. After that, we have (8)y1=g1β1-ρg2xs2︸signal+g1βnr+n1︸noise.Now, the end-to-end signal-to-noise ratio (SNR) at the source nodeS1 can be calculated by (9)γ1=Esignal2Enoise2=β21-ρg12g22Psβ2g12N0+N0=1-ρg12g22Psg12N0+N0/β2.Replacingβ and Pr in (9) we have (10)γ1=1-ρg12g22Psg12N0+N01-ρPsg12+g22+N0/Pr=1-ρg12g22Psg12N0+N01-ρ/κ+N02/Pr.BecauseN0≪Pr and we denote γ0=Ps/N0, (10) can be reduced to(11)γ1≈1-ρg12g22Psg12N0+N01-ρ/κ=1-ρg12g22γ0g12+1-ρ/κ,(12)γ1=1-ρφ1φ2γ0φ1+1-ρ/κ,where φ1=g12 and   φ2=g22.By the same way, the end-to-end signal-to-noise ratio ofγ2 at the source nodeS2 is also given by(13)γ2=1-ρφ1φ2γ0φ2+1-ρ/κ.More details of the analytical mathematical model for delay-limited (DL) transmission and delay-tolerant (DT) transmission modes of the two-way AF relaying system are presented in the following sections [15–22].


## 2.1. Energy Harvesting

In the energy harvesting phase at the relay node, the received signal at the relay node can be calculated as(1)yr=g1xs1+g2xs2+nr.In this equation,xs1,2 is the energy-transmitted signal from sourcesS1 andS2 with Exs12=Exs22=Ps, and nr is the zero-mean additive white Gaussian noise (AWGN) with variance N0. Here E{·} denotes the expectation operation, and g1g2 are the Rician distribution factors.The energy harvested at the relay node is the combination of two components: the first one is the received energy during the first intervalαT (from TSR protocol), while the second one comes from the PSR interval (1-α)T/2. The amount of harvested energy can be calculated as(2)Er=ηαPsg12+ηαPsg22αT+ηρρPsg12+ηρρPsg221-αT2=ηααT+ηρρ1-αT2Psg12+Psg22,where ηα, ηρ represent the EH efficiency of the two stages, TS and PS, respectively, 0<ηα<1, and 0<ηρ<1.Finally, the received power at the relay node can be computed as(3)Pr=Er1-αT/2=ηααT+ηρρ1-αT/2Psg12+Psg221-αT/2=2ηαα1-α+ηρρPsg12+Psg22,Pr=κPsg12+Psg22,where we denote κ=2ηαα/(1-α)+ηρρ.


## 2.2. Information Transmission

In this subsection, the information transmission phase is considered. This interval time is divided into two subintervals. In the first interval, the received signal at the relay can be expressed as(4)yr=1-ρg1xs1+1-ρg2xs2+nr,where nr denotes the AWGN noise at the relay node.In research model, the amplify-and-forward protocol is used. Moreover, the received signal at relay node (which is amplified by a factorβ) is proposed by(5)β=xryr=Pr1-ρPsg12+g22+N0.During the second interval, the sourcesS1 andS2 are receiving the forwarded signal from the relay. Now, because of the symmetry ofS1 andS2, we first provide the expression of y1 for the sourceS1. The received signal at the sourceS1 is given by(6)y1=g1xr+n1.Heren1 is the additive white Gaussian noise (AWGN) at S1 with zero mean and variance N0. Substituting (4) in (6), (6) could be changed to (7)y1=g1xr+n1=g1βyr+n1=g1β1-ρg1xs1+1-ρg2xs2+nr+n1.In (7)S1 wants to extractS2 from y1. Since it knows its own transmitted symbolS1, it can perfectly eliminate the corresponding self-interference term β1-ρg12xs1. After that, we have (8)y1=g1β1-ρg2xs2︸signal+g1βnr+n1︸noise.Now, the end-to-end signal-to-noise ratio (SNR) at the source nodeS1 can be calculated by (9)γ1=Esignal2Enoise2=β21-ρg12g22Psβ2g12N0+N0=1-ρg12g22Psg12N0+N0/β2.Replacingβ and Pr in (9) we have (10)γ1=1-ρg12g22Psg12N0+N01-ρPsg12+g22+N0/Pr=1-ρg12g22Psg12N0+N01-ρ/κ+N02/Pr.BecauseN0≪Pr and we denote γ0=Ps/N0, (10) can be reduced to(11)γ1≈1-ρg12g22Psg12N0+N01-ρ/κ=1-ρg12g22γ0g12+1-ρ/κ,(12)γ1=1-ρφ1φ2γ0φ1+1-ρ/κ,where φ1=g12 and   φ2=g22.By the same way, the end-to-end signal-to-noise ratio ofγ2 at the source nodeS2 is also given by(13)γ2=1-ρφ1φ2γ0φ2+1-ρ/κ.More details of the analytical mathematical model for delay-limited (DL) transmission and delay-tolerant (DT) transmission modes of the two-way AF relaying system are presented in the following sections [15–22].


## 3. The System Performance

Based on the system model on above section, the system performance of the relay network is presented, analyzed, and demonstrated for delay-limited (DL) transmission and delay-tolerant (DT) transmission in detail in this section [7–12]. In addition, the outage probability and throughput performance of the proposed system for delay-limited transmission mode and the ergodic capacity of the system for delay-tolerant mode are proposed and derived.


### 3.1. Delay-Limited (DL) Transmission

#### 3.1.1. Outage Probability

The probability density function (PDF) of random variable (RV)φi, where i=1,2, can be calculated by [21](14)fφix=K+1e-Kλie-K+1x/λiI02KK+1xλi,where λi is the mean value of RV φi, where i=1,2, respectively. K is the Rician K-factor defined as the ratio of the power of the line-of-sight (LOS) component to the scattered components, and I0· is the zeroth order modified Bessel function of the first kind.Usinga=K+1e-K/λi,  b=(K+1)/λi, and I0(x)=∑l=0∞x2l/22ll!2 [23], (14) can be rewritten as follows:(15)fφix=a∑l=0∞bKll!2xle-bx.The cumulative density function (CDF) of RVφi, where i=1,2, can be computed as in [22]. Here, we assume that λ1=λ2:(16)Fφiς=∫0ςfφixdx=1-ab∑l=0∞∑m=0lKlbml!m!ςme-bς.Hence, the outage probability is determined by(17)Pout_1=Fγ1γth=Pr⁡γ1<γth=Pr⁡1-ρφ1φ2γ0φ1+1-ρ/κ<γth.Assume that the source transmits at a constant rateR. Let γth=2R-1 be the lower threshold for SNR at both relay and source nodes. Then the outage probability can reformulate as(18)Pout_1=∫0∞Pr⁡φ2<φ1γth+γth1-ρ/κφ1γ01-ρ∣φ1fφ1φ1dφ1=∫0∞Fφ2φ1γth+γth1-ρ/κφ1γ01-ρfφ1φ1dφ1.Combining (18) with (15) and (16) we have(19)Pout_1=1-∫0∞ab∑l=0∞∑m=0lKlbml!m!φ1γth+γth1-ρ/κφ1γ01-ρm×e-bφ1γth+γth1-ρ/κ/φ1γ01-ρ×a∑l=0∞bKll!2φ1le-bφ1dφ1,(20)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kl!m!k!2γthγ01-ρ+γthκφ1γ0m×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1.Now by applying the equationx+ym=∑n=0mmnxm-nyn to (20), the outage probability can be demonstrated as follows:(21)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kγthml!m!k!2γ0m∑n=0mmn11-ρm-n1κφ1n×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1,Pout_1=1-a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1γthml!n!m-n!k!2γ0m1-ρm-nκn×∫0∞φ1k-n×e-bγth/κφ1γ0×e-bφ1dφ1.Using table of integrals equation [3.471,9] in [23], (22) can reformulated as(22)Pout_1=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0,where Kv(·) is the modified Bessel function of the second kind and vth order.In the same way, the outage probability at the source nodeS2 is also given by(23)Pout_2=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0.


#### 3.1.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(24)τiDL=1-Pouti×R2×1-α,where i∈(1,2).


### 3.2. Delay-Tolerant (DT) Transmission

#### 3.2.1. Ergodic Capacity

In this model, the source transfers at any target rate upper bounded by the ergodic capacity. Hence, the ergodic capacity is given by the following formula:(25)C1=Eg1,g2log2⁡1+γ1C1=∫0∞fγ1γthlog2⁡1+γthdγth=1ln⁡2∫0∞1-Fγ1γth1+γthdγth.Using (22) we have(26)C1=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.By the same way, the ergodic capacity of the source nodeS2 is also given by(27)C2=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.


#### 3.2.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(28)τiDT=Ci×1-α2,where i∈(1,2)


### 3.3. Symbol Error Ratio (SER) Analysis

In this section, the expressions for the symbol error ratio (SER) at the sourcesS1 andS2 of the two-way AF relaying system are obtained. Thus, we have SER equations as below [24]:(29)SERi=EωQ2θγi,where i∈(1,2) and Q(t)=(1/2π)∫t∞e-x2/2dx is the Gaussian Q-function; ω and θ are constants, which is specific for modulation type. Here (ω,θ)=(1,1) for BPSK and (ω,θ)=(1,2) for QPSK. Before obtaining the SER performance, the distribution function of γi is expected. Then, the SER expression is given in (29) directly regarding outage probability at the source by using integration, as follows:(30)SERi=ωθ2π∫0∞e-θxxFγixdx,SER1=ωθ2π∫0∞e-θxxFγ1xdx.Replacing (29) in (32), we have(31)SER1=ωθ2π∫0∞e-θxx1-2a2×e-bx/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×xγ0m+k-n+1/2×Kk-n+12bxκγ0dx,(32)SER1=ωθ2π∫0∞e-θxxdx-a2ωθπ∫0∞∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2γ0m+k+n+1/2×x2m-n+k/2×e-xb/1-ργ0+θKk-n+12bxκγ0dx.Here(33)I1=ωθ2π∫0∞e-θxxdx,I2=∫0∞x2m-n+k/2×e-xb/1-ργ0+θKk-n+12bxκγ0dx.Using equation (3.361,1) in the table of integrals [23], we have(34)I1=ωθ2π×πθ=ω2.Using equation (6.643,3) in table of integrals [23],(35)∫0∞tμ-1/2e-αtK2v2βtdt=α-μ2βΓμ-v+12Γμ+v+12exp⁡β22αW-μ,vβ2α,where Γ(·) is the gamma function and W(·) is the Whittaker function. After that, (35) can be formulated as(36)I2=b1-ργ0+θ-2m-n+k+1/2×κγ02b×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.Substituting (36) in (32), we have(37)SER1=ω2-a2ωθπ×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2γ0m+k-n+1/2×b1-ργ0+θ-2m-n+k+1/2×κγ02b×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.Finally, SER of the sourceS1 can be calculated by the following equation:(38)SER1=ω2-a2ωθ2π×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-2l!n!m-n!k!21-ρm-nκk+n/2γ0m+k-n/2×b1-ργ0+θ-2m-n+k+1/2×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.In the same way as sourceS1, the SER of the sourceS2 can be demonstrated as(39)SER2=ω2-a2ωθ2π×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-2l!n!m-n!k!21-ρm-nκk+n/2γ0m+k-n/2×b1-ργ0+θ-2m-n+k+1/2×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.


## 3.1. Delay-Limited (DL) Transmission

### 3.1.1. Outage Probability

The probability density function (PDF) of random variable (RV)φi, where i=1,2, can be calculated by [21](14)fφix=K+1e-Kλie-K+1x/λiI02KK+1xλi,where λi is the mean value of RV φi, where i=1,2, respectively. K is the Rician K-factor defined as the ratio of the power of the line-of-sight (LOS) component to the scattered components, and I0· is the zeroth order modified Bessel function of the first kind.Usinga=K+1e-K/λi,  b=(K+1)/λi, and I0(x)=∑l=0∞x2l/22ll!2 [23], (14) can be rewritten as follows:(15)fφix=a∑l=0∞bKll!2xle-bx.The cumulative density function (CDF) of RVφi, where i=1,2, can be computed as in [22]. Here, we assume that λ1=λ2:(16)Fφiς=∫0ςfφixdx=1-ab∑l=0∞∑m=0lKlbml!m!ςme-bς.Hence, the outage probability is determined by(17)Pout_1=Fγ1γth=Pr⁡γ1<γth=Pr⁡1-ρφ1φ2γ0φ1+1-ρ/κ<γth.Assume that the source transmits at a constant rateR. Let γth=2R-1 be the lower threshold for SNR at both relay and source nodes. Then the outage probability can reformulate as(18)Pout_1=∫0∞Pr⁡φ2<φ1γth+γth1-ρ/κφ1γ01-ρ∣φ1fφ1φ1dφ1=∫0∞Fφ2φ1γth+γth1-ρ/κφ1γ01-ρfφ1φ1dφ1.Combining (18) with (15) and (16) we have(19)Pout_1=1-∫0∞ab∑l=0∞∑m=0lKlbml!m!φ1γth+γth1-ρ/κφ1γ01-ρm×e-bφ1γth+γth1-ρ/κ/φ1γ01-ρ×a∑l=0∞bKll!2φ1le-bφ1dφ1,(20)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kl!m!k!2γthγ01-ρ+γthκφ1γ0m×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1.Now by applying the equationx+ym=∑n=0mmnxm-nyn to (20), the outage probability can be demonstrated as follows:(21)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kγthml!m!k!2γ0m∑n=0mmn11-ρm-n1κφ1n×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1,Pout_1=1-a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1γthml!n!m-n!k!2γ0m1-ρm-nκn×∫0∞φ1k-n×e-bγth/κφ1γ0×e-bφ1dφ1.Using table of integrals equation [3.471,9] in [23], (22) can reformulated as(22)Pout_1=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0,where Kv(·) is the modified Bessel function of the second kind and vth order.In the same way, the outage probability at the source nodeS2 is also given by(23)Pout_2=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0.


### 3.1.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(24)τiDL=1-Pouti×R2×1-α,where i∈(1,2).


## 3.1.1. Outage Probability

The probability density function (PDF) of random variable (RV)φi, where i=1,2, can be calculated by [21](14)fφix=K+1e-Kλie-K+1x/λiI02KK+1xλi,where λi is the mean value of RV φi, where i=1,2, respectively. K is the Rician K-factor defined as the ratio of the power of the line-of-sight (LOS) component to the scattered components, and I0· is the zeroth order modified Bessel function of the first kind.Usinga=K+1e-K/λi,  b=(K+1)/λi, and I0(x)=∑l=0∞x2l/22ll!2 [23], (14) can be rewritten as follows:(15)fφix=a∑l=0∞bKll!2xle-bx.The cumulative density function (CDF) of RVφi, where i=1,2, can be computed as in [22]. Here, we assume that λ1=λ2:(16)Fφiς=∫0ςfφixdx=1-ab∑l=0∞∑m=0lKlbml!m!ςme-bς.Hence, the outage probability is determined by(17)Pout_1=Fγ1γth=Pr⁡γ1<γth=Pr⁡1-ρφ1φ2γ0φ1+1-ρ/κ<γth.Assume that the source transmits at a constant rateR. Let γth=2R-1 be the lower threshold for SNR at both relay and source nodes. Then the outage probability can reformulate as(18)Pout_1=∫0∞Pr⁡φ2<φ1γth+γth1-ρ/κφ1γ01-ρ∣φ1fφ1φ1dφ1=∫0∞Fφ2φ1γth+γth1-ρ/κφ1γ01-ρfφ1φ1dφ1.Combining (18) with (15) and (16) we have(19)Pout_1=1-∫0∞ab∑l=0∞∑m=0lKlbml!m!φ1γth+γth1-ρ/κφ1γ01-ρm×e-bφ1γth+γth1-ρ/κ/φ1γ01-ρ×a∑l=0∞bKll!2φ1le-bφ1dφ1,(20)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kl!m!k!2γthγ01-ρ+γthκφ1γ0m×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1.Now by applying the equationx+ym=∑n=0mmnxm-nyn to (20), the outage probability can be demonstrated as follows:(21)Pout_1=1-∫0∞a2b∑l=0∞∑k=0∞∑m=0lKl+kbm+kγthml!m!k!2γ0m∑n=0mmn11-ρm-n1κφ1n×e-bγth/γ01-ρ×e-bγth/κφ1γ0×φ1ke-bφ1dφ1,Pout_1=1-a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1γthml!n!m-n!k!2γ0m1-ρm-nκn×∫0∞φ1k-n×e-bγth/κφ1γ0×e-bφ1dφ1.Using table of integrals equation [3.471,9] in [23], (22) can reformulated as(22)Pout_1=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0,where Kv(·) is the modified Bessel function of the second kind and vth order.In the same way, the outage probability at the source nodeS2 is also given by(23)Pout_2=1-2a2×e-bγth/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×γthγ0m+k-n+1/2×Kk-n+12bγthκγ0.


## 3.1.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(24)τiDL=1-Pouti×R2×1-α,where i∈(1,2).


## 3.2. Delay-Tolerant (DT) Transmission

### 3.2.1. Ergodic Capacity

In this model, the source transfers at any target rate upper bounded by the ergodic capacity. Hence, the ergodic capacity is given by the following formula:(25)C1=Eg1,g2log2⁡1+γ1C1=∫0∞fγ1γthlog2⁡1+γthdγth=1ln⁡2∫0∞1-Fγ1γth1+γthdγth.Using (22) we have(26)C1=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.By the same way, the ergodic capacity of the source nodeS2 is also given by(27)C2=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.


### 3.2.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(28)τiDT=Ci×1-α2,where i∈(1,2)


## 3.2.1. Ergodic Capacity

In this model, the source transfers at any target rate upper bounded by the ergodic capacity. Hence, the ergodic capacity is given by the following formula:(25)C1=Eg1,g2log2⁡1+γ1C1=∫0∞fγ1γthlog2⁡1+γthdγth=1ln⁡2∫0∞1-Fγ1γth1+γthdγth.Using (22) we have(26)C1=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.By the same way, the ergodic capacity of the source nodeS2 is also given by(27)C2=1ln⁡2∫0∞2a2∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/21+γth×γthγ0m+k-n+1/2×e-bγth/γ01-ρ×Kk-n+12bγthκγ0dγth.


## 3.2.2. Achievable Throughput

The achievable throughput at the source nodesS1 andS2 can be computed by(28)τiDT=Ci×1-α2,where i∈(1,2)


## 3.3. Symbol Error Ratio (SER) Analysis

In this section, the expressions for the symbol error ratio (SER) at the sourcesS1 andS2 of the two-way AF relaying system are obtained. Thus, we have SER equations as below [24]:(29)SERi=EωQ2θγi,where i∈(1,2) and Q(t)=(1/2π)∫t∞e-x2/2dx is the Gaussian Q-function; ω and θ are constants, which is specific for modulation type. Here (ω,θ)=(1,1) for BPSK and (ω,θ)=(1,2) for QPSK. Before obtaining the SER performance, the distribution function of γi is expected. Then, the SER expression is given in (29) directly regarding outage probability at the source by using integration, as follows:(30)SERi=ωθ2π∫0∞e-θxxFγixdx,SER1=ωθ2π∫0∞e-θxxFγ1xdx.Replacing (29) in (32), we have(31)SER1=ωθ2π∫0∞e-θxx1-2a2×e-bx/γ01-ρ∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2×xγ0m+k-n+1/2×Kk-n+12bxκγ0dx,(32)SER1=ωθ2π∫0∞e-θxxdx-a2ωθπ∫0∞∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2γ0m+k+n+1/2×x2m-n+k/2×e-xb/1-ργ0+θKk-n+12bxκγ0dx.Here(33)I1=ωθ2π∫0∞e-θxxdx,I2=∫0∞x2m-n+k/2×e-xb/1-ργ0+θKk-n+12bxκγ0dx.Using equation (3.361,1) in the table of integrals [23], we have(34)I1=ωθ2π×πθ=ω2.Using equation (6.643,3) in table of integrals [23],(35)∫0∞tμ-1/2e-αtK2v2βtdt=α-μ2βΓμ-v+12Γμ+v+12exp⁡β22αW-μ,vβ2α,where Γ(·) is the gamma function and W(·) is the Whittaker function. After that, (35) can be formulated as(36)I2=b1-ργ0+θ-2m-n+k+1/2×κγ02b×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.Substituting (36) in (32), we have(37)SER1=ω2-a2ωθπ×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-1l!n!m-n!k!21-ρm-nκk+n+1/2γ0m+k-n+1/2×b1-ργ0+θ-2m-n+k+1/2×κγ02b×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.Finally, SER of the sourceS1 can be calculated by the following equation:(38)SER1=ω2-a2ωθ2π×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-2l!n!m-n!k!21-ρm-nκk+n/2γ0m+k-n/2×b1-ργ0+θ-2m-n+k+1/2×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.In the same way as sourceS1, the SER of the sourceS2 can be demonstrated as(39)SER2=ω2-a2ωθ2π×∑l=0∞∑k=0∞∑m=0l∑n=0mKl+kbm+k-2l!n!m-n!k!21-ρm-nκk+n/2γ0m+k-n/2×b1-ργ0+θ-2m-n+k+1/2×Γm+12×Γm-n+k+32×exp⁡b21-ρ2κb+2κθ1-ργ0×W-2m-n+k+1/2,k-n+1/2b21-ρκb+θ1-ργ0.


## 4. Numerical Results and Discussion

In this research, we use the Monte Carlo simulation to investigate the developed analysis in the previous section. For simplicity, in our simulation model, we assume that the source-relay and relay-destination distances are both normalized to unit value. For the delay-limited transmission mode, the outage probability, achievable throughput, and SER are analyzed in detail. On the other hand, the outage probability, the ergodic capacity, and SER for the delay-tolerant transmission mode are proposed and demonstrated. The main simulation parameters are listed in Table1.Table 1
Simulation parameters.
Symbol Name Values ηα=ηρ Energy harvesting efficiency 0.7 λ 1 Mean ofg12 1 λ 2 Mean ofg22 1 K RicianK-factor 3 γ t h SNR threshold 7 P s / N 0 Source power to noise ratio 0–30 dB R Source rate 3 bit/s/HzIn Figures3 and 4, the effect of the ratio on the achievable throughput and the outage probability of the two-way AF relay network system in adaptive relaying (AR), TS, and PS protocols is presented. In this simulation, the DL transmission mode is considered and the ratio Ps/N0 varied from 0 to 30 dB. For AR protocol, α and ρ are set to 0.5. From this setting, the parameters for TS and PS protocols are set accordingly to make sure that the information transmission time is equal between 3 methods. From Figures 3 and 4, the analytical and the simulation results match all possible values of Ps/N0, which confirms that our analysis is reasonable. To be analyzed, the throughput increases and the outage probability decreases when the value of Ps/N0 increases from 0 to 30 dB. It is also observed that the AR protocol can give better performance than PS protocol.Figure 3
The outage probability of the DL transmission mode versusPs/N0.Figure 4
The throughput of the DL transmission mode versusPs/N0.On the other hand, Figures5 and 6 plot the influence of the ratio Ps/N0 on the achievable throughput and the ergodic capacity of the model system in the DT transmission mode. Here, α and ρ are set to 0.5 and the ratio Ps/N0 increased from 0 to 30 dB. The numerical results show that the throughput and ergodic capacity significantly increased with the increase in the ratio Ps/N0. Moreover, the simulation results totally agreed with the mathematical analysis from the above section. Furthermore, Figures 7 and 8 illustrate the throughput of the DL and DT transmission modes versus the system factors α and ρ. In this cases, α and ρ are set to 0.3, 0.5, and 0.7 in each case. From Figure 7, the throughput increased when α increased to the optimal value at the first step, and after that the throughput crucially decreased. These results are similar to the influence of ρ on the system throughput. Moreover, the simulation results totally agreed with the mathematical analysis from the above section. In addition, the throughput in the DL and DT transmission modes in connection with the ratio Ps/N0 is presented in Figure 9. After that, SER of the model system is plotted in Figure 10. In all of them, the simulation and analytical mathematical results agreed well with each other. Moreover, the throughput increased, and the SER decreased in AR, TS, and PS protocols, while the ratio Ps/N0 varied from 0 to 30 dB.Figure 5
The ergodic capacity of the DT transmission mode versusPs/N0.Figure 6
The throughput of the DT transmission mode versusPs/N0.Figure 7
The throughput of the DT and DL transmission modes versusα.Figure 8
The throughput of the DT and DL transmission modes versusρ.Figure 9
The throughput of the DT and DL transmission modes versusPs/N0.Figure 10
SER versusPs/N0.


## 5. Conclusions

In this paper, the system performance of a two-way AF relaying network over the Rician fading channel with the delay-limited and delay-tolerant transmission modes has been proposed, analyzed, and demonstrated. In the delay-limited transmission mode, the closed-form expressions of the outage probability, achievable throughput, and SER are proposed and derived. On the other hand, the closed-form expressions of the ergodic capacity, achievable throughput, and SER of the delay-tolerant transmission mode are presented and derived. The numerical results show that the analytical mathematical expression and the simulation results using Monte Carlo method totally match each other. Moreover, this paper has provided practical insights into the effect of various system parameters on the system performance of the two-way AF relaying network over Rician fading channel. The results could provide the prospective solution for the communication network in the near future.



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*Source: 7693016-2018-04-26.xml*