Half-duplex energy harvesting relay network over different fading environment: System performance with effect of hardware impairment

applied sciences Article Half-Duplex Energy Harvesting Relay Network over Different Fading Environment: System Performance with Effect of Hardware Impairment Duy-Hung Ha 1,2 , Si Thien Chau Dong 2,*, Tan N. Nguyen 1,2 , Tran Thanh Trang 3and Miroslav Voznak 1 1Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava 17, Listopadu 15/2172, 708 33 Ostrava-Poruba, Czech Republic; duy[email protected] (D.-H.H.); [email protected] (T.N.N.); miroslav[email protected] (M.V.) 2Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 70000, Vietnam 3Faculty of Engineering and Technology, Van Hien University, 665-667-669 Dien Bien Phu, Ho Chi Minh City 70000, Vietnam; [email protected] *Correspondence: [email protected] Received: 7 May 2019; Accepted: 24 May 2019; Published: 3 June 2019   Abstract: In this paper, we introduce a half-duplex (HD) energy harvesting (EH) relay network over the different fading environment with the effect of hardware impairment (HI). The model system was investigated with the amplify-and-forward (AF) and the power splitting (PS) protocols. The system performance analysis in term of the outage probability (OP), achievable throughput (AT), and bit error rate (BER) were demonstrated with the closed-form expressions. In addition, the power splitting (PS) factor was investigated. We verified the analytical analysis by Monte Carlo simulation with all primary parameters. From the results, we can state that the analytical and simulation results match well with each other. Keywords: amplify-and-forward (AF); relay network; achievable throughput (AT); outage probability (OP); BER; energy harvesting (EH) 1. Introduction Nowadays, wireless powered communication networks (WPCNs) have shown significant advantages in industry and living. The major benefits of WPCNs mainly come from battery charging operations through the air without physical cable connections and recharging, and thus, replacing the battery. As such, the maintenance, servicing, and charging of many battery-powered devices deployed in networks are crucially simplified, especially for future applications and technology [ 1 – 5 ]. Nowadays, there are three primary wireless energy harvesting (EH) and transfer techniques in the two main types of wireless charging—radiative (or RF based) and non-radiative (or coupling based) are employed in practice in WPCNs—which are feasible using the following techniques. In the first method, inductive coupling based on magnetic field induction can be used for transferring electrical energy over distances ranging from a few millimeters to a few centimeters. The efficiency of this method, from around 6% to 90%, is suitable for cell phone charging, contactless smart cards, and passive RFID cards [ 3 , 4 ]. Magnetic Resonant coupling is the second method, which is based on one vane scent wave coupling by making two separate coils resonate at the same frequency. Its efficiency ranges from 30% to around 90% and is suitable for plug-in hybrid electric vehicles and cell phone charging [ 3 , 4 ]. The last method is RF energy transfer, which is suitable for wireless body and wireless sensor networks. From this point of view, RF energy transfer WPCNs are suitable for long-distance transfers [ 1 – 5 ]. Some papers have Appl. Sci. 2019,9, 2283; doi:10.3390/app9112283 www.mdpi.com/journal/applsci Appl. Sci. 2019,9, 2283 2 of 14 presented the process of EH through the RF signals in cooperative wireless networks using a MIMO relay system, and have investigated multi-user and multi-hop systems for simultaneous information and power transfer with a dual-hop channel [ 6 – 18 ]. In these previous papers, the authors have focused on WPCNs using only the Rayleigh or the Rician fading channel. However, to date not many papers concentrate on using both different fading channels. Hardware impairment (HI) suffers from phase noise, I/Q imbalance, and high-power amplifier nonlinearities [ 19 – 22 ]. HI is rarely studied in the literature on relay WPCNs. In a wireless network, the source and destination may not communicate to each other directly, because the distance between the source and destination is greater than the transmission range of them both, hence the need for an intermediate node(s) to relay. Relaying is an effective way to combat the performance degradation caused by fading, shadowing, and path loss. In relay networks, the relay nodes help to boost the information exchange between source nodes and destination nodes, by forwarding (with or without decoding) the information-bearing radio frequency signals from sources to destinations [ 1 – 5 ]. In this paper, we introduce a half-duplex (HD) energy harvesting (EH) relay network over the different fading environment with the effect of hardware impairment (HI). The model system was investigated with the amplify-and-forward (AF) and the power-splitting (PS) protocols. The system performance analysis, in term of the outage probability (OP), achievable throughput (AT), and the bit error rate (BER), was analyzed and demonstrated with the closed-form expressions. Additionally, the power splitting (PS) factor was investigated. We verified the analytical analysis using a Monte Carlo simulation with all primary parameters. The research results showed that the analytical and simulation results matched well with each other. The main contributions are summarized as follow: (1) An HD EH relay network over the different fading environment (Rayleigh and Rician Fading Channel) with the HI effect of HI is introduced and investigated. (2) The closed form of OP, AT, and BER of the proposed system was analyzed and derived in connection with the main primary system parameters. (3) The correctness of the analytical expression was demonstrated by Monte Carlo simulation. The remainder of this paper is introduced as follows. Section 2introduces the system model with EH and information transmission (IT) phases. OP, AT, and BER are derived in Section 3. Section 4 shows and discusses the numerical results. Finally, Section 5provides some conclusions. 2. System Model Network An HD EH relay network over the different fading environment (Rayleigh and Rician Fading Channel), with the effect of HI, is illustrated in Figure 1. This system model is working in AF mode and PS protocol, in which the source (S) and destination (D) can exchange their signal via the helping Relay (R), as shown in Figure 1. The PS protocol of the system model is plotted in Figure 2. Here, Tdenotes the block time for EH and IT processes in the PS protocol. In the first half interval time T/2, the S simultaneously transfers information and energy to the R with the PS factor ρ∈ (0, 1), ρ Pis used for energy harvesting at the R and (1 −ρ )Pis used for transmitting information to the R node. So, the remaining T/2 interval time is used for transferring information from the R to the D [17–20]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 14 suitable for long-distance transfers [1–5]. Some papers have presented the process of EH through the RF signals in cooperative wireless networks using a MIMO relay system, and have investigated multi-user and multi-hop systems for simultaneous information and power transfer with a dual-hop channel [6–18]. In these previous papers, the authors have focused on WPCNs using only the Rayleigh or the Rician fading channel. However, to date not many papers concentrate on using both different fading channels. Hardware impairment (HI) suffers from phase noise, I/Q imbalance, and high-power amplifier nonlinearities [19–22]. HI is rarely studied in the literature on relay WPCNs. In a wireless network, the source and destination may not communicate to each other directly, because the distance between the source and destination is greater than the transmission range of them both, hence the need for an intermediate node(s) to relay. Relaying is an effective way to combat the performance degradation caused by fading, shadowing, and path loss. In relay networks, the relay nodes help to boost the information exchange between source nodes and destination nodes, by forwarding (with or without decoding) the information-bearing radio frequency signals from sources to destinations [1–5]. In this paper, we introduce a half-duplex (HD) energy harvesting (EH) relay network over the different fading environment with the effect of hardware impairment (HI). The model system was investigated with the amplify-and-forward (AF) and the power-splitting (PS) protocols. The system performance analysis, in term of the outage probability (OP), achievable throughput (AT), and the bit error rate (BER), was analyzed and demonstrated with the closed-form expressions. Additionally, the power splitting (PS) factor was investigated. We verified the analytical analysis using a Monte Carlo simulation with all primary parameters. The research results showed that the analytical and simulation results matched well with each other. The main contributions are summarized as follow: (1) An HD EH relay network over the different fading environment (Rayleigh and Rician Fading Channel) with the HI effect of HI is introduced and investigated. (2) The closed form of OP, AT, and BER of the proposed system was analyzed and derived in connection with the main primary system parameters. (3) The correctness of the analytical expression was demonstrated by Monte Carlo simulation. The remainder of this paper is introduced as follows. Section 2 introduces the system model with EH and information transmission (IT) phases. OP, AT, and BER are derived in Section 3. Section 4 shows and discusses the numerical results. Finally, Section 5 concludes the paper. 2. System Model An HD EH relay network over the different fading environment (Rayleigh and Rician Fading Channel), with the effect of HI, is illustrated in Figure 1. This system model is working in AF mode and PS protocol, in which the source (S) and destination (D) can exchange their signal via the helping Relay (R), as shown in Figure 1. The PS protocol of the system model is plotted in Figure 2. Here, T denotes the block time for EH and IT processes in the PS protocol. In the first half interval time T/2, the S simultaneously transfers information and energy to the R with the PS factor ρ ∈ (0, 1), ρ P is used for energy harvesting at the R and (1 − ρ) P is used for transmitting information to the R node. So, the remaining T/2 interval time is used for transferring information from the R to the D [17–20]. Figure 1. System model. SD h R D h Figure 1. System model. Appl. Sci. 2019,9, 2283 3 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 14 Figure 2. Power splitting (PS) protocol. 2.1. Energy Harvesting (EH) The signal is transferred from the S to the R in the first-half T/2 and can be formulated as () rssr yhx n μ =++ (1) The received RF signal at the input of the energy harvesting unit can be calculated as ,hr r s s r yyhxhn ρρρ μ ρ =×= + + (2) In this equation, xs is the energy-transmitted signal with { } 2 s s x PΕ= , nr is the zero-mean additive white Gaussian noise (AWGN) with variance N0, and µs denotes the distortion error caused by hardware impairment at the source node, which is modeled as a zero-mean Gaussian random variable with variance 2 1s P σ with { } 22 1ss P μ σ Ε= . Here E{.} denotes the expectation operation. From Equation (2), the harvested energy at R in the first interval T/2 can be formulated as the following 2(/2) hs EPhT ηρ = (3) Therefore, the transmitted power at R can be calculated as 2 2 (/2) /2 /2 s h rs Ph T E PPh TT ηρ ηρ == = (4) where η is denoted the energy conversion efficiency of the proposed system. 2.2. Information Transmission (IT) In this model, the IT phase is divided into two equal-length subintervals with the length T/2. In the first interval, we can calculate the received signal as 1( ) 1 1 rssrssr yhxnhxhn ρμ ρ ρμ =− + +=− +− + (5) where xr is the transmitted signal, which satisfies { } 2 rr x PΕ= , µr denotes the distortion error caused by hardware impairment at R, which is modeled as a zero-mean Gaussian random variable with variance 2 2r P σ and { } 22 2rr P μ σ Ε= , and nr is the AWGN noise at R node. Here, we use the amplify-and-forward (AF) protocol for our model. Then, the received signal at R is amplified by a factor β, which is given by Equation (6) 22 2 (1 ) (1 ) rr r s so xP yhP hP N βρρσ == −+− + (6) In the remaining T/2 interval time, R transfers the information to D. Hence, the received signal at D node is formulated by Figure 2. Power splitting (PS) protocol. 2.1. Energy Harvesting (EH) The signal is transferred from the S to the R in the first-half T/2 and can be formulated as yr=h(xs+µs) + nr(1) The received RF signal at the input of the energy harvesting unit can be calculated as yh,r=√ρ×yr=√ρhxs+√ρhµs+√ρnr(2) In this equation, x s is the energy-transmitted signal with En|xs|2o=Ps ,n r is the zero-mean additive white Gaussian noise (AWGN) with variance N 0 , and µs denotes the distortion error caused by hardware impairment at the source node, which is modeled as a zero-mean Gaussian random variable with variance Psσ12with Eµs 2=Psσ12. Here E{.} denotes the expectation operation. From Equation (2), the harvested energy at R in the first interval T/2 can be formulated as the following Eh=ηρPs|h|2(T/2)(3) Therefore, the transmitted power at R can be calculated as Pr=Eh T/2=ηρPs|h|2(T/2) T/2=ηρPs|h|2(4) where ηis denoted the energy conversion efficiency of the proposed system. 2.2. Information Transmission (IT) In this model, the IT phase is divided into two equal-length subintervals with the length T/2. In the first interval, we can calculate the received signal as yr=p1−ρh(xs+µs) + nr=p1−ρhxs+p1−ρhµs+nr(5) where x r is the transmitted signal, which satisfies En|xr|2o=Pr , µr denotes the distortion error caused by hardware impairment at R, which is modeled as a zero-mean Gaussian random variable with variance Prσ22and Eµr 2=Prσ22, and nris the AWGN noise at R node. Here, we use the amplify-and-forward (AF) protocol for our model. Then, the received signal at R is amplified by a factor β,which is given by Equation (6) β=xr yr =sPr (1−ρ)|h|2Ps+ (1−ρ)|h|2Psσ2+No (6) In the remaining T/2 interval time, R transfers the information to D. Hence, the received signal at D node is formulated by Appl. Sci. 2019,9, 2283 4 of 14 yd=g(xr+µr) + nD=gxr+gµr+nd =gβyr+gµr+nd =gβ(p1−ρhxs+p1−ρhµs+nr) + gµr+nD =gβp1−ρhxs | {z } signal +p1−ρhµsgβ+gβnr+gµr+nD | {z } noise (7) Here ndis the noise at the destination, which is assumed to have the same power as nr. The end-to-end signal-to-noise ratio (SNR) at D node can be given by γe2e=En(signal)2o En(noise)2o=(1−ρ)g 2β2|h|2Ps g 2β2|h|2Psσ12+g 2β2No+g 2Prσ22+No (8) We denote ϕ1=|h|2,ϕ2=g 2and replacing this in Equation (8), we have γe2e=(1−ρ)ϕ1ϕ2β2Ps ϕ1ϕ2β2Psσ12+ϕ2β2No+ϕ2Prσ22+No (9) γe2e=(1−ρ)ϕ1ϕ2Ps ϕ1ϕ2Psσ12+ϕ2No+ϕ2Prσ22 β2+N0 β2 = ((1−ρ)ϕ1ϕ2Ps)/(ϕ1ϕ2Psσ12+ϕ2No+ (1−ρ)ϕ1ϕ2Psσ2 2+ (1−ρ)ϕ1ϕ2Psσ2 1σ2 2+ϕ2N0σ2 2+N0(1−ρ) ηρ +N0(1−ρ)σ2 1 ηρ +N2 0 Pr) (10) Because N0<< Pr, then we can reformulate Equation (10) as the following equation γe2e=(1−ρ)ϕ1ϕ2Ps/nϕ1ϕ2Psσ12 +ϕ2No+ (1−ρ)ϕ1ϕ2Psσ2 2+ (1−ρ)ϕ1ϕ2Psσ2 1σ2 2 +ϕ2N0σ2 2+N0(1−ρ) ηρ +N0(1−ρ)σ2 1 ηρ (11) γe2e=(1−ρ)ϕ1ϕ2γ0/nϕ1ϕ2γ0σ12 +ϕ2+ (1−ρ)ϕ1ϕ2γ0σ2 2 +(1−ρ)ϕ1ϕ2γ0σ2 1σ2 2+ϕ2σ2 2+κ+κσ2 1o(12) where we denote κ=1−ρ ηρ ,γ0=Ps N0. In the next section, we analyze the achievable throughput (AT), outage probability (OP) and BER in AF mode with the PS protocol [6–8]. 3. System Model Performance In this section, we investigate the system performance of the relay network with the PS protocol [10–13,23–27] . In this analysis, we consider two Scenarios: (1) S-R link is the Rayleigh Fading Channel and R-D link is Rician Fading Channel, and (2) S-R link is the Rician Fading Channel and R-D link is the Rayleigh Fading Channel. 3.1. Scenario 1: S-R link Is Rayleigh Fading Channel, R-D link Is Rician Fading Channel As in previous studies [ 6 – 9 ], the probability density function (PDF) of a random variable (RV) ϕ1 can be written as the following equation fϕ1(x) = λhe−λhx(13) Here λhis the mean value of RV ϕ1. Appl. Sci. 2019,9, 2283 5 of 14 The cumulative density function (CDF) of RV ϕ1can be written as Fϕ1(x) = 1−e−λhx(14) Similarly, the PDF of RV ϕ2can be obtained as in [26], giving fϕ2(x) = (K+1)e−K λg e−(K+1)x λgI0 2sK(K+1)x λg (15) where λg is the mean value of RV ϕ2 , K denotes the Rician K-factor, and I0(•) is the zero-th order modified Bessel function of the first kind [26]. Then the Equation (14) can be reformulated as the following fϕ2(x) = a∞ X l=0 (bK)l (l!)2xle−bx (16) where we denote a=(K+1)e−K λg,b=K+1 λgand I0(x) = ∞ P l=0 x2l 22l(l!)2[24]. The cumulative density function (CDF) of RV ϕ2can be computed like in [27] Fϕ2(ς) = ς Z 0 fϕ2(x)dx =1−a b ∞ X l=0 l X m=0 Klbm l!m!ςme−bς(17) After that, the OP of the proposed system can be computed as Pout =Fγe2e(γ) = Pr(γe2e< γ)(18) If we denote γ= 2 R− 1 to be the lower threshold for SNR at both R and D, and R is fixed transmission rate at S, then Equation (18) can be reformulated as the following Pout =Pr(ϕ1ϕ2γ0[1−ρ−γσ2 1−γ(1−ρ)σ2 2 −γ(1−ρ)σ2 1σ2 2]< ϕ2(γ+γσ2 2) + γκ +γκσ2 1)(19) Here we denote c1=γ+γσ2 2 , c2=γκ +γκσ2 1 , and c3=γ0[ 1 −ρ−γσ2 1−γ( 1 −ρ)σ2 2−γ( 1 −ρ)σ2 1σ2 2] . We assume that c3is positive, because if c3is negative, the OP of the system is always equal to 1. a. Outage Probability (OP) Pout =Pr(ϕ1<c1ϕ2+c2 c3ϕ2)= ∞ Z 0 Fϕ1 c1ϕ2+c2 c3ϕ2!fϕ2(ϕ2)dϕ2(20) Combining Equation (20) with Equations (14) and (17) we have Pout =1− ∞ Z 0 e−λh c1ϕ2+c2 c3ϕ2×a∞ X l=0 (bK)l (l!)2ϕl 2e−bϕ2dϕ2=1−ae−λhc1 c3∞ X l=0 (bK)l (l!)2 ∞ Z 0 ϕl 2e−bϕ2e−λhc2 c3ϕ2dϕ2(21) Using Table of Integral Equation [3.471,9] in [24], Equation (21) can be given as Pout =1−2ae−λhc1 c3∞ X l=0 (bK)l (l!)2 λhc2 c3b!l+1 2 ×Kl+12rλhc2b c3(22) Appl. Sci. 2019,9, 2283 6 of 14 where Kv(•)is the modified Bessel function of the second kind and vth order. b. Achievable Throughput (AT) Here, the average throughput of the relay network system can be computed regarding the OP as in Equation (23) τ= (1−Pout)R 2=2ae−λhc1 c3∞ X l=0 (bK)l (l!)2 λhc2 c3b!l+1 2 ×Kl+12rλhc2b c3×R 2(23) c. The Bit Error Rate (BER) The BER of the proposed system can be formulated from the expression of the OP as the following equation BER =EhωQ(p2θγi(24) where Q(t) = 1 √2π ∞ Rt e−x2/2dx is the Gaussian Q-function, ω and θ are constants which are specific for modulation type. Here, we use (ω , θ) = ( 1, 2 ) for BPSK and (ω , θ) = ( 1, 1 ) for QPSK. Hence, we begin rewriting the BER expression in Equation (24) directly regarding OP at S by using integration as in the equation below BER =ω√θ 2√π ∞ Z 0 e−θx √xFγe2e(x)dx (25) 3.2. Scenario 2: S-R Link Is the Rician Fading Channel, R-D Link Is the Rayleigh Fading Channel Similar to scenario 1, the CDF of RV ϕ1and PDF of RV ϕ2can be formulated as Fϕ1(ς) = ς Z 0 fϕ1(x)dx =1−a b ∞ X l=0 l X m=0 Klbm l!m!ςme−bς(26) where a=(K+1)e−K λh,b=K+1 λh fϕ2(x) = λge−λgx(27) a. Outage Probability (OP) Pout =Pr(ϕ1<c1ϕ2+c2 c3ϕ2)= ∞ Z 0 Fϕ1 c1ϕ2+c2 c3ϕ2!fϕ2(ϕ2)dϕ2(28) Pout =1− ∞ Z 0 a b ∞ P l=0 l P m=0 Klbm l!m!c1ϕ2+c2 c3ϕ2m e−b(c1ϕ2+c2) c3ϕ2λge−λgϕ2dϕ2 (29) We apply the equation (x+y)m=m P n=0 m n!xm−nyn , and change the variable by setting, and Equation (27) can be reformulated as the following Pout =1−λge−bc1 c3 ∞ Z 0 a b ∞ P l=0 l P m=0 m P n=0 m n!Klbmcm−n 1cn 2 l!m!cm 3tn−2 e−bc2t c3e−λg tdt (30) Appl. Sci. 2019,9, 2283 7 of 14 Pout =1−λge−bc1 c3a b ∞ X l=0 l X m=0 m X n=0 Klbmcm−n 1cn 2 l!n!(m−n)!cm 3 ∞ Z 0 tn−2e−bc2t c3e−λg tdt (31) Using the Table of Integral Equation [3.471,9] in [24], the above equation can be given as Pout =1−2λge−bc1 c3a b ∞ X l=0 l X m=0 m X n=0 Klbmcm−n 1cn 2 l!n!(m−n)!cm 3 λgc3 bc2!n−1 2 Kn−1 2sbc2λg c3 (32) Pout =1−2ae−bc1 c3∞ X l=0 l X m=0 m X n=0 Klb2m−n−3 2λ n+1 2 gcm−n 1c n+1 2 2c n−1−2m 2 3 l!n!(m−n)!×Kn−1 2sbc2λg c3 (33) where Kv(•)is the modified Bessel function of the second kind and vth order. b. Achievable Throughput (AT): τ= (1−Pout)R 2=2ae−bc1 c3∞ X l=0 l X m=0 m X n=0 Klb2m−n−3 2λ n+1 2 gcm−n 1c n+1 2 2c n−1−2m 2 3 l!n!(m−n)!×Kn−1 2sbc2λg c3×R 2(34) c. The Bit Error Rate (BER) Similar to scenario 1, BER can be calculated with the equation below BER =ω√θ 2√π ∞ Z 0 e−θx √xFγe2e(x)dx (35) 3.3. Optimal Power-Splitting (PS) Factor In this section, we can calculate the optimal value ρ∗ by solving the equation dτ(p) dρ= 0, using the AT expression in Equations (23) and (34). Here, we use the Golden section search algorithm as in [25,28–30], which is popularly used in many global optimization problems in communications. 4. Results and Discussion In this section, we investigate the system performance in terms of OP, AT, BER, and the PS factor in connection with the main system parameters: η , ρ ,P s /N 0 and σ1 , σ2 . The primary system simulation parameters are listed in Table 1. Table 1. The main simulation parameters. Symbol Name Values ηEnergy harvesting efficiency 0.7 λhMean of |h|20.5 λgMean of g 20.5 K Rician K-factor 3 γth SNR threshold 7 Ps/N0Source power-to-noise ratio 0–30 dB σ1Distortion error 0.01 σ2Distortion error 0.05 R Source rate 3 bit/s/Hz Appl. Sci. 2019,9, 2283 8 of 14 Figures 3and 4plot the OP and AT versus the energy conversion efficiency η . The effect of η was investigated in both scenarios, and we vary η continuously from 0 to 1. From the figures, we can see that the OP decreased and the AT increased crucially with η varying from 0 to 1; and the OP and AT in the second case is better than in the first case. Furthermore, the results show the correctness of the simulation and analytical expressions. Moreover, Figures 5and 6plot the impact of the ratio P s /N 0 on the OP and AT, while the ratio P s /N 0 increases from 0 to 30 dB. From the research results, we can state that the OP decreased and AT increased with the rising of the ratio P s /N 0 , and the analytical results match very well with the analytical values. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14 Figure 3. Outage probability versus η. Figure 4. The throughput versus η. Figure 5. The outage probability versus ratio of P s /N 0 . Figure 3. Outage probability versus η. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14 Figure 3. Outage probability versus η. Figure 4. The throughput versus η. Figure 5. The outage probability versus ratio of P s /N 0 . Figure 4. The throughput versus η. Appl. Sci. 2019,9, 2283 9 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14 Figure 3. Outage probability versus η. Figure 4. The throughput versus η. Figure 5. The outage probability versus ratio of P s /N 0 . Figure 5. The outage probability versus ratio of Ps/N0. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 14 Figure 6. The throughput versus ratio P s /N 0 . In addition, the OP and AT versus the PS factor ρ are shown in Figures 7 and 8. It can be observed that the AT increased and OP fell at the D with factor ρ from 0 to 0.6. After that, the OP and AT had the opposite effect when ratio ρ from 0.6 to 1.0. We can find the optimal value of factor ρ from 0.6 to 0.7 in this situation. It can be observed that when ρ is too small, R cannot harvest enough energy from S to operate reliably, but when ρ is too large then the reliability of the communications link from S to R is impaired. Furthermore, the OP and AT of the model system versus σ 1 = σ 2 varied from 0 to 0.2 and are plotted in Figures 9 and 10. In the same way, OP increased and AT decreased with the increasing of σ 1 = σ 2 . In all the above figures, the simulation and analytical results agree well with each other. Figure 7. The outage probability versus ρ for the PS protocol. Figure 6. The throughput versus ratio Ps/N0. In addition, the OP and AT versus the PS factor ρ are shown in Figures 7and 8. It can be observed that the AT increased and OP fell at the D with factor ρ from 0 to 0.6. After that, the OP and AT had the opposite effect when ratio ρ from 0.6 to 1.0. We can find the optimal value of factor ρ from 0.6 to 0.7 in this situation. It can be observed that when ρ is too small, R cannot harvest enough energy from S to operate reliably, but when ρ is too large then the reliability of the communications link from S to R is impaired. Furthermore, the OP and AT of the model system versus σ1 = σ2 varied from 0 to 0.2 and are plotted in Figures 9and 10. In the same way, OP increased and AT decreased with the increasing of σ1 =σ2. In all the above figures, the simulation and analytical results agree well with each other.