# Performance Analysis of Triple-Hop Optical Wireless Networks with UAV and
Underwater Relaying Systems

XXX, XXX, and XXX P. Agheli, M. J. Emadi, and H. Beyranvand are with the
Department of Electrical Engineering, Amirkabir University of Technology
(Tehran Polytechnic), Tehran, Iran (E-mails: {pouya.agheli, mj.emadi,
beyranvand}@aut.ac.ir).

###### Abstract

Achieving flexible sea and ocean monitoring systems needs reliable
communication links between terrestrial processing units and underwater
sensors. However, incorporating unmanned aerial vehicles (UAVs) and wireless
networks proposes a novel feature to facilitate robust and reliable
connections. Optical wireless communications (OWCs) are considered promising
technologies for upcoming wireless networks over the air and underwater.
Hence, we study the uplink and downlink of a triple-hop network in which
underwater optical communication (UWOC) links connect sensor nodes (SNs) to an
autonomous underwater vehicle (AUV), and the AUV communicates with a UAV via
free space optical (FSO) links under direct and retro-reflection strategies.
Besides, the UAV is connected to a terrestrial access point (AP) through RF
links. Herein, we propose a tracking system letting the AUV precisely track
the UAV. To acquire the network’s outage probability and average bit error
rate (ABER), we firstly model the channels at each hop and then derive the
cumulative distribution functions (CDFs) of signal-to-noise ratios (SNRs),
where the effects of air-to-water (A2W) and water-to-air (W2A) interfaces are
taken into account. Through numerical results, the outage probability and ABER
for the proposed scenarios are analyzed and compared from various
perspectives. Numerical results are verified using Monte-Carlo simulations.

###### Index Terms:

Sea and ocean monitoring, optical wireless communication, underwater optical
communication link, free space optical link, outage probability, average bit
error rate, air-to-water and water-to-air interface effects.

##  I Introduction

Sea and ocean monitoring systems enable biological observations, safe
navigation, the study of subaquatic animals and plants, and oil spills’
positioning. The aim is to reliably and fast deliver the gathered data via
underwater sensor nodes (SNs) to the terrestrial units in the uplink and the
command data from that units to the SNs in the downlink with the lowest
consumed power. The underwater optical communication (UWOC) is a dependable
technology employed for collecting data of the SNs distributed at the bottom
of the sea, while the radio frequency (RF) and acoustic carriers suffer from
high latency, low data rates, and band limitation [christopoulou2019outage,
kaushal2016underwater, amer2019underwater]. However, taking advantage of
under- and above-water relays supports the line-of-sight (LOS) requirement for
optical links, enables long-distance communications, tackles the high
absorption and scattering of optical signals at the water, and minimizes the
transmission power at the SNs [christopoulou2019outage] and
[vavoulas2014underwater]. In this case, adverse setups and system models have
been proposed to obtain reliable wireless and optical connections between the
SNs and the terrestrial units [christopoulou2019outage, amer2019underwater],
and [naik2020co, anees2019performance, lei2020performance, aneeshybrid,
li2019wdm, jurado2019converging]. Specifically, [amer2019underwater] and
[anees2019performance, naik2020co, lei2020performance] suggest the dual-hop
networks in which RF links are exploited to connect the buoyant relays to the
terrestrial access points (APs). On the other hand, [christopoulou2019outage]
and [aneeshybrid, li2019wdm, jurado2019converging] use optical links to
provide robust and low latency communications between the relays and the APs
based on free space optical (FSO) technology.

The FSO technology offers high data rates with much lower deployment cost,
rapid setup time, easy upgrade, flexibility, freedom form spectrum license
regulations, protocol transparency, and enhanced security
[abdalla2020optical]. However, it comes at the expense of some drawbacks such
as pointing error, the requirement of a LOS connection between nodes, and
sensitivity to atmosphere conditions including rain, snow, fog, and dust
[hassan2017statistical, khalighi2014survey, kaushal2016optical]. To solve the
outage problem of the FSO links in adverse atmospheric conditions, the
combined RF and FSO solution is employed, namely hybrid RF/FSO and relay-based
cooperation [douik2016hybrid, touati2016effects, chen2016multiuser,
usman2014practical, zhang2009soft, jamali2016link, najafi2017optimal].
Furthermore, in [hassan2017statistical], buffer-aided RF/FSO links have been
utilized to enhance the network performance in adverse weather conditions with
the disadvantage of delay. Hassan et. al. [hassan2019joint] have proposed the
optimal power allocations and FSO link selection methods in mm-Wave networks.
The performance of the wireless network based on FSO links has been
investigated in [morra2017impact] and [morra2018mixed], where the impacts of
wireless co-channel interference and FSO pointing error have been taken into
account. The FSO technology is applicable for the indoor and outdoor networks;
lats-mile access, metropolitan fronthaul and backhaul, inter-satellite, inter-
vehicle, point-to-point, and point-to-multipoint links [nabavi2019empirical].

Besides, low cost and highly mobile unmanned aerial vehicles (UAVs) have been
used for not only military and civil applications, including disaster
management and environmental monitoring, but also satellite and cellular
communication networks [ye2018secure, andre2014application, hu2018unmanned].
Thanks to their structures, UAVs enable fast deployment, flexible
reconfiguration, and LOS connections without complex infrastructure
requirements [zeng2016wireless] and [mozaffari2016unmanned]. In terms of
different applications, UAV communication systems must guarantee safe and
dependable connections. [ye2018secure] investigates the secrecy performance of
a UAV-to-UAV system in the presence of eavesdropping UAVs and derives the
expressions for secrecy outage probability and the average secrecy capacity.
On the other hand, UAVs serve as remote sensing and relaying systems gathering
data from multiple sensors via ground-to-air links and, in return, delivering
command data over air-to-ground links [andre2014application]. In this case,
[lei2020performance] analyzes dual-hop RF–UWOC communication systems assisted
by UAVs, buoyant relays, and submarines in the account of ecological effects
on the network’s performance.

For optical wireless and FSO use cases, pointing error is a barrier to have
highly reliable communications in harsh environments or with mobile FSO
transceivers like UAVs [kaymak2018survey]. However, various pointing,
acquisition, and tracking mechanisms have been proposed maintaining stable LOS
connections for FSO mobile applications; gimbal-based, mirror-based, gimbal-
mirror hybrid, adaptive optics, liquid crystal, and RF/FSO hybrid systems
[kaymak2018survey]. Specifically, one promising solution is to take advantage
of a modulating retro-reflection (MRR) system exploited for widespread
applications, such as satellite, UAV, marine, and submarine communication
networks [yang2018performance, yang2017channel, li2017probability,
kaymak2018survey]. In general, the MRR is assembled by an optical modulator
and a passive retro-reflector of which corner cube reflector (CCR) and cat’s
eye reflector (CER) are two frequently-used types [kaymak2018survey].
According to [yang2018performance] and [yang2017channel], the single- or
double-path MRR FSO fading channels are modeled using log-normal and Gamma-
Gamma distributions under weak and strong turbulence conditions, respectively.
Also, [li2017probability] studies the impacts of atmospheric and distance
parameters on the performance of MRR FSO links.

To the best of our knowledge, a triple-hop optical wireless network with UAV
and underwater relaying systems gathering (delivering) data from (to) multiple
subaquatic sensors in the presence of air-water interface effects have not
been investigated in the literature. All previous studies have analyzed the
two-hop networks with buoyant relays that cannot reliably communicate with
deeply located sensors at the bottom of the ocean due to high absorption and
scattering of optical signals at the water. In this paper, we derive the
uplink and downlink outage probability and average bit error rate (ABER) of
the network wherein an autonomous underwater vehicle (AUV) relay collects data
from the SNs via UWOC links, and the AUV is connected to the UAV with FSO
links under direct strategy (DS) and retro-reflection strategy (RRS) taking
into account the air-to-water (A2W) and water-to-air (W2A) interface effects
[nabavi2019empirical] and [nabavi2019performance]. Besides, the UAV
communicates with the terrestrial AP and processing unit through RF links. The
main contributions of the paper are summarized as follows.

  * •

For FSO links, two different transmission strategies are assumed in both
uplink and downlink transmissions, i.e. the DS and the RRS exerting the RRC
technology. In this case, the single- and double-path FSO channels are modeled
based on Birnbaum-Saunders distribution considering the path-loss effect and
the W2A/A2W impacts caused by erratic random and non-random waves in the air-
water interface.

  * •

A gimbal-assisted acquisition-and-tracking algorithm is suggested to minimize
the pointing error by enabling the AUV to precisely acquire and track the UAV
under coarse and fine tracking mechanisms.

  * •

Closed-form uplink and downlink outage probability and ABER expressions are
derived for the proposed scenarios via obtaining the channels’ statistical
properties and signal-to-noise ratios (SNRs) at each hop. Herein, we assume
four different binary modulation techniques computing the end-to-end ABER.

  * •

The uplink and downlink outage probability and ABER of the network under the
DS and RRS are compared from various perspectives. The effects of different
weather conditions and the distance of UWOC and FSO links on the network’s
performance are studied. Besides, the RRC _perfect_ alignment and
_misalignment_ conditions influencing the FSO links’ and network’s
performances are investigated. Finally, the functionality of different binary
modulations are analyzed from the ABER viewpoint. The numerical results are
validated using Monte-Carlo simulations.

Organization: Section II introduces the triple-hop optical wireless networks
with UAV and underwater relaying systems. The proposed tracking system and
performance analysis are represented in Section III. Besides, Section IV
depicts numerical results and discussions. Finally, the paper is concluded in
Section V.

Notation: $\bm{x}\\!\in\\!\mathbb{C}^{n\times 1}$ denotes a vector in an
$n$-dimensional complex space, $\mathbb{E}\\{\cdot\\}$ is the expectation
operator, $[\,\cdot\,]^{T}$ stands for the transpose, and $(\,\cdot\,)^{-1}$
depicts the inverse operator. Also, $y\\!\sim\\!\mathcal{CN}(m,\sigma^{2})$
represents real-valued and complex symmetric Gaussian random variables (RVs)
with mean $m$ and variance $\sigma^{2}$.

##  II System Model

[scale=0.8]Media/SystemModelB.eps

Figure 1: Triple-hop optical wireless network with UAV and underwater relaying
system.

We assume a triple-hop uplink and downlink optical wireless network wherein
$K$ SNs communicate with an AUV relay node via their UWOC links under
wavelength division multiplexing (WDM) technology, c.f. Fig. 1. Also, the AUV
is connected to a UAV sharing node through its FSO link. In this case, both
relay and sharing nodes exploit _decode_ -and-_forward_ (DF) protocol, and to
provide reliable FSO connections, two full-duplex (FDx) transmission schemes
are considered; direct strategy and retro-reflection strategy. Next, it is
assumed that the UAV and a terrestrial AP communicate through a RF link based
on time division duplexing (TDD) method. In this paper, we propose six uplink
and downlink scenarios for transferring data from the SNs to the terrestrial
AP and vice versa, as summarized in Table I; SUD, SUR, SDD, SDR, UDD, and UDR.
The uplink transmission is provided to transfer sensor data to a terrestrial
unit, yet the downlink one delivers command data from that unit to the SNs. A
data stream is transmitted through three time slots in each scenario, every
slot with a length of $\tau$ per hop. Fig. 2 represents the time series of the
$L$-stream double-path data transmission process. In the present case, and for
the SN–AUV and UAV–AP links, each time interval is divide into the uplink
transmission (UT) and downlink transmission (DT) sub-slots with the lengths of
$\tau_{u}$ and $\tau_{d}=\tau-\tau_{u}$, respectively. After establishing the
connections among all nodes, the UAV broadcasts header data (HD) to
synchronize the connected nodes and alert them to transmit their data
immediately after the start slot (SS) and $\tau_{\text{SS}}$ duration.
Finally, after transferring $L$ data streams, the UAV abolishes the
connections via broadcasting trailer data (TD) in the elimination slot (ES)
with a length of $\tau_{\text{ES}}$.

Before investigating outage probability and bit error rates analyses, we delve
into channel modelings and transmission strategies in the following
subsections.

TABLE I: Data transmission scenarios. Scenario | Data communication | AUV–UAV strategy
---|---|---
SUD | Single-path uplink | Direct
SUR | Single-path uplink | Retro-reflection
SDD | Single-path downlink | Direct
SDR | Single-path downlink | Retro-reflection
UDD | Double-path | Direct
UDR | Double-path | Retro-reflection

[scale=0.65]Media/TimeSeries.eps

Figure 2: The time series of the double-path data transmission process.

###  II-A UWOC Channel Model

The UWOC link of the $k$th SN is given by

$h_{k}=h_{l,k}h_{t,k}h_{p,k},~{}\text{for}~{}k=1,2,...,K,$ (1)

which is incorporated of oceanic path-loss ($h_{l,k}$), turbulence
($h_{t,k}$), and pointing error ($h_{p,k}$).

####  II-A1 Oceanic Path-Loss

Under line-of-sight (LOS) conditions, $h_{l,k}$ is modeled by
[christopoulou2019outage]

$h_{l,k}=\frac{A_{ea}\cos(\theta_{k})}{2\pi
d_{a,k}^{2}\left(1-\cos(\theta_{0})\right)}~{}\operatorname{exp}\\!\left(\\!-\frac{c(\lambda_{k})d_{a,k}}{\cos(\theta_{k})}\\!\right)\\!\\!,$
(2)

where $A_{ea}$ depicts aperture area, $\theta_{0}$ is source beam divergence
angle, and $\theta_{k}$ represents the angle between the perpendicular to the
relay and the sensor-relay course. Also, $d_{a,k}$, $c(\lambda_{k})$, and
$\lambda_{k}$ represent perpendicular distance between $k$th SN and the AUV,
extinction coefficient, and transmission wavelength for $k$th sensor,
respectively.

####  II-A2 Oceanic Turbulence

Under weak turbulence conditions, $h_{t,k}$ pursues a log-normal distribution
with probability density function (pdf) given by [christopoulou2019outage]

$f_{h_{t,k}}(h_{t})=\frac{1}{2h_{t}\sqrt{2\pi\sigma_{x_{k}}^{2}}}~{}\operatorname{exp}\\!\left(\\!-\frac{\left(\ln(h_{t})-2\mu_{x_{k}}\right)^{2}}{8\sigma_{x_{k}}^{2}}\right)\\!\\!,$
(3)

where $\mu_{x_{k}}$ and $\sigma_{x_{k}}^{2}$ present mean and variance of
random variable $X_{k}\\!=\\!\frac{1}{2}\ln(h_{t,k})$, respectively. If we
assume that turbulence-induced fading does not amplify or attenuate the
average power, the fading is normalized such that
$\mathbb{E}\\{|h_{t,k}|\\}\\!=\\!1$, which indicates the choice of
$\mu_{x_{k}}\\!=\\!-\sigma_{x_{k}}^{2}$. Besides, the log-amplitude variance
for plane waves is given by
$\sigma_{x_{k}}^{2}\\!=\\!0.307C_{n}^{2}k_{k}^{7/6}d_{a,k}^{11/6}$, while
$C_{n}^{2}$ represents the index of refraction structure, and
$k_{k}={2\pi}/\lambda_{k}$ is the number of optical waves.

####  II-A3 Oceanic Pointing Error

The pdf of $h_{p,k}$ for a circular detection mechanism is modeled as follows
[sandalidis2008ber, naik2020co]

$f_{h_{p,k}}(h_{p})=\frac{\xi^{2}}{h_{0}^{\xi^{2}}}h_{p}^{\xi^{2}-1};~{}~{}0\leq
h_{p}\leq h_{0},$ (4)

where $h_{0}\\!=\\!\left[\operatorname{erf}(\nu)\right]^{2}$, and
$\xi\\!=\\!\frac{1}{2}w_{z_{eq}}\sigma_{s}^{-1}$ implies the ratio between the
equivalent beam radius and the pointing error displacement standard deviation
at the receiver. The parameters
$w_{z_{eq}}^{2}\\!=\\!w_{z}^{2}\sqrt{0.25\pi}\,\operatorname{erf}(\nu)\,\nu^{-1}\operatorname{exp}(\nu^{2})$
and $\nu\\!=\\!\sqrt{0.5\pi}\,w_{z}^{-1}r_{s}$, where $w_{z}$ is beam waist at
distance $z$, and $r_{s}$ depicts radius of the detector.

####  II-A4 UWOC Overall Statistics

To obtain the pdf of $k$th link’s channel coefficient, i.e. $h_{k}$, we have
[farid2007outage]

$\displaystyle f_{h_{k}}(h)\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\int
f_{h_{k}}\\!\left(h|h_{t}\right)f_{h_{t,k}}(h_{t})\,dh_{t}$ (5a)
$\displaystyle=$
$\displaystyle\\!\\!\int\frac{1}{h_{l,k}h_{t}}f_{h_{p,k}}\\!\\!\left(\\!\frac{h}{h_{l,k}h_{t}}\\!\right)f_{h_{t,k}}(h_{t})\,dh_{t}$
(5b) $\displaystyle=$
$\displaystyle\\!\\!\frac{\xi^{2}h^{\xi^{2}-1}}{(h_{0}h_{l,k})^{\xi^{2}}}\int_{h/{h_{0}h_{l,k}}}^{\infty}h_{t}^{-\xi^{2}}f_{h_{t,k}}(h_{t})\,dh_{t}.$
(5c)

By importing (3) into (5), it can be shown that

$f_{h_{k}}(h)=\frac{\xi^{2}h^{\xi^{2}-1}}{2(h_{0}h_{l,k})^{\xi^{2}}}\,\operatorname{erfc}\\!\left(\\!\frac{\ln\\!\left(\\!{\frac{h}{h_{0}h_{l,k}}}\\!\right)+\varphi_{k}}{\sqrt{8\sigma_{x_{k}}^{2}}}\\!\right)\\!\varphi_{k}^{\prime},$
(6)

where $\varphi_{k}=2\sigma_{x_{k}}^{2}\left(1+2\xi^{2}\right)$,
$\varphi_{k}^{\prime}=2\sigma_{x_{k}}^{2}\xi^{2}\left(1+\xi^{2}\right)$, and
$\operatorname{erfc}(x)=1-\operatorname{erf}(x)$ signifies complementary error
function. Now, we model the received signal from the $k$th UWOC link as
follows

$r_{k}=\eta_{k}h_{k}s_{k}+n_{k},$ (7)

where $\eta_{k}$ depicts optical-to-electrical parameter, $s_{k}\in[0,1]$ is
the transmitted symbol, and $n_{k}\\!\sim\\!\mathcal{N}(0,\delta_{k}^{2})$
indicates additive noise. Therefore, the SNR for $k$th UWOC link is derived as

$\gamma_{k}=\frac{\left(\eta_{k}h_{k}s_{k}\right)^{2}}{\delta_{k}^{2}}=\bar{\gamma}_{k}h_{k}^{2},$
(8)

where $\bar{\gamma}_{k}={\eta_{k}^{2}s_{k}^{2}}/{\delta_{k}^{2}}$ represents
average SNR. By using (6), the pdf of $\gamma_{k}$ can be derived as

$f_{\gamma_{k}}(\gamma_{k})=\frac{\xi^{2}\left(\gamma_{k}/\bar{\gamma}_{k}\right)^{\frac{\xi^{2}}{2}-1}}{4(h_{0}h_{l,k})^{\xi^{2}}}\,\operatorname{erfc}\\!\left(\\!\frac{\ln\\!\left(\\!{\frac{\sqrt{\gamma_{k}}}{h_{0}h_{l,k}\sqrt{\bar{\gamma}_{k}}}}\\!\right)+\varphi_{k}}{\sqrt{8\sigma_{x_{k}}^{2}}}\\!\right)\\!\varphi_{k}^{\prime}.$
(9)

######  Proposition 1.

The cumulative distribution function (CDF) of $\gamma_{k}$ is obtained by

$\displaystyle F_{\gamma_{k}}(\gamma_{k})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\bar{\gamma}_{k}\varphi_{k}^{\prime}}{2}\Bigg{[}\operatorname{erfc}\\!\left(\frac{\Theta_{k}(\gamma_{k})}{\sqrt{8\sigma_{x_{k}}^{2}}}\right)\\!\operatorname{exp}\\!\Big{(}{{\Theta_{k}(\gamma_{k})\xi^{2}}}\Big{)}$
(10)
$\displaystyle+\,\operatorname{erfc}\\!\left(\frac{4\sigma_{x_{k}}^{2}\xi^{2}-\Theta_{k}(\gamma_{k})}{\sqrt{8\sigma_{x_{k}}^{2}}}\right)\\!\operatorname{exp}\\!\Big{(}2\sigma_{x_{k}}^{2}\xi^{4}\Big{)}\Bigg{]}\\!\operatorname{exp}\\!\Big{(}\\!\\!-\varphi_{k}\xi^{2}\Big{)},$

where

$\Theta_{k}(\gamma_{k})=\ln\\!\left(\\!{\frac{\sqrt{\gamma_{k}}}{h_{0}h_{l,k}\sqrt{\bar{\gamma}_{k}}}}\\!\right)+\varphi_{k}.$

###### Proof.

The proof is given in Appendix A. ∎

###  II-B FSO Channel Model

The FSO links connecting the AUV and UAV experience air-to-water (A2W) and
water-to-air (W2A) impacts caused by erratic random and non-random waves in
the air-water interface for uplink and downlink transmissions, respectively.
In this case, they reflect and scatter transmitted optical wireless signals
and depict an additive loss component that can exceed absorption loss.
However, we have [nabavi2019empirical, nabavi2019performance]

$I_{j}=I_{l,j}I_{t,j}I_{p,j},$ (11)

where $j\\!=\\!1$ and $j\\!=\\!2$ stand for uplink and downlink transmissions,
respectively. Also, $I_{l,j}$, $I_{t,j}$, and $I_{p,j}$ present atmospheric
path-loss, turbulence, and pointing error.

####  II-B1 Atmospheric Path-Loss

Under line-of-sight (LOS) conditions, $I_{l,j}$ is modeled by
[christopoulou2019outage]

$I_{l,j}=\frac{A_{j}}{\left(\psi_{j}d_{au}\right)^{2}}~{}\operatorname{exp}\\!\left(-\alpha_{au}d_{au}\right)\\!,$
(12)

where $A_{j}$ and $\psi_{j}$ depict aperture area and source beam divergence
angle, respectively. Also, $d_{au}$ represents average link length between the
AUV and UAV, and $\alpha_{au}$ is the attenuation coefficient depending on
atmospheric conditions.

####  II-B2 Atmospheric Turbulence

By taking into account the A2W and W2A effects, $I_{t,j}$ follows a Birnbaum-
Saunders distribution with the pdf of [nabavi2019empirical,
nabavi2019performance]

$f_{I_{t,j}}(I_{t};\alpha,\beta)=\frac{1}{2\sqrt{2\pi}\alpha\beta}\left[\left(\frac{\beta}{I_{t}}\right)^{1/2}\\!+\left(\frac{\beta}{I_{t}}\right)^{3/2}\right]\\!\times\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{I_{t}}{\beta}+\frac{\beta}{I_{t}}-2\right)\right]\\!,$
(13)

where $\alpha\\!>\\!0$ represents shape parameter, and $\beta\\!>\\!0$ depicts
scale parameter.

####  II-B3 Atmospheric Pointing Error

If we consider the circular detection mechanism, the pdf of $I_{p,j}$ is given
by [sandalidis2008ber, naik2020co]

$f_{I_{p,j}}(I_{p})=\frac{\zeta^{2}}{I_{0}^{\zeta^{2}}}I_{p}^{\zeta^{2}-1};~{}~{}0\leq
I_{p}\leq I_{0},$ (14)

where $I_{0}\\!=\\!\left[\operatorname{erf}(\vartheta)\right]^{2}$, and
$\zeta$ represents the ratio between the equivalent beam radius ($w_{x_{eq}}$)
and the pointing error displacement standard deviation ($\sigma_{p}$) at the
receiver. Also, $\vartheta\\!=\\!\sqrt{0.5\pi}\,w_{x}^{-1}r_{p}$, where
$w_{x}$ is beam waist at distance $x$, and $r_{p}$ depicts radius of the
detector.

####  II-B4 FSO Overall Statistics for the DS

With the DS, data are transmitted through two standalone links between the UAV
and AUV in uplink and downlink. Thus, the statistical properties of both
uplink and downlink channels are studied with the similar literature, as in
the following.

######  Proposition 2.

The pdf of the FSO link for the DS is derived as

$f_{I_{j}}(I;\alpha,\beta)=\frac{e^{1/{\alpha^{2}}}}{2\sqrt{\pi}}\frac{\zeta^{2}I^{\zeta^{2}-1}}{\gamma_{0,j}^{\zeta^{2}}}\Bigg{[}\Gamma\\!\left(\frac{1}{2}-\zeta^{2},\frac{I}{\gamma_{0,j}}\right)+\frac{1}{2\alpha^{2}}\,\Gamma\\!\left(\\!-\frac{1}{2}-\zeta^{2},\frac{I}{\gamma_{0,j}}\right)\\!\Bigg{]},$
(15)

where $\gamma_{0,j}=2\alpha^{2}\beta I_{0}I_{l,j}$, and
$\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}dt$ depicts the upper incomplete
Gamma function.

###### Proof.

The proof is given in Appendix B. ∎

The received FSO signal is modeled by

$\hat{r}_{j}=\lambda_{j}I_{j}\hat{s}_{j}+w_{j},$ (16)

where $j\\!=\\!1$ and $j\\!=\\!2$ represent uplink and downlink transmissions,
respectively. Furthermore,
$\hat{s}_{j}\\!=\\!\left[\hat{s}_{j,1},\hat{s}_{j,2},...,\hat{s}_{j,K}\right]^{T}$
indicates the compressed signal, wherein symbol $\hat{s}_{j,k}$ is the
estimated symbol of the $k$th SN, and
$\mathbb{E}\\!\left\\{\hat{s}_{j,k}\hat{s}_{j,k^{\prime}}\right\\}\\!=\\!0$
for $k\\!\neq\\!k^{\prime}$. Also, $\lambda_{j}$ signifies optical-to-
electrical parameter, and $w_{j}\\!\sim\\!\mathcal{N}(0,\Phi_{j}^{2})$ depicts
the independent and identically distributed (i.i.d.) additive noise with
$\Phi_{j}^{2}=\mathbb{E}\\!\left\\{w_{j}w_{j}^{H}\right\\}$. However, the SNR
diagonal matrix is given by

$\hat{\bm{\gamma}}_{j}=\Big{(}{\Phi_{j}^{2}}\Big{)}^{-1}\lambda_{j}^{2}I_{j}^{2}\hat{s}_{j}\hat{s}_{j}^{H}=\tilde{\bm{\gamma}}_{j}I_{j}^{2},$
(17)

where
$\mathbb{E}\\!\left\\{\hat{s}_{j}w_{j}^{H}\right\\}\\!=\\!\mathbb{E}\\!\left\\{\hat{s}_{j}^{H}w_{j}\right\\}\\!=\\!0$,
and
$\tilde{\bm{\gamma}}_{j}\\!=\\!\big{(}{\Phi_{j}^{2}}\big{)}^{-1}\lambda_{j}^{2}\hat{s}_{j}\hat{s}_{j}^{H}$
represents the average SNR diagonal matrix. Thus, by use of (15) and (17), we
have

$\displaystyle
f_{\hat{\gamma}_{j,k}}(\hat{\gamma}_{k};\alpha,\beta)=\frac{e^{1/{\alpha^{2}}}}{4\sqrt{\pi}}\frac{\zeta^{2}(\hat{\gamma}_{k}/\tilde{\gamma}_{k})^{\frac{\zeta^{2}}{2}-1}}{\gamma_{0,j}^{\zeta^{2}}}\Bigg{[}\Gamma\\!\left(\frac{1}{2}-\zeta^{2},\frac{\sqrt{\hat{\gamma}_{k}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)+\frac{1}{2\alpha^{2}}\,\Gamma\\!\left(\\!-\frac{1}{2}-\zeta^{2},\frac{\sqrt{\hat{\gamma}_{k}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)\\!\Bigg{]}\\!.\,$
(18)

######  Proposition 3.

The CDF of $\hat{\gamma}_{j,k}$ for the DS is obtained by

$\displaystyle F_{\hat{\gamma}_{j,k}}(\hat{\gamma}_{k};\alpha,\beta)=$
$\displaystyle\frac{2e^{1/{\alpha^{2}}}}{\sqrt{\pi}}\frac{\zeta^{2}\tilde{\gamma}_{k}^{1-\frac{\zeta^{2}}{2}}}{\gamma_{0,j}^{\zeta^{2}}(\zeta^{2}+6)}\Bigg{[}\Bigg{(}\Gamma\\!\left(\frac{1}{2}-\zeta^{2},\frac{\sqrt{{\hat{\gamma}_{k}}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)+\frac{1}{2\alpha^{2}}\Gamma\\!\left(\\!-\frac{1}{2}-\zeta^{2},\frac{\sqrt{{\hat{\gamma}_{k}}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)\\!\Bigg{)}\hat{\gamma}_{k}^{\frac{\zeta^{2}+6}{8}}$
$\displaystyle~{}~{}~{}+\Bigg{(}\\!\operatorname{\gamma}\\!\left(2-\frac{3}{4}\zeta^{2},\frac{\sqrt{{\hat{\gamma}_{k}}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)+\frac{1}{2\alpha^{2}}\operatorname{\gamma}\\!\left(1-\frac{3}{4}\zeta^{2},\frac{\sqrt{{\hat{\gamma}_{k}}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)\\!\Bigg{)}\Big{(}\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}\Big{)}^{\frac{\zeta^{2}+6}{4}}\Bigg{]}\\!,\,$
(19)

where $\operatorname{\gamma}(s,x)=\int_{0}^{x}t^{s-1}e^{-t}dt$ depicts the
lower incomplete Gamma function.

###### Proof.

The proof is given in Appendix C. ∎

####  II-B5 FSO Overall Statistics for the RRS

With the RRS, we apply the same technique as (18) for uplink transmission.
Nevertheless for downlink transmission, the received uplink signal at the UAV
is reflected via a modulating retro-reflector (MRR) terminal after being
encoded by the passive corner cube retro-reflector (CCR) of the MRR. Fig. 3
represents the RRS’ diagram under the FDx operation, where its details and
tracking system will be represented in the next section. Thanks to the
structure of the RRS, the pointing error is negligible and not taken into
account for this strategy.

[scale=0.55]Media/FSORetroReflector.eps

Figure 3: The RRS’ diagram with the tracking system and under the FDx
operation.

For the uplink transmission of the RRS, we apply the same signal model in (16)
considering $j\\!=\\!1$. However for downlink, we have 111In general, the
equivalent channel model for RRS is obtained by $I\\!=\\!I_{1}I_{2}$ with
correlation coefficient $\rho$. However, we assume that uplink and downlink
channels are entirely correlated, i.e. $\rho\\!=\\!1$, so we have
$I\\!=\\!I_{1}^{2}\\!=\\!I_{2}^{2}$.

$\hat{r}_{2}=R\lambda_{2}I_{2}^{2}\bm{x}_{m}\hat{s}_{1}+w_{2},$ (20)

where
$\bm{x}_{m}\\!=\\!\begin{bmatrix}x_{m,1}&0&\raisebox{3.0pt}{$\scalebox{0.7}{$\cdots$}$}&0\\\
0&x_{m,2}&\raisebox{3.0pt}{$\scalebox{0.7}{$\cdots$}$}&0\\\
\raisebox{3.0pt}{$\scalebox{0.75}{\vdots}$}&\raisebox{3.0pt}{$\scalebox{0.75}{\vdots}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\ddots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{\vdots}$}\\\
0&0&\raisebox{3.0pt}{$\scalebox{0.7}{$\cdots$}$}&x_{m,K}\end{bmatrix}$ is the
i.i.d. modulated signal matrix at MRR with $x_{m,k}\\!\in\\![0,1]$ for $k$th
SN, under on-off keying (OOK) modulation. Also, $R$ depicts the reflection
effect of the CCR. By using (11)–(13), and after some mathematical
manipulations, we acquire

$\displaystyle f_{I_{1}}(I;\alpha,\beta)=\frac{1}{2\sqrt{2\pi}\alpha\beta
I_{l,1}}\left[\left(\frac{\beta
I_{l,1}}{{I}}\right)^{1/2}\\!+\left(\frac{\beta
I_{l,1}}{{I}}\right)^{3/2}\right]\\!\times\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{{I}}{\beta
I_{l,1}}+\frac{\beta I_{l,1}}{{I}}-2\right)\right]\\!,~{}$ (21)

for uplink, and

$\displaystyle
f_{I_{2}}(I;\alpha,\beta)=\frac{I^{-1/2}}{4\sqrt{2\pi}\alpha\beta
I_{l,2}}\left[\left(\frac{\beta
I_{l,2}}{\sqrt{I}}\right)^{1/2}\\!+\left(\frac{\beta
I_{l,2}}{\sqrt{I}}\right)^{3/2}\right]\\!\times\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{\sqrt{I}}{\beta
I_{l,2}}+\frac{\beta I_{l,2}}{\sqrt{I}}-2\right)\right]\\!,~{}$ (22)

for downlink. The SNR diagonal matrix for uplink is similar to (17) with
$j\\!=\\!1$. Nevertheless for downlink, we derive

$\hat{\bm{\gamma}}_{2}=\Big{(}{\Phi_{2}^{2}}\Big{)}^{-1}R^{2}\lambda_{2}^{2}I_{2}^{4}\bm{x}_{m}\hat{s}_{1}\hat{s}_{1}^{H}\bm{x}_{m}^{H}=\tilde{\bm{\gamma}}_{2}I_{2}^{4},$
(23)

where
$\tilde{\bm{\gamma}}_{2}=\big{(}{\Phi_{2}^{2}}\big{)}^{-1}R^{2}\lambda_{2}^{2}\bm{x}_{m}\hat{s}_{1}\hat{s}_{1}^{H}\bm{x}_{m}^{H}$
signifies the average downlink SNR diagonal matrix. Therefore, we have

$\displaystyle f_{\hat{\gamma}_{1,k}}(\hat{\gamma}_{k};\alpha,\beta)\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\frac{1}{4\sqrt{2\pi}\alpha\beta
I_{l,1}}\left[\left(\frac{\beta
I_{l,1}\sqrt{\tilde{{\gamma}}_{1}}}{\sqrt{\hat{{\gamma}}_{1}}}\right)^{\\!\\!1/2}+\left(\frac{\beta
I_{l,2}\sqrt{\tilde{{\gamma}}_{1}}}{\sqrt{\hat{{\gamma}}_{1}}}\right)^{\\!\\!3/2}\right]$
(24)
$\displaystyle\times\,\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{\sqrt{\hat{{\gamma}}_{1}}}{\beta
I_{l,2}\sqrt{\tilde{{\gamma}}_{1}}}+\frac{\beta
I_{l,2}\sqrt{\tilde{{\gamma}}_{1}}}{\sqrt{\hat{{\gamma}}_{1}}}-2\right)\right]\\!,~{}$

for uplink, wherein

$\displaystyle f_{\hat{\gamma}_{2,k}}(\hat{\gamma}_{k};\alpha,\beta)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{\sqrt{\tilde{{\gamma}}_{2}}}{8\sqrt{2\pi}\alpha\beta
I_{l,2}\sqrt{\hat{{\gamma}}_{2}}}\left[\left(\frac{\beta
I_{l,2}\sqrt[4]{\tilde{{\gamma}}_{2}}}{\sqrt[4]{\hat{{\gamma}}_{2}}}\right)^{\\!\\!1/2}+\left(\frac{\beta
I_{l,2}\sqrt[4]{\tilde{{\gamma}}_{2}}}{\sqrt[4]{\hat{{\gamma}}_{2}}}\right)^{\\!\\!3/2}\right]$
(25)
$\displaystyle\times\,\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{\sqrt[4]{\hat{{\gamma}}_{2}}}{\beta
I_{l,2}\sqrt[4]{\tilde{{\gamma}}_{2}}}+\frac{\beta
I_{l,2}\sqrt[4]{\tilde{{\gamma}}_{2}}}{\sqrt[4]{\hat{{\gamma}}_{2}}}-2\right)\right]\\!,~{}$

for downlink. The CDF of $\hat{\gamma}_{j,k}$ for the RRS is respectively
obtained by [kundu2010bivariate]

$\displaystyle
F_{\hat{\gamma}_{j,k}}(\hat{\gamma}_{k};\alpha,\beta)=\operatorname{\Phi}\\!\Bigg{(}\frac{1}{\alpha}\Bigg{[}\bigg{(}\frac{\sqrt[2j]{\hat{{\gamma}}_{j}}}{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}\bigg{)}^{\\!\\!1/2}-\bigg{(}\frac{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}{\sqrt[2j]{\hat{{\gamma}}_{j}}}\bigg{)}^{\\!\\!1/2}\Bigg{]}\Bigg{)}\\!,$
(26)

where $j\\!=\\!1$ and $j\\!=\\!2$ present uplink and downlink transmissions,
respectively. Also, $\operatorname{\Phi}(.)$ depicts the standard normal CDF.

###  II-C RF Channel Model

The wireless link between the UAV and the terrestrial AP, namely $G_{s}$,
pursues a Nakagami-$m$ distribution with the pdf given by
[laurenson1994nakagami]

$f_{G_{s}}(G_{s};m)=\frac{2m^{m}G_{s}^{2m-1}}{\Gamma(m)\Omega^{m}}\,\operatorname{exp}\\!\left(\\!-\frac{mG_{s}^{2}}{\Omega}\right)\\!,$
(27)

where $\Omega\\!=\\!\mathbb{E}\\!\left\\{G_{s}^{2}\right\\}$ represents mean-
square value, $\Gamma(.)$ implies the Gamma function, and
$0.5\\!\leq\\!m\\!\leq\\!\infty$ depicts the Nakagami fading parameter. The
received RF signal is modeled as bellow

$\hat{\hat{r}}=G_{s}\hat{\hat{s}}+v,$ (28)

where
$\hat{\hat{s}}\\!=\\!\left[\hat{\hat{s}}_{1},\hat{\hat{s}}_{2},...,\hat{\hat{s}}_{K}\right]^{T}$
depicts the compressed signal, wherein $\hat{\hat{s}}_{k}$ is the estimated
symbol of the $k$th SN, and
$\mathbb{E}\\!\left\\{\hat{\hat{s}}_{k}\hat{\hat{s}}_{k^{\prime}}\right\\}\\!=\\!0$
for $k\neq k^{\prime}$. Also, $v\\!\sim\\!\mathcal{N}(0,\Lambda^{2})$ depicts
the i.i.d. additive noise with
$\Lambda^{2}=\mathbb{E}\\!\left\\{vv^{H}\right\\}$. Therefore, the SNR
diagonal matrix is derived as

$\hat{\hat{\bm{\gamma}}}=\Big{(}{\Lambda^{2}}\Big{)}^{-1}G_{s}^{2}\hat{\hat{s}}\hat{\hat{s}}^{H}=\tilde{\tilde{\bm{\gamma}}}G_{s}^{2},$
(29)

where
$\tilde{\tilde{\bm{\gamma}}}=\Big{(}{\Lambda^{2}}\Big{)}^{-1}\hat{\hat{s}}\hat{\hat{s}}^{H}$
signifies average SNR diagonal matrix. Hence, by using (27) and (29), we have

$f_{\hat{\hat{\gamma}}_{k}}(\hat{\hat{\gamma}}_{k};m)=\frac{m^{m}{\hat{\hat{\gamma}}}_{k}^{m-1}}{\Gamma(m)\tilde{\tilde{\gamma}}_{k}^{m}}\,\operatorname{exp}\\!\left(\\!-\frac{m\hat{\hat{\gamma}}_{k}}{\tilde{\tilde{\gamma}}_{k}}\\!\right)\\!.$
(30)

In this case, the CDF of $\hat{\hat{\gamma}}_{k}$ is acquired as
[laurenson1994nakagami]

$F_{\hat{\hat{\gamma}}_{k}}(\hat{\hat{\gamma}}_{k};m)=\frac{1}{\Gamma(m)}\operatorname{\gamma}\\!\left(\\!m,\frac{m\hat{\hat{\gamma}}_{k}}{\tilde{\tilde{\gamma}}_{k}}\\!\right)\\!.$
(31)

##  III Tracking System and Performance Analysis

In this section, we propose a AUV–UAV tracking system demanded for a reliable
connection. Next, we obtain the end-to-end outage probability and average bit
error rate (ABER) for each SN under the proposed scenarios to analyze the
network’s performance.

###  III-A AUV–UAV Tracking System

A reliable tracking system between the UAV and AUV nodes is deployed under a
gimbal-mirror hybrid mechanism with two tracking protocols combined to provide
an effective beam tracking; coarse tracking and fine tracking
[kaymak2018survey]. Coarse tracking is performed by exploiting the global
positioning system (GPS) or a local positioning system to acquire the UAV
position and continuously track the trajectory with the motor- and gimbal-
assisted tracker of the AUV. However, the super-accurate GPS has a resolution
of $30\,[\text{cm}]$ not suitable for a reliable high-capacity link. To solve
this issue and precisely track the UAV, the fine tracking is applied
[kaymak2018survey]. In this case, the UAV continuously transmits beacon
signals to the AUV within the transmission interval, and the quadrant
photodiode (QPD) at the AUV uses the 4-quadrant detectors (4QD) algorithm to
analyze the received beacons and drive the fast steering mirror (FSM) locking
to the receiving beam [kaymak2018survey, nakarach2017comparison], c.f. Fig. 3.
The 4QD measures the received signals at four quadrants of the QPD, namely
$\vartheta_{1}$, $\vartheta_{2}$, $\vartheta_{3}$, and $\vartheta_{4}$ as Fig.
4, and derives the error expressions as follows [nakarach2017comparison]

$\displaystyle e_{x}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{(\vartheta_{1}+\vartheta_{4})-(\vartheta_{2}+\vartheta_{3})}{e_{t}},$
(32) $\displaystyle e_{y}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{(\vartheta_{1}+\vartheta_{2})-(\vartheta_{3}+\vartheta_{4})}{e_{t}},$
(33)

where $e_{t}=\vartheta_{1}+\vartheta_{2}+\vartheta_{3}+\vartheta_{4}$. The
tracking controller aims to minimize the errors and achieve a _perfect_
alignment, i.e. $(e_{x},e_{y})\\!=\\!(0,0)$. All in one, the acquisition-and-
tracking mechanism for an FSO link between the UAV and AUV is illustrated in a
flowchart depicted in Fig. 5.

[scale=0.55]Media/QPD.eps

Figure 4: Four quadratic areas of the QPD in which the dashed circle is the
optical beam of the received beacon. Herein, (a) and (b) represent _perfect_
alignment, i.e. $(e_{x},e_{y})\\!=\\!(0,0)$, and _misalignment_ , i.e.
$(e_{x},e_{y})\\!\neq\\!(0,0)$, respectively.

[scale=0.65]Media/Flowchart.eps

Figure 5: Flowchart of the acquisition-and-tracking mechanism for an FSO link
between the UAV and AUV, where we assume a 10-cycle fine tracking protocol.

###  III-B Outage Probability

The end-to-end SNR of the triple-hop network under the DF protocol is
estimated as
$\gamma_{j,k}=\operatorname{min}\\!\left\\{\\!\gamma_{k},\hat{\gamma}_{j,k},\hat{\hat{\gamma}}_{k}\\!\right\\}$
for $k$th SN, with the CDF inspired by [papoulis1991random] as follows

$\displaystyle F_{\gamma_{j,k}}(\gamma;\alpha,\beta,m)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!F_{\gamma_{k}}(\gamma)+F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)+F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)$
(34)
$\displaystyle-\,F_{\gamma_{k}}(\gamma)F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)-F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)$
$\displaystyle-\,F_{\gamma_{k}}(\gamma)F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)+F_{\gamma_{k}}(\gamma)F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)$
$\displaystyle=$
$\displaystyle\\!\\!1-\Big{(}\\!1-F_{\gamma_{k}}(\gamma)\\!\Big{)}\Big{(}\\!1-F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)\\!\Big{)}\Big{(}\\!1-F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)\\!\Big{)}.$

The quality of service (QoS) for the supposed network is ensured by keeping
the instantaneous end-to-end SNR above a threshold $\gamma_{th}$. Thus, the
end-to-end outage probability for $k$th SN is obtained by

$\displaystyle
P_{out,j,k}(\alpha,\beta,m)=\operatorname{P}\\!\left(\gamma_{j,k}\leq\gamma_{th}\right)=F_{\gamma_{j,k}}(\gamma_{th};\alpha,\beta,m).$
(35)

By inserting (34) into (35), we acquire

$\displaystyle
P_{out,j,k}(\alpha,\beta,m)=1-\Big{(}\\!1-F_{\gamma_{k}}(\gamma_{th})\\!\Big{)}\Big{(}\\!1-F_{\hat{\gamma}_{j,k}}(\gamma_{th};\alpha,\beta)\\!\Big{)}\Big{(}\\!1-F_{\hat{\hat{\gamma}}_{k}}(\gamma_{th};m)\\!\Big{)}.$
(36)

###  III-C Average Bit Error Rate

Using the DF protocol, the bit error probability at each network hop is
independent of the other ones. Thus, we can formulate the end-to-end ABER of
the triple-hop network for $k$th SN as below

$\displaystyle P_{e,j,k}(\alpha,\beta,m)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!P_{e_{1},k}\Big{(}1-P_{e_{2},j,k}(\alpha,\beta)\Big{)}\Big{(}1-P_{e_{3},k}(m)\Big{)}+P_{e_{2},j,k}(\alpha,\beta)\Big{(}1-P_{e_{1},k}\Big{)}\Big{(}1-P_{e_{3},k}(m)\Big{)}$
(37)
$\displaystyle+\,P_{e_{3},k}(m)\Big{(}1-P_{e_{1},k}\Big{)}\Big{(}1-P_{e_{2},j,k}(\alpha,\beta)\Big{)}+P_{e_{1},k}P_{e_{2},j,k}(\alpha,\beta)P_{e_{3},k}(m)$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\bigg{[}\Big{(}2P_{e_{1},k}-1\Big{)}\Big{(}2P_{e_{2},j,k}(\alpha,\beta)-1\Big{)}\Big{(}2P_{e_{3},k}(m)-1\Big{)}+1\bigg{]}\\!,$

where

  * a)

$P_{e_{1},k}$ represents the ABER of the UWOC link[naik2020co]

$\displaystyle P_{e_{1},k}$
$\displaystyle=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}F_{{\gamma}_{k}}(\gamma)\,d\gamma$
(38a)
$\displaystyle=\frac{q^{p}\bar{\gamma}_{k}\varphi_{k}^{\prime}}{2\Gamma(p)}\operatorname{exp}\\!\big{(}\\!-\\!\varphi_{k}\xi^{2}\big{)}\Bigg{[}2\operatorname{exp}\\!\Big{(}-\frac{c_{1}(8\sigma_{x_{k}}^{2}\xi^{2}+c_{1})}{8\sigma_{x_{k}}^{2}}\Big{)}$
$\displaystyle~{}~{}\times\Bigg{(}\dfrac{\operatorname{exp}\\!\Big{(}\big{(}8\sigma_{x_{k}}^{2}(\xi^{2}+2p)+2c_{1}\big{)}^{2}/{32\sigma_{x_{k}}^{2}}\Big{)}}{\xi^{2}+2p}-\dfrac{\operatorname{exp}\\!\Big{(}\big{(}8\sigma_{x_{k}}^{2}(\xi^{2}+2p+2)+2c_{1}\big{)}^{2}/{32\sigma_{x_{k}}^{2}}\Big{)}}{\xi^{2}+2p+2}\Bigg{)}$
$\displaystyle~{}~{}+\operatorname{exp}\\!\Big{(}2\sigma_{x_{k}}^{2}\xi^{4}\Big{)}\\!\left(\dfrac{\operatorname{exp}\\!\Big{(}(p+1)\big{(}8\sigma_{x_{k}}^{2}(p+1)+2c_{2}\big{)}\Big{)}}{p+1}-\dfrac{\operatorname{exp}\\!\Big{(}p\big{(}8\sigma_{x_{k}}^{2}p+2c_{2}\big{)}\Big{)}}{p}\right)\Bigg{]}\\!,$
(38b)

where
$c_{1}=\ln\\!\big{(}{{h_{0}h_{l,k}\sqrt{\bar{\gamma}_{k}}}}\big{)}-\varphi_{k}$
and $c_{2}=c_{1}+{4\sigma_{x_{k}}^{2}}\xi^{2}$,

  * b)

$P_{e_{2},j,k}$ depicts the ABER of the FSO link, where

    * 1)

For the DS, we derive

$\displaystyle P_{e_{2},j,k}(\alpha,\beta)$
$\displaystyle=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)\,d\gamma$
(39a)
$\displaystyle=\frac{q^{p}e^{1/{\alpha^{2}}}}{\sqrt{\pi}\Gamma(p)}\frac{\zeta^{2}\tilde{\gamma}_{k}({\gamma_{0,j}^{2}\tilde{\gamma}_{k}})^{-\frac{\zeta^{2}}{2}+p+1}}{(\zeta^{2}+6)}$
$\displaystyle~{}~{}~{}\times\Bigg{[}(\gamma_{0,j}^{2}{\tilde{\gamma}_{k}})^{q}\bigg{(}-\frac{\Gamma(-\xi^{2}+2c_{3}+\frac{9}{2})}{c_{3}+2}-\frac{\Gamma(-\xi^{2}+2c_{3}+\frac{7}{2})}{2\alpha^{2}(c_{3}+2)}$
$\displaystyle~{}~{}~{}+\frac{\Gamma(-\frac{3}{4}\xi^{2}+2p+4)}{p+1}-\frac{\Gamma(-\frac{3}{4}\xi^{2}+2p+3)}{2\alpha^{2}(p+1)}\bigg{)}$
$\displaystyle~{}~{}~{}+\bigg{(}\frac{\Gamma(-\xi^{2}+2c_{3}+\frac{5}{2})}{c_{3}+1}+\frac{\Gamma(-\xi^{2}+2c_{3}+\frac{3}{2})}{2\alpha^{2}(c_{3}+1)}$
$\displaystyle~{}~{}~{}-\frac{\Gamma(-\frac{3}{4}\xi^{2}+2p+2)}{p}+\frac{\Gamma(-\frac{3}{4}\xi^{2}+2p+1)}{2\alpha^{2}p}\bigg{)}\Bigg{]}\\!,$
(39b)

where $c_{3}=\frac{\zeta^{2}-2}{8}+p$,

    * 2)

For the RRS, by applying
$\operatorname{\Phi}(x)\\!=\\!\frac{1}{2}\operatorname{erfc}\\!\left(\frac{x}{\sqrt{2}}\right)$
and some simplifications, we have

$\displaystyle
P_{e_{2},j,k}(\alpha,\beta)=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}F_{\hat{\gamma}_{j,k}}(\gamma;\alpha,\beta)\,d\gamma$
(40a)
$\displaystyle=\frac{q^{p}}{4\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}\operatorname{erfc}\\!\Bigg{(}\frac{1}{\alpha\sqrt{2}}\Bigg{[}\bigg{(}\frac{\sqrt[2j]{{{\gamma}}}}{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}\bigg{)}^{\\!\\!1/2}-\bigg{(}\frac{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}{\sqrt[2j]{{{\gamma}}}}\bigg{)}^{\\!\\!1/2}\Bigg{]}\Bigg{)}d\gamma$
(40b)
$\displaystyle\simeq\frac{q^{p}}{4\Gamma(p)}\Bigg{[}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}\operatorname{erfc}\\!\Bigg{(}\frac{1}{\alpha\sqrt{2}}\bigg{(}\frac{\sqrt[2j]{{{\gamma}}}}{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}\bigg{)}^{\\!\\!1/2}\Bigg{)}d\gamma$
$\displaystyle~{}~{}~{}+\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}\operatorname{erf}\\!\Bigg{(}\frac{1}{\alpha\sqrt{2}}\bigg{(}\frac{\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}{\sqrt[2j]{{{\gamma}}}}\bigg{)}^{\\!\\!1/2}\Bigg{)}d\gamma\Bigg{]}$
(40c)
$\displaystyle=\frac{q^{p}}{4\Gamma(p)}\Bigg{[}\\!-\left(\\!{\alpha\sqrt{2\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}}\right)^{\\!4j(p+1)}\frac{\Gamma\\!\left({2j(p+1)+\frac{1}{2}}\right)}{\sqrt{\pi}(p+1)}$
$\displaystyle~{}~{}~{}+\left(\\!{\alpha\sqrt{2\beta
I_{l,j}\sqrt[2j]{\tilde{{\gamma}}_{j}}}}\right)^{\\!4jp}\frac{\Gamma\\!\left({2jp+\frac{1}{2}}\right)}{\sqrt{\pi}p}+\left(\frac{1}{\alpha}\sqrt{\frac{\sqrt[2j]{\tilde{{\gamma}}_{j}}}{2\beta
I_{l,j}}}\right)^{\\!4j(p+1)}\frac{\Gamma\\!\left({-2j(p+1)+\frac{1}{2}}\right)}{\sqrt{\pi}(p+1)}$
$\displaystyle~{}~{}~{}-\left(\frac{1}{\alpha}\sqrt{\frac{\sqrt[2j]{\tilde{{\gamma}}_{j}}}{2\beta
I_{l,j}}}\right)^{\\!4jp}\frac{\Gamma\\!\left({-2jp+\frac{1}{2}}\right)}{\sqrt{\pi}p}\Bigg{]}\\!,$
(40d)

  * c)

$P_{e_{3},k}$ signifies the ABER of the RF link

$\displaystyle P_{e_{3},k}(m)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}F_{\hat{\hat{\gamma}}_{k}}(\gamma;m)\,d\gamma$
(41a) $\displaystyle=$
$\displaystyle\\!\\!\frac{q^{p}}{2\Gamma(p)\Gamma(m)}\int_{0}^{\infty}e^{-q\gamma}\gamma^{p-1}\operatorname{\gamma}\\!\left(\\!m,\frac{m\gamma}{\tilde{\tilde{\gamma}}_{k}}\\!\right)\,d\gamma$
(41b) $\displaystyle=$
$\displaystyle\\!\\!\frac{q^{p}}{2\Gamma(p)\Gamma(m)}\bigg{[}{\left(\frac{\tilde{\tilde{\gamma}}_{k}}{m}\right)^{\\!p+1}}\frac{\Gamma(m+p+1)}{p+1}-{\left(\frac{\tilde{\tilde{\gamma}}_{k}}{m}\right)^{\\!p}}\frac{\Gamma(m+p)}{p}\bigg{]}\\!,$
(41c)

wherein $p$ and $q$ depict modulation-related parameters of which values for
binary modulation techniques are provided in Table II [naik2020co]. Due to the
small command and sensor data size, binary modulations lower the complexity
and energy consumption.

TABLE II: Parameters for binary modulation techniques. Modulation technique | $p$ | $q$
---|---|---
On-off keying (OOK) | $0.5$ | $0.25$
Binary frequency-shift keying (BFSK) | $1$ | $0.5$
Binary phase-shift keying (BPSK) | $0.5$ | $1$
Differential BPSK (DBPSK) | $1$ | $1$

##  IV Numerical Results and Discussions

In the following section, we compare the performance of the triple-hop optical
wireless network for the proposed scenarios from the outage probability and
ABER perspectives. The UWOC wavelength allocation for each sensor is performed
with the center wavelength of $532$ $\\![\text{nm}]$ and grid sizes of $30$
$\\![\text{mm}]$, c.f. Fig. 6. Likewise, we have

$\dfrac{1}{\lambda_{k}}=\dfrac{1}{532}+\dfrac{(-1)^{k-1}}{30\\!\times\\!10^{6}}\Big{\lceil}\dfrac{k-1}{2}\Big{\rceil},$
(42)

where $\lambda_{k}\,[\text{nm}]$ depicts the wavelength for $k$th SN. The
parameters used for the numerical results are summarized in Table III
[naik2020co, christopoulou2019outage]. In general, we assume clear weather and
seawater conditions, perfect alignment in RRS, and BPSK modulation for ABER
analyses.

TABLE III: Network parameters for numerical results. Parameter | Symbol | Value | Parameter | Symbol | Value
---|---|---|---|---|---
FSO wavelength for the uplink | $\lambda_{u}$ |  $1064$ $\\![\text{nm}]$ | Refraction structure index | $C_{n}^{2}$ |  $10^{-15}$ $\\![\text{m}^{-2/3}]$
FSO wavelength for the downlink | $\lambda_{d}$ |  $1550$ $\\![\text{nm}]$ | UWOC path-loss constant | $h_{0}$ | $0.0764$
Terrestrial radio frequency | $f_{\text{RF}}$ |  $1.9$ $\\![\text{GHz}]$ | FSO path-loss constant | $I_{0}$ | $0.0197$
System bandwidth | $B_{s}$ |  $20$ $\\![\text{MHz}]$ | UWOC pointing error parameter | $\xi$ | $2.35$
UWOC aperture area | $A_{ea}$ |  $1.7\\!\times\\!10^{-4}$ $\\![\text{m}^{-2}]$ | FSO pointing error parameter | $\zeta$ | $4.51$
FSO aperture area | $A_{j}$ |  $7\\!\times\\!10^{-4}$ $\\![\text{m}^{-2}]$ | Birnbaum-Saunders parameters | $\\{\alpha,\beta\\}$ | $\\!\\{0.6866,0.8093\\}$
UWOC source beam divergence angle | $\theta_{0}$ | $60\degree$ | Nakagami fading parameter | $m$ | $0.5$
FSO source beam divergence angle | $\psi_{j}$ | $30\degree$ | UWOC optical-to-electrical parameter | $\eta_{k}$ | $0.25$
Sensor-relay angle | $\theta$ | $5\degree$ | FSO optical-to-electrical parameter | $\lambda_{j}$ | $0.5$
Sensor-relay perpendicular distance | $d_{a}$ |  $0.35$ $\\![\text{km}]$ | UWOC additive noise variance | $\delta_{k}^{2}$ |  $10^{-14}$ $\\![\text{A}^{2}]$
AUV–UAV average link length | $d_{au}$ |  $1$ $\\![\text{km}]$ | FSO additive noise variance | $\Phi_{j,k}^{2}$ |  $10^{-14}$ $\\![\text{A}^{2}]$
Clear seawater condition extinction coefficient | $c(\lambda_{k})$ |  $0.151$ $\\![\text{m}^{-1}]$ | RF additive noise variance | $\Lambda_{k}^{2}$ |  $5.56\\!\times\\!10^{-13}$ $\\![\text{Watt}]$
Clear weather condition attenuation coefficient | $\alpha_{au}$ |  $0.44$ $\\![\text{dB/km}]$ | CCR reflection effect for perfect alignment | $R$ | $1$
Snowy weather condition attenuation coefficient |  $4.53$ $\\![\text{dB/km}]$ | CCR reflection effect for misalignment | $0.5$
Foggy weather condition attenuation coefficient |  $50$ $\\![\text{dB/km}]$ | Number of the sensor nodes | $K$ | $10$
(a)

[scale=0.55]Media/WDM.eps

Figure 6: The wavelength allocation diagram for the UWOC links.

[scale=0.485]Result/1_Monte-Carlo.eps

(a)

[scale=0.485]Result/Copy_of_5_Monte-Carlo.eps

(b)

[scale=0.485]Result/3_Weather.eps

(c)

[scale=0.485]Result/7_Weather.eps

(d)

[scale=0.485]Result/2_RRCResponse.eps

(e)

[scale=0.485]Result/6_RRCResponse.eps

(f)

[scale=0.5]Result/Copy_of_4_Distance.eps

(g)

[scale=0.5]Result/Copy_of_8_Distance.eps

(h)

Figure 7: (a)–(b): The outage probability and ABER versus the average SNR for
all scenarios verified by Monte-Carlo simulations, respectively; (c)–(d): The
outage probability and ABER versus the average SNR for all scenarios with
different weather conditions, respectively; (e)–(f): The outage probability
and ABER versus the average SNR for all scenarios under perfect alignment and
misalignment of CCR, respectively; (g)–(h): The outage probability and ABER
versus the average SNR, respectively, for all scenarios with different lengths
of UWOC and FSO links.

To illustrate and compare the performance of the proposed scenarios, Fig. 7
(a) and Fig. 7 (b) depict the outage probability and ABER versus the average
SNR for all scenarios, respectively. It is shown that the RRS outperforms the
DS and enhances the network’s performance in the uplink and downlink. In this
case, for the uplink, SUR has much better performance than SUD, and for the
downlink, SDR has less outage probability and ABER compared to SDD. Also, UDR
exceeds UDD in double-path transmission. Herein, Monte-Carlo simulations are
applied, confirming the expressions and figures, for which we consider random
data stream and derive the end-to-end outage probability and ABER for each
amount of the average SNR.

Fig. 7 (c) and Fig. 7 (d) study the effects of weather conditions on the
performance of the network, from the outage probability and ABER perspectives,
under the proposed scenarios. As expected, it is represented that clear
weather lets the network show its best performance, while the snowy and foggy
weather conditions degrade the network’s performance. For example, SUR has the
outage probabilities in snowy and foggy weather conditions $10^{3.5}$ and
$100$ times more than that in clear weather, respectively, for the average SNR
of $45\,[\text{dB}]$; similarly, the ABER for SUR in clear weather is $3$ and
$10$ times less than that in snowy and foggy weather states, respectively.

To investigate the CCR reflection effects on the network’s performance in the
RRS, Fig. 7 (e) and Fig. 7 (f) respectively present the outage probability and
ABER versus the average SNR based on perfect alignment and misalignment
conditions. Herein, it is shown that the perfect alignment provides lower
outage probability and ABER in comparison with the misalignment condition.
However, SDR and UDR respectively overcome SDD and UDD even under misalignment
conditions. For instance, SDR has almost $30$ and $10$ times lower outage
probability than that of SDD under perfect alignment and misalignment
conditions, respectively, for the average SNR of $45\,[\text{dB}]$; likewise,
the ABER for SDR is $2$ and $1.5$ times better than that for SDD with perfect
alignment and misalignment conditions, respectively.

Besides, Fig. 7 (g) and Fig. 7 (h) sequentially study the outage probability
and ABER versus the average SNR with variable sets of SN–AUV and AUV–UAV
average distances, i.e. ($d_{a,k}$, $d_{au}$) $\\![\text{km}]$. Without loss
of generality and to take advantage of simple representation, we assume the
same SN–AUV length and sensor relay angle for all SNs, where
$d_{a,k}\\!=\\!d_{a}$ and $\theta_{k}\\!=\\!\theta$ for $k\\!=\\!1,2,...,K$.
In both figures, increasing the distance between SNs and AUV or AUV and UAV
diminishes the network’s performance via increasing the outage probability and
ABER. As an illustrative example for SUR, the outage probability grows from
$10^{-4.5}$ to $10^{-2}$, and the ABER rises from $10^{-1.5}$ to $10^{-1}$ by
doubling the distance set for the average SNR of $45\,[\text{dB}]$.

[scale=0.6]Result/9_Modulation_Uplink.eps

(a)

[scale=0.6]Result/10_Modulation_Downlink.eps

(b)

Figure 8: The ABER versus the average SNR for the (a) uplink and (b) downlink
transmission scenarios based on different binary modulation techniques.

Finally, yet important, to demonstrate the importance of the modulation
selection on the network’s performance, Fig. 8 (a) and Fig. 8 (b) separately
depicting the ABERs for the uplink and downlink transmissions are analyzed
based on different binary modulation techniques, such as BPSK, DBPSK, BFSK,
and OOK. In both figures, BPSK modulation offers the best network performance,
while OOK modulation has the highest ABER compared to the other techniques. In
one case, Fig. 8 (a) confirms that the RRS surpasses the DS in the uplink
under all studied modulations. On the other hand, in the downlink, either the
RRS or the DS can exceed the other one as a function of the selected
modulation technique. Specifically, the RRS certainly provides better
performance than the DS using BPSK and DBPSK modulations at UWOC and RF hops.

##  V Conclusion

We studied the uplink and downlink of a triple-hop optical wireless network
wherein the AUV collects (transmits) data from (to) the SNs via UWOC links,
the AUV is connected to the UAV over FSO links, and the UAV communicates with
the terrestrial AP through RF links. In this case, we proposed and compared
two different strategies for the AUV–UAV FSO communications, i.e. the DS and
RRS, considering the W2A and A2W impacts. Besides, we suggested a heuristic
acquisition-and-tracking mechanism to combat the FSO pointing error and
misalignment in the uplink and downlink. To obtain the network’s end-to-end
outage probability and ABER, we first modeled the channels at each hop and
then derived the CDF of the SNRs. Through numerical results, the outage
probability and ABER for the proposed scenarios were investigated and compared
from different viewpoints, such as weather conditions, UWOC and FSO link
lengths, RRC alignments, and modulation techniques. Furthermore, numerical
results were confirmed by exerting Monte-Carlo simulations.

##  Appendix A CDF Derivation of UWOC Link

By inserting (9) into the CDF formula, we have

$\displaystyle
F_{\gamma_{k}}(\gamma)=\frac{\xi^{2}\bar{\gamma}^{1-\frac{\xi^{2}}{2}}}{4(h_{0}h_{l,k})^{\xi^{2}}}\varphi_{k}^{\prime}\int_{0}^{\gamma}t^{\,\frac{\xi^{2}}{2}-1}\,\operatorname{erfc}\\!\left(\frac{\ln\\!\left({\frac{\sqrt{t}}{h_{0}h_{l,k}\sqrt{\bar{\gamma}}}}\right)+\varphi_{k}}{\sqrt{8\sigma_{x_{k}}^{2}}}\\!\right)dt$
(43a)
$\displaystyle=\frac{\bar{\gamma}\varphi_{k}^{\prime}}{2}\Bigg{[}\operatorname{erfc}\\!\left(\frac{\Theta_{k}(\gamma)}{\sqrt{8\sigma_{x_{k}}^{2}}}\right)\\!\operatorname{exp}\\!\Big{(}\Theta_{k}(\gamma)\xi^{2}\Big{)}+\operatorname{erfc}\\!\left(\\!\frac{4\sigma_{x_{k}}^{2}\xi^{2}-\Theta_{k}(\gamma)}{\sqrt{8\sigma_{x_{k}}^{2}}}\\!\right)\\!\operatorname{exp}\\!\Big{(}2\sigma_{x_{k}}^{2}\xi^{4}\Big{)}\Bigg{]}\\!\operatorname{exp}\\!\Big{(}\\!\\!-\varphi_{k}\xi^{2}\Big{)},$
(43b)

where
$\Theta_{k}(\gamma)=\ln\\!\left(\\!{\frac{\sqrt{\gamma}}{h_{0}h_{l,k}\sqrt{\bar{\gamma}}}}\\!\right)+\varphi_{k}$.
Therefore, we obtain (10).

##  Appendix B pdf Derivation of FSO Link

We utilize the same technique as (5), and replace (13) in the formula. Thus,
we have

$\displaystyle f_{I_{j}}(I;\alpha,\beta)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\zeta^{2}I^{\zeta^{2}-1}}{(I_{0}I_{l,j})^{\zeta^{2}}}\int_{I/{I_{0}I_{l,j}}}^{\infty}I_{t}^{-\zeta^{2}}f_{I_{t,j}}(I_{t};\alpha,\beta)\,dI_{t}$
(44c) $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{2\pi}\alpha\beta}\frac{\zeta^{2}I^{\zeta^{2}-1}}{(I_{0}I_{l,j})^{\zeta^{2}}}\,\Bigg{\\{}\\!\int_{I/{I_{0}I_{l,j}}}^{\infty}I_{t}^{-\zeta^{2}}\left(\frac{\beta}{I_{t}}\right)^{1/2}\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{I_{t}}{\beta}+\frac{\beta}{I_{t}}-2\right)\right]dI_{t}$
$\displaystyle+\int_{I/{I_{0}I_{l,j}}}^{\infty}I_{t}^{-\zeta^{2}}\left(\frac{\beta}{I_{t}}\right)^{3/2}\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(\frac{I_{t}}{\beta}+\frac{\beta}{I_{t}}-2\right)\right]dI_{t}\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2\sqrt{2\pi}\alpha}\frac{\zeta^{2}I^{\zeta^{2}-1}}{(\beta
I_{0}I_{l,j})^{\zeta^{2}}}\,\Bigg{\\{}\\!\int_{I/{\beta
I_{0}I_{l,j}}}^{\infty}\left(\frac{1}{u}\right)^{1/2+\zeta^{2}}\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(u+\frac{1}{u}-2\right)\right]du$
$\displaystyle+\int_{I/{\beta
I_{0}I_{l,j}}}^{\infty}\left(\frac{1}{u}\right)^{3/2+\zeta^{2}}\\!\operatorname{exp}\\!\left[-\frac{1}{2\alpha^{2}}\left(u+\frac{1}{u}-2\right)\right]du\Bigg{\\}}.$

According to the integrals’ limits, $u$ is large enough helping us to apply
the property $u+\frac{1}{u}\simeq u$. By this, we have a negligible error that
is the penalty of approximately solving the integrals. However, after
performing a sequence of mathematical computations, (15) is derived.

##  Appendix C CDF Derivation of FSO Link

By inserting (18) into the CDF formula, we have

$\displaystyle
F_{\hat{\gamma}_{j,k}}(\hat{\gamma}_{k};\alpha,\beta)=\int_{0}^{\hat{\gamma}_{k}}f_{{\gamma}_{j,k}}({\gamma}_{k};\alpha,\beta)d\gamma_{k}$
(45a)
$\displaystyle=\frac{e^{1/{\alpha^{2}}}}{4\sqrt{\pi}}\frac{\zeta^{2}\tilde{\gamma}_{k}^{1-\frac{\zeta^{2}}{2}}}{\gamma_{0,j}^{\zeta^{2}}}\int_{0}^{\hat{\gamma}_{k}}{\gamma}_{k}^{\frac{\zeta^{2}}{2}-1}\Bigg{[}\Gamma\\!\left(\frac{1}{2}-\zeta^{2},\frac{\sqrt{{\gamma}_{k}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)+\frac{1}{2\alpha^{2}}\,\Gamma\\!\left(\\!-\frac{1}{2}-\zeta^{2},\frac{\sqrt{{\gamma}_{k}}}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)\\!\Bigg{]}d\gamma_{k}$
(45b)
$\displaystyle=\frac{e^{1/{\alpha^{2}}}}{2\sqrt{\pi}}\frac{\zeta^{2}\tilde{\gamma}_{k}^{1-\frac{\zeta^{2}}{2}}}{\gamma_{0,j}^{\zeta^{2}}}\int_{0}^{\sqrt{\hat{\gamma}_{k}}}u^{\frac{\zeta^{2}}{4}+\frac{1}{2}}\Bigg{[}\Gamma\\!\left(\frac{1}{2}-\zeta^{2},\frac{u}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)+\frac{1}{2\alpha^{2}}\,\Gamma\\!\left(\\!-\frac{1}{2}-\zeta^{2},\frac{u}{\gamma_{0,j}\sqrt{\tilde{\gamma}_{k}}}\right)\\!\Bigg{]}du.$
(45c) After some mathematical manipulations, (3) is acquired.


0301é

# Spatial-DCCRN: DCCRN Equipped with Frame-level Angle Feature and Hybrid
Filtering for Multi-channel Speech Enhancement

###### Abstract

Recently, multi-channel speech enhancement has drawn much interest due to the
use of spatial information to distinguish target speech from interfering
signal. To make full use of spatial information and neural network based
masking estimation, we propose a multi-channel denoising neural network –
Spatial DCCRN. Firstly, we extend S-DCCRN to multi-channel scenario, aiming at
performing cascaded _sub-channel_ and _full-channel_ processing strategy,
which can model different channels separately. Moreover, instead of only
adopting multi-channel spectrum or concatenating first-channel’s magnitude and
IPD as the model’s inputs, we apply an _angle feature extraction_ module (AFE)
to extract frame-level angle feature embeddings, which can help the model to
apparently perceive spatial information. Finally, since the phenomenon of
residual noise will be more serious when the noise and speech exist in the
same time frequency (TF) bin, we particularly design a _masking and mapping
filtering_ method to substitute the traditional filter-and-sum operation, with
the purpose of cascading coarsely denoising, dereverberation and residual
noise suppression. The proposed model, Spatial-DCCRN, has surpassed EaBNet,
FasNet as well as several competitive models on the L3DAS22 Challenge dataset.
Not only the 3D scenario, Spatial-DCCRN outperforms state-of-the-art (SOTA)
model MIMO-UNet by a large margin in multiple evaluation metrics on the multi-
channel ConferencingSpeech2021 Challenge dataset. Ablation studies also
demonstrate the effectiveness of different contributions.

Index Terms—  multi-channel, Spatial-DCCRN, speech enhancement

##  1 Introduction

Recently, with the tremendous success of deep learning, speech enhancement has
been formulated as a supervised learning problem [1, 2]. Meanwhile, multi-
channel speech enhancement is gaining increasingly interest due to the
utilization of spatial information to distinguish target speech from
interfering signal [3, 4, 5, 6]. Some challenges such as the L3DAS22 challenge
[7] and ConferencingSpeech2021 challenge [8] have been recently organized to
promote research on multi-channel speech processing.

A typical strategy is to combine DNNs with traditional beamforming techniques.
Specifically, TF-mask can be predicted by DNNs, which is used to determine
minimum variance distortionless response (MVDR) [9] and generalized eigenvalue
(GEV) [10] beamforming weights. However, as the second stage is purely based
on statistical theory and is usually irrelevant to the mask estimation, the
pre-estimation error may heavily hamper the subsequent beamforming results
[11]. More recently, neural beamformer method, including FasNet [12], EabNet
[11] and MIMO-Unet [13], have shown outstanding performance. EabNet, MIMO-Unet
and FasNet employ DNNs to estimate beamforming filters and apply filter-and-
sum operation to estimate a single-channel enhanced complex spectrum or
waveform.

In frequency domain, for EaBNet [11], two core modules are designed
accordingly, namely EM and BM. The embedding module (EM) learns the 3D
spectral and spatial embedding tensor, while the beamforming module (BM)
estimates the beamforming weights to implement filter-and-sum operation.
Moreover, MIMO-Unet [13] uses a convolutional U-Net to estimate beamforming
filters and applies filter-and-sum operation to estimate a single-channel
enhanced complex spectrum. Working in the time domain, FasNet [12] estimates
linear spatial filters for filter-and-sum beamforming.

The spatial information is of vital importance for the multi-channel scenario.
However, without any explicite spatial features as input, the above approaches
only adopt the multi-channel spectrum as the input of the model and let the
network learn spatial information implicitly. Some other strategies adopt
reference channel’s magnitude and the inter-channel phase difference (IPD) as
the input of the network [14]. However, only using phase information to
reflect channel correlation will lead to apparent information lost.
Furthermore, the above methods mostly estimate a group of filters and apply
filter-and-sum operation. Yet, when the noise and speech exist in the same TF-
bin, the phenomenon of residual noise will be more serious as well.

In this paper, to address the above problems, we propose Spatial-DCCRN for
multi-channel speech enhancement. The contribution of this work is three-fold,
summarized as follows.

  * •

We extend the cascaded sub-band and full-band processing strategy of Super
wide band DCCRN (S-DCCRN) [15] to multi-channel scenario to execute sub-
channel and full-channel processing, aiming at benefiting from both local and
global channel information processing. Different from the oracle S-DCCRN, we
use LSTM in sub/full-channel DCCRN to accept the concatenation of angle
feature embedding and encoder’s output simultaneously. In addition, the
complex mask is replaced with masking and mapping filter (MMF), aiming at
denoising and dereverberation simultaneously. Compared with monaural
processing, sub/full-channel processing strategy in multi-channel scenario can
take advantage of the spatial information meticulously and model different
channels separately.

  * •

We design an angle feature extraction (AFE) module, which only adopts the
cosIPDs feature as input, aiming at extracting frame-level angle feature. With
the help of convolution layers and denseblock, the channel- and time-
correlation information can be effectively modeled by the AFE module, which
helps the network perceive spatial information apparently.

  * •

We design a masking and mapping filtering (MMF) method to replace the typical
filter-and-sum operation. Using such cascading strategy, the masking operation
aims at dereverberation and coarsely denoising while the mapping operation is
designed to further remove residual noise.In detail, the masking operation
aims at dereverberation and coarsely denoising while mapping operation is
designed to remove residual noise. Specifically, we apply sub/full-channel
DCCRN to estimate the mapping filters. After that the masking filters are
estimated by a group of conv3d blocks which receive the stack of noisy
spectrum and mapping filters as the inputs, due to the mapping filters contain
the masking filters. With the assistance of conv3d, the magnitude mask of the
target channel can be estimated by the input of the MMF module. After applying
masking filters to magnitude, the mapping filters are employed on the coarse
real / imaginary part to acquire the enhanced speech.

After combining the contributions above, Spatial-DCCRN has surpassed several
SOTA models, including FasNet [12] and EaBNet [11], and obtains 0.956 metric
score which are composed of STOI and WER and ranks forth on the L3DAS22
challenge dataset. Moreover, with the experiments on ConferencingSpeech2021
challenge dataset, our system outperforms baseline by a large margin and also
surpasses MIMO-Unet [13] which ranked first in that challenge.

##  2 Proposed System

###  2.1 Signal Model

Assuming ${x^{p}(t)}$, with ${p=0,...,P-1}$, denotes the time domain noisy and
reverberant speech signal at the ${p}$th microphone. The signal model of
multi-channel speech enhancement in the short-time Fourier transform (STFT)
domain can be given by:

$\displaystyle X_{f,t}=c_{f}S_{f,t}+N_{f,t}$ (1)

where {${X_{f,t},S_{f,t},N_{f,t}}$} denote noisy signal, clean signal and
noise respectively with frequency index of ${f\in{1,...,F}}$ and time index of
${t\in{1,...,T}}$. Furthermore, ${c_{f}\in\mathbb{C}^{P\times 1}}$ denotes the
relative transfer function (RTF) of the source speech.

###  2.2 Multi-Channel DCCRN

Previously, S-DCCRN [15] was proposed for super wide-band speech enhancement.
The S-DCCRN is equipped with a cascaded sub-band and full-band (SAF)
processing module, aiming at benefiting from both local and global frequency
information processing. In detail, the SAF module consists of cascaded
sub/full-band DCCRN, which substitutes the complex convolution of original
DCCRN with group complex convolution. In this paper, inspired by the idea of
local and global modeling, we extend S-DCCRN to multi-channel scenario by
applying sub-channel and full-channel processing to fully utilize the spatial
information. The overall architecture of the proposed Spatial-DCCRN is shown
in Fig. 1. Similar to the oracle S-DCCRN, the learnable spectrum compression
(LSC) module, which consists of a group of trainable compression ratios, is
applied to adjust the energy of different frequency bands. The motivation of
this design is due to the fact that higher frequency band of the far-field
audio is likely to have low energy components. Moreover, the complex feature
encoder/decoder module (CFE/CFD) is adopted to extract information from multi-
channel complex spectrum. With the help of convolution layers and denseblock
[16], the CFE/CFD block can refine channel-correlation and time-correlation
information. More details of LSC and CFE/CFD can be found in [15].

Different from the original S-DCCRN, we substitute the complex masking
strategy with our proposed masking and mapping filtering (MMF) method, aiming
at denoising and dereverberation simultaneously as reverberation is a key
issue for far-field speech. Furthermore, we design a learnable angle feature
extraction (AFE) module to extract frame-level angle features. Then a modified
LSTM layer of sub/full-band DCCRN is applied to accept the concatenation of
angle features embedding and encoder’s output as input for joint modeling.
With the help of the AFE module, the model can model the spatial information
apparently. The AFE and MMF modules in our Spatial-DCCRN are introduced in
detail in the following.

###  2.3 Angle Feature Extraction

Angle feature is critical for multi-channel enhancement. A widespread strategy
is to adpot cosIPDs and reference channel’s magnitude as the input’s features.
However, concatenating those features may increase the difficulty of modeling
– the neural network has to model angle features and magnitude features
simultaneously. Another method is to employ a module to estimate the direction
of arrival (DOA) of the source speech and noise. However, due to the complex
acoustic conditions in real scenario, it is extremely difficult to accurately
estimate such DOA features. To this end, we propose a block which only adopts
the cosIPDs features as the input to estimate frame-level angle features.

As shown in Fig 2, the angle feature extraction module is similar to CFE. We
employ conv2d with a kernel size of 1 to extract high-dimensional information.
Then a dilated dense blockwhose depth is 2 is used to capture long-term
contextual angle features from time scale. Finally, a conv2d is adopted to
extract local angle features. LayerNorm and PReLU activation are placed after
each convolution layer. Afterwards, the LSTM layer of sub/full-channel DCCRN
receives the frame-level time-variant angle feature embedding as the input for
temporal dependency modeling. With the help of the angle feature extraction
module, the complex spectrum enhancement module can apparently perceive the
angle information of every frame.

Fig. 1: Network structure of the proposed Spatial-DCCRN. ”LSC” denotes
learnable spectrum compression, ”AFE” denotes angle feature extraction, ”CFE”
denotes complex feature encoder, ”CFD” denotes complex feature decoder, ”MMF”
denotes masking and mapping filtering and ”CAT” denotes concatenate. Fig. 2:
Network structure of angle feature extraction and sub/full channel DCCRN.
‘GCC‘ denotes group complex convolution, ‘GCTC‘ denotes group complex
transpose convolution and ‘CP‘ denotes convolution pathway – a convolution
layer among the encoder and decoder [17]. ”AF” denotes angle feature
embedding. Fig. 3: Network structure of masking and mapping filtering.

###  2.4 Masking and Mapping Filtering

Previous studies usually estimate a group of filter weights to do mask based
beamforming in frequency or time domain. However, when the noise and speech
exist in same TF-bin, residual noise components are relatively hard to be
removed. Aiming to suppress the noise and dereverberation simultaneously, we
design a novel masking and mapping filtering method. Specifically, we first
define the estimated clean speech as

$\displaystyle S_{f,t}={M}^{\prime}_{f,t}X_{f,t}+M^{\prime\prime}_{f,t}$ (2)

where $M^{\prime}$ and $M^{\prime\prime}$ denote the masking metrix and
mapping metrix respectively. Combining Eq. (1) and Eq. (2), ${M^{\prime}}$ and
${M^{\prime\prime}}$ can be described as

$\displaystyle M^{\prime}_{f,t}$ $\displaystyle=1/c_{f,t}$ (3) $\displaystyle
M^{\prime\prime}_{f,t}$ $\displaystyle=-N_{f,t}/c_{f,t}$

As a result, the masking operation aims to dereverb while the mapping
operation works for removing residual noise. The detailed design of MMF is
shown in Figure 3. Inspired by the collaborative reconstruction module (CRM)
in [18], we first employ masking on the magnitude domain and then apply
mapping residual operation on the complex domain. The difference between CRM
and MMF is that we apply the priority information ($M^{\prime\prime}$) to
estimate $M^{\prime}$.

As shown in Eq. (3), ${M^{\prime\prime}}$ consists of noise and the RTF of
speech, which should be estimated by a complex neural network. Therefore, the
sub/full-channel processing block is employed to estimate $M^{\prime\prime}$.
Due to ${M^{\prime\prime}}$ involves ${M^{\prime}}$, we stack
${M^{\prime\prime}}$ with the noisy complex spectrum to estimated
${M^{\prime}}$. Specifically, we first rearange the input channel to ensure
that the real and imaginary parts of the input are alternately placed along
channel axis. Then we stack the noisy spectrum and the output of SAF block as
the input of the conv3d layer. On the one hand, with the conv3d, the spectral
information exists in noisy and ${M^{\prime\prime}}$ can be well grasped. For
another, we set the stride of the dim of mic channel to 2, which can maintain
that magnitude mask of the target channel is estimated by its corresponding
real and imaginary components. After generating ${M^{\prime}}$, we apply it to
noisy magnitude. Finally, ${M^{\prime\prime}}$ is applied to the coarse
real/imaginary part to remove the residual noise.

###  2.5 Loss Function

For the learning objective, we adopt a hybrid loss function strategy.
Specifically, besides taking SI-SNR [19] as the time-domain loss function, we
also adopt the PHASEN loss to emphasize the phase of T-F bins with higher
amplitude, which can help the network to focus on the high amplitude T-F bins
where most speech signals are located [20].

Finally, the STOI loss [21] is applied to directly improve the objective
results [22]. Finally, the three losses are optimized jointly by

$\begin{cases}\mathcal{L}_{\textbf{PHASEN}}&=\left||S|^{p}-|\hat{S}|^{p}\right|^{2}+\left||S|^{p}e^{j\varphi(S)}-|\hat{S}|^{p}e^{j\varphi(\hat{S})}\right|^{2}\\\
\mathcal{L}&=\mathcal{L}_{\textbf{SI-
SNR}}+\mathcal{L}_{\textbf{STOI}}+\mathcal{L}_{\textbf{PHASEN}}\\\
\end{cases}$ (4)

where ${\hat{S}}$ and ${S}$ denote the network output and clean spectrum
respectively. Hyper-parameter ${p}$ is a spectral compression factor
empirically set to 0.3. Operator ${\varphi}$ calculates the argument of a
complex number.

Table 1: Results of various models and ablation experiments of the proposed model on L3DAS22 dataset. ‘++MMF‘ denotes applying AFE and MMF, ‘Cau‘ denotes causal, ‘Ch.‘ denotes the channel number of inputs, ‘SR‘ denotes SI-SNR, ‘PN‘ denotes PHASEN and ‘ST‘ denotes STOI. # | Model | Cau. | Ch. | Loss Function | Para.(M) | STOI | WER | Metric
---|---|---|---|---|---|---|---|---
1 | Spatial-DCCRN∗ | ✓ | 4 | SR+PN+ST | 2.34 | 0.913 | 0.086 | 0.914
2 |  \+ MMF | ✓ | 4 | SR+PN+ST | 2.34 | 0.921 | 0.08 | 0.921
3 |  \+ AFE | ✓ | 4 | SR+PN+ST | 2.61 | 0.926 | 0.075 | 0.925
4 |  \+ + MMF | ✓ | 4 | SR+PN+ST | 2.61 | 0.931 | 0.071 | 0.930
5 |  \+ + MMF | ✓ | 8 | ST | 2.61 | 0.890 | 0.671 | 0.609
6 |  \+ + MMF | ✓ | 8 | SR | 2.61 | 0.837 | 0.270 | 0.783
7 |  \+ + MMF | ✓ | 8 | PN | 2.61 | 0.941 | 0.057 | 0.941
8 |  \+ + MMF | ✓ | 8 | PN+ST | 2.61 | 0.941 | 0.053 | 0.944
9 |  \+ + MMF | ✓ | 8 | PN+SR | 2.61 | 0.946 | 0.056 | 0.945
10 |  \+ + MMF | ✓ | 8 | SR+ST | 2.61 | 0.916 | 0.118 | 0.899
11 |  \+ + MMF | ✓ | 8 | SR+PN+ST | 2.61 | 0.946 | 0.055 | 0.946
12 |  \+ + MMF | $\times$ | 8 | SR+PN+ST | 8.86 | 0.957 | 0.045 | 0.956
13 | MIMO-UNet [23] | $\times$ | 4 | L1 | 5.52 | 0.889 | 0.175 | 0.857
14 | EaBNet [11] | ✓ | 4 | SR+PN+ST | 2.84 | 0.877 | 0.110 | 0.884
15 | FasNet [12] | ✓ | 4 | SR+PN+ST | 3.7 | 0.832 | 0.241 | 0.795

##  3 Experiments

###  3.1 Datasets

We first conduct experiments to prove the effectiveness of each proposed sub-
modules on the L3DAS22 challenge dataset [7]. The objective of 3D speech
enhancement task in this challenge is to enhance speech signals immersed in
the spatial sound field of a reverberant office environment collected by 1st
order Ambisonics microphone. Specifically, the dataset contains more than
40000 virtual 3D audio environments with a duration up to 12 seconds each,
reaching a total duration of more than 80 hours. Clean utterances are selected
from the clean subset of Librispeech [24] (approximately 53$\%$ male and
47$\%$female speech) while the monophonic noise signals come from FSD50K [25].
There are a total of 1440 noise sound files, which include 14 transient noise
classes and 4 continuous noise classes. The 3D audio signals are generated by
convolving the monophonic audio signals with RIRs which is obtained by
performing a circular convolution between the recorded sound and the time-
inverted analytic signal [26]. Totally, the fixed training and validation
sets, with SNR ranging from 6 to 16 db, contain 37,398 utterances (81 h) and
2362 utterances (4 h), respectively.

Meanwhile, the proposed Spatial-DCCRN is trained and evaluated on the
ConferencingSpeech 2021 challenge dataset [8] to show its robustness on video
conferencing multi-talker scenario. In the dataset, the source speech data
comes from AISHELL-1 [27], AISHELL-3 [28], VCTK [29] and Librispeech(train-
clean-360) [24]. The speech utterances with SNR larger than 15dB are selected
for training. The total duration of clean training speech is around 550 hours.
The noise dataset is composed of MUSAN [30] and Audioset [31], with total
duration of about 120 hours. Besides these two open source datasets, 98 real
meeting room noise files recorded by high fidelity devices are also used. The
training data are generated on-the-fly and segmented into 8 s chunks in one
batch with SNR ranging from -5 to 25 dB.

###  3.2 Training setup and baselines

For the proposed models, the window length, frame shift and future frame are
25 ms, 6.25 ms and 6.25 ms, respectively, resulting in a 37.5 ms processing
time. The STFT length is 512. For the 3D speech enhancement task on L3DAS22,
all models are trained for 40 epochs with totally 3200h training data. For the
models trained on ConferencingSpeech, the total data ’seen’ by the model is
more than 9900 h after 18 epochs of training. The initial learning rate of all
models is 0.001 and will get halved if there is no loss decrease on the
validation set. We also compare the proposed Spatial-DCCRN and its ablation
components with other SOTA models on the L3DAS22 dataset.

The configuration of our Spatial-DCCRN is described as follows. The number of
channels for the sub-channel DCCRN is {32,64,64,64,128,128}, and the
convolution kernel size and stride are set to (5,2) and (2,1) respectively. In
addition, the configuration of the full-channel DCCRN is similar to the sub-
channel DCCRN except that the channel number of the first layer is 64. One
LSTM layer which consists 256 nodes and follows by a 256 $\times$ 256 fully
connected layer is adopted to process the concatenation of the encoder outputs
of the sub/full-channel DCCRN and the angle feature embeddings. Each
encoder/decoder module handles the current frame and one previous frame. The
output number of channel of the complex feature encoder/decoder module is 32,
and the depth of denseblock is 5. LayerNorm and PReLU are performed after each
convolution, except for the last layer of the CFD module. For the AFE module,
the number of hidden channels is 16 and the depth of denseblock is 2. For the
MMF module, the kernel size, the number of output channels and the step size
of the first conv3d are set to (2, 5, 3), 8 and (2, 1, 1) respectively. In
addition, the kernel size, output channel and step size of the second conv3d
are (1, 5, 3), 1 and (1, 1, 1) respectively.

###  3.3 Experimental results and discussion

As presented in Table LABEL:tab:l3d, ablation studies are conducted to
evaluate the effectiveness of different model components of Spatial-DCCRN,
including a) Spatial-DCCRN without AFE and MMF (Spatial-DCCRN∗), b) Spatial-
DCCRN without AFE, c) Spatial-DCCRN without MMF, d) different input channels
of the observed signal, e) the non-causal version of Spatial-DCCRN and f) the
ablation on different loss function. For the non-causal model, we substitute
the LSTM in Spatial-DCCRN with BLSTM and look ahead one frame in each
convolution layer. The evaluation metric is a combination of STOI [21] and
WER, according to the official challenge rule [7]:

$\text{Metric}=(\text{STOI}+(1-\text{WER}))/2$ (5)

The WER is computed based on the transcription of the estimated target signal
and that of the reference signal, both decoded by a pre-trained Wav2Vec2.0
based ASR model [32].

It can be seen from the results that the performance of Spatial-DCCRN∗ (the
base version) is obviously better than MIMO Unet, resulting in 0.057 metric
improvement with a smaller model and causal inference. Compared with frequency
domain SOTA model EaBNet and time domain SOTA model FasNet, Spatial-DCCRN∗
yields 0.03 and 0.119 metric improvement respectively, which can prove that
our model is competitive. Adding the MMF module and the AFE module leads to
0.007 and 0.011 metric gain respectively. Furthermore, the AFE module yields
relatively better performance over the MMF module. This is because the angle
information in multi-channel scenario is more essential. Combining AFE and MMF
together, we achieve 0.016 metric improvement over the base version Spatial-
DCCRN∗. Moreover, when more channels are available (4$\rightarrow$8),
considerable improvements for those metrics can be achieved. Finally, when we
extend Spatial-DCCRN to its non-causal version, the best metric score is
obtained.

By comparing different loss functions, several observations can be made. 1)
From experiment No. 1, 7, and 8, we can observe that when only the STOI Loss
is adopted, the performance is bad. However, when combining it with other loss
functions, better metric scores are obtained. It can be considered that the
STOI Loss is an useful auxiliary loss function. 2) From experiment No. 6 and
7, it is found that the PHASEN loss yields better performance over SI-SNR.
This is because WER and STOI are sensitive to spectrum distortion, while time
domain loss is unstable for those metrics. 3) From experiment No. 7 and 9,
compared with the single PHASEN Loss, when the SI-SNR loss is further
employed, considerable improvement can be achieved. This shows that optimizing
the model from both time and frequency domains is beneficial.

Table 2: Results of various models on ConferencingSpeech2021 challenge development set. # | Model | Cau. | PESQ | STOI | E-STOI | SI-SNR
---|---|---|---|---|---|---
1 | Noisy | - | 1.515 | 0.823 | 0.690 | 4.474
2 | Baseline | ✓ | 1.999 | 0.888 | 0.780 | 9.159
3 |  MIMO-Unet [13] | ✓ | 2.215 | 0.908 | 0.817 | 9.287
4 | Spatial-DCCRN | ✓ | 2.523 | 0.923 | 0.847 | 10.167

Figure 4 illustrates an example on the proposed masking and mapping operation
(MMF). It can be observed that the noisy spectrum was coarsely denoised and
dereverbed after the masking operation. As expressed in Eq. (3),
${M^{\prime}_{f,t}}$ is the inverse of ${c_{f,t}}$. As a result, the masking
operation does good to dereverberation. It should be noted that we do not
guide the learning of the mask operation during training. As shown in Figure 4
(c), after the masking operation, the mapping operation focuses on the lost
detail from the complex-domain perspective and removes the residual noise.

Fig. 4: Masking and mapping filtering results on a testing noisy clip.

We further evaluate Spatial-DCCRN on the development set provided by
ConferencingSpeech2021 challenge task 1. The official baseline system of
ConferencingSpeech2021 is a network composed of 3 LSTM layers and a dense
layer, and the model’s input is the complex spectrum of the first channel
together with IPD features. Here we also take the first rank system MIMO-Unet
[13] for comparison as well measured inPESQ [33], STOI, E-STOI [34] and SI-
SNR. As shown in table LABEL:tab:conference, Spatial-DCCRN outperforms the
baseline by a large margin and clearly surpasses MIMO-Unet in all metrics.

##  4 Conclusions

In this paper, we propose a novel multi-channel complex domain denosing
network – Spatial-DCCRN, which is extended from S-DCCRN [15]. With the help of
the cascaded sub-channel and full-channel processing strategy, the model can
benefit from both local and global channel information processing.
Importantly, an angle feature extraction module is adopted to extract frame-
level angle feature, aiming at assisting the network to perceive spatial
information more apparently. Finally a masking and mapping filtering method is
employed to replace the traditional filter-and-sum operation. The proposed
Spatial-DCCRN model obtains excellent performance with 0.956 metric score on
the L3DAS22 dataset. Furthermore, Spatial-DCCRN surpasses the first-rank model
MIMO-Unet on the task1 development set provided by the ConferencingSpeech 2021
challenge. 111Enhanced clips can be found from
https://imybo.github.io/Spatial-DCCRN/

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