Performance analysis of a satellite–multi-terrestrial relay network with hardware impairments using switch-and-stay combining scheme

Abstract

Space information networks are proposed to broaden the observation area and realize continuous information acquisition using satellites and high-altitude platform stations. Space information networks are able to enhance detection and transmission capabilities compared to the current single Earth observation satellite. The satellite–terrestrial network is an important component of the space information networks. Hence, the major contribution of this article is to propose a new transmitted link selection scheme in a satellite–terrestrial network. Switch-and-stay combining scheme is used by destination to have a trade-off between the complexity and efficiency. The satellite, terrestrial relays, and destination user are influenced by hardware impairments, respectively. The closed-form expressions for the outage probability and throughput are derived. In order to analyze the system performance at high signal-to-noise rate, the asymptotic expressions for the system performance are also derived to give a special view on the impact of hardware impairments on the considered network. What is more, numerical results are derived to verify the correctness of our analytical results.

Introduction

Recently, with the fast development of space exploration, the space information network (SIN) becomes an attractive and hot research field.^1–6 The concept of SIN was first proposed in 1998. To spread the terrestrial Internet to the space, the Defense Advanced Research Projects Agency (DARPA) supported the Jet Propulsion Laboratory (JPL) to investigate space Internet to realize end-to-end communications. ¹ The National Aeronautics and Space Administration (NASA) began to sponsor this project in 2002 and then the concepts were extended. In 2006, NASA proposed the NASA space communications data networking architecture in a NASA Technical Report. ³ From then on, more efforts were done on SIN. For example, Hefei and Yuan ⁴ presented a feasible mesh-based network architecture to suppose the scalability and compatibility of SIN. And some key technologies of SIN were discussed in Dai and Zhu. ⁵ Nishiyama et al. ⁶ proposed a promising multilayered satellite network to provide global ubiquitous broadband communication and discussed the corresponding routing scheme for the proposed network. In general, a SIN is a network that is capable of achieving real-time acquisition, transmission, and processing space information using different space platforms, that is, geostationary Earth orbit (GEO) satellites, middle Earth orbit (MEO) satellites, low Earth orbit (LEO) satellites, stratospheric balloons, manned and unmanned aircraft, and so on. The SIN plays an important role in many areas, such as ocean-going voyages, emergency rescue, navigation positioning, air transport, and aerospace measurement and control. Moreover, it can support real-time transmission to meet the requirements of both satellite–terrestrial communications and deep space exploration and communications. Considering the appealing features of SIN, it has become an important research field for universal scientists, which leads to many new topics in information theory, communications theory, and network theory.^7,8

SINs have been proposed in recent years, the satellite–terrestrial network is an important part of the SIN. The communication between long distance mainly relies on the satellite–terrestrial networks. As for this reasons, satellite–terrestrial networks attain a great attention because of its utility in several applications such as broadcasting, disaster recovery, defense, and so on. ⁹ There are two main challenges in satellite systems. One is masking effect and another is latency. In masking effect, the line of sight (LOS) between satellite and terrestrial user is blocked by the obstacles. To resolve this issue, the hybrid satellite–terrestrial cooperative systems have been proposed in Evans et al.; ¹⁰ in these systems, masked earth station (ES) received the signals of satellite through relay ES, situated at the ground. Furthermore, in order to overcome the masking effect, several satellite diversity techniques have been proposed in Schodorf and Wu ¹¹ and Chini et al. ¹² Disaster recovery is another important issue which requires establishment of broadband access from a disaster area to the rest of the world. Geostationary satellites can be used to provide broadband communication links in the disaster area when terrestrial infrastructure is destroyed.

On the contrary, spatial diversity with multi-antenna nodes has been considered to further improve the performance of the satellite communication systems.^13–15 In Arti ¹⁵ and Arti and Manav, ¹⁶ the beamforming and combining method for satellite–terrestrial networks were analyzed. In Lin et al., ¹⁷ multi-terrestrial satellite–terrestrial networks were studied, and outage probability and symbol error rate were analyzed. In Nikolaos et al., ¹⁸ multi-antenna relay node was considered in the satellite network, in order to derive better system performance, maximum ratio transmission (MRT), and maximum ratio combining (MRC) beamforming technologies were used at source and destination, respectively. The closed-form expression of the outage probability was derived in this article. Recently, An et al. ¹⁹ have researched a multiuser satellite–terrestrial network with opportunistic user scheduling. Especially, outage probability of the system was got by the authors. In Prabhat and Pankaj, ²⁰ max–max user relay selection scheme was used in multiuser and multi-relay hybrid satellite–relay system. The analytical and asymptotic expressions for the outage probability were provided. In Iqbal and Ahmed, ²¹ the authors analyzed the hybrid satellite–terrestrial networks under independent and nonideal shadowed Rician and Nakagami-m channel. Amplify-and-forward (AF) and MRC protocols were used in the system, respectively. Especially, the analytical approach was derived to evaluate the performance of the system in terms of outage probability and symbol. The authors just discussed the system in ideal hardware case, which ignored the hardware impairments.

Apart from the conventional diversity combining techniques, several variants of the switched-diversity combining techniques, such as switch-and-stay combining (SSC), switch-and-examine combining (SEC), and SEC with post-selection (SECps), were proposed in Yang and Alouini, ²² Tan et al., ²³ and Yang et al. ²⁴ due to their simplicity of implementation. In particular, the multibranch SSC scheme was first addressed in Hong and Alouini, ²⁵ which reduced the implementation complexity of multichannel communication scenarios. In the multibranch SSC scheme, if the channel quality of the currently connected branch exceeds a predetermined threshold, then this branch is kept. Otherwise, no matter what the channel quality of the switch to branch is, the scheduler settles on that branch for the next transmission burst. ²⁵ In Diomidis and George, ²⁶ the authors proposed and analyzed the performance of a distributed switch and stay combining (DSSC) scheme, which utilizes a single decode and forward (DF) relay. The outage probability and spectral efficiency of the considered system was analyzed by the authors. In Katiyar et al., ²⁷ SSC scheme was used in the relay networks, where the relay was a multi-antenna-input–single-antenna-output (MISO) node. The analytical expression for the outage probability was derived for the considered network. However, SSC scheme was rarely studied in the satellite–terrestrial networks. Hence, the work of this article is necessary.

Common to all these works dealing with SSC scheme is the assumption of perfect transceiver hardware (i.e. ideal hardware) of the terminals. However, in practice, the transceiver hardware is imperfect due to phase noise, I/Q imbalance, and amplifier nonlinearities.^28–30 Few works have investigated the effect of hardware impairments on dual-hop cooperative networks, and they are briefly discussed next.

In Bjornson et al., ³¹ the authors summarized the former works and proposed a normal hardware impairment model which was easy to follow. The closed-form expressions of the outage probability for the AF and DF relay networks were derived, which provided fast ways to evaluate the impact of hardware impairments on the transceiver networks. In Bjornson et al., ³² the authors analyzed the multiple-input and multiple-output (MIMO) networks with hardware impairments. An optimal beamforming scheme was used to derive better system performance. In Matthaiou et al., ³³ the authors analyzed the impact of hardware impairments on two-way relay networks, and the accurate and asymptotic expressions were derived to investigate the effect of hardware impairments on the considered networks. In Pankaj and Prabhat, ³⁴ the impact of hardware impairments was first considered in the cognitive networks. In Duy et al., ³⁵ partial relay selection and opportunistic relay selection schemes were used in the multi-relay networks with hardware impairments to obtain better system performance. In Hung et al., ³⁶ the impact of hardware impairments on the secrecy networks was first discussed. The authors discussed the system performance in three conditions and gave the analytical expressions for the secrecy outage probability. In Guo et al., ³⁷ the authors first analyzed the effect of hardware impairments on the outage probability of the satellite–relay networks. To its regret, the authors just considered the classical one source, one relay, and one destination model which just had relay link and ignored the direct transmitted link between source and destination. From the above analysis, we know that hardware impairments are a hot topic to be studied.

To the best knowledge of the authors, the satellite–terrestrial network with hardware impairments using SSC scheme has not been reported in the literature thus far. We are, therefore, motivated to examine the outage probability and throughput of the considered network. The main contributions of this article are summarized as follows:

• First, we propose a transmitted link selection scheme between the satellite–user link and the satellite–relay–user link. The scheme is that the satellite will always use one transmitted link when the signal-to-noise and distortion rate (SNDR) of this transmitted link is above the given threshold. Otherwise, the satellite will use the other transmitted link no matter what the quality of that transmitted link is. In the relay link, partial terrestrial relay selection scheme is used to choose a suitable relay to gain better performance.

• Second, the closed-form expressions for the outage probability and throughput are derived, which shows that the outage probability will have an upper bound when the given threshold increases to a fixed value, and the expressions also provide a fast way to evaluate the impact of hardware impairments on the throughput.

• Finally, the asymptotic expressions of the outage probability and throughput at high signal-to-noise rates (SNRs) are also derived. We can find the impact of hardware impairments on the considered network through the asymptotic analysis.

The rest of this article is organized as follows: The system model is described in section “The system model.” In section “The system performance,” closed-form expressions and asymptotic expressions for the outage probability and throughput are derived. Simulation results are presented in section “Numerical results” along with representative numerical results. Finally, this article is concluded in section “Conclusion.”

The system model

As shown in Figure 1, we consider a satellite–terrestrial relay user network. A signal antenna satellite (S), $N_1$ terrestrial relays (R) with $M_1$ antennas for transmitting and receiving signals, and a signal user (D) with single antenna were considered in the network. In particular, D can receive the signal both directly from S and indirectly through the virtual satellite–relay–destination (S-R-D) link. In addition, popular transmit and receive diversity techniques were considered in the analysis, namely, MRT at the transmitter side and MRC at the receiver side by the relay. SSC scheduling scheme is used at the destination to combine the signals.

Figure 1.

The system model.

SSC scheme is introduced as follows:

1. S broads its signal both to D.

2. Then S examines the SNDR of satellite–destination (S-D) transmitted link ${\gamma }_{\mathrm{sd}}$ with the switching threshold ${\gamma }_T$ .

3. If the received SNDR of S-D transmitted link is larger than the switching threshold ${\gamma }_T$ , S will always use this transmitted link until its quality drops below ${\gamma }_T$ .

4. If the quality of S-D link drops below ${\gamma }_T$ , then S will change the transmitted link to S-R-D transmitted link, no matter what the quality of the S-R-D link is.

5. In the next transmitted stage, when switching S-R-D transmitted link, if the SNDR of S-R-D transmitted link is larger than ${\gamma }_T$ , S will always use this S-R-D link until its quality falls below ${\gamma }_T$ .

In S-R-D transmitted link, transmission takes two time slots. In the first slot, S broads its signal to all R, as partial relay selection scheme is used in the relay selection. Hence, in the second slot, only R which has the largest SNDR of S-R link will forward the received signal to D. In particular, DF protocol is used by R. MRC and MRT technologies are used by the relay in the first and second slots, respectively.

Remark 1

In partial relay selection scheme, the satellite can select a relay Earth terrestrial (ET) based on the knowledge of the instantaneous SNRs of the satellite–relay ET and relay–destination ET links. For practical application of such a complex scheme, channel state information (CSI) of all relay ES-destination ES links is required at the satellite, which is provided using feedback from the destination ES to the satellite. Since in satellite systems, latency is very high, it is very difficult to provide CSI of all links at the satellite. Therefore, we consider a simple relay ES selection scheme, where the satellite only requires the CSI feedback of the satellite–relay ET links from the relays. In this article, we demonstrate analytically that using a simple relay ET selection scheme, it is possible to significantly lower the outage probability and achieve improvement in the diversity gain and consequently the performance of the considered system. In satellite systems, this performance gain is very important because by achieving this gain, satellite needs to transmit less power (approximately half for 3 dB SNR improvement), which can help in reducing the dimensions and weight and subsequently cost of the satellite system. Therefore, the proposed scheme is very useful for hybrid satellite–terrestrial communication systems.

The system performance

In this section, first, we derive the SNDR of the system; second, we obtain the closed-form expressions for the outage probability and the throughput of the considered system; finally, we derive the asymptotic expressions for the outage probability and throughput.

The end-to-end SNDR of the system

In this subsection, the end-to-end SNDR of the system is derived. As SSC scheme is used at destination of the system, at first, we analyze the direct transmitted link. S broads its signal to D. Hence, the received signal of direct transmitted link at D is given by

$$y_{\mathrm{sd}}=h_{\mathrm{sd}}(x_s+{\eta }_{\mathrm{sdt}})+{\eta }_{\mathrm{sdr}}+n_d$$

(1)

where $y_{\mathrm{sd}}$ is the received signal at D; $h_{\mathrm{sd}}$ is the shadowed Rician random variable (RV) between S and D; $x_s$ is the transmitted signal from S with power $P_s=E[{\left|x_s\right|}^2]$ ; ${\eta }_{\mathrm{sdt}}$ and ${\eta }_{\mathrm{sdr}}$ are the distortion noise with zero mean and variance $k_{\mathrm{sdt}}^2{P}_s$ and $k_{\mathrm{sdr}}^2{\left|h_{\mathrm{sd}}\right|}^2{P}_s$ , respectively; and $n_d$ is the additive white Gaussian noise (AWGN) at D with power $n_d~\mathcal{C}\mathcal{N}(0,{\delta }_d^2)$ . From equation (1), the SNDR of ${\gamma }_{\mathrm{sd}}$ of $y_{\mathrm{sd}}$ is given by

$${\gamma }_{\mathrm{sd}}=\frac{{\lambda }_{\mathrm{sd}}}{{\lambda }_{\mathrm{sd}}{k}_{\mathrm{sd}}^2+1}$$

(2)

where $k_{\mathrm{sdt}}$ and $k_{\mathrm{sdr}}$ are the impairment levels at S and D, respectively. ${\lambda }_{\mathrm{sd}}={\left|h_{\mathrm{sd}}\right|}^2{P}_s/{\delta }_d^2$ and $k_{\mathrm{sd}}^2=k_{\mathrm{sdt}}^2+k_{\mathrm{sdr}}^2$ .

Next, if S selects the S-R-D transmitted link according to SSC selection scheme, then in the first slot, the received signal at ith R is given by

$$y_{1i}={\mathbf{w}}_1^{†}{\mathbf{h}}_{1i}(x_s+{\eta }_{\mathrm{srt}})+{\eta }_{\mathrm{srr}}+{\mathbf{w}}_1^{†}{\mathrm{n}}_r$$

(3)

where ${\mathbf{w}}_1^{†}$ is the beamforming vector of S-ith R channel; $(·{)}^{†}$ is the conjugate transpose operator; because MRC is used at ith R, ${\mathrm{w}}_1=h_{1i}/\Vert h_{1i}{\Vert }_F$ , where $\Vert ·{\Vert }_F$ denotes the Frobenius norm of a matrix, $h_{1i}$ denotes the channel coefficient vector of S-ith R channel which is formed as shadowed Rician RV, and ${\mathrm{n}}_r$ is the $M_1\times 1$ AWGN vector which can be expressed as $n_r~\mathcal{C}\mathcal{N}(0,{\delta }_r^2{\mathbf{I}}_{M_1\times M_1})$ , where ${\mathbf{I}}_{M_1\times M_1}$ is the $M_1\times M_1$ unit matrix; and ${\eta }_{\mathrm{srt}}$ and ${\eta }_{\mathrm{srr}}$ are the additional distortion noise terms which can be written as ${\eta }_{\mathrm{srt}}~\mathcal{C}\mathcal{N}(0,{\Upsilon }_{\mathrm{srt}})$ and ${\eta }_{\mathrm{srr}}~\mathcal{C}\mathcal{N}(0,{\Upsilon }_{\mathrm{srr}})$ , with

$$\begin{array}{c}{\Upsilon }_{\mathrm{srt}}=k_{\mathrm{srt}}^2{P}_s\hfill \\ {\Upsilon }_{\mathrm{srr}}=k_{\mathrm{srr}}^2{P}_s{\left|{\mathbf{h}}_{1i}{\mathbf{w}}_1^{†}\right|}^2\hfill \end{array}$$

(4)

where $k_{\mathrm{srt}}$ and $k_{\mathrm{srr}}$ are sufficient to characterize the aggregate level of impairments of S-ith R link, respectively.

From equation (3), the SNDR of $y_{1i}$ is given by

$${\gamma }_{1i}=\frac{{\lambda }_{1i}}{{\lambda }_{1i}{k}_{1i}^2+1}$$

(5)

where ${\lambda }_{1i}=\Vert h_{1i}{\Vert }_F^2/{\delta }_r^2$ and $k_{1i}^2=k_{\mathrm{srt}}^2+k_{\mathrm{srr}}^2$ .

As mentioned above, partial relay selection scheme is used in the first relay link; hence, the SNDR of the first hop is given by

$${\gamma }_1=\underset{i=1,\dots ,N_1}{max}\left({\gamma }_{1i}\right)$$

(6)

In the second slot, the selected R will forward the received signal to D. The same with equation (3), the received signal at D is given by

$$y_d={\mathbf{h}}_2^T({\mathbf{w}}_2{x}_r+{\eta }_{\mathrm{rt}})+{\eta }_d+n_d$$

(7)

where ${\mathbf{h}}_2^T$ is the channel coefficient vector between the selected R and D, $(·{)}^T$ is the transpose function, $x_r$ is the forward signal from the selected R with power $P_r=E[{\left|x_r\right|}^2]$ , and ${\mathbf{w}}_2$ is the beamforming vector of the selected R-D link. Since MRT is used by R, ${\mathbf{w}}_2={\mathbf{h}}_2^T/\Vert {\mathbf{h}}_2^T{\Vert }_F$ , ${\eta }_{\mathrm{rt}}$ and ${\eta }_d$ are the additional distortion noise terms which can be written as ${\eta }_{\mathrm{rt}}~\mathcal{C}\mathcal{N}(0,{\Upsilon }_{\mathrm{rt}})$ and ${\eta }_d~\mathcal{C}\mathcal{N}(0,{\Upsilon }_d)$ . Theoretical investigations and several measurements have suggested that

$${\Upsilon }_{\mathrm{rt}}=k_{\mathrm{rt}}^2{P}_r\mathrm{diag}({\left|w_{21}\right|}^2,\dots ,{\left|w_{2M_1}\right|}^2)$$

(8)

$${\Upsilon }_d=k_{\mathrm{rr}}^2{P}_r{\left|{\mathbf{h}}_2^T{\mathbf{w}}_2\right|}^2$$

where $k_{\mathrm{rt}}$ and $k_{\mathrm{rr}}$ are sufficient to characterize the aggregate level of impairments of the selected R-D link.

From equation (7), the SNDR of the second hop is given by

$${\gamma }_2=\frac{{\lambda }_2}{{\lambda }_2{k}_2^2+1}$$

(9)

where ${\lambda }_2=P_r\Vert h_2^T{\Vert }_F^2/{\delta }_d^2$ and $k_2^2=k_{\mathrm{rt}}^2+k_{\mathrm{rr}}^2$ .

As DF protocol is used by R, the SNDR of the S-R-D link is given by

$${\gamma }_{\mathrm{srd}}=min({\gamma }_1,{\gamma }_2)$$

(10)

Preliminaries

Before we derived the outage probability of the considered system, first, with the help of Nikolaos et al., ¹⁸ the probability density function (PDF) of ${\lambda }_{\mathrm{sd}}$ is given by

$$f_{{\lambda }_{\mathrm{sd}}}\left({\lambda }_{\mathrm{sd}}\right)={\alpha }_0\sum _{k_0=0}^{m_0-1}\frac{{(1-m_0)}_{k_0}{(-{\delta }_0)}^{k_0}}{{(k_0!)}^2{\left({\overline{\lambda }}_{\mathrm{sd}}\right)}^{k_0+1}}{\lambda }_{\mathrm{sd}}^{k_0}{e}^{-{\Delta }_0{\lambda }_{\mathrm{sd}}}$$

(11)

where ${\Delta }_0={\beta }_0-{\delta }_0/{\overline{\lambda }}_{\mathrm{sd}}$ , ${\overline{\lambda }}_{\mathrm{sd}}$ is the average SNR of the direct link. ${\alpha }_0\stackrel{\Delta }{=}(2b_0{m}_0/2b_0{m}_0+{\Omega }_0{)}^{m_0}/2b_0$ , ${\beta }_0\stackrel{\Delta }{=}1/2b_0$ , ${\delta }_0\stackrel{\Delta }{=}{\Omega }_0/2b_0(2b_0{m}_0+{\Omega }_0)$ , ${\Omega }_0$ , $2b_0$ , and $m_0\ge 0$ correspond to the average power of the LOS component, the average power of the multipath component, and the fading severity parameter ranging from $0$ to $\infty$ , respectively. $(·{)}_{k_0}$ is the Pochhammer symbol. ³⁸

Second, in the S-R transmitted link, we assume that each antenna link has the same channel parameters, from Nikolaos et al., ¹⁸ the PDF of ${\lambda }_{1i}$ can be expressed as

$$f_{{\lambda }_{1i}}\left({\lambda }_{1i}\right)=\sum _{k_1=0}^{m_1-1}\cdots \sum _{k_{M_1}=0}^{m_1-1}\Xi \left(M_1\right){\lambda }_{1i}^{\Lambda -1}{e}^{-{\Delta }_1{\lambda }_{1i}}$$

(12)

where

$$\Xi \left(M_1\right)\stackrel{\Delta }{=}\underset{\tau =1}{\overset{M_1}{\Pi }}\xi \left(k_{\tau }\right){\alpha }_1^{M_1}\underset{\upsilon =1}{\overset{M_1-1}{\Pi }}B(\sum _{l=1}^{\upsilon }{k}_l+\upsilon ,k_{\upsilon +1}+1)$$

and $\Lambda \stackrel{\Delta }{=}\sum _{\tau =1}^{M_1}{k}_{\tau }+M_1$ , $\xi (k_{\tau })=(1-m_{\tau }{)}_{k_{\tau }}(-{\delta }_{\tau }{)}^{k_{\tau }}/(k_{\tau }!{)}^2({\overline{\lambda }}_{1i}{)}^{k_{\tau }+1}$ , while $B(.,.)$ denotes the beta function. ³⁸

Third, the PDF of ${\lambda }_2$ is derived. As MRT is used by R, the PDF of ${\lambda }_2$ can be expressed as

$$f_{{\lambda }_2}\left({\lambda }_2\right)=\sum _{i=1}^{\rho \left({\mathrm{A}}_1\right)}\sum _{j=1}^{{\tau }_i\left({\mathrm{A}}_1\right)}{\chi }_{i,j}\left({\mathrm{A}}_1\right)\frac{{\mu }_{〈i〉}^{-j}}{(j-1)!}{\lambda }_2^{j-1}{e}^{-{\lambda }_2/{\mu }_{〈i〉}}$$

(13)

where ${\mathrm{A}}_1=\mathrm{diag}({\mu }_1,\dots ,{\mu }_{M_1})$ , $\rho ({\mathrm{A}}_1)$ is the number of distinct diagonal elements of ${\mathrm{A}}_1$ , ${\mu }_{〈1〉}>{\mu }_{〈2〉}>\dots >{\mu }_{〈\rho ({\mathrm{A}}_1)〉}$ are the multiplicity of ${\mu }_{〈i〉}$ , and ${\chi }_{i,j}({\mathrm{A}}_1)$ is the $(i,j)\mathrm{th}$ characteristic coefficient of ${\mathrm{A}}_1$ .

Outage probability of the system

In wireless systems, the outage probability is an important quality-of-service (QoS) performance measure, which is defined as the probability that the output instantaneous SNDR ${\gamma }_e$ falls below an acceptable threshold $x_0$ , namely

$$P_{\mathrm{out}}({\gamma }_e\le x_0)=F_{{\gamma }_e}\left(x_0\right)$$

(14)

From Yong et al., ³⁹ we know that the outage probability of the considered system is given by

$$P\left({\gamma }_e\le x_0\right)=\{\begin{array}{c}\mathrm{Pr}\left[\left({\gamma }_{sd}\le {\gamma }_T\right)\mathrm{and}\left({\gamma }_{srd}\le {\gamma }_T\right)\right],\gamma <{\gamma }_T\\ \mathrm{Pr}\left[\begin{array}{c}\left({\gamma }_T\le {\gamma }_{sd}<{\gamma }_{th}^{sd}\right)\\ \mathrm{or}\left({\gamma }_{sd}\le {\gamma }_T\mathrm{and}{\gamma }_{srd}\le {\gamma }_{th}^{srd}\right)\end{array}\right],\gamma \ge {\gamma }_T\end{array}$$

(15)

Suppose that $r$ is the target spectral efficiency, then ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=x_0=2^r-1$ . As transmission takes two slots in S-R-D link, the outage threshold ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}=2^{2r}-1=x_0(x_0+2)$ . After some simplifications, equation (15) is given by

$$P_{out}\left({\gamma }_e\le x_0\right)=F_{{\gamma }_e}\left(x_0\right)=\{\begin{array}{c}\frac{F_{{\gamma }_{sd}}\left({\gamma }_T\right)F_{{\gamma }_{srd}}\left({\gamma }_T\right)}{F_{{\gamma }_{sd}}\left({\gamma }_T\right)+F_{{\gamma }_{srd}}\left({\gamma }_T\right)}\left[F_{{\gamma }_{sd}}\left({\gamma }_{th}^{sd}\right)+F_{{\gamma }_{srd}}\left({\gamma }_{th}^{srd}\right)\right],\gamma <{\gamma }_T\\ \frac{F_{{\gamma }_{sd}}\left({\gamma }_T\right)F_{{\gamma }_{srd}}\left({\gamma }_T\right)}{F_{{\gamma }_{sd}}\left({\gamma }_T\right)+F_{{\gamma }_{srd}}\left({\gamma }_T\right)}\left[F_{{\gamma }_{sd}}\left({\gamma }_{th}^{sd}\right)+F_{{\gamma }_{srd}}\left({\gamma }_{th}^{srd}\right)-2\right]+\frac{F_{{\gamma }_{sd}}\left({\gamma }_{th}^{sd}\right)F_{{\gamma }_{srd}}\left({\gamma }_T\right)+F_{{\gamma }_{sd}}\left({\gamma }_T\right)F_{{\gamma }_{srd}}\left({\gamma }_{th}^{srd}\right)}{F_{{\gamma }_{sd}}\left({\gamma }_T\right)+F_{{\gamma }_{srd}}\left({\gamma }_T\right)}\gamma \ge {\gamma }_T\end{array}$$

(16)

Equation (16) can be written as a simple expression, which is given by

$$\begin{array}{cc}\multicolumn{1}{c}{F_{{\gamma }_e}\left(x_0\right)=}& p_{\mathrm{sd}}[(1-F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right))\mathrm{Pr}\{o|(B=\mathrm{sd},{\gamma }_{\mathrm{sd}}>{\gamma }_T)\}\hfill \\ \multicolumn{1}{c}{}& +F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)\mathrm{Pr}\{o|B=\mathrm{srd}\}]\hfill \\ \multicolumn{1}{c}{}& +p_{\mathrm{srd}}[(1-F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right))\mathrm{Pr}\{o|(B=\mathrm{srd},{\gamma }_{\mathrm{srd}}>{\gamma }_T)\}\hfill \\ \multicolumn{1}{c}{}& +F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)\mathrm{Pr}\{o|B=\mathrm{sd}\}]\hfill \end{array}$$

(17)

where $p_{\mathrm{sd}}$ and $p_{\mathrm{srd}}$ denote the steady-state selection probabilities of activating S-D transmitted link and S-R-D transmitted link, respectively. In addition, they can be re-expressed as equations (18) and (19), respectively, which have been defined in Diomidis and George ²⁶ and Katiyar et al. ²⁷

$$p_{\mathrm{sd}}=\mathrm{Pr}\{B=\mathrm{sd}\}=\frac{F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)}{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)+F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)}$$

(18)

$$p_{\mathrm{srd}}=\mathrm{Pr}\{B=\mathrm{srd}\}=\frac{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)}{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)+F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)}$$

(19)

Note that the conditional outage probabilities in equation (17) are taken from the general form

$$\mathrm{Pr}\{o|(B=Z,{\gamma }_Z>\omega )\}=\{\begin{array}{cc}\frac{F_{{\gamma }_Z}\left({\gamma }_{\mathrm{th}}^Z\right)-F_{{\gamma }_Z}\left(\omega \right)}{1-F_{{\gamma }_Z}\left(\omega \right)},\hfill & \omega \le {\gamma }_{\mathrm{th}}^Z\hfill \\ \multicolumn{1}{c}{0,}& \omega >{\gamma }_{\mathrm{th}}^Z\hfill \end{array}$$

(20)

By substituting $Z$ and $\omega$ with the appropriate values, that is, $Z\in \{\mathrm{sd},\mathrm{srd}\}$ , $\omega \in \{{\gamma }_T,0\}$ . Next, $F_{{\gamma }_{\mathrm{sd}}}({\gamma }_{\mathrm{th}}^{\mathrm{sd}})$ and $F_{{\gamma }_{\mathrm{srd}}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ are derived, respectively.

Firstly, the cumulative distribution function (CDF) of $F_{{\gamma }_{\mathrm{sd}}}({\gamma }_{\mathrm{th}}^{\mathrm{sd}})$ is derived, with the help of equations (11) of (2), $F_{{\gamma }_{\mathrm{sd}}}({\gamma }_{\mathrm{th}}^{\mathrm{sd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}<1/k_{\mathrm{sd}}^2$ is given by

$$\begin{array}{cc}\multicolumn{1}{c}{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_{\mathrm{th}}^{\mathrm{sd}}\right)}& =F_{{\gamma }_{\mathrm{sd}}}({\gamma }_{\mathrm{sd}}\le {\gamma }_{\mathrm{th}}^{\mathrm{sd}})\hfill \\ \multicolumn{1}{c}{}& =F_{{\gamma }_{\mathrm{sd}}}(\frac{{\lambda }_{\mathrm{sd}}}{{\lambda }_{\mathrm{sd}}{k}_{\mathrm{sd}}^2+1}\le {\gamma }_{\mathrm{th}}^{\mathrm{sd}})\hfill \\ \multicolumn{1}{c}{}& ={\alpha }_0\sum _{k_0=0}^{m_0-1}\frac{{(1-m_0)}_{k_0}{(-{\delta }_0)}^{k_0}}{{(k_0!)}^2{({\beta }_0-{\delta }_0)}^{k_0+1}}\gamma \hfill \\ \multicolumn{1}{c}{}& (k_0+1,\frac{{\Delta }_0{\gamma }_{\mathrm{th}}^{\mathrm{sd}}}{1-k_{\mathrm{sd}}^2{\gamma }_{\mathrm{th}}^{\mathrm{sd}}})\hfill \end{array}$$

(21)

Next, the CDF of $F_{{\gamma }_{\mathrm{srd}}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ is derived. From equation (10), we know that

$$F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)=F_{{\gamma }_1}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)+F_{{\gamma }_2}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)-F_{{\gamma }_1}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)F_{{\gamma }_2}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)$$

(22)

From equation (22), we know that we should first obtain $F_{{\gamma }_1}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ and $F_{{\gamma }_2}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ . Next, $F_{{\gamma }_1}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ is derived. As defined in equation (6), we know that before deriving $F_{{\gamma }_1}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ , we should obtain $F_{{\gamma }_{1i}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ . With the help of equations (12) and (5), $F_{{\gamma }_{1i}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}<1/k_{1i}^2$ is given by equation (23)

$$\begin{array}{cc}\multicolumn{1}{c}{F_{{\gamma }_{1i}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}& =F_{{\gamma }_{1i}}({\gamma }_{1i}\le {\gamma }_{\mathrm{th}}^{\mathrm{srd}})=F_{{\gamma }_{1i}}(\frac{{\lambda }_{1i}}{{\lambda }_{1i}{k}_{1i}^2+1}\le {\gamma }_{\mathrm{th}}^{\mathrm{srd}})\hfill \\ \multicolumn{1}{c}{}& =\sum _{k_1=0}^{m_1-1}\cdots \sum _{k_{M_1}=0}^{m_1-1}\frac{\Xi \left(M_1\right)}{{\Delta }_1^{\Lambda }}\gamma (\Lambda ,\frac{{\Delta }_1{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{1-k_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}})\hfill \\ \multicolumn{1}{c}{}& =1-\sum _{k_1=0}^{m_1-1}\cdots \sum _{k_{M_1}=0}^{m_1-1}\Xi \left(M_1\right)\sum _{s=0}^{\Lambda -1}{e}^{-\frac{{\Delta }_1{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{1-k_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}}\hfill \\ \multicolumn{1}{c}{}& \frac{(\Lambda -1)!{(1-k_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{\Lambda -s}}{s!{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{\Lambda -s}}\hfill \end{array}$$

(23)

Because partial relay selection is used in the first hop of S-R-D link, with the help of equation (6), the CDF of ${\gamma }_1$ is given by equation (24)

$$\begin{array}{c}{F}_{{\gamma }_1}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)={\left[F_{{\gamma }_{1i}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)\right]}^{N_1}\hfill \\ =\sum _{v=1}^{N_1}{N}_1\begin{array}{c}\hfill \\ \multicolumn{1}{c}{v}\end{array}{(-1)}^v\hfill \\ {[\sum _{k_1=0}^{m_1-1}\cdots \sum _{k_{M_1}=0}^{m_1-1}\Xi \left(M_1\right)\sum _{s=0}^{\Lambda -1}{e}^{-\frac{{\Delta }_1{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{1-k_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}}\frac{(\Lambda -1)!{(1-k_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{\Lambda -s}}{s!{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{\Lambda -s}}]}^v\hfill \end{array}$$

(24)

Next, $F_{{\gamma }_2}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ is derived in the same way. With the help of equations (9) and (13), $F_{{\gamma }_2}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}<1/k_2^2$ is given by

$$\begin{array}{c}{F}_{{\gamma }_2}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)=F_{{\gamma }_2}({\gamma }_2\le {\gamma }_{\mathrm{th}}^{\mathrm{srd}})\hfill \\ =\sum _{i=1}^{\rho \left({\mathrm{A}}_1\right)}\sum _{j=1}^{{\tau }_i\left({\mathrm{A}}_1\right)}\frac{{\chi }_{i,j}\left({\mathrm{A}}_1\right)}{(j-1)!}\gamma (j,\frac{{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{(1-k_2^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}){\mu }_{〈i〉}})\hfill \end{array}$$

(25)

Substituting equations (25) and (24) into equation (22), equation (22) is given by equation (26)

$$F_{{\gamma }_{srd}}\left({\gamma }_{th}^{srd}\right)=(\begin{array}{l}{\displaystyle \sum _{v=1}^{N_1}}\left(\begin{array}{c}{N}_1\\ v\end{array}\right){\left(-1\right)}^v{\left[{\displaystyle \sum _{k_1=0}^{m_1-1}}\cdots {\displaystyle \sum _{k_{M_1}=0}^{m_1-1}}\Xi \left(M_1\right){\displaystyle \sum _{s=0}^{\Lambda -1}}{e}^{-\frac{{\Delta }_1{\gamma }_{th}^{srd}}{1-k_{1i}^2{\gamma }_{th}^{srd}}}\frac{\left(\Lambda -1\right)!{\left(1-k_{1i}^2{\gamma }_{th}^{srd}\right)}^{\Lambda -s}}{s!{\left({\gamma }_{th}^{srd}\right)}^{\Lambda -s}}\right]}^v\hfill \\ +{\displaystyle \sum _{i=1}^{\rho \left({\mathrm{A}}_1\right)}}{\displaystyle \sum _{j=1}^{{\tau }_i\left({\mathrm{A}}_1\right)}}\frac{{\chi }_{i\mathrm{,}j}\left({\mathrm{A}}_1\right)}{\left(j-1\right)!}\gamma \left(j\mathrm{,}\frac{{\gamma }_{th}^{srd}}{\left(1-k_2^2{\gamma }_{th}^{srd}\right){\mu }_{〈i〉}}\right)-{\displaystyle \sum _{i=1}^{\rho \left({\mathrm{A}}_1\right)}}{\displaystyle \sum _{j=1}^{{\tau }_i\left({\mathrm{A}}_1\right)}}\frac{{\chi }_{i\mathrm{,}j}\left({\mathrm{A}}_1\right)}{\left(j-1\right)!}\gamma \left(j\mathrm{,}\frac{{\gamma }_{th}^{srd}}{\left(1-k_2^2{\gamma }_{th}^{srd}\right){\mu }_{〈i〉}}\right)\hfill \\ \times {\displaystyle \sum _{v=1}^{N_1}}{N}_1\begin{array}{c}\\ v\end{array}{\left(-1\right)}^v{\left[{\displaystyle {\sum }_{k_1=0}^{m_1-1}\cdots {\displaystyle {\sum }_{k_{M_1}=0}^{m_1-1}\Xi \left(M_1\right){\displaystyle \sum _{s=0}^{\Lambda -1}}{e}^{-\frac{{\Delta }_1{\gamma }_{th}^{srd}}{1-k_{1i}^2{\gamma }_{th}^{srd}}}\frac{\left(\Lambda -1\right)!{\left(1-k_{1i}^2{\gamma }_{th}^{srd}\right)}^{\Lambda -s}}{s!{\left({\gamma }_{th}^{srd}\right)}^{\Lambda -s}}}}\right]}^v\mathrm{,}{\gamma }_{th}^{srd}<\mathrm{in}\left(\frac{1}{k_{1i}^2}\mathrm{,}\frac{1}{k_2^2}\right)\hfill \\ \mathrm{1,}{\gamma }_{th}^{srd}\ge \min \left(\frac{1}{k_{1i}^2}\mathrm{,}\frac{1}{k_2^2}\right)\hfill \end{array}$$

(26)

By substituting equations (18)–(21) and (26) into equation (17), the outage probability of the system is derived.

Throughput of the system

As mentioned in Diomidis and George, ²⁶ the throughput is defined as

$$T_e=p_{\mathrm{sd}}r+p_{\mathrm{srd}}\frac{r}{2}$$

(27)

where $r$ is the expected transmission rate mentioned before. By substituting equations (18) and (19) into equation (27), equation (27) is given by

$$T_e=\frac{F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)r}{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)+F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)}+\frac{F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)r}{2[F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_T\right)+F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_T\right)]}$$

(28)

Since the equation is too long, it is left out here.

Asymptotic analysis of the system

In this subsection, the asymptotic analysis of the considered system is derived. From the above analysis, we know that the key point of the analysis is deriving $p_{\mathrm{sd}}$ , $p_{\mathrm{srd}}$ , and $\mathrm{Pr}\{o|(B=Z,{\gamma }_Z>\omega )\}$ at high SNRs.

Before deriving the asymptotic expressions, we first give the PDF for the S-D link, S-R link, and R-D link, respectively. From Prabhat and Pankaj, ²⁰ the PDF of ${\lambda }_{\mathrm{sd}}$ is given by

$$f_{{\lambda }_{\mathrm{sd}}}\left({\lambda }_{\mathrm{sd}}\right)\approx \frac{{\alpha }_0}{{\overline{\lambda }}_{\mathrm{sd}}}+o\left({\lambda }_{\mathrm{sd}}\right)$$

(29)

Substituting equation (29) into equation (21), $F_{{\gamma }_{\mathrm{sd}}}({\gamma }_{\mathrm{th}}^{\mathrm{sd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}<1/k_{\mathrm{sd}}^2$ is given by

$$F_{{\gamma }_{\mathrm{sd}}}\left({\gamma }_{\mathrm{th}}^{\mathrm{sd}}\right)\approx \frac{{\alpha }_0}{{\overline{\lambda }}_{\mathrm{sd}}}\left(\frac{{\gamma }_{\mathrm{th}}^{\mathrm{sd}}}{1-k_{\mathrm{sd}}^2{\gamma }_{\mathrm{th}}^{\mathrm{sd}}}\right)+o\left({\lambda }_{\mathrm{sd}}\right)$$

(30)

From Prabhat and Pankaj, ²⁰ the PDF of ${\lambda }_{1i}$ at high SNRs is given by

$$f_{{\lambda }_{1i}}\left({\lambda }_{1i}\right)\approx \frac{{\alpha }_1^{M_1}{\lambda }_{1i}^{M_1-1}}{(M_1-1)!{\overline{\lambda }}_{1i}^{M_1}}+o\left({\lambda }_{1i}^{M_1}\right)$$

(31)

With the help of equations (5) and (23), $F_{{\gamma }_{1i}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}<1/k_{1i}^2$ is given by

$$F_{{\gamma }_{1i}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)=\frac{{\alpha }_1^{M_1}{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{M_1}}{\left(M_1\right)!{({\overline{\lambda }}_{1i}-{\overline{\lambda }}_{1i}{k}_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{M_1}}$$

(32)

As partial relay selection is used by the system, the final CDF of ${\gamma }_1$ with ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}<1/k_{1i}^2$ is given by

$$\begin{array}{c}{F}_{{\gamma }_1}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)={\left[F_{{\gamma }_{1i}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)\right]}^{N_1}\hfill \\ \multicolumn{1}{c}{=\frac{{\alpha }_1^{N_1{M}_1}{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{N_1{M}_1}}{{(M_1!)}^{N_1}{({\overline{\lambda }}_{1i}-{\overline{\lambda }}_{1i}{k}_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{N_1{M}_1}}}\end{array}$$

(33)

From Bletsas et al., ⁴⁰ the PDF of ${\lambda }_2$ at high SNRs which is given by

$$f_{{\lambda }_2}\left({\lambda }_2\right)=\frac{1}{{\overline{\lambda }}_2(M_1-1)!}{\left(\frac{{\lambda }_2}{{\overline{\lambda }}_2}\right)}^{M_1-1}$$

(34)

With the help of equations (9) and (34), $F_{{\gamma }_2}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ with ${\gamma }_{\mathrm{th}}^{\mathrm{srd}}<1/k_2^2$ is given by

$$F_{{\gamma }_2}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)=\frac{1}{M_1!}{\left(\frac{{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{{\overline{\lambda }}_2-{\overline{\lambda }}_2{k}_2^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}\right)}^{M_1}$$

(35)

Substituting equations (33) and (35) into equation (22), the asymptotic expression of $F_{{\gamma }_{\mathrm{srd}}}({\gamma }_{\mathrm{th}}^{\mathrm{srd}})$ is given by equation (36)

$$F_{{\gamma }_{\mathrm{srd}}}\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)=(\begin{array}{c}\frac{{\alpha }_1^{N_1{M}_1}{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{N_1{M}_1}}{{(M_1!)}^{N_1}{({\overline{\lambda }}_{1i}-{\overline{\lambda }}_{1i}{k}_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{N_1{M}_1}}+\frac{1}{M_1!}{\left(\frac{{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}{{\overline{\lambda }}_2-{\overline{\lambda }}_2{k}_2^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}}}\right)}^{M_1}\hfill \\ \multicolumn{1}{c}{-\frac{{\alpha }_1^{N_1{M}_1}{\left({\gamma }_{\mathrm{th}}^{\mathrm{srd}}\right)}^{(N_1+1)M_1}}{{(M_1!)}^{N_1+1}{({\overline{\lambda }}_{1i}-{\overline{\lambda }}_{1i}{k}_{1i}^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{N_1{M}_1}{({\overline{\lambda }}_2-{\overline{\lambda }}_2{k}_2^2{\gamma }_{\mathrm{th}}^{\mathrm{srd}})}^{M1}},{\gamma }_{\mathrm{th}}^{\mathrm{srd}}<min(\frac{1}{k_{1i}^2},\frac{1}{k_2^2})}\\ \multicolumn{1}{c}{1,{\gamma }_{\mathrm{th}}^{\mathrm{srd}}\ge min(\frac{1}{k_{1i}^2},\frac{1}{k_2^2})}\end{array}$$

(36)

In the same way, substituting equations (18)–(20), (30), and (36) into equation (17), the asymptotic expression for the outage probability is derived.

Substituting equations (30) and (36) into equation (28), the asymptotic expression for the throughput is derived. Above all are the asymptotic analysis for the considered system.

Numerical results

This section provides numerical results to validate the theoretical analysis and show the effects of key parameters on the system outage performance and the throughput. Here, the simulation results are obtained by performing ${10}^8$ channel realizations, with different shadowing scenarios of the satellite links (S-D, S-R), including frequent heavy shadowing (FHS) and average shadowing (AS) being given in Table 1. The channel parameters are located in Table 1.

Table 1.

Channel parameters.

Shadowing	$m_1$	$b_1$	${\Omega }_1$
Frequent heavy shadowing (FHS)	1	0.063	0.0007
Average shadowing (AS)	5	0.251	0.279

Some illustrations for the simulations and figures:

• The impairments level of the system $k_{\mathrm{sd}}^2=k_{1i}^2=k_2^2=k^2$ is assumed to be the same;

• $N_1=3$ and $M_1=3$ ;

• ${\overline{\lambda }}_{\mathrm{sd}}={\overline{\lambda }}_{1i}={\overline{\lambda }}_2=\overline{\lambda }$ is assumed to be the same for simulations;

• Case 1: $k=0$ ;

• Case 2: $k=0.1$ ;

• Case 3: $k=0.2$ .

Figure 2 plots the outage probability of the system with ${\gamma }_T=2\mathrm{dB}$ and $x_0=3\mathrm{dB}$ . First, we can find that the theory results are sufficiently across the simulation results which verify the correctness of the theory results. The asymptotic results are nearly the same with the simulation results at high SNRs, which show our asymptotic analysis is right. Second, we also can find that the system will have worse performance when the channel is under heavy channel fading at low SNRs. Finally, the outage probability of two channel conditions is same at high SNRs, which just has the relation with the impairments level. The larger the impairment level, the worse the system performance.

Figure 2.

Outage probability of the system with ${\gamma }_T=3\mathrm{dB}$ and $x_0=1\mathrm{dB}$ .

Figure 3 examines the outage probability of the system with ${\gamma }_T=1\mathrm{dB}$ and $x_0=3\mathrm{dB}$ . First, the simulation results are also sufficiently across the theory results. The asymptotic results are also nearly the same with the simulation results at high SNRs. Second, we also find that the system has worse performance when the channel is under heavy fading. Finally, compared with Figure 2, we know that the system will have better performance when ${\gamma }_T$ is larger.

Figure 3.

Outage probability of the system with ${\gamma }_T=1\mathrm{dB}$ and ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=1\mathrm{dB}$ .

Figure 4 depicts that the outage probability of the system with ${\gamma }_T=3\mathrm{dB}$ versus different ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}$ . First, the outage threshold will have a bound, which is testified by equation (17). When the threshold is larger than the bound, the outage probability will be always 1. But when the system has ideal hardware, there is no threshold bound and the outage probability will increase to 1 with the threshold increasing to infinite. Second, we can find that the system will have worse system performance when the channel is under heavy fading. Finally, we find that the threshold bound just only has relation with the impairments level. The larger the level, the lower the bound.

Figure 4.

Outage probability of the system with versus different ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}$ with ${\gamma }_T=3\mathrm{dB}$ .

Figure 5 illustrates the outage probability of the system with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=3\mathrm{dB}$ versus different ${\gamma }_T$ . First, from Figure 5, the outage probability will be a fixed value when $\overline{\lambda }$ is 30 dB. The reason is that when $\overline{\lambda }$ is larger enough, S does not need to change the transmitted link, and the current transmitted link is good enough to be stay. This is the character of the SSC transmitted scheme. Second, Figure 5 also tells us that the lower the impairment level, the larger the outage probability. Finally, the impact of hardware impairments will be weaker when the channel is under heavy channel fading in this situation.

Figure 5.

Outage probability of the system versus different ${\gamma }_T$ with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=3\mathrm{dB}$ .

Figure 6 shows the throughput of the system with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=3\mathrm{dB}$ . First, the simulation results are sufficient across the theory results, which show the correctness of our analytical analysis. Second, the throughput of the system will be larger when ${\gamma }_T$ grows larger which is proved in equation (27). Third, the throughput will have the maximal value when the $\overline{\lambda }$ increases to a suitable value. What is more, the larger the impairment level, the larger the optimal $\overline{\lambda }$ . Fourth, the heavier the channel fading, the lower the throughput; the larger the impairment level, the lower the throughput. Finally, from the simulation results, we find that the throughput of the system would be half the value of $r$ at high SNRs which means that S just uses the relay link to transmit the signal.

Figure 6.

The throughput of the system with ${\gamma }_{\mathrm{th}}^{\mathrm{sd}}=3\mathrm{dB}$ .

Conclusion

This article first introduced the SIN, and from the introduction, we knew that satellite–terrestrial network was an important component of SIN. The major contributions of this article were that this article first analyzed the system performance of the satellite multi-terrestrial relay network with hardware impairments using SSC scheme. Then, we proposed a new transmitted link selection scheme in a satellite–terrestrial network. SSC scheme was used by destination to have a balance between the complexity and efficiency. Especially, we derived the accurate and asymptotic closed-form expressions for the outage probability and throughput of the system, respectively. The results revealed that the outage floor appeared when the hardware impairments existed. In addition, a system threshold bound occurred once the system suffered from hardware impairments. In other words, the outage probability would be always 1 when the threshold was larger than the bound. What is more, from the simulation results, we found that when the transmitted SNR became larger enough, the outage probability would be a fixed value, which just had the relation with the impairment level. The throughput of the system would be half the value of $r$ at high SNRs which meant that S just used the relay link to transmit the signal. The results could make great contributions to the SIN networks in transmitted link selection scheme.