Outage Probability Analysis of Decode-and-Forward Two-Way Relaying System with Energy Harvesting Relay

In this paper, we evaluate a two-way relay system consisting of two terminals and an intermediate relay. In this model, two terminals do not have a direct link but exchange data with each other via the relay in three phases. The relay utilizes energy harvesting technology to collect energy from the received signals of two terminals in the first two phases and then uses the obtained energy for signal transmission in the third phase. Each node is equipped with a single antenna and operates under a half-duplex mode. All wireless channels are influenced by reversible independent flat Rayleigh fading. Using analytical methods, we provide the exact and approximate closed-form expressions of user outage probability and system outage probability. The approximate expressions of these outage probabilities are more explicit and straightforward, providing a better understanding of the influences of network parameters on the system quality. Monte-Carlo simulations are used to confirm the correctness of mathematical analyses.


Introduction
Nowadays, energy harvesting (EH) technology has become a hot research trend and attracted increasing interest from many research groups around the world [1][2][3]. It is a research trend towards green information in which network nodes can harvest the energy from different sources, such as solar, wind, vibration, thermoelectric effects, and other physical phenomena [4][5][6]. Based on the fact that radio frequency (RF) signals can carry energy and information at the same time [7], a new emerging solution for a wireless network is to avail ambient RF signals, in which wireless nodes can harvest energy and process the information simultaneously. Therefore, with EH technology, network nodes can collect energy from received signals in the radio band and use it for next operations [8,9]. This operation not only helps to extend the operation life time of wireless nodes but also reduces their battery usage, resulting in a reduction of toxic wastage. Generally, based on the operation of receiver, EH technology is divided into two main techniques: (i) time switching (TS), where the received signal is divided into two time phases to harvest energy and decode the signal, and (ii) power splitting (PS), where the received signal power is split into two parts for energy harvesting and signal decoding [9]. Based on receiver architecture, EH systems are divided into two main categories: (i) harvest-use, where all harvested energy is used immediately, and (ii) harveststore-use, where harvested energy can be stored for the next operation [10]. For the receiver applying the TS scheme, its antenna is successively connected to the energy harvesting block and the signal processing block. The timing mechanism controls the signals coming to these blocks. On the contrary, for the receiver using the PS scheme, its antenna is connected to both the energy harvesting block and data processing block; thus, the received data are shared between them. Therefore, the EH phase and the information processing phase may concurrently occur in a given time slot, which shortens the transmission cycle. However, from the receiver's complexity perspective, the TS scheme is superior to the PS scheme because the commercially available circuits information in two time phases. The system performance in terms of outage probability and average throughput was analyzed. However, because the SNR expression was too complicated, they could not give closed-form expressions of these performance metrics. In [24], the authors consider relay beamforming and power-domain nonorthogonal multiple access for a wireless-powered multipair two-way relay network. Their objective was to optimize the energy transfer beamforming matrix by maximizing the minimum of the achievable rates among all the users. Applying a piecewise linear energy harvesting model for the users and the relay in dual-hop wireless powered two-way communication, the authors in [25] investigated the problem of total throughput maximization of both AF and DF relaying models. They concluded that it is essential to study a realistic EH model as the impractical linear EH model causes tremendous performance loss.
Motivated by the above issues, in this paper, we will derive the exact and approximate closed-form expression of overall system outage probability using the Taylor series expansion and Gaussian-Chebyshev quadrature approach. The approximation expressions are presented in a more concise form, allowing us to see the effects of key parameters such as the average transmission power, the channel gain, the time allocation of signal phase, and the power allocation coefficient on the system outage performance more easily. Particularly in this paper, the time allocation ratio of the signal phase and the power allocation coefficient are the distinguished parameters of the considered two-way DF relay system with EH relay; thus, it is necessary to clarify these parameters' influences on the outage performance of the considered system. Moreover, using the Taylor series expansion and Gaussian-Chebyshev quadrature approach allows us to adjust necessary precision by changing the number of terms in finite sum.
The rest of this paper is organized as follows. Section 2 outlines the proposed system model. The outage probability of this system model is studied in Section 3. Section 4 presents numerical results. Section 5 concludes the main findings of this paper.

System Model
In this paper, we consider a two-way relay system as shown in Figure 1. This system comprises two terminals S 1 and S 2 exchanging their data via an intermediate relay node R. The relay R is assumed to have a limited power and therefore has to harvest the energy from the radio frequency signals of two terminals to forward the information of these two terminals by using the DF three-phase two-way relay protocol. The motivation for using the DF relay are as follows: (i) the DF relay is found to be of more practical interest; (ii) compared with the AF relaying strategy, DF relaying avoids noise amplification and can be easily combined with coding technologies. All wireless channels are assumed to be reciprocal and undergo independent flat Rayleigh fading.
The three-phase two-way relay protocol can be described as follows. A cycle T for signal transmission between two terminals is divided into three phases. In the first phase t 1 , after 2 Wireless Communications and Mobile Computing R receives signals from S 1 , a power divider is used to divide the received power into two parts: one part is used to process signal and another part is used to convert energy. In the second phase t 2 , S 2 transmits its signal to R. Then, R divides the power of the received signal similarly to the first phase. It is assumed that t 1 = t 2 = ρT, where 0 < ρ < 0:5 is the time allocation ratio of the signal phase. In the third phase t 3 = ð1 − 2ρÞT, the relay broadcasts the reencoding signals to two terminals. We assume that the global channel state information (CSI) and partial CSI for the relay and terminals can be acquired perfectly. Specifically, for the relay R, the CSI of h 1 and h 2 have to be acquired to correctly decode the signal from two users simultaneously. For the terminal S 1 (S 2 ), the CSI of h 1 (h 2 ) have to be acquired to correctly decode the desired signals. In our considered relay system, the channel coefficients h 1 and h 2 can be obtained as follows. Terminal S 1 broadcasts a ready-to-send (RTS) message before information transmission. After receiving the RTS message from S 1 , terminal S 2 replies with a clear-to-send (CTS) message. Then, the relay R can estimate the channel coefficients of both h 1 and h 2 by overhearing the RTS and CTS messages. Finally, terminals S 1 and S 2 are informed of the corresponding channel coefficients through the feedbacks from the relay R [26].
Denote h i , i ∈ f1, 2g, as the channel coefficients between S i and R. When S i transmits with power P, the received signal at R in the ith phase is where n i is an additive white Gaussian noise at the relay in the ith phase time, x i is the transmitted signal symbol from S i , and Efx 2 i g = 1. In the first two phases, R uses the divider to divide the received signal power into two parts: ffiffi ffi δ p y i R to harvest energy and ffiffiffiffiffiffiffiffiffi ffi 1 − δ p y i R to signal processing, where δ is the power allocation coefficient, 0 < δ < 1. The received power at R from S i is given by where η is the energy conversion efficiency, 0 < η < 1. Consequently, we have the total energy which the relay collects in two phases as It is worth noticing that the receiver consumes a certain amount of energy for CSI acquisition and circuitry. However, in our considered relay system, we assume the power required for CSI acquisition and circuitry is not supplied by the harvester energy but from an independent battery [10]. Thus, when all energy harvested in two phases is used for the signal transmission power of R in the third phase, the transmission power of R in the third phase is According to the DF relaying protocol, R receives signals from S 1 and S 2 in two phases and decodes the received signals y 1 R and y 2 R into symbols x 1 and x 2 , respectively. Then, R encodes these two decoded symbols by applying XOR operation and obtains the normalized symbol x R , i.e., where Efx 2 R g = 1 and ⊕ is the bitwise XOR operation. In the third phase, R broadcast x R to S 1 and S 2 . Terminal S i receives this symbol and decodes it by using XOR operation with its transmitted symbol. Consequently, the received Relay R forwards information to S 1 and S 2 Figure 1: System model of the two-way DF relay system with EH relay. 3 Wireless Communications and Mobile Computing signal at S i in the third phase is given by where n S i is the AWGN at S i . Since the noise power at all nodes in the system is equal to N 0 , the signal-to-noise ratios (SNRs) at R in the first and second phases are, respectively, calculated as In the third phase, the SNR of the received signal at S i , i ∈ f1, 2g, can be calculated as

User Outage Probability (UOP)
3.1.1. Exact Expression. In the three-phase two-way relay protocol, the user outage probability of terminal S i is the probability that S i cannot successfully decode the intended received signal from S j . In other words, the user outage probability of S i is the probability that the SNR of the received signal at S i is less than a threshold γ th , i.e., where γ ji = min ðγ j , γ 3i Þ with i, j ∈ f1, 2g, i ≠ j and γ th = 2 3R − 1; R is the desired data transfer rate of S i . We can rewrite (9) as For clarity, we set X = jh i j 2 , Y = jh j j 2 , ϕ = ηδρ/ð1 − 2ρÞ. Then, X and Y are exponential distributed random variables with mean parameters λ x and λ y , respectively. Hence, the UOP of S i is computed as More specifically, the user outage probability at the terminal node S i is defined in the following Theorem 1.

Theorem 1.
The user outage probability of S i in the considered two-way DF communication system with EH relay is given by where where x v = ððb − aÞ/2Þy v + ðb + aÞ/2, y v = cos ððð2v − 1Þ/2VÞ πÞ, and ω V = π/V. Then, the approximate value of I 1a in (A.5) is where y l = cos ððð2l − 1Þ/2LÞπÞ, ω L = π/L, x l = x 0 /2ðy l + 1Þ, and L is a parameter that determines the tradeoff between complexity and accuracy. Finally, the approximate expression of the UOP of S i is determined in the following corollary.

Corollary 2. The approximate expression of the UOP of S i is given by
Thanks to a more straightforward form than in (12), we can easily see the impacts of system parameters on each terminal's outage probability.

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System Outage Probability (SOP)
3.2.1. Exact Expression. The outage probability of a two-way relay system is the probability that the SNR in at least one phase is lower than a given threshold γ th .
Denote γ e = min ðγ 3i, γ 3j Þ, the system outage probability is calculated as where Then, the system outage probability of the considered system is defined in the following Theorem 3.

Theorem 3.
The exact closed-form expression of the system outage probability of the considered two-way DF relay system with EH relay is given by where Proof. See Appendix B.

Approximate Expression in High SNR Regime.
Recalling the Gaussian-Chebyshev quadrature approach in (13), the approximate value of I 2a in (B.7) is as where x k = ð1/2Þy k ðx 1 − ðγ th N 0 /Pð1 − δÞÞÞ + ð1/2Þðx 1 + ðγ th N 0 /Pð1 − δÞÞÞ, y k = cos ððð2k − 1Þ/2KÞπÞ, ω K = π/K with K is a parameter that determines the tradeoff between complexity and accuracy. Applying a similar method for I 3 , we obtain an approximate expression of the SOP in the following corollary.

Corollary 4. The approximate expression of the SOP is given by
It is noted that higher K results in a smaller difference in the approximate SOP expression. However, the value of K cannot be arbitrarily large because of the computational complexity. Fortunately, for our considered system model, even a small value of K ensures that the approximate expression well fits the exact expression, as demonstrated in the next section.
Remark 5. In this paper, the two-way relay system with one relay node is considered. In the case of a two-way multiple relay system, we must first model the relay selection scheme mathematically to find the outage probability (OP) expression. Then, the OP expression will contain the components representing the relay selection algorithm. In the case of a two-way relay system where the relay is equipped with multiple antennas, the channels between the relay and two terminals are random matrices. Furthermore, in the third phase time, the relay may employ a beamforming or transmit antenna selection (TAS) technique to transmit its signals. Thus, the OP expression will contain the components representing the beamforming vector at the relay or TAS algorithm. To sum up, it is challenging to find the OP expressions in these two scenarios.

Numerical Results
In this section, we use the exact and approximation expressions of the UOP and SOP obtained in the previous section to evaluate the two-way DF system's outage performance with the EH relay. Various scenarios are carried out to reveal

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h 2 which are random variables with Rayleigh distribution with mean λ x and λ x , respectively. Then, we compute the SNR at the receivers. If this SNR is less than the predetermined threshold γ th , then the outage happens. Such a process is repeated for a number of iterations of 10 6 . The outage probability is determined as the ratio of the number of times the outage events occur to the number of samples. It is assumed that network nodes are located on a 2D plane. The distance between the two terminals S 1 and S 2 is normalized to 1. Specifically, the locations of all nodes are S 1 (0,0), S 2 (1,0), and R(0.4,0). We denote d 1 and d 2 as the physical distances from R to S 1 and from R to S 2 , respectively. For freespace path-loss transmission, we have λ x = d −β , where β, 2 ≤ β ≤ 6, is the path loss exponent. Unless otherwise stated, we set the following parameters β = 3, γ th = 7, η = 0:8, δ = 0:6, N 0 = 1, N = 5, and K = L = 5. Figure 2 plots the exact and approximate analysis results and simulation results of the UOP of S 1 versus the average SNR P/N 0 for ρ = 0:2, 0.3, and 0.4. We can see that the analysis results are in excellent agreement with the simulation ones, proving the correctness of the derived mathematical expressions. Moreover, the approximate results obtained by using the Gaussian-Chebyshev quadrature approach as in (15) are very close to the exact results. In addition, we observe that as ρ increases, the UOP of S 1 is reduced. It is because the increased ρ makes the transmission power in the third phase higher, thus improving the probability of successfully decoding signal at S 1 .
All theoretical analyses in this paper are based on the Taylor approximation method for exponential function such as in (A.4). For this approximation method, the accurateness can be enhanced by using more terms in the Taylor series. However, it may affect the computation time. To study the accurateness matter, we conduct an evaluation of the UOP of S 1 for a different number of terms in Taylor series expansion, i.e., N = 1, N = 3, and N = 5, and give the results in Figure 3. We can see that a small number of terms in Taylor series expansion can provide a good match between the analysis results and simulation results. Particularly, even when N = 1, the analysis results are very similar to the simulation results. It means that Taylor series expansion is an efficient approximation method in this paper. Figure 4 presents the system outage probability versus the average SNR for three cases: (1) ρ = 0:2, δ = 0:5; (2) ρ = 0:3, δ = 0:7; and (3) ρ = 0:4, δ = 0:8. It can be seen from Figure 4 that the simulation results and analysis results are coincident, confirming the correctness of the analysis steps in this paper. We can observe that when increasing ρ and δ, the SOP decreases. It is because as ρ and δ gets higher, the harvested energy of R in two phases increases. Consequently, its transmission power in the third phase increases, resulting in higher SNR. Figure 5 shows the SOP versus the time allocation ratio of signal phase ρ for three cases of the transmission power, i.e., P = 20, 25, and 30 dB. As observed in Figure 5, as ρ increases, the SOP decreases, indicating that the communication quality between S 1 and S 2 is better.
To further investigate the influence of the power-splitting ratio δ on the system performance, we evaluate the SOP as δ ranges from 0 to 1 and depict the result in Figure 6. We can see that based on the derived mathematical expression, the optimal δ at which the system outage probability is smallest can be determined. Specifically, the optimal value of δ in Figure 6 corresponding to P = 20 dB, P = 25 dB, and P = 30 dB are 0.34, 0.4, and 0.45, respectively.

Conclusion
In this paper, we have derived the exact and approximate closed-form expressions of the user outage probability and the system outage probability of a two-way DF relay system where the relay is not powered by a separated power supply but harvests the energy from the received radio frequency signals in two phases to convert it into the transmission power in the third phase. The approximate expressions of these two kinds of outage probabilities are in explicit and simplified forms, providing a better understanding of the effects of parameters on the quality of the system. Moreover, the accuracy of the approximate expressions can be adjusted by changing the number of terms in Taylor series expansion and the coefficients in the Gaussian-Chebyshev quadrature approach. All analysis results are verified by Monte-Carlo simulation results. Numerical results show that when the time allocation of signal phase ρ increases, the system outage probability decreases. Furthermore, using numerical results can determine the optimal power-splitting ratio δ at which the system outage probability is smallest. The exact and approximate closed-form expressions of the UOP and SOP in this paper provide a solid foundation for analyzing the performance of two-way relay systems with EH relay under other performance metrics such as ergodic capacity, energy efficiency, and average symbol error probability.
we have where N is the number of truncated terms in the series expansion.

B. Proof of Theorem 3
First, we find a closed-form expression of I 2 . Substituting (6), (7), and (8) into (17), we have Considering the case when two terminals have transmission power satisfying the condition P < 2ϕγ th N 0 /ð1 − δÞ 2 , the integral domain for I 2 in the low power region, I low 2 , is presented in Figure 8, where y 2 = ðγ th N 0 /ϕPxÞ − x and y 3 = x, and x 1 is the root of the equation y 2 = y 3 .
Based on Figure 8, we have where f X,Y ðx, yÞ is the joint probability density function of two random variables X and Y given in (A.2). Substituting (A.2) into (B.2), we can calculate I low 2 as Similarly, we can also calculate I 3 in the low transmission power region, I low 3 , as Plugging (B.3) and (B.2) into (16), we obtain the SOP in the low SNR region as in (19).
B.2. Case 2: The Transmission Power P > 2ϕγ th N 0 /ð1 − δÞ 2 . Considering the case when two terminals have transmission power satisfying the condition P > 2ϕγ th N 0 /ð1 − δÞ 2 , from (B1), we split the integral domain of I 2 in the high SNR region into two small regions I 2a and I 2b as in Figure 9.

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Wireless Communications and Mobile Computing Based on Figure 9, I 2 can be computed as ðB:5Þ The first part of (B.5), I 2a , can be calculated as ðB:6Þ Recalling the Taylor series expansion in (A.4), we can rewrite (B.6) as x n e − γ th N 0 /ϕPλ y x ð Þ dx x n e − γ th N 0 /ϕPλ y x ð Þ dx: ðB:7Þ Applying [27] (Eq. (3.471.2)), we obtain The second part of (B.5), I 2b , can be calculated as x 0 y y 2 x 1 th N 0 P (1 -) Figure 9: The integral calculation domain for I 2 in the high power region. 11 Wireless Communications and Mobile Computing By using the above analysis steps for I 3 , the closed-form expression of I 3 is given by ðB:11Þ Then, substituting (B.10) and (B.11) into (16), we obtain the closed-form expression of the SOP in the high SNR region as in (20).

Data Availability
Data are available on request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.