# On the Performance of RIS-Assisted Dual-Hop Mixed RF-UWOC Systems

Sai Li, Liang Yang, Daniel Benevides da Costa, Marco Di Renzo, and Mohamed-
Slim Alouini 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).M. Di Renzo is with
Université Paris-Saclay, CNRS and CentraleSupélec, Laboratoire des Signaux et
Systèmes, Gif-sur-Yvette, France. (email:
marco.direnzo@centralesupelec.fr).M.-S. Alouini is with the CEMSE Division,
King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900,
Saudi Arabia (email: slim.alouini@kaust.edu.sa).

###### Abstract

In this paper, we investigate the performance of a reconfigurable intelligent
surface (RIS)-assisted dual-hop mixed radio-frequency underwater wireless
optical communication (RF-UWOC) system. An RIS is an emerging and low-cost
technology that aims to enhance the strength of the received signal, thus
improving the system performance. In the considered system setup, a ground
source does not have a reliable direct link to a given marine buoy and
communicates with it through an RIS installed on a building. In particular,
the buoy acts as a relay that sends the signal to an underwater destination.
In this context, analytical expressions for the outage probability (OP),
average bit error rate (ABER), and average channel capacity (ACC) are derived
assuming fixed-gain amplify-and-forward (AF) and decode-and-forward (DF)
relaying protocols at the marine buoy. Moreover, asymptotic analyses of the OP
and ABER are carried out in order to gain further insights from the analytical
frameworks. In particular, the system diversity order is derived and it is
shown to depend on the RF link parameters and on the detection schemes of the
UWOC link. Finally, it is demonstrated that RIS-assisted systems can
effectively improve the performance of mixed dual-hop RF-UWOC systems.

###### Index Terms:

Underwater wireless optical communications, reconfigurable intelligent
surface, relaying systems, mixed dual-hop transmission schemes.

##  I Introduction

Underwater wireless optical communication (UWOC) is a promising technique for
beyond fifth-generation (B5G) wireless networks because it provides higher
data rates and better confidentiality than traditional underwater wireless
communication systems [1, 2]. In UWOC, however, the scattering and absorption
of the optical signal caused by various components in the seawater may have an
adverse effect on the transmission of the optical signal, and these issues
need to be properly evaluated. Along the years, mixed dual-hop transmissions
assuming radio-frequency (RF) and UWOC links have been widely investigated.
Specifically, the secrecy performance of mixed RF-UWOC systems was
investigated in [3]. In [4], mixed RF-UWOC relaying networks with both decode-
and-forward (DF) and amplify-and-forward (AF) capabilities were analyzed. In
[5], the authors proposed a unified UWOC channel model that accounts for the
fluctuations of the beam irradiance in the presence of both air bubbles and
temperature gradients. Such a distribution was called the Exponential-
Generalized Gamma (EGG) distribution and it encompasses as a special case the
Exponential Gamma (EG) fading model at a uniform temperature. Furthermore, in
[5], closed-form expressions for the outage probability (OP), average bit
error rate (ABER), and average channel capacity (ACC) were derived. Using the
EGG distribution, the performance of dual-hop UWOC systems with a fixed-gain
AF relay was studied in [6]. Finally, relying on the EGG fading model, the
performance of mixed RF-UWOC systems was investigated in [7, 8, 9].

Reconfigurable intelligent surface (RIS) technology is a promising efficient
solution to overcome the negative effects of wireless channels, by making the
wireless environment programmable. By smarting controlling the polarization,
scattering, reflection, and refraction of the impinging electromagnetic waves,
the propagation environment can be customized in order to meet the desired
application requirements [10, 11, 12, 13, 14, 15, 16]. In [16], a detailed
overview of the modeling and application of RISs in wireless communications
was provided, in which the differences between RISs and other related
techniques, such as relays, were discussed and the Non-Central Chi-Square
(NCCS) distribution was employed for performance analysis. Channel modeling
and beamforming optimization design issues of RIS-assisted systems were
studied in [17, 18, 19, 20, 21]. The performance of RIS-assisted mixed free-
space optical-RF (FSO-RF) systems was examined in [22], while in [23] the
authors studied the coverage and signal-to-noise ratio (SNR) gain of RIS-
assisted communication systems. In addition, the secrecy performance of RIS-
assisted secure wireless communication systems was studied in [24, 25].

Although the NCCS distribution has recently emerged as an efficient
statistical model for RIS-assisted systems, such a model has some limitations.
In the high SNR regime, for example, the deviation between simulation and
theoretical results is large. In addition, the NCCS model is only applicable
to scenarios with a large number of reflecting elements $N$. In order to
overcome the limitations of using the NCCS distribution, a more accurate
distribution was proposed in [26] to approximate the end-to-end channel
distribution of RIS-aided wireless systems, which is referred to as the
squared generalized-K ($K_{G}$) distribution. By relying upon the $K_{G}$
distribution, analytical and simulation results well match for an arbitrary
number of reflecting elements $N$. The approach proposed in [23] was applied
to the performance evaluation of unmanned aerial vehicle (UAV) relaying
systems [27] and indoor mixed visible light communication/RF (VLC/RF) systems
[28].

As discussed in previous text, RISs have been leveraged for enhancing the
performance of various emerging technologies, such as FSO, UAV, and VLC
networks [21, 22, 27, 28]. Moreover, RISs have bee applied for improving the
performance of multiple-input multiple/single-output (MIMO/MISO) networks [20,
29, 30] and millimeter wave (mmWave) communications [31, 32]. Furthermore,
dual-hop mixed RF-UWOC systems have been employed in applications like ocean
monitoring [4]. However, the application of RISs in dual-hop mixed RF-UWOC
communication systems has not been studied yet. Motivated by these
considerations, in this paper we investigate the scenario in which an RIS is
used to reflect the signals transmitted from a terrestrial source to a buoy
relay located on the sea surface, which in turn relays the signal to an
underwater destination by using fixed gain AF or DF relaying protocols. To
evaluate the impact of key system parameters on the overall performance, which
include the number of reflecting elements, as well as the bubble level,
temperature gradient, water type, and detection technology, a comprehensive
performance analysis is carried out in terms of OP, ABER, and ACC. The main
contributions of this paper can be summarized as follows:

$\bullet$ We study an RIS-aided dual-hop mixed RF-UWOC system, where the RF
and UWOC links are modeled by using a squared $K_{G}$ distribution and an EGG
distribution, respectively. Both heterodyne detection (HD) and intensity
modulation/direct detection (IM/DD) techniques are considered for the optical
link.

$\bullet$ Closed-form expressions for the cumulative distribution function
(CDF) and probability density function (PDF) of the end-to-end (e2e) SNR under
both fixed-gain AF and DF relaying protocols are derived. Based on these
distributions, a performance analysis is carried out and closed-form
expressions are derived for the OP, ABER, and ACC.

$\bullet$ To get further insights from the analytical frameworks, asymptotic
expressions for the OP and ABER are derived in the form of simple elementary
functions, which enables us to determine the system diversity order. It is
shown that the achievable diversity order depends on the RF link parameters as
well as on the type of the detection technique at the UWOC link.

$\bullet$ Insightful discussions are provided. For instance, it is found that
when the bubble level and temperature gradient are low, that is, when the
underwater turbulence intensity is small, the overall system performance is
better. In addition, it is shown that the e2e performance of the UWOC link is
better if the HD scheme is used. Finally, it is demonstrated through several
illustrative numerical examples that RIS-assisted schemes can effectively
improve the performance of mixed dual-hop RF-UWOC systems.

The remainder of this paper is organized as follows. Section II introduces the
system and channel models along with some useful statistics for the RF and
UWOC links. In Section III, a more in-depth statistical characterization is
performed for both fixed-gain AF and DF relaying schemes. Based on the
preliminary results obtained in the previous sections, Section IV derives
exact closed-form expressions for key performance metrics and an asymptotic
analysis is conducted as well. Section VI provides some illustrative numerical
examples based on which insightful findings are obtained. Finally, Section VII
concludes the paper.

Figure 1: The RIS-assisted dual-hop mixed RF-UWOC system model.

##  II System and Channel Models

We consider an RIS-assisted dual-hop mixed RF-UWOC system that consists of a
base station (S), an RIS with $N$ reflecting elements that is installed on a
building, a buoy relay (R) that is located on the water surface, and an
underwater destination (D), as shown in Fig. 1. It is assumed that R can
operate under both fixed-gain AF and DF relaying protocols. All nodes are
equipped with a single antenna, and there is no direct link between S and R
due to the long distance and the presence of environmental obstructions.
Therefore, the RF signal is first sent from S to the RIS, and then the RIS
reflects the incoming signal to R. After the electro-optical conversion at R,
the optical signal is transmitted to D via the UWOC channel. In Fig. 1,
$h_{i}=\alpha_{i}e^{-j\theta_{i}}$ and $g_{i}=\beta_{i}e^{-j\varphi_{i}}$ are,
respectively, the fading channels of the S-RIS and RIS-R links related to the
$i$th reflecting element ($i=1,2,\ldots,N$), where $\alpha_{i}$ and
$\theta_{i}$ denote the channel amplitude and phase of $h_{i}$, respectively,
while $\beta_{i}$ and $\varphi_{i}$ represent the channel amplitude and phase
of $g_{i}$, respectively. The RIS provides adjustable phase shifts and, in our
analysis, the channel phases of $h_{i}$ and $g_{i}$ at the RIS are assumed to
be known, which corresponds to the best scenario in terms of system operation.
In addition, from a practical point of view, due to the high probability of
line-of-sight signals on the S-RIS link, it is assumed that the S-RIS link
obeys a Nakagami-$m$ fading distribution. Furthermore, since the RIS-R link is
usually impaired by complex near-ground environmental factors (e.g., the
presence of trees or buildings), we assume that the RIS-R link follows a
Rayleigh fading.

###  II-A The SNR of the RF Link

From [22, Eq. (1)], the instantaneous SNR at R, $\gamma_{1}$, can be
formulated as follows

$\displaystyle\gamma_{1}=\Big{|}\sum_{i=1}^{N}h_{i}v_{i}g_{i}\Big{|}^{2}\overline{\gamma}_{1},$
(1)

where $\overline{\gamma}_{1}$ is the average SNR of the RF link,
$v_{i}=\rho_{i}(\phi_{i})e^{\phi_{i}}$ is the reflection coefficient
introduced by the $i$th reconfigurable element of the RIS, and
$\rho_{i}(\phi_{i})=1$, $\forall i$, for ideal reflecting elements. When
$\rho_{i}(\phi_{i})=1$, (1) reduces to
$\gamma_{1}=|\sum_{i=1}^{N}\alpha_{i}\beta_{i}e^{(\phi_{i}-\theta_{i}-\varphi_{i})}|^{2}\overline{\gamma}_{1}$.
In order to obtain the maximum value of $\gamma_{1}$, the reflected signals
need to be co-phased by setting $\phi_{i}=\theta_{i}+\varphi_{i}$, for
$i=1,2,\ldots,N$.

Therefore, the maximum instantaneous SNR at R can be written as [16, 26]

$\displaystyle\gamma_{1}=Z^{2}\overline{\gamma}_{1},$ (2)

where $Z=\sum_{i=1}^{N}\alpha_{i}\beta_{i}$, $\alpha_{i}$ is a Nakagami-$m$
random variable (RV), with $m\geq\frac{1}{2}$ being the fading parameter of
the distribution, and $\beta_{i}$ is a Rayleigh RV with mean $\sqrt{\pi}/2$
and variance $(4-\pi)/4$. By letting $\chi_{i}=\alpha_{i}\beta_{i}$, $Z$ turns
out to be the sum of $N$ independent and identically distributed (i.i.d.) RVs
$\chi_{i}$. The PDF of $\chi_{i}$ can be proved to be
$f_{\chi_{i}}(x)=\frac{4m^{\frac{1+m}{2}}x^{m}}{\Gamma(m)}K_{m-1}(2\sqrt{m}x)$
[27, Eq. (5)], [33, Eq. (8.432.7)], where $K_{v}(\cdot)$ is the modified
Bessel function [33, Eq. (8.432)] and $\Gamma(\cdot)$ denotes the Gamma
function [33, Eq. (8.310)]. Comparing the PDF of $\chi_{i}$ with [34, Eq.
(1)], one can see that the PDF of $\chi_{i}$ is a special case of the $K_{G}$
distribution. In [34], it is stated that the PDF of the sum of multiple
$K_{G}$ RVs can be well-approximated by the PDF of $\sqrt{W}$, with
$W=\sum_{j=1}^{N}\chi_{i}^{2}$. Therefore, the PDF of $Z$ can be formulated as

$f_{Z}(x)=\frac{4\widetilde{\Xi}^{\frac{k_{w}+m_{w}}{2}}}{\Gamma(k_{w})\Gamma(m_{w})}x^{k_{w}+m_{w}-1}K_{k_{w}-m_{w}}(2\sqrt{\widetilde{\Xi}}x),$
(3)

where $k_{w}=\frac{-b_{w}+\sqrt{b_{w}^{2}-4a_{w}c_{w}}}{2a_{w}}$ and
$m_{w}=\frac{-b_{w}-\sqrt{b_{w}^{2}-4a_{w}c_{w}}}{2a_{w}}$ are the the shaping
parameters, $\widetilde{\Xi}=k_{w}m_{w}/\Omega_{w}$, and
$\Omega_{w}=\mu_{R}(2)$ is the mean power of $Z$. Moreover, the parameters
$a_{w}$, $b_{w}$, and $c_{w}$ are defined in [34] and they are related to the
moment $\mu_{Z}(n)$ of $Z$, namely,

$\displaystyle\mu_{Z}(n)$
$\displaystyle=\sum_{n_{1}=0}^{n}\sum_{n_{2}=0}^{n_{1}}...\sum_{n_{N-1}=0}^{n_{N-2}}\binom{n}{n_{1}}\binom{n_{1}}{n_{2}}...\binom{n_{N-2}}{n_{N-1}}$
$\displaystyle\times\mu_{\chi_{1}}(n-n_{1})\mu_{\chi_{2}}(n_{1}-n_{2})...\mu_{\chi_{N-1}}(n_{N-1}),$
(4)

where
$\mu_{\chi_{i}}(n){=}\frac{2^{m{+}n{+}1}m^{\frac{1{+}m}{2}}}{\Gamma(m)}\Gamma\left(\frac{n}{2}\right)\Gamma\left(m{+}\frac{n}{2}\right)$
is the $n$th moment of $\chi_{i}$. Note that $k_{w}$ and $m_{w}$ are required
to be real numbers. But, in some cases, when $k_{w}$ and $m_{w}$ are
calculated by using the moment-based estimators, they are conjugate complex
numbers. In this case, they are set to the estimated modulus values of the
conjugate complex number.

From (2) and (3), the PDF of the instantaneous SNR, $\gamma_{1}$, is given by

$\displaystyle
f_{\gamma_{1}}(\gamma_{1})=\frac{2\Xi^{\frac{k_{w}{+}m_{w}}{2}}\gamma_{1}^{\left(\frac{k_{w}{+}m_{w}}{2}{-}1\right)}}{\Gamma(k_{w})\Gamma(m_{w})}K_{k_{w}{-}m_{w}}\left(2\sqrt{\Xi\gamma_{1}}\right),$
(5)

where $\Xi=\widetilde{\Xi}/\overline{\gamma}_{1}$. With the aid of [35, Eq.
(8.4.23.1)] and [33, Eq. (9.31.5)], and after some algebraic manipulations,
the PDF in (5) can be rewritten as

$\displaystyle
f_{\gamma_{1}}(\gamma_{1})=\frac{\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}\mathrm{G}_{0,2}^{2,0}\left[\left.\begin{matrix}\frac{\widetilde{\Xi}\gamma_{1}}{\overline{\gamma}_{1}}\end{matrix}\right|\begin{matrix}-\\\
k_{w}{-}1,m_{w}{-}1\end{matrix}\right],$ (6)

where $\mathrm{G}_{\cdot,\cdot}^{\cdot,\cdot}[\cdot]$ denotes the Meijer’s
G-function [33, Eq. (9.301)]. Thus, the CDF of $\gamma_{1}$ can be formulated
as

$\displaystyle
F_{\gamma_{1}}(\gamma_{1})=\frac{1}{\Gamma(m_{w})\Gamma(k_{w})}\,{\mathrm{G}}_{1,3}^{2,1}\left[\left.\begin{matrix}\frac{\widetilde{\Xi}\gamma_{1}}{\overline{\gamma}_{1}}\end{matrix}\right|\begin{matrix}1\\\
k_{w},m_{w},0\end{matrix}\right]\\!.$ (7)

According to (4), we find that $m_{w}$ and $k_{w}$ depend on $N$ and $m$, with
$k_{w}>m_{w}$. After a careful inspection, when $m\geq 0.5$ and $m_{w}\geq 1$,
it follows that $m_{w}$ increases as $N$ and $m$ increase. Therefore, the CDF
and PDF of $\gamma_{1}$ are directly related to $N$ and $m$.

###  II-B The SNR of the UWOC Link

The UWOC link is modelled as a mixture EGG distribution which is equivalent to
a weighted sum of the Exponential and Generalized Gamma (GG) distributions
[5]. The instantaneous SNR of the UWOC link, $\gamma_{2}$, is given by

$\displaystyle\gamma_{2}=\mu_{r}I^{r},$ (8)

where $r$ is the parameter that depends on the detection technique used, i.e.,
$r=1$ if the HD technique is considered and $r=2$ if the IM/DD technique is
considered, $I$ denotes the channel coefficient that is distributed according
to a mixture EGG distribution, and $\mu_{r}$ is the average electrical SNR. As
for the HD technique, $\mu_{1}=\overline{\gamma}_{2}$. As for the IM/DD
technique,
$\mu_{2}=\frac{\overline{\gamma}_{2}}{2\omega\lambda^{2}+b^{2}(1-\omega)\Gamma(a+2/c)/\Gamma(a)}$
[5, Eq. (19)], where $\overline{\gamma}_{2}$ represents the average SNR of the
UWOC link, $0<\omega<1$ is the mixture weight or mixture coefficient of the
distributions, $\lambda$ is the fading parameter of the Exponential
distribution, and $a$, $b$, and $c$ represent the fading parameters associated
with the GG distribution. These parameters vary with the water temperature,
water salinity, and bubble level. Typical parameters of the EGG distribution
are given in Table I and Table II.

From [5, Eqs. (21) and (22)], the PDF and CDF of $\gamma_{2}$ can be written
in terms of Meijer’s G-function. Departing from this representation and
relying on [36, Eq. (07.34.26.0008.01)] and [37, Eq. (1.59)], the unified PDF
and CDF expressions of $\gamma_{2}$ can be formulated in terms of Fox’s
H-function as

$\displaystyle f_{\gamma_{2}}(\gamma_{2})=\frac{\omega}{\gamma_{2}}$
$\displaystyle\,{\mathrm{H}}_{0,1}^{1,0}\left[{{\frac{\gamma_{2}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{-}\\\
{(1,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)}{\gamma_{2}\Gamma(a)}\,{\mathrm{H}}_{0,1}^{1,0}\left[{{\frac{\gamma_{2}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{-}\\\
{(a,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!,$ (9) $\displaystyle
F_{\gamma_{2}}(\gamma_{2})=\omega r$
$\displaystyle\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{2}}{\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{\Gamma(a)c}\,{\mathrm{H}}_{1,2}^{1,1}\left[{{\frac{\gamma_{2}}{b^{r}\mu_{r}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!,$ (10)

where $\mathrm{H}_{\cdot,\cdot}^{\cdot,\cdot}[\cdot]$ is the Fox’s H-function
[37, Eq. (1.2)].

TABLE I: Parameters of the EGG Distribution for Different Bubble Levels BL (L/min) and Temperature Gradient $\Delta\mathrm{T}$ (${}^{\circ}\mathrm{C}.cm^{-1}$) [5] BL (L/min) |  $\Delta\mathrm{T}$ $(^{\circ}\mathrm{C}.cm^{-1})$ |  $\omega$ |  $\lambda$ |  a |  b |  c
---|---|---|---|---|---|---
2.4 |  0.05 |  0.2130 |  0.3291 |  1.4299 |  1.1817 |  17.1984
2.4 |  0.20 |  0.1665 |  0.1207 |  0.1559 |  1.5216 |  22.8754
4.7 |  0.05 |  0.4580 |  0.3449 |  1.0421 |  1.5768 |  35.9424

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

In this section, we derive closed-form expressions for the e2e statistics
(i.e., CDF and PDF of the SNR) of RIS-assisted dual-hop mixed RF-UWOC system
assuming both fixed-gain AF and DF relaying protocols. These expressions are
useful for the performance analysis carried out in the next section.

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

As for a fixed-gain AF relaying protocol, the overall e2e instantaneous SNR at
D can be written as [38]

$\displaystyle\gamma^{AF}=\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C},$ (11)

where $C$ is a constant related to the relay gain.

####  III-A1 Cumulative Distribution Function

The CDF of the e2e SNR can be expressed as

$\displaystyle F_{\gamma^{AF}}(\gamma)$
$\displaystyle=\int_{0}^{\infty}P\left[\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C}<\gamma\Big{|}\gamma_{1}\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}$
$\displaystyle=F_{\gamma_{1}}(\gamma)+\underbrace{\int_{\gamma}^{\infty}F_{\gamma_{2}}\left(\frac{C\gamma}{\gamma_{1}-\gamma}\right)f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}}_{I_{1}},$
(12)

where $F_{\gamma_{1}}(\gamma)$ is the CDF of $\gamma_{1}$. By substituting
(6), (7) and (10) into (12), and after some algebraic manipulations, we obtain

$\displaystyle
F_{\gamma^{AF}}(\gamma)=\frac{1}{\Gamma(m_{w})\Gamma(k_{w})}\,{\rm{G}}_{1,3}^{2,1}\left[{\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\left|{\begin{matrix}{1}\\\
{k_{w},m_{w},0}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{\omega
r\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}$
$\displaystyle\times{\rm{H}_{1,0:3,1:2,1}^{0,1:1,2:0,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (0,r)(0,1)(1,r)\\\ (0,r)\\\ (2{-}k_{w},1)(2{-}m_{w},1)\\\
(1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\widetilde{\Xi}\gamma}\\!\\!}\right]$
$\displaystyle+\frac{(1{-}\omega)r\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)c}$
$\displaystyle\times{\rm{H}_{1,0:3,1:2,1}^{0,1:1,2:0,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(0,1)(1,\frac{r}{c})\\\ (0,\frac{r}{c})\\\
(2{-}k_{w},1)(2{-}m_{w},1)\\\
(1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\widetilde{\Xi}\gamma}\\!\\!}\right],$
(13)

where
$\rm{H}_{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}^{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}[\cdot,\cdot]$
is the Extended Generalized Bivariate Fox’s H-Function (EGBFHF), which is also
known as the Fox’s H-function of two variables [37, Eq. (2.56)].

Proof: See Appendix A.

A peculiar characteristic of this transcendental function is that many special
functions can be formulated in terms of a Fox’s H-function, including the
Extended Generalized Fivariate Meijer’s G-Function (EGBMGF)
$\rm{G}_{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}^{\cdot,\cdot:\cdot,\cdot:\cdot,\cdot}[\cdot,\cdot]$,
defined in [36, Eq. (07.34.21.0081.01)]. It is known that the solution of many
practical problems cannot be formulated in terms of traditional integral
transforms, such as the Fourier, Laplace, and Mellin transforms. However, such
problems can be efficiently characterized by integral transforms with
specified functions as kernels [39, 40, 41]. The EGBFHF, in particular, can be
computed efficiently by, e.g., using the MATLAB implementation available in
[42].

$\displaystyle
P_{out}^{AF,\infty}\approx\frac{\Gamma(|m_{w}-k_{w}|)}{\Gamma(m_{w})\Gamma(k_{w})t}\left(\frac{\widetilde{\Xi}\gamma_{th}}{\overline{\gamma}_{1}}\right)^{t}+\frac{w\Gamma\left(k_{w}-\frac{1}{r}\right)\Gamma\left(m_{w}-\frac{1}{r}\right)}{\Gamma(m_{w})\Gamma(k_{w})}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{1}{r}}$
$\displaystyle+\frac{wr\Gamma\left(1-k_{w}r\right)\Gamma\left(m_{w}-k_{w}\right)\Gamma(k_{w}r)}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+k_{w}r)}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{k_{w}}+\frac{wr\Gamma\left(1-m_{w}r\right)\Gamma\left(k_{w}-m_{w}\right)\Gamma(m_{w}r)}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+m_{w}r)}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{m_{w}}$
$\displaystyle+\frac{(1-w)\Gamma\left(k_{w}-\frac{ac}{r}\right)\Gamma\left(m_{w}-\frac{ac}{r}\right)}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+a)}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{b^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{ac}{r}}+\frac{(1-w)\Gamma\left(a-\frac{k_{w}r}{c}\right)\Gamma\left(m_{w}-k_{w}\right)\Gamma\left(\frac{k_{w}r}{c}\right)r}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)\Gamma\left(1+\frac{k_{w}r}{c}\right)c}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{b^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{k_{w}}$
$\displaystyle+\frac{(1-w)\Gamma\left(a-\frac{m_{w}r}{c}\right)\Gamma\left(k_{w}-m_{w}\right)\Gamma\left(\frac{m_{w}r}{c}\right)r}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)\Gamma\left(1+\frac{m_{w}r}{c}\right)c}\left(\frac{C\widetilde{\Xi}\gamma_{th}}{b^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{m_{w}}.$
(22)

####  III-A2 Probability Density Function

Taking the derivative of (12), the PDF of the e2e SNR can be formulated as

$\displaystyle
f_{\gamma^{AF}}(\gamma)=\frac{d}{d\gamma}\int_{0}^{\infty}P\left[\frac{\gamma_{1}\gamma_{2}}{\gamma_{2}+C}<\gamma\Big{|}\gamma_{1}\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}.$
(14)

Using the method in [43], and after some algebraic manipulations, the final
expression of the PDF is

$\displaystyle f_{\gamma^{AF}}$
$\displaystyle(\gamma)=\frac{\omega\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}$
$\displaystyle\times{\rm{H}_{1,0:2,0:2,1}^{0,1:0,2:0,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (0,r)(1,1)\\\ -\\\ (2{-}k_{w},1)(2{-}m_{w},1)\\\
(2,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\widetilde{\Xi}\gamma}\\!\\!}\right]$
$\displaystyle+\frac{(1{-}\omega)\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)}$
$\displaystyle\times{\rm{H}_{1,0:2,0:2,1}^{0,1:0,2:0,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(1,1)\\\ -\\\ (2{-}k_{w},1)(2{-}m_{w},1)\\\
(2,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}}{\widetilde{\Xi}\gamma}\\!\\!}\right].$
(15)

Proof: See Appendix B.

TABLE II: Parameters of the EGG Distribution for Different Bubble Levels BL (L/min) for Fresh Water and Salty Water [5] Salinity |  BL (L/min) |  $\omega$ |  $\lambda$ |  a |  b |  c
---|---|---|---|---|---|---
Salty Water |  4.7 |  0.2064 |  0.3953 |  0.5307 |  1.2154 |  35.7368
Salty Water |  7.1 |  0.4344 |  0.4747 |  0.3935 |  1.4506 |  77.0245
Salty Water |  16.5 |  0.4951 |  0.1368 |  0.0161 |  3.2033 |  82.1030
Fresh Water |  4.7 |  0.2190 |  0.4603 |  1.2526 |  1.1501 |  41.3258
Fresh Water |  7.1 |  0.3489 |  0.4771 |  0.4319 |  1.4531 |  74.3650
Fresh Water |  16.5 |  0.5117 |  0.1602 |  0.0075 |  2.9963 |  216.8356

###  III-B DF Relaying Protocol

As for the DF relaying protocol, the e2e instantaneous SNR at D is defined as

$\displaystyle\gamma^{DF}=\mathrm{min}\\{\gamma_{1},\gamma_{2}\\}.$ (16)

####  III-B1 Cumulative Distribution Function

The CDF of $\gamma^{DF}$ can be written as [44]

$\displaystyle F_{\gamma^{DF}}(\gamma)$
$\displaystyle=\mathrm{Pr}(\mathrm{min}\\{\gamma_{1},\gamma_{2}\\}<\gamma)$
$\displaystyle=F_{\gamma_{1}}(\gamma)+F_{\gamma_{2}}(\gamma)-F_{\gamma_{1}}(\gamma)F_{\gamma_{2}}(\gamma),$
(17)

where $F_{\gamma_{1}}(\gamma)$ and $F_{\gamma_{2}}(\gamma)$ denote the CDFs of
$\gamma_{1}$ and $\gamma_{2}$, respectively. By substituting (7) and (10) in
(17), the CDF can be formulated in closed-form.

####  III-B2 Probability Density Function

By taking the derivative of (17) with respect to $\gamma$, the following PDF
is obtained

$\displaystyle f_{\gamma^{DF}}(\gamma)=$ $\displaystyle
f_{\gamma_{1}}(\gamma)+f_{\gamma_{2}}(\gamma)-f_{\gamma_{1}}(\gamma)F_{\gamma_{2}}(\gamma)$
$\displaystyle-f_{\gamma_{2}}(\gamma)F_{\gamma_{1}}(\gamma),$ (18)

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

##  IV Performance Analysis

In this section, exact closed-form expressions for the OP, ABER, and ACC are
derived assuming fixed-gain AF and DF relays. Moreover, in order to have a
better understanding of the impact of different system and fading parameters
on the e2e performance, tight asymptotic expressions in the high SNR regime
are derived, based on which the diversity order is computed.

$\displaystyle\overline{P}_{out}^{AF,\infty}\approx$
$\displaystyle\frac{\Gamma(|m_{w}-k_{w}|)\Gamma(p+t)}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})t}\left(\frac{\widetilde{\Xi}}{\overline{\gamma}_{1}q}\right)^{t}+\frac{w\Gamma\left(k_{w}-\frac{1}{r}\right)\Gamma\left(m_{w}-\frac{1}{r}\right)\Gamma\left(p+\frac{1}{r}\right)}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})}\left(\frac{C\widetilde{\Xi}}{\lambda^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{1}{r}}$
$\displaystyle+\frac{wr\Gamma\left(1-k_{w}r\right)\Gamma\left(m_{w}-k_{w}\right)\Gamma(k_{w}r)\Gamma(p+k_{w})}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+k_{w}r)}\left(\frac{C\widetilde{\Xi}}{\lambda^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{k_{w}}$
$\displaystyle+\frac{wr\Gamma\left(1-m_{w}r\right)\Gamma\left(k_{w}-m_{w}\right)\Gamma(m_{w}r)\Gamma(p+m_{w})}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+m_{w}r)}\left(\frac{C\widetilde{\Xi}}{\lambda^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{m_{w}}$
$\displaystyle+\frac{(1-w)\Gamma\left(k_{w}-\frac{ac}{r}\right)\Gamma\left(m_{w}-\frac{ac}{r}\right)\Gamma\left(p+\frac{ac}{r}\right)}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(1+a)}\left(\frac{C\widetilde{\Xi}}{b^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{ac}{r}}$
$\displaystyle+\frac{(1-w)\Gamma\left(a-\frac{k_{w}r}{c}\right)\Gamma\left(m_{w}-k_{w}\right)\Gamma\left(\frac{k_{w}r}{c}\right)\Gamma(p+k_{w})r}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)\Gamma\left(1+\frac{k_{w}r}{c}\right)c}\left(\frac{C\widetilde{\Xi}}{b^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{k_{w}}$
$\displaystyle+\frac{(1-w)\Gamma\left(a-\frac{m_{w}r}{c}\right)\Gamma\left(k_{w}-m_{w}\right)\Gamma\left(\frac{m_{w}r}{c}\right)\Gamma(p+m_{w})r}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)\Gamma\left(1+\frac{m_{w}r}{c}\right)c}\left(\frac{C\widetilde{\Xi}}{b^{r}q\mu_{r}\overline{\gamma}_{1}}\right)^{m_{w}}.$
(28)

###  IV-A Fixed-Gain AF Relaying

####  IV-A1 Outage Probability

The OP, $P_{out}$, is defined as the probability that the instantaneous SNR
$\gamma$ falls below a given threshold $\gamma_{th}$. Therefore, the exact OP
expression of the considered system with a fixed-gain AF relay can be easily
obtained from (13), i.e.,

$\displaystyle
P_{out}^{AF}=\Pr[\gamma<\gamma_{th}]=F_{\gamma^{AF}}(\gamma_{th}).$ (19)

Since (19) is written in terms of EGBFHF, it is hard to get engineering
insights from it. Therefore, to obtain more useful insights, an asymptotic
analysis is carried out by setting
$\overline{\gamma}_{1}=\overline{\gamma}_{2}=\overline{\gamma}\rightarrow\infty$,
which yields

$\displaystyle P_{out}^{AF,\infty}\approx
F_{\gamma_{1}}^{\infty}+I_{1}^{\infty},$ (20)

where $F_{\gamma_{1}}^{\infty}$ is the asymptotic OP of $\gamma_{1}$ and
$I_{1}^{\infty}$ is the asymptotic result of $I_{1}$ given in (12). By using
[36, Eq. (07.34.06.0040.01)], $F_{\gamma_{1}}^{\infty}$ can be formulated as

$\displaystyle
F_{\gamma_{1}}^{\infty}(\gamma_{th})=\frac{\Gamma(|m_{w}-k_{w}|)}{\Gamma(k_{w})\Gamma(m_{w})t}\left(\frac{E_{w}\gamma_{th}}{\overline{\gamma}_{1}}\right)^{t},$
(21)

where $t=\min(m_{w},k_{w})$. From [45, Eq. (1.1)], [46, Th. 1.11] and [46, Eq.
(1.8.4)], the OP of fixed-gain AF relaying, $P_{out}^{AF,\infty}$, in the high
SNR regime is given in (22), shown at the bottom of this page. Note that the
asymptotic expression in (22) is written in terms of elementary functions that
can be computed by using any computer software.

Proof: See Appendix C.

From (22), in addition, we evince that, the diversity order of the considered
system setup is

$\displaystyle
G_{d}^{AF}=\min\left(m_{w},k_{w},\frac{2}{r},\frac{2ac}{r}\right).$ (23)

From (23), we observe that the diversity order is a function of $m_{w}$,
$k_{w}$, $a$, $c$, and $r$. Based on Section II, in addition, $m_{w}$ and
$k_{w}$ depend on $N$ and $m$. Therefore, the diversity order is determined by
$m$, $N$, and by the detection technique (i.e., $r$) employed.

####  IV-A2 Average Bit Error Rate

The ABER of several binary modulation schemes can be expressed as [44, Eq.
(25)]

$\displaystyle\overline{P}_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\exp(-q\gamma)\gamma^{p-1}F_{\gamma}(\gamma)d\gamma,$
(24)

where $p$ and $q$ depend on the modulation scheme. For example, binary phase
shift keying (BPSK) is obtained by setting $p=0.5$ and $q=1$. Then, by
substituting (12) into (24), the ABER of fixed-gain AF relaying can be
formulated as

$\displaystyle\overline{P}_{e}^{AF}{=}\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p{-}1}\exp({-}q\gamma)F_{\gamma_{1}}(\gamma)d\gamma{+}\frac{q^{p}}{2\Gamma(p)}$
$\displaystyle{\times}\int_{0}^{\infty}\gamma^{p{-}1}\exp(-q\gamma)\int_{\gamma}^{\infty}F_{\gamma_{2}}\left(\frac{C\gamma}{\overline{\gamma}_{1}{-}\gamma}\right)f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}d\gamma$
$\displaystyle=\overline{P}_{e,1}+I_{2},$ (25)

where $\overline{P}_{e,1}$ is the ABER of the RF link. By replacing (7) into
(24), expressing $\exp(-q\gamma)$ in terms of Meijer’s G-function [35, Eq.
(8.4.3.1)], and utilizing [36, Eq. (07.34.21.0011.01)], we obtain

$\displaystyle\overline{P}_{e,1}=\frac{1}{2\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})}\,{\mathrm{G}}_{2,3}^{2,2}\left[{{\frac{\widetilde{\Xi}}{q\overline{\gamma}_{1}}}\left|{\begin{matrix}{1,1{-}p}\\\
{k_{w},m_{w},0}\\\ \end{matrix}}\right.}\right]\\!.$ (26)

Similar to (13), $I_{2}$ can be formulated as

$\displaystyle I_{2}=\frac{\omega
r\widetilde{\Xi}}{2\overline{\gamma}_{1}\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})q}$
$\displaystyle\times{\rm{H}_{1,0:3,1:2,2}^{0,1:1,2:1,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (0,r)(0,1)(1,r)\\\ (0,r)\\\ (2{-}k_{w},1)(2{-}m_{w},1)\\\
(1{+}p,1)(1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{q\overline{\gamma}_{1}}{\widetilde{\Xi}}\\!\\!}\right]$
$\displaystyle+\frac{(1{-}\omega)r\widetilde{\Xi}}{2\overline{\gamma}_{1}\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)cq}$
$\displaystyle\times{\rm{H}_{1,0:3,1:2,2}^{0,1:1,2:1,2}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(0,1)(1,\frac{r}{c})\\\ (0,\frac{r}{c})\\\
(2{-}k_{w},1)(2{-}m_{w},1)\\\
(1{+}p,1)(1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{q\overline{\gamma}_{1}}{\widetilde{\Xi}}\\!\\!}\right].$
(27)

Proof: Please, see Appendix D.

By letting
$\overline{\gamma}_{1}=\overline{\gamma}_{2}=\overline{\gamma}\rightarrow\infty$
and employing the same method presented in Appendix C, the asymptotic ABER can
be obtained as formulated in (28), shown at the bottom of this page. Similar
to the asymptotic outage analysis, the diversity order calculated through the
asymptotic ABER coincide, as expected, with (23).

####  IV-A3 Average Channel Capacity

The ACC of the considered system setup operating under both HD and IM/DD can
be formulated as [38, Eq. (25)] [47, Eq. (15)]

$\displaystyle\overline{C}=\frac{1}{2\ln(2)}\int_{0}^{\infty}\ln(1+\tau\gamma)f_{\gamma^{AF}}(\gamma)d\gamma,$
(29)

where $\tau$ is a constant such that $\tau=e/(2\pi)$ holds for the IM/DD
technique (i.e., when $r=2$) and $\tau=1$ holds for the HD technique (i.e.,
when $r=1$) [38, 44]. It is worth noting that the expression in (29) is exact
if $r=1$ while it is a lower-bound if $r=2$ [47]. Using [35, Eq. (8.4.6.5)]
and [36, Eq. (07.34.26.0008.01)], $\ln(1+\tau\gamma)$ can be represented as

$\displaystyle\ln(1+\tau\gamma)=\,{\rm{H}}_{2,2}^{1,2}\left[{{\tau\gamma}\left|{\begin{matrix}{(1,1)(1,1)}\\\
{(1,1)(0,1)}\\\ \end{matrix}}\right.}\right]\\!.$ (30)

By inserting (15) and (30) into (29), and then applying [37, Eqs. (2.8) and
(2.57)], the ACC of fixed-gain AF relaying can be derived in closed-form as

$\displaystyle\overline{C}^{AF}=\frac{\omega\widetilde{\Xi}}{2\ln(2)\tau\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}$
$\displaystyle\times{\rm{H}_{1,0:2,0:3,2}^{0,1:0,2:1,3}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (0,r)(1,1)\\\ -\\\ (2{-}k_{w},1)(2{-}m_{w},1)(2,1)\\\
(2,1)(1,1)\end{matrix}}\right|\\!\frac{\lambda^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}\tau}{\widetilde{\Xi}}\\!\\!}\right]$
$\displaystyle+\frac{(1-\omega)\widetilde{\Xi}}{2\ln(2)\tau\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)}$
$\displaystyle\times{\rm{H}_{1,0:2,0:3,2}^{0,1:0,2:1,3}}\left[{\\!\\!\left.{\begin{matrix}\left({2;-1,1}\right)\\\
-\\\ (1{-}a,\frac{r}{c})(1,1)\\\ -\\\ (2{-}k_{w},1)(2{-}m_{w},1)(2,1)\\\
(2,1)(1,1)\end{matrix}}\right|\\!\frac{b^{r}\mu_{r}}{C},\\!\frac{\overline{\gamma}_{1}\tau}{\widetilde{\Xi}}\\!\\!}\right].$
(31)

When $m=1$, $r=1$, and $c=1$, (31) reduces to a special case where both the
S-RIS and RIS-R links undergo Rayleigh fading, while the UWOC link undergoes
EG fading and employs the HD technique.

###  IV-B DF Relaying

####  IV-B1 Outage Probability

As for the DF relaying, the OP is obtained from (17) as

$\displaystyle P_{out}^{DF}=F_{\gamma^{DF}}(\gamma_{th}).$ (32)

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

$\displaystyle P_{out}^{DF,\infty}$
$\displaystyle=F_{\gamma_{1}}^{\infty}(\gamma_{th})+F_{\gamma_{2}}^{\infty}(\gamma_{th})-F_{\gamma_{1}}^{\infty}(\gamma_{th})F_{\gamma_{2}}^{\infty}(\gamma_{th})$
$\displaystyle\approx
F_{\gamma_{1}}^{\infty}(\gamma_{th})+F_{\gamma_{2}}^{\infty}(\gamma_{th}),$
(33)

where $F_{\gamma_{1}}^{\infty}(\cdot)$ is given in (21). When
$\mu_{r}\rightarrow\infty$, $F_{\gamma_{2}}^{\infty}(\gamma_{th})$ can be
simplified by using [46, Eq. (1.8.4)], which yields

$\displaystyle
F_{\gamma_{2}}^{\infty}(\gamma_{th})=\omega\left(\frac{\gamma_{th}}{\lambda^{r}\mu_{r}}\right)^{\frac{1}{r}}+\frac{(1-\omega)}{\Gamma(a+1)}\left(\frac{\gamma_{th}}{b^{r}\mu_{r}}\right)^{\frac{ac}{r}},$
(34)

Therefore, by using (21) and (34), the diversity order is

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

Similar to the AF relaying, the diversity order of the DF relaying is
determined by $m$, $N$, and by the detection scheme.

####  IV-B2 Average Bit Error Rate

As for the DF relaying, 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 $\overline{P}_{e,1}$ and $\overline{P}_{e,2}$ are the ABER of the RF and
UWOC links, respectively. In particular, $\overline{P}_{e,1}$ is given in
(26). By substituting (10) into (24) and applying the identity
$\exp(-q\gamma){=}\,{\rm{H}}_{0,1}^{1,0}\left[{q\gamma\left|{\begin{matrix}{-}\\\
{(0,1)}\\\ \end{matrix}}\right.}\right]\\!$ [35, Eq. (8.4.3.1)], [36, Eq.
(07.34.26.0008.01)], and then utilizing [35, Eq. (2.25.1.1)], we obtain

$\displaystyle\overline{P}_{e,2}$ $\displaystyle=\frac{\omega
r}{2\Gamma(p)}\,{\rm{H}}_{2,2}^{1,2}\left[{{\frac{1}{\lambda^{r}\mu_{r}q}}\left|{\begin{matrix}{(1,r)(1{-}p,1)}\\\
{(1,r)(0,r)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle+\frac{(1-\omega)r}{2\Gamma(p)\Gamma(a)c}\,{\rm{H}}_{2,2}^{1,2}\left[{{\frac{1}{b^{r}\mu_{r}q}}\left|{\begin{matrix}{(1,\frac{r}{c})(1{-}p,1)}\\\
{(a,\frac{r}{c})(0,\frac{r}{c})}\\\ \end{matrix}}\right.}\right]\\!.$ (37)

Finally, from (26), (36) and (37), a closed-form expression for the ABER can
be derived. It is noteworthy that an efficient MATHEMATICA implementation for
evaluating the Fox’s H-function can be found in [48].

By using [36, Eq. (07.34.06.0040.01)] and [46, Eq. (1.8.4)], an asymptotic
expression for the ABER is given as

$\displaystyle\overline{P}_{e}^{DF,\infty}\approx\overline{P}_{e,1}^{\infty}+\overline{P}_{e,1}^{\infty},$
(38)

where

$\displaystyle\overline{P}_{e,1}^{\infty}=\frac{\Gamma(|m_{w}-k_{w}|)\Gamma(p+t)}{2\Gamma(p)\Gamma(k_{w})\Gamma(m_{w})t}\left(\frac{E_{w}}{\overline{\gamma}_{1}q}\right)^{t},$
(39)

and

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

####  IV-B3 Average Channel Capacity

From (29) and using (18) and (30), 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 [35, Eq. (2.25.1.1)] and after some algebraic operations, we
have

$\displaystyle I_{C1}$
$\displaystyle=\frac{\widetilde{\Xi}}{2\ln(2)\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\tau}$
$\displaystyle\times$
$\displaystyle\,{\rm{H}}_{2,4}^{4,1}\left[{{\frac{\widetilde{\Xi}}{\overline{\gamma}_{1}\tau}}\left|{\begin{matrix}{(-1,1)(0,1)}\\\
{(k_{w}{-}1,1)(m_{w}{-}1,1)(-1,1)(-1,1)}\\\ \end{matrix}}\right.}\right]\\!.$
(42)

Similarly, $I_{C2}$ is 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, from (6) and (10), and utilizing [36, Eq. (07.34.26.0008.01)] and
[45, Eqs. (1.1) and (2.3)], $I_{C3}$ can be formulated as

$\displaystyle I_{C3}{=}\frac{wr}{2\ln(2)\Gamma(m_{w})\Gamma(k_{w})}$
$\displaystyle\times{\rm{H}_{2,0:2,2:1,2}^{0,2:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(1{-}k_{w};1,1)(1{-}m_{w};1,1)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,r)\\\
(1,r)(0,r)\end{matrix}}\right|\\!\frac{\overline{\gamma}_{1}\tau}{\widetilde{\Xi}},\\!\frac{\overline{\gamma}_{1}}{\lambda^{r}\mu_{r}\widetilde{\Xi}}\\!\\!}\right]$
$\displaystyle{+}\frac{(1-w)r}{2\ln(2)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)}$
$\displaystyle\times{\rm{H}_{2,0:2,2:1,2}^{0,2:1,2:1,1}}\left[{\\!\\!\left.{\begin{matrix}(1{-}k_{w};1,1)(1{-}m_{w};1,1)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,\frac{r}{c})\\\
(a,\frac{r}{c})(0,\frac{r}{c})\end{matrix}}\right|\\!\frac{\overline{\gamma}_{1}\tau}{\widetilde{\Xi}},\\!\frac{\overline{\gamma}_{1}}{b^{r}\mu_{r}\widetilde{\Xi}}\\!\\!}\right],$
(44)

while with the help of (7), (9) and (41), $I_{C4}$ is given as follows

$\displaystyle I_{C4}=\frac{w}{2\ln(2)\Gamma(m_{w})\Gamma(k_{w})}$
$\displaystyle\times{\rm{H}_{1,0:2,2:1,3}^{0,1:1,2:2,1}}\left[{\\!\\!\left.{\begin{matrix}(0;r,r)\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,1)\\\
(k_{w},1)(m_{w},1)(0,1)\end{matrix}}\right|\\!\tau\lambda^{r}\mu_{r},\\!\frac{\lambda^{r}\mu_{r}\widetilde{\Xi}}{\overline{\gamma}_{1}}\\!\\!}\right]$
$\displaystyle{+}\frac{(1-w)}{2\ln(2)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)}$
$\displaystyle\times{\rm{H}_{1,0:2,2:1,3}^{0,1:1,2:2,1}}\left[{\\!\\!\left.{\begin{matrix}(1{-}a;\frac{r}{c},\frac{r}{c})\\\
-\\\ (1,1)(1,1)\\\ (1,1)(0,1)\\\ (1,1)\\\
(k_{w},1)(m_{w},1)(0,1)\end{matrix}}\right|\\!\tau
b^{r}\mu_{r},\\!\frac{b^{r}\mu_{r}\widetilde{\Xi}}{\overline{\gamma}_{1}}\\!\\!}\right].$
(45)

By setting $r=1$ and $c=1$, as a special case for the DF relaying, (45) is
simplified to the setup when the UWOC link follows a EG distribution and
employs the HD technology.

##  V Numerical Results and Discussions

In this section, illustrative numerical examples are provided in order to
investigate the impact of key system parameters on the e2e system performance.
Without loss of generality, we set the threshold $\gamma_{th}=2$ dB, the
fading parameter $m=2$, and the fixed relay gain $C=1.5$. The parameters of
the UWOC link are given in Table I and Table II. As for the system without an
RIS, we assume that S is equipped with one antenna and communicates directly
with R over a Rayleigh fading channels. As it will be observed from the
figures, the simulation results match perfectly with the analytical results,
thus confirming the accuracy of our analytical derivation.

Figure 2: OP versus $\overline{\gamma}$ for fixed-gain AF relaying under the
IM/DD technique and by varying $N$. Figure 3: OP versus $\overline{\gamma}$
for fixed-gain AF relaying under both HD and IM/DD technique and by setting
$m=2,3$.

Fig. 2 shows the OP versus $\overline{\gamma}$ of the considered fixed-gain AF
relay-aided system with and without the RIS, where $\overline{\gamma}$ is the
average SNRs of both links, i.e.,
$\overline{\gamma}_{1}=\overline{\gamma}_{2}=\overline{\gamma}$. Moreover, we
assume that the system employs the IM/DD technique and operates under variable
turbulence conditions. Compared to the scheme without the RIS, one can observe
that the system outage performance in the presence of an RIS is improved. In
particular, the performance the system outage performance gets better with
$N$. In addition, in Table I, the higher the level of the air bubbles and/or
the temperature gradient is, the higher the intensity of turbulence is.
Therefore, it can be seen from Fig. 2 that the bubble level and the
temperature gradient affect the performance of the system. As the level of the
air bubbles and temperature gradients increase, the turbulence conditions are
more severe, thereby resulting in performance deterioration. Furthermore,
compared with the setup $\mathrm{BL}=2.4$ L/min,
$\Delta\mathrm{T}=0.05^{\circ}\mathrm{C}.cm^{-1}$, the outage performance
under the other two turbulent conditions shows that the effect of the level of
air bubbles is more severe than the level of the temperature gradient. The
asymptotic results derived in (22) are in very tight agreement with the exact
analysis in the high SNR regime.

Figure 4: OP versus $\overline{\gamma}_{1}$ for both fixed-gain AF relaying
and DF relaying under IM/DD technique. Figure 5: ABER versus
$\overline{\gamma}$ for fixed-gain AF relaying under both HD and IM/DD
techniques.

Fig. 3 depicts the system outage performance versus $\overline{\gamma}$ by
considering both HD and IM/DD techniques and by setting $m=2,3$. This figure
demonstrates that the HD technique provides better outage performance than the
IM/DD technique. This improvement in performance is due to the fact that the
HD technique can better overcome the turbulence effect compared to the IM/DD
technique. When $N$ is small, the system provides better outage performance if
$m=3$. This is because a large value of $m$ indicates weaker fading on the RF
link. When $N$ is large, the parameter $m$ has little effect on the system
performance. This is because with the help of an RIS, the increase of $N$
reduces the OP of the RF link. Therefore, at high SNR, the OP of the overall
system is mainly determined by the UWOC link. Furthermore, we observe that the
curves have the same slopes when the system utilizes the same detection
technology. Therefore, it can be observed that the diversity order depends on
the detection technique of the UWOC link, which is consistent with (23).

In Fig. 4, we plot the OP versus $\overline{\gamma}_{1}$ for both fixed-gain
AF and DF relays and by considering the IM/DD technique. We assume
$\overline{\gamma}_{2}=15$ dB. As expected, the OP of the DF relaying is worse
than the OP of fixed-gain AF relaying system. For large values of
$\overline{\gamma}_{1}$, in particular, the OP of DF relaying is mainly
determined by the UWOC link, and, therefore, it is almost independent of the
RF link. The outage performance improves by using an RIS and the OP decreases
as $\overline{\gamma}_{1}$ and $N$ increase.

Figure 6: ABER versus $\overline{\gamma}$ for DF relaying under various levels
of air bubbles and gradient temperatures. Figure 7: Average channel capacity
versus $\overline{\gamma}_{1}$ for fixed-gain AF relaying and considering
three different levels of air bubbles using both salty and fresh waters in the
case of the IM/DD technique.

Fig. 5 depicts the ABER versus $\overline{\gamma}$ for fixed-gain AF relaying
and for both HD and IM/DD detection techniques. It is assumed that the
turbulence condition corresponds to $\mathrm{BL}=2.4$ L/min,
$\Delta\mathrm{T}=0.05^{\circ}\mathrm{C}.cm^{-1}$. In this setup, the ABER
performance improves as $N$ increases. As expected, the HD technique provides
better error performance than the IM/DD technique. The asymptotic results of
the ABER at high SNR are also shown. The asymptotic results are in a perfect
agreement with the analytical results in the high SNR regime. This justifies
the accuracy and tightness of (28).

In Fig. 6, the ABER versus $\overline{\gamma}_{1}$ is depicted for DF relaying
and the HD technique by considering various levels of air bubbles and gradient
temperatures. The error performance of the proposed system is better than the
system without RISs, as expected. For systems with the same number of
reflecting elements, the error performance is improved under weak turbulence
conditions. In general, the larger the number of reflecting elements is the
lower the ABER is. In addition, under a given type of turbulence, for a fixed
$\overline{\gamma}_{1}$, the ABER tends to be constant when
$\overline{\gamma}_{2}=\infty$. This is because for a fixed
$\overline{\gamma}_{1}$ and a large $\overline{\gamma}_{2}$, the overall
system SNR $\gamma^{DF}$ is determined by $\gamma_{1}$ due to the fact that
$\gamma^{DF}=\min\\{\gamma_{1},\gamma_{2}\\}$. Furthermore, the asymptotic
ABER in (38) is in agreement with (36) in the high SNR regime.

Figure 8: Average channel capacity versus $\overline{\gamma}_{1}$ for both
fixed-gain AF and DF relaying and assuming HD technique.

Fig. 7 shows the ACC versus $\overline{\gamma}_{1}$ for fixed-gain AF relaying
and assuming different levels of air bubbles, for both salty and fresh waters,
and considering the IM/DD technique. We assume $N=5$. We observe that, for a
given type of water, the ACC decreases as the turbulence conditions are more
severe. For instance, when the levels of air bubbles increase from 4.7 L/min
to 16.5 L/min, the ACC decrease to a greater extent. The effect of water
salinity is analyzed in Fig. 7. We note that its impact is less significant
than the impact of air bubbles. Under the assumption of weak turbulence
intensity, e.g., BL$=4.7$ L/min, the impact of water salinity is almost
negligible. In addition, we observe that the ACC of a system without RIS is
always worse than that of a system with RIS.

Assuming the HD technique, Fig. 8 shows the ACC versus $\overline{\gamma}_{1}$
for both fixed-gain AF and DF relaying, and for different values of $N$. We
assume $\overline{\gamma}_{2}=20$ dB. As for DF relaying, the ACC reaches an a
asymptote for large values of $\overline{\gamma}_{1}$, which is similar as the
observations made in Fig. 4. Clearly, this figure also demonstrates the ACC of
the fixed-gain AF relaying is better than the ACC of DF relaying.

##  VI Conclusion

In this work, we investigated the performance of RIS-assisted dual-hop mixed
RF-UWOC systems. Under different detection techniques, exact closed-form
expressions for the OP, ABER, and ACC were formulated in terms of Meijer’s
G-function and Fox’s H-function for both fixed-gain AF and DF relaying
schemes. The impact of RISs on the system performance was investigated and it
was shown that an RIS can effectively improve the overall system performance.
Also, when $N$ or $m$ reaches a large value, the performance of the considered
system setup mainly depends on the UWOC link. Furthermore, we have derived
tight asymptotic results for the obtained performance metrics, which are
written in terms of simple elementary functions. The system diversity order
was determined and it is shown to be determined by the RF link parameters
(including $m$ and $N$), and by the detection technology of the UWOC link.
Furthermore, the impact of temperature gradients, air bubbles, fresh and salt
water under different turbulent conditions were analyzed. Our results
demonstrated that the performance in the presence of lower bubbles level and
temperature gradient is better than that with a higher bubbles level and
temperature gradient. Moreover, it was shown that the water salinity has an
effect on the performance of the mixed RF-UWOC system, but its effect is less
pronounced than that of the air bubbles. If the bubble level is high, the
performance in fresh water is better than in salt water. Compared to a DF
relaying, an RIS-assisted dual-hop mixed RF-UWOC system with a fixed-gain AF
relay attains better performance.

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

From (12), performing the change of variable $x=\gamma_{1}-\gamma$, with the
help of (6) and (10), using the primary definition of Fox’s H-function in [37]
with the integral identity [33, Eq. (3.194.3)], and then utilizing [33, Eq.
(8.384.1)], $I_{1}$ can be written as

$\displaystyle I_{1}{=}\frac{\omega
r\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}\frac{\gamma}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}\Gamma(t{-}s{-}1)$
$\displaystyle{\times}\frac{\Gamma(1{+}rs)\Gamma({-}rs)\Gamma(1{+}s)}{\Gamma(1{-}rs)}\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)}{\Gamma(t)}$
$\displaystyle{\times}\left(\frac{C}{\lambda^{r}\mu_{r}}\right)^{-s}\left(\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\right)^{-t}dsdt$
$\displaystyle{+}\frac{(1{-}\omega)r\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)c}\frac{\gamma}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}\Gamma(t{-}s{-}1)$
$\displaystyle{\times}\frac{\Gamma\left(a{+}\frac{r}{c}s\right)\Gamma({-}\frac{r}{c}s)\Gamma(1{+}s)}{\Gamma(1{-}\frac{r}{c}s)}\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)}{\Gamma(t)}$
$\displaystyle{\times}\left(\frac{C}{b^{r}\mu_{r}}\right)^{-s}\left(\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\right)^{-t}dsdt,$
(46)

where $\ell_{1}$ and $\ell_{2}$ are two integral contours in the complex
domain. Then, based on the definition of the EGBFHF [37, eq. (2.57)], the CDF
can be derived as given in (13).

##  Appendix B PDF of The End-to-End SNR for Fixed-Gain AF Relaying

Based on the transformation in (14) and using the same method as in [43], we
have

$\displaystyle f_{\gamma^{AF}}(\gamma)=$
$\displaystyle\frac{d}{d\gamma}\left[{\int_{0}^{\gamma}\Pr\left[\gamma_{2}(\gamma_{1}-\gamma)<C\gamma\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}}\right.$
$\displaystyle\left.{+\int_{\gamma}^{\infty}\Pr\left[\gamma_{2}(\gamma_{1}-\gamma)<C\gamma\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}}\right],$
(47)

where $f_{\gamma_{1}}(\gamma_{1})$ is the PDF of $\gamma_{1}$. When
$0<x<\gamma$, $\Pr\left[\gamma_{2}(\gamma_{1}-\gamma)<C\gamma\right]=1$.
Hence, (47) can be written as

$\displaystyle f_{\gamma^{AF}}(\gamma)=$
$\displaystyle\frac{d}{d\gamma}\left[{\int_{0}^{\gamma}f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}}\right.$
$\displaystyle\left.{+\int_{\gamma}^{\infty}\Pr\left[\gamma_{2}(\gamma_{1}-\gamma)<C\gamma\right]f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}}\right].$
(48)

After calculating the derivative, we obtain

$\displaystyle
f_{\gamma^{AF}}(\gamma)=f_{\gamma_{1}}(\gamma)-\lim_{\gamma_{1}\rightarrow\gamma^{+}}\Pr\left[\gamma_{2}<\frac{C\gamma}{(\gamma_{1}-\gamma)}\right]f_{\gamma_{1}}(\gamma)$
$\displaystyle+\int_{\gamma}^{\infty}\frac{C\gamma_{1}}{(\gamma_{1}-\gamma)^{2}}f_{\gamma_{2}}\left(\frac{C\gamma}{\gamma_{1}-\gamma}\right)f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}.$
(49)

Since
$\lim_{\gamma_{1}\rightarrow\gamma^{+}}\Pr[\gamma_{2}<C\gamma/(\gamma_{1}-\gamma)]=\Pr[\gamma_{2}<+\infty]=1$,
we have

$\displaystyle
f_{\gamma^{AF}}(\gamma)=\int_{\gamma}^{\infty}\frac{C\gamma_{1}}{(\gamma_{1}-\gamma)^{2}}f_{\gamma_{2}}\left(\frac{C\gamma}{\gamma_{1}-\gamma}\right)f_{\gamma_{1}}(\gamma_{1})d\gamma_{1}.$
(50)

By using (6) and (9), (50) simplifies to

$\displaystyle
f_{\gamma^{AF}}(\gamma){=}\frac{\omega\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}\frac{1}{(2\pi
i)^{2}}\int\limits_{\ell_{1}}\int\limits_{\ell_{2}}\Gamma(t{-}s{-}1)$
$\displaystyle\times\Gamma(1{+}rs)\Gamma(s)\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)}{\Gamma(t{-}1)}$
$\displaystyle\times\left(\frac{C}{\lambda^{r}\mu_{r}}\right)^{-s}\left(\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\right)^{-t}dsdt$
$\displaystyle+\frac{(1-\omega)\widetilde{\Xi}}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)}\frac{1}{(2\pi
i)^{2}}\int\limits_{\ell_{1}}\int\limits_{\ell_{2}}\Gamma(t{-}s{-}1)$
$\displaystyle\times\Gamma\left(a+\frac{r}{c}s\right)\Gamma(s)\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)}{\Gamma(t{-}1)}$
$\displaystyle\times\left(\frac{C}{b^{r}\mu_{r}}\right)^{-s}\left(\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\right)^{-t}dsdt.$
(51)

Using the same method as in Appendix A, (15) is obtained.

##  Appendix C Asymptotic Outage for Fixed-Gain AF Relaying

By using the definition of the EGBFHF in [45, Eq. (1.1)], $I_{1}$ can be
written as

$\displaystyle I_{1}{=}\frac{\omega
r\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})}\frac{1}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}$
$\displaystyle{\times}\,{\mathrm{H}}_{1,3}^{3,0}\left[{{\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}}\left|{\begin{matrix}{(0,1)}\\\
{({-}s{-}1,1)(k_{w}{-}1,1)(m_{w}{-}1,1)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle\times\frac{\Gamma(1{+}rs)\Gamma({-}rs)\Gamma(1{+}s)}{\Gamma(1{-}rs)}\left(\frac{C}{\lambda^{r}\mu_{r}}\right)^{-s}ds$
$\displaystyle{+}\frac{(1{-}\omega)r\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)c}\frac{1}{(2\pi
i)^{2}}{\int\limits_{\ell_{1}}}{\int\limits_{\ell_{2}}}$
$\displaystyle{\times}\,{\mathrm{H}}_{1,3}^{3,0}\left[{{\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}}\left|{\begin{matrix}{(0,1)}\\\
{({-}s{-}1,1)(k_{w}{-}1,1)(m_{w}{-}1,1)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle\times\frac{\Gamma\left(a{+}\frac{r}{c}s\right)\Gamma\left({-}\frac{r}{c}s\right)\Gamma(1{+}s)}{\Gamma\left(1{-}\frac{r}{c}s\right)}\left(\frac{C}{b^{r}\mu_{r}}\right)^{-s}ds$
$\displaystyle{=}\Lambda_{1}+\Lambda_{2}.$ (52)

By letting $\overline{\gamma}_{1}\rightarrow\infty$, the Fox’s H-functions in
(52) can be approximated by using its Taylor expansion in [46, Eq. (1.8.4)] as

$\displaystyle\,{\mathrm{H}}_{1,3}^{3,0}\left[{{\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}}\left|{\begin{matrix}{(0,1)}\\\
{({-}s{-}1,1)(k_{w}{-}1,1)(m_{w}{-}1,1)}\\\ \end{matrix}}\right.}\right]\\!$
$\displaystyle\approx\frac{\Gamma(k_{w}{+}s)\Gamma(m_{w}{+}s)}{\Gamma(1{+}s)}\left(\frac{\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}}\right)^{({-}s{-}1)}.$
(53)

By substituting (53) into (52) and applying [45, Eq. (1.1)], $\Lambda_{1}$ and
$\Lambda_{2}$ can be rewritten as

$\displaystyle\Lambda_{1}{=}\frac{wr}{\Gamma(m_{w})\Gamma(k_{w})}\,{\mathrm{H}}_{1,4}^{3,1}\left[{{\frac{C\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}\lambda^{r}\mu_{r}}}\left|{\begin{matrix}{(1,r)}\\\
{(1,r)(k_{w},1)(m_{w},1)(0,r)}\\\ \end{matrix}}\right.}\right]\\!,$ (54)

and

$\displaystyle\Lambda_{2}$
$\displaystyle{=}\frac{(1{-}w)r}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)c}$
$\displaystyle\times\,{\mathrm{H}}_{1,4}^{3,1}\left[{{\frac{C\widetilde{\Xi}\gamma}{\overline{\gamma}_{1}b^{r}\mu_{r}}}\left|{\begin{matrix}{(1,\frac{r}{c})}\\\
{(a,\frac{r}{c})(k_{w},1)(m_{w},1)(0,\frac{r}{c})}\\\
\end{matrix}}\right.}\right]\\!,$ (55)

By letting $\mu_{r}\rightarrow\infty$ and utilizing [46, Eq. (1.8.4)],
$\Lambda_{1}$ can be further simplified as

$\displaystyle\Lambda_{1}\approx\frac{w\Gamma\left(k_{w}{-}\frac{1}{r}\right)\Gamma\left(m_{w}{-}\frac{1}{r}\right)}{\Gamma(m_{w})\Gamma(k_{w})}\left(\frac{C\widetilde{\Xi}\gamma}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{\frac{1}{r}}$
$\displaystyle+\frac{wr\Gamma\left(1{-}k_{w}r\right)\Gamma\left(m_{w}{-}k_{w}\right)\Gamma(k_{w}r)}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(1{+}k_{w}r)}\left(\frac{C\widetilde{\Xi}\gamma}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{k_{w}}$
$\displaystyle+\frac{wr\Gamma\left(1{-}m_{w}r\right)\Gamma\left(k_{w}{-}m_{w}\right)\Gamma(m_{w}r)}{\Gamma(m_{w})\Gamma(k_{w})\Gamma(1{+}m_{w}r)}\left(\frac{C\widetilde{\Xi}\gamma}{\lambda^{r}\mu_{r}\overline{\gamma}_{1}}\right)^{m_{w}}.$
(56)

Following the same approach as for $\Lambda_{1}$ and applying [45, Eq. (1.1)]
and [46, Eq. (1.8.4)] with the aid of some algebraic manipulations, we get the
asymptotic expression of $\Lambda_{2}$. With the help of (21) and the
asymptotic expressions of $\Lambda_{1}$ and $\Lambda_{2}$, we obtain the
closed-form expression for the OP at high SNR in (22).

##  Appendix D ABER for Fixed-Gain AF Relaying

Based on (46) and utilizing [33, Eq. (3.326.2)], we have

$\displaystyle I_{2}=\frac{\omega
r\widetilde{\Xi}}{2\overline{\gamma}_{1}\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})q}\frac{1}{(2\pi
i)^{2}}\int\limits_{\ell_{1}}\int\limits_{\ell_{2}}\Gamma(t{-}s{-}1)$
$\displaystyle\times\frac{\Gamma(1{+}rs)\Gamma({-}rs)\Gamma(1{+}s)}{\Gamma(1{-}rs)}\left(\frac{C}{\lambda^{r}\mu_{r}}\right)^{-s}$
$\displaystyle\times\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)\Gamma(p{+}1{-}t)}{\Gamma(t)}\left(\frac{\widetilde{\Xi}}{\overline{\gamma}_{1}q}\right)^{-t}dsdt$
$\displaystyle+\frac{(1-\omega)r\widetilde{\Xi}}{2\overline{\gamma}_{1}\Gamma(p)\Gamma(m_{w})\Gamma(k_{w})\Gamma(a)cq}\frac{1}{(2\pi
i)^{2}}\int\limits_{\ell_{1}}\int\limits_{\ell_{2}}\Gamma(t{-}s{-}1)$
$\displaystyle\times\frac{\Gamma\left(a+\frac{r}{c}s\right)\Gamma\left(-\frac{r}{c}s\right)\Gamma(1{+}s)}{\Gamma\left(1-\frac{r}{c}s\right)}\left(\frac{C}{b^{r}\mu_{r}}\right)^{-s}$
$\displaystyle\times\frac{\Gamma(t{+}k_{w}{-}1)\Gamma(t{+}m_{w}{-}1)}{\Gamma(t)}\left(\frac{\widetilde{\Xi}}{\overline{\gamma}_{1}q}\right)^{-t}dsdt.$
(57)

By using [37, Eq. (2.57)], $I_{2}$ is obtained as in (27). Finally, with the
help of (26) and (27), the ABER can be formulated in closed-form in terms of
the EGBHFH.

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