# Performance Analysis of UAV-based Mixed RF-UWOC Transmission Systems

Sai Li, Liang Yang, and Daniel Benevides da Costa S. Li and L. Yang are with
the College of Computer Science and Electronic Engineering, Hunan University,
Changsha 410082, China<EMAIL_ADDRESS>liangy@hnu.edu.cn).D. B.
da Costa is with the Department of Computer Engineering, Federal University of
Ceará, Sobral 62010-560, Brazil (email: danielbcosta@ieee.org).

###### Abstract

In this paper, we investigate the performance of a mixed radio-frequency-
underwater wireless optical communication (RF-UWOC) system where an unmanned
aerial vehicle (UAV), as a low-altitude mobile aerial base station, transmits
information to an autonomous underwater vehicle (AUV) through a fixed-gain
amplify-and-forward (AF) or decode-and-forward (DF) relay. Our analysis
accounts for the main factors that affect the system performance, such as the
UAV height, air bubbles, temperature gradient, water salinity variations, and
detection techniques. Employing fixed-gain AF relaying and DF relaying, we
derive closed-form expressions for some key performance metrics, e.g., outage
probability (OP), average bit error rate (ABER), and average channel capacity
(ACC). In addition, in order to get further insights, asymptotic analyses for
the OP and ABER are also carried out. Furthermore, assuming DF relaying, we
derive analytical expressions for the optimal UAV altitude that minimizes the
OP. Simulation results show that the UAV altitude influences the system
performance and there is an optimal altitude which ensures a minimum OP.
Moreover, based on the asymptotic results, it is demonstrated that the
diversity order of fixed-gain AF relaying and DF relaying are respectively
determined by the RF link and by the detection techniques of the UWOC link.

###### Index Terms:

Mixed dual-hop transmission schemes, performance analysis, radio-frequency
(RF) links, underwater wireless optical communications, unmanned aerial
vehicles (UAVs).

##  I Introduction

Unmanned aerial vehicles (UAVs)-based wireless communication systems have
received extensive attention, especially as aerial base stations (BSs) and/or
as relays to assist existing communication systems [1, 2, 3]. The actual
scenarios of UAVs application include real-time monitoring, wireless coverage,
and safety supervision. Compared with the ground-based Internet of Things
(IoT) communication platforms, UAV wireless communication systems have the
advantages of easy deployment, low cost, and better communication channel
quality brought by short-range line-of-sight (LoS) connections. In addition,
unlike fixed relays, UAV-assisted relays can flexibly adjust their locations
according to the changes of urban environment to improve communication
reliability and achieve long-distance data transmission [4, 5, 6, 7, 8]. For
instance, an UAV-based mixed radio-frequency/free-space optical (RF/FSO)
communication network was considered in [4] with the purpose of facilitating
the uplink communication among multiple users and a ground station. In [5], a
multi-hop low-altitude UAV based RF/FSO communication system was proposed and
analytical expressions for the outage probability (OP) and optimal altitude
were derived. The secrecy performance of UAV-assisted relaying systems was
investigated in [6, 7]. In [8], the energy harvesting technology was applied
to an UAV-based communication system and the system OP was evaluated. The
works in [9, 10] investigated the coverage quality in areas without
infrastructure coverage or when the infrastructure is damaged, such as low
altitude marine monitoring and emergency communications under extreme
conditions, by deploying UAVs as aerial BSs. Other works focused on addressing
the UAV deployment optimization and the air-to-ground (A2G) channel modeling
[10, 11, 12]. For instance, in [10] closed-form expressions for the optimal
UAV altitude that minimizes the OP of an arbitrary A2G link have been
obtained, while in [11] the authors derived the optimal altitude that allows
the UAV to reach its maximum coverage radius.

On another front, underwater wireless optical communication (UWOC) systems
have widely been investigated in the literature due to their promising gains,
such as wider bandwidth, higher data transmission rate, and more secure
transmission [13, 14, 15]. However, in UWOC links, in addition to absorption
and scattering, underwater optical turbulence (UOT) caused by temperature
fluctuations, salinity variations, and air bubbles arise as nature impediments
that may degrade the system performance. To evaluate more properly their
impacts on the system performance, the performance of UWOC links in the
presence of different air bubble populations and/or underwater channels with
temperature or salinity gradients was evaluated in [14]. Specifically, in
[15], by taking into account the effect of the air bubbles, temperature
gradients and salinity on the water channel, the authors presented a unified
UWOC turbulence model relying on the mixture Exponential-Generalized Gamma
(EGG) distribution. Recently, similar to previous RF/FSO research works,
cooperative communication technology has been applied to UWOC systems in order
to improve the overall reliability. In [16], employing EGG distribution, the
performance of a dual-hop UWOC system was analyzed and it was shown that the
proposed system is more effective and robust to underwater turbulence than a
single UWOC link. The uplink scenario was considered in [17], in which the
data collected by sensors located underwater was first transmitted through an
UWOC link to a static relay deployed on the sea surface, and then the data was
transmitted to land users through a high-speed FSO link. Assuming a downlink
transmission and a terrestrial base station, the authors in [18] derived
closed-from expressions for the OP and average bit error rate (ABER) of mixed
dual-hop RF-UWOC relaying systems. Very recently [19], the performance of RF-
UWOC systems was examined assuming that the RF link and the UWOC link
experience Generalized-$K$ distribution and mixture EGG distribution,
respectively.

Despite the importance of all previous works, the studies related to RF-UWOC
systems have mainly focused on terrestrial and underwater cooperative
networks. Due to this fact, efficient coverage service of static BSs largely
depends on their locations with respect to the sea surface. On the other hand,
deploying UAVs as mobile aerial BSs can effectively alleviate and surpass
coverage issues since UAVs are able to flexibly move to the required coverage
area. Motivated by this and relying on air-to-sea cooperative communications
[20], in this paper we investigate the performance of mixed RF-UWOC systems
where an UAV is employed as a mobile BS and transmits information to an
autonomous underwater vehicle (AUV) through a fixed-gain amplify-and-forward
(AF) or decode-and-forward (DF) relay. We assume that the RF link follows
Rician fading while the UWOC link undergoes mixture EGG fading distribution.
To the best of the authors’ knowledge, there is no work in the literature
performing this kind of investigation so that our paper can serve as a
benchmark for future studies. The contributions of this paper is summarized as
follows:

  * $\bullet$

Using fixed-gain AF and DF relaying, Closed-form expressions for the
cumulative distribution function (CDF) and probability density function (PDF)
of the end-to-end (E2E) signal-to-noise ratio (SNR) were derived. Based on
them, accurate analytical expressions for the OP, ABER, and average channel
capacity (ACC) are presented. These expressions are written in terms of
Extended Generalized Bivariate Fox’s H-Function (EGBFHF), which can be easily
implemented relying on results published elsewhere in the literature.

  * $\bullet$

The effect of the UAV altitude and coverage on the overall system performance
are examined. To this end, we derive analytical expressions for the optimal
elevation angle and for the optimal UAV altitude that minimize the OP.

  * $\bullet$

In the UWOC link, heterodyne detection (HD) and intensity modulation/direct
detection (IM/DD) techniques are employed. Considering the case in which the
absorption and scattering effect are not significant, as well as the UOT
dominates the fading characteristics of the channel, we analyze the effects of
air bubbles, temperature and salinity gradient on the performance of the
proposed system setup.

  * $\bullet$

Asymptotic expressions at high SNR regime for the OP and ABER are presented in
order to get further insights. Our results reveal that the diversity order of
fixed-gain AF and DF relaying are determined by the RF link and by the
detection techniques of the UWOC link, respectively.

The remainder of this paper can be organized as follows. The system and
channel models as well as the E2E SNRs of the fixed-gain AF and DF relaying
are presented in Section II. The CDF and PDF of the two relaying strategies
are derived In Section III, which will be useful for calculating the key
performance metrics. Section IV derives exact expressions for the OP, ABER,
and ACC along with asymptotic results. Section V provides the optimal altitude
analysis for the DF relaying system. In order to obtain insightful findings,
some illustrative numerical examples are conducted in Section VI. Section VII
concludes the paper.

##  II System and Channel Models

As shown in Fig. 1, we consider a dual-hop UAV-based RF-UWOC system where a
UAV serves as an aerial BS to communicate with an AUV through an over-sea
surface relay (R). At R, both fixed-gain AF and DF strategies are considered.
In the first slot, UAV broadcasts the signal to R via the RF link (UAV-R link)
and, in the second slot, the received RF signal is converted into an optical
signal and then forwarded to the AUV through the UWOC link (R-AUV link). We
assume that the UAV-R and R-AUV links undergo Rician fading and the mixture
EGG fading, respectively.

Figure 1: The dual-hop mixed UAV-based RF-UWOC system model.

###  II-A UAV-R Link

The received signal at R is expressed as

$\displaystyle y_{UR}=\frac{1}{\sqrt{L_{UR}}}h_{UR}x+n_{1},$ (1)

where $h_{UR}$ denotes the small-scale fading channel coefficient between the
UAV and R, and $n_{1}\sim\mathcal{C}\mathcal{N}(0,N_{01})$ stands for the
additive white Gaussian noise (AWGN) term with zero mean and variance
$N_{01}$. Moreover, $L_{UR}=Ad_{UR}^{\alpha}$ represents the path loss, $A$ is
a constant associated with the signal frequency and transmission environment,
$d_{UR}=\sqrt{h_{1}^{2}+r_{1}^{2}}$ is the distance between UAV and R, where
$h_{1}$ denotes the altitude of the UAV, $r_{1}$ stands for the horizontal
distance between the UAV and R, $\alpha$ holds for the path loss exponent,
which satisfies $\alpha(\theta)=a_{1}P_{LoS}(\theta)+b_{1}$ [10], with
$\theta$ denoting the angle between the UAV and R, $P_{LoS}$ is the LoS
probability, and $a_{1}$ and $b_{1}$ depend on the environment and the
transmission frequency.

Therefore, the instantaneous SNR of the UAV-R link is given by
$\gamma_{1}=\frac{|h_{UR}|^{2}}{L_{UR}}\overline{\gamma}_{1}$, where
$\overline{\gamma}_{1}$ is the average SNR of the UAV-R link. Since the UAV-R
link undergoes Rician fading, the PDF of the instantaneous SNR, $\gamma_{1}$,
is given by [8, 21]

$\displaystyle f_{\gamma_{1}}(\gamma_{1})=\frac{\vartheta
e^{-K}}{\overline{\gamma}_{1}}\exp\left(-\frac{\vartheta\gamma_{1}}{\overline{\gamma}_{1}}\right)I_{0}\left(2\sqrt{\frac{K\vartheta\gamma_{1}}{\overline{\gamma}_{1}}}\right),$
(2)

where $\vartheta=(1+K)L_{UR}$, $I_{0}(\cdot)$ is defined as the zero-order
modified Bessel function of the first kind [22, Eq. (8.431)], $K$ means the
Rician fading factor, which is modeled as a function of $\theta$. From [23]
and [10, Eqs. (34) and (35)], it follows that
$K(\theta)=a_{2}e^{b_{2}\theta}$, where $a_{2}=K(0)$ and
$b_{2}=\frac{2}{\pi}\ln\left(\frac{K(\frac{\pi}{2})}{K(0)}\right)$. In
addition, the CDF of $\gamma_{1}$ can be written as

$\displaystyle
F_{\gamma_{1}}(\gamma_{1})=1-Q_{1}\left(\sqrt{2K},\sqrt{2\gamma_{1}\vartheta/\overline{\gamma}_{1}}\right),$
(3)

where $Q_{1}(\cdot,\cdot)$ is the Marcum Q-function of the first order [24,
Eq. (86)].

###  II-B R-AUV Link

####  II-B1 Amplify-and-Forward

For the AF relaying, the received signal at AUV is expressed as

$\displaystyle y_{RA}=\eta IGy_{UR}+n_{2},$ (4)

where $I$ stands for the normalized irradiance, $\eta$ denotes the optical-to-
electrical conversion coefficient, $G$ represents the fixed gain, and
$n_{2}\sim\mathcal{C}\mathcal{N}(0,N_{02})$ denotes the AWGN term with zero
mean and variance $N_{02}$.

TABLE I: Parameters of the EGG Distribution for Different Bubble Levels BL (L/min) and Temperature Gradient (${}^{\circ}\mathrm{C}.cm^{-1}$) [15] BL |  Temperature Gradient |  $\sigma_{I}^{2}$ |  $\omega$ |  $\lambda$ |  a |  b |  c
---|---|---|---|---|---|---|---
2.4 |  0.05 |  0.1484 |  0.2130 |  0.3291 |  1.4299 |  1.1817 |  17.1984
2.4 |  0.20 |  0.2178 |  0.1665 |  0.1207 |  0.1559 |  1.5216 |  22.8754
4.7 |  0.05 |  0.4201 |  0.4580 |  0.3449 |  1.0421 |  1.5768 |  35.9424
TABLE II: Parameters of the EGG Distribution for Different Bubble Levels BL (L/min) for Fresh Water and Salty Water [15] Salinity |  BL |  $\sigma_{I}^{2}$ |  $\omega$ |  $\lambda$ |  a |  b |  c
---|---|---|---|---|---|---|---
Salty Water |  2.4 |  0.1006 |  0.1770 |  0.4687 |  0.7736 |  1.1372 |  49.1773
Salty Water |  4.7 |  0.1308 |  0.2064 |  0.3953 |  0.5307 |  1.2154 |  35.7368
Salty Water |  7.1 |  0.3111 |  0.4344 |  0.4747 |  0.3935 |  1.4506 |  77.0245
Salty Water |  16.5 |  1.1273 |  0.4951 |  0.1368 |  0.0161 |  3.2033 |  82.1030
Fresh Water |  2.4 |  0.1088 |  0.1953 |  0.5273 |  3.7291 |  1.0721 |  30.3214
Fresh Water |  4.7 |  0.1233 |  0.2190 |  0.4603 |  1.2526 |  1.1501 |  41.3258
Fresh Water |  7.1 |  0.3150 |  0.3489 |  0.4771 |  0.4319 |  1.4531 |  74.3650
Fresh Water |  16.5 |  1.0409 |  0.5117 |  0.1602 |  0.0075 |  2.9963 |  216.8356

####  II-B2 Decode-and-Forward

As for the DF relaying system, the received signal at AUV can be written as

$\displaystyle y_{RA}=\eta I\hat{x}+n_{2},$ (5)

where $\hat{x}$ denotes the decoded signal at R.

The instantaneous SNR of the R-AUV link, $\gamma_{2}$, is formulated as
$\gamma_{2}=(I^{r}/\mathbb{E}[I]^{r})\mu_{r}$, where $r$ stands for the
parameter that depends on the detection technique used, i.e., $r=1$ when the
HD technique is employed and $r=2$ when the IM/DD technique is adopted. The
parameter $I$ is modelled by the mixture EGG distribution, whose the PDF can
be given by [15, Eq. (1)]

$\displaystyle
f_{I}(I){=}\frac{w}{\lambda}\exp\left({{-}\frac{I}{\lambda}}\right){+}(1{-}w)\frac{cI^{ac{-}1}}{b^{ac}}\frac{\exp\left({{-}\left({\frac{I}{b}}\right)^{c}}\right)}{\Gamma(a)},$
(6)

where $\omega$, $\lambda$, $a$, $b$, and $c$ represent the fading parameters
related to the mixture EGG distribution. From (6), the $n$-th moment of $I$
can be derived as
$\mathbb{E}[I^{n}]=\omega\lambda^{n}n!+(1-\omega)b^{n}\Gamma(a+\frac{n}{c})/\Gamma(a)$
[15, Eq. (4)]. Based on it, the scintillation index $\sigma_{I}^{2}$
represents the turbulence intensity and can be formulated as
$\sigma_{I}^{2}=\frac{\mathbb{E}[I^{2}]-\mathbb{E}[I]^{2}}{\mathbb{E}[I]^{2}}$.
Moreover, the average electrical SNR $\mu_{r}$ is defined as
$\mu_{r}=\eta^{r}\mathbb{E}[I]^{r}/N_{02}$ and the relation between $\mu_{r}$
and the average SNR $\overline{\gamma}_{2}$ of the R-AUV link is given by
$\overline{\gamma}_{2}=(\mu_{r}\mathbb{E}[I^{r}])/\mathbb{E}[I]^{r}$. For the
HD technique, $\mu_{1}=\overline{\gamma}_{2}$, while for the IM/DD technique,
$\mu_{2}=\frac{\overline{\gamma}_{2}}{2\omega\lambda^{2}+b^{2}(1-\omega)\Gamma(a+2/c)/\Gamma(a)}$
[15, Eq. (19)]. These parameters vary with the water temperature, water
salinity, as well as bubble level. The parameters of the EGG distribution are
given in Table I and Table II.

From [19], the PDF of $\gamma_{2}$ can be formulated in terms of Fox’s
H-function as

$\displaystyle f_{\gamma_{2}}(\gamma_{2})=\frac{\omega}{\gamma_{2}}$
$\displaystyle\,{\mathrm{H}}_{0,1}^{1,0}\left[{{\frac{\gamma_{2}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{-}\\\
{(1,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)}{\gamma_{2}\Gamma(a)}\,{\mathrm{H}}_{0,1}^{1,0}\left[{{\frac{\gamma_{2}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{-}\\\
{(a,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!,$ (7)

and the CDF expression can be formulated as

$\displaystyle F_{\gamma_{2}}(\gamma_{2})=\omega r$
$\displaystyle\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{2}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{\Gamma(a)c}\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{2}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!,$ (8)

where $\mathrm{H}_{\cdot,\cdot}^{\cdot,\cdot}[\cdot]$ is the Fox’s H-function
[25, Eq. (1.2)]. Note that a valid MATHEMATICA implementation code for
evaluating the Fox’s H-function has been provided in [26].

From (4), the E2E SNR when AF relaying scheme is considered can be expressed
as [27, Eq. (6)]

$\displaystyle\gamma^{AF}=\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C},$ (9)

where $C$ is a constant that depends on the relay gain, i.e.,
$C=1/(G^{2}N_{01})$ [27, Eq. (5)].

By its turn, for the DF relaying scheme, the E2E SNR can be expressed as

$\displaystyle\gamma^{DF}=\min(\gamma_{1},\gamma_{2}).$ (10)

##  III Closed-Form End-to-End Statistics

###  III-A Fixed-Gain AF Relaying Protocol

####  III-A1 Cumulative Distribution Function

For fixed-gain AF relaying, the CDF of the E2E SNR can be expressed as

$\displaystyle F_{\gamma^{AF}}(\gamma)$
$\displaystyle=\int_{0}^{\infty}P\left[\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C}<\gamma\Big{|}\gamma_{1}\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1},$
(11)

where $F_{\gamma_{1}}(\gamma)$ is the CDF of $\gamma_{1}$. From [28, Eq.
(A.3)] and taking the following change of variables $x=\gamma_{1}-\gamma$,
(11) can be rewritten as

$\displaystyle
F_{\gamma^{AF}}(\gamma)=F_{\gamma_{1}}(\gamma)+\underbrace{\int_{0}^{\infty}F_{\gamma_{2}}\left(\frac{C\gamma}{x}\right)f_{\gamma_{1}}(x+\gamma)dx}_{\mathcal{I}_{1}}.$
(12)

By replacing (2) into (12) to calculate $\mathcal{I}_{1}$, a closed-form
expression for the CDF of the fixed-gain AF relaying is hard to attain. In
order to arrive at a closed-form expression for $\mathcal{I}_{1}$, we rely on
[29, Eqs. (13) and (19)] so that (12) can be simplified as

$\displaystyle f_{\gamma_{1}}(\gamma_{1})\approx\sum_{m=0}^{n}$ $\displaystyle
a_{1}\gamma_{1}^{m}\exp\left(-\frac{\vartheta\gamma_{1}}{\overline{\gamma}_{1}}\right),$
(13)

where $a_{1}=\varpi\left(\frac{\vartheta}{\overline{\gamma}_{1}}\right)^{m+1}$
with $\varpi=\frac{(n-m-1)!n^{1-2m}K^{m}e^{-K}}{m!(n-m)!\Gamma(m{+}1)}$. Then,
applying [22, Eq. (3.351.1)], the CDF of $\gamma_{1}$ can be approximately
derived as

$\displaystyle F_{\gamma_{1}}(\gamma_{1})\approx\sum_{m=0}^{n}$
$\displaystyle\varpi\Upsilon\left(m+1,\frac{\vartheta\gamma_{1}}{\overline{\gamma}_{1}}\right),$
(14)

where $\Upsilon(\cdot,\cdot)$ is the lower incomplete Gamma function [22, Eq.
(8.350.1)]. Finally, by substituting (8) and (13) into $\mathcal{I}_{1}$, and
after some algebraic operations, we have

$\displaystyle\mathcal{I}_{1}=\sum_{m=0}^{n}\epsilon_{1}\gamma^{m+1}{\rm{H}_{1,0:3,1:1,1}^{0,1:1,2:0,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m+2;-1,1}\right)\\\
-\\\ (0,r)(0,1)(1,r)\\\ (0,r)\\\ (1,1)\\\
(m+1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\\!\\!}\right]$
$\displaystyle+\sum_{m=0}^{n}\epsilon_{2}\gamma^{m+1}{\rm{H}_{1,0:3,1:1,1}^{0,1:1,2:0,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m+2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(0,1)(1,\frac{r}{c})\\\ (0,\frac{r}{c})\\\ (1,1)\\\
(m+1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\\!\\!}\right],$
(15)

where $\epsilon_{1}=a_{1}\omega r$,
$\epsilon_{2}=\frac{a_{1}(1-\omega)r}{\Gamma(a)c}$ and
$\rm{H}_{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}^{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}[\cdot,\cdot]$
denotes the EGBFHF [25, Eq. (2.56)]. It is worth noting that the EGBFHF can be
efficiently calculated by using the MATLAB implementation code available in
[30].

Proof: See Appendix A.

Thus, with the help of (3) and (15), the CDF of the fixed-gain AF relaying
system is finally formulated as

$\displaystyle
F_{\gamma^{AF}}(\gamma)=1-Q_{1}\left(\sqrt{2K},\sqrt{2\gamma\vartheta/\overline{\gamma}_{1}}\right)+\mathcal{I}_{1}.$
(16)

####  III-A2 Probability Density Function

By making the derivative of (11), the PDF of the E2E SNR is formulated as

$\displaystyle
f_{\gamma^{AF}}(\gamma)=\frac{d}{d\gamma}\int_{0}^{\infty}P\left[\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C}<\gamma\Big{|}\gamma_{1}\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}.$
(17)

Using [19, Eq. (25)] and after some algebraic operations, the final expression
for the PDF of $\gamma^{AF}$ can be formulated as

$\displaystyle
f_{\gamma^{AF}}(\gamma)=\sum_{m=0}^{n}\epsilon_{3}\gamma^{m}{\rm{H}_{1,0:2,0:1,1}^{0,1:0,2:0,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (0,r)(1,1)\\\ -\\\ (1,1)\\\
(m{+}2,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\\!\\!}\right]$
$\displaystyle+\sum_{m=0}^{n}\epsilon_{4}\gamma^{m}{\rm{H}_{1,0:2,0:1,1}^{0,1:0,2:0,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(1,1)\\\ -\\\ (1,1)\\\
(m{+}2,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\\!\\!}\right],$
(18)

where $\epsilon_{3}=a_{1}\omega$,
$\epsilon_{4}=\frac{a_{1}(1-\omega)}{\Gamma(a)}$. Similar derivations can also
be found in [31].

###  III-B DF Relaying Protocol

####  III-B1 Cumulative Distribution Function

As for the DF relaying protocol, the CDF of $\gamma^{DF}$ can be written as

$\displaystyle F_{\gamma^{DF}}(\gamma)$
$\displaystyle=F_{\gamma_{1}}(\gamma)+F_{\gamma_{2}}(\gamma)-F_{\gamma_{1}}(\gamma)F_{\gamma_{2}}(\gamma),$
(19)

where $F_{\gamma_{1}}(\gamma)$ and $F_{\gamma_{2}}(\gamma)$ denote the CDFs of
$\gamma_{1}$ and $\gamma_{2}$, respectively. By substituting (3) and (8) in
(19), the CDF of $\gamma^{DF}$ can be formulated in closed-form.

####  III-B2 Probability Density Function

By taking the derivative of (19) with respect to $\gamma$, the following PDF
is obtained

$\displaystyle f_{\gamma^{DF}}(\gamma)=$ $\displaystyle
f_{\gamma_{1}}(\gamma)+f_{\gamma_{2}}(\gamma)-f_{\gamma_{1}}(\gamma)F_{\gamma_{2}}(\gamma)$
$\displaystyle-f_{\gamma_{2}}(\gamma)F_{\gamma_{1}}(\gamma),$ (20)

which can be obtained from the PDFs and CDFs of $\gamma_{1}$ and $\gamma_{2}$.

##  IV Performance Analysis

In this section, we derive exact closed-form expressions for the OP, ABER, and
ACC of the considered system setup assuming fixed-gain AF and DF relays. In
addition, in order to clearly see the impact of fading parameters on the
overall performance, tight asymptotic expressions at high SNR regime are
derived, based on which the respective diversity orders are computed.

###  IV-A Fixed-Gain AF Relaying

####  IV-A1 Outage Probability

The exact closed-form expression for the OP of dual-hop RF-UWOC systems with
fixed-gain AF relaying system is obtained from (16), and it is given by

$\displaystyle
P_{out}^{AF}=\Pr[\gamma<\gamma_{th}]=F_{\gamma^{AF}}(\gamma_{th}),$ (21)

where $P_{out}^{AF}$ holds for the CDF of $\gamma^{AF}$ evaluated at the
threshold $\gamma_{th}$.

In order to get more useful performance insights, an asymptotic outage
expression can be derived as

$\displaystyle P_{out}^{AF,\infty}\approx
F_{\gamma_{1}}^{\infty}(\gamma_{th})+\mathcal{I}_{1}^{\infty}(\gamma_{th}),$
(22)

where $F_{\gamma_{1}}^{\infty}(\cdot)$ is the asymptotic OP of $\gamma_{1}$
and $\mathcal{I}_{1}^{\infty}(\cdot)$ denotes the asymptotic result of
$\mathcal{I}_{1}$. From [32] and [21, Eq. (18)], when
$\overline{\gamma}_{1}\rightarrow\infty$, the $F_{\gamma_{1}}^{\infty}$ can be
written as

$\displaystyle F_{\gamma_{1}}^{\infty}\approx\frac{\vartheta
e^{-K}\gamma_{th}}{\overline{\gamma}_{1}}.$ (23)

Additionally, assuming that $\overline{\gamma}_{1}\rightarrow\infty$ and
employing [25, Eq. (1.2)], $\mathcal{I}_{1}^{\infty}$ can be derived as

$\displaystyle\mathcal{I}_{1}^{\infty}\underset{\overline{\gamma}_{1}\gg
1}{=}\omega
r\sum_{m=0}^{n}\varpi\,{\mathrm{H}}_{1,3}^{2,1}\left[{{\frac{C\vartheta\gamma_{th}}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(m+1,1)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{\Gamma(a)c}\sum_{m=0}^{n}\varpi\,{\mathrm{H}}_{1,3}^{2,1}\left[{{\frac{C\vartheta\gamma_{th}}{b^{r}\mu_{r}\overline{\gamma}_{1}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(m+1,1)(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!.$
(24)

When $\overline{\gamma}_{2}\rightarrow\infty$, $\mathcal{I}_{1}$ can be
further asymptotically expressed by using [33, Eq. (1.8.4)] as

$\displaystyle\mathcal{I}_{1}^{\infty}\underset{\overline{\gamma}_{1},~{}\mu_{r}\gg
1}{\approx}\omega r\sum_{m=0}^{n}\sum_{j=1}^{2}\varpi$
$\displaystyle\times\frac{\Pi_{i=1,i\neq
j}^{2}\Gamma\left(b_{1,i}{-}\frac{b_{1,j}\beta_{1,i}}{\beta_{1,j}}\right)\Gamma\left(\frac{rb_{1,j}}{\beta_{1,j}}\right)}{\beta_{1,j}\Gamma\left(1{+}\frac{rb_{1,j}}{\beta_{1,j}}\right)}\left(\frac{C\vartheta\gamma_{th}}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{b_{1,j}}{\beta_{1,j}}}$
$\displaystyle+\frac{(1-\omega)r}{\Gamma(a)c}\sum_{m=0}^{n}\sum_{j=1}^{2}\varpi$
$\displaystyle\times\frac{\Pi_{i=1,i\neq
j}^{2}\Gamma\left(b_{2,i}{-}\frac{b_{2,j}\beta_{2,i}}{\beta_{2,j}}\right)\Gamma\left(\frac{rb_{2,j}}{c\beta_{2,j}}\right)}{\beta_{2,j}\Gamma\left(1{+}\frac{rb_{2,j}}{c\beta_{2,j}}\right)}\left(\frac{C\vartheta\gamma_{th}}{b^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{b_{2,j}}{\beta_{2,j}}},$
(25)

where $\frac{b_{1,j}}{\beta_{1,j}}=\\{m{+}1,\frac{1}{r}\\}$ and
$\frac{b_{2,j}}{\beta_{2,j}}=\\{m{+}1,\frac{ac}{r}\\}$. Therefore, by
substituting (23) and (25) into (22), the asymptotic OP can be obtained. From
Table I and Table II, it is found that $ac>1$. As such, one can conclude that
the achievable diversity order of the considered fixed-gain AF relaying system
is given by

$\displaystyle G_{d}^{AF}=\min\left(1,\frac{2}{r}\right)=1.$ (26)

Furthermore, regardless the detection technology used by the R-AUV link, i.e.,
HD (i.e. $r=1$) or IMDD (i.e. $r=2$), $G_{d}^{AF}$ is equal to 1. This easy-
to-handle result is particularly importance since it reveals that the
diversity order of the fixed-gain AF relaying system under study depends
solely of the UAV-R link.

####  IV-A2 Average Bit Error Rate

The ABER of various binary modulation schemes can be given by [28, Eq. (25)]

$\displaystyle\overline{P}_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p-1}e^{-q\gamma}F_{\gamma}(\gamma)d\gamma,$
(27)

where the values of $p$ and $q$ determine various modulation scheme. For
instance, $\\{p=\frac{1}{2},q=1\\}$ stands for binary phase shift keying
(BPSK). Then, by substituting (12) into (27), the ABER of fixed-gain AF
relaying is formulated as

$\displaystyle\overline{P}_{e}^{AF}{=}$
$\displaystyle\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p{-}1}e^{-q\gamma}F_{\gamma_{1}}(\gamma)d\gamma$
$\displaystyle+\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p{-}1}e^{-q\gamma}\mathcal{I}_{1}d\gamma=\overline{P}_{e,1}+\mathcal{I}_{2},$
(28)

where $\overline{P}_{e,1}$ is the ABER of the RF link. By replacing (14) into
(27) and utilizing [22, Eq. (6.455.2)], we obtain

$\displaystyle\overline{P}_{e,1}=\sum_{m=0}^{n}$ $\displaystyle
a_{1}\frac{q^{p}\Gamma(m{+}1{+}p)}{2\Gamma(p)(m{+}1)\left(q{+}\frac{\theta}{\overline{\gamma}_{1}}\right)^{m{+}1{+}p}}$
$\displaystyle\times{}_{2}F_{1}\left(1,m{+}1{+}p;m{+}1;\frac{\theta}{q\overline{\gamma}_{1}{+}\theta}\right).$
(29)

Based on (15), utilizing [22, Eq. (3.326.2)] with the same method in Appendix
A, $\mathcal{I}_{2}$ can be formulated as

$\displaystyle\mathcal{I}_{2}=\sum_{m=0}^{n}\xi_{1}{\rm{H}_{1,0:3,1:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (0,r)(0,1)(1,r)\\\ (0,r)\\\ (1,1)\\\
(m{+}1{+}p,1)(1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{q\overline{\gamma}_{1}}{\vartheta}\\!\\!}\right]$
$\displaystyle+\sum_{m=0}^{n}\xi_{2}{\rm{H}_{1,0:3,1:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(0,1)(1,\frac{r}{c})\\\ (0,\frac{r}{c})\\\ (1,1)\\\
(m{+}1{+}p,1)(1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{q\overline{\gamma}_{1}}{\vartheta}\\!\\!}\right].$
(30)

where $\xi_{1}=\frac{a_{1}\omega r}{2\Gamma(p)q^{m+1}}$ and
$\xi_{2}=\frac{a_{1}(1-\omega)r}{2\Gamma(p)\Gamma(a)cq^{m+1}}$.

When
$\overline{\gamma}_{1}=\overline{\gamma}_{2}=\overline{\gamma}\rightarrow\infty$,
an asymptotic expression for the ABER is derived as

$\displaystyle\overline{P}_{e}^{AF,\infty}\approx\overline{P}_{e,1}^{\infty}+\mathcal{I}_{2}^{\infty},$
(31)

where $\overline{P}_{e,1}^{\infty}$ is the asymptotic ABER of the UAV-R link
and $\mathcal{I}_{2}^{\infty}$ denotes the asymptotic result of the
$\mathcal{I}_{2}$. By inserting (23) into (27) and using [22, Eq. (3.326.2)],
we obtain

$\displaystyle\overline{P}_{e,1}^{\infty}\approx\frac{\vartheta
e^{-K}\Gamma(p+1)}{2\Gamma(p)q\overline{\gamma}_{1}}.$ (32)

Moreover, using the same method of $\mathcal{I}_{1}^{\infty}$,
$\mathcal{I}_{2}^{\infty}$ can be derived as

$\displaystyle\mathcal{I}_{2}^{\infty}\underset{\overline{\gamma}_{1},~{}\mu_{r}\gg
1}{\approx}\frac{\omega
r}{2\Gamma(p)}\sum_{m=0}^{n}\sum_{j=1}^{2}\varpi\Gamma\left(p{+}\frac{b_{1,j}}{\beta_{1,j}}\right)$
$\displaystyle\times\frac{\Pi_{i=1,i\neq
j}^{2}\Gamma\left(b_{1,i}{-}\frac{b_{1,j}\beta_{1,i}}{\beta_{1,j}}\right)\Gamma\left(\frac{rb_{1,j}}{\beta_{1,j}}\right)}{\beta_{1,j}\Gamma\left(1{+}\frac{rb_{1,j}}{\beta_{1,j}}\right)}\left(\frac{C\vartheta}{\lambda^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{b_{1,j}}{\beta_{1,j}}}$
$\displaystyle+\frac{(1-\omega)r}{2\Gamma(p)\Gamma(a)c}\sum_{m=0}^{n}\sum_{j=1}^{2}\varpi\Gamma\left(p{+}\frac{b_{2,j}}{\beta_{2,j}}\right)$
$\displaystyle\times\frac{\Pi_{i=1,i\neq
j}^{2}\Gamma\left(b_{2,i}{-}\frac{b_{2,j}\beta_{2,i}}{\beta_{2,j}}\right)\Gamma\left(\frac{rb_{2,j}}{c\beta_{2,j}}\right)}{\beta_{2,j}\Gamma\left(1{+}\frac{rb_{2,j}}{c\beta_{2,j}}\right)}\left(\frac{C\vartheta}{b^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{b_{2,j}}{\beta_{2,j}}}.$
(33)

Therefore, the diversity order calculated through the asymptotic ABER
coincide, as expected, with (26).

####  IV-A3 Average Channel Capacity

The ACC of the considered system setup operating under both HD and IM/DD can
be expressed as [34, Eq. (26)], [35, Eq. (25)]

$\displaystyle\overline{C}=\frac{1}{2\ln(2)}\int_{0}^{\infty}\ln(1+\tau\gamma)f_{\gamma}(\gamma)d\gamma,$
(34)

where $\tau$ has two potential values such that $\tau=e/(2\pi)$ stands for the
IM/DD technique and $\tau=1$ represents the HD technique.

For fixed-gain AF relaying, by inserting (18) into (34), using [36, Eq.
(8.4.6/5)] followed by [25, Eq. (2.9)], then applying [25, Eq. (2.57)], the
ACC can be formulated in closed-form as

$\displaystyle\overline{C}^{AF}=\sum_{m=0}^{n}\kappa_{1}{\rm{H}_{1,0:2,0:2,2}^{0,1:0,2:1,2}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (0,r)(1,1)\\\ -\\\ (1,1)(m{+}2,1)\\\
(m{+}2,1)(m{+}1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}\tau}{\vartheta}\\!\\!}\right]$
$\displaystyle+\sum_{m=0}^{n}\kappa_{2}{\rm{H}_{1,0:2,0:2,2}^{0,1:0,2:1,2}}\left[{\\!\\!\left.{\begin{matrix}\left({m{+}2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(1,1)\\\ -\\\ (1,1)(m{+}2,1)\\\
(m{+}2,1)(m{+}1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}\tau}{\vartheta}\\!\\!}\right].$
(35)

where $\kappa_{1}=\frac{a_{1}\omega}{2\ln(2)\tau^{m+1}}$ and
$\kappa_{2}=\frac{a_{1}(1-\omega)}{2\ln(2)\Gamma(a)\tau^{m+1}}$.

###  IV-B DF Relaying

####  IV-B1 Outage Probability

For DF relaying, the OP is obtained from (19) as

$\displaystyle P_{out}^{DF}=F_{\gamma^{DF}}(\gamma_{th}).$ (36)

Then, the asymptotic CDF of $\gamma^{DF}$ can be expressed as

$\displaystyle P_{out}^{DF,\infty}\approx
F_{\gamma_{1}}^{\infty}(\gamma_{th})+F_{\gamma_{2}}^{\infty}(\gamma_{th}),$
(37)

where $F_{\gamma_{1}}^{\infty}(\gamma_{th})$ is the asymptotic CDF of the
UAV-R link, which is given by (23). When
$\overline{\gamma}_{2}\rightarrow\infty$,
$F_{\gamma_{2}}^{\infty}(\gamma_{th})$ can be simplified by using [33, Eq.
(1.8.4)], which yields

$\displaystyle
F_{\gamma_{2}}^{\infty}(\gamma_{th})=\omega\left(\frac{\gamma_{th}}{\lambda^{r}\mu_{r}}\right)^{\frac{1}{r}}+\frac{(1-\omega)}{\Gamma(a+1)}\left(\frac{\gamma_{th}}{b^{r}\mu_{r}}\right)^{\frac{ac}{r}},$
(38)

Therefore, with the aid of (23) and (38), the diversity order is

$\displaystyle G_{d}^{DF}=\min\left(1,\frac{1}{r}\right).$ (35)

One can see that the diversity order of the DF relaying is determined by
detection schemes used in the R-AUV link.

####  IV-B2 Average Bit Error Rate

The ABER can be expressed as

$\displaystyle\overline{P}_{e}^{DF}$
$\displaystyle=\overline{P}_{e,1}+\overline{P}_{e,2}-2\overline{P}_{e,1}\overline{P}_{e,2},$
(36)

where $\overline{P}_{e,2}$ denotes the ABER of the R-AUV link. In particular,
$\overline{P}_{e,1}$ is given in (26). By substituting (8) into (27) and
representing $\exp(-q\gamma)$ in terms of the Fox’s H-function using [36, Eq.
(8.4.3/1)] and [37, Eq. (07.34.26.0008.01)], i.e.,
$\exp(-q\gamma){=}\,{\rm{H}}_{0,1}^{1,0}\left[{q\gamma\left|{\begin{matrix}{-}\\\
{(0,1)}\\\ \end{matrix}}\right.}\right]\\!$, and then utilizing [35, Eq.
(2.25.1/1)], we obtain

$\displaystyle\overline{P}_{e,2}$ $\displaystyle=\frac{\omega
r}{2\Gamma(p)}\,{\rm{H}}_{2,2}^{1,2}\left[{{\frac{1}{\lambda^{r}\mu_{r}q}}\left|{\begin{matrix}{(1,r)(1{-}p,1)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{2\Gamma(p)\Gamma(a)c}\,{\rm{H}}_{2,2}^{1,2}\left[{{\frac{1}{b^{r}\mu_{r}q}}\left|{\begin{matrix}{(1,\frac{r}{c})(1{-}p,1)}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!.$ (39)

Finally, with the help of (26), (39) and (36), a analytical expression for the
ABER can be derived.

Similarly, the asymptotic ABER expression can be formulated as

$\displaystyle\overline{P}_{e}^{DF,\infty}\approx\overline{P}_{e,1}^{\infty}+\overline{P}_{e,2}^{\infty},$
(37)

where $\overline{P}_{e,1}^{\infty}$ is given by (32). Using [33, Eq. (1.8.4)],
the asymptotic expression of $\overline{P}_{e,2}^{\infty}$ is obtained as

$\displaystyle\overline{P}_{e,2}^{\infty}=\frac{\omega}{2\Gamma(p)}$
$\displaystyle\Gamma\left(p+\frac{1}{r}\right)\left(\frac{1}{\lambda^{r}\mu_{r}q}\right)^{\frac{1}{r}}$
$\displaystyle+\frac{(1-\omega)\Gamma\left(p+\frac{ac}{r}\right)}{2\Gamma(p)\Gamma(a+1)}\left(\frac{1}{b^{r}\mu_{r}q}\right)^{\frac{ac}{r}}.$
(40)

Thus, one can see that the diversity order of the considered DF relay is
consistent with (35).

####  IV-B3 Average Channel Capacity

Substituting (20) into (34) and applying the identity
$\ln(1+\tau\gamma)=\,{\rm{H}}_{2,2}^{1,2}\left[{{\tau\gamma}\left|{\begin{matrix}{(1,1)(1,1)}\\\
{(1,1)(0,1)}\\\ \end{matrix}}\right.}\right]\\!$ [36, Eq. (8.4.6/5)] [37, Eq.
(07.34.26.0008.01)], the ACC can be formulated as

$\displaystyle\overline{C}^{DF}=$
$\displaystyle\frac{1}{2\ln(2)}\int_{0}^{\infty}\,{\rm{H}}_{2,2}^{1,2}\left[{{\tau\gamma}\left|{\begin{matrix}{(1,1)(1,1)}\\\
{(1,1)(0,1)}\\\ \end{matrix}}\right.}\right]\\!f_{\gamma}^{DF}(\gamma)d\gamma$
$\displaystyle=$ $\displaystyle I_{C1}+I_{C2}-I_{C3}-I_{C4}.$ (41)

Then, by applying [36, Eq. (2.25.1/1)] and after some algebraic operations, we
have

$\displaystyle I_{C1}$
$\displaystyle=\frac{\varpi}{2\ln(2)}\sum_{m=0}^{n}\,{\rm{H}}_{3,2}^{1,3}\left[{{\frac{\overline{\gamma}_{1}\tau}{\vartheta}}\left|{\begin{matrix}{(-m,1)(1,1)(1,1)}\\\
{(1,1)(0,1)}\\\ \end{matrix}}\right.}\right]\\!.$ (42)

Similarly, $I_{C2}$ can be derived as

$\displaystyle I_{C2}=\frac{\omega
r}{2\ln(2)}\,{\rm{H}}_{1,2}^{2,1}\left[{{\frac{1}{\lambda^{r}\mu_{r}\tau}}\left|{\begin{matrix}{(0,1)}\\\
{(0,r)(0,1)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{2\ln(2)\Gamma(a)c}\,{\rm{H}}_{2,3}^{3,1}\left[{{\frac{1}{b^{r}\mu_{r}\tau}}\left|{\begin{matrix}{(0,1)(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})(0,1)}\\\ \end{matrix}}\right.}\right]\\!.$
(43)

In addition, with the help of (8) and (13), and utilizing the identity
$\exp\left(-\frac{\vartheta\gamma_{1}}{\overline{\gamma}_{1}}\right){=}\,{\rm{H}}_{0,1}^{1,0}\left[{\frac{\vartheta\gamma_{1}}{\overline{\gamma}_{1}}\left|{\begin{matrix}{-}\\\
{(0,1)}\\\ \end{matrix}}\right.}\right]\\!$ along with [37, Eqs. (1.1) and
(2.3)], $I_{C3}$ can be formulated as

$\displaystyle
I_{C3}{=}\sum_{m=0}^{n}\delta_{1}{\rm{H}_{1,0:2,2:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(-m;1,1)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,r)\\\
(1,r)(0,r)\end{matrix}}\right|\\!\frac{\overline{\gamma}_{1}\tau}{\vartheta},\\!\frac{\overline{\gamma}_{1}}{\lambda^{r}\mu_{r}\vartheta}\\!\\!}\right]$
$\displaystyle{+}\sum_{m=0}^{n}\delta_{2}{\rm{H}_{1,0:2,2:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(-m;1,1)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,\frac{r}{c})\\\
(a,\frac{r}{c})(0,\frac{r}{c})\end{matrix}}\right|\\!\frac{\overline{\gamma}_{1}\tau}{\vartheta},\\!\frac{\overline{\gamma}_{1}}{b^{r}\mu_{r}\vartheta}\\!\\!}\right],$
(44)

where $\delta_{1}=\frac{\varpi\omega r}{2\ln(2)}$ and
$\delta_{2}=\frac{\varpi(1-\omega)r}{2\ln(2)\Gamma(a)c}$. With the help of
(7), (14) and (41), $I_{C4}$ can be derived as

$\displaystyle
I_{C4}=\sum_{m=0}^{n}\delta_{3}{\rm{H}_{1,0:2,2:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(0;r,r)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,1)\\\
(m+1,1)(0,1)\end{matrix}}\right|\\!\tau\lambda^{r}\mu_{r},\\!\frac{\lambda^{r}\mu_{r}\vartheta}{\overline{\gamma}_{1}}\\!\\!}\right]$
$\displaystyle{+}\sum_{m=0}^{n}\delta_{4}{\rm{H}_{1,0:2,2:1,2}^{0,1:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(1{-}a;\frac{r}{c},\frac{r}{c})\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,1)\\\
(m+1,1)(0,1)\end{matrix}}\right|\\!\tau
b^{r}\mu_{r},\\!\frac{b^{r}\mu_{r}\vartheta}{\overline{\gamma}_{1}}\\!\\!}\right].$
(45)

where $\delta_{3}=\frac{\varpi\omega}{2\ln(2)}$ and
$\delta_{4}=\frac{\varpi(1-\omega)}{2\ln(2)\Gamma(a)}$.

##  V Optimal Altitude Analysis

In UAV-based communication networks, although high altitude can increase the
coverage, it may cause a large path loss. As such, a trade-off exists between
link quality and coverage which required a more in-depth investigation. In
this section, we take the DF relaying as an example to analyze the optimal
deployment of the UAV with the aim to minimize the OP.

Following (19), (21) and (36), the OP of the DF relaying $P_{out}^{DF}$ can be
written by using (3) and (8) as

$\displaystyle P_{out}^{DF}=1-\left({1-\omega
r\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{th}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!}\right.$
$\displaystyle\left.{-\frac{(1-\omega)r}{\Gamma(a)c}\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{th}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!}\right)$
$\displaystyle\times
Q_{1}\left(\sqrt{2K(\theta)},\sqrt{2\gamma_{th}(1+K(\theta))d_{UR}^{\alpha(\theta)}/\overline{\gamma}_{1}}\right).$
(46)

Note that the OP depends on the UAV altitude $h_{1}$ and the horizontal
distance $r_{1}$ [10]. Thus, (46) is formulated as
$P_{out}^{DF}=P_{out}^{DF}(r_{1},h_{1})$. At a given parameter $r_{1}$, the
optimal altitude can be obtained as

$\displaystyle\widetilde{h}=\arg\underset{h\in[0,\infty)}{\min}P_{out}^{DF}(r_{1},h_{1}).$
(47)

Therefore, to minimize the OP, we take the derivative of (46) with respect to
$\theta$ as

$\displaystyle\frac{\partial}{\partial\theta}P_{out}^{DF}(r_{1},h_{1})=0,$
(48)

Then, by substituting (46) into (48), we get

$\displaystyle\Lambda_{1}\frac{\partial Q(x,y)}{\partial\theta}=0,$ (49)

where $x=\sqrt{2K(\theta)}$ and
$y=\sqrt{2\gamma_{th}(1+K(\theta))d_{UR}^{\alpha(\theta)}/\overline{\gamma}_{1}}$
represent the first and second arguments of the Marcum Q-function in (46),
respectively. From (49), one can obtain the following expression

$\displaystyle\Lambda_{1}=$ $\displaystyle\omega
r\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{th}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{\Gamma(a)c}\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{th}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!-1,$ (50)

which shows that $\Lambda_{1}$ is independent of $\theta$. In fact, by letting
$\frac{\partial Q(x,y)}{\partial\theta}=0$, the optimal solution can be
obtained. Therefore, taking the required derivative, it follows that [5, Eq.
(38)]

$\displaystyle\frac{\partial
Q(x,y)}{\partial\theta}=ye^{\frac{x^{2}+y^{2}}{2}}\left[{I_{1}(xy)\frac{K^{\prime}(\theta)}{x}-I_{0}(xy)}\right.$
$\displaystyle\left.{\times\frac{y}{2}\left[\frac{K^{\prime}(\theta)}{1+K(\theta)}+\alpha^{\prime}(\theta)\ln\left(\frac{r_{1}}{\cos\theta}\right)+\alpha(\theta)\tan\theta\right]}\right],$
(51)

where $I_{1}(\cdot)$ denotes the first-order modified Bessel function.

Therefore, for a given $r_{1}$, the optimal altitude $\widetilde{h}$ can be
expressed as

$\displaystyle\widetilde{h}_{1}=r_{1}\tan(\widetilde{\theta}),$ (52)

where $\widetilde{\theta}$ is obtained from (49).

##  VI Numerical Results and Discussions

In this section, Monte Carlo simulation is used to verify the correctness of
our analysis. Without loss of generality, we set the threshold
$\gamma_{th}=1.5$ dB and the relaying gain is $C=1.3$. Also, we set the
average SNR of the UAV-R link and the R-AUV link to be equal, i.e,
$\overline{\gamma}_{1}=\overline{\gamma}_{2}=\overline{\gamma}$. The
parameters of the UAV-R link are $A=1$, $a_{1}=-1.5$, $b_{1}=3.5$, $K(0)=5$
dB, and $K(\frac{\pi}{2})=15$ dB. In addition, the parameters setup used in
the R-AUV link is listed in Table I and Table II.

TABLE III: The Optimal Elevation Angle $\widetilde{\theta}$[∘] and Optimal Altitude $\widetilde{h}_{1}$[m] at Given $r_{1}$[m] in The Case of Different Turbulence Conditions. $r_{1}$[m] |  $\widetilde{\theta}$ [∘], |  $\widetilde{h}_{1}$[m], |  $\widetilde{h}_{1}$[m],
---|---|---|---
|  $\sigma_{I}^{2}=0.1484$ |  $\sigma_{I}^{2}=0.1484$ |  $\sigma_{I}^{2}=1.1273$
500 |  70.9 |  1443.9 |  1443.9
600 |  69.0 |  1563.1 |  1563.1
700 |  67.1 |  1657.1 |  1657.1
800 |  65.3 |  1793.3 |  1793.3
900 |  63.6 |  1813.0 |  1813.0
1000 |  62.0 |  1880.7 |  1880.7
1100 |  60.4 |  1936.4 |  1936.4
1200 |  58.8 |  1981.4 |  1981.4
1300 |  57.3 |  2025.0 |  2025.0
1400 |  55.8 |  2060.0 |  2060.0
1500 |  54.3 |  2087.5 |  2087.5

Fig. 2 depicts the OP versus $h_{1}$ of the considered fixed-gain AF relaying
with $\overline{\gamma}=15$ dB and various values of $r_{1}$. Results show
that by increasing $h_{1}$, the OP first decreases and then increases. This is
because the positioning of the UAV determines the LoS probability and the
increase of the UAV altitude implies a higher LoS contribution. As such, the
system has an optimal altitude, enabling to minimize the OP. However, if
$h_{1}$ continues to increase, the path loss affects the system performance
and reduces the link reliability. In addition, one can notice that reducing
$r_{1}$ increases the coverage probability of the UAV and improves the system
performance. Additionally, Fig. 3 studies the effect of $h_{1}$ on the
obtained OP of the DF relaying, where a three-dimensional (3D) plot shows the
relation between $h_{1}$, $r_{1}$, and the resulting OP. In this figuer, we
can clear see that different distances imply in different optimal altitudes to
guarantee the minimum OP. Furthermore, for a given $r_{1}$, the optimal
elevation angles and optimal UAV altitudes $\widetilde{h}_{1}$ enabling to
minimize the OP are presented in Table III, considering different turbulence
conditions. As can be seen, $\widetilde{\theta}$ reduces with $r_{1}$ since
the link length is more susceptible to the elevation angle at large $r_{1}$.
Moreover, one can observe that UWOC link parameters have no effect on the
optimal altitude.

Figure 2: The OP versus $h_{1}$ of the considered fixed-gain AF relaying
system with different values of $r_{1}$. Figure 3: The OP versus $h_{1}$ and
$r_{1}$ of the considered fixed-gain AF relaying system. Figure 4: The OP
versus $\overline{\gamma}$ of the fixed-gain AF relaying system for different
turbulence conditions and detection techniques, along with the asymptotic
results at high SNR. Figure 5: The OP versus $\overline{\gamma}$ of the DF
relaying system for different turbulence conditions and detection techniques,
along with the asymptotic results at high SNR.

Fig. 4 shows the OP versus $\overline{\gamma}$ of the considered fixed-gain AF
relaying system under different types of water and various levels of air
bubbles. In this setup, we assume that $h_{1}=1.5$ km, $r_{1}=1.2$ km. For a
fixed type of water, as the bubble level increases, the value of
$\sigma_{I}^{2}$ increases, which indicates that the system is experiencing
stronger turbulence. In this figure, we assume that BL $=4.7$ L/min
corresponds to weak UOT and BL $=16.5$ L/min represents strong UOT. One can
observe that for given a type of water, the OP of the system in weak
turbulence is significantly lower than that in strong turbulence. If the
bubble level is the same, such as BL $=4.7$ L/min, the effect of water
salinity on system performance is low. Especially in the case of higher bubble
level, i.e, BL $=16.5$ L/min, the influence of salinity is smaller. Therefore,
it can be inferred that the impact of bubble level on the considered system is
much higher than that of water salinity due to the rapid intensity
fluctuations caused by air bubbles. In addition, due to the use of the HD
technique can better reduce the performance loss caused by underwater
turbulence, note that the outage performance under the HD technique is higher
than IM/DD. At high SNR, the asymptotic results are depicted, which reveal
that the system diversity order using the two detection techniques is equal.
This is because the overall system outage performance is solely determined by
the UAV-R link, which is consistent with the result in (26).

Figure 6: The ABER versus $h_{1}$ of the fixed-gain AF relaying system with
different values of the $r_{1}$. Figure 7: The ABER versus $\overline{\gamma}$
of the DF relaying system for different altitude of the UAV with $r_{1}=0.8$
km.

Fig. 5 shows the OP versus $\overline{\gamma}$ of the considered DF relaying
system under various levels of air bubbles and gradient temperatures for the
HD and IM/DD techniques. As can be observed, the HD technique outperforms
IM/DD technique, as expected. Moreover, Fig. 5 demonstrates that the bubble
levels and gradient temperatures have a serious impact on system outage
performance. In particular, the increase in bubble level and gradient
temperature causes an increase of the value of $\sigma_{I}^{2}$ and therefore
the system suffers from stronger underwater turbulence, which increases the
OP. Note also that, under low SNR, air bubbles have a greater adverse effect
on outage performance than gradient temperatures. Similarly, the asymptotic
results are also shown in this figure, but, different from the fixed-gain AF
relaying system, the system diversity order depends on the detection
techniques, which is consistent with (35).

In Fig. 6, the ABER versus $h_{1}$ is presented for the fixed-gain AF relaying
and HD technique by considering different altitude of the UAV. The ABER
performance first improves as $h_{1}$ increases and then decreases as the
altitude continues to increase. Furthermore, as the $r_{1}$ increases, the
error performance gets worse. This is because increasing $r_{1}$ reduces the
angle between the UAV and the relay, thereby reducing the probability of the
LoS signals.

Figure 8: The ACC versus $\overline{\gamma}$ of the fixed-gain AF relaying
system for different levels of air bubbles under IM/DD techniques. Figure 9:
The ACC versus $\overline{\gamma}$ of the fixed-gain AF relaying system for
different altitude of the UAV with $r_{1}=2$ km.

In Fig. 7, the ABER versus $\overline{\gamma}$ is plotted along with the
asymptotic results for DF relaying by considering various values of $h_{1}$
and by setting $r_{1}=0.8$ km. Moreover, we assume that the system employs
both IM/DD and HD techniques, and it is considered a weak turbulence
corresponding to BL $=2.4$ L/min and a temperature gradient of
$0.05^{\circ}\mathrm{C}.cm^{-1}$. As can be clearly seen, $h_{1}$ can effect
the ABER performance at low SNRs. For $r_{1}=0.8$ km, as the UAV altitude
drops from 2 km to 1 km, the path loss is reduced, which improves error
performance. Furthermore, one can observe that a system with small $h_{1}$ has
the same ABER at high SNRs. This is because with the increase of the average
SNR and decrease of $h_{1}$, the R-AUV link becomes dominant at high SNR
regions.

Assuming various levels of air bubbles and employing salty water and fresh
water, Fig. 8 shows the ACC versus $\overline{\gamma}$ of the fixed-gain AF
relaying system in the case of IM/DD technique. As can be clearly seen, as the
air bubbles level increases, the ACC decreases. As expected, the performance
in slaty water is not as good as that in fresh water, but compared with bubble
levels, the water salinity has little effect on performance, especially in the
case of weak turbulence, such as BL $=2.4$ L/min.

In Fig. 9, the ACC of the fixed-gain AF relaying is presented versus
$\overline{\gamma}$ for various altitude of the UAV and assuming both IM/DD
and HD techniques. In this setup, we set $r_{1}=2$ km, the turbulence
condition corresponds to a BL $=2.4$ L/min, and the temperature gradient is
$0.05^{\circ}\mathrm{C}.cm^{-1}$. As expected, the ACC decreases as $h_{1}$
increases. This is because that increasing the height of the UAV increases the
LoS propagation probability of the signal and then improves the ACC.
Furthermore, as $h_{1}$ continues to increase, the ACC reduces due to the
serious path loss, which is also reflected in Figs. 2 and 6.

##  VII Conclusions

In this paper, we have investigated the performance of the UAV-based dual-hop
RF-UWOC system employing fixed-gain AF relaying and DF relaying. In the UWOC
link, both HD and IM/DD modulation schemes were considered. We derived closed-
form expressions of the for the OP, ABER, and ACC. For DF relaying, analytical
expressions for optimal elevation angle and optimal height were derived.
Results showed that UAV can be flexibly deployed as a base station, which
brings better performance for dual-hop RF-UWOC systems. It was found that
different horizonal distances have different optimal altitudes that guarantee
better performance. In addition, temperature fluctuations, salinity
variations, and the presence of bubbles all affect the system performance.
Specifically, the increase of bubbles levels may cause significant
deterioration of the performance compared to changes in temperature and
salinity. Furthermore, asymptotic results for the OP and ABER of fixed-gain AF
and DF relays were derived and some useful insights were obtained. It was
demonstrated that the diversity order of the fixed-gain AF relaying is
determined by the RF link, while the diversity order of the DF relaying is
determined by the type of detection technique being used in UWOC link.

##  Appendix A CDF of The E2E SNR for Fixed-Gain AF Relaying

Making use of [25, Eq. (1.2)] and then [23, Eqs. (3.194.3) and (8.384.1)], we
obtain

$\displaystyle\mathcal{I}_{1}=\sum_{m=0}^{n}a_{1}\omega
r\frac{\gamma^{m+1}}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}\Gamma(t{-}s{-}m{-}1)$
$\displaystyle\times\frac{\Gamma(1{+}rs)\Gamma(-rs)\Gamma(1{+}s)}{\Gamma(1{-}rs)}\frac{\Gamma(t)}{\Gamma(t{-}m)}\left(\frac{\lambda^{r}\mu_{r}}{C}\right)^{s}\left(\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\right)^{t}dsdt$
$\displaystyle+\sum_{m=0}^{n}a_{1}\frac{(1-\omega)r}{\Gamma(a)c}\frac{\gamma^{m+1}}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}\Gamma(t{-}s{-}m{-}1)$
$\displaystyle\times\frac{\Gamma(a{+}\frac{r}{c}s)\Gamma(-\frac{r}{c}s)\Gamma(1{+}s)}{\Gamma(1{-}\frac{r}{c}s)}\frac{\Gamma(t)}{\Gamma(t{-}m)}\left(\frac{b^{r}\mu_{r}}{C}\right)^{s}\left(\frac{\overline{\gamma}_{1}}{\vartheta\gamma}\right)^{t}dsdt,$
(53)

where $\ell_{1}$ and $\ell_{2}$ are the $s-$plane and the $t-$plane contours
in the complex domain, respectively. By utilizing [25, Eqs. (2.56) and
(2.57)], (53) can be expressed in closed-form as (15).

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