1

Relayed FSO Communication with Aperture

Averaging Receivers and Misalignment Errors
Prabhat Kumar Sharma, Ankur Bansal, Parul Garg, Theodoros A. Tsiftsis, and R. Barrios

Abstract

In this paper, the performance of decode-and-forward relay-assisted free-space-optical (FSO) com-

munication systems under atmospheric turbulence induced fading and misalignment errors is investi-
gated. To mitigate the adverse effects of the atmospheric turbulence, the aperture averaging receivers
are considered both at the relay and destination sides. The atmospheric turbulence induced fading is
modeled via the exponentiated-Weibull distribution, which has recently been proposed to characterize
an FSO link in the presence of finite-sized receiver aperture. The expression for the moment generating
function (MGF) of the instantaneous signal-to-noise ratio is derived. Further, new closed form expression
for the outage probability is obtained. Moreover, the new expression for the average symbol error rate
of the subcarrier intensity modulated M-ary phase shift keying is obtained using MGF-based approach.
Finally, numerical examples are discussed and all the derived analytical results are corroborated by
Monte-Carlo simulations.

I. I NTRODUCTION

The cooperative free-space optical (FSO) communication has emerged as a potential research
trend in last few years. It provides the combined advantages of the wireless optics and co-

Part of this work has been published in proceedings of IEEE International Conference on Communications 2015.
Prabhat Kumar Sharma is with the Department of Electronics and Communication Engineering, Visvesvaraya National Institute
of Technology, Nagpur India.(e-mail: prabhatmadhavec1@gmail.com).
Ankur Bansal and Parul Garg are with the Division of Electronics and Communication Engineering, Netaji Subhas Institute
of Technology, New Delhi, India. (e-mail: bansal.ankur143@gmail.com, parul_saini@yahoo.co.in).
Theodoros A. Tsiftsis is with the Technological Educational Institute of Central Greece, Lamia 35100, Greece,
and with the School of Engineering, Nazarbayev University, Astana 010000, Kazakhstan, (e-mail: tsiftsis@teiste.gr;
theodoros.tsiftsis@nu.edu.kz).
R. Barrios is with the Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Wessling,
Germany. (e-mail: Ricardo.Barrios@dlr.de)
 2

operative communication, which lead to larger bandwidth and improved performance. Several
cooperative protocols, such as amplify-and-forward (AF) [1], decode-and-forward (DF) [2], and
two-way relaying [3] have been studied recently in the relay-assisted FSO communication. The
performance of the above cooperative relaying protocols, however, is highly affected by various
perturbations in the propagating laser beam caused by the atmospheric turbulence. The above
perturbations include the beam wander that represents the random movement of the instantaneous
centre of the beam at receiving aperture. However, as the beam wander-induced fluctuations are
very slow, they can be combatted through tracking schemes [4]. Moreover the optical radiation
traversing the atmosphere spreads out beyond the diffraction limit of the beam radius, and hence
produces the beam spreading [4]. Beam divergence can be minimized by employing a very
narrow coherent laser source. In addition, the perturbations in the laser beam produced by the
turbulence can cause the scintillation which accounts for the random fluctuations in the beam
irradiance. The degrading effects of irradiance fluctuations become more dominant in the FSO
systems which employ the point size receive apertures.
The misalignment of laser beam in the receiver plane is another challenge which significantly
affects the communication over FSO links, and is a major concern in urban areas where the
FSO equipments are mounted on top of high rise buildings. The misalignment errors have
been incorporated in the literature as a fading effect. Based on the assumed distribution for the
horizontal and vertical displacements in the receiver plane, the misalignment errors are considered
to be of two types, zero boresight and non-zero boresight. In zero boresight misalignment, the
horizontal and vertical displacements in the receiver planes are characterized as zero mean Gaus-
sian random variable, whereas, the non-zero mean Gaussian distributed horizontal and vertical
displacements result into non-zero boresight misalignment. The Rayleigh distribution has been
used for modelling the radial displacement in zero boresight [5] misalignment fading, whereas
the Rician distribution characterizes the radial displacement in non-zero boresight misalignment
[6].
Recently, several techniques have been proposed in the literature [7], [8] to improve the
performance of a communication system operating in atmospheric turbulence. Aperture averaging
is one of the most widely used alternative technique due to its simplicity and lower cost. In
aperture averaging a collecting aperture is placed at the end of the FSO link in the receiver side
to combat the adverse effects of the atmospheric turbulence induced fading. The focussing lens
 3

concentrates the greater portion of the incoming irradiance flux into the photodetector of the point
receiver, and hence virtually acts like the finite-sized receiver. The FSO communication system
with aperture averaging in lognormal and Gamma-Gamma distributed atmospheric turbulence
is analyzed in [9]. In [10], a cooperative FSO system has been studied with AF and DF
relaying over lognormal turbulence-induced fading channel which is actually appropriate for
weak turbulence conditions. The Authors in [10] considered the intensity modulation direct-
detection (IM/DD) employing binary pulse position modulation (BPPM). However, it has been
shown in [11], [12] that the exponentiated-Weibull (EW) distribution is a more generalized
distribution as compared to lognormal and Gamma-Gamma distributions, and provides better fit
to simulation and experimental data under all aperture averaging conditions.
The EW distribution [11], [12] assumes that the observed field at the receiver consists
of an on-axis component and a weak multipath component which includes several scattered
components via different independent off-axis paths. As the number of independent and correlated
components in the observed field is unknown, to provide the necessary degrees of freedom to
uncorrelated terms, the received irradiance is assumed to be a generalized average of several
mutually independent and weighted irradiance random variables. To find the values of parameters
appear in EW distribution which provide an excellent fit to all aperture averaging conditions, the
probability density function (PDF) is obtained using simulation data for different aperture sizes
and turbulence conditions. Thus, the EW distribution captures the effect of aperture-averaging
through its constituent parameters as these parameters depend on the scintillation index of the
received irradiance. Thus, the EW distribution captures the effect of aperture-averaging through
its constituent parameters as these parameters depend on the scintillation index of the received
irradiance.
The performance of an FSO communication system over EW channels has been analyzed in
several works [13]- [16]. The approximate expressions for the bit error rate (BER) have been
derived in [13]- [15]. The average capacity of the optical wireless communication systems over
EW distribution turbulence channels has been introduced in [16]. However, the analysis in all
above works utilized the PDF of the irradiance, and none of them derived and utilized the PDF of
the signal-to-noise ratio (SNR). The statistical analysis through SNR-based approach is less com-
plex and more general as can be extended to different modulation techniques directly. Recently,
the performance of a non-cooperative FSO system has been evaluated in terms of the outage
 4

probability and average BER for various modulation schemes in [17]. The exact expressions for
outage probability and the average BER for on-off keying modulation in non-cooperative scenario
have been derived in [18] with EW distributed turbulence and non-zero boresight misalignment.
Authors in [19] presented the bit-error rate of binary pulse position modulation for multihop
DF FSO communication over EW channels with pointing errors. The M-ary phase shift keying
(MPSK) along with subcarrier intensity modulation (SIM) has been considered in [20]- [21] for
FSO communication systems in Gamma-Gamma atmospheric turbulence. The consideration of
SIM facilitates the use of several proven tools and techniques of radio-frequency communication
in the analysis of FSO communication systems.
To the best of authors’ knowledge the DF cooperative relaying in aperture averaged FSO
communication has not been considered over EW distributed turbulence links in the literature.
The main contribution of this paper is the performance analysis of a dual-hop DF cooperative
system. We extend the analytical framework presented in [17] to the DF cooperative scenario
where the source transmits its information to the destination with the help of a DF relay. The
detailed contributions through this work are:
• A DF relay assisted FSO communication system with aperture-averaged receivers is studied
under the presence of atmospheric turbulence and misalignment errors. In turbulence induced
effects, however, only irradiance fluctuations due to scintillation are considered, and the
beam wander and beam spreading are ignored.
• New expression for the moment generating function (MGF) of the SNR over EW-distributed
atmospheric turbulence channel is derived.
• New expression for the outage probability for the considered DF cooperative system is
derived.
• Average symbol error rate (SER) is obtained using the MGF based approach for SIM M-ary
phase shift keying (MPSK).
The rest of this paper is organized as follows: A detailed description of channel model is
given in section II. In section III, the PDF, CDF and MGF of the instantaneous SNR are derived.
Performance analysis metrics such as the outage probability and average SER are obtained in
Section IV. Section V provides the numerical results and conclusions are given in section VI.
 5

II. S YSTEM AND C HANNEL M ODEL

A. Propagation Model and Aperture Averaging

For characterization of wireless optical links mainly three propagation models have been used
in the literature (i) plane-wave propagation model, (ii) spherical-wave propagation model, and
(iii) Gaussian beam model. The plane-wave propagation model is suitable for the communication
from space to ground whereas the spherical-wave propagation model is used for ground to
space communication in absence of the beam wander effect. However, in this paper, we assume
that the Gaussian beam model which characterizes the FSO links approximately in terrestrial
communication systems.
The spatial coherence of the information carrying Gaussian laser beam reduces when it
propagates through the turbulent media. The degree of coherence is measured by the coherence
radius of the beam represented as ρ. For the ith link, the coherence radius can be given as:
ρ0,i = (0.55Cn2 κ2 Li )−3/5 , where Cn2 is the refractive index structure parameter (in m−2/3 ) and
characterizes the strength of atmospheric turbulence, κ is the wave number, and Li is the link
length.
The aperture averaging is used to reduce the variance of intensity fluctuations at the receiver
end. The effect of aperture averaging is quantified by a parameter known as aperture averaging
factor, which can be defined as
σI2 (Di )
Ai = , i ∈ {1, 2} (1)
σI2 (0)
where σI2 (Di ) represents the variance of intensity fluctuations i.e. scintillation index for a receiver
with diameter Di and σI2 (0) denotes the scintillation index for a point receiver [8].
Remark 1: The Gaussian beam model has been customarily approximated by spherical-wave
model in the literature when the atmospheric conditions makes the beam size at the receiver
plane comparatively larger than the collecting aperture size. The aperture averaging factor (Ai )
for spherical beam can also be expressed as [22],
[ ( ) 5 ]− 75
κDi2 6
Ai = 1 + 0.33 (2)
4Li
where κ is the wave number, D1 and D2 are the diameter of receive apertures at relay and
destination, respectively, and L1 is the length of source to relay link and L2 is the length of
relay to destination link.
 6

Fig. 1. System Model.

Remark 2: For a point receiver the aperture averaging factor Ai approaches unity, and the
aperture diameter is comparable with the atmospheric coherence radius. For the receivers with
aperture averaging the receive diameter should considerably be larger than the coherence radius
i.e. Di >> ρ0i .

B. System Model

The system model consists of a source, a DF relay, and a destination. The source has one
transmit antenna aperture to send its information to the relay. The relay is equipped with one
transmit and one receive antenna apertures, and the destination has one receive antenna aperture.
The receiving antennas at the relay and the destination have aperture averaged receivers. The
diameter of the receive aperture at relay is D1 and destination receive aperture diameter is D2 .
It is assumed that the relay is placed in such a way that it can maintain the line-of-sight (LOS)
communication with the source as well as with the destination. This kind of cooperative relay
placement is practically suitable in the scenarios where either the source and the destination are
situated at a long distance apart or they are at non-LOS from each other.
 7

C. Channel Model

We consider a composite model for the FSO channels between the source and the relay,
represented by the channel coefficient H1 , and between the relay and the destination, represented
by the channel coefficient H2 . The channel coefficient Hi , where i ∈ {1, 2}, is composed of
atmospheric turbulence induced fading represented by the coefficient hati , and the misalignment
fading denoted as hmi . Thus composite channel coefficient can be given as

Hi = hati hmi . (3)

The atmospheric turbulence induced fading is modelled under EW distribution. The PDF of
the coefficient hati (hati > 0), is given by [11]
( )β −1 [ ( )βi]{ [ ( ) ]}αi−1
β
αi βi hati i hati hati i
fhai(hati)= exp − 1−exp − , (4)
ηi ηi ηi ηi

where βi > 0 and αi > 0 are the shape parameters, and ηi > 0 is the scale parameter of the
turbulence channel represented by the channel coefficient hati .
The EW distribution incorporates the aperture averaging through its parameters as the pa-
rameters depend on the scintillation index (and hence on the dimension of receive aperture and
atmospheric turbulence condition of the link) [23] as per the following relation,
2
7.22σI3 (Di )
αi ≃ 2
Γ(2.487σI6 (Di ) − 0.104)
≃ 1.012(αi σI2 (Di ))− 25 + 0.142,
13
βi
1
ηi = ( ) , where
1
αi Γ 1 + βi ξ(αi , βi )
∞
∑ (−1)j Γ(αi )
ξ(αi , βi ) = 1+ β1
(5)
j=0 j!(1 + j) i Γ(αi − j)

Remark 3: The parameter definition in (5) [23, Eq 21] has not been tested experimentally for
weak turbulence channel conditions, however, the validity of EW parameters in all turbulence
conditions (including the weak atmospheric turbulence) has been confirmed through best fitting
process using the Levenberg-Marquardt least-square algorithm [23, section 5]. Moreover, the
parameter definition in (5) has been used widely in [31, section 4.5.6 ] for the purpose of
performance evaluation of FSO communication systems in all atmospheric turbulence scenarios.
The misalignment fading hmi has been characterized statistically in [5]. When horizontal and
vertical jitter in the receiver plane are independent of each other, and are distributed with identical
 8

zero mean Gaussian random variable, then the radial displacement at the receiver follows the
Rayleigh distribution. Hence the PDF of hmi can be given, as
µ2i µ2 −1
fhmi (hmi ) = µ2i
hmii , 0 ≤ hmi ≤ A0i (6)
A0i

where, for the misalignment errors corresponding to channel Hi , the term A0i = [erf(νi )]2
√
represents the fraction of the collected optical power, νi = πa2i /2wb2i , ai is the radius of
receiver aperture, and wbi is normalized beam waist. Further, µi = 2σωsi , ωi is the equivalent
√ i

beam width at the receiver and can be given as ωi = [ A0,i πwb2i /(2νi exp(−νi2 ))]1/2 , and σs2i is
the variance of pointing error displacement characterized by the horizontal sway and elevation
[5].

D. Transmission Protocol

The DF transmission protocol, which takes place in two time slots, is considered. In the first
time slot, the source transmits its information to the relay, and the relay receives the transmission
at its receiving antenna aperture. The relay then decodes the information received, and after
decoding, forwards the decoded signal to the destination, in second time slot.
The received signal at relay in first time slot is

yr = H1 Rx + nr (7)

where x is the transmitted symbol from source, R is the responsivity of the photodetector, nr
is additive white Gaussian noise with zero mean and N0 variance at the relay.
Similarly, the signal received at the destination in second time slot

yd = H2 Rx̂ + nd (8)

where x̂ is the decoded version of x and nd is additive white Gaussian noise with zero mean
and N0 variance at destination. Without loss of generality we assume that responsivity is same
for the photodetectors at both relay and destination.

III. S TATISTICS OF THE INSTANTANEOUS SNR

In this section we derive the PDF, CDF and MGF of the instantaneous SNR over EW-
distributed FSO channels. The SNR over channel Hi is given by γi = γ̄0 |Hi |2 , where γ̄0 =
 9

(Pt Rζ)2
N0
. The term Pt is the average transmit power, N0 is the average noise power of the additive
white Gaussian noise at the receiver, and ζ is the modulation index [13, eq. (3)]. The average
SNR can be given as γ̄i = γ̄0 E[|Hi |2 ], where E[·] is the expectation operator [24].

Theorem 1. The PDF of the instantaneous SNR γi can be given by

∞
∑ µ2
[ βi
]
i −1
fγi (γi ) = B1i Ψi (j)γi 2 Γ τi , B2i (j)γi 2 , (9)
j=0

(−1)j Γ(αi ) αi µ2i µ2

where Ψi (j) = µ2
, B1i = √ 2, τi = 1 − βii , and B2i (j) = 1+j
√
(ηi A0i γ̄0 )βi
, Γ(·)
1− i 2(ηA0i γ̄0 )µi
j!Γ(αi −j)(1+j) βi
is the gamma function, and Γ(·, ·) is the upper incomplete gamma function [25]

Proof. Proof of (9) is given in Appendix I. Moreover, the validity of the PDF defined in (9) is
proved in Appendix II.

The expression of the PDF in (9) consists of infinite summation which results from the use of
Newton’s generalized binomial expansion, however it can easily be verified using MATLAB or
MATHEMATICA that the infinite summation is convergent and ten to fifteen terms are sufficient
for this series to converge [14]. Further this can also be verified from Fig. 2, where we plotted
the PDF for j = 0 to 10, and j = 0 to 100 terms, and it can be observed that the plots of the
PDF in both the cases are perfectly matched.

Lemma 1. The CDF of received SNR can be given as,

  
∞
∑  µ2i
2B1i µ2 βi  1 − β , 1
Ψi (j)z G2,3 B2i (j)z 2 
2,1  ,
i i
Fγ (z) = 2
µ2
(10)
βi j=0  0, τi , − βii

Proof. The CDF of the SNR is defined as,

∫ z
Fγi (z) = fγi (γi )dγi ,
0
∞
∑ ∫ z µ2
[ βi
]
i −1
= B1i Ψi (j) γi 2 Γ τi , B2i (j)γi 2 . (11)
j=0 0

On replacing the upper incomplete gamma function in (11) with its Meijer’s-G equivalent [26,
βi
eq. (06.06.26.0005.01)] and substituting γi 2 = t we get,
∞ ∫ z β2i µ2 (  )
2B1i ∑ i −1  1

Fγ (z)= Ψi (j) t βi G2,0
1,2 B 2i (j)t  0, τi dt, (12)
βi j=0 0
( · )
where Gm,n
pq ·  · is the Meijer’s G function [26, eq. (07.34.02.0001.01)]. Using [26, eq.
(07.34.21.0084.01)] the integral in (12) can be reduced to (10).
 10

0.05

0.045

0.04

0.035 {α, β, η, µ} = {2.1,3.8,0.89,1.45}

PDF, f (γ)
0.03
γ
0.025 100 summation terms
10 summation terms
0.02
Simulations

0.015

0.01

0.005

0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
Variable γ

Fig. 2. The PDF of the instantaneous SNR γi given in (9) .

Lemma 2. MGF of the instantaneous received SNR can be given as

µ2
i −1 µ2
∞
∑ τ −1/2
s−
i
ki i li 2 2
Mγi(s)=B1,i Ψi (j) li +2ki −3
j=0 (2π) 2

(  )
(B2i (j)) i lili ∆(li , 1− µ22 ), ∆(ki ,1)
k
×G2k i ,2li
li +ki ,2ki  ∆ (ki , τi ) ∆ (ki , 0) , (13)
sli kiki
li
where li and ki are integer constants so that βi = ki
, and ∆(m, n) = (n/m)((n + 1)/m)....((n +
m − 1)/m).
∫∞
Proof. The MGF of the SNR is defined as Mγi (s) = 0
fγi (γi )exp(−sγi )dγi . Using (9), [26,
eq. (06.06.26.0005.01)] and [27, eq. (2.24.1.1)] the MGF, Mγi (s) can be written as given in
(13).

IV. P ERFORMANCE ANALYSIS

A. Outage Analysis
We now analyze the outage behavior of the given dual hop DF communication system
over EW-distributed FSO channels. The outage event is said to be occurred when received
instantaneous SNR falls below some specified threshold SNR. For the dual hop DF relaying
 11

system, the outage occurs if either source to relay or relay to destination link falls in outage.
Thus the outage probability in terms of received instantaneous SNRs can be given as,

Pout = Pr(γ1 < γth ) + (1−Pr(γ1 < γth )) Pr(γ2 < γth ), (14)

where γth is the threshold SNR and γ1 and γ2 are the received instantaneous SNRs over channels
H1 and H2 , respectively.
The probability terms Pr(γ1 < γth ) and Pr(γ2 < γth ) can be evaluated using (10). Thus, the
expression of outage probability can be given by substituting (10) into (14) as,

Pout =Fγ1 (γth ) + (1−Fγ1 (γth ))Fγ2 (γth ). (15)

B. Average SER for MPSK

The equal energy MPSK constellation is considered where transmission of each symbol is
equiprobable. In SIM, the signal z(t) is represented in terms of its in-phase component zI (t)
∑
and quadrature component zQ (t) i.e. z(t) = zI (t) + jzQ (t), where zI (t) = ∞n=0 xI [n]g(t − nTs )
∑∞
and zQ (t) = n=0 xQ [n]g(t − nTs ), respectively; g(t) represents the pulse shaping function and
Ts is the symbol duration. Further, xI [n] and xQ [n] are the in-phase and quadrature components of
the n-th data symbol x[n], and can be written as xI [n] = cos ϕn and xQ [n] = sin ϕn , respectively,
2(M −1)π
where ϕn ∈ {θ, θ + 2π
M
, ...., θ + M
} and θ ∈ {0, 2π}.
For a symmetric MPSK constellation with equidimensional decision regions, the probability
M P SK
of error PSR (e) can be given as [28], [29]
∑
M
M P SK
PSR (e) =1− Pk (γ1 )Pk (γ2 ) (16)
k=1

where Pk (γ1 ) is the probability that the source transmits xj and the relay receives it as xk i.e.
Pk (γ1 ) = Pr[H1 xj + NR ∈ Dk ] and Pk (γ2 ) is the probability that the relay transmits xk and it is
received at the destination as xj i.e. Pk (γ2 ) = Pr[H2 sk + NB ∈ Dj ], j, k ∈ {1, 2, ..., M }, j ̸= k,
Dj and Dk are the decision regions coresponding to symbol xj and xk , respectively, and the
terms NR and NB represent the AWGN with zero mean and variance N0 at the relay and the
destination, respectively.
 12

The probabilities Pk (γ1 ), and Pk (γ2 ), can be given as [28]

 ( 2 π )
 ∫ (MM
−1)π

1 − 1
M
sin ( M )
dϕ k = 1

 π 0 γ i sin2 (ϕ)

 ∫ ( 2 π )

1
(M −1)π
sin ( M )
π 0
M
Mγi sin2 (ϕ) dϕ k=M 2 +1
Pk (γi ) =
 ∫ π−ak−1 ( ) (17)

 1
Mγi sinsin(a
2
k−1 )
dϕ

 2π 0 2 (ϕ)

 ∫ ( 2 )

− 1 π−akM sin (ak ) dϕ
2π 0 γi sin2 (ϕ) otherwise,

where i ∈ {1, 2}.

Remark 4: The occurrence of error for the DF cooperative strategy, as given in (16), can
be explained as follows: suppose source transmits the symbol xj ∈ A, where A is an arbitrary
M-PSK constellation. Now the relay erroneously decodes it as another symbol xk , k ̸= j and
j, k ∈ {1, 2, ..., M }. Now it is quite possible that destination erroneously decides the symbol xk
as xj . In this way the error made by the relay is compensated by another error made by the
destination, and hence overall transmission
( becomes
) error free.
∫
1 Θ
The integrals of the form π 0 Mγi sin2 (ϕ) dϕ can be accurately approximated as [30],
g

∫ Θ ( ( ) ) ( )
1 Θg 1 1 4g
Mγi dϕ ≈ 2 − Mγi (g) + Mγi
0 π 2π
sin (ϕ) 6 4 3
( ) ( )
Θ 1 g
+ − Mγi (18)
2π 4 sin2 Θ
{ } { }
−1)π
where Θ ∈ (MM , π − ak−1 , π − ak , and g ∈ sin2 ( M
π
), sin2 (ak−1 ), sin2 (ak ) .
The probability term Pk (γi ) as defined in (17), can be rewritten using (18) as follows,
 [ ( )
 ( 2 ( π )) 1 4 sin2 ( Mπ
)

 1− c1 Mγi sin M + 4 Mγi

 3

 ( 2 )]


 ( )
π
sin

 +c2 Mγi sin2 ϕM2 k=1

 ( )

 ( 2 ( π )) 1

 4 sin2 ( M
π
)

 c M sin + M


1 γi M 4M γi 3
( 2 )
Pk (γi ) ≈ +c M sin ( M ) π
k =M

 2 γi sin2 ϕ2 2 +1



 [ ( ) 1 ( )

 4 sin2 ϕ1
 2 c3 Mγi sin ϕ1 + 4 Mγi
1 2

 3

 ( 2 ) ( )



 +c 4 M γ
sin ϕ1
− c6 Mγi sin2 ϕ0

 (
i sin2 ϕ2
) ( 2 )]

 1
 2
+ 4 Mγi 4 sin3 ϕ0 +c5 Mγi sin ϕ0
otherwise,
sin2 ϕ2

(19)
( 2M −3 ) ( M −1 ) ( ) ( ) ( π−2a ) ( 2π−3a )
2π−3ak−1 π−2ak−1
where c1 = , c2 = , c3 = , c4 = , c5 = k
, c6 = k
,
6M 4M
( 6π ) 4π 4π 6π
(M −1)π
ϕ0 = π − ak ϕ1 = π− ak−1 , and ϕ2 = M
.
M P SK
The new expression of the average SER PSR (e) for MPSK signalling can be derived by
substituting (19) in (16).
 13

V. N UMERICAL R ESULTS

In this section, numerical results are derived using Monte-Carlo simulations. Specifically, the
simulation of EW channels is implemented by following the same approach as the one presented
in [23], [31]. An EW random variable with parameter {α, β, η} is approximately equal to the
random variable obtained from max(w1 , w2 , ...wm ), where all wi {i = 1, 2, ..m}, are identical
Weibull distributed random variables with parameters {β, η} and m is the nearest integer to α.
To evaluate the effect of aperture averaging in different atmospheric turbulence conditions,
three different values of refractive index structure parameter Cn2 are considered i.e. Cn2 =
7.2 × 10−15 m−2/3 for weak turbulence, Cn2 = 5.0 × 10−14 m−2/3 for moderate turbulence and
Cn2 = 3.6 × 10−13 m−2/3 for strong turbulence conditions. The wavelength of optical signals
is taken as 780 nm. The link lengths are considered to be 1 km i.e. L1 = L2 = 1 km. For
the assumption of spherical wave the coherence radius and Rytov variances are calculated as
11
ρ0,i = (0.55Cn2 κ2 Li )−3/5 and σrv
7
2
= 1.23Cn2 κ 6 Li6 . Thus for weak turbulence ρ0,1 = ρ0,2 = 3.58
2 2
cm, σrv = 0.3194; for moderate turbulence ρ0,1 = ρ0,2 = 1.112 cm, σrv = 2.2181, and for
2
strong turbulence ρ0,1 = ρ0,2 = 0.34 cm, σrv = 15.97. The misalignment fading is characterized
Di
as follows: the beam radius at receiver wb1 = wb2 = 2m, aperture radius ai = 2
. Moreover,
without loss of generality we consider D1 = D2 = D, ρ0,1 = ρ0,2 = ρ, A01 = A02 = A0 and
σs1 = σs2 = σs . Additionally, for numerical analysis purpose (for Fig. 3, Fig. 4 and Fig. 5 ) we
consider that the source to relay and relay to destination channels are identical i.e. β1 = β2 = β,
α1 = α2 = α, η1 = η2 = η. The case of non-identical s − r and r − d links is discussed in Fig.
6.
In Fig. 3, the outage probability is plotted for different values of aperture averaging diameter
considering moderate atmospheric turbulence regime and σs = 30 cm. Three aperture diameters
are considered (i.e. D = 200 mm, 100 mm, 50 mm) along with the case when there is no
aperture averaging (when the point receiver is deployed). The threshold SNR for the plot is
taken as γth = 2 dB. For point receivers, values of the parameters {α, β, η} are obtained using
[31, Ch. 2 Eq. 59] and value of Rytov variance corresponding to the given turbulence scenario.
It can be observed from this figure that the aperture averaging significantly improves the system
performance as compared to the case of point receivers. Moreover, as the diameter of receive
aperture is increased, the outage probability deteriorates further.
 14

The outage behaviour of the considered dual-hop DF system for aperture averaging effect
in different fading situations can be observed in Fig. 4. Here the outage probability is plotted
against the average SNR for four different values of aperture averaging diameter (point receiver,
D = 200 mm, 100 mm, 50 mm) in weak and strong atmospheric turbulence conditions. For
misalignment fading we consider σs = 20 cm. It can be seen from this figure that the outage
performance is intuitively better in lower atmospheric turbulence, and it improves further as the
size of receive aperture is increased from point toD = 200 mm. Further, it can be observed
from this plot that the shift in the outage probability curves for point aperture to the 50 mm
diameter aperture is more significant than the shift in the curves from 50 mm diameter to 100
mm diameter or from 100 mm diameter to 200 mm diameter apertures. Additionally, it can also
be seen that the improvement in the system outage performance with respect to the atmospheric
turbulence conditions, is more prominent for D = 50 mm. It indicates the existence of some
receive aperture size (say Doptimum ), for a given turbulence and misalignment conditions, which
limits the further improvement in the performance.
The effect of receive aperture diameter and misalignment fading is observed on the SER
performance for QPSK and 8-PSK signalling schemes in Fig. 5. We consider two possibilities
of receiver, one the point receiver and the other is receiver with 50 mm aperture diameter. Two
values for misalignment jitter standard deviation σs are considered i.e. σs = 20 mm and σs = 40
mm. It should be noted that higher jitter standard deviation indicates severity of misalignment.
The SER performance shown in this figure reveals that the reduction in the jitter standard
deviation improves the performance for both point receiver and aperture averaged receivers.
However, it can be observed that change in the performance with misalignment errors is less
significant for point receivers as compared to 50 mm diameter receiver. This is because of the
fact that the intensity of information carrying laser beam collected by the point receiver is heavily
affected by even smaller misalignment errors, however the small misalignment in the receiver
plane can be compensated by aperture averaged receiver so as to facilitate larger collection of
optical power in the receiver.
The effect of non-identical s − r and r − d links in weak and strong turbulence scenarios is
shown in Fig 6, where the average error rate for binary phase shift keying (BPSK) modulation
is plotted with the average SNR (γ̄rd ) of r − d link. The average SNR (γ̄sr ) of s − r link is
taken as, γ̄sr = γ̄sr − 5 dB,γ̄sr = γ̄sr , and γ̄sr = γ̄sr + 5 dB. The D is considered to be 50 mm.
 15

0
10

−1
10
Outage Probabilty

−2
10

−3 D = 200 mm
10
D = 100 mm
D = 50 mm
Point Receiver
Simulations
−4
10
0 5 10 15 20 25 30 35 40
Average SNR, [dB]

Fig. 3. Outage probability for different aperture averaging conditions.

The increase in γ̄sr improves the error performance for strong as well as weaker atmospheric
turbulence conditions. Moreover in weaker turbulence, the betterment in the s − r link statistics
results in more better performance as compared to that in stronger turbulence.

VI. C ONCLUSIONS

In this paper, the performance of an aperture averaging-based DF cooperative communication

system is investigated over EW distributed FSO channels. The PDF, CDF and MGF of the SNR,
over the composite channel which includes turbulence induced fading, and misalignment fading,
were derived. Using the statistics of the SNR, closed form expressions for the outage probability
and average SER were obtained for SIM-MPSK modulation.
 16

0
10

−1
10
Outage probability

−2
10
Weak turb., D = 200 mm
Strong turb., D = 200 mm
Weak turb., D = 100 mm
Strong turb., D = 100 mm
−3 Weak turb., D = 50 mm
10
Strong turb., D = 50 mm
Weak turb., Point receiver
Strong turb., Point receiver
Simulations

−4
10
0 5 10 15 20 25 30 35
Average SNR, [dB]

Fig. 4. Outage probability for different turbulence and aperture averaging conditions.

It was shown that for a given aperture size, the turbulence induced fading is more dominant
in lower misalignment, and at high misalignment fading the effect of turbulence induced fading
becomes less severe. Further, the performance of FSO communication system in atmospheric
turbulence can be improved using the aperture averaging. In moderate misalignment and high
turbulence induced fading it was observed that any increase in receive aperture diameter signif-
icantly reduces the error probability.

A PPENDIX I
 17

0
10

Point Receiver

−1
10
D = 50 mm
Average SER

−2
10

QPSK, σs = 20 cm
QPSK, σ = 40 cm
s
8−PSK, σ = 20 cm
s
−3 8−PSK, σ = 40 cm
10 s
Simulations

0 5 10 15 20 25
Average SNR, [dB]

Fig. 5. Average BER for MPSK and 8PSK in varying turbulence and misalignment errors.

P ROOF OF T HEOREM 1
In (3), the terms hati and hmi are random variables with PDFs given in (4) and (6), respectively,
and the term is the deterministic factor. To derive the PDF of Hi , we first obtain the PDF of
the RV X , hati hmi . The PDF of X can be written as,
∫ ∞
fX (x) = fhmi (x|hati )fhati (hati )dhati . (20)
x
A0
i
 18

0
10

γ̄sr = γ̄rd -5 [dB], Strong turb.

γ̄sr = γ̄rd , Strong turb.

γ̄sr = γ̄rd +5 [dB], Strong turb.
−1
10 γ̄sr = γ̄rd -5 [dB], Weak trub.

γ̄sr = γ̄rd , Weak trub.

γ̄sr = γ̄rd +5 [dB], Weak trub.
Average BER

Simulations
−2
10

−3
10

−4
10
5 10 15 20 25
Averge SNR of r − d link, γ̄rd [dB]

Fig. 6. The effect of non-identical s − r and r − d links on the BER.

Using (4) and (6), (20) can be written as,

∫ ∞ ( )β −1 [ ( )β ]
µ2i µ2i −1 −µ2i αi βi hati i hati i
fX (x)= µ2
x hati exp −
A0ii x
A0
ηi ηi ηi
i
{ [ ( )β ]}αi −1
hati i
× 1 − exp − dhati . (21)
ηi
| {z }
A1

To solve the integration in (21), we expand the A1 using the Newton’s generalized binomial
∑
theorem i.e. (1 + y)t = ∞ Γ(t+1)y j
j=0 Γ(t−j+1)j! . After some mathematical manipulations (21) can be
 19

rewritten as,
∞
∑
αi βi µ2i µ2i −1 (−1)j Γ(αi )
fX (x) = x
µ2
ηiβi A0ii j=0
j!Γ(αi − j)
∫ ∞ ( )
−(j + 1)
hati βi −1−µi exp
2
× βi
hβatii dhati . (22)
x
A0
ηi
i

The PDF of the channel coefficient Hi can be deduced by evaluating the integral in (22) using
[25, eq. 381.3], as
∞
[ ]
αi µ2i ∑ µ2i −1 µ2i 1+ j
fHi (h)= µ2i
Ψi (j)h Γ 1− , h , h ≥ 0.
βi
(ηi A0i ) j=0
βi (ηi A0i )βi
(23)

Now the PDF of instantaneous SNR γ defined can easily be derived using [24, eq. (5-8)].

A PPENDIX II
VALIDATION OF PDF IN (9)

For the expression in (23) to be a valid PDF it must be non-negative and area under this PDF
should be unity. The non-negativity of PDF fh (h) in (23) can be observed from Table I as the
range of fHi (h) in the possible domain of h is positive.

TABLE I

h 0 0.1 0.2 0.3 0.4 0.5 0.6

fHi (h) 0 .0284 .0609 .0948 .1279 .1547 .1656
h 0.7 0.8 0.9 1.0 1.1 1.2 1.3
fHi (h) .1513 .1124 .0645 .0273 .0082 .0017 .002

As the PDF given in (4) is a valid PDF [11], area under this PDF should be one i.e.
∫∞
0
fhati (hati )dhati = 1. Rewriting the PDF fhati (hati ) using Newton’s generalized binomial
theorem, we get
∞
αi βi∑(−1)j Γ(αi )
ηi j=0j!Γ(αi − j)
∫ ∞( )β −1 [ ( )β ]
hati i hati i
× exp −(1 + j) dhati = 1. (24)
0 ηi ηi

After solving this integral, followed by some mathematical rearrangements (24) reduces to
∞
∑ (−1)j Γ(αi ) 1
= . (25)
j=0
j!Γ(α i − j)(1 + j) αi
 20

∫∞
The area under the PDF fHi (h) is S = 0
fHi (h)dh. Using (23) we get,
∞ ∫ [ ]
αi µ2 ∑ ∞
µ2 −1 µ2 1+ j
S= µ2
Ψ(j) h Γ 1− , hβi
dh.
(ηi A0 ) j=0 0 βi (ηi A0 )βi
(26)

On substituting hβi = r in (26), and using [25, eq. (6.455.1)], we get

∑∞ ( )
Ψ(j) µ2
S= µ2
F
2 1 1, 1; + 1; 0 , (27)
βi
j=0 (1 + j) βi
where 2 F1 (·, ·; ·; ·) is Gauss hypergeometric function [26, eq. (07.23.02.0001.01)]. Now using
[26, eq. (07.23.03.0001.01)] and (25) we get S = 1.

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