# Performance of UAV assisted Multiuser Terrestrial-Satellite Communication
System over Mixed FSO/RF Channels

Praveen Kumar Singya and Mohamed-Slim Alouini P. K. Singya and M.-S. Alouini
are with the Computer, Electrical, and Mathematical Science and Engineering
(CEMSE) Division, King Abdullah University of Science and Technology (KAUST),
Thuwal 23955-6900, Saudi Arabia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, performance of a multi-antenna multiuser unmanned aerial vehicle
(UAV) assisted terrestrial-satellite communication system over mixed free
space optics (FSO)/ radio frequency (RF) channels is analyzed. Downlink
transmission from the satellite to the UAV is completed through FSO link which
follows Gamma-Gamma distribution with pointing error impairments. Both the
heterodyne detection and intensity modulation direct detection techniques are
considered at the FSO receiver. To avail the antenna diversity, multiple
transmit antennas are considered at the UAV. Selective decode-and-forward
scheme is assumed at the UAV and opportunistic user scheduling is performed
while considering the practical constraints of outdated channel state
information (CSI) during the user selection and transmission phase. The RF
links are assumed to follow Nakagami-m distribution due to its versatile
nature. In this context, for the performance analysis, analytical expressions
of outage probability, asymptotic outage probability, ergodic capacity,
effective capacity, and generalized average symbol-error-rate expressions of
various quadrature amplitude modulation (QAM) schemes such as hexagonal-QAM,
cross-QAM, and rectangular QAM are derived. A comparison of various modulation
schemes is presented. Further, the impact of pointing error, number of
antennas, delay constraint, fading severity, and imperfect CSI are highlighted
on the system performance. Finally, all the analytical results are verified
through the Monte-Carlo simulations.

###### Index Terms:

Satellite, multiantenna UAV, mixed FSO/RF, Gamma-Gamma, pointing errors,
outdated CSI, average symbol-error-rate (ASER), hexagonal QAM (HQAM), cross
QAM (XQAM), rectangular QAM (RQAM).

##  I Introduction

Recently, readily deployable flying wireless access devices like unmanned
aerial vehicles (UAVs) have gain an increased attention for robust and
reliable communication in various applications including military operations,
temporary social events or disastrous situations to provide improved coverage,
capacity, and data-rates to meet the requirements of $5^{th}$ generation (5G)
and beyond communications [1, 2]. After incredible efforts from the industries
and academia, it has been found that the terrestrial networks alone are not
able to provide the desired data-rates targeted for the 5G and beyond
communication systems and satellite communication has emerged as an essential
approach [3]. However, in practice, masking occurs between the satellite and
the terrestrial users and line-of-sight (LoS) communication becomes difficult.
This can be solved by deploying the terrestrial ground relays which can
transmit the satellite signal to improve the throughput and reliability of the
satellite communication system. However, such infrastructures may damaged in
some disastrous situations such as in flood conditions or in earthquake
situations. Also, in case of some temporary events, readily deployable
infrastructures such as UAVs are very helpful to provide reliable wireless
communication over the hybrid satellite-terrestrial networks (HSTNs) [4]. On
the basis of operating range and size, platforms for UAV are categorized into
low altitude platform (LAP), suitable for only few hundred meters and high
altitude platform (HAP), ranges in kilometers [5]. Various applications and
technological trends of HAP can be seen in [6]. Application of LAP in aerial-
terrestrial communication can be seen in [7]. Recently, UAVs have gained an
increased attention as flying base stations. In [8, 9, 10], the authors have
considered the UAVs as the mobile relay nodes for the communication. In [5],
the authors have proposed the mobility modeling of random 3-dimensional (3D)
mobile UAV networks and investigated their coverage probability. However, the
above mentioned works have not been considered in HSTNs. The UAV assisted
HSTNs have received an increased attention and European ABSOLUTE project has
adopted this in emergency situations for communication [11]. In [4], 3D mobile
UAVs assisted HSTN is considered and outage probability is derived for the
performance analysis. In [12], a UAV assisted multiuser HSTN is considered and
outage and asymptotic outage probability are derived. However, [4, 12]
consider only the radio-frequency (RF) links for communication.

Recently, the high data-rate requirement with increased capacity and bandwidth
direct us towards the free space optics (FSO) as an efficient replacement to
the RF links. This is because of its easy deployment, low cost, and point-to-
point high data-rate communication which provides high bandwidth and operates
in license free band [13, 14]. However, some challenges such as atmospheric
turbulence, modulation, and pointing error need to be considered during the
satellite to UAV optical communication [15]. For the modeling of FSO link with
atmospheric turbulence, Gamma-Gamma distribution is preferred commonly due to
its suitability to consider both the large and small scale atmospheric
fluctuations.

There are only few works which consider an optical link between the satellite
and UAV. In [16], feasibility of a satellite to HAP FSO link for the relay
purpose is analyzed by obtaining the capacity and efficiency. In [17], authors
have proposed an FSO based UAV to satellite optical communication system and
the PDF expression and bit-error-rate (BER) expression for binary phase shift
keying (BPSK) are derived for both the uplink and downlink scenario. In [18],
a multiuser UAV assisted decode-and-forward (DF) based HSTN is considered,
where Gamma-Gamma fading for FSO link and correlated Rayleigh fading for RF
links are considered and only the ergodic capacity is obtained. In [19], a
single user UAV assisted amplify-and-forward (AF) based HSTN is considered,
where Gamma-Gamma fading for FSO link with pointing error and correlated
Rayleigh fading for RF link with imperfect channel state information (CSI) are
considered and only outage probability is obtained. In [20], a HAP based
uplink FSO/RF ground to satellite communication system is presented, where
Gamma-Gamma fading for FSO link with pointing error and shadowed-Rician fading
for RF link is considered and the outage probability, asymptotic outage
probability, and BER for BPSK is obtained.

However, a considerable works on mixed RF/FSO or FSO/RF relay systems have
been reported in the literature. In [21], an AF based hybrid RF/FSO satellite-
terrestrial communication system is considered, where Shadowed-Rician and
Gamma-Gamma distributions are considered for the RF and FSO links,
respectively. For the performance analysis, analytical average symbol-error-
rate (ASER) expression for M-ary PSK (MPSK) is derived and asymptotic analysis
is performed on it. In [22] and [23], DF and AF based asymmetric RF/FSO
communication systems are considered, respectively, where FSO link is
considered to have atmospheric turbulence influenced Gamma-Gamma fading with
pointing error impairments and RF link is Nakagami-m distributed. For the
performance analysis, analytical expressions of outage probability, average
channel capacity, and average BER (ABER) for various MPSK and M-ary quadrature
amplitude modulation (MQAM) are derived. In [24], AF based mixed RF/FSO system
is considered, where FSO link is considered to have atmospheric turbulence
influenced Gamma-Gamma fading with pointing error impairments and RF link is
Nakagami-m distributed. For the performance analysis, average channel capacity
for optimum simultaneous power and rate adaptation and truncated channel
inversion with fixed rate schemes are derived. In [25], an AF based hybrid
FSO/RF system is considered where Gamma-Gamma and Nakagami-m distributions are
considered for the FSO and RF links, respectively. For the performance
analysis, outage probability, ergodic capacity, and BER of BPSK are obtained.
In [26], for a mixed FSO/RF system, energy harvesting is performed and outage
probability expression is derived. In [27], authors have consider a mixed
FSO/RF satellite communication system where an uplink optical feeder link is
considered to satellite and downlink RF links are considered to terrestrial
users. For the performance analysis, outage probability, ergodic capacity, and
ABER expressions are derived. In [28], authors have considered a mixed FSO/RF
satellite communication system where an uplink optical feeder link is
considered to satellite which is Gamma-Gamma distributed and downlink RF links
are considered to terrestrial users following the shadowed-Rician fading.
Further, outage probability, ergodic capacity, and ABER expressions are
derived for the performance analysis. In [29], a LAP based multihop RF/FSO/RF
system is proposed and only the outage probability is obtained for the
considered system. In [30], capacity of an AF based multihop FSO/RF system is
calculated. In [31], for a nonorthogonal multiple access based multiuser
hybrid FSO/RF system, outage probability and ergodic sum-rate are obtained.
Performance of a switching-based hybrid FSO/RF DF relay system with maximal-
ratio combining (MRC) at the destination is analyzed in [32]. In [33], for a
UAV assisted hybrid FSO/RF system with finite size buffer, data throughput is
maximized.

On the other hand, the family of QAMs has attracted significant attraction in
present and future wireless communication systems due to its capability to
provide bandwidth efficient high data-rate communication. The family mainly
includes the square QAM (SQAM), rectangular QAM (RQAM), cross QAM (XQAM), and
hexagonal QAM (HQAM) constellations. SQAM is valid only for the even power of
2 constellations. However, odd power of 2 constellations such as RQAM are also
required for better channel adaptation. From average power point of view, RQAM
is not a good choice and instead XQAM is preferred which has considerably
lower peak and average powers than RQAM [34]. However, the demand of high
data-rates leads the research for more compact two dimensional (2D)
constellations and a hexagonal lattice based constellation, HQAM comes into
existence. HQAM has the densest 2D packing for a given Euclidean distance
between the constellation points, hence, has the lower peak and average powers
and provides optimum SER performance than the other QAM constellations [35,
36, 37]. HQAM is categorized into regular and irregular HQAM constellations,
however, in this work, we have considered only the irregular HQAM
constellations due to its optimum performance [38, 13]. A significant work on
the ASER performance of various QAM constellations including the HQAM have
been reported in the literature [35, 36, 37, 39, 40, 41, 38]. However, such
works focus on the RF systems only. For the very first time, authors in [13]
have investigated the ASER performance of various QAM constellations in a
mixed RF/FSO system.

Motivated with this, in this work, a UAV (HAP) assisted multiuser multiantenna
HSTN is considered, where the downlink transmission from the satellite to UAV
is completed through the FSO link. A selective DF scheme is adopted at the UAV
and the decoded signal is then broadcasted to the terrestrial users via RF
links. It is assumed that the FSO link is modeled with the atmospheric
turbulence influenced Gamma-Gamma fading with pointing error impairments and
both the intensity modulation direct detection (IM/DD) and heterodyne
detection techniques are considered at the FSO receiver. The RF links are
assumed to follow the Nakagami-m distribution. Further, opportunistic user
scheduling is performed for the multiuser diversity and outdated CSI is
considered during the user selection and signal transmission due to the time
varying nature of the channel statistics. Practical path-loss modeling is also
considered for both the FSO and RF links. From this prospective, the major
contributions of this work are as follow:

  * •

For the considered system model, analytical outage probability expression is
derived. The impact of FSO receiver’s type, pointing error, correlation
coefficient, fading severity of the RF link, and number of transmitting
antennas at the UAV are shown on the performance of the outage probability.

  * •

To obtain the diversity order of the considered system, asymptotic outage
probability is also obtained by performing the high signal-to-noise-ratio
(SNR) approximation on the outage probability and the combined impact of
number of users, number of transmit antennas, outdated CSI, fading
coefficient, pointing error, atmospheric turbulence, and the type of FSO
detector are observed on the system performance.

  * •

Based on the instantaneous received rate at the UEs, ergodic capacity
expression is derived. The impact of FSO receiver’s type, pointing error,
correlation coefficient, fading severity of the RF link, and number of
transmitting antennas at the UAV are shown on the performance of the ergodic
capacity.

  * •

To guarantee a minimum delay and to maintain an efficient constant data-rate,
effective capacity is also derived and the impact of delay constraint
($\Theta$) is observed on the performance of effective capacity.

  * •

Based on the derived outage probability expression, a CDF based ASER analysis
is performed for various generalized higher order QAM constellations including
the general order HQAM, RQAM, and XQAM constellations.

  * •

For such complex higher order QAM constellation heterodyne detection at the
FSO receiver is considered and the impact of pointing error, correlation
coefficient, fading severity of RF link, number of transmitting antennas at
the UAV, and number of terrestrial users are shown on the ASER performance of
the general order HQAM, SQAM, RQAM, and XQAM constellations.

  * •

The ASER performances of various QAM constellations are compared which
validate the superiority of the HQAM constellation over the others.

  * •

Finally, all the analytical results are verified through the simulation
results.

The rest of the paper is organized as: Section II discusses the system and
channel model in details. Based on the end-to-end (e2e) instantaneous received
SNR, outage and asymptotic outage probability are derived in Section III. In
Section IV, ergodic capacity and effective capacity are analyzed. Section V
discusses the theoretical and simulation results in details. Finally,
conclusions from the obtained results are drawn in Section VI.

##  II System and Channel Model

Figure 1: Considered system model.

In this work, a downlink asymmetric system model is considered where multiples
terrestrial user equipments (UEs) received information signal through the
satellite (S) via a UAV as shown in Fig. 1. The N terrestrial UEs
($U_{1},U_{2},...,U_{N}$) with single antenna are considered and multiple
antennas are deployed at the UAV. A significant path loss between the S and
UAV exist and hence, a selective DF approach (threshold based DF relaying) is
considered. An asymmetric FSO/RF communication model is considered for e2e
communication, where an FSO link between the satellite and UAV is considered
and RF links between the UAV and multiple UEs are considered. The FSO link is
assumed to be atmospheric turbulence induced Gamma-Gamma distributed with
pointing error impairments. RF links are assumed to be Nakagami-m distributed.
The e2e communication completes in two time phases. In the first time phase,
satellite source (S) transmits an optical signal $x(t)$
($\mathbb{E}[|x(t)|^{2}=1]$) to the UAV. Here, $\mathbb{E}[\cdot]$ represents
the statistical expectation operator. We have considered a 1550nm window for
optical beam which is the most suitable one because it is eye safe and the
Doppler effect is minimal than the other frequencies. At the LEO satellite,
semiconductor laser is embarked and a lens collimates its beam to a 0.15m
diameter telescope. At the receiver, a telescope with a lens diameter 0.25m is
considered to receive the optical beam which is then passed through a filter
for optical background noise reduction. We consider that a 10 Gbps signal is
transmitted in optical domain with 20 GHz bandwidth 111In [14], authors have
shown a diverse range of data-rates (in tens of Gbps) of various commercial
FSO systems for a particular distance (in km). Further, [16] proposes a low
earth orbit (LEO) satellite to HAP optical link setup capable of providing
Gbps data-rates.. Let suppose the Doppler shift may increase the bandwidth by
10 GHz, then we required nearly 30 GHz bandwidth for the filter. Hence, the
optical bandwidth $B_{0}=30$ GHz is considered. At the UAV, photodetector
converts the optical signal into electrical signal which results in

$\displaystyle y_{R}(t)=({P_{s}}\zeta_{R}\eta I)^{i/2}x(t)+v_{R}(t),$ (1)

where $I=I_{p}I_{a}$ represents the FSO channel fading coefficient wherein
$I_{p}$ is the pointing error and $I_{a}$ is the atmospheric turbulence
induced fading. $\eta$ represents the optical-to-electrical conversion
coefficient and $v_{R}(t)$ represents the additive white Gaussian noise (AWGN)
with 0 mean and $\sigma^{2}_{R}$ variance, associated with the S-UAV link. The
noise variance is $\sigma^{2}_{R}=\kappa B_{o}T$, where $\kappa=1.38\times
10^{-23}$ represents the Boltzmann constant, $B_{o}$ is the optical receiver
bandwidth, and $T$ is the noise temperature in kelvin. Also, $i$ represents
the type of optical detector at the FSO receiver, where $i=1$ stands for the
heterodyne detection and $i=2$ stands for the IM/DD detection. Further,
$\zeta_{R}=1/2(G_{S}+G_{R}-A_{atm}-A_{FS}-L_{lenses}-M_{s})$ dB is the
amplitude of the path-loss between the satellite and UAV, where $G_{S}$ and
$G_{R}$ are the transmitter and receiver gains at the satellite and UAV,
respectively, $A_{FS}$ and $A_{atm}$ are the free-space and atmospheric
attenuations, respectively, $L_{lenses}$ is the lenses loss, and $M_{s}$ is
the system margin [16, 18]. The optical transmitter and receiver gains at the
satellite and the UAV are $G_{S}=\frac{\pi^{2}D_{S}^{2}}{\lambda_{f}}$ and
$G_{R}=\frac{\pi^{2}D_{R}^{2}}{\lambda_{f}}$, respectively, where $D_{S}$ and
$D_{R}$ are the lens diameters and $\lambda_{f}$ is the wavelength of the FSO
link. The instantaneous received SNR at the UAV is given as

$\displaystyle\gamma_{R}=\frac{({P_{s}}\zeta_{R}\eta I)^{i}}{\sigma^{2}_{R}}.$
(2)

Selective DF relaying is preferred at the UAV. Hence, information signal at
the UAV is first decoded successfully and is forwarded to terrestrial UEs only
if $\gamma_{R}$ is greater than or equal to a predefined threshold SNR
($\delta_{th}$). If $\gamma_{R}<\delta_{th}$, UAV cannot decode anything from
the received signal and no information is forwarded to the UEs. To avoid this,
transmit signal power is maintained in such a way that the received SNR will
always be greater than or equal to $\delta_{th}$, so that the system will not
be in outage. Hence, $\phi(\gamma_{R})$ is defined as

$\displaystyle\phi(\gamma_{R})=\begin{cases}0,&\text{for
}\gamma_{R}<\delta_{th}\\\ 1,&\text{for }\gamma_{R}\geq\delta_{th}\end{cases}$
(3)

During the second communication phase, decoded electrical signal at UAV is
transmitted to the terrestrial UEs. To improve the transmission reliability
and to improve the channel capacity, $N$ transmit antennas are considered at
the UAV. Further, single antenna is considered at the UEs due to size limit
constraint. Thus, after transmit beamforming, received signal at the $n^{th}$
UE is

$\displaystyle
y_{n}(t)=\phi(\gamma_{R})\sqrt{P_{R}}\zeta_{n}\textbf{h}^{H}_{n}\textbf{w}_{n}x(t)+v_{2}(t),$
(4)

where $P_{R}$ is the transmit power at the UAV.
$\textbf{w}_{n}\in\mathbb{C}^{N_{t}\times 1}$ is the transmit beamforming
weight vector and $\textbf{h}^{H}_{n}$ is the $N_{t}\times 1$ channel vector
between the UAV and the $n^{th}$ UE. Here, $(\cdot)^{H}$ represents the
Hermitian operator. The beamforming weight vector
$\textbf{w}_{n}=\frac{\textbf{h}_{n}}{||\textbf{h}_{n}||_{F}}$ is chosen
according to the maximal ratio transmission (MRT) principle, where
$||(\cdot)||_{F}$ represents the Frobenius norm. Also, $v_{2}(t)$ denotes the
AWGN with 0 mean and $\sigma^{2}_{n}$ variance, associated with the
UAV-$U_{n}$ link. Here, $\sigma^{2}_{n}=\kappa B_{r}T$, where $B_{r}$ is the
noise bandwidth at the RF UEs. Further,
$\zeta^{2}_{n}=\frac{1}{2}\big{(}20~{}\text{log}(\lambda_{RF})-20\alpha_{t}~{}\text{log}(d_{k})-20~{}\text{log}(4\pi)\big{)}$
dB represents the path loss between the UAV and the $n^{th}$ UE, where $d_{k}$
is the distance of the $n^{th}$ UE from the UAV, $\lambda_{RF}$ is the
wavelength of the RF link, and $\alpha_{t}\in[2,4]$ is the path loss factor of
the RF link [42].

Thus, the e2e instantaneous SNR at the $n^{th}$ UE is given as

$\displaystyle\gamma_{U_{n}}=\phi(\gamma_{R})\frac{P_{R}\zeta_{n}^{2}|\textbf{w}^{H}_{n}\textbf{h}_{n}|^{2}}{\sigma^{2}_{n}}=\phi(\gamma_{R})\gamma_{RU_{n}}.$
(5)

The satellite to UAV link is an FSO link, its fading channel coefficient is
given as $I=I_{p}I_{a}$. The atmospheric turbulence $I_{a}$ is modeled with
Gamma-Gamma distribution, its probability density function (PDF) is given as

$\displaystyle
f_{I_{a}}(x)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta)}x^{(\alpha+\beta)/2-1}\text{K}_{\alpha-\beta}\Big{(}2\sqrt{\alpha\beta
x}\Big{)},$ (6)

where $\text{K}_{v}(\cdot)$ represents the $v^{th}$ order modified Bessel
function of second kind, and $\alpha$ and $\beta$ are the fading coefficients
of the atmospheric turbulence. Further, the pointing error can be modeled as

$\displaystyle
f_{I_{p}}(x)=\frac{\xi^{2}}{A_{0}^{\xi^{2}}}x^{\xi^{2}-1},~{}~{}~{}~{}\text{for
}0\leq x\leq A_{0}$ (7)

where $\xi$ is the ratio of the equivalent beam radius to the standard
deviation of jitter at the receiver. Further, $A_{0}$ represents the fraction
of the receiver power at $r=0$, where $r$ is the radial displacement at the
receiver.

As the FSO link is assumed to follow the Gamma-Gamma distribution with
pointing error impairments, the PDF of its SNR ($\gamma_{R}$) for both the
IM/DD and heterodyne detection techniques at the FSO receiver is given as

$\displaystyle f_{\gamma_{R}}(x)$
$\displaystyle=\frac{\xi^{2}}{i\Gamma(\alpha)\Gamma(\beta)x}\text{G}^{3,0}_{1,3}\Big{[}\alpha\beta\Big{(}\frac{x}{\mu_{i}}\Big{)}^{1/i}\Big{|}^{\xi^{2}+1}_{\xi^{2},\alpha,\beta}\Big{]}.$
(8)

Now (8) is substituted in
$F_{{\gamma}_{R}}(x)=\int_{0}^{x}f_{{\gamma}_{R}}(\gamma)d\gamma$ and (9) [43,
(07.34.21.0084.01)] is used as

$\displaystyle\int_{0}^{a}x^{c_{1}-1}\text{G}^{m,n}_{p,q}\Big{[}c_{2}x^{l/i}\Big{|}^{a_{1},...,a_{n},a_{n+1},...,a_{p}}_{b_{1},...,b_{m},b_{m+1},...,b_{q}}\Big{]}dx=\frac{i^{a^{*}}}{(2\pi)^{c^{*}(i-1)}a^{-c_{1}}}$
$\displaystyle\text{G}^{mi,ni+l}_{pi+l,qi+l}\Big{[}\frac{c_{2}^{i}a^{l}}{i^{i(q-p)}}\Big{|}^{\frac{1-c_{1}}{l},...,\frac{l-c_{1}}{l},\frac{a_{1}}{i},...,\frac{a_{1}+i-1}{i},...,\frac{a_{n}}{i},...,\frac{a_{n}+i-1}{i},\frac{a_{n+1}}{i},...,\frac{a_{n+1}+i-1}{i},...,\frac{a_{p}}{i},...,\frac{a_{p}+i-1}{i}}_{\frac{b_{1}}{i},...,\frac{b_{1}+i-1}{i},...,\frac{b_{m}}{i},...,\frac{b_{m}+i-1}{i},\frac{b_{m+1}}{i},...,\frac{b_{m+1}+i-1}{i},...,\frac{b_{q}}{i},...,\frac{b_{q}+i-1}{i},\frac{-c_{1}}{l},...,\frac{l-c_{1}-1}{l}}\Big{]},$
(9)

where $c^{*}=m+n-\frac{p+q}{2}$, and
$a^{*}=\sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}+\frac{p-q}{2}+1$. Solving (9),
the cumulative distribution function (CDF) of the SNR of the FSO link for both
the detection techniques is given as

$\displaystyle
F_{\gamma_{R}}(x)=A\text{G}^{3i,1}_{i+1,3i+1}\Big{[}\frac{B}{\mu_{i}}x\Big{|}^{1,\tau_{1}}_{\tau_{2},0}\Big{]},$
(10)

where
$A=\frac{i^{(\alpha+\beta-2)}\xi^{2}}{(2\pi)^{i-1}\Gamma(\alpha)\Gamma(\beta)}$,
$B=\frac{(\alpha\beta)^{i}}{i^{2i}}$, and $\mu_{i}$ represents the average SNR
of the FSO link for both the detection techniques. For heterodyne detection,
$i=1$, and hence, $\mu_{1}=\bar{\gamma}_{R}$. For the IM/DD detection, $i=2$,
and hence,
$\mu_{2}=\frac{\xi^{2}\alpha\beta(\xi^{2}+2)}{(\alpha+1)(\beta+1)(\xi^{2}+1)^{2}}\bar{\gamma}_{R}$.
Further, $\tau_{1}=\big{[}\frac{\xi^{2}+1}{i},...,\frac{\xi^{2}+i}{i}\big{]}$
and
$\tau_{2}=\Big{[}\frac{\xi^{2}}{i},...,\frac{\xi^{2}+i-1}{i},\frac{\alpha}{i},...,\frac{\alpha+i-1}{i},\frac{\beta}{i},...,\frac{\beta+i-1}{i}\Big{]}$.

The UAV is equipped with $N_{t}$ antennas and single antenna is considered at
each of the N terrestrial UEs. Further, UAV to the $n^{th}$ terrestrial UE
link is assumed to follow Nakagami-m distribution with fading severity $m$ and
average power $\Omega$. Thus, the PDF and CDF of the SNR at the $n^{th}$ UE
are given as [44]

$\displaystyle f_{\gamma_{RU_{n}}}(x)$
$\displaystyle=\frac{1}{\Gamma(mN_{t})}\Big{(}\frac{m}{\bar{\gamma}_{U}}\Big{)}^{mN_{t}}x^{mN_{t}-1}e^{-\frac{m}{\bar{\gamma}_{U}}x}$
$\displaystyle F_{\gamma_{RU_{n}}}(x)$
$\displaystyle=1-\frac{1}{\Gamma(mN_{t})}\Gamma\Big{(}mN_{t},\frac{m}{\bar{\gamma}_{U}}x\Big{)},$
(11)

respectively, where $\bar{\gamma}_{U}=\Omega\bar{\gamma}_{RU_{n}}$ and
$\Gamma(\cdot)$ is the complete gamma function. It is assumed that the UEs are
in closed proximity and following the independent and identical distribution
(i.i.d.). For multiuser diversity, opportunistic scheduling of the UEs is
performed. Thus, the channel quality between the UAV and the $n^{th}$ UE is
estimated for the transmission. For this, first the strongest link between the
UAV and the set of UEs is selected by the UAV and the index of the selected UE
is feedback to the satellite. Therefore, after opportunistic scheduling, the
instantaneous SNR of the UAV to terrestrial UE link is given as
$\gamma_{RU}=\underset{\overset{n=1,2,...,N}{}}{\text{Max}}\gamma_{RU_{n}}$.
Applying the order statistics, the PDF of $\gamma_{RU}$ is given as

$\displaystyle
f_{\gamma_{RU}}(x)=N(F_{\gamma_{RU_{n}}}(x))^{N-1}f_{\gamma_{RU_{n}}}(x).$
(12)

Substituting the PDF and CDF of the SNR of UAV to $n^{th}$ UE link from (II)
in (12), utilizing the series form of the upper incomplete Gamma function, and
expanding the multinomial using [45, (0.314)], the PDF of $\gamma_{RU}$ is
given as

$\displaystyle f_{\gamma_{RU}}(x)$
$\displaystyle=N\sum_{k=0}^{N-1}\binom{N-1}{k}(-1)^{k}\sum_{l=0}^{k(mN_{t}-1)}\varphi_{l}^{k}\Big{(}\frac{m}{\bar{\gamma}_{U}}\Big{)}^{mN_{t}+l}\frac{1}{\Gamma(mN_{t})}x^{mN_{t}+l-1}e^{-\frac{m}{\bar{\gamma}_{U}}(k+1)x},$
(13)

where $\varphi_{l}^{k}$ is recursively calculated as
$\varphi_{0}^{k}=(\delta_{0})^{k}$, $\varphi_{1}^{k}=k(\delta_{1})$,
$\varphi_{l}^{k}=\frac{1}{l\delta_{0}}\sum_{q=1}^{l}\Big{[}(qk-l+q)\delta_{q}\varphi^{k}_{l-q}\Big{]}$
for $2\leq l\leq(mN_{t}-1)$,
$\varphi_{l}^{k}=\frac{1}{l\delta_{0}}\sum_{q=1}^{mN_{t}-1}\Big{[}(qk-l+q)\delta_{q}\varphi^{k}_{l-q}\Big{]}$
for $mN_{t}\leq l<k(mN_{t}-1)$, and
$\varphi_{k(mN_{t}-1)}^{k}=(\delta_{mN_{t}-1})^{k}$, wherein
$\delta_{l}=\frac{1}{l!}$ [36].

In practice, the channel statistics changes due to its time varying nature and
hence, outdated CSI is obtained at the UAV. Delay occurs during the user
selection and signal transmission from the UAV. Let, $\hat{\gamma}_{RU}$ is
the delayed version of the ${\gamma}_{RU}$ since $\hat{\gamma}_{RU_{n}}$ is
the delayed version of the ${\gamma}_{RU_{n}}$. Let, $\hat{\gamma}_{RU_{n}}$
is correlated to ${\gamma}_{RU_{n}}$ with $\rho=J_{0}(2\pi f_{d}\tau)$, where
$f_{d}$ is the Doppler frequency, $\tau$ is the time delay, and $J_{0}(\cdot)$
is the zeroth order Bessel function of first kind. According to the order
statistics, $\hat{\gamma}_{RU}$ will be the induced order statistics of
${\gamma}_{RU}$ [44]. Therefore, the PDF of $\hat{\gamma}_{RU}$ can be given
as

$\displaystyle
f_{\hat{\gamma}_{RU}}(x)=\int_{0}^{\infty}f_{\hat{\gamma}_{RU}|\gamma_{RU}}(x|y)f_{{\gamma}_{RU}}(y)dy,$
(14)

where $f_{\hat{\gamma}_{RU}|\gamma_{RU}}(x|y)$ is the conditional probability
of $\hat{\gamma}_{RU}$ given ${\gamma}_{RU}$. As $\hat{\gamma}_{RU}$ and
${\gamma}_{RU}$ are the correlated Gamma distributed random variables, their
conditional probability is given as

$\displaystyle f_{\hat{\gamma}_{RU}|\gamma_{RU}}(x|y)$
$\displaystyle=\frac{1}{1-\rho}\Big{(}\frac{m}{\bar{\gamma}_{U}}\Big{)}\Big{(}\frac{x}{\rho
y}\Big{)}^{\frac{(mN_{t}-1)}{2}}e^{-\frac{m(x+\rho
y)}{\bar{\gamma}_{U}(1-\rho)}}{I}_{mN_{t}-1}\Big{(}\frac{2m\sqrt{\rho
xy}}{\bar{\gamma}_{U}(1-\rho)}\Big{)},$ (15)

where $I_{v}(\cdot)$ represents the $v^{th}$ order modified Bessel function of
first kind. Substituting the values from (13) and (15) in (14), considering
the series expansion of $I_{v}(\cdot)$, and after some mathematical
computations, we get

$\displaystyle f_{\hat{\gamma}_{RU}}(x)$
$\displaystyle=N\sum_{k=0}^{N-1}\binom{N-1}{k}\frac{(-1)^{k}}{\Gamma(mN_{t})}\sum_{l=0}^{k(mN_{t}-1)}\varphi_{l}^{k}\sum_{j=0}^{l}\binom{l}{j}\Big{(}\frac{m}{\bar{{\gamma}}_{U}}\Big{)}^{(mN_{t}+j)}\frac{\Gamma(mN_{t}+l)}{\Gamma(mN_{t}+j)}$
$\displaystyle\times\Big{(}\frac{\rho^{j}(1-\rho)^{l-j}}{(1+k(1-\rho))^{(mN_{t}+l+j)}}\Big{)}x^{mN_{t}+j-1}e^{-\big{(}\frac{m(k+1)x}{\bar{{\gamma}}_{U}(1+k(1-\rho))}\big{)}}.$
(16)

Substituting (II) in
$F_{\hat{\gamma}_{RU}}(x)=\int_{0}^{x}f_{\hat{\gamma}_{RU}}(\gamma)d\gamma$,
the CDF of the outdated SNR of the UAV to selected terrestrial UE link is
given as

$\displaystyle F_{\hat{\gamma}_{RU}}(x)$
$\displaystyle=N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\Upsilon\Big{(}\mathbb{C}_{1},\mathbb{C}_{2}x\Big{)},$
(17)

where $\Upsilon(\cdot,\cdot)$ represents the lower incomplete Gamma function,
$\underset{\overset{k,l,j}{}}{\sum}=\sum_{k=0}^{N-1}\sum_{l=0}^{k(mN_{t}-1)}\sum_{j=0}^{l}$,
$\mathbb{C}_{0}=\binom{N-1}{k}\binom{l}{j}\frac{(-1)^{k}}{\Gamma(mN_{t})}\varphi_{l}^{k}\Big{(}\frac{m}{\bar{{\gamma}}_{U}}\Big{)}^{mN_{t}+j}\frac{\rho^{j}(1-\rho)^{l-j}\Gamma(mN_{t}+l)}{(1+k(1-\rho))^{(mN_{t}+l+j)}\Gamma(mN_{t}+j)}$,
$\mathbb{C}_{1}=mN_{t}+j$, and
$\mathbb{C}_{2}=\frac{m(k+1)}{\bar{{\gamma}}_{U}(1+k(1-\rho))}$.

##  III Outage Performance

###  III-A Outage Probability

Outage probability is defined as the probability that the e2e instantaneous
SNR reaches below a predefined threshold level ($\gamma_{th}$). Selective DF
protocol is used at the UAV. After applying the order statistics to select the
strongest user and considering the outdated CSI, the e2e instantaneous
received SNR can be expressed as
$\gamma_{e2e}=\phi(\gamma_{R})\hat{\gamma}_{RU}$. Thus, the outage probability
is given as

$\displaystyle\mathcal{P}_{o}(\gamma_{th})$
$\displaystyle=\mathcal{P}[\phi(\gamma_{R})\hat{\gamma}_{RU}<\gamma_{th}],$
$\displaystyle=\mathcal{P}[\phi(\gamma_{R})=0]+\mathcal{P}[\phi(\gamma_{R})=1]\mathcal{P}[\hat{\gamma}_{RU}<\gamma_{th}],$
(18)

where,

$\displaystyle\mathcal{P}[\phi(\gamma_{R})=0]$
$\displaystyle=\mathcal{P}[\phi(\gamma_{R})<\delta_{th}]=F_{\gamma_{R}}(\delta_{th}),$
$\displaystyle\mathcal{P}[\phi(\gamma_{R})=1]$
$\displaystyle=\mathcal{P}[\phi(\gamma_{R})\geq\delta_{th}]=1-F_{\gamma_{R}}(\delta_{th}).$
(19)

Further,
$\mathcal{P}[\hat{\gamma}_{RU}<\gamma_{th}]=F_{\hat{\gamma}_{RU}}(\gamma_{th})$
which is given in (17). Finally, substituting the values from (17) and (III-A)
in (III-A), the outage probability for the considered system is given as

$\displaystyle\mathcal{P}_{o}(\gamma_{th})$
$\displaystyle=F_{\gamma_{R}}(\delta_{th})+(1-F_{\gamma_{R}}(\delta_{th}))F_{\hat{\gamma}_{RU}}(\gamma_{th}),$
$\displaystyle=A\text{G}^{3i,1}_{i+1,3i+1}\Big{[}\frac{B\delta_{th}}{\mu_{i}}\Big{|}^{1,\tau_{1}}_{\tau_{2},0}\Big{]}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\Big{(}1-A\text{G}^{3i,1}_{i+1,3i+1}\Big{[}\frac{B\delta_{th}}{\mu_{i}}\Big{|}^{1,\tau_{1}}_{\tau_{2},0}\Big{]}\Big{)}\Upsilon(\mathbb{C}_{1},\mathbb{C}_{2}\gamma_{th}).$
(20)

###  III-B Diversity Order

The outage probability obtained in (III-A) highlights various insights about
the performance of the considered system. However, obtaining the diversity
order from (III-A) is quite tricky due to its complex nature. Therefore, in
this subsection, an asymptotic outage probability expression is derived by
performing the high SNR approximation on the outage probability. For this, the
transmit SNR tends to infinity, and hence,
$\gamma_{R},\gamma_{U}\rightarrow\infty$. Thus, the outage probability (III-A)
can be approximated as

$\displaystyle\mathcal{P}_{o}^{Asym}(\gamma_{th})\approx
F_{\gamma_{R}}(\delta_{th})+F_{\hat{\gamma}_{RU}}(\gamma_{th}).$ (21)

It is to be noted that the product term from (III-A) is neglected at high SNR
as it tends to zero.

Corollary 1: The approximate expression of the asymptotic outage probability
can be given as

$\displaystyle\mathcal{P}_{o}^{Asym}(\gamma_{th})$ $\displaystyle\approx
A\sum_{p=1}^{3i}\Big{(}\frac{B\delta_{th}}{\mu_{i}}\Big{)}^{\tau_{4},p}\frac{\prod_{\underset{q\neq
p}{q=1}}^{3i}\Gamma(\tau_{4,q}-\tau_{4,p})\prod_{\underset{q=1}{}}^{1}\Gamma(1-\tau_{3,q}+\tau_{4,p})}{\prod_{\underset{q=2}{}}^{i+1}\Gamma(\tau_{3,q}-\tau_{4,p})\prod_{\underset{3i+1}{}}^{3i+1}\Gamma(1-\tau_{4,q}+\tau_{4,p})}$
$\displaystyle+\begin{cases}F_{\hat{\gamma}_{RU1}}(\gamma_{th})\text{ for
$\rho=1$}\\\ F_{\hat{\gamma}_{RU2}}(\gamma_{th})\text{ for
$\rho<1$}\end{cases}$ (22)

Here, $F_{\hat{\gamma}_{RU1}}(\gamma_{th})$ represents the approximate high
SNR CDF expression of the terrestrial link for the perfect CSI case which is
given as

$\displaystyle\vspace{-1em}F_{\hat{\gamma}_{RU1}}(\gamma_{th})\approx\Big{(}\frac{1}{\Gamma(mN_{t}+1)}\Big{)}^{N}\Big{(}\frac{m\gamma_{th}}{\bar{\gamma}_{U}}\Big{)}^{NmN_{t}}.$
(23)

Further, $F_{\hat{\gamma}_{RU2}}(\gamma_{th})$ represents the approximate high
SNR CDF expression of the terrestrial link in case of outdated CSI which is
given as

$\displaystyle F_{\hat{\gamma}_{RU2}}(\gamma_{th})\approx
N\underset{\overset{k,l}{}}{\sum}~{}\mathbb{D}_{0}\Big{(}\frac{m\gamma_{th}}{\bar{\gamma}_{U}}\Big{)}^{mN_{t}},$
(24)

where
$\underset{\overset{k,l}{}}{\sum}=\sum_{k=0}^{N-1}\sum_{l=0}^{k(mN_{t}-1)}$
and
$\mathbb{D}_{0}=\binom{N-1}{k}\frac{(-1)^{k}}{(\Gamma(mN_{t}))^{2}}\varphi_{l}^{k}\frac{\Gamma(mN_{t}+l)}{mN_{t}}(1+k(1-\rho))^{-(mN_{t}+l)}(1-\rho)^{l}.\vspace{1em}$
Proof: See Appendix.

From (47), it is observed that the dominant term of meijer-G function is
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i})$ (defined by the
smallest negative power of the transmit SNR term (consequently $\gamma_{R}$)).

Case-1: For the perfect CSI case, using (23) in (III-B), the diversity order
for the considered system can be obtained as
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i},NmN_{t})$.

Case-2: For the outdated CSI case, using (24) in (III-B), the diversity order
for the considered system can be obtained as
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i},mN_{t})$.

From the above, it can be concluded that the diversity order depends upon the
atmospheric turbulence, pointing error, FSO detection type, number of users,
fading parameter, and number of transmit antennas for the perfect CSI case,
however, the multiuser diversity effect is vanished in case of outdated CSI.

##  IV Capacity Analysis

In this section, we will derive the ergodic and the effective capacity (in
bits/s/Hz) of the considered network.

###  IV-A Ergodic Capacity

Statistical expectation of the mutual information between the end users gives
us the ergodic capacity (bits/s/Hz) of the considered system which can be
expressed as [28, 46, 47, 42]

$\displaystyle\mathcal{C}_{e}$
$\displaystyle=\frac{1}{2}\mathbb{E}[\text{log}_{2}(1+\varrho\,\gamma_{e2e})]$
$\displaystyle=\frac{1}{2}\mathbb{E}[\text{log}_{2}(1+\varrho\,\phi(\gamma_{R})\hat{\gamma}_{RU})]$
$\displaystyle=\frac{1}{2}{\mathcal{P}[\phi(\gamma_{R})=1]}~{}\underset{\overset{I_{1}}{}}{\underbrace{\mathbb{E}[\text{log}_{2}(1+\varrho\,\hat{\gamma}_{RU})]}}.$
(25)

Here, $\varrho=1$ for the heterodyne detection ($i=1$) and
$\varrho=\frac{e}{2\pi}$ for the IM/DD ($i=2$). It is to be noted that (IV-A)
is exact for heterodyne detection ($i=1$) and is the lower-bound for the IM/DD
($i=2$). From (IV-A), $I_{1}$ can be solved as

$\displaystyle I_{1}=$
$\displaystyle\int_{0}^{\infty}\text{log}_{2}(1+\varrho\,\gamma)f_{\hat{\gamma}_{RU}}(\gamma)d\gamma.$
(26)

Invoking (II) in (26), and solving the required integral with the help of [48,
(8.4.6.5))], we get

$\displaystyle I_{1}=$
$\displaystyle\frac{N}{\text{ln}(2)}\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\text{G}^{1,3}_{3,2}\Big{[}\frac{\varrho}{\mathbb{C}_{2}}\Big{|}^{-\mathbb{C}_{1}+1,1,1}_{1,0}\Big{]}.$
(27)

Substituting the identities from (III-A) and (27) in (IV-A), final expression
of ergodic capacity is derived as

$\displaystyle\mathcal{C}_{e}$
$\displaystyle=\frac{N}{2\,\text{ln}(2)}\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\text{G}^{1,3}_{3,2}\Big{[}\frac{\varrho}{\mathbb{C}_{2}}\Big{|}^{-\mathbb{C}_{1}+1,1,1}_{1,0}\Big{]}\Big{(}1-A\text{G}^{3i,1}_{i+1,3i+1}\Big{[}\frac{B\delta_{th}}{\mu_{i}}\Big{|}^{1,\tau_{1}}_{\tau_{2},0}\Big{]}\Big{)}.$
(28)

###  IV-B Effective Capacity

The emerging next generation wireless communication applications such as
multimedia streaming, interactive gaming, autonomous vehicles, mobile video
telephony, etc require low latency. The extensively used ergodic capacity
metric is unable to measure the quality of service (QoS) of such emerging
real-time applications. Hence, effective capacity, a QoS aware link-layer
channel model is used in the wireless networks for performance analysis under
such delay constraints [49]. Effective capacity guarantees system’s minimum
delay, and maintains an efficient constant data-rate. Higher value of
effective capacity guarantees higher QoS and lower delay of the considered
system. Effective capacity can be expressed as

$\displaystyle\mathcal{C}_{eff}(\Theta)=-\frac{1}{\Theta}\text{ln}(\mathbb{E}[e^{-\Theta
r_{e2e}}]),$ (29)

where $r_{e2e}$ is the target rate of the communication system and $\Theta$ is
the quality parameter. Increase in $\Theta$ limits the effective capacity of
the communication system. The target rate (in bits/s/Hz) of the considered
communication system can be given as
$r_{e2e}=\frac{1}{2}\text{log}_{2}(1+\varrho\,\gamma_{e2e})$. Substituting
$r_{e2e}$ in (29), we obtain

$\displaystyle\mathcal{C}_{eff}(\Theta)=-\frac{1}{\Theta}\text{ln}\Big{(}\mathbb{E}[e^{-\frac{\Theta}{2\text{ln}(2)}\;\text{ln}(1+\varrho\,\gamma_{e2e})}]\Big{)}.$
(30)

After mathematical simplification, we get

$\displaystyle\mathcal{C}_{eff}(\Theta)=-\frac{1}{\Theta}\text{ln}\Big{(}\mathbb{E}[(1+\varrho\,\gamma_{e2e})^{-\hat{\Theta}}]\Big{)},$
(31)

where $\hat{\Theta}=\frac{\Theta}{2\text{ln}(2)}$.

Substituting the $\gamma_{e2e}$ in (31), $\mathcal{C}_{eff}(\Theta)$ can be
written as

$\displaystyle\mathcal{C}_{eff}(\Theta)$
$\displaystyle=-\frac{1}{\Theta}\text{ln}\Big{(}\mathbb{E}[(1+\varrho\,\phi(\gamma_{R})\hat{\gamma}_{RU})^{-\hat{\Theta}}]\Big{)}.$
(32)

Further, (32) can be written as

$\displaystyle\mathcal{C}_{eff}(\Theta)$
$\displaystyle=-\frac{1}{\Theta}{\mathcal{P}[\phi(\gamma_{R})=1]}~{}\text{ln}\Big{(}\mathbb{E}[(1+\varrho\,\hat{\gamma}_{RU})^{-\hat{\Theta}}]\Big{)},$
$\displaystyle=-\frac{1}{\Theta}{\mathcal{P}[\phi(\gamma_{R})=1]}~{}\text{ln}\Big{(}\int_{0}^{\infty}(1+\varrho\,\gamma)^{-\hat{\Theta}}f_{\hat{\gamma}_{RU})}(\gamma)d\gamma\Big{)}.$
(33)

Substituting $f_{\hat{\gamma}_{RU})}(\gamma)$ from (II) in (IV-B), effective
capacity in integral form can be expressed as

$\displaystyle\mathcal{C}_{eff}(\Theta)$
$\displaystyle=-\frac{1}{\Theta}{\mathcal{P}[\phi(\gamma_{R})=1]}\text{ln}\Big{(}N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\int_{0}^{\infty}(1+\varrho\,\gamma)^{-\hat{\Theta}}~{}\gamma^{\mathbb{C}_{1}-1}e^{-\mathbb{C}_{2}\gamma}d\gamma\Big{)}.$
(34)

After change of variables and with some mathematical computations, we get

$\displaystyle\mathcal{C}_{eff}(\Theta)$
$\displaystyle=-\frac{1}{\Theta}{\mathcal{P}[\phi(\gamma_{R})=1]}~{}\text{ln}\Big{(}N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}e^{\mathbb{C}_{2}}\sum_{z1=0}^{\mathbb{C}_{1}-1}\binom{\mathbb{C}_{1}-1}{z1}(-1)^{\mathbb{C}_{1}-1-z1}\int_{1}^{\infty}t^{z1-\hat{\Theta}}~{}e^{-\mathbb{C}_{2}t}dt\Big{)}.$
(35)

Solving the required integral with the help of [45, (3.351)], analytical
expression of effective capacity can be derived as

$\displaystyle\mathcal{C}_{eff}(\Theta)$
$\displaystyle=-\frac{1}{\Theta}\Big{(}1-A\text{G}^{3i,1}_{i+1,3i+1}\Big{[}\frac{B\delta_{th}}{\mu_{i}}\Big{|}^{1,\tau_{1}}_{\tau_{2},0}\Big{]}\Big{)}\text{ln}\Big{(}N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\Big{(}\frac{1}{\varrho}\Big{)}^{\mathbb{C}_{1}-z_{1}+\hat{\Theta}-1}e^{\frac{\mathbb{C}_{2}}{\varrho}}\sum_{z1=0}^{\mathbb{C}_{1}-1}\binom{\mathbb{C}_{1}-1}{z1}$
$\displaystyle\times(-1)^{\mathbb{C}_{1}-1-z1}\mathbb{C}_{2}^{-(z1-\hat{\Theta}+1)}\Gamma\Big{(}(z1-\hat{\Theta}+1),\mathbb{C}_{2}\Big{)}\Big{)}.$
(36)

##  V ASER Analysis

For any modulation scheme, the CDF based generalized ASER expression for a
given instantaneous e2e SNR is given as

$\mathcal{P}_{e}=-\int_{0}^{\infty}\mathcal{P}_{s}^{{}^{\prime}}(e|\gamma)\mathcal{P}_{o}(\gamma)d\gamma,$
(37)

where $\mathcal{P}_{o}(\gamma)$ represents the outage probability of the
considered system and $\mathcal{P}_{s}^{{}^{\prime}}(e|\gamma)$ is the first
order derivative of the conditional SEP ($\mathcal{P}_{s}(e|\gamma)$) of the
received SNR.

###  V-A Hexagonal QAM

For the AWGN channel, generalized conditional SEP expression of $M$-ary HQAM
constellation is given as

$\displaystyle\mathcal{P}^{H}_{s}(e|\gamma)$
$\displaystyle=KQ(\sqrt{\theta\gamma})+\frac{2}{3}K_{c}Q^{2}\Big{(}\sqrt{\frac{2\theta\gamma}{3}}\Big{)}-2K_{c}Q(\sqrt{\theta\gamma})Q\Big{(}\sqrt{\frac{\theta\gamma}{3}}\Big{)},$
(38)

where $\theta$, $K$, and $K_{c}$ are the constants and their values for
various irregular HQAM (optimum) constellations are given in Table I. The
first order derivative of ($\mathcal{P}_{s}(e|\gamma)$) can be given as [36,
35]

$\displaystyle\mathcal{P^{{}^{\prime}}}_{s}^{H}(e|\gamma)$
$\displaystyle=\frac{1}{2}\sqrt{\frac{\theta}{2\pi}}(K_{c}-K)\gamma^{-\frac{1}{2}}e^{-\frac{\theta\gamma}{2}}-\frac{K_{c}}{3}\sqrt{\frac{\theta}{3\pi}}\gamma^{-\frac{1}{2}}e^{-\frac{\theta\gamma}{3}}+\frac{K_{c}}{2}\sqrt{\frac{\theta}{6\pi}}\gamma^{-\frac{1}{2}}e^{-\frac{\theta\gamma}{6}}$
$\displaystyle+\frac{2K_{c}\theta}{9\pi}e^{-\frac{2\theta}{3}\gamma}\;_{1}F_{1}\Big{(}1;\frac{3}{2};\frac{\theta}{3}\gamma\Big{)}-\frac{K_{c}\theta
e^{-\frac{2\theta}{3}\gamma}}{2\sqrt{3}\pi}\Big{[}\;_{1}F_{1}\Big{(}1;\frac{3}{2};\frac{\theta}{2}\gamma\Big{)}+\;_{1}F_{1}\Big{(}1;\frac{3}{2};\frac{\theta}{6}\gamma\Big{)}\Big{]}.$
(39)

Invoking $\mathcal{P}_{o}(\gamma)$ from (III-A) and
$\mathcal{P^{{}^{\prime}}}_{s}^{H}(e|\gamma)$ from (V-A) into (37), and using
the identities from [45, (6.455), (7.813)] to solve the required integrals,
the generalized ASER expression of HQAM constellation is derived as

$\displaystyle\mathcal{P}^{H}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{\theta}{2\pi}}(K-K_{c})\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{2}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{\theta}{2}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{2}\Big{)}$
$\displaystyle+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{\theta}{2}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}+\frac{K_{c}}{3}\sqrt{\frac{\theta}{3\pi}}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{3}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{\theta}{3}\Big{)}$
$\displaystyle-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{3}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{\theta}{3}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}-\frac{K_{c}}{2}\sqrt{\frac{\theta}{6\pi}}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{6}\Big{)}$
$\displaystyle+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{\theta}{6}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{\theta}{6}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{\theta}{6}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle-\sum_{z_{1}=0}^{\infty}\mathbb{C}_{3}\bigg{[}A\mathbb{F}\Big{(}{-z_{1}-1},\frac{2\theta}{3}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}z_{1}+1,\frac{2\theta}{3}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}-z_{1}-1,\frac{2\theta}{3}\Big{)}$
$\displaystyle+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-z_{1}-1,\mathbb{C}_{2}+\frac{2\theta}{3}\Big{)}\bigg{)}\bigg{\\}}\bigg{]},$
(40)

where
$\mathbb{C}_{3}=\frac{(1)_{z_{1}}}{(1.5)_{z_{1}}z_{1}!}\Big{(}\frac{2K_{c}\theta}{9\pi}\Big{(}\frac{\theta}{3}\Big{)}^{z_{1}}-\frac{K_{c}\theta}{2\sqrt{3}\pi}\Big{(}\Big{(}\frac{\theta}{2}\Big{)}^{z_{1}}+\Big{(}\frac{\theta}{6}\Big{)}^{z_{1}}\Big{)}\Big{)}$,
$\mathbb{F}(\psi_{1},\psi_{2})=(\psi_{2})^{\psi_{1}}\text{G}^{3i,2}_{i+2,3i+1}\Big{[}\frac{B}{\psi_{2}\mu_{i}}\Big{|}^{\psi_{1}+1,1,\tau_{1}}_{\tau_{2},0}\Big{]}$,
and
$\mathbb{G}(\psi_{1},\psi_{2})=\frac{\mathbb{C}_{2}^{\mathbb{C}_{1}}\Gamma(\mathbb{C}_{1}+\psi_{1})}{\mathbb{C}_{1}(\mathbb{C}_{2}+\psi_{2})^{(\mathbb{C}_{1}+\psi_{1})}}\;_{2}F_{1}(1,\mathbb{C}_{1}+\psi_{1},\mathbb{C}_{1}+1,\frac{\mathbb{C}_{2}}{\mathbb{C}_{2}+\psi_{2}})$.

TABLE I: Various selection parameters for irregular HQAM (optimum) constellations [13]. | Irregular HQAM (optimum)
---|---
M | $\theta$ | $K$ | $K_{c}$
4 | 1 | $\frac{5}{2}$ | $\frac{3}{2}$
8 | $\frac{32}{69}$ | $\frac{7}{2}$ | $\frac{21}{8}$
16 | $\frac{8}{35}$ | $\frac{33}{8}$ | $\frac{27}{8}$
32 | $\frac{512}{4503}$ | $\frac{75}{16}$ | $\frac{33}{8}$
64 | $\frac{8}{141}$ | $\frac{163}{32}$ | $\frac{75}{16}$
128 | $\frac{2}{70.56}$ | $\frac{343}{64}$ | $\frac{81}{16}$
256 | $\frac{2}{141}$ | $\frac{711}{128}$ | $\frac{171}{32}$
512 | $\frac{200}{28217}$ | $\frac{2911}{512}$ | $\frac{5667}{1024}$
1024 | $\frac{100}{28227}$ | $\frac{2955}{512}$ | $\frac{1449}{256}$

###  V-B RQAM

For the AWGN channel, the conditional SEP expression for $M_{i}\times
N_{q}$-QAM constellation is given in

$\displaystyle\mathcal{P}^{R}_{s}(e|\gamma)=2\Big{[}p_{0}Q(a_{0}\sqrt{\gamma})(1-2q_{0}Q(b_{0}\sqrt{\gamma}))+q_{0}Q(b_{0}\sqrt{\gamma})\Big{]},$
(41)

where $p_{0}=1-\frac{1}{M_{i}}$, $q_{0}=1-\frac{1}{N_{q}}$,
$a_{0}=\sqrt{\frac{6}{(M_{i}^{2}-1)+(N_{q}^{2}-1)\beta_{R}}}$, and
$b_{0}=\beta_{R}a_{0}$. Here $\beta_{R}=d_{Q}/d_{I}$ is the ratio of the
quadrature and in-phase decision distances and $M_{i}$ and $N_{q}$ are the
constellation points of in-phase and quadrature phase, respectively. The first
order derivative of (41) can be obtained as

$\displaystyle\mathcal{P^{{}^{\prime}}}^{R}_{s}(e|\gamma)$
$\displaystyle=\gamma^{-\frac{1}{2}}\Big{[}\frac{a_{0}p_{0}(q_{0}-1)}{\sqrt{2\pi}}e^{-\frac{a_{0}^{2}\gamma}{2}}+\frac{b_{0}(p_{0}-1)q_{0}}{\sqrt{2\pi}}e^{-\frac{b_{0}^{2}\gamma}{2}}\Big{]}$
$\displaystyle-\frac{a_{0}b_{0}p_{0}q_{0}}{\pi}e^{-\frac{(a_{0}^{2}+b_{0}^{2})\gamma}{2}}\Big{[}\,_{1}F_{1}\Big{(}1;\frac{3}{2};\frac{a_{0}^{2}\gamma}{2}\Big{)}+\,_{1}F_{1}\Big{(}1;\frac{3}{2};\frac{b_{0}^{2}\gamma}{2}\Big{)}\Big{]}.$
(42)

Substituting the respective values of
$\mathcal{P^{{}^{\prime}}}_{s}^{R}(e|\gamma)$ and $\mathcal{P}_{0}(\gamma)$
from (V-B) and (III-A) into (37), solving the required integral with the help
of [45, (6.455), (7.813)] with some mathematical computations, the generalized
ASER expression of RQAM constellation is derived as

$\displaystyle\mathcal{P}^{R}$
$\displaystyle=\frac{a_{0}p_{0}(1-q_{0})}{\sqrt{2\pi}}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{a_{0}^{2}}{2}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{a_{0}^{2}}{2}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{a_{0}^{2}}{2}\Big{)}$
$\displaystyle+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{a_{0}^{2}}{2}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}+\frac{b_{0}q_{0}(1-p_{0})}{\sqrt{2\pi}}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{b_{0}^{2}}{2}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{b_{0}^{2}}{2}\Big{)}$
$\displaystyle-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{b_{0}^{2}}{2}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{b_{0}^{2}}{2}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}+\sum_{z_{1}=0}^{\infty}\mathbb{C}_{4}\bigg{[}A\mathbb{F}\Big{(}{-z_{1}-1},\frac{a_{0}^{2}+b_{0}^{2}}{2}\Big{)}$
$\displaystyle+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}z_{1}+1,\frac{a_{0}^{2}+b_{0}^{2}}{2}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}-z_{1}-1,\frac{a_{0}^{2}+b_{0}^{2}}{2}\Big{)}$
$\displaystyle+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-z_{1}-1,\mathbb{C}_{2}+\frac{a_{0}^{2}+b_{0}^{2}}{2}\Big{)}\bigg{)}\bigg{\\}}\bigg{]},$
(43)

where
$\mathbb{C}_{4}=\frac{a_{0}b_{0}p_{0}q_{0}}{\pi}\frac{(1)_{z_{1}}}{(1.5)_{z_{1}}z_{1}!}\Big{(}\Big{(}\frac{a_{0}^{2}}{2}\Big{)}^{z_{1}}+\Big{(}\frac{b_{0}^{2}}{2}\Big{)}^{z_{1}}\Big{)}$.

###  V-C XQAM Constellations

Generalized conditional SEP expression of XQAM constellation for the AWGN
channel is given as [50]

$\displaystyle\mathcal{P}_{s}^{X}(e|\gamma)$
$\displaystyle=N_{n}Q\Big{(}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}+\frac{8}{M_{i}N_{q}}\Big{[}\sum_{l_{1}=1}^{\frac{a_{2}}{2}-1}Q\Big{(}2l_{1}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}+Q\Big{(}a_{2}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}$
$\displaystyle-2\sum_{l_{1}=1}^{\frac{a_{2}}{2}-1}Q\Big{(}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}Q\Big{(}2l_{1}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}-Q\Big{(}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}Q\Big{(}a_{2}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)}\Big{]}-a_{3}Q^{2}\Big{(}\sqrt{\frac{2\gamma}{a_{1}}}\Big{)},$
(44)

where $N_{n}=4-\frac{2(M_{i}+N_{q})}{M_{i}N_{q}}$,
$a_{1}=\frac{2}{3}\Big{(}\frac{31M_{i}N_{q}}{32}-1\Big{)}$,
$a_{2}=\frac{M_{i}-N_{q}}{2}$, and
$a_{3}=4-\frac{4(M_{i}+N_{q})}{M_{i}N_{q}}+\frac{8}{M_{i}N_{q}}$.

The first order derivative of $\mathcal{P}_{s}^{X}(e|\gamma)$ is given as [41]

$\displaystyle\mathcal{P^{{}^{\prime}}}_{s}^{X}(e|\gamma)$
$\displaystyle=\frac{1}{2\sqrt{\pi
a_{1}}}\Big{(}-N_{n}+\frac{12}{M_{1}N_{q}}+a_{3}\Big{)}\gamma^{-\frac{1}{2}}e^{-\frac{\gamma}{a_{1}}}-\frac{a_{4}}{\sqrt{\pi
a_{1}}}\gamma^{-\frac{1}{2}}e^{-a_{2}^{2}\frac{\gamma}{a_{1}}}$
$\displaystyle-\frac{16}{M_{i}N_{q}}\sum_{l_{1}=1}^{\frac{a_{2}}{2}-1}\Big{[}\frac{l_{1}}{\pi
a_{1}}e^{-a_{5}\gamma}\Big{\\{}\,_{1}F_{1}\Big{(}1,\frac{3}{2},\frac{\gamma}{a_{1}}\Big{)}+\,_{1}F_{1}\Big{(}1,\frac{3}{2},\frac{4l_{1}^{2}\gamma}{a_{1}}\Big{)}\Big{\\}}\Big{]}-2\frac{a_{4}}{\pi
a_{1}}$ $\displaystyle\times
e^{-\frac{(1+a_{2}^{2})}{a_{1}}\gamma}\Big{\\{}\,_{1}F_{1}\Big{(}1,\frac{3}{2},\frac{\gamma}{a_{1}}\Big{)}+\,_{1}F_{1}\Big{(}1,\frac{3}{2},\frac{a_{2}\gamma}{a_{1}}\Big{)}\Big{\\}}-\frac{a_{3}}{\pi
a_{1}}e^{-\frac{2\gamma}{a_{1}}}\,_{1}F_{1}\Big{(}1,\frac{3}{2},\frac{\gamma}{a_{1}}\Big{)},$
(45)

where $a_{4}=\frac{M_{i}-N_{q}}{M_{i}N_{q}}$,
$a_{5}=\frac{4l_{1}^{2}+1}{a_{1}}$. Substituting $\mathcal{P}_{o}(\gamma)$ and
$\mathcal{P^{{}^{\prime}}}_{s}^{X}(e|\gamma)$ from (III-A) and (V-C),
respectively in (37), solving the necessary integrations and after some
mathematical computations, the final generalized SEP expression of XQAM
constellation is given as

$\displaystyle\mathcal{P}^{X}$
$\displaystyle=-a_{6}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{1}{a_{1}}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{1}{a_{1}}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{1}{a_{1}}\Big{)}$
$\displaystyle+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{1}{a_{1}}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}+\frac{a_{4}}{\sqrt{\pi
a_{1}}}\bigg{[}A\mathbb{F}\Big{(}\frac{-1}{2},\frac{a_{2}^{2}}{a_{1}}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}$
$\displaystyle\times\bigg{\\{}\mathbb{G}\Big{(}\frac{1}{2},\frac{a_{2}^{2}}{a_{1}}\Big{)}-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}\frac{-1}{2},\frac{a_{2}^{2}}{a_{1}}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-\frac{1}{2},\mathbb{C}_{2}+\frac{a_{2}^{2}}{a_{1}}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle+\frac{16}{M_{i}N_{q}}\sum_{l_{1}=1}^{\frac{a_{2}}{2}-1}\frac{l_{1}}{\pi
a_{1}}\sum_{z_{1}=0}^{\infty}\mathbb{C}_{5}\bigg{[}A\mathbb{F}\Big{(}{-z_{1}-1},a_{5}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}z_{1}+1,a_{5}\Big{)}$
$\displaystyle-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}-z_{1}-1,a_{5}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-z_{1}-1,\mathbb{C}_{2}+a_{5}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle+\frac{2a_{4}}{\pi
a_{1}}\sum_{z_{1}=0}^{\infty}\mathbb{C}_{6}\bigg{[}A\mathbb{F}\Big{(}{-z_{1}-1},\frac{(1+a_{2}^{2})}{a_{1}}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}z_{1}+1,\frac{(1+a_{2}^{2})}{a_{1}}\Big{)}$
$\displaystyle-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}-z_{1}-1,\frac{(1+a_{2}^{2})}{a_{1}}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-z_{1}-1,\mathbb{C}_{2}+\frac{(1+a_{2}^{2})}{a_{1}}\Big{)}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle+\frac{a_{3}}{\pi
a_{1}}\sum_{z_{1}=0}^{\infty}\mathbb{C}_{7}\bigg{[}A\mathbb{F}\Big{(}{-z_{1}-1},\frac{2}{a_{1}}\Big{)}+N\underset{\overset{k,l,j}{}}{\sum}\mathbb{C}_{0}\mathbb{C}_{2}^{-\mathbb{C}_{1}}\bigg{\\{}\mathbb{G}\Big{(}z_{1}+1,\frac{2}{a_{1}}\Big{)}$
$\displaystyle-A\Gamma(\mathbb{C}_{1})\bigg{(}\mathbb{F}\Big{(}-z_{1}-1,\frac{2}{a_{1}}\Big{)}+\sum_{z=0}^{\mathbb{C}_{1}-1}\frac{\mathbb{C}_{2}^{z}}{z!}\mathbb{F}\Big{(}-z-z_{1}-1,\mathbb{C}_{2}+\frac{2}{a_{1}}\Big{)}\bigg{)}\bigg{\\}}\bigg{]},$
(46)

where $a_{6}=\frac{1}{2\sqrt{\pi
a_{1}}}\Big{(}-N_{n}+\frac{12}{M_{1}N_{q}}+a_{3}\Big{)}$,
$\mathbb{C}_{5}=\frac{(1)_{z_{1}}}{(1.5)_{z_{1}}z_{1}!}\Big{(}\Big{(}\frac{1}{a_{1}}\Big{)}^{z_{1}}+\Big{(}\frac{4l_{1}^{2}}{a_{1}}\Big{)}^{z_{1}}\Big{)}$,
$\mathbb{C}_{6}=\frac{(1)_{z_{1}}}{(1.5)_{z_{1}}z_{1}!}\Big{(}\Big{(}\frac{1}{a_{1}}\Big{)}^{z_{1}}+\Big{(}\frac{a_{2}}{a_{1}}\Big{)}^{z_{1}}\Big{)}$,
and
$\mathbb{C}_{7}=\frac{(1)_{z_{1}}}{(1.5)_{z_{1}}z_{1}!}\Big{(}\frac{1}{a_{1}}\Big{)}^{z_{1}}$.

##  VI Theoretical and Simulation Results

In this Section, the numerical results are validated with the simulation
results obtained through the Monte-Carlo simulations. For the simulation
results, $10^{7}$ realizations have been performed. For the FSO link, product
of two random variables which follow generalized Gamma distributions is used
to generate the samples of atmospheric turbulence. The pointing error is
generated by considering Rayleigh distribution for radial displacement (7).
For the analysis, a low earth orbit (LEO) satellite at an altitude of 705 km
is considered and the UAV (HAP) is considered in the stratosphere at 17 km
altitude from the earth surface. For the FSO link, parameters are taken from
[16, 18] which are given in Table II. Parameters for the RF links are also
mentioned in Table II. As the HAP is considered for UAV, the satellite to HAP
FSO link experiences weak atmospheric turbulence, hence, the atmospheric
turbulence parameters $\alpha=2.902$ and $\beta=2.51$ are considered. Further,
the UAV is a moving relay node and hence, and two different values of pointing
error i.e., $\xi=1.1$ for severe case and $\xi=6.7$ for negligible pointing
error case are considered. Electrical to optical conversion coefficient
$\eta=1$ is considered. The ASER expressions of HQAM, RQAM, and XQAM ((V-A),
(V-B), and (V-C), respectively) consist of one infinite series “$z_{1}$" which
needs to be truncated to a finite value to get the numerical results from
these mathematical expressions. Hence, it is truncated to 80 values which give
us acceptable accuracy with reduced complexity.

TABLE II: Various parameters related to the FSO and RF links. FSO link | RF link
---|---
$\lambda_{f}$ | 1550 nm | $f_{RF}$ | 2 GHz
$D_{S}$ | 0.15 m | $\alpha_{t}$ | 2
$D_{r}$ | 0.25 m | H | 17 km
$A_{atm}$ | 0.5 dB | $R_{n}$ | 500 m
$A_{FS}$ | 268 dB | $B_{r}$ | 20 MHz
$L_{lenses}$ | 3 dB | $T$ | 300 k
$M_{s}$ | 3 dB |  |
$B_{o}$ | 30 GHz |  |

To show the impact of number of transmit antennas and the fading severity,
different values of $N_{t}$ and $m$ are considered for the RF link. Further,
various values of $N$ and $\rho$ are considered to highlight the impact of
user selection and the channel outdatedness on the system performance.
Furthermore, $\delta_{th}=\gamma_{th}=5$ dB is considered for the analysis.

(a) Heterodyne detection with negligible pointing error (b) Heterodyne
detection with severe pointing error

Figure 2: Comparison of theoretical and simulation results of outage
probability versus transmit power.

In Fig. 2, theoretical and simulation results of outage probability are
compared while heterodyne detection with negligible pointing error ($\xi=6.7$)
and severe pointing error ($\xi=1.1$) are considered, respectively. Further,
$N=2$ is fixed and the impact of $N_{t}$, $m$, and $\rho$ are observed on the
performance. In Fig. 2(a) with $\xi=6.7$, a clear improvement in the outage
performance is observed by increasing $\rho$ from 0.2 to 0.8 for all the
investigated cases as $\rho=1$ corresponds to the perfect CSI case. To achieve
an outage probability of $10^{-4}$, with the increase in $\rho$, nearly 3.7 dB
gain is obtained for all the investigated cases except the case when
$m=2,N_{t}=2,N=2$, where a gain of $2.6$ dB is obtained. The gain is reduced
due to the fact that with $m=2,N_{t}=2$ the system reaches to its maximum
performance because the performance is limited to the minimum of both the
link’s SNR. Further, it is observed that the increase in $N_{t}$ provides more
gain as compared to the increase in $m$. Considering $\rho=0.8,m=1,N_{t}=1$ as
a reference case, approximately 3 dB gain is observed with the increase in
$N_{t}$ from 1 to 2 as compared to the increase in $m$ from 1 to 2 to obtain
$10^{-4}$ outage probability. Monte-Carlo simulations are performed for all
the investigated cases which validate the correctness of all the theoretical
results. In Fig. 2(b) with $\xi=1.1$, a clear improvement in the outage
performance is observed with the increase in $\rho$ from 0.2 to 0.8 for all
the cases except for $m=2,N_{t}=2,N=2$ throughout the transmit power range and
similar trends can be seen as seen in Fig. 2(a). For $m=2,N_{t}=2,N=2$, with
the increase in $\rho$, the improvement in outage performance is observed only
till mid transmit power range. At high transmit powers, the performance is
limited as it is reached to the maximum performance and no significant
performance improvement can be observed. This is due to the fact that the FSO
link’s performance is limited due to the severe pointing error which limits
the overall system performance. Further, the simulations validate the derived
theoretical results for all the cases.

Figure 3: Comparison of theoretical results of outage and asymptotic outage
probability for both the IM/DD and heterodyne detection techniques.

In Fig. 3, theoretical results of outage probability are compared for both the
IM/DD ($i=2$) and heterodyne detection ($i=1$) techniques. For the analysis,
$\xi=6.7,\rho=0.8,N=2$ are fixed and the impact of $m$ and $N_{t}$ are
observed for both the detection techniques. It is observed, that the
heterodyne detection outperforms the IM/DD technique for all the investigated
cases because the heterodyne detection handles the turbulence effects in more
efficient manner despite its implementation complexity [25]. Considering the
IM/DD technique and $m=1,N_{t}=1$ as a reference case, considerable
improvement in outage performance is observed only till the mid transmit power
range with the increase in $m$ or $N_{t}$ or both. No considerable improvement
at high transmit powers is noticed because the performance is dominated with
the FSO link. On the other hand, for heterodyne technique with $m=1,N_{t}=1$
as a reference case, for an outage of $10^{-2}$ order, significant performance
gain is observed with the increase in $m$ (7.5 dB) or $N_{t}$ (10.5 dB) or
both (13.5 dB) which is clear from Fig. 3. Further, to find out the diversity
order of the communication system, asymptotic outage probability results are
also obtained by performing the high SNR approximation. From Fig. 3, it can be
seen that the asymptotic results match the outage probability results at high
transmit powers which justifies the high SNR analysis. The diversity order of
the considered system is
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i},NmN_{t})$ for the
perfect CSI case which depends upon the atmospheric turbulence, pointing
error, FSO detection type, number of users, fading parameter, and number of
transmit antennas. However, with outdated CSI, the impact of user selection is
vanished and the diversity order reduced to
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i},mN_{t})$.

Figure 4: Comparison of theoretical and simulation results of ergodic capacity
versus transmit power.

In Fig. 4, theoretical and simulation results of ergodic capacity are compared
while heterodyne detection with negligible pointing error ($\xi=6.7$) and
severe pointing error ($\xi=1.1$) are considered, respectively. Further, $N=2$
is fixed and the impact of $N_{t}$, $m$, and $\rho$ are observed on the
performance. In Fig. 4 (left), a clear improvement in the ergodic capacity is
observed with the increase in $\rho$ from 0.2 to 0.8 for all the investigated
cases. Further, it is observed that the increase in $N_{t}$ improves the
performance more as compared to the increase in $m$. Let us consider the case
$\rho=0.2,m=1,N_{t}=1,N=2$ as benchmark, and fixing the transmit power to 20
dBm, it is observed that with the increase in $m$ from 1 to 2, a gain of 0.18
bits/s/Hz is obtained in the ergodic capacity. On the other hand, with the
increase in $N_{t}$ from 1 to 2, a gain of 0.67 bits/s/Hz is obtained.
Further, with the increase in $\rho$ from 0.2 to 0.8, a gain of 0.26 bits/s/Hz
is obtained. Similar observations can be seen for the severe pointing error
case as shown in Fig. 4 (right), however, a little decrement in performance is
observed for all the investigated cases. Magnified portions of both the
figures are also highlighted to show the change in performance with the above
considered parameters ($\rho,m,N_{t}$). Further, all the theoretical results
present in Fig. 4 are validated through the simulation results.

Figure 5: Comparison of theoretical results of ergodic capacity for both the
IM/DD and heterodyne detection techniques.

In Fig. 5, theoretical results of ergodic capacity are compared for both the
IM/DD and heterodyne detection techniques. For the analysis,
$\xi=6.7,\rho=0.8,N=2$ are fixed and the impact of $m$ and $N_{t}$ are
observed for both the detection techniques. From Fig. 5, a significant gain in
ergodic capacity is observed for the heterodyne detection as compared to the
IM/DD technique throughout the transmit power range. Fixing transmit power at
15 dBm, nearly 0.55 bits/s/Hz gain in capacity is observed for the heterodyne
detection as compared to the IM/DD for all the considered cases. Considering
$m=1,N_{t}=1$ as the reference case, considerable improvement in ergodic
capacity is obtained with the increase in $m$ or $N_{t}$ or both for both the
detection techniques. However, with the increase in $N_{t}$, ergodic capacity
improve more as compared to the increase in $m$. Further, negligible
improvement in the ergodic capacity is observed while moving from
$m=1,N_{t}=2$ to $m=2,N_{t}=2$ for both the detection techniques. This is due
to the fact that the overall system capacity reached to its maximum value as
the performance is limited by the weaker link (FSO link in this case).

Figure 6: Effective capacity versus delay constraint $\Theta$ for both the
IM/DD and heterodyne detection techniques.

In Fig. 6, a comparison of effective capacity against the delay constraint
$\Theta$ is shown for both the IM/DD and heterodyne techniques. For the
analysis, $\xi=6.7$, $\rho=0.8$, and $N=2$ are fixed and the impact of $m$,
$N_{t}$, and detection techniques are highlighted. Further, transmit power is
fixed to 10 dBm for analysis. From Fig. 6, it is observed that effective
capacity improves with the increase in $m$ or $N_{t}$ or both. Also, effective
capacity is better for $m=1,N_{t}=2$ case as compared to the $m=2,N_{t}=1$
case for both the detection techniques. For all the combinations of $m$ and
$N_{t}$, effective capacity for the heterodyne detection is better than the
IM/DD. Further, it is observed that the effective capacity reduces with the
increase in delay constraint $\Theta$ and becomes almost 0 for very high
values of $\Theta$ for both the detection techniques.

Figure 7: Theoretical and simulation results of 16-HQAM for heterodyne
detection. Figure 8: Theoretical and simulation results of $4\times 2$-QAM for
heterodyne detection.

In Fig. 7, theoretical and simulation results of 16-HQAM are compared for
various cases. For the analysis, heterodyne detection is considered at the FSO
receiver and $\rho=0.8$ is fixed for the analysis. Further, the impact of
$\xi$, $m$, and $N_{t}$ are shown on the ASER performance for $N=2$ users. It
can be seen that the ASER performance improves significantly by changing
$\xi=1.1$ (severe pointing error) to $\xi=6.7$ (negligible pointing error).
Considering $\xi=1.1$ and $m=1,N_{t}=1,N=2$ as the reference case, it is
observed that the performance improved significantly with the increase in $m$
or $N_{t}$ or both in the low and medium transmit power regimes. However, no
significant improvement in ASER performance is observed at high transmit
powers. This is due to the fact that there exist a severe pointing error in
FSO link which limits the performance of the FSO link, and hence, the overall
system performance is dominated by the FSO link. On the other hand,
considering $\xi=6.7$ and $m=1,N_{t}=1,N=2$ as the reference case, we observe
a significant improvement in ASER performance with the increase in $m$ or
$N_{t}$ or both, throughout the transmit power range. To achieve an ASER of
$10^{-2}$ order, we obtain approximately 5.4 dB, 8.4 dB, and 10.5 dB
respective gain when increasing $m$ from 1 to 2, or $N_{t}$ from 1 to 2, or
both. All the theoretical results shown in Fig. 7 are validated through the
simulations. The results shown in Fig. 7 consider $\rho=0.8$, however, with
$\rho=0.2$, similar outcomes will come with little decrement in the system
performance.

In Fig. 8, theoretical and simulation results of $4\times 2$-QAM are compared
for various cases. For the analysis, heterodyne detection is considered at the
FSO receiver and $\rho=0.8$ is fixed for the analysis. Further, the impact of
$\xi$, $m$, and $N_{t}$ are shown on the ASER performance for $N=2$ users. It
is observed that the ASER performance improves significantly with the change
in $\xi$ from 1.1 to 6.7. Considering $\xi=1.1$ and $m=1,N_{t}=1,N=2$ as the
reference case, significant performance improvement is observed with the
increase in $m$ or $N_{t}$ or both in the low and medium transmit power
regimes. At high transmit powers, no significant improvement in ASER
performance is observed due to the severe pointing error in FSO link which
limits the performance of the FSO link, and hence, the overall system
performance is dominated by the FSO link. On the other hand, considering
$\xi=6.7$ and $m=1,N_{t}=1,N=2$ as the reference case, a significant
improvement in ASER performance is observed throughout the transmit power
range with the increase in $m$ or $N_{t}$ or both. All the theoretical results
shown in Fig. 8 are validated through the simulations. The results shown in
Fig. 8 consider $\rho=0.8$, however, with $\rho=0.2$, similar outcomes will
come with little decrement in the system performance.

Figure 9: Comparison of the theoretical results of even power of 2 QAM
constellations. Figure 10: Comparison of the theoretical results of odd power
of 2 QAM constellations.

In Fig. 9, theoretical results of even power of 2 QAM constellations are
compared up to a constellation size 1024. For the the analysis $\xi=6.7$,
$\rho=0.8$, $m=1$, $N_{t}=2$, and $N=2$ are considered. It is observed that
the performance of the HQAM is better than the SQAM for all the constellation
size except for $M=4$. This is because both the constellations have same
average power but 4-SQAM has less number of nearest neighborhood than the
4-HQAM. For an ASER of $10^{-3}$, 4-SQAM provides nearly 0.14 dB gain over
4-HQAM. Further, with the increase in $M$, the HQAM performance improves
accordingly which is clear from Fig. 9. For an ASER of $10^{-3}$, 16-HQAM,
64-HQAM, and 256-HQAM, provide nearly 0.3 dB, 0.5 dB, 0.65 dB gain over
respective SQAM which will further increase with the increase in constellation
order.

In Fig. 10, theoretical results of odd power of 2 QAM constellations are
compared. For the the analysis $\xi=6.7$, $\rho=0.8$, $m=1$, $N_{t}=2$, and
$N=2$ are considered. It is observed that the XQAM provides significant
performance gain than the RQAM for all the constellation orders. This is
because the XQAM has low peak and average powers as compared to the RQAM due
to its more compact design than the RQAM. Further, HQAM provides considerable
SNR gain over the XQAM which increases with the increase in M. Hence, from the
above results, it is justified that the HQAM is the optimum constellation than
others.

##  VII Conclusions

In this work, a multiuser multi-antenna UAV assisted DF based downlink
terrestrial-satellite communication system was considered which operates over
the mixed FSO/RF channels. Gamma-Gamma distribution with pointing error
impairments was considered for the FSO channel modeling and both the IM/DD and
heterodyne detection techniques were studied. Nakagami-m distribution was
preferred for the modeling of RF links. Opportunistic user scheduling was
performed for the terrestrial users to attain the multiuser diversity and
channel outdatedness was considered during the user selection and signal
transmission phase. For the performance analysis, outage probability,
asymptotic outage probability, ergodic capacity, effective capacity, and ASER
results of HQAM, RQAM, and XQAM constellations were obtained. The impact of
pointing error impairment, FSO detector, channel outdatedness, number of
users, number of transmit antennas, fading severity of RF links, and delay
constraint were highlighted on all the performance metrics. Finally, all the
results were validated with Monte-Carlo simulations. From the diversity
analysis it was concluded that in case of outdated CSI, no impact of user
selection is observed on the system performance. Increase in number of
antennas provides improved performance than the increase in fading severity.
It was also observed that the heterodyne detection outperforms the IM/DD
technique because it handles the turbulence effects more efficiently despite
its implementation complexity. Further, it was observed that the effective
capacity reduces with the increase in delay constraint and becomes almost zero
for the high values of delay constraint. From the ASER results it was
concluded that the HQAM provides improved performance than the other QAM
schemes and the performance improves further with the increased constellation
orders. This justifies HQAM as the optimum constellation.

Meijer-G is the key function in the expression of outage probability because
the satellite to UAV link (FSO link) is Gamma-Gamma distributed. At the high
SNR, the Meijer-G function can be approximated in (III-A) in terms of the
basic elementary function as [43, (07.34.06.0001.01)]

$\displaystyle\text{G}^{a,b}_{c,d}\Big{[}x\Big{|}^{\tau_{3}}_{\tau_{4}}\Big{]}$
$\displaystyle\approx\sum_{p=1}^{a}x^{\tau_{4},p}\frac{\prod_{\underset{q\neq
p}{q=1}}^{a}\Gamma(\tau_{4,q}-\tau_{4,p})\prod_{\underset{q=1}{}}^{b}\Gamma(1-\tau_{3,q}+\tau_{4,p})}{\prod_{\underset{q=b+1}{}}^{c}\Gamma(\tau_{3,q}-\tau_{4,p})\prod_{\underset{a+1}{}}^{d}\Gamma(1-\tau_{4,q}+\tau_{4,p})},$
(47)

where $\tau_{3}=[1,\tau_{1}]$ and $\tau_{4}=[\tau_{2},0]$. From (47), it is
observed that the dominant term of meijer-G function is
$\min(\frac{\xi^{2}}{i},\frac{\alpha}{i},\frac{\beta}{i})$.

To obtain the approximate CDF expression of the terrestrial link for the
perfect CSI case ($\rho=1$), first the PDF and CDF of the SNR of the
terrestrial link (II) are approximated at high SNR by considering the high SNR
approximation of
$\Upsilon(m,x)\underset{\overset{x=0}{}}{\approx}\frac{x^{m}}{m}$. Thus, the
approximate PDF and CDF are respectively given as
$f_{\gamma_{RU_{n}}}(x)=\frac{1}{\Gamma(mN_{t})}\Big{(}\frac{m}{\bar{\gamma}_{U}}\Big{)}^{mN_{t}}x^{mN_{t}-1}$
and
$F_{\gamma_{RU_{n}}}(x)=\frac{1}{\Gamma(mN_{t})}\Big{(}\frac{m}{\bar{\gamma}_{U}}x\Big{)}^{mN_{t}}$.
Substituting them in (12), the approximate CDF expression of the terrestrial
link for the perfect CSI case is obtained as (23).

To obtain the approximate CDF expression of the terrestrial link for the
outdated CSI case ($\rho<1$), first (15) is approximated by considering the
high SNR approximation of $I_{v}(x)=\frac{1}{\Gamma(v+1)}(\frac{x}{2})^{v}$,
and then is substituted in (14) along with $f_{\gamma_{RU}}(x)$ (13). Thus,
$f_{\hat{\gamma}_{RU}}(x)$ is calculated as

$\displaystyle f_{\hat{\gamma}_{RU}}(x)$ $\displaystyle\approx
N\sum_{k=0}^{N-1}\binom{N-1}{k}\frac{(-1)^{k}}{(\Gamma(mN_{t}))^{2}}\sum_{l=0}^{k(mN_{t}-1)}\varphi_{l}^{k}\Big{(}\frac{m}{\bar{\gamma}_{U}(1-\rho)}\Big{)}^{mN_{t}}\Big{(}(k+1)+\frac{\rho}{1-\rho}\Big{)}^{-(mN_{t}+l)}$
$\displaystyle\times\Gamma(mN_{t}+l)~{}x^{mN_{t}-1}e^{-\frac{m}{\bar{\gamma}_{U}(1-\rho)}x}.$
(48)

Performing
$F_{\hat{\gamma}_{RU}}(\gamma_{th})=\int_{0}^{\gamma_{th}}f_{\hat{\gamma}_{RU}}(x)dx$,
and using the high SNR approximation of
$\Upsilon(m,x)\underset{\overset{x=0}{}}{\approx}\frac{x^{m}}{m}$, the
approximate CDF expression of the terrestrial link for the outdated CSI case
is obtained as (24).

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|  Praveen K. Singya (M’20) received his B.E. in Electronics and Communication
Engineering from Jabalpur Engineering College, Jabalpur, India in 2012. He
received his M.Tech. in Communication System Engineering from VNIT, Nagpur,
India in 2014. He received his Ph.D. degree from the Indian Institute of
Technology Indore, India, in 2019. He is currently a postdoctoral fellow at
King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah
Province, Saudi Arabia. His research interest includes design and performance
analysis of various wireless networks over different fading channels,
satellite communication, HAP communication, and free-space optics.
---|---
|  Mohamed-Slim Alouini (S’94-M’98-SM’03-F’09) was born in Tunis, Tunisia. He
received the Ph.D. degree in Electrical Engineering from the California
Institute of Technology (Caltech), Pasadena, CA, USA, in 1998. He served as a
faculty member at the University of Minnesota, Minneapolis, MN, USA, then in
the Texas A&M University at Qatar, Education City, Doha, Qatar before joining
King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah
Province, Saudi Arabia as a Professor of Electrical Engineering in 2009. His
current research interests include modeling, design, and performance analysis
of wireless communication systems.
---|---