Hybrid TSR–PSR Alternate Energy Harvesting Relay Network over Rician Fading Channels: Outage Probability and SER Analysis

In this research, we investigate a hybrid time-switching relay (TSR)–power-splitting relay (PSR) alternate energy harvesting (EH) relaying network over the Rician fading channels. For this purpose, the amplify-and-forward (AF) mode is considered for the alternative hybrid time TSR–PSR. The system model consists of one source, one destination and two alternative relays for signal transmission from the source to the destination. In the first step, the exact and asymptotic expressions of the outage probability and the symbol errors ratio (SER) are derived. Then, the influence of all system parameters on the system performance is investigated, and the Monte Carlo simulation verifies all results. Finally, the system performances of TSR–PSR, TSR, and PSR cases are compared in connection with all system parameters.


Introduction
Currently, energy harvesting (EH) from green environmental sources and the conversion of this energy into the electrical energy used to supply communication network devices is considered to be a leading research direction. Furthermore, this solution can provide not only environmentally friendly energy supplies, but also self-maintained, long-lived, and autonomous communication systems. In the series of primary environmentally green energy sources, such as solar, wind, geothermal, and mechanical energy, radio frequency (RF) signals can be considered as a prospective energy source in the future. The RF sources can be used independently from time and location in urban areas and can be produced cheaply in small dimensions, which could be a significant advantage in the manufacturing of small and low-cost communication devices such as sensor nodes. Moreover, RF signals could provisionally fill the role of information transmission or energy harvesting in the sensor nodes. Wireless power transfer and the harvesting of electrical energy from a power source to one or more electrical loads is a well-known technique in communication networks. Thus, this research direction for RF-powered mobile networks has received significant attention during the last decade in wireless sensor networks (WSNs) and cooperative communication systems from both academia Moreover, Figure 2 shows the division of the transmission time. In the first interval time T α with 0 1 α ≤ < , S transfers energy by signal to R1 and R2. After that, S transfers the part of energy interval time, R1 transfers information to D and R2 harvests energy from R1 [14,25,26,29]. In this model, R1 forwards the source information data to D by using its energy harvested in the current T blocks and the previous T blocks. Please note that R1 and R2 always harvest energy from the received RF signals in all of the first T blocks. In the following T blocks, R2 works as a helping relay, while R1 harvests energy in all T blocks by overhearing the transmissions from S and R2. The EH and data relay of R2 are performed similarly to the above procedure for R1. Thus, R1 and R2 will alternately forward source data in every T block. Compared with the TS-EH or PS-EH-based singlerelay system, more energy can be harvested by relays in our protocol for the DT [14,26,29].   Moreover, Figure 2 shows the division of the transmission time. In the first interval time αT with 0 ≤ α < 1, S transfers energy by signal to R 1 and R 2 . After that, S transfers the part of energy ρP s (0 ≤ ρ < 1) to R 1 in the next (1 − α)T/2 interval time and uses (1 − ρ)P s energy to transfer information to R 1 . In the same interval time, R 2 harvests energy from S. In the final (1 − α)T/2 interval time, R 1 transfers information to D and R 2 harvests energy from R 1 [14,25,26,29].
In this model, R 1 forwards the source information data to D by using its energy harvested in the current T blocks and the previous T blocks. Please note that R 1 and R 2 always harvest energy from the received RF signals in all of the first T blocks. In the following T blocks, R 2 works as a helping relay, while R 1 harvests energy in all T blocks by overhearing the transmissions from S and R 2 . The EH and data relay of R 2 are performed similarly to the above procedure for R 1 . Thus, R 1 and R 2 will alternately forward source data in every T block. Compared with the TS-EH or PS-EH-based single-relay system, more energy can be harvested by relays in our protocol for the DT [14,26,29]. Moreover, Figure 2 shows the division of the transmission time. In the first interval time T α with 0 1 α ≤ < , S transfers energy by signal to R1 and R2. After that, S transfers the part of energy interval time, R1 transfers information to D and R2 harvests energy from R1 [14,25,26,29]. In this model, R1 forwards the source information data to D by using its energy harvested in the current T blocks and the previous T blocks. Please note that R1 and R2 always harvest energy from the received RF signals in all of the first T blocks. In the following T blocks, R2 works as a helping relay, while R1 harvests energy in all T blocks by overhearing the transmissions from S and R2. The EH and data relay of R2 are performed similarly to the above procedure for R1. Thus, R1 and R2 will alternately forward source data in every T block. Compared with the TS-EH or PS-EH-based singlerelay system, more energy can be harvested by relays in our protocol for the DT [14,26,29].

System Performance
In this section, we analyze and investigate the energy harvesting and data transmission processes in the two relays in the hybrid TSR-PSR protocol. To increase the understanding of the readers, we show all symbols used in Table 1. Table 1. All symbols used.

Symbol
Definition Time-switching factor 0 ≤ ρ < 1 Power-splitting factor P s /N 0 Source-power-to-noise ratio K Rician K-factor λ 1 Mean of |h 1 | 2 λ 2 Mean of |g 1 | 2 R Source rate E r1 Harvested energy at relay 1 P r1 Average transmit power of relay 1 E r2 Harvested energy at relay 2 P r2 Average transmit power of relay 2 P out Outage probability γ e2e1 End to end signal to noise ratio Hypergeometric function SER Symbol error ratio β Amplifying factor Q(t) Gaussian Q-function P s Transmit power of the source T Total time of processing In the system model, the inter-relay channel is assumed to be symmetric, i.e., h 12 = h 21 . Rician block fading is assumed, so all the channels are circularly-symmetric jointly-Gaussian complex random and denoted as h i ≈ C(0, 1), g i ≈ C(0, 1) and h 12 ≈ C(0, 1), where i ∈ (1, 2).
In the hybrid TSR-PSR alternative relaying, the source provides an energy signal to both R 1 and R 2 in αT and (1 − α)T/2 blocks. In the (1 − α)T/2 block, R 1 allocates 0 ≤ ρ < 1 (ρ is the power splitting factor) as part of the received source signal for the energy harvesting (EH). Therefore, the total harvested energy at R 1 can be given by where (E |h i | 2 = 1 and h i ≈ C(0, 1), g i ≈ C(0, 1), h 12 ≈ C(0, 1)), 0 < η < 1 and 0 ≤ α < 1 are the energy conversion efficiency and time-switching factor, respectively. In this model, the average EH amount by omitting the small-scale channel fading is proposed and considered.
In a similar way, the total harvested energy at R 2 can be given by the equation below: When S provides the data to R 1 , after splitting the ρ part of the received signal for the EH at the relays, the remaining signal at R 1 can be obtained as where n r1 is the additive white Gaussian noise (AWGN) with variance N 0 at R 1 , and E |x s | 2 = P s in which E{•} is expectation operator. Furthermore, R 1 amplifies and forwards the signal to D in the next stage. The transmitted signal from R 1 can be expressed as is the amplifying factor.
Then, the received signal at D can be formulated as the following expression: where n 1d is the additive white Gaussian noise (AWGN) with variance N 0 at D, E |x r1 | 2 = P r1 , and P r1 is the average transmitted power of R 1 .
Replacing (3) and (4) into (5), the received signal at D can be obtained as: In this case, when R 1 performs the delay-tolerant (DT) transmission mode, the end to end signal to noise ratio (SNR) at D can be calculated as After algebra calculation and using the fact that N 0 << P r , the end to end SNR can be obtained: In this T block time, R 2 can harvest energy from S in αT + (1 − α)T/2 blocks, i.e., T 2 (1 + α) blocks, and R 2 can also harvest energy from R 1 in (1 − α)T/2 blocks. Therefore, the total harvested energy at R 2 when R 1 joins in the data transmission (DT) can be calculated by Similar to R 2 , the total harvested energy at R 1 when R 2 joins in the DT can be obtained as where P r2 is the average transmitted power of R 2 . From the EH process at R 1 in the previous T blocks and current T blocks, the total harvested energy of R 1 for DT can be obtained as because of the symmetry property in our proposed system. Similar to R 1 , the total harvested energy of R 2 also can be obtained as the following: From (11), the average transmitted power of R 1 can be calculated as From (12), the average transmitted power of R 2 also can be obtained as Substituting (14) into (13), we obtain where we denote Ψ = (3α−αρ+1)P s +ρ 1−α . Finally, the SNR of the proposed system in (7) can be rewritten as the following: where ϕ 1 = |h 1 | 2 , ϕ 2 = |g 1 | 2 and P r1 is defined by (15).

Exact Outage Probability Analysis
The probability density function (PDF) of random variable (RV) ϕ i (where i = 1, 2) as in [28] is where we denote , in which λ i is the unit mean value of RV ϕ i where i = 1, 2, respectively, because we consider the small-scale power fading |h 1 | 2 , |g 1 | 2 in the derivation. Therefore, a and b can be re-denoted as a = (K + 1)e −K , b = K + 1, K is the Rician K-factor defined as the ratio of the power of the line-of-sight (LOS) component to the scattered components. The cumulative density function (CDF) of RV ϕ i , where i = 1, 2, can be computed as in [17]: Theorem 1 (Exact Outage Probability). The expression of the exact outage probability of the proposed system can be formulated by the following: where K v (•) is the modified Bessel function of the second kind and vth order.
Proof of Theorem 1. See Appendix A.

Asymptotic Outage Probability Analysis
From (16), at the high SNR regime, the end to end SNR can be approximated as Then, the asymptotic outage probability can be formulated as

SER (Symbol Error Ratio) Analysis
In this section, we obtain new expressions for the symbol error ratio (SER) at the destination. We first consider the outage probability, which was obtained in [30,31]. Thus, we obtain ∞ t e −x 2 /2 dx is the Gaussian Q-function, ω and θ are constants which are specific for the modulation type. (φ, θ) = (1, 1) for BPSK and (φ, θ) = (1, 2) for QPSK. For this purpose, the distribution function of γ e2e1 is considered for analyzing the SER performance. Then, Equation (22) can be rewritten directly regarding the outage probability at the source by using integration as follows: Theorem 2 (Exact SER). The exact SER can be calculated by the below expression: where Γ(•) is the gamma function, and F(υ, β; γ; z) is a hypergeometric function.

Proof of Theorem 2. See Appendix B.
Theorem 3 (Asymptotic SER Analysis). The asymptotic SER can be formulated by the below equation: Proof of Theorem 3. See Appendix C.

Numerical Results and Discussion
For validation of the correctness of the derived outage probability and SER expressions, as well as the investigation of the effect of various parameters on the system performance, a set of Monte Carlo simulations are conducted and presented in this section. For each simulation, we first provide the graphs of the outage probability and SER obtained by the analytical formulas. Secondly, we plot the same outage probability and SER curves that result from the Monte Carlo simulation. To do this, we generate 10 6 random samples of each channel gain, which are Rician distributed. The analytical curve and the simulation curve should match to verify the correctness of our analysis. All simulation parameters are listed in Table 2.

Symbol
Values Figure 3 shows the outage probability of the model system versus η in three cases-TSR, PSR, TSR-PSR. In this model, we set P s /N 0 = 10 dB, ρ = 0.5 and α = 0.5. From the results, we see that the outage probability decreases remarkably while η varies from 0 to 1. The research results show that the numerical results and simulation results match exactly with each other, validating the correctness of the theoretical analysis in the above section. Furthermore, the function of the outage probability to K is presented in Figure 4. Similarly, we set P s /N 0 = 10 dB, ρ = 0.5 and α = 0.5, and the outage probability decreases remarkably while K varies from 0 to 4. Once again, the simulation results and theoretical results agree well with each other. Figure 5 plots the numerical and simulation results of the system outage probability in connection with the ratio P s /N 0 . In Figure 5, both the exact and asymptotic outage probability in cases TSR, PSR, and TSR-PSR are illustrated. The main parameters are set as R = 0.5, ρ = 0.3 and α = 0.3. From the results, the exact outage probability decreases and comes close to the asymptotic line when the ratio P s /N 0 increases from 0 to 20 dB. On another hand, the influence of R on the outage probability in three cases-TSR, PSR, TSR-PSR-is investigated in Figure 6 with P s /N 0 = 15 dB, ρ = 0.7 and α = 0.3. The outage probability significantly increases with R from 0 to 4. From Figures 5 and 6, the analytical results and the simulation results match well with each other for all values of R and P s /N 0 . Figure 7 illustrates the numerical and simulation results of the system outage probability concerning α and ρ with P s /N 0 = 10 dB. It is clearly shown that the outage probability increases with increasing α and ρ, and the minimum outage probability can be obtained with α = 0 and ρ = 1. Moreover, SER versus the ratio P s /N 0 in three cases-TSR, PSR, and TSR-PSR-is shown in Figure 8. Furthermore, Figure 9 plots the comparison of the exact and asymptotic outage probability of three cases-TSR, PSR, and TSR-PSR-versus P s /N 0 . The results indicate that all the simulation and analytical values agreed well with each other.              Figure 7 illustrates the numerical and simulation results of the system outage probability concerning α and ρ with Ps/N0 = 10 dB. It is clearly shown that the outage probability increases with increasing α and ρ, and the minimum outage probability can be obtained with α = 0 and ρ = 1. Moreover, SER versus the ratio Ps/N0 in three cases-TSR, PSR, and TSR-PSR-is shown in Figure 8. Furthermore, Figure 9 plots the comparison of the exact and asymptotic outage probability of three cases-TSR, PSR, and TSR-PSR-versus Ps/N0. The results indicate that all the simulation and analytical values agreed well with each other.    Figure 7 illustrates the numerical and simulation results of the system outage probability concerning α and ρ with Ps/N0 = 10 dB. It is clearly shown that the outage probability increases with increasing α and ρ, and the minimum outage probability can be obtained with α = 0 and ρ = 1. Moreover, SER versus the ratio Ps/N0 in three cases-TSR, PSR, and TSR-PSR-is shown in Figure 8. Furthermore, Figure 9 plots the comparison of the exact and asymptotic outage probability of three cases-TSR, PSR, and TSR-PSR-versus Ps/N0. The results indicate that all the simulation and analytical values agreed well with each other.

Conclusions
In this research, the hybrid TSR-PSR alternate EH relay network over AF-based Rician fading channels is presented and investigated. The system model consists of one source, one destination and two alternative relays for signal transmission from the source to the destination. We derive the exact and asymptotic expressions of the outage probability and SER and investigate the influence of all system parameters on the system performance using the Monte Carlo simulation. The research results indicate that the alternative hybrid TSR-PSR has better performance in comparison with the TSR and PSR cases. The research results can provide essential recommendations for communication network research and practice directions.

Conclusions
In this research, the hybrid TSR-PSR alternate EH relay network over AF-based Rician fading channels is presented and investigated. The system model consists of one source, one destination and two alternative relays for signal transmission from the source to the destination. We derive the exact and asymptotic expressions of the outage probability and SER and investigate the influence of all system parameters on the system performance using the Monte Carlo simulation. The research results indicate that the alternative hybrid TSR-PSR has better performance in comparison with the TSR and PSR cases. The research results can provide essential recommendations for communication network research and practice directions.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Proof of Theorem 1-Exact Outage Probability
The outage probability of the model system can be calculated as where γ th = 2 2R − 1 is threshold and R is the source rate. The Equation (A2) can be rewritten as From (18), the outage probability can be formulated as the following Furthermore, from (17) we obtain By changing a variable t = P r1 ϕ 2 − γ th N 0 in to (25), we obtain Now, by applying the equation (x + y) n = n ∑ k=0 n k x n−k y k to (20), the outage probability can be demonstrated as follows: Apply (3.471,9) of [32]:

Appendix C. Proof of Theorem 3-Asymptotic SER Analysis
We obtain We consider We use the table of integral Equation (3.381,4) in [32]: where Γ(•) is the gamma function.