Diversity scheme based hybrid FSO/RF systems in satellite-aerial-ground integrated network with outdated CSI

Abstract

Satellite-aerial-ground integrated network (SAGIN) confronts significant challenges from weather-induced impairments and transmission attenuation. To address this issue, this paper proposes a weather-dependent hybrid FSO/RF switching scheme for SAGIN, accounting for both weather effects and outdated channel state information (CSI). Specifically, three different working modes and two diversity schemes are designed to further mitigate the impact of adverse weather. System performance is evaluated using outage probability (OP), average bit-error rate (BER), and ergodic capacity (EC). Asymptotic results in the high signal-to-noise ratio (SNR) regime and under perfect CSI conditions are also provided to offer further engineering insights. Numerical results reveal that the proposed scheme achieves over 8.8 dB gain in OP and more than 8.1 dB gain in average BER.

Introduction

Recently, with the rapid advancement of the Internet of Things (IoT) and the explosive demand for ubiquitous access, terrestrial cellular networks have struggled to meet requirements in terms of coverage and network capacity^1,2. Fortunately, the rapid development of low Earth orbit (LEO) satellites has enabled the proposal of a new satellite-aerial-ground integrated network (SAGIN) for seamless coverage^3,4. However, terabits/s throughput satellite feeder links are required to provide high data rates at affordable costs, posing significant challenges to traditional radio frequency (RF) transmission. Limited by spectrum resources and bandwidth constraints, more than 50 gateways are required to achieve 1 Terabit/s throughput with traditional RF links⁵, which is economically unacceptable and geographically unattainable.

Due to advantages of enormous available bandwidth, electromagnetic interference immunity, and license-free operation, free-space optical (FSO) communication emerges as an attractive solution for terabits/s feeder links^6,7. However, FSO links require strict line-of-sight (LOS) alignment, and attenuation caused by atmospheric turbulence and weather conditions (e.g. fog and rain) can significantly degrade FSO link performance⁸. An effective solution is to deploy the RF link as a backup, as FSO and RF links are typically affected by different weather conditions and simultaneous impairments are rare⁹.

Hybrid FSO/RF systems have been studied in terrestrial networks to enable high-speed and reliable transmission^10,11,12,13. In¹², the performance of hybrid FSO/RF three-dimensional heterogeneous networks was investigated, with reconfigurable intelligent surface (RIS)-assisted technology adopted in the RF link to enhance system secrecy performance. The authors of¹³ proposed a non-orthogonal multiple access (NOMA)-based hybrid FSO/RF system, incorporating transmit antenna selection (TAS) and nonlinear energy harvesting techniques. However, due to the limited coverage of terrestrial networks, integrating hybrid FSO/RF systems with satellites has gained increasing research attention^14,15,16,17. In¹⁴, the impact of weather conditions on hybrid FSO/RF systems in low Earth orbit (LEO) satellite scenarios was first explored, where aperture averaging technique was proposed to mitigate atmospheric turbulence and pointing errors¹⁵. analyzed high-throughput satellite (HTS) systems, demonstrating that performance was determined by the weakest FSO or RF link. The authors of¹⁶ studied deep space mixed RF/FSO systems, considering solar scintillation effects. In¹⁷, the performance of hybrid FSO/RF systems with high-altitude platform (HAP) relay nodes was investigated, revealing that HAP selection based on satellite-to-HAP channel conditions can further optimize system performance.

Considering the attenuation caused by long-distance transmission between LEO satellites and ground stations (GS), relay technology is an effective way to alleviate the attenuation in both FSO and RF links. This technology can also increase session duration between LEO satellites and GS¹⁸. Owing to environmental friendliness and cost-effectiveness, HAPs are widely employed as relay nodes. Considering relaying technology, hybrid FSO/RF systems integrated into SAGIN have been preliminarily studied in the existing literature^19,20,21,22. The authors of¹⁹ studied hybrid FSO/RF systems under weather and atmospheric influences, deploying reconfigurable intelligent surface (RIS) assisted unmanned aerial vehicle (UAV) to achieve spatial diversity²⁰. proposed a rate adaptation design for hybrid FSO/RF systems, enhancing system performance by reducing frequently switching between FSO and RF links. In²¹, a SAGIN-based hybrid FSO/RF system for uplink satellite communication was presented, where system reliability was improved through the deployment of backup RF links and HAPs. In our previous work²², we proposed hybrid FSO/RF systems with RIS-assisted UAVs, introducing three relay schemes to alleviate weather-induced impairments.

In practical scenarios, when considering variable gain relay, the instantaneous channel state information (CSI) of RF link is required at relay node to determine the amplification gain²³. Thus, the relay node should continuously monitor the source-to-relay link and periodically estimate the CSI. However, channel estimation errors and quantization errors lead to imperfect CSI estimation at the relay node. Moreover, due to long-distance transmission and the time-varying characteristics of channel statistics, the relay node often receives outdated CSI²⁴. Previous research has shown that outdated CSI significantly impacts hybrid FSO/RF systems²⁵, highlighting the need to incorporate outdated CSI into system models for enhanced practicality and precision. In²⁶, the impact of outdated CSI at the relay node was investigated, demonstrating that it can induce severe performance degradation²⁷. studied the performance of UAV-assisted multiuser mixed FSO/RF systems, proving that outdated CSI can seriously deteriorate system performance. The authors of²⁸ proposed a mixed FSO/RF relaying system, discussing the effects of feedback delay and interference signal strength²⁹. analyzed feedback delay and interference in mixed FSO/RF full-duplex relay systems, revealing the system performance was highly dependent on outdated CSI and interfering signals³⁰. explored system physical layer security with NOMA techniques applied to both FSO and RF links, showing that imperfect CSI significantly compromised secrecy performance.

Table 1 The comparison of some related FSO/RF systems.

On the other hand, when considering the influence of complex weather conditions, diversity technology is an effective approach to adapt to different weather conditions. The diversity scheme-based hybrid FSO/RF system achieves an optimal balance among reliability, adaptability, and implementation complexity in SAGIN scenarios, especially when addressing challenges posed by outdated CSI and environmental unpredictability. Compared to SSK which offers simplicity at the cost of limited capabilities³, MIMO that delivers higher performance but requires accurate CSI², and UAV solutions that provide mobility with added complexity³¹. Aerial diversity was firstly investigated in^19,20. For satellite-to-GS downlink communication scenarios, preliminary studies on deploying RIS-assisted UAVs for spatial diversity were conducted in^19,22, where system performance was significantly enhanced through diversity techniques. The authors of³² studied hybrid FSO/RF systems for satellite communications, deploying a HAP to implement spatial diversity. In²², considering satellite-to-GS uplink communication, the authors proposed diversity schemes to mitigate weather impacts, which were achieved by deploying an additional GS or HAP. Table 1 summarizes the existing research landscape. As shown, only a few studies^1,22,32,33 have addressed weather conditions, primarily analyzing their impact rather than designing specific mitigation schemes. Additionally, most research assumed perfect CSI scenario, with imperfect CSI scenario only explored in^24,30.

However, a critical gap identified in the literature is the absence of studies designing specific switching schemes to mitigate weather impacts. In practice, flexible weather-dependent switching schemes represent an urgent research need for alleviating the effects of complex weather conditions. In addition, while most studies focus on downlink scenarios, only a few works^21,32,33 have addressed uplink feeder links. Notably, these studies exclusively adopted decode-and-forward (DF) relay schemes and lacked comprehensive analysis of system performance. In fact, AF relay strategy is a better choice for reducing the hardware complexity of relay node. Furthermore^21,32,33, overlooked the impact of outdated CSI, which is a factor critical for accurate performance evaluation. Motivated by this research gaps, we proposed a flexible weather-dependent switching scheme in SAGIN, which is for the purpose of reducing the effect of weather conditions. The main contributions are as follows.

(1) In order to relieve the impact of complex weather conditions on hybrid FSO/RF links, we propose a flexible weather-dependent switching scheme in SAGIN, which is achieved by designing three different weather-dependent transmission modes. Specifically, we conduct further research on the influence of weather factors and validate the robustness of our proposed scheme under different weather conditions.

(2) A practical and accurate link model based on the AF scheme is proposed, accounting for the outdated CSI in the dual-hop RF/FSO relay link. We also derive a novel and exact expression for the cumulative distribution function (CDF) of dual-hop RF/FSO relay link under outdated CSI conditions.

(3) We derive closed form expressions of outage probability (OP), average bit-error rate (BER) and ergodic capacity (EC) for the proposed systems. The perfect CSI case is also derived as a benchmark to quantify the impact of outdated CSI. Based on these results, we conduct asymptotic analysis and diversity gain to provide engineering insights for link performance compensation.

(4) Based on the analysis, we present a comprehensive performance evaluation of the proposed system. Additionally, we conduct performance comparisons between the proposed schemes and the existing schemes. Simulation results show that the proposed scheme achieves more than an 8.8 dB gain in OP and over an 8.1 dB gain in average BER. Furthermore, the tradeoff between performance improvement and complexity for the two proposed diversity schemes is analyzed.

The remainder of this paper is organized as follows. In “System and channel models”, the system and channel models as well as site diversity scheme design are discussed in detail. The closed-form expressions for system performance are presented in “Performance evaluation”. Section “Asymptotic analysis” gives the asymptotic results in the high SNR region and the perfect CSI case of RF link is also provided. Section “Numerical results and discussion” gives the numerical and simulation results. Finally, concluding remarks and future work are drawn in “Conclusion”.

System and channel models

We consider the uplink communication between GS and LEO satellite, with HAPs deployed as relay nodes. The purpose of deploying the HAP is to mitigate attenuation caused by complex weather conditions and extend the session duration between GS and LEO satellite. Two system models are presented in Fig. 1. First, we consider the relay scheme with site diversity, as shown in Fig. 1a. HAP₁ and HAP₂ are deployed directly above the GS, which are connected via a terrestrial fiber optic link. The main GS is equipped with both FSO and RF transmitters while the backup GS is only equipped with an FSO transmitter. The FSO link between the main GS and HAP₁ is prioritized due to its superior capacity performance. The FSO link between backup GS and HAP₂ is adopted when the FSO link between main GS and HAP₁ is unavailable due to rainy weather. When both FSO links are unavailable due to foggy weather, the RF link between main GS and HAP₁ is activated. However, employing the RF link reduces system throughput because of its limited bandwidth.

The HAPs are assumed to operate at an altitude of 20 km. The zenith angles between main GS and HAP₁, as well as between backup GS and HAP₂ are assumed as 0°. The distance between main GS and backup GS as well as HAP₁ and HAP₂ is set as $20\sqrt 3$ km. The distance between two GS is assumed large enough to experience uncorrelated weather conditions. As shown in Fig. 1b, we consider a scheme with lower complexity when considering the increased economic cost and complexity of building an additional GS. Unlike Fig. 1a, when the FSO link between GS and HAP₁ is unavailable, the FSO link between GS and HAP₂ is used instead. The RF link is given lower priority owing to its limited capacity. The zenith angles between GS and HAP₁ is assumed as 0° while the zenith angles between GS and HAP₂ is assumed as 60°. The distance between HAP₁ and HAP₂ is calculated as $20\sqrt 3$ km and the distance between GS and HAP₂ is calculated as 40 km. Similarly, the uncorrelated weather conditions are assumed between HAP₁ and HAP₂. Both proposed schemes require only supplementary deployment of GS and HAP, maintaining low system complexity while enabling practical implementation. The key notations of this paper are also plotted in Fig. 1.

Fig. 1

Hybrid FSO/RF systems in SAGIN network: (a) relay scheme with site diversity, (b) low-complexity relay scheme with multiple HAPs.

GS-HAP-satellite signal models

At the HAP, the received signal and instantaneous SNR from GS over the FSO link during uplink transmission are given as

$${y_{{G_i}{H_k}}}=\sqrt {{P_{{G_i}}}} {I_{{G_i}{H_k}}}{x_{{G_i}}}+{n_{{H_k}}}$$

(1)

$${\gamma _{{G_i}{H_k}}}=\frac{{{P_{{G_i}}}I_{{{G_i}{H_k}}}^{2}}}{{\sigma _{{{H_k}}}^{2}}}=I_{{{G_i}{H_k}}}^{2}{\bar {\gamma }_{{G_i}{H_k}}}$$

(2)

where $i=1$ indicates the main GS, $i=2$ indicates the backup GS. Similarly, $k=1$ means the HAP₁ while $k=2$ means the HAP₂. ${P_{{G_i}}}$ is the transmit power of GS_i, ${I_{{G_i}{H_k}}}$ is the channel irradiance of GS_i-to-HAP_k FSO link, ${x_{{G_i}}}$ is the transmit signal, ${n_{{H_k}}}$ is the additive white Gaussian noise (AWGN) with variance $\sigma _{{{H_k}}}^{2}$, and ${\bar {\gamma }_{{G_i}{H_k}}}$ is the average SNR of GS-to-HAP FSO link defined as ${\bar {\gamma }_{{G_i}{H_k}}}={P_{{G_i}}}/\sigma _{{{H_k}}}^{2}$. Similarly, the received signal and instantaneous SNR at the HAP from GS over the RF link during uplink transmission is given as

$${y_{GHr}}=\sqrt {{P_{{G_i}}}{h_l}} {h_{GH}}{x_{{G_i}}}+{n_r}$$

(3)

$${\gamma _{RF}}=\frac{{{P_{{G_i}}}{h_l}|{h_{GH}}{|^2}}}{{\sigma _{r}^{2}}}=|{h_{GH}}{|^2}{\bar {\gamma }_{RF}}$$

(4)

where ${h_{GH}}$ is the channel gain of the RF link, ${n_r}$ is the AWGN with variance $\sigma _{r}^{2}$, and ${\bar {\gamma }_{RF}}$ is the average SNR of the RF link given as ${\bar {\gamma }_{RF}}={P_{{G_i}}}{h_l}/\sigma _{r}^{2}$. ${h_l}$ is path loss of the RF link defined as¹⁴.

$${h_l}\left[ {dB} \right]=G_{T}^{r}+G_{R}^{r} - 20{\log _{10}}(\frac{{4\pi {L_1}}}{f}) - {\omega _{oxy}}{L_1} - {\omega _{rain}}{L_1}$$

(5)

where $G_{T}^{r}$ and $G_{R}^{r}$ are the gains of the transmitting antenna and receiving antenna, ${L_1}$ is the length the RF link, f is the frequency of the RF link, ${\omega _{oxy}}$ and ${\omega _{rain}}$ are attenuation coefficients caused by oxygen and rainfall scattering, respectively. Due to channels estimation errors and delays, outdated CSI is obtained in the RF link. Considering the outdated CSI, the actual received SNR ${\hat {\gamma }_{RF}}$ of the RF link is a delayed version of ${\gamma _{RF}}$, with a correlation coefficient $\rho$²⁷. $\rho$ is defined as$\rho =J_{0}^{2}\left( {2\pi {f_d}\tau } \right)$³⁴, where ${J_0}( \cdot )$ is the zeroth-order Bessel function of first kind, ${f_d}$ is the Doppler frequency, and $\tau$ is the time delay.

When the FSO signal is received at the HAP, it is amplified with a fixed relay gain ${G_f}$ and then forwarded to the LEO satellite. During uplink transmission, the received signal and instantaneous SNR at the satellite from the HAP over the FSO link are given by

$${y_{{H_k}{S_f}}}=\sqrt {{P_{{H_k}}}} {G_f}\eta {I_{{H_k}S}}{x_{{H_k}S}}+{n_{sf}}$$

(6)

$${\gamma _{{H_k}{S_f}}}=\frac{{{P_{{H_k}}}G_{f}^{2}{\eta ^2}I_{{{H_k}S}}^{2}}}{{\sigma _{{sf}}^{2}}}=I_{{{H_k}S}}^{2}{\bar {\gamma }_{{H_k}{S_f}}}$$

(7)

where ${P_{{H_k}}}$ is the transmit power of HAP, $\eta$ is the optical-to-electrical conversion coefficient, ${I_{{H_k}S}}$ is the channel irradiance from HAP to satellite, ${x_{{H_k}S}}$ is the amplified signal, ${n_{sf}}$ is the AWGN with variance $\sigma _{{sf}}^{2}$, and ${\bar {\gamma }_{{H_k}{S_f}}}$ is the average SNR of the HAP-to-satellite FSO link defined as ${\bar {\gamma }_{{H_k}{S_f}}}=G_{f}^{2}{\eta ^2}{P_{{H_k}}}/\sigma _{{sf}}^{2}$.

When the RF signal is received at the HAP, it is firstly converted into optical signal by an electro-optic converter, and then forwarded to the LEO satellite with a variable gain ${G_r}$. The variable gain ${G_r}$ is bounded by²⁴

$$G_{r}^{2}=\frac{1}{{{P_{{G_1}}}{h_l}|{{\hat {h}}_{GH}}{|^2}}}$$

(8)

During uplink transmission, the received signal and instantaneous SNR at the satellite over the RF/FSO relay link are given as

$${y_{{H_k}{S_r}}}=\sqrt {{P_{{H_k}}}} {G_r}\eta {I_{{H_k}S}}{x_{{H_k}S}}+{n_{sr}}$$

(9)

$${\gamma _{{H_k}{S_r}}}=\frac{{{P_{{H_k}}}G_{r}^{2}{\eta ^2}I_{{{H_k}S}}^{2}}}{{\sigma _{{sr}}^{2}}}=I_{{{H_k}S}}^{2}{\bar {\gamma }_{{H_k}{S_r}}}$$

(10)

where ${n_{sr}}$ is the AWGN with variance $\sigma _{{sr}}^{2}$, and ${\bar {\gamma }_{{H_k}{S_r}}}$ is the average SNR defined as ${\bar {\gamma }_{{H_k}{S_r}}}=G_{r}^{2}{\eta ^2}{P_{{H_k}}}/\sigma _{{sr}}^{2}$.

GS-to-HAP FSO channel model

Since imperfect CSI in FSO links can be effectively compensated via short-term average optical power measurements, and its uncertainty imposes a substantially weaker impact on system performance compared to RF links. In this paper, we primarily focus on analyzing the effect of weather effects and outdated CSI on the RF link²⁶. Consequently, we adopt the assumption of perfect CSI for FSO links to decouple the cross-link imperfect CSI interactions. The GS-to-HAP FSO link is assumed to experience Málaga distribution with atmospheric attenuation and pointing error. The probability density function (PDF) of ${I_{{G_i}{H_k}}}$ is given as (³⁵ Eq. 5)

$${f_{{I_{{G_i}{H_k}}}}}({I_{{G_i}{H_k}}}) = \frac{{{\xi _1}^2{A_1}}}{{2{I_{{G_i}{H_k}}}}}\sum\limits_{{m_1} = 1}^{{\beta _1}} {{b_{{m_1}}}} G_{1,3}^{3,0}\left[ {\frac{{{\alpha _1}{\beta _1}}}{{(g{\beta _1} + {\Omega ^{\prime}})}}\frac{{{I_{{G_i}{H_k}}}}}{{{A_0}{I_l}}}\left| {\begin{matrix}{{\xi _1}^2 + 1} \\ {{\xi _1}^2,{\alpha _1},{m_1}} \\ \end{matrix} } \right.} \right]$$

(11)

where ${\alpha _1}$ is the scattering parameter which indicates the large-scale irradiance fluctuations, ${\beta _1}$ is a natural number which indicates the amount of fading parameter, and ${A_0}$ is the fraction of power collected at the beam center. ${I_l}$ is the path loss coefficient, which can be calculated by exponential Beers-Lambert Law as ${I_l}=\exp \left( { - {\alpha _0}{L_2}} \right)$ (¹⁹, Eq. 16), ${\alpha _0}$ is the weather dependent attenuation coefficient, ${L_2}$ is the length of FSO link, and ${b_{{m_1}}}={a_{{m_1}}}[{\alpha _1}{\beta _1}/(g{\beta _1}+{\Omega ^{\prime}})]$. The expressions for ${A_1}$ and ${a_{{m_1}}}$ are defined as

$$\begin{aligned}&{A_1} = \frac{{2{\alpha _1}^{{\alpha _1}/2}}}{{{g^{1 + {\alpha _1}/2}}\Gamma \left( {{\alpha _1}} \right)}}{\left( {\frac{{g{\beta _1}}}{{g{\beta _1} + \Omega^{\prime}}}} \right)^{{\beta _1} + {\alpha _1}/2}}\nonumber\\ & {a_{{m_1}}} = \left( {\begin{matrix}{{\beta _1} - 1} \\ {{m_1} - 1} \\ \end{matrix} } \right)\frac{{{{(g{\beta _1} + {\Omega ^{\prime}})}^{1 - {m_1}/2}}}}{{({m_1} - 1)!}}{\left( {\frac{{{\Omega ^{\prime}}}}{g}} \right)^{{m_1} - 1}}{\left( {\frac{{{\alpha _1}}}{{{\beta _1}}}} \right)^{{m_1}/2}}\end{aligned}$$

(12)

where g is the average power of the scattering component, ${\Omega ^{\prime}}$ is the average power from the coherent contributions. ${\xi _1}$ in (11) is the pointing errors coefficient which is defined as ${\xi _1}={\omega _{Leq}}/2{\sigma _j}$, ${\omega _{Leq}}$ is the equivalent beam waist, and ${\sigma _j}$ is the jitter standard deviation. ${\omega _{Leq}}$ is given by (⁹, Eq. 1)

$$\omega _{{Leq}}^{2}=\frac{{\omega _{L}^{2}\sqrt \pi erf\left( v \right)}}{{2v\exp ( - {v^2})}},{\omega _L} \approx {\omega _0}\sqrt {1+{\Theta _0}\left( {\frac{{\lambda L}}{{\pi \omega _{0}^{2}}}} \right)}$$

(13)

where $v=\frac{{\sqrt \pi {r_0}}}{{\sqrt 2 {\omega _L}}}$, ${r_0}$ is the receiver aperture radius, ${A_0}$ in Eq. (11) is defined as ${A_0}=er{f^2}\left( v \right)$, ${\omega _0}$ is the beam waist at $L=0$ and L is the link distance in meters, ${\Theta _0}=1+2\omega _{0}^{2}/\rho _{0}^{2}\left( L \right)$, ${\rho _0}\left( L \right)={\left( {0.55C_{n}^{2}k_{f}^{2}L} \right)^{ - 3/5}}$, $C_{n}^{2}$ is the refractive index parameter, ${k_f}$ is the optical wave number defined as ${k_f}=2\pi /\lambda$, and $\lambda$ is the wavelength of FSO link. Considering both heterodyne detection (HD) and intensity modulation with direct-detection (IM/DD), the PDF of ${\gamma _{{G_i}{H_k}}}$ can be derived from (11) given as (³⁵, Eq. 9)

$${f_{{\gamma _{{G_i}{H_k}}}}}\left( \gamma \right)=\frac{{{\xi _1}^{2}{A_1}}}{{{2^b}\gamma }}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {{b_{{m_1}}}} G_{{1,3}}^{{3,0}}\left[ {{B_1}{{\left( {\frac{\gamma }{{{\mu _1}}}} \right)}^{\frac{1}{b}}}\left| {\begin{array}{*{20}{c}} {{\xi _1}^{2}+1} \\ {{\xi _1}^{2},{\text{ }}{\alpha _1},{\text{ }}{m_1}} \end{array}} \right.} \right]$$

(14)

where ${B_1}=\xi _{1}^{2}{\alpha _1}{\beta _1}(g+{\Omega ^{\prime}})/[(\xi _{1}^{2}+1)(g{\beta _1}+{\Omega ^{\prime}})]$, ${\mu _1}={\bar {\gamma }_{{G_i}{H_k}}}$ for HD, ${\mu _1}=\frac{{\xi _{1}^{2}(\xi _{1}^{2}+2){\alpha _1}(g+{\Omega ^{\prime}})}}{{{{(\xi _{1}^{2}+1)}^2}\left( {{\alpha _1}+1} \right)[2g(g+2{\Omega ^{\prime}})+{\Omega ^{\prime}}^{2}(1+1/{\beta _1})]}}{\bar {\gamma }_{{G_i}{H_k}}}$ for IM/DD, $b=1$ for HD, and $b=2$ for IM/DD. The CDF of ${\gamma _{{G_i}{H_k}}}$ can be obtained from (14) using (³⁶, Eq. 07.34.21.0084.01) given by

$${F_{{\gamma _{{G_i}{H_k}}}}}(\gamma )={F^1}\sum\limits_{{{m_1}}}^{{{\beta _1}}} {{c_{{m_1}}}} G_{{b+1,3b+1}}^{{3b,{\text{ }}1}}\left( {\frac{{B_{1}^{b}\gamma }}{{{b^{2b}}{\mu _1}}}\left| {\begin{array}{*{20}{c}} 1&{{E_1}} \\ {{E_2}}&0 \end{array}} \right.} \right)$$

(15)

where ${F^1}=\frac{{\xi _{1}^{2}{A_1}}}{{{{\left( {2\pi } \right)}^{b - 1}}{2^b}}}$, ${c_{{m_1}}}={b_{{m_1}}}{b^{{\alpha _1}+{m_1} - 1}}$, ${E_1}=\left\{ {\frac{{{\xi _1}^{2}+1}}{b},...,\frac{{{\xi _1}^{2}+b}}{b}} \right\}$, ${E_2}=\left\{ {\frac{{{\xi _1}^{2}}}{b},...,\frac{{{\xi _1}^{2}+b - 1}}{b},...,\frac{{{\alpha _1}}}{b},...,\frac{{{\alpha _1}+b - 1}}{b},\frac{{{m_1}}}{b},...,\frac{{{m_1}+b - 1}}{b}} \right\}$.

GS-to-HAP RF channel model

The GS-to-HAP RF link is assumed to follow Nakagami-m fading and ${N_t}$ antennas are equipped to mitigate channel fading effects. The PDF and CDF of ${\gamma _{RF}}$ are given as²⁷, Eq. (11)

$${f_{{\gamma _{RF}}}}\left( \gamma \right)=\frac{1}{{\Gamma \left( {m{N_t}} \right)}}{\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right)^{m{N_t}}}{\gamma ^{m{N_t} - 1}}\exp \left( { - \frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right)$$

(16)

$${F_{{\gamma _{RF}}}}\left( \gamma \right)=1 - \frac{1}{{\Gamma \left( {m{N_t}} \right)}}\Gamma \left( {m{N_t},\frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right)$$

(17)

where m is the fading severity parameter. As ${\hat {\gamma }_{RF}}$ and ${\gamma _{RF}}$ are two correlated Gamma-distribution random variables, the PDF of ${\hat {\gamma }_{RF}}$ is given as²⁷, Eq. (15)

$${f_{{{\hat {\gamma }}_{RF}}|{\gamma _{RF}}}}\left( {x|y} \right)=\frac{1}{{1 - \rho }}\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right){\left( {\frac{x}{{\rho y}}} \right)^{\frac{{\left( {m{N_t} - 1} \right)}}{2}}}\exp \left( { - \frac{{m\left( {x+\rho y} \right)}}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right){I_{m{N_t} - 1}}\left( {\frac{{2m\sqrt {\rho xy} }}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)$$

(18)

Therefore, the joint PDF of ${\gamma _{RF}}$ and ${\hat {\gamma }_{RF}}$ can be calculated as

$${f_{{{\hat {\gamma }}_{RF}},{\gamma _{RF}}}}\left( {x,y} \right)={f_{{{\hat {\gamma }}_{RF}}|{\gamma _{RF}}}}\left( {x|y} \right){f_{{\gamma _{RF}}}}\left( y \right)$$

(19)

Substituting (16) and (18) into (19), the final expression of ${f_{{{\hat {\gamma }}_{RF}},{\gamma _{RF}}}}\left( {x,y} \right)$ is given as

$${f_{{{\hat {\gamma }}_{RF}},{\gamma _{RF}}}}\left( {x,y} \right)=\frac{1}{{1 - \rho }}\frac{1}{{\Gamma (m{N_t})}}{\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right)^{m{N_t}+1}}{\left( {\frac{{xy}}{\rho }} \right)^{\frac{{\left( {m{N_t} - 1} \right)}}{2}}}\exp \left( { - \frac{{m\left( {x+y} \right)}}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right){I_{m{N_t} - 1}}\left( {\frac{{2m\sqrt {\rho xy} }}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)$$

(20)

HAP-to-satellite FSO channel model

The HAP-to-satellite FSO link is also assumed to follow Málaga distribution with atmospheric attenuation and pointing error. We use ${\gamma _{{H_k}S}}$ to represent ${\gamma _{{H_k}{S_f}}}$ and ${\gamma _{{H_k}{S_r}}}$ uniformly and we also use ${\bar {\gamma }_{{H_k}S}}$ to represent ${\bar {\gamma }_{{H_k}{S_f}}}$ and ${\bar {\gamma }_{{H_k}{S_r}}}$ uniformly. Therefore, the CDF of ${\gamma _{{H_k}S}}$ can be obtained similarly to (15) expressed as

$${F_{{\gamma _{{H_k}S}}}}(\gamma )={F^2}\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{c_{{m_2}}}} G_{{b+1,3b+1}}^{{3b,{\text{ }}1}}\left( {\frac{{B_{2}^{b}\gamma }}{{{b^{2b}}{\mu _2}}}\left| {\begin{array}{*{20}{c}} 1&{{E_3}} \\ {{E_4}}&0 \end{array}} \right.} \right)$$

(21)

where${F^2}=\frac{{\xi _{2}^{2}{A_2}}}{{{{\left( {2\pi } \right)}^{b - 1}}{2^b}}}$, ${c_{{m_2}}}={b_{{m_2}}}{b^{{\alpha _2}+{m_2} - 1}}$, ${E_3}=\left\{ {\frac{{{\xi _2}^{2}+1}}{b},...,\frac{{{\xi _2}^{2}+b}}{b}} \right\}$, ${E_4}=\left\{ {\frac{{{\xi _2}^{2}}}{b},...,\frac{{{\xi _2}^{2}+b - 1}}{b},...,\frac{{{\alpha _2}}}{b},...,\frac{{{\alpha _2}+b - 1}}{b},\frac{{{m_2}}}{b},...,\frac{{{m_2}+b - 1}}{b}} \right\}$. ${A_2}$ and ${b_{{m_2}}}$ can be obtained from (12) by substituting ${\alpha _1}$, ${\beta _1}$, ${\xi _1}$ with ${\alpha _2}$, ${\beta _2}$, ${\xi _2}$, ${B_2}=\xi _{2}^{2}{\alpha _2}{\beta _2}(g+{\Omega ^{\prime}})/[(\xi _{2}^{2}+1)(g{\beta _2}+{\Omega ^{\prime}})]$, ${\mu _2}=\frac{{\xi _{2}^{2}(\xi _{2}^{2}+2){\alpha _2}(g+{\Omega ^{\prime}})}}{{{{(\xi _{2}^{2}+1)}^2}\left( {{\alpha _2}+1} \right)[2g(g+2{\Omega ^{\prime}})+{\Omega ^{\prime}}^{2}(1+1/{\beta _2})]}}{\bar {\gamma }_{{H_k}S}}$ for IM/DD, and ${\mu _2}={\bar {\gamma }_{{H_k}S}}$ for HD. We assume ${\alpha _1}<{\alpha _2}$, ${\beta _1}<{\beta _2}$ and ${\xi _1}<{\xi _2}$ considering that the GS-to-HAP FSO link is more affected by weather conditions compared to HAP-to-satellite FSO link²¹.

Site diversity scheme design

In this section, we provide a detailed description for the two schemes. For the relay scheme with site diversity illustrated in Fig. 1a, the system owns three working modes, as summarized in Table 2. For mode 0, we consider clear weather condition where the FSO link is in good condition, and the FSO/FSO relay is adopted to transmit signals from the main GS to the satellite. For mode 1, the foggy weather is considered where both the main GS FSO link and the backup GS FSO link are unavailable, and the RF/FSO relay is adopted to transmit signals from the main GS to the satellite. It should be noted that the data transmission rate decreases when mode 1 is adopted. For mode 2, we consider the rainy weather, where the FSO link of the main GS are in worse condition, and the backup FSO link is considered. Thus, for mode 2, the signal is first transmitted from the main GS to the backup GS via the ground optical network, and then forwarded from the backup GS to the satellite through relay node HAP₂. As for the proposed low-complexity relay scheme with multiple HAPs shown in Fig. 1b, it has the same working mechanism as site diversity scheme. The key distinction is that the signal is first transmitted to the backup HAP₂ rather than the backup GS, and then forwarded to satellite.

Table 2 Weather-dependent link selection scheme.

The HAP is deployed with weather sensors to detect the weather conditions and a 2-bit feedback signal is sent to the main GS to determine the selected transmission mode. The feedback signal is based on the weather conditions and instantaneous SNR received at the HAP.

Performance evaluation

In this section, we evaluate the comprehensive performance of the proposed hybrid FSO/RF relay systems in SAGIN. The AF relay scheme is assumed and the outdated CSI in RF link is also considered.

End-to-end CDF of hybrid FSO/RF systems

When both the GS-to-HAP link and the HAP-to-satellite link adopting FSO links, the end-to-end SNR ${\gamma _f}$ is expressed as

$${\gamma _f}=\frac{{{\gamma _{{G_i}{H_k}}}{\gamma _{{H_k}S}}}}{{{\gamma _{{H_k}S}}+{G_f}}}$$

(22)

Therefore, the closed-form expression for the end-to-end CDF of FSO/FSO relay systems is written as

$${F_{{\gamma _f}}}\left( \gamma \right)=\Pr \left[ {\frac{{{\gamma _{{G_i}{H_k}}}{\gamma _{{H_k}S}}}}{{{\gamma _{{H_k}S}}+{G_f}}}<\gamma } \right]$$

(23)

The final expression for ${F_{{\gamma _f}}}\left( \gamma \right)$ can be obtained in terms of the bivariate Fox’s H function shown as

$$\begin{gathered} {F_{{\gamma _f}}}\left( \gamma \right)={F_{{\gamma _{{G_i}{H_k}}}}}(\gamma ) - \frac{{{F^2}\xi _{1}^{2}{A_1}}}{{{2^b}}}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{b_{m1}}{c_{m2}}} } H_{{1,0:3,2:b,3b+1}}^{{0,1:0,3:3b+1,0}} \hfill \\ \left[ {\begin{array}{*{20}{c}} {(1,1,\frac{1}{b})} \\ - \end{array}} \right.\left| {\begin{array}{*{20}{c}} {(1 - \xi _{1}^{2},1),(1 - {\alpha _1},1),(1 - {m_1},1)} \\ {( - \xi _{1}^{2},1),{\text{ }}(0,\frac{1}{b})} \end{array}} \right.\left| {\begin{array}{*{20}{c}} {({E_3},1)} \\ {({E_4},1),(0,1)} \end{array}} \right.\left. {\left| {\frac{1}{{{B_1}}}{{\left( {\frac{{{\mu _1}}}{\gamma }} \right)}^{\frac{1}{b}}}} \right.,\frac{{{B_2}{G_f}}}{{{b^{2b}}{\mu _2}}}} \right] \hfill \\ \end{gathered}$$

(24)

Proof

Please refer to Appendix A.

When the GS-to-HAP link adopts RF link and the HAP-to-satellite link adopts FSO link, and considering the outdated CSI of RF link, the end-to-end SNR ${\gamma _r}$ is expressed as (²⁴, Eq. 5)

$${\gamma _r}=\frac{{{\gamma _{RF}}{\gamma _{{H_k}S}}}}{{{\gamma _{{H_k}S}}+{{\hat {\gamma }}_{RF}}}}$$

(25)

Therefore, the end-to-end CDF of RF/FSO relay systems can be obtained as (²⁴, Eq. 21)

$${F_{{\gamma _r}}}\left( \gamma \right)=\Pr \left[ {\frac{{{\gamma _{RF}}{\gamma _{{H_k}S}}}}{{{\gamma _{{H_k}S}}+{{\hat {\gamma }}_{RF}}}}<\gamma } \right]$$

(26)

The final expression for ${F_{{\gamma _r}}}\left( \gamma \right)$ is given as

$$\begin{gathered} {F_{{\gamma _r}}}\left( \gamma \right)=1 - \frac{1}{{\Gamma \left( {m{N_t}} \right)}}\Gamma \left( {m{N_t},\frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right)+\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{{k+m{N_t} - 1}} {\frac{{{\rho ^k}{{(1 - \rho )}^{n+1 - k}}}}{{k!\Gamma \left( {k+m{N_t}} \right)}}\frac{{{F^2}{c_{{m_2}}}}}{{\Gamma \left( {m{N_t}} \right)}}} } } \hfill \\ {\text{ }} \times \left( {\begin{array}{*{20}{c}} {k+m{N_t} - 1} \\ n \end{array}} \right){\left( {\frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right)^{k+m{N_t} - n - 1}}\exp \left( { - \frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)G_{{b+2,3b+2}}^{{3b+1,2}}\left( {\left. {\frac{{B_{2}^{b}\gamma }}{{{b^{2b}}{\mu _2}}}} \right|\begin{array}{*{20}{c}} {1,1 - k - m{N_t},{E_3}} \\ {{E_4},{\text{ }}n+1,{\text{ }}0} \end{array}} \right) \hfill \\ \end{gathered}$$

(27)

Proof

Please refer to Appendix B.

where ${F_{{\gamma _f}}}\left( \gamma \right)$ and ${F_{{\gamma _r}}}\left( \gamma \right)$ mean the CDF of mode 0 and mode 1, respectively. The CDF for mode 2 can be obtained similar to mode 0, but with different average SNR ${\mu _b}$, ${\alpha _b}$,${\beta _b}$, and ${\xi _b}$, which is due to the different weather conditions and link loss. ${\alpha _b}$, ${\beta _b}$, ${\xi _b}$ and ${\mu _b}$ are the parameters for the backup GS-to-HAP FSO relay link.

Outage probability

The FSO link and RF link will be in outage when the received SNR are lower than the predefined SNR. To simplify the analysis, we assume that both the FSO and RF links have the same outage threshold ${\gamma _{th}}$. The outage probability of the proposed hybrid FSO/RF relay systems is defined as

$$P_{{out}}^{{GS}}\left( {{\gamma _{th}}} \right)=\sum\limits_{{i=0}}^{2} {{\pi _i}} {P_{out,i}}\left( {{\gamma _{th}}} \right)$$

(28)

where ${P_{out,i}}\left( {{\gamma _{th}}} \right)={F_{{\gamma _i}}}\left( {{\gamma _{th}}} \right)$ means the outage probability of mode i $(i=0,1,2)$, and ${\pi _i}$ is the probability of mode i. $P_{{out}}^{{GS}}\left( {{\gamma _{th}}} \right)$ is the outage probability of GS-to-HAP hybrid FSO/RF systems, which means that both the main relay scheme and the backup relay scheme are in outage. The outage probability of mode 0 can be obtained by substituting ${\gamma _{th}}$ into (24) and the outage probability of mode 1 can be obtained by substituting ${\gamma _{th}}$ into (27). The outage probability of mode 2 can be obtained similar to mode 0 using the corresponding parameters ${\alpha _b}$, ${\beta _b}$, ${\xi _b}$ and ${\mu _b}$.

Average bit-error rate performance

In this section, we derive the average BER of the proposed hybrid FSO/RF relay systems. The average BER for different modulation schemes is defined as [5, Eq. (46)]

$${P_{e,i}}=\frac{\sigma }{{2\Gamma \left( p \right)}}\sum\limits_{{u=1}}^{{{n_0}}} {q_{u}^{p}\int_{0}^{\infty } {{x^{p - 1}}{e^{ - {q_u}x}}{F_{{\gamma _i}}}\left( x \right)dx} }$$

(29)

where $\sigma$, p, ${n_0}$ and ${q_u}$ are parameters related to different modulation techniques and detection methods⁵. The average BER of mode 0 is obtained by substituting (24) into (29) derived as

$$\begin{gathered} {P_{e,0}}=\frac{\sigma }{{2\Gamma \left( p \right)}}{F_p} - \frac{{{F^2}\xi _{1}^{2}{A_1}\sigma }}{{{2^{b+1}}\Gamma \left( p \right)}}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{b_{{m_1}}}{c_{{m_2}}}} } } H_{{1,0:3,3:b,3b+1}}^{{0,1;1,3:3b+1,0}} \hfill \\ \left[ {\begin{array}{*{20}{c}} {(1,1,1/b)} \\ - \end{array}\left| {\begin{array}{*{20}{c}} {(1 - \xi _{1}^{2},1),\left( {1 - {\alpha _1},1} \right),\left( {1 - {m_1},1} \right)} \\ {\left( {p,1/b} \right),\left( {0,1/b} \right),\left( { - \xi _{1}^{2},1} \right)} \end{array}\left| {\begin{array}{*{20}{c}} {({E_3},1)} \\ {({E_4},1),\left( {0,1} \right)} \end{array}\left| {\frac{1}{{{B_1}}}{{({\mu _1}{q_u})}^{1/b}},\frac{{B_{2}^{b}{G_f}}}{{{b^{2b}}{\mu _2}}}} \right.} \right.} \right.} \right] \hfill \\ \end{gathered}$$

(30)

Proof

Please refer to Appendix C.

Considering the foggy weather, where RF/FSO relay link is adopted, the average BER of mode 1 is derived by substituting (27) into (29) obtained as

$$\begin{gathered} {P_{e,1}}=\frac{{\sigma {n_0}}}{2} - \frac{\sigma }{{2\Gamma \left( p \right)\Gamma \left( {m{N_t}} \right)}}\sum\limits_{{u=1}}^{{{n_0}}} {G_{{2,2}}^{{2,1}}} \left( {\left. {\frac{m}{{{{\bar {\gamma }}_{RF}}{q_u}}}} \right|\begin{array}{*{20}{c}} {1 - p,{\text{ 1}}} \\ {0,{\text{ }}m{N_t}} \end{array}} \right)+\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{P} {\sum\limits_{{u=1}}^{{{n_0}}} {{Q^{P - n+p}}\left( {\begin{array}{*{20}{c}} P \\ n \end{array}} \right)} } } } {\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right)^{P - n}} \hfill \\ {\text{ }} \times \frac{{{\rho ^k}{{(1 - \rho )}^{n+1 - k}}\sigma {F^2}{c_{{m_2}}}q_{u}^{p}}}{{2k!\Gamma \left( {k+m{N_t}} \right)\Gamma (p)\Gamma \left( {m{N_t}} \right)}}G_{{b+3,3b+2}}^{{3b+1,3}}\left( {\left. {\frac{{B_{2}^{b}Q}}{{{b^{2b}}{\mu _2}}}} \right|\begin{array}{*{20}{c}} {1,{\text{ }} - P,{\text{ 1}}+n - P - p,{\text{ }}{E_3}} \\ {{E_4},{\text{ }}n+1,{\text{ }}0} \end{array}} \right) \hfill \\ \end{gathered}$$

(31)

Proof

Please refer to Appendix D.

Similarly, the average BER of mode 2 can be obtained similar to mode 0 with the corresponding parameters ${\mu _b}$, ${\alpha _b}$, ${\beta _b}$, and ${\xi _b}$. Therefore, the average BER of our proposed hybrid FSO/RF relay systems is given as${\bar {P}_e}=\sum\limits_{{i=0}}^{2} {{\pi _i}{P_{e,i}}}$.

Ergodic capacity

Ergodic capacity is an important index which determines the transmission capacity of the backbone link in SAGIN. For the proposed hybrid FSO/RF relay systems, the ergodic capacity for mode 0 can be expressed by (⁹, Eq. 31)

$${\bar {C}_{EC,0}}={{\text{W}}_f}\int_{0}^{\infty } {{{\log }_2}(1+\tau \gamma ){f_{{\gamma _f}}}(\gamma )} d\gamma$$

(32)

where ${W_f}$ is the bandwidth of FSO link, $\tau =e/2\pi$ for IM/DD technique and $\tau =1$ for heterodyne technique. ${f_{{\gamma _f}}}(\gamma )$ is the PDF of ${\gamma _f}$ ,which can be obtained by taking the derivative of (24) shown as

$$\begin{gathered} {f_{{\gamma _f}}}\left( \gamma \right)={f_{{\gamma _{{G_i}{H_k}}}}}(\gamma ) - \frac{{{F^2}\xi _{1}^{2}{A_1}}}{{{2^b}\gamma }}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{b_{m1}}{c_{m2}}} } H_{{1,0:3,2:b,3b+1}}^{{0,1:0,3:3b+1,0}} \hfill \\ \left[ {\begin{array}{*{20}{c}} {(1,1,\frac{1}{b})} \\ - \end{array}} \right.\left| {\begin{array}{*{20}{c}} {(1 - \xi _{1}^{2},1),(1 - {\alpha _1},1),(1 - {m_1},1)} \\ {( - \xi _{1}^{2},1),{\text{ }}(1,\frac{1}{b})} \end{array}} \right.\left| {\begin{array}{*{20}{c}} {({E_3},1)} \\ {({E_4},1),(0,1)} \end{array}} \right.\left. {\left| {\frac{1}{{{B_1}}}{{\left( {\frac{{{\mu _1}}}{\gamma }} \right)}^{\frac{1}{b}}}} \right.,\frac{{{B_2}{G_f}}}{{{b^{2b}}{\mu _2}}}} \right] \hfill \\ \end{gathered}$$

(33)

By applying (³⁶, Eq. 07.34.03.0456.01), the logarithm function in (32) can be written in form of Meijer G function shown as

$${\log _2}\left( {1+\tau \gamma } \right)=\frac{1}{{\ln 2}}G_{{{\text{ 2,2}}}}^{{1,2}}\left( {\left. {\tau \gamma } \right|\begin{array}{*{20}{c}} {1,1} \\ {1,0} \end{array}} \right)$$

(34)

Therefore, the ergodic capacity for mode 0 can be obtained as

$$\begin{gathered} {{\bar {C}}_{EC,0}}={F_s} - \frac{{{W_f}{F^2}\xi _{1}^{2}{A_1}}}{{{2^b}\ln 2}}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{b_{m1}}{c_{m2}}} } H_{{1,0:4,3:b,3b+1}}^{{0,1:1,4:3b+1,0}} \hfill \\ \left[ {\begin{array}{*{20}{c}} {(1,1,\frac{1}{b})} \\ - \end{array}} \right.\left| {\begin{array}{*{20}{c}} {{E_5},{\text{ }}(1,\frac{1}{b})} \\ {(1,\frac{1}{b}),( - \xi _{1}^{2},1),(0,\frac{1}{b})} \end{array}} \right.\left| {\begin{array}{*{20}{c}} {({E_3},1)} \\ {({E_4},1),{\text{ }}(0,1)} \end{array}} \right.\left. {\left| {\frac{1}{{{B_1}}}{{\left( {\tau {\mu _1}} \right)}^{\frac{1}{b}}}} \right.,\frac{{{B_2}{G_f}}}{{{b^{2b}}{\mu _2}}}} \right] \hfill \\ \end{gathered}$$

(35)

where ${E_5}=\left\{ {(1 - \xi _{1}^{2},1),(1 - {\alpha _1},1),(1 - {m_1},1)} \right\}$, which is for the simplicity of (35).

Proof

Please refer to Appendix E.

Similar to mode 0, the ergodic capacity for mode 1 can be calculated by (⁵, Eq. 49)

$${\bar {C}_{EC,1}}=\frac{{{W_r}\tau }}{{\ln (2)}}\int_{0}^{\infty } {{{(1+\tau \gamma )}^{ - 1}}F_{{{\gamma _r}}}^{c}(\gamma )} d\gamma$$

(36)

where $F_{{{\gamma _r}}}^{c}(\gamma )$ is the complementary CDF of ${F_{{\gamma _r}}}\left( \gamma \right)$ defined as $F_{{{\gamma _r}}}^{c}(\gamma )=1 - {F_{{\gamma _r}}}\left( \gamma \right)$, ${W_r}$ is the bandwidth of RF link. The final expression for mode 1 is derived as

$$\begin{gathered} {{\bar {C}}_{EC,1}}=\frac{{{W_r}}}{{\ln 2\Gamma \left( {m{N_t}} \right)}}G_{{2,3}}^{{3,1}}\left( {\left. {\frac{m}{{{{\bar {\gamma }}_{RF}}\tau }}} \right|\begin{array}{*{20}{c}} {0,{\text{ }}1} \\ {m{N_t},0,0} \end{array}} \right) - \sum\limits_{{k=0}}^{\infty } {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{P} {\frac{{{W_r}{F^2}{\rho ^k}{{(1 - \rho )}^{m{N_t}+1}}{c_{{m_2}}}\tau {{\bar {\gamma }}_{RF}}}}{{k!\Gamma \left( {k+m{N_t}} \right)\Gamma \left( {m{N_t}} \right)m\ln 2}}} } } \hfill \\ {\text{ }} \times \left( {\begin{array}{*{20}{c}} P \\ n \end{array}} \right)G_{{1,0:1,1:b+2,3b+2}}^{{0,1:1,1:3b,2}}\left( {\left. {\begin{array}{*{20}{c}} {n - P} \\ - \end{array}} \right|\begin{array}{*{20}{c}} 0 \\ 0 \end{array}\left| {\begin{array}{*{20}{c}} {1,{\text{ }} - P,{\text{ }}{E_3}} \\ {{E_4},{\text{ }}n+1,{\text{ }}0} \end{array}\left| {\frac{{\tau {{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}{m},\frac{{B_{2}^{b}{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}{{{b^{2b}}m{\mu _2}}}} \right.} \right.} \right) \hfill \\ \end{gathered}$$

(37)

Proof

Please refer to Appendix F.

Similarly, the ergodic capacity ${\bar {C}_{EC,2}}$ of mode 2 can be obtained similar to mode 0 with the corresponding parameters ${\mu _b}$, ${\alpha _b}$,${\beta _b}$, and ${\xi _b}$. Therefore, the ergodic capacity of our proposed hybrid FSO/RF relay systems is given as ${\bar {C}_{EC}}=\sum\limits_{{i=0}}^{2} {{\pi _i}} {\bar {C}_{EC,i}}$.

Asymptotic analysis

In this section, asymptotic analysis in the high SNR region is provided, where the coding gain and diversity order are obtained to get more engineering insights. Besides, perfect CSI case is also analyzed to reveal the impact of outdated CSI on system performance.

High SNR analysis of FSO/FSO relay

The outage probability of FSO/FSO relay in (28) is expressed by (24) in terms of bivariate Fox’s H function, which is a complex mathematical function. Therefore, we perform asymptotic analysis of (24) in the high SNR region. The asymptotic result for (24) can be obtained by using (³⁷, Eq. 1.1), and then applying (³⁸, Eq. 1.8.4), obtained as

$${P_{out,0}}\left( {{\gamma _{th}}} \right)\mathop \approx \limits^{{{\mu _1},{\mu _2} \gg 1}} {F^1}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {{c_{{m_1}}}\sum\limits_{{{k_0}=0}}^{{3b}} {\frac{{\prod\nolimits_{\begin{subarray}{l} j=1 \\ j \ne {k_0} \end{subarray} }^{{3b}} {\Gamma \left( {{E_{2j}} - {E_{2{k_0}}}} \right)} }}{{{E_{2{k_0}}}\prod\nolimits_{{j=2}}^{{b+1}} {\Gamma \left( {{E_{1j}} - {E_{2{k_0}}}} \right)} }}} } {\left( {\frac{{B_{1}^{b}{\gamma _{th}}}}{{{b^{2b}}{\mu _1}}}} \right)^{{E_{2{k_0}}}}}+\sum\limits_{{i=1}}^{6} {\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{X_i}{{\left( {\frac{{B_{1}^{b}B_{2}^{b}{\gamma _{th}}{G_f}}}{{{\mu _1}{\mu _2}}}} \right)}^{G_{{d0}}^{i}}}} } }$$

(38)

where ${E_{2j}}$ and ${E_{2{k_0}}}$ are the jth and k₀th terms of ${E_2}$, ${E_{1j}}$ means the jth of ${E_1}$, ${X_i}$ is the polynomial about ${\alpha _i}$, ${m_i}$,${\xi _i}$ (i.e., $i=1,2$) and b, and the expression for ${X_i}$ is not given here for simplicity. $G_{{d0}}^{i}$ means the ith of ${G_{d0}}$ with ${G_{d0}}=\left\{ {\frac{{{\alpha _1}}}{b},\frac{{{m_1}}}{b},\frac{{\xi _{1}^{2}}}{b},\frac{{{\alpha _2}}}{b},\frac{{{m_2}}}{b},\frac{{\xi _{2}^{2}}}{b}} \right\}$. From (38), we can find that ${P_{out,0}}\left( {{\gamma _{th}}} \right) \propto {\left( {{G_c}{\mu _1}} \right)^{ - {G_d}}}$ and ${P_{out,0}}\left( {{\gamma _{th}}} \right) \propto {\left( {{G_c}{\mu _2}} \right)^{ - {G_d}}}$, where ${G_c}$ means the coding gain given by ${G_c}=\frac{1}{{B_{1}^{b}B_{2}^{b}{\gamma _{th}}{G_f}}}{\left( {\sum\limits_{{i=1}}^{6} {\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{X_i}} } } } \right)^{ - \frac{1}{{{G_d}}}}}$ and ${G_d}$ is the diversity order defined as ${G_d}=\hbox{min} \left\{ {\frac{{{\alpha _1}}}{b},\frac{{{m_1}}}{b},\frac{{\xi _{1}^{2}}}{b},\frac{{{\alpha _2}}}{b},\frac{{{m_1}}}{b},\frac{{\xi _{2}^{2}}}{b}} \right\}$. When considering ${\alpha _1}<{\alpha _2}$, ${\beta _1}<{\beta _2}$ and ${\xi _1}<{\xi _2}$, ${G_d}$ can be further expressed as ${G_d}=\hbox{min} \left\{ {\frac{{{\alpha _1}}}{b},\frac{{{m_1}}}{b},\frac{{\xi _{1}^{2}}}{b}} \right\}$. From the expression of diversity order, we can easily find that system outage probability is mainly affected by the detection technology, turbulence parameters and pointing errors. Similar to the analysis in (38), the asymptotic result for (30) can be derived as

$${P_{e,0}}\mathop \approx \limits^{{{\mu _1},{\mu _2} \gg 1}} {F^1}\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {{c_{{m_1}}}\sum\limits_{{u=1}}^{{{n_0}}} {\sum\limits_{{{k_0}=0}}^{{3b}} {\frac{{\sigma \prod\nolimits_{\begin{subarray}{l} j=1 \\ j \ne {k_0} \end{subarray} }^{{3b}} {\Gamma \left( {{E_{2j}} - {E_{2{k_0}}}} \right)\Gamma \left( {p+{E_{2{k_0}}}} \right)} }}{{2\Gamma \left( p \right){E_{2{k_0}}}\prod\nolimits_{{j=3}}^{{b+2}} {\Gamma \left( {{E_{1j}} - {E_{2{k_0}}}} \right)} }}} } } {\left( {\frac{{B_{1}^{b}}}{{{b^{2b}}{q_u}{\mu _1}}}} \right)^{{E_{2{k_0}}}}}+\sum\limits_{{i=1}}^{6} {\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{Y_i}{{\left( {\frac{{B_{1}^{b}B_{2}^{b}{G_f}}}{{{q_u}{\mu _1}{\mu _2}}}} \right)}^{G_{{d0}}^{i}}}} } } }$$

(39)

where ${Y_i}$ is the polynomial about ${\alpha _i}$, ${m_i}$,${\xi _i}$ (i.e., $i=1,2$) and b. Similar to the analysis for (38), we have ${P_{e,0}} \propto {\left( {{G_c}{\mu _1}} \right)^{ - {G_d}}}$ and ${P_{e,0}} \propto {\left( {{G_c}{\mu _2}} \right)^{ - {G_d}}}$, where ${G_c}$ is the coding gain defined as ${G_c}=\frac{{{q_u}}}{{B_{1}^{b}B_{2}^{b}{G_f}}}{\left( {\sum\limits_{{i=1}}^{6} {\sum\limits_{{{m_1}=1}}^{{{\beta _1}}} {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{Y_i}} } } } } \right)^{ - \frac{1}{{{G_d}}}}}$. Therefore, the diversity order of ${P_{e,0}}$ is given as ${G_d}=\hbox{min} \left\{ {\frac{{{\alpha _1}}}{b},\frac{{{m_1}}}{b},\frac{{\xi _{1}^{2}}}{b}} \right\}$.

High SNR analysis of RF/FSO relay

When ${\bar {\gamma }_{RF}}$ is large enough with small value of $\rho$, and ${\mu _2}$ is high enough under weak turbulence conditions, the expression for ${P_{out,1}}\left( {{\gamma _{th}}} \right)$ in (27) can be validly approximated using the first term ($k=0$), shown as

$$\begin{gathered} {P_{out,1}}\left( {{\gamma _{th}}} \right)\mathop \approx \limits^{{{{\bar {\gamma }}_{RF}},{\mu _2} \gg 1}} 1 - \frac{1}{{\Gamma \left( {m{N_t}} \right)}}\Gamma \left( {m{N_t},\frac{{m{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}}}} \right)+\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{{m{N_t} - 1}} {\frac{{{{(1 - \rho )}^{n+1}}}}{{\Gamma \left( {m{N_t}} \right)}}} } \frac{{{F^2}{c_{{m_2}}}}}{{\Gamma \left( {m{N_t}} \right)}}\left( {\begin{array}{*{20}{c}} {m{N_t} - 1} \\ n \end{array}} \right){\left( {\frac{{m{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}}}} \right)^{m{N_t} - n - 1}} \hfill \\ {\text{ }} \times \exp \left( { - \frac{{m{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)\sum\limits_{{{k_1}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_1}=1 \\ {j_1} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_1}}} - {E_{6{k_1}}}} \right)\Gamma \left( {m{N_t}+{E_{6{k_1}}}} \right)} }}{{{E_{6{k_1}}}\prod\nolimits_{{j=3}}^{{b+2}} {\Gamma \left( {{E_{3{j_1}}} - {E_{6{k_1}}}} \right)} }}} {\left( {\frac{{B_{2}^{b}{\gamma _{th}}}}{{{b^{2b}}{\mu _2}}}} \right)^{{E_{6{k_1}}}}} \hfill \\ \end{gathered}$$

(40)

where ${E_6}=\left\{ {{E_4},n+1} \right\}$, ${E_{6{j_1}}}$ and ${E_{6{k_1}}}$ are the j₁th and k₁th terms of ${E_6}$, and ${E_{3{j_1}}}$ means the j₁th of ${E_3}$. Similar to the analysis in (38), we have ${P_{out,1}}\left( {{\gamma _{th}}} \right) \propto {\left( {{G_c}{\mu _2}} \right)^{ - {G_d}}}$, where the diversity order is given by ${G_d}=\hbox{min} \left\{ {m{N_t},\frac{{{\alpha _2}}}{b},\frac{{{m_2}}}{b},\frac{{\xi _{2}^{2}}}{b}} \right\}$. The coding gain ${G_c}$ is given by

$$\begin{gathered} {G_c}=\frac{{{b^{2b}}}}{{B_{2}^{b}{\gamma _{th}}}}\left[ {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{{m{N_t} - 1}} {\frac{{{{(1 - \rho )}^{n+1}}}}{{\Gamma \left( {m{N_t}} \right)}}} } \times \frac{{{F^2}{c_{{m_2}}}}}{{\Gamma \left( {m{N_t}} \right)}}\left( {\begin{array}{*{20}{c}} {m{N_t} - 1} \\ n \end{array}} \right){{\left( {\frac{{m{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}}}} \right)}^{m{N_t} - n - 1}}} \right. \hfill \\ {\left. { \times \exp \left( { - \frac{{m{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)\sum\limits_{{{k_1}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_1}=1 \\ {j_1} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_1}}} - {E_{6{k_1}}}} \right)\Gamma \left( {m{N_t}+{E_{6{k_1}}}} \right)} }}{{{E_{6{k_1}}}\prod\nolimits_{{j=3}}^{{b+2}} {\Gamma \left( {{E_{3{j_1}}} - {E_{6{k_1}}}} \right)} }}} } \right]^{ - \frac{1}{{{G_d}}}}} \hfill \\ \end{gathered}$$

(41)

Furthermore, when considering the case $m=1$ and ${N_t}=1$, (40) can be further simplified to

$${P_{out,1}}\left( {{\gamma _{th}}} \right)\mathop \approx \limits^{{{{\bar {\gamma }}_{RF}},{\mu _2} \gg 1}} 1 - \Gamma \left( {1,\frac{{{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}}}} \right)+\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {(1 - \rho )} {F^2}{c_{{m_2}}}\exp \left( { - \frac{{{\gamma _{th}}}}{{{{\bar {\gamma }}_{RF}}\left( {1 - \rho } \right)}}} \right)\sum\limits_{{{k_1}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_1}=1 \\ {j_1} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_1}}} - {E_{6{k_1}}}} \right)\Gamma \left( {{E_{6{k_1}}}} \right)} }}{{\prod\nolimits_{{j=3}}^{{b+2}} {\Gamma \left( {{E_{3{j_1}}} - {E_{6{k_1}}}} \right)} }}} {\left( {\frac{{B_{2}^{b}{\gamma _{th}}}}{{{b^{2b}}{\mu _2}}}} \right)^{{E_{6{k_1}}}}}$$

(42)

From (40) and (42), we can find that when the value of ${\bar {\gamma }_{RF}}$ is very high, the value of the second term in (40) and (42) approaches 1, and the value of $\exp ( \cdot )$ also tends to 1. Therefore, the outage performance is mainly determined by the FSO link. However, when the value of ${\mu _2}$ is very large, the Meijer’s G function in (27) tends to zero, and the outage probability is mainly determined by the value of the second term in (40) and (42). In this case, the outage performance is not affected by outdated CSI, but determined by the average SNR of RF link and the outage threshold ${\gamma _{th}}$. Similar to the analysis in (40), the asymptotic result for (31) can be derived as

$$\begin{gathered} {P_{e,1}}\mathop \approx \limits^{{{{\bar {\gamma }}_{RF}},{\mu _2} \gg 1}} \frac{{\sigma {n_0}}}{2} - \frac{\sigma }{{2\Gamma \left( p \right)\Gamma \left( {m{N_t}} \right)}} \times \sum\limits_{{u=1}}^{{{n_0}}} {\sum\limits_{{{k_2}=0}}^{2} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_2}=1 \\ {j_2} \ne {k_2} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{7{j_2}}} - {E_{7{k_2}}}} \right)} }}{{\Gamma \left( {1 - {E_{7{k_2}}}} \right)}}} } \Gamma \left( {p+{E_{7{k_2}}}} \right){\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}{q_u}}}} \right)^{{E_{7{k_2}}}}} \hfill \\ {\text{ }}+\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{{{P_1}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{Q^{{P_1} - n+p}}\frac{{{{(1 - \rho )}^{n+1}}\sigma {F^2}{c_{{m_2}}}q_{u}^{p}}}{{{\Gamma ^2}\left( {m{N_t}} \right)\Gamma (p)}}} } } \times \left( {\begin{array}{*{20}{c}} {{P_1}} \\ n \end{array}} \right){\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right)^{{P_1} - n}}\Gamma \left( {p+{P_1} - n+{E_{6{k_1}}}} \right) \hfill \\ {\text{ }} \times \sum\limits_{{{k_3}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_3}=1 \\ {j_3} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_3}}} - {E_{6{k_3}}}} \right)\Gamma \left( {1+{P_1}+{E_{6{k_3}}}} \right)} }}{{{E_{6{k_3}}}\prod\nolimits_{{{j_3}=4}}^{{b+3}} {\Gamma \left( {{E_{3{j_3}}} - {E_{6{k_3}}}} \right)} }}} {\left( {\frac{{B_{2}^{b}Q}}{{{b^{2b}}{\mu _2}}}} \right)^{{E_{6{k_3}}}}} \hfill \\ \end{gathered}$$

(43)

where ${E_7}=\left\{ {0,m{N_t}} \right\}$, ${E_{7{j_2}}}$ and ${E_{7{k_2}}}$ are the j₂th and k₂th terms of ${E_7}$, ${P_1}=m{N_t} - 1$. Similar to the analysis in (40), we have ${P_{e,1}} \propto {\left( {{G_c}{\mu _2}} \right)^{ - {G_d}}}$, where the diversity order is given by ${G_d}=\hbox{min} \left\{ {m{N_t},\frac{{{\alpha _2}}}{b},\frac{{{m_2}}}{b},\frac{{\xi _{2}^{2}}}{b}} \right\}$. The coding gain ${G_c}$ is given as

$$\begin{gathered} {G_c}=\frac{{{b^{2b}}}}{{B_{2}^{b}Q}}\left[ {\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{n=0}}^{{{P_1}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{Q^{{P_1} - n+p}}\frac{{{{(1 - \rho )}^{n+1}}\sigma {F^2}{c_{{m_2}}}q_{u}^{p}}}{{{\Gamma ^2}\left( {m{N_t}} \right)\Gamma (p)}}} } } \left( {\begin{array}{*{20}{c}} {{P_1}} \\ n \end{array}} \right){{\left( {\frac{m}{{{{\bar {\gamma }}_{RF}}}}} \right)}^{{P_1} - n}}} \right. \hfill \\ {\left. { \times \sum\limits_{{{k_3}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_3}=1 \\ {j_3} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_3}}} - {E_{6{k_3}}}} \right)\Gamma \left( {1+{P_1}+{E_{6{k_3}}}} \right)} }}{{{E_{6{k_3}}}\prod\nolimits_{{{j_3}=4}}^{{b+3}} {\Gamma \left( {{E_{3{j_3}}} - {E_{6{k_3}}}} \right)} }}} \Gamma \left( {p+{P_1} - n+{E_{6{k_1}}}} \right)} \right]^{ - \frac{1}{{{G_d}}}}} \hfill \\ \end{gathered}$$

(44)

Furthermore, when considering $m=1$ and ${N_t}=1$, (43) can be further simplified to

$$\begin{gathered} {P_{e,1}}\mathop \approx \limits^{{{{\bar {\gamma }}_{RF}},{\mu _2} \gg 1}} \frac{{\sigma {n_0}}}{2} - \frac{\sigma }{{2\Gamma \left( p \right)}}\sum\limits_{{u=1}}^{{{n_0}}} {\sum\limits_{{{k_2}=0}}^{2} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_2}=1 \\ {j_2} \ne {k_2} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{7{j_2}}} - {E_{7{k_2}}}} \right)} }}{{\Gamma \left( {1 - {E_{7{k_2}}}} \right)}}} } \Gamma \left( {p+{E_{7{k_2}}}} \right){\left( {\frac{1}{{{{\bar {\gamma }}_{RF}}{q_u}}}} \right)^{{E_{7{k_2}}}}} \hfill \\ +\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {\sum\limits_{{u=1}}^{{{n_0}}} {{Q^p}\frac{{(1 - \rho )\sigma {F^2}{c_{{m_2}}}q_{u}^{p}}}{{\Gamma (p)}}} } \sum\limits_{{{k_3}=0}}^{{3b+1}} {\frac{{\prod\nolimits_{\begin{subarray}{l} {j_3}=1 \\ {j_3} \ne {k_1} \end{subarray} }^{{3b+1}} {\Gamma \left( {{E_{6{j_3}}} - {E_{6{k_3}}}} \right)\Gamma \left( {{E_{6{k_3}}}} \right)\Gamma \left( {p+{E_{6{k_1}}}} \right)} }}{{\prod\nolimits_{{{j_3}=4}}^{{b+3}} {\Gamma \left( {{E_{3{j_3}}} - {E_{6{k_3}}}} \right)} }}} {\left( {\frac{{B_{2}^{b}Q}}{{{b^{2b}}{\mu _2}}}} \right)^{{E_{6{k_3}}}}} \hfill \\ \end{gathered}$$

(45)

From (43) and (45), we can find that when ${\bar {\gamma }_{RF}}$ tends to infinity, ${P_{e,1}}$ is mainly determined by ${\mu _2}$, correlation coefficient $\rho$and the detection technique adopted. However, when ${\mu _2}$ tends to infinity, the floor of ${P_{e,1}}$ is mainly limited by ${\bar {\gamma }_{RF}}$ and the modulation format applied, and it is unaffected by the outdated CSI.

RF/FSO relay under perfect CSI case

Considering the case $\rho =1$, which means that ${\gamma _{RF}}$ and ${\hat {\gamma }_{RF}}$ in (25) are fully-correlated. Therefore, the overall SNR can be rewritten as ${\gamma _N}=\frac{{{\gamma _{RF}}{\gamma _{{H_k}S}}}}{{{\gamma _{{H_k}S}}+{\gamma _{RF}}}}$, and ${\gamma _N}$ can be approximated as ${\gamma _N}=\hbox{min} \left( {{\gamma _{RF}},{\gamma _{{H_k}S}}} \right)$³⁹. Hence, the CDF of RF/FSO relay under perfect CSI case is written as

$$\begin{gathered} {F_N}\left( \gamma \right)=1 - {{\bar {F}}_{RF}}\left( \gamma \right){{\bar {F}}_{{H_k}S}}\left( \gamma \right) \hfill \\ {\text{ }}={F_{RF}}\left( \gamma \right)+{F_{{H_k}S}}\left( \gamma \right) - {F_{RF}}\left( \gamma \right){F_{{H_k}S}}\left( \gamma \right) \hfill \\ \end{gathered}$$

(46)

where ${\bar {F}_{RF}}\left( \gamma \right)$and ${\bar {F}_{{H_k}S}}\left( \gamma \right)$ are the complementary CDF ${F_{RF}}\left( \gamma \right)$ and ${F_{{H_k}S}}\left( \gamma \right)$. Substituting (17) and (21) into (46), ${F_N}\left( \gamma \right)$ is obtained as

$${F_N}\left( \gamma \right)=1 - \frac{1}{{\Gamma \left( {m{N_t}} \right)}}\Gamma \left( {m{N_t},\frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right)+\frac{1}{{\Gamma \left( {m{N_t}} \right)}}\Gamma \left( {m{N_t},\frac{{m\gamma }}{{{{\bar {\gamma }}_{RF}}}}} \right){F^2}\sum\limits_{{{m_2}=1}}^{{{\beta _2}}} {{c_{{m_2}}}} G_{{b+1,3b+1}}^{{3b,{\text{ }}1}}\left( {\frac{{B_{2}^{b}\gamma }}{{{b^{2b}}{\mu _2}}}\left| {\begin{array}{*{20}{c}} 1&{{E_3}} \\ {{E_4}}&0 \end{array}} \right.} \right)$$

(47)

The outage probability of RF/FSO relay under prefect CSI case can be defined as $P_{{out}}^{{\rho =1}}={F_N}\left( {{\gamma _{th}}} \right)$. Similar to the derivation process of (31), the average BER of RF/FSO relay under perfect CSI case is obtained as

$$\begin{gathered} P_{e}^{{\rho =1}}=\frac{{\sigma {n_0}}}{2} - \frac{\sigma }{{2\Gamma \left( p \right)\Gamma \left( {m{N_t}} \right)}}\sum\limits_{{u=1}}^{{{n_0}}} {G_{{2,2}}^{{2,1}}\left( {\left. {\frac{m}{{{\gamma _{RF}}{q_u}}}} \right|\begin{array}{*{20}{c}} {1 - p,1} \\ {m{N_t},0} \end{array}} \right)} +\frac{{\sigma {F^2}{c_{m2}}}}{{2\Gamma \left( p \right)\Gamma \left( {m{N_t}} \right)}} \hfill \\ {\text{ }} \times \sum\limits_{{u=1}}^{{{n_0}}} {G_{{1,0:1,2:b+1,3b+1}}^{{0,1:2,0:3b,1}}\left( {\begin{array}{*{20}{c}} {1 - p} \\ - \end{array}\left| {\begin{array}{*{20}{c}} 1 \\ {m{N_t},0} \end{array}\left| {\begin{array}{*{20}{c}} {1,{E_3}} \\ {{E_4},0} \end{array}\left| {\frac{m}{{{{\bar {\gamma }}_{RF}}{q_u}}},\frac{{B_{2}^{b}}}{{{b^{2b}}{\mu _2}{q_u}}}} \right.} \right.} \right.} \right)} \hfill \\ \end{gathered}$$

(48)

Numerical results and discussion

In this section, we present the performance evaluation of our proposed switching scheme in terms of outage probability, average BER and ergodic capacity. Monte Carlo simulations are adopted to compare with the obtained analytical results over ${10^7}$ realizations⁴⁰. Moreover, comparison schemes are also demonstrated to prove the superiority of the proposed schemes. The system parameters are presented in Table 3 and the weather dependent variables are listed in Table 4. In the simulations for outage probability, the threshold is assumed to ${\gamma _{th}}=5dB$ and we set ${\mu _1}={\mu _2}={\bar {\gamma }_{RF}}$ in all the simulations. ${\omega _{oxy}}=0.1$ dB/km is assumed and 50 terms of k are used for calculating system performance. We consider three different pointing errors case with ${\sigma _j}/{r_0}=4.5$, ${\sigma _j}/{r_0}=10$ and ${\sigma _j}/{r_0}=15$, which indicate the weak point error, moderate point error and strong point error, respectively. The values of $\xi$ are calculated by using (13). Without loss of generality, we set that different modes have the same probability of occurrence. Furthermore, the parameter settings for atmospheric turbulence are the same as those in^5,41, where the GS-to-HAP FSO link is assumed to experience moderate turbulence while the HAP-to-satellite FSO link is assumed to experience weak turbulence.

The comparison schemes are defined as follows. The SAG-FSO/RF relay scheme is proposed in²¹, where the links between the GS and the HAP are hybrid FSO/RF links, while the link between the HAP and the satellite is FSO link. The SAG-FSO relay scheme is proposed in⁴², where the link between the GS and the HAP and the link between the HAP and the satellite are both FSO links. The SAG-FSO relay with site diversity is proposed in³³, where an additional GS is deployed as a backup. In simulations, the average SNR of FSO/RF link is defined as the ratio of power to noise. We use this as a benchmark, and the specific average SNR values under different weather conditions can be calculated based on attenuation formulas.

Table 3 System parameters.

Table 4 Weather-dependent variables⁴³.

The effect of outdated CSI on RF/FSO relay

The outage probability of RF/FSO relay scheme versus SNR for different values of correlation coefficient $\rho$ and antennas number ${N_t}$ is plotted in Fig. 2, where both IM/DD and HD techniques are considered. The pointing error is assumed to be weak pointing error with $\xi =3.67$. The theoretical analysis results match well with the simulation results, which prove the accuracy of our derivation. From Fig. 2, it can be observed that outdated CSI deteriorates system outage performance, especially in the case $\rho =0.2$. As shown in Fig. 2a, considering the case $m=1$, ${N_t}=1$, and to achieve an outage probability of $5 \times {10^{ - 3}}$, additional 2.3 dB is needed for $\rho =0.8$ while additional 5.5 dB is needed for $\rho =0.2$, compared to the perfect CSI case. Similarly, as shown in Fig. 2b, considering the case $m=2$, ${N_t}=2$, and to achieve an outage probability of $1 \times {10^{ - 3}}$, additional 1 dB is needed for $\rho =0.8$ while additional 2.5 dB is needed for $\rho =0.2$, compared to the perfect CSI case. Interestingly, the performance degradation caused by outdated CSI is not significant in the case $m=2$, ${N_t}=2$, compared to $m=1$, ${N_t}=1$, especially in the high SNR region. It is speculated that outage performance is mainly limited by the FSO link rather than RF link in this case. Furthermore, the performance of HD technique is superior to IM/DD technique due to its better spectral efficiency and higher sensitivity, but at the expense of higher complexity^44,45,46.

Fig. 2

Outage probability versus SNR for different values of correlation coefficient $\rho$ and antennas number ${N_t}$. (a) IM/DD detection. (b) Heterodyne detection.

Figure 3 shows the average BER versus SNR for different values of correlation coefficient $\rho$ and antennas number ${N_t}$. The pointing error coefficient is set as $\xi =3.67$. We can observe that compared with the perfect CSI case, outdated CSI degrades system average BER performance, especially in the case $\rho =0.2$. As shown in Fig. 3a, the degradation of system average BER performance is particularly evident in case $m=1$, ${N_t}=2$. For the case $m=1$, ${N_t}=1$, to achieve an average BER of $1 \times {10^{ - 3}}$, additional 2.5 dB is needed for the case $\rho =0.8$, and extra 5.5 dB is required for the case $\rho =0.2$. Moreover, considering the case $m=1$, ${N_t}=2$ with HD technology as shown in Fig. 3b, to achieve an average BER of $1 \times {10^{ - 5}}$, additional 6.1 dB is needed for the case $\rho =0.8$, and extra 9.4 dB is required for the case $\rho =0.2$. Furthermore, it can be clearly observed that HD technique outperforms IM/DD technique in terms of average BER in any case. In a nutshell, outdated CSI deteriorates system performance, which can be improved by increasing the number of antennas and using HD technique in the FSO link.

Fig. 3

Average Bit-Error Rate versus SNR for different values of correlation coefficient $\rho$ and antennas number ${N_t}$. (a) IM/DD detection. (b) Heterodyne detection.

Outage performance

In Fig. 4, system outage probability of the proposed relay scheme with site diversity under different pointing errors is depicted, where both IM/DD and HD techniques are taken into consideration. ${\xi _1}$ means the pointing error of GS-to-HAP FSO link and ${\xi _2}$ is the pointing error of HAP-to-satellite FSO link. It can be clearly observed that system outage performance can obtain a significant improvement with larger values of $\xi$, which can be attributed to the fact that a larger values of $\xi$ means smaller impact of pointing error, and both the GS-to-HAP and HAP-to-satellite FSO link are improved. More specifically, considering HD technique and to achieve an average BER of $1 \times {10^{ - 3}}$, a 7.8 dB gain can be obtained by the weak pointing error case compared to the strong pointing error case. Moreover, for the weak pointing error case and to achieve an average BER of $1 \times {10^{ - 3}}$, HD technique can obtain a 16.7 dB gain compared to IM/DD technique, which proves the effectiveness of HD technique in improving system outage performance.

Fig. 4

Outage probability of proposed scheme under different pointing errors using both IM/DD and HD techniques.

System outage probability under varying weather conditions for different relay schemes is plotted in Fig. 5, where IM/DD technique is adopted and $\rho$ is set as 0.8. The proposed scheme in Fig. 5 is the proposed scheme with site diversity. As we can see, the proposed scheme can achieve better outage performance in different weather conditions, which is due to that the propose scheme can flexibly select transmission modes based on different weather conditions. In foggy weather, the proposed scheme can obtain a 14.5 dB gain compared to SAG-FSO scheme at an outage probability $5 \times {10^{ - 2}}$. Considering rainy weather condition and to achieve an outage probability of $1 \times {10^{ - 2}}$, additional 9.3 dB is needed for SAG-FSO/RF scheme compared with the proposed scheme. Moreover, benefiting from flexible weather mode selection, the proposed scheme can achieve a reliable and excellent outage performance in varying weather conditions. Furthermore, from Fig. 1 and Table II, we can find the main difference between mode 0 and mode 2 is the HAP-to-satellite FSO link, but the performance of FSO/FSO relay is mainly determined by the relatively worse GS-to-HAP FSO link. Therefore, the proposed scheme can achieve similar outage performance in rainy weather and clear weather conditions.

Fig. 5

Outage probability performance under varying weather conditions for different relay schemes.

System outage probability with varying average SNR for different relay schemes is presented in Fig. 6, where IM/DD technique is adopted and correlation coefficient $\rho$ is set as 0.8. We can find that the SAG-FSO scheme owns the worst outage performance which is because that the GS-to-HAP FSO link is affected by both foggy and rainy weather conditions. Limited by the huge attenuation caused by rainy weather, the SAG-FSO/RF scheme also exhibits poorer outage performance than the proposed schemes. With the help of diversity technology, the SAG-FSO with site diversity scheme has a better outage performance compared to SAG-FSO scheme. This is due to that site diversity can reduce the adverse effects of rainy weather through GS diversity. When the average SNR exceeds 12.5 dB, the performance of the SAG-FSO with site diversity scheme is inferior to SAG-RF/FSO scheme. It is speculated that under the high SNR condition, the influence of rainy weather on RF link is less than that of foggy weather on FSO link. Moreover, the proposed scheme with site diversity owns the best outage performance which is attribute to the fact that it can flexibly choose RF link or backup FSO link to minimize the impact of complex weather conditions. It can obtain a gain of 8.8 dB compared to SAG-FSO/RF scheme and a gain of 13.4 dB compared to SAG-FSO with site diversity scheme, when achieving an outage probability of $1 \times {10^{ - 2}}$. Furthermore, compared to the proposed scheme with site diversity, the proposed low-complexity scheme only causes a 2.1 dB performance degradation when achieving an outage probability of $1 \times {10^{ - 2}}$, but with lower economic cost and complexity. The performance degradation is due to that the GS-to-HAP FSO link is more affected compared to the proposed scheme with site diversity. Thus, we should tradeoff performance improvement with economic cost and complexity.

Fig. 6

Outage probability performance with varying average SNR for different relay schemes.

In Fig. 7, we perform an asymptotic analysis for outage probability under IM/DD technique for different values of $\alpha$, $\beta$ and $\xi$, which are set as shown in Table 5. FSO/FSO-1 and FSO/FSO-2 are corresponding to different parameter sets $\alpha$, $\beta$ and $\xi$,which mean that both the GS-to-HAP and HAP-to-satellite are FSO links. RF/FSO-1 and RF/FSO-2 mean that the GS-to-HAP is RF link and HAP-to-satellite is FSO link. As shown in Fig. 7, the four curves have the same slope in the high SNR region, which means that they have the same diversity order. The reasons are analyzed as follows. For FSO/FSO-1 and FSO/FSO-2, the diversity order is determined by ${G_0}=\hbox{min} \left\{ {{\alpha _1}/2,{m_1}/2,\xi _{1}^{2}/2} \right\}$, which can be obtained from Table 5. For RF/FSO-1 and RF/FSO-2, the diversity order is determined by ${G_1}=\hbox{min} \left\{ {m{N_t},{\alpha _2}/2,{m_2}/2,\xi _{2}^{2}/2} \right\}$, which can also be obtained from Table 5. It can be clearly found that they have the same diversity order. Therefore, the four curves have the same slope, which also verifies the correctness and tightness of our asymptotic analysis.

Table 5 Values of different $\alpha$, $\beta$ and $\xi$ for different cases

Fig. 7

Asymptotic outage performance analysis under IM/DD technique for different values of $\alpha$, $\beta$ and $\xi$

Average BER performance

Figure 8 presents system average BER of the proposed scheme under different pointing errors using both IM/DD and HD techniques. It can be clearly observed that HD technique outperforms IM/DD technique, especially under high SNR conditions. For instance, at the SNR of 25 dB and ${\xi _1}=2.75$, ${\xi _2}=3.67$, HD technique can achieve an average BER of $5.27 \times {10^{ - 5}}$ while IM/DD technique only achieves an average BER of $2.63 \times {10^{ - 3}}$. It can also be inferred that the larger the value of $\xi$ is, the smaller the influence of pointing error is, and better average BER performance can be obtained. More specifically, HD technique can obtain a gain of 8.2 dB when ${\xi _1}$, ${\xi _2}$ increases from 1.10 to 1.65 to 2.75 and 3.67 at an average BER of $1 \times {10^{ - 4}}$.

Fig. 8

Average BER of proposed scheme under different pointing errors using both IM/DD and HD techniques.

In Fig. 9, system average BER performance under varying weather conditions for different relay schemes is presented, where IM/DD technique is adopted and $\rho$ is assumed as 0.8. The proposed scheme in Fig. 9 is the proposed scheme with site diversity. It can clearly see that the proposed scheme is superior to SAG-FSO scheme and SAG-FSO/RF scheme in varying weather conditions. Compared to SAG-FSO scheme, the proposed scheme can obtain a 14.2 dB gain in foggy weather, at an average BER of $1 \times {10^{ - 2}}$. Besides, considering rainy weather and to achieve an average BER of $1 \times {10^{ - 3}}$, an 8.7 dB gain can be achieved for the proposed scheme compared to SAG-RF/FSO scheme. Moreover, the proposed scheme can achieve a stable and reliable average BER performance through flexible modes selection. Hence, the proposed scheme can be applied to the feeder links in SAGIN considering its practicality and low complexity.

Fig. 9

Average BER performance under varying weather conditions for different relay schemes.

System average BER performance with varying average SNR for different relay schemes is plotted in Fig. 10. It can be observed that the proposed scheme with site diversity owns the best average BER performance and additional 1.8 dB is needed for the proposed low-complexity scheme to achieve an average BER of $1 \times {10^{ - 3}}$. Due to the huge attenuation of the FSO link caused by rainy weather and foggy weather conditions, the SAG-FSO scheme has the worst average BER performance. Owing to the use of diversity technology to reduce the impact of rainy weather, SAG-FSO with site diversity scheme is superior to SAG-FSO scheme. The performance of SAG-FSO/RF scheme is superior to SAG-FSO with site diversity scheme when then average SNR exceeds 27.5 dB. It is speculated that under the high SNR condition, the influence of rainy weather on RF link is less than that of foggy weather on FSO link. Furthermore, to achieve an average BER of $5 \times {10^{ - 3}}$, the proposed scheme with site diversity can obtain a gain of 8.1 dB compared to SAG-RF/FSO scheme and a gain of 11.8 dB compared to SAG-FSO with site diversity scheme.

Fig. 10

Average BER performance with varying average SNR for different relay schemes.

As illustrated in Fig. 11, the average BER performance with varying average SNR for different modulation schemes is given, where the pointing error is assumed to be weak pointing error with ${\xi _1}=2.75$, ${\xi _2}=3.67$. It can be clearly observed that the 64-QAM owns the worst average BER performance compared to the other three modulation schemes, which is consistent with the analysis in^4,18,22 and can also be proven mathematically. And 16-QAM outperforms 16-PSK which is consistent with the analysis in⁴⁷. Moreover, the BPSK owns the best average BER performance as expected, and to achieve an average BER of $1 \times {10^{ - 4}}$, 9.2 dB, 12.2 dB and 13.7 dB gains can be obtained compared to the 16-QAM, 16-PSK, and 64-QAM modulation schemes.

The asymptotic analysis for average BER under IM/DD technique for different values of $\alpha$, $\beta$ and $\xi$ is given in Fig. 12. Similar to the analysis in Fig. 7, and for FSO/FSO-1 and FSO/FSO-2, the diversity order of average BER is determined by ${G_{e,0}}=\hbox{min} \left\{ {{\alpha _1}/2,{m_1}/2,\xi _{1}^{2}/2} \right\}$, which can be obtained from Table 5. For RF/FSO-1 and RF/FSO-2, the diversity order of average BER is determined by ${G_{e,0}}=\hbox{min} \left\{ {m{N_t},{\alpha _1}/2,{m_1}/2,\xi _{1}^{2}/2} \right\}$, which can also be obtained from Table 5. It can also be clearly found that they have the same diversity order. From Fig. 12, we can observe that the four curves have the same slope, which also verifies the rigor and correctness of our asymptotic average BER analysis. Furthermore, through asymptotic analysis, we can observe that the system diversity order is determined by the worst of the RF fading parameters m, number of antennas ${N_t}$, turbulence parameters $\alpha$,$\beta$, and pointing error parameters $\xi$. This also provides engineering guidance for us to compensate for link performance and analyze bottleneck performance of the hybrid FSO/RF systems.

Fig. 11

Average BER performance with varying average SNR for different modulation schemes.

Fig. 12

Asymptotic average BER analysis under IM/DD technique for different values of $\alpha$, $\beta$ and $\xi$

Ergodic capacity performance

Figure 13 shows the ergodic capacity of the proposed site diversity scheme under different pointing errors using both IM/DD and HD techniques. For GS-to-HAP RF link, correlation coefficient $\rho$ is set as 0.8, and $m=2$, ${N_t}=2$ are assumed. As can be seen, HD technique has better ergodic capacity performance compared to IM/DD technique, and even under strong pointing error case, HD technique can achieve comparable ergodic capacity compared to IM/DD technique with weak pointing error. More specifically, for the case ${\xi _1}=1.10$, ${\xi _2}=1.65$, and to obtain an ergodic capacity of 3 Gbps, HD technique can obtain a 1.2 dB gain compared to IM/DD technique. Moreover, increasing the pointing error will reduce the ergodic capacity, and for the IM/DD technique to achieve an ergodic capacity of 3 Gbps, additional 1 dB is required for the strong pointing error case.

Fig. 13

Ergodic capacity of proposed scheme under different pointing errors using both IM/DD and HD techniques.

System ergodic capacity with varying average SNR for different relay schemes is shown in Fig. 14, where $\rho =0.8$, $m=2$, and ${N_t}=2$ are considered. As shown in Fig. 14, the SAG-FSO scheme has the worst ergodic capacity performance, which is due to the attenuation cause by foggy weather and rainy weather conditions. Compared to SAG-FSO scheme with site diversity, the SAG-FSO/RF scheme has a 2.1 dB performance loss when achieving a capacity of 2 Gbps, which is due to the large bandwidth of FSO link and the attenuation of RF links caused by rainy weather. Moreover, the proposed scheme with site diversity owns the best ergodic capacity and to achieve an ergodic capacity of 2 Gbps, it can obtain a 2.0 dB gain compared to SAG-FSO scheme with site diversity and a 4.2 dB gain compared to SAG-FSO/RF scheme. Furthermore, the proposed low-complexity scheme also has a better ergodic capacity performance, which only makes a 1.2 dB performance loss compared to the proposed scheme with site diversity when achieve a capacity of 2 Gbps.

Fig. 14

Ergodic capacity with varying average SNR for different relay schemes.

Conclusion

In this paper, we propose a weather-dependent link switching scheme for SAGIN, aiming to alleviate the impact of complex weather conditions on hybrid FSO/RF systems, which also considers the effect of outdated CSI in the RF link. Numerical results reveal that outdated CSI degrades system performance, and such degradation can be reduced by mitigating RF fading and increasing the number of antennas. Owing to flexible weather-adaptive switching, the proposed scheme outperforms existing benchmark schemes across diverse weather conditions. Moreover, considering economic costs and implementation complexity, the low-complexity scheme may represent a preferable choice with acceptable performance trade-offs. However, to specifically highlight the impact of complex weather on FSO links and outdated CSI on RF links, this study deliberately omits considering outdated CSI effects in FSO links. Future research will focus on investigating the impact of outdated CSI on both FSO and RF links. Additionally, the security performance of hybrid FSO/RF systems in SAGIN under complex weather and malicious jamming conditions represents a critical research priority for future work.