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

Abstract

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.

1. 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 and industry [1,2,3,4,5,6,7,8,9]. Furthermore, EH wireless communication can substantially prolong the network lifetime for wireless sensor networks with low power nodes, and RF signals can recharge nodes more controllably. Wireless nodes can harvest RF energy either in the time domain before data reception, or in the power domain by dividing the received RF signals for the EH and information decoding (ID) [9,10,11,12]. In a cooperative network, references [13,14,15,16] developed two new relaying protocols based on the receiver structures adopted at R, termed time-switching-based relaying (TSR) and power-splitting-based relaying (PSR). From [14,15,16], the TSR and PSR protocols have some drawbacks; for instance, TSR loses information in the switching process to the harvesting mode, and PSR has a low coverage area. Furthermore, PSR requires a complicated hardware structure to make sure that a proper portion of energy from the source signal is extracted for energy harvesting. In contrast, TSR can simplify the hardware at the expense of the throughput or achievable rate of the system. Based on the fact that both TSR and PSR protocols have their drawbacks, the proposed method combines these two protocols to get the best out of them. This solution is obtained in this paper by using an adaptive relaying protocol [17,18]. From the other point of view, the authors highlighted the problem of maximizing the throughput in connection with power allocation and the degree of channel state information (CSI) in [19], and secrecy systems operating over spatially correlated composite fading channels are presented in [20]. Also, the performance of dynamic relays in different types of cellular networks under the presence of inter-cell interference (ICI) is investigated in [21], and a novel approach for maintaining the gains of relaying and keeping the signaling overheads at a low level is proposed in [22]. Furthermore, maximizing harvested power in MIMO SWIPT systems with PSR is investigated in [23], the joint design of spatial channel assignment and power allocation in MIMO systems is studied in [24], and ref. [25] considers a dual-hop cognitive inter-vehicular relay-assisted communication system by the double Rayleigh fading distribution.

The main objective of this paper is to provide a system performance analysis (in terms of the outage probability and symbol error ratio (SER)) of hybrid TSR–PSR alternate energy harvesting relaying networks over Rician fading channels. In the analysis process, we analyze and derive the exact and asymptotic expressions of the outage probability and SER. After that, the Monte Carlo simulation is used to validate the analytical analysis in connection with all system parameters. The main contributions of this research are as follows:

(1) We propose and investigate the alternative hybrid TSR–PSR energy harvesting relaying networks over Rician fading channels. In this model, two relays R1 and R2 are alternatively used for energy harvesting and data transmission process from S to D;

(2) The exact and asymptotic expressions of the outage probability and SER are proposed, analyzed and derived. Moreover, the comparison of the exact and asymptotic expressions of the outage probability and SER in three cases—TSR, PSR, and TSR–PSR—is demonstrated;

(3) The influence of all system parameters on the outage probability and SER is investigated and discussed;

(4) The Monte Carlo simulation is used to verify all the research results.

The rest of this manuscript is organized as follows. In Section 2, we present the proposed system model. Section 3 investigates the exact and asymptotic analysis of the outage probability and SER. Numerical results and a discussion are given in Section 4. Section 5 concludes this manuscript.

2. System Model

Consider an AF relaying system with four nodes: the source S, two relays R₁, R₂ and the destination D as shown in Figure 1. Each device works with a single antenna and in a half-duplex (HD) mode, and there is no direct link between R and D. In this model, S, and D have their own stable power supplies, while R1 and R2 operate with EH and alternately forward source data according to the AF protocol. We denote that h₁ and h₂ are the fading channel gains from the source to relays, g₁ and g₂ are the fading channel gain from the relays R₁ and R₂ to the destination D, and h₁₂ and h₂₁ are the gain factors between R₁ and R₂, respectively [26,27,28,29].

Moreover, Figure 2 shows the division of the transmission time. In the first interval time $\alpha T$ with $0\le \alpha <1$, S transfers energy by signal to R₁ and R₂. After that, S transfers the part of energy $\rho P_s$ ($0\le \rho <1$) to R₁ in the next $(1-\alpha )T/2$ interval time and uses $(1-\rho )P_s$ energy to transfer information to R₁. In the same interval time, R₂ harvests energy from S. In the final $(1-\alpha )T/2$ 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 single-relay system, more energy can be harvested by relays in our protocol for the DT [14,26,29].

3. 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. All symbols used.

Symbol	Definition
$0<\eta <1$	Energy conversion efficiency
$0\le \alpha <1$	Time-switching factor
$0\le \rho <1$	Power-splitting factor
Ps/N0	Source-power-to-noise ratio
K	Rician K-factor
${\lambda }_1$	Mean of ${\left|h_1\right|}^2$
${\lambda }_2$	Mean of ${\left|g_1\right|}^2$
R	Source rate
Er1	Harvested energy at relay 1
Pr1	Average transmit power of relay 1
Er2	Harvested energy at relay 2
Pr2	Average transmit power of relay 2
Pout	Outage probability
${\gamma }_{e2e1}$	End to end signal to noise ratio
$K_v(•)$	Modified Bessel function of the second kind and vth order
$\mathsf{\Gamma }(•)$	Gamma function
$F\left(\upsilon ,\beta ;\gamma ;z\right)$	Hypergeometric function
SER	Symbol error ratio
$\beta$	Amplifying factor
$Q(t)$	Gaussian Q-function
Ps	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 $h_i\approx C{\rm N}\left(0,1\right)$, $g_i\approx C{\rm N}\left(0,1\right)$ and $h_{12}\approx C{\rm N}\left(0,1\right)$, where $i\in (1,2)$.

In the hybrid TSR–PSR alternative relaying, the source provides an energy signal to both R₁ and R₂ in $\alpha T$ and $(1-\alpha )T/2$ blocks. In the $(1-\alpha )T/2$ block, R₁ allocates $0\le \rho <1$ ($\rho$ is the power splitting factor) as part of the received source signal for the energy harvesting (EH). Therefore, the total harvested energy at R₁ can be given by

$$E_{r1a}=\eta \alpha T{P}_s+\frac{\eta \rho (1-\alpha )T{P}_s}{2}$$

(1)

where ($E\left\{{\left|h_i\right|}^2\right\}=1$ and $h_i\approx C{\rm N}\left(0,1\right)$, $g_i\approx C{\rm N}\left(0,1\right)$, $h_{12}\approx C{\rm N}\left(0,1\right)$), $0<\eta <1$ and $0\le \alpha <1$ 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 R₂ can be given by the equation below:

$$E_{r2a}=\eta \alpha T{P}_s+\frac{\eta \rho (1-\alpha )T{P}_s}{2}$$

(2)

When S provides the data to R₁, after splitting the $\rho$ part of the received signal for the EH at the relays, the remaining signal at R₁ can be obtained as

$$y_{r1}=\sqrt{1-\rho }{h}_1{x}_s+n_{r1}$$

(3)

where $n_{r1}$ is the additive white Gaussian noise (AWGN) with variance N₀ at R₁, and $E\left\{{\left|x_s\right|}^2\right\}=P_s$ in which $E\{•\}$ is expectation operator.

Furthermore, R₁ amplifies and forwards the signal to D in the next stage. The transmitted signal from R₁ can be expressed as

$$x_{r1}=\beta y_{r1}$$

(4)

where $\beta =\frac{\sqrt{P_{r1}}}{\sqrt{(1-\rho )P_s{\left|h_1\right|}^2+N_0}}$ is the amplifying factor.

Then, the received signal at D can be formulated as the following expression:

$$y_{1d}=g_1{x}_{r1}+n_{1d}$$

(5)

where $n_{1d}$ is the additive white Gaussian noise (AWGN) with variance N₀ at D, $E\left\{{\left|x_{r1}\right|}^2\right\}=P_{r1}$, and $P_{r1}$ is the average transmitted power of R₁.

Replacing (3) and (4) into (5), the received signal at D can be obtained as:

$$y_{1d}=\beta g_1\left[\sqrt{1-\rho }{h}_1{x}_s+n_{r1}\right]+n_{1d}=\underset{signal}{\underbrace{\beta g_1\sqrt{1-\rho }{h}_1{x}_s}}+\underset{noise}{\underbrace{\beta g_1{n}_{r1}+n_{1d}}}$$

(6)

In this case, when R₁ performs the delay-tolerant (DT) transmission mode, the end to end signal to noise ratio (SNR) at D can be calculated as

$${\gamma }_{e2e1}=\frac{\mathrm{E}\left\{{\left|signal\right|}^2\right\}}{\mathrm{E}\left\{{\left|noise\right|}^2\right\}}=\frac{{\beta }^2{\left|g_1\right|}^2{\left|h_1\right|}^2{P}_s(1-\rho )}{{\beta }^2{\left|g_1\right|}^2{N}_0+N_0}$$

(7)

After algebra calculation and using the fact that N₀ << P_r, the end to end SNR can be obtained:

$${\gamma }_{e2e1}=\frac{(1-\rho )P_s{P}_{r1}{\left|h_1\right|}^2{\left|g_1\right|}^2}{{\left|g_1\right|}^2{N}_0+(1-\rho )P_s{\left|h_1\right|}^2{N}_0}$$

(8)

In this T block time, R₂ can harvest energy from S in $\alpha T+(1-\alpha )T/2$ blocks, i.e., $\frac{T}{2}(1+\alpha )$ blocks, and R₂ can also harvest energy from R₁ in $(1-\alpha )T/2$ blocks. Therefore, the total harvested energy at R₂ when R₁ joins in the data transmission (DT) can be calculated by

$$E_{r2b}=\frac{\eta (1+\alpha )T{P}_s}{2}+\frac{\eta (1-\alpha )T{P}_{r1}}{2}$$

(9)

Similar to R₂, the total harvested energy at R₁ when R₂ joins in the DT can be obtained as

$$E_{r1b}=\frac{\eta (1+\alpha )T{P}_s}{2}+\frac{\eta (1-\alpha )T{P}_{r2}}{2}$$

(10)

where $P_{r2}$ is the average transmitted power of R₂.

From the EH process at R₁ in the previous T blocks and current T blocks, the total harvested energy of R₁ for DT can be obtained as

$$\begin{array}{cc}\hfill E_{r1}& =E_{r1a}+E_{r1b}=\eta \alpha T{P}_s+\frac{\eta \rho (1-\alpha )T{P}_s}{2}+\frac{\eta (1+\alpha )T{P}_s}{2}+\frac{\eta (1-\alpha )T{P}_{r2}}{2}\hfill \\ & =\frac{\eta T\left[(3\alpha -\alpha \rho +1)P_s+\rho +(1-\alpha )P_{r2}\right]}{2}\hfill \end{array}$$

(11)

because of the symmetry property in our proposed system. Similar to R₁, the total harvested energy of R₂ also can be obtained as the following:

$$E_{r2}=E_{r2a}+E_{r2b}=\frac{\eta T\left[(3\alpha -\alpha \rho +1)P_s+\rho +(1-\alpha )P_{r1}\right]}{2}$$

(12)

From (11), the average transmitted power of R₁ can be calculated as

$$P_{r1}=\frac{E_{r1}}{(1-\alpha )T/2}=\frac{\eta T\left[(3\alpha -\alpha \rho +1)P_s+\rho +(1-\alpha )P_{r2}\right]}{(1-\alpha )T}=\eta \left[\frac{(3\alpha -\alpha \rho +1)P_s+\rho }{1-\alpha }+P_{r2}\right]$$

(13)

From (12), the average transmitted power of R₂ also can be obtained as

$$P_{r2}=\frac{E_{r2}}{(1-\alpha )T/2}=\eta \left[\frac{(3\alpha -\alpha \rho +1)P_s+\rho }{1-\alpha }+P_{r1}\right]$$

(14)

Substituting (14) into (13), we obtain

$$P_{r1}=\frac{\eta \mathsf{\Psi }}{1-\eta }$$

(15)

where we denote $\mathsf{\Psi }=\frac{(3\alpha -\alpha \rho +1)P_s+\rho }{1-\alpha }$.

Finally, the SNR of the proposed system in (7) can be rewritten as the following:

$${\gamma }_{e2e1}=\frac{(1-\rho )P_s{P}_{r1}{\left|h_1\right|}^2{\left|g_1\right|}^2}{{\left|g_1\right|}^2{N}_0+(1-\rho )P_s{\left|h_1\right|}^2{N}_0}=\frac{(1-\rho )P_s{P}_{r1}{\phi }_1{\phi }_2}{{\phi }_2{N}_0+(1-\rho )P_s{\phi }_1{N}_0}$$

(16)

where ${\phi }_1={\left|h_1\right|}^2, {\phi }_2={\left|g_1\right|}^2$ and $P_{r1}$ is defined by (15).

3.1. Exact Outage Probability Analysis

The probability density function (PDF) of random variable (RV) ${\phi }_i$ (where i = 1, 2) as in [28] is

$$f_{{\phi }_i}(x)=a{\displaystyle \sum _{l=0}^{\infty }\frac{{(bK)}^l}{{(l!)}^2}{x}^l{e}^{-bx}}$$

(17)

where we denote ${\phi }_1={\left|h_1\right|}^2, {\phi }_2={\left|g_1\right|}^2$, $a=\frac{(K+1)e^{-K}}{{\lambda }_i}, b=\frac{K+1}{{\lambda }_i}$, in which ${\lambda }_i$ is the unit mean value of RV ${\phi }_i$ where i = 1, 2, respectively, because we consider the small-scale power fading ${\left|h_1\right|}^2, {\left|g_1\right|}^2$ 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 ${\phi }_i$, where i = 1, 2, can be computed as in [17]:

$$F_{{\phi }_i}(\varsigma )={\displaystyle \underset{0}{\overset{\varsigma }{\int }}{f}_{{\phi }_i}(x)dx=1-\frac{a}{b}{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n}{l!n!}}}}{\varsigma }^n{e}^{-b\varsigma }=1-{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n{e}^{-K}}{l!n!}}}{\varsigma }^n{e}^{-b\varsigma }$$

(18)

$$F_{{\phi }_i}(\varsigma )={\displaystyle \underset{0}{\overset{\varsigma }{\int }}{f}_{{\phi }_i}(x)dx=1-\frac{a}{b}{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n}{l!n!}}}}{\varsigma }^n{e}^{-b\varsigma }=1-{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n{e}^{-K}}{l!n!}}}{\varsigma }^n{e}^{-b\varsigma }$$

(19)

Theorem 1 (Exact Outage Probability).

The expression of the exact outage probability of the proposed system can be formulated by the following:

$$\begin{array}{l}{P}_{out}^1=1-2a{e}^{-K}{e}^{-\frac{b{\gamma }_{th}{N}_0}{P_{r1}{P}_s(1-\rho )}}{e}^{-\frac{b{\gamma }_{th}{N}_0}{P_{r1}}}{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{m=0}^{\infty }{\displaystyle \sum _{n=0}^l{\displaystyle \sum _{k=0}^{n+m}\frac{K^{l+m}{b}^{n+m}{\left({\gamma }_{th}{N}_0\right)}^{n+m+1}(n+m)!}{l!{(m!)}^{2}n!k!(n+m-k)!P_{r1}{}^{n+m+1}{\left[P_s(1-\rho )\right]}^{\frac{2n+m-k+1}{2}}}}}}}\\ \times K_{m-k+1}\left(\frac{2b{\gamma }_{th}{N}_0}{P_{r1}\sqrt{P_s(1-\rho )}}\right)\end{array}$$

(20)

where $K_v(•)$ is the modified Bessel function of the second kind and vth order.

Proof of Theorem 1.

See Appendix A. □

3.2. Asymptotic Outage Probability Analysis

From (16), at the high SNR regime, the end to end SNR can be approximated as

$${\gamma }_{{\gamma }_{e2e1}}^{\infty }=\frac{(1-\rho )P_s{P}_{r1}{\phi }_1{\phi }_2}{{\phi }_2{N}_0+(1-\rho )P_s{\phi }_1{N}_0}\approx \frac{P_{r1}{\phi }_2}{N_0}$$

(21)

Then, the asymptotic outage probability can be formulated as

$$P_{out}^{1,\infty }=\mathrm{Pr}\left(\frac{P_{r1}{\phi }_2}{N_0}<{\gamma }_{th}\right)=\mathrm{Pr}\left({\phi }_2<\frac{{\gamma }_{th}{N}_0}{P_{r1}}\right)=1-{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n{e}^{-K}}{l!n!}}}{\left(\frac{{\gamma }_{th}{N}_0}{P_{r1}}\right)}^n{e}^{-\frac{b{\gamma }_{th}{N}_0}{P_{r1}}}$$

(22)

3.3. 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

$$SE{R}_1=\mathrm{E}\left[\varphi Q\sqrt{2\theta {\gamma }_{e2e1}}\right]$$

(23)

$Q(t)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{t}{\overset{\infty }{\int }}{e}^{-x^2/2}dx}$ is the Gaussian Q-function, $\omega$ and $\theta$ are constants which are specific for the modulation type. $(\varphi ,\theta )=(1,1)$ for BPSK and $(\varphi ,\theta )=(1,2)$ for QPSK. For this purpose, the distribution function of ${\gamma }_{e2e1}$ 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:

$$SE{R}_1=\frac{\varphi \sqrt{\theta }}{2\sqrt{\pi }}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{e^{-\theta x}}{\sqrt{x}}{F}_{{\gamma }_{e2e1}}(x)dx}$$

(24)

Theorem 2 (Exact SER).

The exact SER can be calculated by the below expression:

$$\begin{array}{l}SE{R}_1=\frac{\varphi }{2}-a{e}^{-K}\varphi \sqrt{\theta }{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{m=0}^{\infty }{\displaystyle \sum _{n=0}^l{\displaystyle \sum _{k=0}^{n+m}\frac{4^{m-k+1}{K}^{l+m}{b}^{n+2m-k+1}(n+m)!{\left(N_0\right)}^{n+2m-k+2}}{l!{(m!)}^{2}n!k!(n+m-k)!P_{r1}{}^{n+2m-k+2}{\left[P_s(1-\rho )\right]}^{\frac{2n+3m-3k+3}{2}}}}}}}\\ \times \frac{1}{{\left[\theta +\frac{3b{N}_0}{P_{r1}\sqrt{P_s(1-\rho )}}+\frac{b{N}_0}{P_{r1}}\right]}^{n+2m-k+5/2}}\times \frac{\mathsf{\Gamma }\left(n+2m-k+\frac{5}{2}\right)\mathsf{\Gamma }\left(n+k+\frac{1}{2}\right)}{\mathsf{\Gamma }\left(n+m+2\right)}\\ \times F\left(n+2m-k+\frac{5}{2},m-k+\frac{3}{2};n+m+2;\frac{\theta -\frac{b{N}_0}{P_{r1}\sqrt{P_s(1-\rho )}}+\frac{b{N}_0}{P_{r1}}}{\theta +\frac{3b{N}_0}{P_{r1}\sqrt{P_s(1-\rho )}}+\frac{b{N}_0}{P_{r1}}}\right)\end{array}$$

(25)

where $\mathsf{\Gamma }(•)$ is the gamma function, and $F\left(\upsilon ,\beta ;\gamma ;z\right)$ 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:

$$SE{R}_1^{\infty }=\frac{\varphi }{2}-\frac{e^{-K}\varphi \sqrt{\theta }}{2\sqrt{\pi }}{\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{n=0}^l\frac{K^l{b}^n}{l!n!}}}{\left(\frac{N_0}{P_{r1}}\right)}^n{\left(\theta +\frac{b{N}_0}{P_{r1}}\right)}^{-n-1/2}\mathsf{\Gamma }\left(n+\frac{1}{2}\right)$$

(26)

Proof of Theorem 3.

See Appendix C. □

4. 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 10⁶ 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. Simulation parameters.

Symbol	Values
$0<\eta \le 1$	0.7
${\lambda }_1$	1
${\lambda }_2$	1
Ps/N0	0:20 dB
K	3
R	0.5 bps

.

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/N₀ = 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/N₀ = 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/N₀. 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/N₀ 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/N₀ = 15 dB, ρ = 0.7 and α = 0.3. The outage probability significantly increases with R from 0 to 4. From Figure 5 and Figure 6, the analytical results and the simulation results match well with each other for all values of R and P_s/N₀.

Figure 7 illustrates the numerical and simulation results of the system outage probability concerning α and ρ with P_s/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 P_s/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 P_s/N₀. The results indicate that all the simulation and analytical values agreed well with each other.

5. 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.