Performance Analysis of Parallel Free-Space Optical/Radio Frequency Transmissions in Satellite–Aerial–Ground Integrated Network with Power Allocation

Abstract

Satellite–aerial–ground integrated networks (SAGINs) combined with hybrid free-space optical (FSO) and radio frequency (RF) transmissions have shown great potential in improving service throughput and reliability. The coverage mismatch and rate limitation of traditional hybrid FSO/RF design have restricted its development. In this paper, we investigate the performance of parallel FSO/RF transmissions in SAGIN, taking into account the effect of weather conditions and the quality of service (QoS) of ground users. A three-hop relay system is proposed, where the FSO and RF links jointly provide services to ground users in remote areas. Specifically, considering the limited transmit power of the relay node, we have studied the optimal power allocation between parallel FSO and RF links to further improve system energy efficiency. The performance of the proposed system is evaluated in terms of capacity outage probability, weighted average bit-error rate (BER), and energy efficiency. Moreover, asymptotic capacity outage probability is also derived to obtain more engineering insights. Finally, numerical results show that the energy efficiency of the proposed parallel scheme improves by 30.9% compared to only the FSO scheme at a total transmit power of 15 dBW.

1. Introduction

Recently, with the popularity of low Earth orbit (LEO) satellites and high-altitude platforms (HAPs), satellite–aerial–ground integrated networks (SAGINs) have been widely envisioned as an attractive solution for sixth-generation (6G) and beyond-6G (B6G) communications [1]. Due to the congested spectrum and limited bandwidth, traditional radio frequency (RF) links are unable to meet the massive bandwidth demand in next-generation communications [2]. Free-space optical (FSO) communication has become an effective alternative due to its large unlicensed spectrum and high data rates [3]. In SAGIN, the Terabits/s FSO links can provide high data rates and affordable transmission, while the RF links can achieve ubiquitous coverage. Despite this advantage, the hybrid FSO/RF system is sensitive to complex weather conditions (e.g., fog, rain, and snow) and is greatly affected by attenuation caused by long-distance transmission [4]. Relay technology is an effective way to relieve the impact of attenuation, where the decode-and-forward (DF) relaying technology can enhance the overall signal-to-noise ratio (SNR) while the amplify-and-forward (AF) relaying technology owns low processing complexity [5].

The link connecting the ground station (GS) to the satellite is called the feeder link, while the link between the satellite and ground users is referred to as the user link [2,6]. Considering the Terabits/s feeder link between GS and satellite, a single FSO feeder link is usually adopted due to its immense available bandwidth, along with the advantages of being immune to interference and high security [7]. As for the user link from satellite to ground users, a HAP is often deployed as a relay node to relieve atmospheric attenuation and increase each session time between satellite and ground users [8]. Due to the different beam characteristics of FSO and RF links, the FSO link can only cover less than a few kilometers, while the RF link can range up to several hundred kilometers. Therefore, we can combine the large capacity of the FSO link with the wide coverage of the RF link to improve the throughput and reliability of access areas while achieving ubiquitous coverage [9].

Attracted by the enormous potential of hybrid FSO/RF systems, its implementation has been widely studied in terrestrial networks [10,11]. The authors of [10] proposed a switching scheme for a hybrid FSO/RF network with a selective DF-relaying protocol, assuming a maximal ratio combining (MRC) scheme at the destination. In [11], the authors analyzed the performance of dual-hop multiuser relay systems with mixed FSO/RF links, in which nonorthogonal multiple access (NOMA) technology was adopted to improve the system ergodic sum rate. Considering the limited range of terrestrial networks, the hybrid FSO/RF system has recently been extended to the HAP-based SAGIN [8,12,13,14]. The authors of [7] investigated the performance of dual-hop hybrid FSO/RF links for HAP-based SAGIN, and the optimal switching threshold value between FSO and RF links was also obtained. The comprehensive performance of a dual-hop SAGIN-based hybrid FSO/RF switching system was investigated in [12], where the authors derived the closed-form expressions for the optimum switching threshold. Considering the multiple-hop mixed RF/FSO systems, the authors of [13] proposed two practical cases for multicast networks and analyzed the tradeoff between outage probability and energy efficiency. In [14], the authors analyzed the performance of a multiuser terrestrial-satellite system with mixed FSO/RF links, where the outdated channel state information (CSI) in the RF link was also considered.

Due to the advantages of low complexity and ease of implementation, the hard-switching scheme is widely adopted in the existing hybrid FSO/RF systems [15,16,17]. The performance of the communication systems that integrated space–air–ground FSO relay links and single-hop hybrid FSO/RF links was investigated in [15], and the authors revealed that the integrated system could achieve higher throughput and reliability compared to existing solutions. Considering the combination of a hybrid FSO/RF system and space–air–ground FSO system, the authors of [16] have proven that the integrated system can achieve about a 10 dB performance gain compared to existing systems. In [7], the performance of very high-throughput satellite feeder links was studied, where the effect of atmospheric turbulence and weather conditions was greatly reduced by performing site diversity. Considering the effect of various weather conditions, the authors of [17] proposed three AF-based relay schemes under hybrid FSO/RF systems in reconfigurable intelligent surface (RIS)-assisted SAGIN, and a weather-dependent hard-switching scheme was adopted among different relay schemes.

However, due to the required link alignment time and setup time, the frequent hard-switching between FSO and RF links may result in service interruptions, reducing the system’s reliability. Motivated by this, the adaptive combining scheme was recently studied by researchers [18,19,20]. The hybrid FSO/RF system with an adaptive combining scheme was first studied in [18], in which the FSO link had a higher priority, and when the received SNR of the FSO link was lower than the threshold SNR, both the FSO and RF links would be activated simultaneously. The authors of [19] studied the ergodic capacity performance of an adaptive combining hybrid FSO/RF system, where the authors revealed that the adaptive combining scheme was superior to the maximal-ratio combining (MRC) scheme and hard-switching scheme. In [20], the authors have derived the closed-form expression for the optimum beam waist of the FSO link and also obtained the optimum switching threshold SNR by using a numerical optimization technique. Considering the frequent switching between FSO and RF links caused by fixed-rate design, a rate adaptation design scheme was proposed in [21,22]. The authors of [21] proposed a rate adaptation scheme based on the instantaneous received SNR, revealing that it was superior to traditional fixed-rate design. Considering the impact of weather conditions and atmospheric turbulence in SAGIN, a RIS-assisted unmanned aerial vehicle (UAV) was adopted in [22] to further improve the performance of hybrid FSO/RF systems. In order to fully utilize the resources of FSO and RF links, the soft-switching strategy was studied in [23], which was superior to traditional design but with a higher decoding complexity.

In practical scenarios, due to the limited coverage of the FSO link, it usually needs to be combined with access points or RF links to provide services for ground users. The above literature has not considered the problem of how the FSO link can access the widely distributed ground users. Actually, it is of great necessity to combine the large capacity of FSO links with the wide coverage of RF links. Moreover, when considering the limited power of the relay node and also for the purpose of matching the energy efficiency of terrestrial networks, the power optimization problem of hybrid FSO/RF systems has also attracted the attention of researchers [9,24,25,26,27]. The authors of [24,25] conducted research on power distribution from the perspectives of materials and technology, respectively. In our previous work [26], the power allocation between FSO and RF links with a certain proportion was preliminarily investigated. Considering the influence of weather conditions and the limitations of the satellite transmit power, the authors of [27] studied the problem of power allocation and link switching between FSO and RF links. In [9], the authors have studied the parallel FSO-RF transmissions, in which the satellite can adaptively allocate power between FSO and RF links according to the weather conditions. In a nutshell, the overall status of the research work is presented in

Table 1. The Comparison of Some Related Hybrid FSO/RF Systems.

Ref.	Weather	Parallel Links	Power Allocation	QoS	Metrics	Relay
[2]	No	No	No	No	OP, EC, BER	AF
[9]	Yes	Yes	Yes	No	COP	No
[13]	No	No	No	No	OP, EC	DF
[16]	Yes	No	No	No	OP, SEP	DF
[17]	Yes	No	No	No	OP, EC, BER	AF
[20]	Yes	No	No	No	OP, SEP	No
[22]	Yes	No	No	No	OP, EC, SE	AF
[26]	Yes	No	Yes	No	OP, BER	AF
[27]	Yes	No	Yes	No	OP	No

.

As revealed in Table 1, the power allocation problem of hybrid FSO/RF systems in SAGIN with parallel FSO and RF relay links was rarely studied, especially considering the limited transmission power of the relay node. Only the authors of [9] have considered the problem of parallel FSO/RF transmissions in satellite–terrestrial integrated networks with power allocation. However, the system performance in [9] was only evaluated in terms of capacity outage probability (COP) with unacceptable computational complexity, and the differences in quality of service (QoS) among ground users were not considered. Additionally, to the best of our knowledge, the study on the weighted average bit-error rate (BER) performance of parallel FSO/RF links in which the FSO link and RF link transmit signals simultaneously is limited, and this is a very important link performance index. Furthermore, the energy efficiency optimization problem should be further considered, which is of great significance for improving the endurance of relay nodes and matching the energy efficiency of ground networks. Hence, the main contributions of this paper are summarized as follows:

(1)

We have proposed a three-hop-based hybrid FSO/RF system in SAGIN, in which the parallel FSO/RF scheme is considered to serve users simultaneously. Specifically, considering the impacts of complex weather conditions and the limited transmission power of the HAP, we have further studied the power allocation problem between FSO and RF links.

(2)

In order to better evaluate the average BER performance of parallel FSO/RF links, we have proposed a novel formula for the weighted average BER, which is calculated by using the weighted ergodic capacity of parallel FSO and RF links.

(3)

To maximize system energy efficiency, we have proposed an energy efficiency optimization model under the constraints of QoS among different users and obtained the optimum power allocation coefficient by using a numerical optimization technique.

(4)

Based on the proposed model, we have derived the closed-form expressions for system capacity outage probability, weighted average BER, and energy efficiency. Asymptotic capacity outage probability and diversity order at high SNR regions are provided to reveal more engineering insights.

(5)

The simulation results indicate that our proposed parallel FSO/RF scheme has superior capacity outage and energy efficiency performance compared to only the FSO and only the RF scheme, with acceptable performance loss in average BER. Furthermore, limited by the GS-satellite and satellite-HAP FSO links, performance saturation appears in the high SNR region.

The remainder of this article is organized as follows. In Section 2, system and channel models are discussed in detail. The closed expressions for capacity outage probability, weighted average BER, and the energy efficiency optimization model are provided in Section 3. Section 4 shows the asymptotic analysis at high SNR regions, in which the diversity order and coding gain are also provided. The numerical and simulation results are given in Section 5. Finally, concluding remarks and future work are drawn in Section 6.

2. System and Channel Models

In this article, satellite communication systems are applied to provide service for remote areas, where ground networks are unavailable due to economic challenges and physical barriers. We consider a three-hop hybrid FSO/RF system with parallel FSO/RF transmissions, which consists of the end-to-end links from the optical ground station (OGS) to different ground users in the remote areas. Specifically, as shown in Figure 1, the three-hop hybrid FSO/RF system consists of the high-throughput FSO feeder link between OGS and the LEO satellite, the FSO relay link between the LEO satellite and HAP, and the parallel FSO/RF links between HAP and ground users. The optical signal received at the LEO satellite is firstly amplified and then transmitted to the HAP through an FSO relay link during the downlink transmission. The optical signal received at the HAP is decoded and forwarded to ground users through the parallel FSO/RF links. Considering the large capacity of the FSO link, we assume that data transmission can be completed within the overhead time of a single LEO satellite, and the case of multiple LEOs is not within the scope of this work’s discussion. The parallel FSO/RF links are adopted to achieve high speed and reliable transmission, where the limited power of HAP is allocated to FSO and RF links in a variable coefficient $\rho$. The RF link provides extensive coverage and basic services for all users, while the FSO link provides enhanced services to the users in hotspot areas through ground access points (AP). The proposed system is quite practical as it can overcome the rate limitation caused by traditional hybrid FSO/RF design and reduce link outage caused by frequent switching.

2.1. Signal Models

The signal is firstly modulated at the OGS and then transmitted to the LEO satellite. At the LEO satellite, the received signal over the FSO feeder link during the uplink transmission is expressed by

$$y_{Sf}={\left({\displaystyle \frac{P_G{G}_f^{Gt}{G}_f^{Sr}}{F{L}_G}}{I}_{GS}\right)}^{{\displaystyle \frac{b}{2}}}{x}_f+n_s$$

(1)

where $P_G$ is the transmit power at OGS, $G_f^{Gt}$ is the transmit telescope gain at OGS, $G_f^{Sr}$ is the receive telescope gain at LEO satellite. $F{L}_G$ is the free-space loss defined as $F{L}_G={\displaystyle \frac{4\pi L_{GS}}{{\lambda }_f}}$ [9], $L_{GS}$ is the link length between OGS and LEO satellite, ${\lambda }_f$ is the optical wavelength. $I_{GS}$ denotes the irradiance of FSO link calculated by $I_{GS}=I_{GS}^l{I}_{GS}^a{I}_{GS}^p$, $I_{GS}^l$ denotes the atmospheric attenuation due to complex weather conditions, $I_{GS}^a$ denotes atmospheric turbulence, $I_{GS}^p$ denotes pointing errors. $b=1$ is associated with heterodyne detection (HD) and $b=2$ is associated with intensity modulation with direct-detection (IM/DD). $x_f$ denotes the transmit signal at OGS, $n_s$ is the additive white Gaussian noise (AWGN) with variance ${\sigma }_s^2$. Then, the instantaneous received SNR at the LEO satellite is given by

$${\gamma }_{GS}^f={\displaystyle \frac{{\left(P_G{G}_f^{Gt}{G}_f^{Sr}\right)}^b}{F{L}_G^b{\sigma }_s^2}}{I}_{GS}^b={\overline{\gamma }}_{GS}^f{I}_{GS}^b$$

(2)

where ${\overline{\gamma }}_{GS}^f={\displaystyle \frac{{\left(P_G{G}_f^{Gt}{G}_f^{Sr}\right)}^b}{F{L}_G^b{\sigma }_s^2}}$ is the average received SNR at LEO satellite. The received signal at the LEO satellite is amplified and then forwarded to the HAP. The received signal at the HAP over the FSO relay link is written as

$$y_{Hf}={\left({\displaystyle \frac{\eta G_f^{St}{G}_f^{Hr}}{F{L}_H}}{I}_{SH}\right)}^{{\displaystyle \frac{b}{2}}}{y}_{Sf}+n_H$$

(3)

where $\eta$ is the photoelectric conversion ratio, $G_f^{St}$ is the transmit telescope gain at satellite and $G_f^{Hr}$ is the receive telescope gain at HAP. $F{L}_H$ is the free-space loss, $I_{SH}$ denotes the irradiance of satellite-to-HAP FSO relay link, $n_H$ is the AWGN with variance ${\sigma }_H^2$. The instantaneous received SNR at HAP is defined as

$${\gamma }_{SH}^f={\displaystyle \frac{{\left(P_G{G}_f^{Gt}{G}_f^{Sr}\eta G_f^{St}{G}_f^{Hr}\right)}^b}{\eta G_f^{St}{G}_f^{Hr}{I}_{SH}^{b}F{L}_H^b{\sigma }_s^2+F{L}_H^b{\sigma }_H^2}}{I}_{GS}^b{I}_{SH}^b$$

(4)

The received signal at the HAP is firstly decoded and then forwarded to ground users through the parallel FSO/RF links. The received signal at the AP through the FSO link is expressed as

$$y_A={\left({\displaystyle \frac{\rho \eta P_H{G}_f^{Ht}{G}_f^{Ar}}{F{L}_A}}{I}_{HA}\right)}^{{\displaystyle \frac{b}{2}}}{\widehat{y}}_{Hf}+n_f$$

(5)

where $\rho$ is the power allocation coefficient, $\eta$ is the photoelectric conversion ratio, $P_H$ is the total transmit power of HAP, $G_f^{Ht}$ is the transmit telescope gain at HAP, $G_f^{Ar}$ is the receive telescope gain at AP. $F{L}_A$ is the free-space loss, $I_{HA}$ is the irradiance of the HAP-to-AP FSO link, ${\widehat{y}}_{Hf}$ is the decoded version $y_{Hf}$, $n_f$ is the AWGN with variance ${\sigma }_f^2$. The instantaneous received SNR at the AP is given as

$${\gamma }_f={\displaystyle \frac{{\left(\rho \eta P_H{G}_f^{Ht}{G}_f^{Ar}\right)}^b}{F{L}_A^b{\sigma }_f^2}}{I}_{HA}^b={\overline{\gamma }}_f{I}_{HA}^b$$

(6)

where ${\overline{\gamma }}_f={\displaystyle \frac{{\left(\rho \eta P_H{G}_f^{Ht}{G}_f^{Ar}\right)}^b}{F{L}_A^b{\sigma }_f^2}}$ is the average received SNR at AP. Considering the HAP-to-users RF link, the received RF signal at ground users is shown by

$$y_u={\left({\displaystyle \frac{\eta \left(1-\rho \right)P_H{G}_r^{Ht}{G}_r^{Ur}}{F{L}_r{h}_l}}\right)}^{{\displaystyle \frac{1}{2}}}{h}_{RF}{\widehat{y}}_{Hf}+n_r$$

(7)

where $G_r^{Ht}$ is the transmit antenna gain, $G_r^{Ur}$ is the receive antenna gain, $F{L}_r$ is the free space loss defined as $F{L}_r={\displaystyle \frac{4\pi L_{HU}}{{\lambda }_r}}$, $L_{HU}$ is the length of RF link, ${\lambda }_r$ is the wavelength of RF signal, $h_l$ is the link attenuation due to weather conditions defined as $h_l={\omega }_{oxy}{L}_{HU}+{\omega }_{rain}{L}_{HU}$ [27], ${\omega }_{oxy}$ and ${\omega }_{rain}$ are the attenuation coefficients cause by the oxygen and rain scattering, respectively, $h_{RF}$ is the channel coefficient, $n_r$ is the AWGN with variance ${\sigma }_r^2$. The instantaneous received SNR at the end users is given as

$${\gamma }_{RF}={\displaystyle \frac{\eta \left(1-\rho \right)P_H{G}_r^{Ht}{G}_r^{Ur}}{F{L}_r{h}_l{\sigma }_r^2}}{h}_{RF}^2={\overline{\gamma }}_{RF}{h}_{RF}^2$$

(8)

where ${\overline{\gamma }}_{RF}={\displaystyle \frac{\eta \left(1-\rho \right)P_H{G}_r^{Ht}{G}_r^{Ur}}{F{L}_r{h}_l{\sigma }_r^2}}$ is the average received SNR of RF signal at end users.

2.2. Channel Models

The OGS-to-satellite FSO link is assumed to follow a Gamma–Gamma distribution with atmospheric attenuation and pointing error. The probability density function (PDF) of instantaneous SNR ${\gamma }_{GS}^f$ is given by ([2], Equation (30))

$$f_{{\gamma }_{GS}^f}\left(\gamma \right)={\displaystyle \frac{{{\xi }_1}^2}{b\gamma \Gamma ({\alpha }_1)\Gamma ({\beta }_1)}}{G}_{1,3}^{3,0}\left[{\displaystyle \frac{{\alpha }_1{\beta }_1{{\xi }_1}^2}{{{\xi }_1}^2+1}}{\left({\displaystyle \frac{\gamma }{{\mu }_1}}\right)}^{{\displaystyle \frac{1}{b}}}|\begin{array}{c}{{\xi }_1}^2+1\\ {{\xi }_1}^2, {\alpha }_1, {\beta }_1\end{array}\right]$$

(9)

where ${\alpha }_1$ and ${\beta }_1$ are parameters that indicate large-scale and small-scale irradiance fluctuations, ${\xi }_1$ is the pointing error coefficient, which is used to measure the severity of beam misalignment, ${\mu }_1$ is the average electrical SNR defined as ${\mu }_1={\overline{\gamma }}_{GS}^f$ for HD and ${\mu }_1={\displaystyle \frac{{\alpha }_1{\beta }_1{\xi }_1^2\left({\xi }_1^2+2\right)}{\left({\alpha }_1+1\right)\left({\beta }_1+1\right){\left({\xi }_1^2+1\right)}^2}}{\overline{\gamma }}_{GS}^f$ for IM/DD technology. The cumulative distribution function (CDF) of ${\gamma }_{GS}^f$ can be obtained by using ([28], Equation (07.34.21.0084.01)) shown as

$$F_{{\gamma }_{GS}^f}(\gamma )=F^1{G}_{b+1,3b+1}^{3b,\hspace{1em}1}\left({\displaystyle \frac{{D_1}^b\gamma }{b^{2b}{\mu }_1}}|\begin{array}{cc}1& E_1\\ E_2& 0\end{array}\right)$$

(10)

where $F^1={\displaystyle \frac{{{\xi }_1}^2{b}^{{\alpha }_1+{\beta }_1-2}}{{(2\pi )}^{b-1}\Gamma ({\alpha }_1)\Gamma ({\beta }_1)}}$, $D_1={\displaystyle \frac{{\alpha }_1{\beta }_1{\xi }_1^2}{{\xi }_1^2+1}}$, $E_1=\left\{{\displaystyle \frac{{{\xi }_1}^2+1}{b}},\dots ,{\displaystyle \frac{{{\xi }_1}^2+b}{b}}\right\}$, $E_2=\left\{{\displaystyle \frac{{{\xi }_1}^2}{b}},\dots ,{\displaystyle \frac{{{\xi }_1}^2+b-1}{b}},\dots ,{\displaystyle \frac{{\alpha }_1}{b}},\dots ,{\displaystyle \frac{{\alpha }_1+b-1}{b}},{\displaystyle \frac{{\beta }_1}{b}},\dots ,{\displaystyle \frac{{\beta }_1+b-1}{b}}\right\}$. The satellite-to-HAP FSO relay link is assumed to follow Gamma–Gamma distribution with angle-of-arrival (AOA) fluctuations and the PDF of ${\gamma }_{SH}^f$ is given by ([17], Equation (23))

$$\begin{array}{l}{f}_{{\gamma }_{SH}^f}\left(\gamma \right)=\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\delta \left(\gamma \right)+\left[1-\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\right]\\ \hspace{1em}\hspace{1em}\times {\displaystyle \frac{{{\xi }_2}^2}{b\gamma \Gamma ({\alpha }_2)\Gamma ({\beta }_2)}}{G}_{1,3}^{3,0}\left[{\displaystyle \frac{{\alpha }_2{\beta }_2{{\xi }_2}^2}{{{\xi }_2}^2+1}}{\left({\displaystyle \frac{\gamma }{{\mu }_2}}\right)}^{{\displaystyle \frac{1}{b}}}|\begin{array}{c}{{\xi }_2}^2+1\\ {{\xi }_2}^2, {\alpha }_2, {\beta }_2\end{array}\right]\end{array}$$

(11)

where ${\theta }_{FOV}$ is the field-of-view angle, ${\sigma }_{angle,u}^2$ is the variance of AOA, ${\mu }_2={\overline{\gamma }}_{SH}^f$ for HD technology, ${\mu }_2={\displaystyle \frac{{\alpha }_2{\beta }_2{{\xi }_2}^2({{\xi }_2}^2+2)}{\left({\alpha }_2+1\right)\left({\beta }_2+1\right){({{\xi }_2}^2+1)}^2}}{\overline{\gamma }}_{SH}^f$ for IM/DD technology. Similar to (10), the CDF of ${\gamma }_{SH}^f$ is obtained as

$$F_{{\gamma }_{SH}^f}(\gamma )=\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)+\left[1-\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\right]F^2{G}_{b+1,3b+1}^{3b,\hspace{1em}1}\left({\displaystyle \frac{{D_2}^b\gamma }{b^{2b}{\mu }_2}}|\begin{array}{cc}1& E_3\\ E_4& 0\end{array}\right)$$

(12)

where $F^2={\displaystyle \frac{{{\xi }_2}^2{b}^{{\alpha }_2+{\beta }_2-2}}{{(2\pi )}^{b-1}\Gamma ({\alpha }_2)\Gamma ({\beta }_2)}}$, $D_2={\displaystyle \frac{{\alpha }_2{\beta }_2{\xi }_2^2}{{\xi }_2^2+1}}$, $E_3=\left\{{\displaystyle \frac{{{\xi }_2}^2+1}{b}},\dots ,{\displaystyle \frac{{{\xi }_2}^2+b}{b}}\right\}$, $E_4=\left\{{\displaystyle \frac{{{\xi }_2}^2}{b}},\dots ,{\displaystyle \frac{{{\xi }_2}^2+b-1}{b}},\dots ,{\displaystyle \frac{{\alpha }_2}{b}},\dots ,{\displaystyle \frac{{\alpha }_2+b-1}{b}},{\displaystyle \frac{{\beta }_2}{b}},\dots ,{\displaystyle \frac{{\beta }_2+b-1}{b}}\right\}$. Similar to the OGS-to-satellite FSO link, the PDF of ${\gamma }_f$ for HAP-to-AP FSO link is written as

$$f_{{\gamma }_f}\left(\gamma \right)={\displaystyle \frac{{{\xi }_3}^2}{b\gamma \Gamma ({\alpha }_3)\Gamma ({\beta }_3)}}{G}_{1,3}^{3,0}\left[{\displaystyle \frac{{\alpha }_3{\beta }_3{{\xi }_3}^2}{{{\xi }_3}^2+1}}{\left({\displaystyle \frac{\gamma }{{\mu }_3}}\right)}^{{\displaystyle \frac{1}{b}}}|\begin{array}{c}{{\xi }_3}^2+1\\ {{\xi }_3}^2, {\alpha }_3, {\beta }_3\end{array}\right]$$

(13)

where ${\mu }_3={\displaystyle \frac{{\alpha }_3{\beta }_3{{\xi }_3}^2({{\xi }_3}^2+2)}{\left({\alpha }_3+1\right)\left({\beta }_3+1\right){({{\xi }_3}^2+1)}^2}}{\overline{\gamma }}_f$ for IM/DD technique, ${\mu }_3={\overline{\gamma }}_f$ for HD technique. Similar to (10), the CDF of ${\gamma }_f$ can be obtained from (13) and is given by

$$F_{{\gamma }_f}(\gamma )=F^3{G}_{b+1,3b+1}^{3b,\hspace{1em}1}\left({\displaystyle \frac{{D_3}^b\gamma }{b^{2b}{\mu }_3}}|\begin{array}{cc}1& E_5\\ E_6& 0\end{array}\right)$$

(14)

where $F^3={\displaystyle \frac{{{\xi }_3}^2{b}^{{\alpha }_3+{\beta }_3-2}}{{(2\pi )}^{b-1}\Gamma ({\alpha }_3)\Gamma ({\beta }_3)}}$, $D_3={\displaystyle \frac{{\alpha }_3{\beta }_3{\xi }_3^2}{{\xi }_3^2+1}}$, $E_5=\left\{{\displaystyle \frac{{{\xi }_3}^2+1}{b}},\dots ,{\displaystyle \frac{{{\xi }_3}^2+b}{b}}\right\}$, $E_6=\left\{{\displaystyle \frac{{{\xi }_3}^2}{b}},\dots ,{\displaystyle \frac{{{\xi }_3}^2+b-1}{b}},\dots ,{\displaystyle \frac{{\alpha }_3}{b}},\dots ,{\displaystyle \frac{{\alpha }_3+b-1}{b}},{\displaystyle \frac{{\beta }_3}{b}},\dots ,{\displaystyle \frac{{\beta }_3+b-1}{b}}\right\}$. The HAP-to-users RF link is assumed to follow Nakagami-m fading with $N_t$ antennas. the PDF and CDF of ${\gamma }_{RF}$ are given by ([13], Equation (11))

$$f_{{\gamma }_{RF}}\left(\gamma \right)={\displaystyle \frac{1}{\Gamma \left(m{N}_t\right)}}{\left({\displaystyle \frac{m}{{\overline{\gamma }}_{RF}}}\right)}^{m{N}_t}{\gamma }^{m{N}_t-1}\exp \left(-{\displaystyle \frac{m\gamma }{{\overline{\gamma }}_{RF}}}\right)$$

(15)

$$F_{{\gamma }_{RF}}\left(\gamma \right)=1-{\displaystyle \frac{1}{\Gamma \left(m{N}_t\right)}}\Gamma \left(m{N}_t,{\displaystyle \frac{m\gamma }{{\overline{\gamma }}_{RF}}}\right)$$

(16)

where $m$ is the fading severity parameter. The instantaneous capacity of HAP-to-AP FSO link can be calculated by

$$C_f=W_f{\log }_2\left(1+\tau {\gamma }_f\right)$$

(17)

where $W_f$ is the bandwidth of FSO link, $\tau =e/2\pi$ for IM/DD technique and $\tau =1$ for HD technique. Based on (13) and (17), and using the power transformation of random variables, the PDF of instantaneous capacity for HAP-to-AP FSO link can be obtained as

$$f_{C_f}\left(c\right)={\displaystyle \frac{{{\xi }_3}^2{2}^{c/W_f}\ln 2}{b{W}_f\Gamma ({\alpha }_3)\Gamma ({\beta }_3)}}{\left(2^{c/W_f}-1\right)}^{-1}{G}_{1,3}^{3,0}\left[{\displaystyle \frac{{\alpha }_3{\beta }_3{{\xi }_3}^2}{{{\xi }_3}^2+1}}{\left({\displaystyle \frac{2^{c/W_f}-1}{\tau {\mu }_3}}\right)}^{{\displaystyle \frac{1}{b}}}|\begin{array}{c}{{\xi }_3}^2+1\\ {{\xi }_3}^2, {\alpha }_3, {\beta }_3\end{array}\right]$$

(18)

Applying series expansion to $\exp (\cdot )$ function in Equation (15), the PDF of ${\gamma }_{RF}$ is further expressed by

$$f_{{\gamma }_{RF}}\left(\gamma \right)={\displaystyle \frac{1}{\Gamma \left(m{N}_t\right)}}{A}^{m{N}_t}{\displaystyle \sum _{i=0}^{\infty }\frac{{\left(-A\right)}^i}{i!}}{\gamma }^{m{N}_t+i-1}$$

(19)

where $A=m/{\overline{\gamma }}_{RF}$. The instantaneous capacity of HAP-to-user RF link is calculated by

$$C_{RF}=W_r{\log }_2\left(1+{\gamma }_{RF}\right)$$

(20)

where $W_r$ is the bandwidth of the HAP-to-users RF link. According to (19) and (20), the PDF of instantaneous capacity for HAP-to-users RF link is given by

$$f_{C_{RF}}\left(c\right)={\displaystyle \frac{2^{c/W_r}\ln 2}{W_r\Gamma \left(m{N}_t\right)}}{A}^{m{N}_t}{\displaystyle \sum _{i=0}^{\infty }\frac{{\left(-A\right)}^i}{i!}}{\left(2^{c/W_r}-1\right)}^{m{N}_t+i-1}$$

(21)

The CDF of instantaneous capacity of HAP-to-users RF link is obtained by

$$F_{C_{RF}}\left(c\right)={\displaystyle {\int }_0^c{f}_{C_{RF}}\left(x\right)}dx$$

(22)

Substituting (21) into (22), $F_{C_{RF}}\left(c\right)$ can be further obtained as

$$F_{C_{RF}}\left(c\right)={\displaystyle \frac{A^{m{N}_t}}{\Gamma \left(m{N}_t\right)}}{\displaystyle \sum _{i=0}^{\infty }\frac{{(-A)}^i{(2^{c/W_r}-1)}^{m{N}_t+i}}{i!\left(m{N}_t+i\right)}}$$

(23)

The instantaneous sum capacity of parallel FSO/RF links is defined as $C_t=C_f+C_{RF}$. According to (18) and (21), the PDF of $C_t$ can be calculated as follows:

$$f_{C_t}\left(c\right)={\displaystyle {\int }_0^c{f}_{C_f}}\left(x\right)f_{C_{RF}}\left(c-x\right)dx$$

(24)

The final expression of $f_{C_t}\left(c\right)$ is given by

$$\begin{array}{l}{f}_{C_t}\left(c\right)={\displaystyle \frac{{\xi }_3^2\Lambda b^{\alpha +\beta -2}}{\Gamma \left({\alpha }_3\right)\Gamma \left({\beta }_3\right){\left(2\pi \right)}^{b-1}}}{\displaystyle \sum _{m_0=0}^k\left(\begin{array}{c}k\\ m_0\end{array}\right)}{\left({\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n}}\right)}^{k-m_0}\\ \hspace{1em}\hspace{1em}\times {\left({\displaystyle \frac{W_f}{\ln 2}}\right)}^{k-m_0}{c}^{m_0}{T}^{kn-m_{0}n}{G}_{b+1,3b+1}^{3b,1}\left[{\displaystyle \frac{D_3^{b}T}{b^{2b}\mu \tau }}|\begin{array}{c}{m}_{0}n+1-kn, E_5\\ E_6,m_{0}n-kn\end{array}\right]\end{array}$$

(25)

Proof. 

Please refer to Appendix A. □

The CDF of $C_t$ is calculated by

$$F_{C_t}(c)={\displaystyle {\int }_0^c{f}_{C_t}}\left(x\right)dx$$

(26)

The final expression of $F_{C_t}\left(c\right)$ is given by

$$\begin{array}{l}{F}_{C_t}\left(c\right)={\displaystyle \frac{{\xi }_3^2\Lambda b^{{\alpha }_3+{\beta }_3-2}}{\Gamma \left({\alpha }_3\right)\Gamma \left({\beta }_3\right){\left(2\pi \right)}^{b-1}}}{\displaystyle \sum _{m_0=0}^k\left(\begin{array}{c}k\\ m_0\end{array}\right)}{\left({\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n}}\right)}^{k-m_0}{\left({\displaystyle \frac{W_f}{\ln 2}}\right)}^{k+1}{\displaystyle \sum _{h=0}^{\infty }\frac{{\left(-1\right)}^h}{h!}}\\ \times {\left({\displaystyle \sum _{z=1}^{\infty }\frac{{(-1)}^{z-1}}{z}}\right)}^{m_0+h}{T}^{kn-m_{0}n+m_{0}z+hz+1}{G}_{b+2,3b+2}^{3b,2}\left[{\displaystyle \frac{D_3^{b}T}{b^{2b}\mu \tau }}|\begin{array}{c}{m}_{0}n-kn-m_{0}z-hz,m_{0}n+1-kn,E_5\\ E_6,m_{0}n-kn,m_{0}n-kn-m_{0}z-hz-1\end{array}\right]\end{array}$$

(27)

Proof. 

Please refer to Appendix A for the derivation for $F_{C_t}\left(c\right)$ and please refer to Appendix B for the interval of convergence for $F_{C_t}\left(c\right)$. □

3. Performance Evaluation

In this section, we derive new exact closed-form expressions for performance metrics, which include the capacity outage probability and weighted average BER. In addition, system energy efficiency, which considers the QoS constraints of users inside and outside the hotspot areas, is also analyzed.

3.1. Capacity Outage Probability

The end-to-end CDF of the dual-hop FSO link from OGS to HAP under the AF scheme with a fixed relay gain $Q$ is given by our previous work ([17], Equation (33))

$$\begin{array}{l}{F}_{{\gamma }_1}\left(\gamma \right)=\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)+\left(1-\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\right)(F_{{\gamma }_{GS}^f}\left(\gamma \right)-{\displaystyle \frac{{\xi }_1^2{F}^2}{b\Gamma \left({\alpha }_1\right)\Gamma \left({\beta }_1\right)}}\\ \hspace{1em}\hspace{1em}\times G_{1,0:3,2: b, 3b+1}^{0,1:0,3:3b+1, 0}\left[\begin{array}{c}1\\ -\end{array}|\begin{array}{c}1-{\xi }_1^2,1-{\alpha }_1,1-{\beta }_1\\ -{\xi }_1^2,0\end{array}|\begin{array}{c}{E}_3\\ E_4,0\end{array}|{\displaystyle \frac{1}{D_1}}{\left({\displaystyle \frac{{\mu }_1}{\gamma }}\right)}^{{\displaystyle \frac{1}{b}}},{\displaystyle \frac{D_2^{b}Q}{b^{2b}{\mu }_2}} \right])\end{array}$$

(28)

Capacity outage probability is defined as the probability that the instantaneous link capacity falls below a predefined capacity threshold $C_{th}$. The capacity outage probability of our proposed three-hop hybrid FSO/RF system is defined as ([8], Equation (23))

$$P_{out}=P_{out1}+P_{out2}-P_{out1}{P}_{out2}$$

(29)

where $P_{out2}$ is the capacity outage probability of HAP-to-AP parallel FSO/RF links, which can be obtained by using (27) shown as $P_{out2}=F_{C_t}\left(C_{th}\right)$. $P_{out1}$ defines the capacity outage probability of a dual-hop FSO link from OGS to HAP and can be calculated by $P_{out1}=F_{{\gamma }_1}\left({\gamma }_{th}\right)$. ${\gamma }_{th}$ is the predefined SNR, which is determined capacity threshold $C_{th}$. The instantaneous capacity of the dual-hop FSO link from OGS to HAP is shown as

$$C_{GSH}=\min \left\{C_{GS}, C_{SH}\right\}$$

(30)

where $C_{GS}$ is the instantaneous capacity of the OGS-to-satellite FSO link, $C_{SH}$ is the instantaneous capacity of the satellite-to-HAP FSO link. When considering that the OGS-to-satellite FSO link is more affected by weather conditions compared to the satellite-to-HAP FSO link [8,16], we have the relationship ${\alpha }_1<{\alpha }_2$, ${\beta }_1<{\beta }_2$ and ${\xi }_1<{\xi }_2$. Thus, the relationship $C_{GS}<C_{SH}$ holds true. $C_{GSH}$ can be rewritten as $C_{GSH}=C_{GS}$, which is calculated by

$$C_{GS}=W_f{\log }_2\left(1+\tau {\gamma }_{GS}^f\right)$$

(31)

Therefore, ${\gamma }_{th}$ can be calculated from (31) with a predefined threshold $C_{th}$ and the capacity outage probability in (29) can be obtained.

3.2. Weighted Average BER

The average BER of our proposed three-hop hybrid FSO/RF system can be calculated by ([8], Equation (36))

$${\overline{P}}_{eSRD}={\overline{P}}_{eSR}+{\overline{P}}_{eRD}-{\overline{P}}_{eSR}{\overline{P}}_{eRD}$$

(32)

where ${\overline{P}}_{eSR}$ is the average BER of the dual-hop FSO link from OGS to HAP, ${\overline{P}}_{eRD}$ means the weighted average BER of parallel FSO/RF links for HAP-to-ground users. Here, we propose the weighted average BER to evaluate the average BER performance of parallel FSO/RF links, which is calculated by using the weighted ergodic capacity of parallel FSO and RF links. The generalized expression for average BER under different modulation schemes is shown as ([2], Equation (46))

$${\overline{P}}_e={\displaystyle \frac{\sigma }{2\Gamma \left(p\right)}}{\displaystyle \sum _{u=1}^{n_0}{q}_u^p}{\displaystyle {\int }_0^{\infty }{x}^{p-1}{e}^{-q_{u}x}}F\left(x\right)dx$$

(33)

where $n_0$, $\sigma$, $p$, and $q_u$ are parameters depending on the type of detection and modulation techniques shown in

Table 2. Parameters for Different Modulation Schemes.

Modulation	$\mathit{\sigma }$	$\mathit{p}$	${\mathit{q}}_{\mathit{u}}$	${\mathit{n}}_0$	Detection Type
OOK	1	0.5	0.5	1	IM/DD
BPSK	1	0.5	1	1	HD
M-PSK	${\displaystyle \frac{2}{\max ({\log }_{2}M,2)}}$	0.5	${\sin }^2\left({\displaystyle \frac{\left(2u-1\right)\pi }{M}}\right)$	$\max \left({\displaystyle \frac{M}{4}},1\right)$	HD
M-QAM	${\displaystyle \frac{4}{{\log }_{2}M}}\left(1-{\displaystyle \frac{1}{\sqrt{M}}}\right)$	0.5	${\displaystyle \frac{3{\left(2u-1\right)}^2}{2\left(M-1\right)}}$	${\displaystyle \frac{\sqrt{M}}{2}}$	HD

[3].

The average BER of the dual-hop FSO link from OGS to HAP is obtained by substituting (28) into (33) shown as

$$\begin{array}{l}{\overline{P}}_{eSR}=\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)+\left(1-\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\right)({\displaystyle \frac{\sigma F_p}{2\Gamma \left(p\right)}}-{\displaystyle \frac{\sigma {\xi }_1^2{F}^2}{2b\Gamma \left({\alpha }_1\right)\Gamma \left({\beta }_1\right)\Gamma \left(p\right)}}\\ \hspace{1em}\hspace{1em}\times {\displaystyle \sum _{u=1}^{n_0}{G}_{1,0:3,3:b, 3b+1}^{0,1:1,3:3b+1,0}}\left[\begin{array}{c}1\\ -\end{array}|\begin{array}{c}1-{\xi }_1^2,1-{\alpha }_1,1-{\beta }_1\\ p,-{\xi }_1^2,0\end{array}|\begin{array}{c}{E}_3\\ E_4,0\end{array}|{\displaystyle \frac{1}{D_1}}{\left(q_u{\mu }_1\right)}^{{\displaystyle \frac{1}{b}}},{\displaystyle \frac{D_2^{b}Q}{b^{2b}{\mu }_2}}\right])\end{array}$$

(34)

where

$$F_p=F^1{\displaystyle \sum _{u=1}^{n_0}{G}_{b+2,3b+1}^{3b,\hspace{1em}2}\left(\frac{{D_1}^b}{b^{2b}{\mu }_1{q}_u}|\begin{array}{ccc}1& 1-p& E_1\\ E_2& & 0\end{array}\right)}$$

(35)

The average BER of the HAP-to-AP single FSO link can be derived by substituting (14) into (33) and applying ([28], Equation (07.34.21.0013.01)) shown as

$${\overline{P}}_{ef}={\displaystyle \frac{\sigma }{2\Gamma \left(p\right)}}{F}^3{\displaystyle \sum _{u=1}^{n_0}{G}_{b+2,3b+1}^{3b,\hspace{1em}2}\left(\frac{{D_3}^b}{b^{2b}{\mu }_3{q}_u}|\begin{array}{c}1,1-p,E_5\\ E_6,0\end{array}\right)}$$

(36)

Similarly, the average BER of HAP-to-users single RF link is obtained by substituting (16) into (33) and using ([28], Equation (07.34.03.0613.01)), then ([28], Equation (07.34.21.0013.01)) given by

$${\overline{P}}_{er}={\displaystyle \frac{\sigma n_0}{2}}-{\displaystyle \frac{\sigma }{2\Gamma \left(p\right)\Gamma \left(m{N}_t\right)}}{\displaystyle \sum _{u=1}^{n_0}{G}_{2,2}^{2,1}}\left({\displaystyle \frac{m}{{\overline{\gamma }}_{RF}{q}_u}}|\begin{array}{c}1, 1-p\\ m{N}_t, 0\end{array}\right)$$

(37)

Since it is difficult to evaluate the average BER performance of parallel FSO/RF links, the weighted average BER is adopted, which is defined as

$${\overline{P}}_{eRD}={\displaystyle \frac{C_f}{C_t}}{\overline{P}}_{ef}+{\displaystyle \frac{C_{RF}}{C_t}}{\overline{P}}_{er}$$

(38)

From (38), we can find that the weighted average BER is determined by the average BER of the FSO link, the average BER of the RF link, and their weighted instantaneous link capacity. Substituting (34) and (38) into (32), the average BER of our proposed system can be obtained.

3.3. Energy Efficiency

Energy efficiency is an essential index for the HAP relay node, which can improve its endurance time and realize energy efficiency matching with ground networks. The ergodic capacity of FSO and RF links can be calculated by

$$\begin{array}{l}\overline{C}=E\left[W{\log }_2(1+\tau \gamma )\right]\\ \hspace{1em}={\displaystyle \frac{W}{\ln 2}}{\displaystyle {\int }_0^{\infty }\ln (1+\tau \gamma )f(\gamma )}d\gamma \end{array}$$

(39)

where $W=W_f$ is the link bandwidth of FSO and $W=W_r$ is the bandwidth of RF link. The ergodic capacity of the dual-hop FSO link from OGS to HAP is obtained by using the differential of (28) and (39) expressed as

$$\begin{array}{l}{\overline{C}}_{EC}=\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)+\left(1-\exp \left(-{\displaystyle \frac{{\theta }_{FoV}^2}{2{\sigma }_{angle,u}^2}}\right)\right)\times (F_s+{\displaystyle \frac{{\xi }_1^2{F}^2}{b\Gamma \left({\alpha }_1\right)\Gamma \left({\beta }_1\right)}}\\ \hspace{1em}\times G_{1,0:4,3:b,3b+1}^{0,1:1,4:3b+1,0}\left[\begin{array}{c}1\\ -\end{array}|\begin{array}{c}1-{\xi }_1^2,1-{\alpha }_1,1-{\beta }_1,1\\ 1,-{\xi }_1^2,0\end{array}|\begin{array}{c}1+{\xi }_2^2\\ {\xi }_2^2,{\alpha }_2,{\beta }_2,0\end{array}|{\displaystyle \frac{{\xi }_1^2+1}{{\alpha }_1{\beta }_1{\xi }_1^2}}{\left(\tau {\mu }_1\right)}^{{\displaystyle \frac{1}{b}}},{\displaystyle \frac{D_2^{b}Q}{b^{2b}{\mu }_2}}\right])\end{array}$$

(40)

where $F_s$ is given by ([16], Equation (E.3)) shown as

$$F_s={\displaystyle \frac{{\xi }_1^2{b}^{{\alpha }_1+{\beta }_1-2}}{{\left(2\pi \right)}^{b-1}\Gamma \left({\alpha }_1\right)\Gamma \left({\beta }_1\right)}}{G}_{b+2,3b+2}^{3b+2,1}\left[{\displaystyle \frac{D_1^b}{\tau b^{2b}{\mu }_1}}|\begin{array}{c}0, 1, E_1\\ E_2, 0, 0\end{array}\right]$$

(41)

The ergodic capacity of the HAP-to-AP single FSO link can be obtained by substituting (13) into (39) and using ([28], Equation (07.34.21.0013.01)) shown as

$${\overline{C}}_{RDf}={\displaystyle \frac{W_f{\xi }_3^2{b}^{{\alpha }_3+{\beta }_3-2}}{{\left(2\pi \right)}^{b-1}\Gamma \left({\alpha }_3\right)\Gamma \left({\beta }_3\right)}}{G}_{b+2,3b+2}^{3b+2,1}\left[{\displaystyle \frac{D_3^b}{\tau b^{2b}{\mu }_3}}|\begin{array}{c}0, 1, E_5\\ E_6, 0, 0\end{array}\right]$$

(42)

Similarly, the ergodic capacity of the HAP-to-AP single RF link is derived by substituting (15) into (39) and applying ([28], Equation (07.34.21.0013.01)) given by

$${\overline{C}}_{RDr}={\displaystyle \frac{W_r}{\Gamma \left(m{N}_t\right)\ln 2}}{G}_{3,2}^{1,3}\left({\displaystyle \frac{\tau {\overline{\gamma }}_{RF}}{m}}|\begin{array}{c}1,1,1-m{N}_t\\ 1,0\end{array}\right)$$

(43)

Therefore, considering the QoS constraints of ground users inside and outside the hotspot areas, the optimization model with maximizing system energy efficiency ${\eta }_s$ can be expressed as

$$\begin{array}{l}\underset{\rho }{\max } {\eta }_s={\displaystyle \frac{\min \left\{{\overline{C}}_{EC}, {\overline{C}}_t\left(1-F_{{\gamma }_1}\left({\gamma }_{th}\right)\right)\right\}}{P_H}}\\ \hspace{1em}\mathrm{s}.\mathrm{t}. C1: 0\le \rho \le 1\\ \hspace{1em}\hspace{1em} C2 : F_{C_{RF}}\left(c\le C_{th}^1\right)\le {\delta }_{th1}\\ \hspace{1em}\hspace{1em} C3 : F_{C_t}\left(c\le C_{th}^2\right)\le {\delta }_{th2}\end{array}$$

(44)

where ${\overline{C}}_t={\overline{C}}_{RDf}+{\overline{C}}_{RDr}$ is the total capacity of HAP-to-user parallel FSO/RF links. The objective function in (44) means that the system ergodic capacity is determined by the smaller one between ${\overline{C}}_{EC}$ and ${\overline{C}}_t\left(1-F_{{\gamma }_1}\left({\gamma }_{th}\right)\right)$, where ${\overline{C}}_t\left(1-F_{{\gamma }_1}\left({\gamma }_{th}\right)\right)$ is the ergodic capacity of parallel FSO/RF links during the non-outage period. $C_1$ is the range constraint of power distribution coefficient $\rho$. $C_2$ is the QoS constraint of users outside the hotspot areas, where ${\delta }_{th1}$ is the maximum acceptable capacity outage probability and $C_{th}^1$ is the corresponding capacity threshold. $C_3$ is the QoS constraint of users inside the hotspot areas, in which ${\delta }_{th2}$ is the maximum acceptable capacity outage probability and $C_{th}^2$ is the corresponding capacity threshold. Due to the complex expressions of ${\overline{C}}_{EC}$, ${\overline{C}}_t$ and $F_{{\gamma }_1}\left({\gamma }_{th}\right)$, it is difficult to obtain the theoretical optimal $\rho$ of the optimization model. Therefore, we use the numerical optimization techniques to solve the optimization model in the simulation part.

4. Asymptotic Analysis

In this section, we perform the asymptotic analysis for system capacity outage probability in the high SNR region, as the expression of (29) is too complex and reveals limited engineering insights. Since we tend to focus on the performance of parallel FSO/RF links, a fixed SNR is assumed for the dual-hop FSO link from OGS to HAP, ensuring an outage probability less than $1\times {10}^{-3}$. Thus, the asymptotic capacity outage probability in (29) is mainly determined by $P_{out2}$. Applying ([28], Equation (07.34.06.0040.01)) and ignoring the high-order terms, the asymptotic capacity outage probability can be expressed by

$$\begin{array}{l}{P}_{out}\stackrel{{\mu }_3,{\overline{\gamma }}_{RF}\gg 1}{\approx }{\displaystyle \frac{W_f{\xi }_3^2{b}^{{\alpha }_3+{\beta }_3-2}}{W_r\Gamma \left(m{N}_t\right)\Gamma \left({\alpha }_3\right)\Gamma \left({\beta }_3\right){\left(2\pi \right)}^{b-1}}}{\left({\displaystyle \frac{m}{{\overline{\gamma }}_{RF}}}\right)}^{m{N}_t}\left(2^{c/W_f}-1\right)\\ \hspace{1em}\hspace{1em}\times {\displaystyle \sum _{k_0=1}^{3b}\frac{{\displaystyle \prod _{j_0=1,j_0\ne k_0}^{3b}\Gamma \left(E_{6,j_0}-E_{6,k_0}\right)}}{{\displaystyle \prod _{j_0=3}^{b+2}\Gamma \left(E_{5,j_0}-E_{6,k_0}\right)}{E}_{6,k_0}\left(1+E_{6,k_0}\right)}}{\left[{\displaystyle \frac{D_3^b\left(2^{c/W_f}-1\right)}{b^{2b}{\mu }_3\tau }}\right]}^{E_{6,k_0}}\end{array}$$

(45)

where $E_{6,j_0}$ and $E_{6,k_0}$ are the $j_0$th and $k_0$th terms of $E_6$; $E_{5,j_0}$ means the $j_0$th term of $E_5$. From (45), we can find that $P_{out}\propto {\left({\overline{\gamma }}_{RF}\right)}^{-m{N}_t}{\left({\mu }_3\right)}^{-E_{6,k_0}}$. Hence, the diversity order of system capacity outage probability is given by

$$G_d=m{N}_t+\min \left\{{\displaystyle \frac{{\alpha }_3}{b}},{\displaystyle \frac{{\beta }_3}{b}},{\displaystyle \frac{{\xi }_3^2}{b}}\right\}$$

(46)

It is to be noted from (46) that the proposed parallel FSO/RF links can obtain full gain value from both FSO and RF links. Coding gain ($G_c$) can be obtained from (45) and (46) when the asymptotic expression is in the form of $P_{out}\approx {\left(G_c\overline{\gamma }\right)}^{-G_d}$. By comparing (45) to $P_{out}\approx {\left(G_c\overline{\gamma }\right)}^{-G_d}$, we obtain the expression of coding gain as

$$\begin{array}{l}{G}_c={\displaystyle \sum _{k_0=1}^{3b}[\frac{W_f{\xi }_3^2{b}^{{\alpha }_3+{\beta }_3-2}{m}^{m{N}_t}}{W_r\Gamma \left(m{N}_t\right)\Gamma \left({\alpha }_3\right)\Gamma \left({\beta }_3\right){\left(2\pi \right)}^{b-1}}\left(2^{c/W_f}-1\right)}\\ {\hspace{1em}\times {\displaystyle \frac{{\displaystyle \prod _{j_0=1,j_0\ne k_0}^{3b}\Gamma \left(E_{6,j_0}-E_{6,k_0}\right)}}{{\displaystyle \prod _{j_0=3}^{b+2}\Gamma \left(E_{5,j_0}-E_{6,k_0}\right)}{E}_{6,k_0}\left(1+E_{6,k_0}\right)}}]}^{-{\displaystyle \frac{1}{G_d}}}{\left[{\displaystyle \frac{D_3^b\left(2^{c/W_f}-1\right)}{b^{2b}\tau }}\right]}^{-{\displaystyle \frac{E_{6,k_0}}{G_d}}}\end{array}$$

(47)

5. Numerical Results

In this section, we provide performance evaluation for our proposed system in terms of capacity outage probability, weighted average BER, and energy efficiency. The optimal power allocation coefficients of parallel FSO/RF links under different weather conditions are also obtained. Additionally, we have discussed the performance of our proposed parallel FSO/RF scheme with the existing schemes, including the only FSO scheme and only RF scheme. System parameters are listed as shown in

Table 3. System Parameters.

Parameter	Symbol	Value
Satellite altitude	$h_S$	550 km
HAP altitude	$h_H$	20 km
AP height	$h_G$	10 m
Wind speed	${\omega }_d$	21 m/s
Capacity threshold	$C_{th}^1$, $C_{th}^2$	0.3 Gbps, 0.5 Gbps
FSO subsystem
Link bandwidth	$W_f$	1 GHz
Wavelength	$\lambda$	1550 nm
Conversion coefficient	$\rho$	0.8
Beam radius	$w_0$	0.05 m
Fixed relay gain	$Q$	1.2
RF subsystem
Carrier frequency	$f$	28 GHz
Link bandwidth	$W_r$	300 MHz
Fading parameter	$m$	1, 2
Antenna number	$N_t$	1

unless otherwise specified, and the weather-dependent attenuation coefficients are given in

Table 4. Weather-Dependent Variables [29].

Weather Conditions	${\mathit{\omega }}_{\mathit{f}\mathit{o}\mathit{g}}$ (dB/km)	${\mathit{\omega }}_{\mathit{r}\mathit{a}\mathit{i}\mathit{n}}$ (dB/km)
Clear Weather	0.43	0
Haze weather	3.34	0
Moderate rain	5.84	5.69
Heavy rain	9.29	10.09
Light fog	16.67	0
Moderate fog	35.38	0

[29]. Three types of turbulence conditions are assumed similar to [2]. Without loss of generality, we assume that the maximum acceptable capacity outage probability of users inside or outside the hotspot areas is $1\times {10}^{-3}$. Note that ten terms in (27) are sufficient to achieve a relative error of $1\times {10}^{-7}$. Monte Carlo simulations are also adopted over ${10}^7$ realizations to validate the correctness of analytical results [30].

In Figure 2, the capacity outage probability of our proposed system under different weather conditions is plotted as a function of the power allocation coefficient $\rho$. We assume that the average SNR of both OGS-to-satellite and satellite-to-HAP FSO links is 40 dB. The total transmit power of HAP is 14 dBW, and moderate turbulence is assumed. It can be clearly observed that severe weather conditions can seriously deteriorate the capacity outage probability of parallel FSO/RF links. For a given weather condition and target capacity threshold, there exists an optimal power allocation coefficient that can minimize the system capacity outage probability. Specifically, $\rho *$ is 0.75 for clear weather while $\rho *$ also equal to 0.75 for moderate rain and light fog weather conditions. It is worth noting that $\rho *$ is not sensitive to the target capacity threshold and a larger capacity threshold, resulting in a worse capacity outage performance. Compared to clear weather, the performance of parallel FSO/RF links decreases in moderate rain weather, which is due to the fact that both the FSO and the RF links are affected by the attenuation caused by rainfall. The light fog weather owns the worst capacity outage performance due to the large attenuation of the FSO link caused by foggy weather. Moreover, we have also marked the minimum value of $\rho$ in Figure 2, which satisfies the maximum acceptable capacity outage probability of users inside the hotspot areas. Therefore, we should flexibly set the value of $\rho$ according to different weather conditions to optimize system capacity outage probability.

The capacity outage probability of our proposed scheme under different capacity thresholds is plotted in Figure 3. The turbulence level is assumed to be weak turbulence and $\rho$ is set as 0.75. As shown in Figure 3, when considering clear weather conditions, the parallel FSO/RF scheme outperforms only the FSO scheme and only the RF scheme in different capacity thresholds. The parallel FSO/RF scheme has a similar performance to only the FSO scheme when the capacity threshold is 0.95 Gbps. The only FSO scheme is superior to only the RF scheme in clear weather conditions due to its large bandwidth. Considering clear weather conditions, the parallel FSO/RF scheme can achieve an outage probability of $8.70\times {10}^{-5}$, compared to $4.15\times {10}^{-3}$ for only RF scheme and $2.23\times {10}^{-4}$ for only the FSO scheme, at a capacity threshold of 0.5 Gbps. Limited by the outage probability of the GS-satellite and satellite-HAP FSO links, the capacity outage probability of parallel FSO/RF links cannot be further improved in a small value of capacity threshold in clear weather conditions. However, when considering the light fog weather, the performance of only the FSO scheme deteriorates sharply, which is due to the fact that the FSO link is largely affected by fog weather. Compared to only the FSO scheme, the parallel FSO/RF scheme has a smaller performance degradation, which is because the use of an RF link can reduce the effect of foggy weather. Interestingly, considering light fog weather, when $\rho$ is no more than 0.20, the parallel FSO/RF scheme even has a better capacity outage performance than only the RF scheme. This can be attributed to the fact that the FSO link can effectively supplement the capacity of the parallel FSO/RF links with a very small portion of power. As such, the parallel FSO/RF scheme is a desirable choice in clear weather and can also be adopted in light fog with a small value of $\rho$.

The effect of atmospheric turbulence under different detection techniques was depicted in Figure 4, and three nominal ground turbulence levels were assumed for parallel FSO/RF links, which are $C_n^2\left(0\right)=1\times {10}^{-12}{m}^{-{\displaystyle \frac{2}{3}}}$, $C_n^2\left(0\right)=5\times {10}^{-13}{m}^{-{\displaystyle \frac{2}{3}}}$ and $C_n^2\left(0\right)=2.5\times {10}^{-13}{m}^{-{\displaystyle \frac{2}{3}}}$. According to ([15], Equation (54)) and ([15], Equation (61)), the scintillation parameters are calculated as $\alpha =4.51$, $\beta =2.73$ for strong turbulence, $\alpha =6.06$, $\beta =4.48$ for moderate turbulence, $\alpha =8.98$, $\beta =7.47$ for weak turbulence. The average SNR for OGS-to-satellite and satellite-to-HAP FSO links is set as 40 dB, and $\rho$ is set as 0.75. The capacity threshold is assumed as 0.5 Gbps and $m=1$, $N_t=1$ are also assumed. It can be obviously seen that the HD technique owns better outage performance than the IM/DD technique in different turbulence levels. Considering moderate turbulence and to achieve a capacity outage probability of $1\times {10}^{-4}$, the HD technique can achieve an 11.6 dB gain compared to the IM/DD technique. This is because the HD technique has better spectral efficiency and higher sensitivity than the IM/DD technique [31,32,33]. The HD technique suffers a 1.6 dB and 4.8 dB performance loss when the turbulence level changes from weak turbulence to moderate turbulence and strong turbulence at an outage probability of $1\times {10}^{-6}$. Limited by the GS-satellite and satellite-HAP FSO links, performance saturation appears in the IM/DD technique under weak turbulence when SNR exceeds 22 dB.

When considering that the diversity order in Figure 4 is mainly limited by the value of ${\xi }_3$, we give a further analysis of asymptotic analysis in Figure 5, assuming ${\xi }_3=5.2$. The black dashed line in Figure 5 represents the asymptotic analysis at a high SNR region, and its slope is defined as the diversity order. The capacity outage probabilities for the IM/DD technique under strong turbulence levels are $3.37\times {10}^{-7}$ and $2.35\times {10}^{-8}$ at a total transmit power of 35 dBW and 40 dBW. By taking the logarithm of capacity outage probabilities, the diversity order is obtained as $2.32\approx G_d=2.37$. Then we obtain $2.22\times {10}^{-7}$ and $3.13\times {10}^{-9}$ for HD technique under strong turbulence level at transmit power of 20 dBW and 25 dBW. Therefore, the diversity order is calculated as $3.70\approx G_d=3.73$. Similarly, the diversity order equals to $3.17\approx G_d=3.24$ for IM/DD technique under moderate turbulence level. We also obtain the diversity as $5.40\approx G_d=5.48$ for the HD technique under moderate turbulence levels. The above analysis further verifies the accuracy and rigor of asymptotic analysis in (46). As the system diversity order is mainly determined by the smallest values of ${\alpha }_3$, ${\beta }_3$, and ${\xi }_3$, we should flexibly set the values of beam waist in different turbulence levels to obtain the maximum returns.

The weighted average BER under different weather conditions and turbulence levels is presented in Figure 6. We assume that the average SNR of the OGS-to-satellite and satellite-to-HAP FSO link is 40 dB, and the total transmit power of HAP is 14 dBW. As can be seen, rainy weather and foggy weather can deteriorate the system-weighted average BER performance, and there exists an optimal value of $\rho$ to minimize the system-weighted average BER. Specifically, considering clear weather and moderate rain weather, the values of ${\rho }^{*}$ increases in heavy turbulence compared to moderate turbulence. It can be inferred that the average BER of the FSO link becomes the main factor that affects the weighted average BER of parallel FSO/RF links. Hence, more power should be allocated to the FSO link to obtain the optimal weighted average BER. In addition, considering the light fog case, the weighted average BER first increases and then decreases with the increase of $\rho$. Therefore, considering the optimal $\rho$ in Figure 2, a trade-off should be made between weighted average BER and capacity outage probability to select the optimal $\rho$. Moreover, the effect of turbulence on the weighted average BER becomes smaller when $\rho$ is larger than 0.75 in clear weather and moderate rain. In this case, the system-weighted average BER may be mainly dominated by the worse average BER performance of the RF link.

In Figure 7, the weighted average BER under different transmission schemes is depicted where the weak turbulence is assumed with an optimal $\rho$. It can be observed that the parallel FSO/RF scheme owns a worse weighted average BER performance compared with only the FSO scheme and only the RF scheme, especially in moderate rain weather. This is because the parallel FSO/RF links have a smaller average SNR in both FSO and RF links, which is caused by the power allocation between FSO and RF links. The only RF scheme owns a better average BER performance in both clear and rainy weather. Moreover, with the increase in average SNR, the average BER performance gap between only the FSO and only the RF scheme gradually decreases, and only the FSO scheme outperforms only the RF scheme in clear weather. Considering the advantages in capacity outage performance and service flexibility, the parallel FSO/RF links can be a better choice with an acceptable average BER performance loss.

Figure 8 depicts the weighted average BER under different turbulence levels, in which both IM/DD and HD techniques are considered. The average SNR for OGS-to-satellite and satellite-to-HAP FSO links is assumed to be 40 dB, and clear weather conditions are considered with $\rho =0.75$. We can observe that strong turbulence can deteriorate weighted average BER performance, which is due to the fact that strong turbulence will reduce the average BER performance of the FSO link. For the IM/DD technique, to achieve a weighted average BER of $1\times {10}^{-2}$, an additional 5.9 dB is needed for strong turbulence compared to weak turbulence. In addition, the HD technique can outperform the IM/DD technique in different turbulence levels. Considering weak turbulence conditions, a 7.1 dB gain can be achieved by the HD technique compared to the IM/DD technique at a weighted average BER of $1\times {10}^{-3}$.

The weighted average BER for different modulation schemes is shown in Figure 9, where we assume the average SNR for OGS-to-satellite and satellite-to-HAP FSO links as 40 dB and $\rho =0.75$ is adopted. It can be observed that the BPSK modulation can achieve the best weighted average BER performance; 10.2 dB, 7.0 dB, and 12.5 dB gains can be achieved compared to 16PSK modulation, 16QAM modulation, and 64QAM modulation at a weighted average BER of $1\times {10}^{-3}$. The 16QAM owns a better weighted average BER performance than 16PSK, which is consistent with [34]. The 16PSK modulation can outperform OOK modulation when the total transmit power is larger than 15 dBW, while the 64QAM modulation can outperform OOK modulation when the total transmit power is larger than 20 dBW. Limited by the average BER of the RF link, the weighted average BER performance gap between BPSK and 16-QAM becomes smaller in a high total transmit power region.

Figure 10 presents the system’s energy efficiency under different weather conditions, in which the optimal values of $\rho$ are also marked. In addition, the capacity outage probability of users outside the hotspot area under different weather conditions is given in Figure 11. From Figure 11, we can find that the value of $\rho$ should not exceed 0.94 in clear weather to satisfy the maximum acceptable capacity outage probability of $1\times {10}^{-3}$ for users outside the hotspot area. Similarly, the values of $\rho$ should not exceed 0.77 and 0.37 in moderate rain and heavy rain weather, respectively. From Figure 2, the value of $\rho$ should not be less than 0.05 in clear weather to ensure the QoS of users inside the hotspot area and $\rho$ should not be less than 0.20 and 0.35 in moderate rain and light fog weather, respectively.

From Figure 10, it can be clearly seen that the largest energy efficiency can be achieved in clear weather with an optimal $\rho$ of 0.80, which also satisfies the QoS of users inside and outside the hotspot area. Compared to clear weather, hazy weather has a smaller energy efficiency, and the optimal $\rho$ equals 0.80. When considering moderate fog weather, system energy efficiency decreases with the increase of $\rho$, and more power should be allocated to RF links to achieve a higher energy efficiency. This is due to the fact that the FSO link has poor availability in moderate fog weather. Compared to moderate fog, optimal energy efficiency can be obtained in light fog weather with $\rho =0.70$, which also satisfies the QoS of users inside and outside the hotspot area. Moreover, $\rho$ equals 0.80 for both moderate rain and heavy rain weather; however, considering the QoS of users, $\rho$ should be set as 0.77 and 0.37 in moderate rain weather and heavy rain weather, respectively. The optical values of $\rho$ are also given in

Table 5. The optimal value of $\rho$ in different weather conditions.

Weather Conditions	Value Range	Optimal Value
Clear weather	$0.05\le \rho \le 0.94$	0.80
Light fog	$0.35\le \rho \le 0.94$	0.70
Moderate rain	$0.20\le \rho \le 0.77$	0.77

. Furthermore, when $\rho$ is less than 0.55, the light fog weather can achieve a higher energy efficiency than heavy rain weather, while instead when $\rho$ is larger than 0.55. In a nutshell, we should flexibly set the value of $\rho$ according to different weather conditions to satisfy the QoS of users and achieve the best energy efficiency.

The system energy efficiency under both IM/DD and HD techniques is plotted in Figure 12, in which three different weather conditions are considered. We assume the average SNR for OGS-to-satellite and satellite-to-HAP FSO links as 40 dB, and $\rho =0.75$ is assumed. As we can see, system energy efficiency decreases with the increase in total transmit power, which is due to the fact that the logarithmic increase in capacity is less than the linear increase in total transmit power. In addition, the HD technique can achieve a higher energy efficiency than the IM/DD technique due to its better spectral efficiency. Considering the clear weather conditions, the HD scheme can achieve an energy efficiency of 593.2 Mbit/J, while the IM/DD scheme can only achieve an energy efficiency of 460.9 Mbit/J at a total transmit power of 10 dBW. Moreover, the rainy weather and foggy weather will reduce system energy efficiency. Considering the HD technique and a total transmit power of 15 dBW, the energy efficiency is 238.8 Mbit/J in clear weather, while it is 179.5 Mbit/J and 126.1 Mbit/J in moderate rain weather and light fog weather, respectively. Furthermore, the energy efficiency in moderate rainy weather is larger than that in light fog, especially at a small total transmit power.

In Figure 13, we plot the system energy efficiency under different transmission schemes using the IM/DD technique, and both clear weather and moderate rain weather conditions are considered. The average SNR for OGS-to-satellite and satellite-to-HAP FSO links is assumed to be 40 dB and $\rho =0.75$ is also assumed. It can be observed that the parallel FSO/RF scheme can achieve the best energy efficiency in different weather conditions. Due to the limited bandwidth of the RF link, the only RF scheme owns the worst energy efficiency performance. Considering clear weather with a total transmit power of 15 dBW, the parallel FSO/RF scheme can achieve an energy efficiency of 194.7 Mbit/J, while the only FSO scheme and only RF scheme can achieve an energy efficiency of 148.7 Mbit/J and 82.3 Mbit/J. Therefore, the parallel FSO/RF scheme can improve energy efficiency by 30.9% compared to only the FSO scheme. The parallel FSO/RF scheme has significant advantages in energy efficiency at low total transmission power regions. Furthermore, the parallel FSO/RF scheme can achieve the balance between enhanced coverage and extensive coverage.

6. Discussion

From the result of this study, we can find that there exists an optimal value of $\rho$ that can minimize system capacity outage probability and maximize system energy efficiency. And the optimal power allocation coefficient $\rho$ varies with different weather conditions. Additionally, our proposed parallel FSO/RF transmission scheme has obvious advantages in capacity outage probability and energy efficiency compared with only the FSO scheme and only the RF scheme. Furthermore, the heterodyne detection technique has better capacity outage probability, weighted average BER, and energy efficiency than intensity modulation with direct-detection technique. The conclusion of this study can be further applied to SAGIN to achieve wide coverage, high reliability, and high-speed transmission. In the future, the performance of parallel FSO/RF links under the imperfect channel station information can be further investigated.

7. Conclusions

In this paper, we have studied the performance of parallel FSO/RF transmissions in SAGIN, where the effect of weather conditions and the QoS of ground users are analyzed. Considering the widely distributed and varying densities of ground users, the FSO and RF links are jointly adopted to provide services for users, in which the RF link provides extensive coverage while the FSO link provides enhanced services. Based on the proposed system, we investigate the system performance in terms of capacity outage probability, weighted average BER, and energy efficiency. Asymptotic capacity outage analysis and diversity order at high SNR regions are also provided to show more engineering insights. Numerical results demonstrate that there exists an optimal power allocation coefficient to minimize capacity outage probability and weighted average BER and maximize the system energy efficiency. Moreover, limited by the GS-satellite and satellite-HAP FSO links, there exists a bottleneck for capacity outage probability. Furthermore, parallel FSO/RF links can outperform only FSO and only RF links in capacity outage probability and system energy efficiency, with acceptable performance loss in weighted average BER. Finally, as prospective future research, the co-channel interference in RF links and the dynamic characteristics of the parallel FSO/RF links can be further investigated.