Amplify-and-Forward Strategy Using MRC Reception Over FSO Channels with Pointing Errors

Abstract

Free-space optical (FSO) systems offer ultrahigh data rates for a number of terrestrial applications. This paper presents what we believe is the first characterization of average bit error rate (BER) for a cooperative FSO system that uses channel side information-assisted amplify-and-forward (AF) relaying when line-of-sight (LOS) is available. This paper uses maximum ratio combining reception at the destination node to maximize the output signal-to-noise ratio under pointing error effects. An asymptotic closed-form expression for the BER is obtained over gamma–gamma fading channels with pointing errors. The developed asymptotic expression is used to analyze the diversity order gain in relation to the direct path link. The maximum diversity order is determined to be β SD + min ( β SR , β RD ) , where β SD , β SR , and β RD are atmospheric turbulence parameters corresponding to the source–destination, source-relay, and relay-destination links, respectively. Moreover, optimal relay placement, which plays an important role in the performance, is derived by taking full advantage of asymptotic expressions without the need to use optimization methods. Obtained results corroborate that a higher diversity order gain and, hence, a better BER performance, is achieved compared to the direct path link and equivalent AF dual-hop transmissions.

Full text

Amplify-and-Forward Strategy Using
MRC Reception Over FSO Channels
with Pointing Errors
Rubén Boluda-Ruiz, Antonio García-Zambrana, Carmen Castillo-Vázquez,
Beatriz Castillo-Vázquez, and Steve Hranilovic
Abstract—Free-space optical (FSO) systems offer ultra-
high data rates for a number of terrestrial applications.
This paper presents what we believe is the first characteri-
zation of average bit error rate (BER) for a cooperative FSO
system that uses channel side information-assisted am-
plify-and-forward (AF) relaying when line-of-sight (LOS)
is available. This paper uses maximum ratio combining
reception at the destination node to maximize the output
signal-to-noise ratio under pointing error effects. An
asymptotic closed-form expression for the BER is obtained
over gamma–gamma fading channels with pointing errors.
The developed asymptotic expression is used to analyze
the diversity order gain in relation to the direct path
link. The maximum diversity order is determined to be
βSD minβSR,βRD , where βSD,βSR , and βRD are atmospheric
turbulence parameters corresponding to the source–desti-
nation, source-relay, and relay-destination links, respec-
tively. Moreover, optimal relay placement, which plays an
important role in the performance, is derived by taking full
advantage of asymptotic expressions without the need to
use optimization methods. Obtained results corroborate
that a higher diversity order gain and, hence, a better
BER performance, is achieved compared to the direct
path link and equivalent AF dual-hop transmissions.
Index Terms—Amplify-and-forward (AF); Bit error rate
(BER); Cooperative diversity; Free-space optical (FSO);
Line-of-sight (LOS); Maximum ratio combining (MRC).
I. INTRODUCTION
Free-space optical (FSO) communication systems have
been thoroughly investigated for decades, and have
traditionally been presented as an alternative solution to
radio frequency (RF) links. Now, FSO technology is further
considered as an emerging broadband wireless access sol-
ution. A significant number of applications can be provided
by FSO technology, such as last-mile access networks, high
data-rate links between buildings, next-generation wire-
less broadband networks, and backhaul for wireless cellu-
lar networks, among others [1]. Despite the advantages of
FSO links, atmospheric turbulence, pointing errors, and
atmospheric effects such as fog greatly deteriorate system
performance. Different techniques have widely been pro-
posed in the literature, such as multiple-input/multiple-
output (MIMO) systems and channel coding to mitigate
the impact of atmospheric turbulence on terrestrial FSO
links under the presence of pointing errors. Although
MIMO techniques have been considered in FSO systems,
these techniques may not be practical because of hardware,
size, or even cost limitations. An alternative approach is
the deployment of cooperative communications to extend
the coverage area and achieve spatial diversity, which
avoids some issues related to MIMO FSO communication
systems.
Cooperative strategies such as amplify-and-forward
(AF) and detect-and-forward (DF) have been considered
in-depth by different authors in both serial and parallel re-
laying [2–14]. Even though AF relaying has been a high-
lighted research topic in the last several years, most
contributions have been focused on multihop transmis-
sions [i.e., without line-of-sight (LOS)]. As demonstrated
in [3,6,11,14] using DF relaying, the use of LOS can notably
improve the performance in cooperative FSO systems even
when the FSO link is strongly affected by atmospheric tur-
bulence and pointing errors. In [4], a relay-assisted system
that uses optical AF relaying and LOS is proposed to evalu-
ate the error-rate performance over log-normal (LN) fading
channels and equal gain combining (EGC) reception by
using the photon-counting method without considering
pointing errors. Unlike [4], this paper provides a more
realistic performance analysis in terms of error probability
for practical cooperative FSO networks based on AF re-
laying with LOS and maximum ratio combining (MRC)
reception under the presence of pointing errors. An MRC
combiner has already been adopted in [15] to analyze
the performance of RF systems over Nakagami-m fading
channels using AF relaying.
In this paper, we characterize, for what we believe is the
first time, the average bit error rate (BER) for a cooperative
FSO system that uses channel side information assisted
(CSI-assisted) AF relaying under pointing error effects.
https://doi.org/10.1364/JOCN.10.000545
Manuscript received January 17, 2018; revised March 21, 2018; accepted
March 23, 2018; published April 19, 2018 (Doc. ID 319513).
R. Boluda-Ruiz (e-mail: rbr@ic.uma.es), A. García-Zambrana, and B.
Castillo-Vázquez are with the Department of Communications Engineering,
Universidad de Málaga, Málaga, Spain.
C. Castillo-Vázquez is with the Department of Mathematical Analysis,
Statistics and Operations Research and Applied Mathematics, Universidad
de Málaga, Málaga, Spain.
S. Hranilovic is with the Department of Electrical and Computer
Engineering, McMaster University, Hamilton, Ontario, Canada.
Boluda-Ruiz et al. VOL. 10, NO. 5/MAY 2018/J. OPT. COMMUN. NETW. 545
1943-0620/18/050545-08 Journal © 2018 Optical Society of America

To the best of our knowledge, there is no other work regard-
ing AF relaying scenarios that uses LOS to improve the
BER performance affected by pointing errors. In this
way, an asymptotic closed-form expression for the BER
is obtained using MRC reception at the destination, and
electrical amplification at the relay terminal. The MRC
combiner is considered since this combining technique
maximizes the output signal-to-noise ratio (SNR), which
is especially important in cooperative FSO systems. The
reason behind this is that cooperative communication sys-
tems need to maximize the SNR due to the fact that the
SNR at the destination is significantly deteriorated as a
consequence of the source-relay-destination path. In light
of the obtained results, the relay location plays an essential
role in the diversity order. On other hand, a greater robust-
ness against larger amounts of misalignment is achieved
when LOS is deployed. Monte Carlo simulations are fur-
ther included to confirm the analytical results.
This paper has a total of five sections. System and chan-
nel models are described in Section II. In Section III,we
obtain the asymptotic closed-form expression for the aver-
age BER corresponding to the AF cooperative protocol by
using MRC reception over GG atmospheric turbulence
channels with pointing errors. This asymptotic closed-form
expression is totally valid for practical cooperative FSO
networks and FSO scenarios with larger amounts of mis-
alignment. We present some numerical results and discus-
sions in Section IV, and conclude the paper in Section V.
II. SYSTEM AND CHANNEL MODELS
A. System Model
Following Fig. 1, which shows a three-way FSO commu-
nications setup, a CSI-assisted AF cooperative protocol us-
ing MRC reception at the destination (D) node and LOS,
called the AF-LOS cooperative protocol, is proposed here.
Each of the FSO links are based on intensity modulation
and direct detection (IM/DD), and on–off keying (OOK)
modulation, as shown in Fig. 1. The AF-LOS cooperative
protocol works in two stages. In the first stage, the source
(S) node sends its own data to the destination node and
the relay (R) node (i.e., the source node transmits the same
information to the relay node and destination node). In
the second stage, the relay node sends the received data
from source node in the first stage to the destination node.
The relay node uses an electrical amplifier with gain Gto
amplify the received signal from source node and then for-
wards it to the destination node. It must be mentioned that
in a full-CSI relaying case, instantaneous CSI of the S-R
link is assumed to be available at the relay node to calculate
the corresponding gain before forwarding the received
signal. Also note that the CSI is only known at the receiver
side of the relay and destination nodes. Finally, the received
signals from source node and relay node are combined at the
destination node. Then, the combined output is detected.
Note that the same information rate is obtained at destina-
tion node compared to the direct transmission (DT) or the
direct path link without using a cooperative strategy.
As observed in Fig. 1, synchronization mechanisms
are required at the destination to detect data received from
the source and the relay. In this way, data received from the
source can be stored in a buffer at the destination for
further detection. Without a loss of generality, possible
synchronization errors on BER performance are beyond
the main objective of this study.
B. Channel Model
The received electrical signal for each FSO link of
this cooperative system is given by
ymhmRx zm, (1)
where Ris the detector responsivity, assumed hereinafter
set to be the unity, xis the transmitted optical signal, hmis
the gain of the channel between the source and the receiver,
and zmis additive white Gaussian noise (AWGN) with zero
mean and variance σ2
nN0∕2A2∕Hz. The transmitted
optical signal is either 0 or 2Ptwhere Ptis the average
optical power. In the following, the subscript mis used
to represent the different FSO links considered here (i.e.,
S-D, S-R, and R-D). The channel gain is a product of several
factors (i.e., hmLm·ha·hp, atmospheric path loss Lm,
atmospheric turbulence ha, and geometric spread and
pointing errors hp). Note that both atmospheric turbulence
and pointing errors can be assumed as statistically
independent random variables. The path loss, Lm, is deter-
mined by the exponential Beers–Lambert law as
Lme−Φdm, where dmis the link distance and Φis the
atmospheric attenuation coefficient [16].
With regard to the statistical channel model, atmos-
pheric turbulence is modeled according to the GG distribu-
tion to consider a wide range of turbulence conditions [17].
Pointing errors at the receiver are modeled assuming a
model of misalignment where the effect of beam width,
Fig. 1. Block diagram of a three-node FSO system, where dSD is
S-D link distance, (xR,yR) is the relay location, Gamplifies the
received signal from S, and PD stands for photodetector.
546 J. OPT. COMMUN. NETW./VOL. 10, NO. 5/MAY 2018 Boluda-Ruiz et al.

detector size, and jitter variance are considered [18].
The attenuation due to geometric spread and pointing
errors is approximated assuming a Gaussian spatial inten-
sity profile of a beam waist radius, shown in [18]: Eq. (9),
as hpr;z≈A0exp−2r2∕ωz2
eq , where v
π
pa∕
2
pωz,A0
erfv2is the fraction of the collected power at r0,
ω2
zeq ω2
z
π
perfv∕2vexp−v2is the equivalent beam
width, and ais the radius of a circular detection aperture.
The beam width ωzcan be approximated by ωzθz·dm,
where θzis the divergence angle defining the increase in
beam radius with a link distance.
An exact closed-form expression for the composite
fading channel was obtained in [19], Eq. (12), in terms of
the Meijer G-function, shown in [20], Eq. (9.301). Here,
an asymptotic closed-form expression for this probability
density function (PDF) is adopted, noting that the first
term of the Maclaurin series expansion of fhmhis
fhmhamhbm−1Ohbm. This approximation is required
to give a deeper understanding of the influence of atmos-
pheric turbulence and pointing errors on BER performance
corresponding to AF relay-assisted FSO systems. As pre-
sented in [21,22], the PDF is approximated by a single
polynomial term as
fhmh≐amhbm−1
8
>
>
>
<
>
>
>
:
φ2αββΓα−β
A0LmβΓαΓβφ2−βhβ−1,φ2>β
φ2αβφ2Γα−φ2Γβ−φ2
A0Lmφ2ΓαΓβhφ2−1,φ2<β
, (2)
where bmminβm,φ2
mand φ2ω2
zeq ∕4σ2
sis the ratio be-
tween the equivalent beam radius at the receiver, and the
corresponding pointing error displacement standard
deviation (jitter) at the receiver. It is noteworthy to men-
tion that the asymptotic expression given in Eq. (2)is
dominated by bm−1. Moreover, Γ·is the Gamma function,
and (α,β) can be directly linked to physical parameters
through [17]
αexp0.49σ2
R∕11.11σ12∕5
R7∕6−1−1, (3a)
βexp0.51σ2
R∕10.69σ12∕5
R5∕6−1−1, (3b)
where σ2
R1.23C2
nκ7∕6d11∕6
mis the Rytov variance assum-
ing plane wave propagation, which is a measure of optical
turbulence strength. Here, κ2π∕λis the optical wave
number and C2
nstands for the altitude-dependent index
of the refractive structure parameter. It is demonstrated
that the relationship α>βis always satisfied, and βis
lower bounded above 1 as the turbulence strength in-
creases [23]. Here, we assume that hSD ,hSR, and hRD
are statistically independent.
III. ASYMPTOTIC BER ANALYSIS
Asymptotic closed-form expressions are obtained to
quantify the average BER at high SNR for this cooperative
FSO system by taking full advantage of the asymptotic
expression given in Eq. (2). According to MRC reception,
the combined output can easily be expressed assuming
CSI at the receiver as
yAF
Tw1ySD w2ySRD
x
2w1hSD w2hSRhRD G
w1zSD w2zSRhRD GzRD, (4)
where w1and w2are the optimum combining weights
corresponding to the received SNR at the destination
node, and they can readily be derived as w1hSD and
w2hSRhRD G∕1h2
RDG2. Due to the fact that the
source node transmits the same information to the relay
node and destination node during the first stage, the di-
vision by 2 is considered in Eq. (4) to hold the average op-
tical power at a constant level of Pt. In this sense, each of
lasers corresponding to the source node transmits an aver-
age optical power of Pt∕2. The variable gain, G, is selected
at the relay node to satisfy its power constraint, and is
given by
G2
P2
t
P2
th2
SR N0
s≈2
hSR
:(5)
The multiplication by 2 is also considered in Eq. (5)to
ensure an average optical power of Ptsince only one laser
is used at the relay node. By using the optimum combining
weights, the instantaneous received SNR at the destina-
tion node is given by
γAF-LOS
Tγh2
SD 4h2
SRh2
RD
h2
SR 4h2
RDγh2
T, (6)
where γP2
tTb∕N0is the received electrical SNR in
absence of turbulence and pointing errors, where Tbis
the bit period. In this way, the asymptotic BER perfor-
mance corresponding to the AF-LOS cooperative protocol
is obtained as
PAF-LOS
b≐Z∞
0
Qγopt

2
phTfhThTdhT, (7)
where Q·is the Gaussian Q-function, and γopt is the opti-
cal SNR in the absence of turbulence given by
γopt 
γ
pPt
Tb
p∕
N0
p. Next, the asymptotic closed-
form expression for the PDF of h2
Tis derived. Notice that
h2
Tis expressed as in Eq. (6)as
h2
Th2
SD 4h2
SRh2
RD
h2
SR 4h2
RD h2
SD h2
SRD:(8)
According to Eq. (2), we first derive the asymptotic
PDF of h2
SD as
fh2
SD h≐aSD
2hbSD
2−1:(9)
Boluda-Ruiz et al. VOL. 10, NO. 5/MAY 2018/J. OPT. COMMUN. NETW. 547

Now, we derive the asymptotic PDF of h2
SRD. To the
best of the authors’knowledge, a closed-form analytical
derivation of the h2
SRD statistics given in Eq. (8) is math-
ematically intractable. Therefore, we approximate h2
SRD
as in [24]as
h2
SRD 4h2
SRh2
RD
h2
SR 4h2
RD
≈minh2
SR,4h2
RD:(10)
The asymptotic PDF of h2
SRD is derived from its cumula-
tive density function (CDF) as fh2
SRD h d
dh Fh2
SRD h. Note
that an asymptotic expression for the CDF can easily be
derived from Eq. (2)asFhmh≐am∕bmhbm. Then, the
CDF of minh2
SR,4h2
RDcan be written as in ([24], Eq. (4)) as
Fh2
SRD h≃Fh2
SR hFh2
RD h∕4−Fh2
SR hFh2
RD h∕4, (11)
where Fh2
SR hand Fh2
RD hare the CDFs of h2
SR and h2
RD,
respectively. Similar to Eq. (9), fh2
SR hand fh2
RD hare
derived, then Fh2
SR hand Fh2
RD hare also derived. After
some algebraic manipulations, we can obtain the asymp-
totic expression for the CDF of h2
SRD. Finally, the corre-
sponding PDF is obtained via fh2
SRD hd
dh Fh2
SRD h as
fh2
SRD h≐aSR
2hbSR
2−1aRD
2bRD1hbRD
2−1
aSRaRD bSR bRDhbSR
2bRD
2−1
2bRD1bSR bRD
:(12)
Since h2
SD and h2
SRD are statistically independent, the
resulting PDF of h2
Th2
SD h2
SRD is derived as
fh2
ThTL
−1fMh2
SD −t·Mh2
SRD −tg, (13)
where Mh2
SD tand Mh2
SRD tare the moment-generating
functions (MGF) of h2
SD and h2
SRD, respectively. Note that
Mh2
SD −tand Mh2
SRD −tare the two-sided Laplace trans-
forms of fh2
SD hand fh2
SRD h, respectively. Hence, the PDF
of h2
Tis obtained via inverse Laplace transform. Finally,
taking into account that the PDF of hTis derived
as fhTh≐d
dh Fh2
Th2, the corresponding asymptotic
closed-form expression for the PDF of hT,fhTh,is
given by
fhTh≐ΓbSD∕2ΓbSR ∕2hbSDbSR −1
2aSDaSR −1ΓbSD bSR∕2
ΓbSD∕2ΓbRD ∕2hbSDbRD −1
2bRD1aSD aRD−1ΓbSD bRD ∕2:(14)
Clearly, the above asymptotic PDF is dominated by
bSD minbSR,bRD −1. It is noteworthy to mention that
amand bmdepend on the relationship between φ2and β
for any FSO link, as can be observed in Eq. (2), corroborat-
ing that the diversity order does not depend on pointing
errors when the condition φ2>βholds. Now, substituting
Eq. (14) into Eq. (7), we can evaluate the integral in Eq. (7)
making use of the relationship erfcx2Q
2
pxthat is in
[20], Eq. (6.287). Then, we compute the integral with the
help of [20], Eq. (6.281), obtaining an asymptotic closed-
form solution for the BER corresponding to the proposed
AF-LOS cooperative protocol, as can be seen in Eq. (15).
Interestingly, it is simple to prove that the average BER
behaves asymptotically as Gcγ−Gd, where Gcand Gdde-
note coding gain and diversity order, respectively [25].
At a high SNR, the diversity order represents the slope
of the BER versus the SNR curve and the coding gain
the shift of the curve in SNR. According to Eq. (15), the di-
versity order gain, GAF-LOS
d, corresponding to the AF-LOS
cooperative protocol in relation to the direct transmission
is given by
PAF-LOS
b≐aSRaSD ΓbSR∕2ΓbSD ∕2ΓbSD bSR 1∕2
23−bSD−bSR ΓbSD bSR 2∕2
π
pγ−bSDbSR 
opt ,bSR <b
RD, (15a)
PAF-LOS
b≐aRDaSD ΓbRD2ΓbSD ∕2ΓbSD bRD 1∕2
23−bSD ΓbSD bRD 2∕2
π
pγ−bSDbRD 
opt ,bSR >b
RD, (15b)
GAF-LOS
dbSD minbSR,bRD 
bSD
1minbSR,bRD 
bSD
:(16)
Note that the above diversity gain is always greater
than 1 regardless of the pointing error effects and atmos-
pheric turbulence considered in this study.
At this point, it can be interesting to compare and con-
trast the BER performance derived here for cooperative
strategies based on AF-LOS using MRC reception with a
DH transmission based on AF relaying, which we call
548 J. OPT. COMMUN. NETW./VOL. 10, NO. 5/MAY 2018 Boluda-Ruiz et al.

AF-DH transmission, where LOS is not available. It should
be noted that the asymptotic closed-form expression ob-
tained for AF-DH transmission is not the contribution of
this paper and is reproduced here for convenience to make
a fair comparison between both cooperative strategies. A
multihop transmission employing CSI-assisted AF re-
laying over GG fading channels with pointing errors was
analyzed in [9,10], where closed-form asymptotic expres-
sions were not obtained to study the impact of this
three-way communications setup on the diversity order
gain. Similar to the proposed AF-LOS cooperative protocol,
the asymptotic closed-form expression for the PDF of hT
corresponding to the AF-DH transmission is directly re-
lated to hSRD, but taking into account that both S-R and
R-D links transmit the same average optical power (i.e.,
Pt). Hence, the variable gain at relay node is redefined
as G1∕hSR for an AF-DH transmission. Therefore, we
can readily obtain the asymptotic closed-form solution
for the BER corresponding to the AF-DH transmission as
PAF-DH
b≐aminΓ1bmin ∕2
2bmin 
π
pγ−bmin
opt , (17)
where bmin minbSR,bRD . As in Eq. (15), it can be de-
duced from Eq. (17) that the diversity order gain corre-
sponding to the AF-DH transmission is given by
GAF-DH
dminbSR,bRD ∕bSD:(18)
Unlike Eq. (16), the above diversity order gain is not al-
ways greater than 1. It is noteworthy to mention that both
the asymptotic closed-form expression in Eq. (17) and di-
versity order gain in Eq. (18) had not been derived in
any earlier work to the best of our knowledge [9,10].
Finally, we can conclude that the asymptotic closed-form
expressions obtained in this work allow us to carefully
study the BER performance of practical FSO networks
based on AF relaying and affected by atmospheric turbu-
lence and pointing errors. These expressions are very use-
ful and accurate to compute the BER performance due to
the fact that a typical BER performance target is set to 10−6
for most practical FSO links, as we will see in the section.
IV. NUMERICAL RESULTS
In this section, we show some numerical results for the
asymptotic BER performance corresponding to the pro-
posed cooperative strategy under clear visibility conditions
of 16 km with C2
n1.7 × 10−14 m−2∕3. The parameters α
and βare calculated from Eq. (3). Pointing errors are
present here, assuming a transmit divergence (θz)at
1∕e2of 1 mrad [26], and different jitter angles (θs), which
can take values up to 0.4 mrad [27]. Note that the system
configuration adopted in Table Iis used in most practical
terrestrial FSO links [26].
To illustrate the importance of the AF-LOS cooperative
strategy on BER performance, the diversity order gain in
Eq. (16) is plotted in Fig. 2(a) as a function of the relay
location xR,yRfor a source–destination link distance of
dSD 3 km. The source node is located at the origin of a
Cartesian system. Note that the condition φ2
m>βmis
satisfied for each FSO link in Fig. 2(a) and, hence, these
results are independent of pointing errors [i.e., atmos-
pheric turbulence is the dominant effect when pointing
error values of θz,θs1, 0.1mrad are considered]. It
must be mentioned that most practical FSO systems oper-
ate under this desirable condition [26]. These curves are
obtained from the intersection of two expressions: βSD 
βRD∕βSD and βSD βSR ∕βSD, as deduced from Eq. (16).
At the same time, the diversity order gain corresponding
TABLE I
FSO SYSTEM SETTINGS
Parameter Symbol Value
S-D link distance dSD 3km
Index of the refractive
structure parameter
C2
n1.7 × 10−14 m−2∕3
Visibility (clear sky) V16 km
Wavelength λ1550 nm
Responsivity R1 A/W
Receiver aperture diameter D2a10 cm
Transmit divergence θz1 mrad
Normalized beam width ωz∕a≃20 · dm(km)
Jitter angle θs0.1–0.4 mrad
Normalized jitter σs∕a≃f2−8g·dm(km)
Fig. 2. (a) Diversity order gain, and (b) BER performance for a
S-D link distance of dSD 3 km and a relay location of xR,yR
1.2, 0.5km under different pointing error values.
Boluda-Ruiz et al. VOL. 10, NO. 5/MAY 2018/J. OPT. COMMUN. NETW. 549

to the AF-DH transmission obtained in Eq. (18) is also in-
cluded in Fig. 2(a) to make a fair comparison and highlight
the advantages of using LOS in cooperative FSO systems,
as demonstrated in [6,11,14] using DF relaying. It can be
observed that the maximum diversity order is obtained
when dSR dRD dSD∕2. As expected, the available
diversity order strongly depends on the relay placement,
achieving a much higher diversity order gain than the
AF-DH transmission as a consequence of using LOS.
Additionally, these results are totally valid for any pair
of θz,θsas long as the condition φ2
m>βmholds for each
link. The latter will be checked in Fig. 2(b). A notable im-
provement in BER performance is observed when com-
pared to the two-transmitter case (diversity order gain is
always two) as well as an alternative to other cooperative
protocols based on AF relaying [9,10], presenting a diver-
sity order gain quite superior to 2 or even 3 for some relay
locations. It must also be mentioned that the diversity or-
der gain corresponding to the AF-LOS cooperative protocol
tends to a constant level equals 1 βSR ∕βSD on the right
side (i.e., when xR>d
SD∕2), and 1 βRD ∕βSD on the left
side (i.e., when xR<d
SD∕2). This constant level is greater
in strong turbulence than moderate turbulence due to the
fact that βtends to 1 faster as the strength of the atmos-
pheric turbulence increases and, hence, the relations
βSR∕βSD and βRD ∕βSD are approximately equal to 1 in
strong turbulence.
The results corresponding to this asymptotic BER per-
formance analysis are depicted in Fig. 2(b) as a function
of the normalized optical SNR γopt dB. A source–destina-
tion link distance of dSD 3 km is considered for a relay
location of xR,yR1.2, 0.5km together with pointing
error values of θz,θsf1, 0.1,2, 0.3g mrad. To confirm
the accuracy and usefulness of the derived expression,
Monte Carlo simulation results, where the FSO link is
modeled using Eq. (1), are included by generating the cor-
responding random variables from the exact combined
PDF and no approximations including turbulence condi-
tions and pointing errors. Due to the long time involved,
Monte Carlo simulation results only up to 10−7are consid-
ered in this analysis. As can be seen in Fig. 2(b), the asymp-
totic expression derived in Eq. (15) for the average BER
corresponding to the AF-LOS cooperative protocol is in
good agreement with these simulation results as well as
the asymptotic expression derived in Eq. (17) for the
AF-DH transmission. At the same time, we also consider
the BER performance for DT to establish a benchmark,
whose asymptotic BER performance was derived in [12],
Eq. (8), as
PDT
b≐aSDΓbSD 1∕2
2bSD 
π
pγ−bSD
opt :(19)
Importantly, it can be proven that these BER results are
also in good agreement with the previous results shown in
Fig. 2(a) in relation to the diversity order gain. Diversity
gains of 2.6 and 1.6 (indicated by black “×”) can be seen
when the relay node is located at xR,yR1.2, 0.5km
for AF-LOS cooperative protocol and AF-DH transmission,
respectively. As mentioned before, these diversity gain
values can be obtained for any pair of θz,θsas long as
the condition φ2
m>βmholds for each FSO link (i.e., for
θz,θs1, 0.1mrad and θz,θs2, 0.3mrad), where
the only difference is that pointing error values of θz,θs
1, 0.1mrad offer a coding gain of ≈12 dB in relation to
θz,θs2, 0.3mrad in this case.
To observe how the diversity order gain derived in
Eq. (16) corresponding to the AF-LOS cooperative protocol
is affected by larger amounts of misalignment, the results
are repeated in Fig. 3(a) over a variety of θsvalues for a
source–destination link distance of dSD 3 km and a ver-
tical displacement of yR0.25 km. A transmit divergence
value of θz1 mrad together with jitter angle values of
θsf0.25, 0.3, 0.35, 0.4gmrad are considered. Unlike
Fig. 2(a), the condition φ2
m>βmmight not be always satis-
fied for each FSO link as jitter values increase. Actually,
this condition is not satisfied when the relay node is located
at a nearby placement between the source node and desti-
nation node for jitter angle values of θs≥0.3 (blue, cyan,
and green colors) in S-R and R-D links. The diversity order
gain depends on pointing errors (i.e., 1 φ2
SR∕βSD or
1φ2
RD∕βSD ), and remains constant when the condition
φ2
m>βmdoes not hold. Notice that the range of xRvalues
for which the condition φ2
m>βmis not satisfied, increases
as θs. Moreover, bSD βSD for all jitter values considered in
Fig. 3(a) since a greater normalized beam width is obtained
in S-D link and, hence, the relationship φ2
SD >βSD always
(a)
(b)
Fig. 3. (a) Diversity order gain and (b) BER performance for a S-D
link distance of dSD 3 km and a relay location of xR,yR
2, 0.25km under larger amounts of misalignment.
550 J. OPT. COMMUN. NETW./VOL. 10, NO. 5/MAY 2018 Boluda-Ruiz et al.

holds. As expected, the BER performance is considerably
decreased when pointing errors become dominant in rela-
tion to atmospheric turbulence. These comments are con-
trasted in Fig. 3(b) for BER performance when pointing
error values of θz,θsf1, 0.25,1, 0.4g mrad are as-
sumed for a relay location of xR,yR2, 0.25km. It
must be highlighted that the AF-LOS cooperative protocol
is able to keep a much greater robustness than the AF-DH
transmission for larger amounts of misalignment, such as
θz,θs1, 0.4mrad. As in previous figures (using black
marks), diversity gains of 2.46 and 2.02 can be seen for the
AF-LOS cooperative protocol as well as 1.46 and 1.02 for
AF-DH transmission when pointing errors values of
θz,θs1, 0.25mrad and θz,θs1, 0.4mrad are
assumed, respectively.
V. CONCLUSIONS
Research into a three-way FSO communications setup
with AF relaying using MRC reception is carried out over
GG atmospheric turbulence channels with zero boresight
pointing errors. What we believe is a novel asymptotic
closed-form solution for the BER is derived, which has been
validated through Monte Carlo simulations with very high
precision.
We can conclude that a cooperative strategy based on AF
relaying and making use of LOS is an interesting approach
to mitigate the combined effect of atmospheric turbulence
and pointing errors. This kind of relaying technique can be
applied to extend the coverage area and achieve spatial di-
versity without investing in extra hardware (i.e., to achieve
a much higher diversity order as well as a greater robust-
ness against larger amounts of misalignment). Hence, LOS
always helps to achieve a better BER performance regard-
less of atmospheric turbulence and pointing errors. These
gains in BER performance occur despite the fact that the
atmospheric turbulence and pointing errors are more se-
vere due to the increased distances traversed compared
to S-R and R-D links. A remarkable improvement is
obtained not only compared to the equivalent AF-DH
transmission but also to the non-cooperative case with
two transmitters.
In light of the asymptotic closed-form expression ob-
tained in this analysis, it is also concluded that the diver-
sity order is strongly dependent on the relay placement. As
expected and proved in [8] using optimization methods, the
optimal relay location is obtained when the relation dSR 
dRD dSD∕2 holds. This latter is drawn from a diversity
order gain point of view, which was derived by taking full
advantage of the asymptotic behavior of FSO systems with-
out the need to use optimization methods. In this case, the
diversity order is maximum for the AF-LOS cooperative
protocol and the equivalent AF-DH transmission, with fur-
ther atmospheric turbulence being the dominant effect in
relation to pointing errors. Once again, it is corroborated
that practical FSO networks operating under this desirable
condition provide not only a higher diversity order but also
a higher coding gain.
From the relevant results obtained in this paper, we be-
lieve that investigating the impact of adding more relays on
the diversity order of AF relaying combined with tech-
niques based on optical path selection using LOS can be
interesting topics for future research.
ACKNOWLEDGMENT
This work was partially performed at McMaster University
during a research stay of R. Boluda-Ruiz. The authors
would like to acknowledge support received from the
FOCAL lab at this university. The authors are also grateful
for financial support from the Universidad de Málaga and
the Junta de Andalucía [research group, Communications
Engineering (TIC-0102)].
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