Performance analysis of dual-hop optical wireless communication systems over kdistribution turbulence channel with pointing error
Neha Mishra, Sriram Kumar D., and Pranav Kumar Jha

Citation: AIP Conference Proceedings 1849, 020011 (2017); doi: 10.1063/1.4984158
View online: http://dx.doi.org/10.1063/1.4984158
View Table of Contents: http://aip.scitation.org/toc/apc/1849/1
Published by the American Institute of Physics

Performance Analysis of Dual-Hop Optical Wireless
Communication Systems over K-distribution Turbulence
Channel with Pointing Error
Neha Mishra1,a), Sriram Kumar D.2,b) and Pranav Kumar Jha3)
1,2,3

Department of Electronics and Communication Engineering,
National Institute of Technology, Tiruchirappalli, Tamil Nadu, India
a)
nehamishrabe@gmail.com
b)
srk@nitt.edu
Abstract. In this paper, we investigate the performance of the dual-hop free space optical (FSO) communication systems
under the effect of strong atmospheric turbulence together with misalignment effects (pointing error). We consider a relay
assisted link using decode and forward (DF) relaying protocol between source and destination with the assumption that
Channel State Information is available at both transmitting and receiving terminals. The atmospheric turbulence channels
are modeled by k-distribution with pointing error impairment. The exact closed form expression is derived for outage
probability and bit error rate and illustrated through numerical plots. Further BER results are compared for the different
modulation schemes.

INTRODUCTION
Transmission of optical signals through the atmosphere in a line of sight manner is known as optical wireless
communication (OWC). It has gained significant research interest in recent years, however its performance is highly
ill-protected to unfavorable atmospheric conditions which occur mainly due to turbulence and pointing errors. The
turbulence arises due to the inhomogeneities in temperature and pressure of the atmosphere. This causes a change in
refractive index and leads to the fluctuation of amplitude and intensity in the transmitted optical signals. The pointing
error occur due to the sway of high rise buildings as a result of weak earthquakes, dynamic wind loads and thermal
expansions. This causes misalignment between source and destination transceivers.
To evaluate the impact of atmospheric turbulence various channel model are developed such as gamma-gamma,
negative exponential and log normal model. The log normal distribution is found suitable for weak turbulence
regimes were as gamma- gamma distribution is suitable for moderate to strong turbulence regimes. Another channel
model is a k distribution channel model which is very popularly used for strong turbulence regimes. Further in
literature, we study two types of relays i.e. amplify and forward (AF) and decode and forward relay (DF) relay. In this
paper, we consider Decode and Forward relay assisted FSO system over k-distribution channel model considering the
combine effect of atmospheric turbulence and misalignment error.
In [2] authors have analyzed the behavior of pointing errors and derived the probability density function (pdf) for
normalized irradiance due to boresight and jitter. The effect of pointing errors on the performance of the FSO
communication system over log normal and gamma-gamma channel models have been studied in [3]-[5]. In [6]-[8]
authors, did outage and BER analysis for relay assisted FSO system over strong atmospheric turbulence channel with
spatial diversity and misalignment errors for different modulation scheme. [9] Analyzed the performance of free space
optical (FSO) system employing subcarrier intensity modulation with differential phase shift keying described by the
lognormal atmospheric turbulence and the pointing error effects. In [10] authors, investigate the outage capacity under
the influence of atmospheric turbulence and pointing error over gamma distribution model.
The rest of the paper is arranged as follows: Section second discusses the system model and channel model used
for proposed work. In section third, the outage probability of relay assisted OWC system is discussed. Section fourth,

Let There be Light
AIP Conf. Proc. 1849, 020011-1–020011-8; doi: 10.1063/1.4984158
Published by AIP Publishing. 978-0-7354-1521-8/$30.00

020011-1

discusses about bit error rate performance of the system. Section fifth, describes the numerical results and graphical
analysis for those results. Finally in section sixth, concluding remarks are highlighted.

SYSTEM MODEL AND CHANNEL MODEL
System Model
In this paper we consider a dual-hop FSO system as shown in figure 1. The system model consists of a source (s)
and the destination (d) which is communicating with the help of the relay (r). To enable line of sight communication
the transceivers are located on the high rise buildings. We assume that due to practical reasons direct communication
between source and destination is not feasible which may be either greater distance between source and destination or
non-LOS condition. The system employs a full duplex DF relaying protocol. We assume that channel state
information (CSI) is present at source, destination and the relay. The laser beam propagates through k-distributed
turbulence channel which is corrupted by additive white Guassian noise (AWGN). Further, we consider that channel
model is aggregated which takes into account the combined effects of atmospheric turbulence induced fading and
misalignment fading (pointing error).

FIGURE 1. System Model.

Channel Model
The k distribution is a widely accepted model for strong atmospheric turbulence. It is considered as a product of
gamma distribution and exponential distribution which are two independent models. The probability density function
(pdf) of the normalized irradiance ‘I’ for K distribution is given by,
݂ூ ሺ‫ܫ‬ሻ ൌ

ଶఈ

ഀశభ
ቀ మ ቁ ሺഀషభሻ

మ

‫ܫ‬

୻ሺఈሻ

‫ܭ‬௔ିଵ ൫ʹξߙ‫ܫ‬൯ǡ ‫ ܫ‬൐ Ͳ

(1)

where channel parameter ߙis related to the effective no. of discrete scatters, ȞሺǤ ሻ is the gamma function [15,
eq.(8.310.1)], and ‫ܭ‬௩ ሺǤ ሻ is the vth-order modified Bessel function of the second kind [15, eq.(8.432.2)]. By
combining pdf for turbulence given in equation (1) and misalignment fading the final pdf of I is given as,
݂ூ ሺ‫ܫ‬ሻ ൌ 

ఈ௚మ
஺బ ୻ሺఈሻ

ଷǡ଴
‫ܩ‬ଵǡଷ
ቀ

ఈ
஺బ

మ

‫ܫ‬ቚ ିଵା௚௚మǡఈିଵǡ଴ቁ

(2)

where the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard
deviation (jitter) at the receiver is given by ݃ ൌ ߱௭೐೜ Τʹߪ௦ . Using the relations ߴ ൌ ξߨ‫ݎ‬Τξʹ ߱௭ , ‫ܣ‬଴ ൌ ሺԂሻଶ
and߱௭ଶ೐೜ = ߱௭ଶ ξߨሺߴ) /2ߴ ሺെߴ ଶ ሻǡthe parameter ߱௭೐೜ can be calculated where erf(.) is the error function and ߱௭
is the beam waist (radius calculated as ݁ ିଶ ) at distance z .The pdf of electrical SNR, ݂ఊ ሺߛሻ can be expressed as,
݂ఊ ሺߛሻ ൌ 

ఈ௚మ
஺బ ୻ሺఈሻఓ భȀమ ఊభȀమ

ଷǡ଴
‫ܩ‬ଵǡଷ
ቀ

ఈ
஺బ ξఓ

020011-2

మ

ߛ ଵȀଶ ቚ ିଵା௚௚మ ǡఈିଵǡ଴ቁ

(3)

where instantaneous electrical signal-to-noise ratio (SNR) can be defined asߛ ൌ ሺߟ‫ܫ‬ሻଶ Ȁܰ଴ . The electrical average
SNR is defined as ߤ ൌ ሺߟ‫ܧ‬ሾ‫ܫ‬ሿଶ ሻȀܰ଴ ǤE[1]=1 since I is normalized. The CDF of ߛ can be expressed as,
‫ܨ‬ఊ ሺߛሻ ൌ 

ఈ௚మ

‫׬‬

ఊ೟೓

஺బ ୻ሺఈሻఓ భȀమ ଴

భ

ଷǡ଴
ߛ మିଵ ‫ܩ‬ଵǡଷ
ቀ

మ

ఈ
஺బ ξఓ

Let,
ଷǡ଴
ܰ ൌ  ‫ܩ‬ଵǡଷ
ቀ

ߛ ଵȀଶ ቚ ିଵା௚௚మ ǡఈିଵǡ଴ቁ ݀ߛ

(4)

మ

ఈ
஺బ ξఓ

ߛ ଵȀଶ ቚ ିଵା௚௚మ ǡఈିଵǡ଴ቁ

(5)

By solving (5), N can be expressed as,
ܰ ൌ

ଶഀషమ
ଶగ

଺ǡ଴
‫ܩ‬ଶǡ଺
൭

೒మ ೒మశభ

ǡ

ఈమ
ଵ଺஺బ మ ఓ

మ
ߛอ షభశ೒మమ೒మ ഀషమ
ഀ

ǡ

మ

మ

ǡ

మ

ǡ ǡ଴ǡ
మ

൱

(6)

భ
మ

substitute (6) in (4),
െͳ ݃ଶ ݃ଶ ൅ ͳ
ǡ ǡ
ߙ݃ ʹ
ߙ
ʹ
ʹ ʹ
ଵȀଶ ଺ǡଵ
ߛ
‫ܩ‬
൮
ߛ
ተ
‫ܨ‬ఊ ሺߛሻ ൌ 
ଶ
ଶ
௧௛
௧௛
ଷǡ଻
ଶ
݃
ߙ
െ
ͳ
ߙ
ͳ െͳ൲
െͳ
൅
݃
ʹߨ‫ܣ‬଴ Ȟሺߙሻߤ ଵȀଶ
ͳ͸‫ܣ‬଴ ߤ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
ʹ ʹ
ʹ ʹ
ʹ
ʹ
ଶ ఈିଶ

ଶ

(7)
Equation (7) represents the final CDF of k-distributed turbulence model with misalignment fading (pointing error).

OUTAGE ANALYSIS
The outage probability is defined as the probability when the instantaneous SNR falls below a specified
threshold ߛ௧௛ and is given as,
ఊ

ሺߛ൏ߛ௧௛ ሻൌ‫׬‬଴ ೟೓ ݂ఊ ሺߛሻߛ(8)
ሺߛ൏ߛ௧௛ ሻൌ‫ܨ‬ఊ ሺߛ௧௛ ሻ(9)





where ߛ௧௛ is the threshold SNR. The outage probability ܲ௢௨௧ at the destination is obtained in terms of received SNR
given as,
ܲ௢௨௧ ൌ ܲሺߛଵ ൏ ߛ௧௛ ሻ൅ܲሺߛଶ ൏ ߛ௧௛ ሻǦܲሺߛଵ ൏ ߛ௧௛ ሻܲሺߛଶ ൏ ߛ௧௛ ሻ(10)

ܲ௢௨௧ ൌ ‫ܨ‬ఊభ ሺߛ௧௛ ሻ൅‫ܨ‬ఊమ ሺߛ௧௛ ሻǦ‫ܨ‬ఊభ ሺߛ௧௛ ሻ‫ܨ‬ఊమ ሺߛ௧௛ ሻ(11)
where ߛଵ is the electrical SNR from the source (s) to relay (r) and ߛଶ is the electrical SNR from relay (r) to
destination (d). Substituting (7) in (11), the outage probability of s to r link can be written as,
ܲ௢௨௧ ൌ


ఈభ ௚మ ଶഀభ షమ

భ
మ

଺ǡଵ
భ ߛ௧௛ ‫ܩ‬ଷǡ଻ ൭

ଶగ஺బ ௰ሺఈభ ሻఓభ మ
ఈమ ௚మ ଶഀమ షమ

భ

భ
ଶగ஺బ ௰ሺఈమ ሻఓమ మ

ఈభ ఈమ ௚ర ଶഀభ షమ ଶഀమ షమ
భ

଺ǡଵ
భ ߛ௧௛ ‫ܩ‬ଷǡ଻

ସగమ ஺బ మ ௰ሺఈభ ሻ௰ሺఈమ ሻఓభ మ ఓమ మ

൭

ఈభ మ

ଵ଺஺బ మ ఓభ

଺ǡଵ
ߛ௧௛ మ ‫ܩ‬ଷǡ଻
൭

ఈభ మ

ଵ଺஺బ మ ఓభ
ఈమ మ

ଵ଺஺బ మ ఓమ

షభ ೒మ ೒మ శభ

ǡ

ǡ

ߛ௧௛ อ షభశ೒మ ೒మమ ഀమభషభమഀభ

൱൅

ߛ อ

൱െ

భ షభ
ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ మ
షభ ೒మ ೒మ శభ
ǡ
ǡ
మ మ
మ
௧௛ షభశ೒మ ೒మ ഀమ షభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ
మ

షభ ೒మ ೒మ శభ
ǡ ǡ
మ మ
మ
௧௛ షభశ೒మ ೒మ ഀభ షభ ഀభ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

൱

020011-3

ǡ

଺ǡଵ
‫ܩ‬ଷǡ଻

൭

ఈమ మ

ଵ଺஺బ మ ఓమ

షభ ೒మ ೒మశభ
ǡ ǡ
మ మ
మ
௧௛ షభశ೒మ ೒మ ഀమ షభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

൱(12)

Equation (12) represents the closed form expression of outage probability for dual-hop FSO system. Considering the
special case with ߙଵ ൌ ߙଶ ൌ ߙǡ ߤଵ ൌ ߤଶ ൌ ߤ, we will get closed form expression of outage probability as,
ܲ௢௨௧ ൌ


ఈ௚మ ଶഀషమ

గ஺బ ୻ሺఈሻఓ

ଵȀଶ ଺ǡଵ
‫ܩ‬ଷǡ଻ ൭
భȀమ ߛ௧௛

ఈమ
ଵ଺஺బ మ ఓ

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀషభ ഀ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ మ మ మ మ

ߛ อ

൱െቀ

ଶ

ఈ௚మ ଶഀషమ
ଶగ஺బ ୻ሺఈሻఓ

ଶ

షభ ೒మ ೒మ శభ

ǡ

ఈమ

଺ǡଵ
భȀమ ቁ ߛ௧௛ ൥‫ܩ‬ଷǡ଻ ൭

ଵ଺஺బ మ ఓ

ǡ

ߛ௧௛ อ షభశ೒మమ೒మమഀషభమഀ
మ

ǡ

మ

ǡ

మ

భ షభ൱൩

ǡ ǡ଴ǡ ǡ
మ

మ మ

(13)
Equation (13) represents the outage probability for single relay assisted FSO systems with pointing error.

BER ANALYSIS
The average BER can be written as,
ܲ௘ ൌ 

ஶ
௤ು
‫ ׬‬ሺെ‫ߛݍ‬ሻ ߛ ௉ିଵ ‫ܨ‬ሺߛሻ݀ߛ
ଶ୻ሺ௉ሻ ଴

(14)

where the parameter p and q in (14) account for different modulation schemes. Substituting the value of ‫ܨ‬ఊ ሺߛሻfrom
equation (12) to equation (14), we get,
ܲ௘ ൌ
௤ು

ஶ

‫ ׬‬ሺെ‫ߛݍ‬ሻ ߛ ௉ିଵ ൥
ଶ୻ሺ௉ሻ ଴



ఈభ ௚మ ଶഀభ షమ

ఈమ ௚మ ଶഀమ షమ

ଶగ஺బ ୻ሺఈభ ሻఓభ మ
భ
మ

଺ǡଵ
భ ߛ௧௛ ‫ܩ‬ଷǡ଻

ଶగ஺బ ୻ሺఈమ ሻఓమ మ
ఈభ ఈమ ௚ర ଶഀభ షమ ଶഀమ షమ
భ

଺ǡଵ
భ ߛ௧௛ ‫ܩ‬ଷǡ଻

ସగమ ஺బ మ ୻ሺఈభ ሻ୻ሺఈమ ሻఓభ మ ఓమ మ

൭

ఈభ మ

ଵ଺஺బ మ ఓభ

భ

଺ǡଵ
భ ߛ௧௛ మ ‫ܩ‬ଷǡ଻ ൭

൭

ఈమ మ

ଵ଺஺బ మ ఓమ

ǡ

ଵ଺஺బ మ ఓభ

మ

ߛ อ

଺ǡଵ
൱ ‫ܩ‬ଷǡ଻

൭

ǡ

ߛ௧௛ อ షభశ೒మ ೒మమ ഀమభషభమഀభ
ǡ

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀమషభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀభషభ ഀభ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

షభ ೒మ ೒మ శభ

ఈభ మ

ఈమ మ

ଵ଺஺బ మ ఓమ

మ

ǡ

మ

ǡ

మ

൱൅

భ షభ
మ మ

ǡ଴ǡ ǡ

൱െ

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀమ షభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

൱൩ ݀ߛ
(15)

Further simplification,
െͳ ݃ଶ ݃ଶ ൅ ͳ
ஶ
‫ۍ‬
ǡ ǡ
‫ې‬
ଵ
ߙଵ ݃ଶ ʹఈభିଶ
ߙଵ ଶ
ʹ ʹ
ʹ
ቀ௉ା ቁିଵ ଺ǡଵ
ଶ
‫ێ‬
‫ۑ‬
‫ܩ‬ଷǡ଻ ൮
ߛ௧௛ ተ െͳ ൅ ݃ଶ ݃ଶ ߙ െ ͳ ߙ
൲ ݀ߛ
ଶ
ଵ න ሺെ‫ߛݍ‬ሻ ߛ௧௛
ͳ
െͳ
ͳ͸‫ܣ‬଴ ߤଵ
‫ێ‬
‫ۑ‬
ǡ ǡ ଵ
ǡ ଵ ǡ Ͳǡ ǡ
ʹߨ‫ܣ‬଴ Ȟሺߙଵ ሻߤଵ ଶ ଴
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
‫ێ‬
‫ۑ‬
ଶ
ଶ
െͳ ݃ ݃ ൅ ͳ
‫ێ‬
‫ۑ‬
ஶ
ǡ ǡ
ଵ
ߙଶ ݃ଶ ʹఈమିଶ
ߙଶ ଶ
ʹ ʹ
ʹ
‫ێ‬
‫ۑ‬
ቀ௉ା ቁିଵ ଺ǡଵ
ଶ
൅
න
ሺെ‫ߛݍ‬ሻ
ߛ
‫ܩ‬
൮
ߛ
ተ
൲
݀ߛ
ଶ
ଶ
௧௛
௧௛ െͳ ൅ ݃ ݃ ߙ െ ͳ ߙ
ଷǡ଻
ଶ
ଵ
ͳ െͳ
‫ێ‬
‫ۑ‬
ଶ
ଶ
ͳ͸‫ܣ‬
ߤ
଴
଴
ଶ
ǡ
ǡ
ǡ
ǡ
Ͳǡ
ǡ
ʹߨ‫ܣ‬଴ Ȟሺߙଶ ሻߤଶ ଶ
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
‫ݍ‬௉ ‫ێ‬
‫ۑ‬
ܲ௘ ൌ
‫ۑ‬
ʹȞሺܲሻ ‫ێ‬
െͳ ݃ଶ ݃ଶ ൅ ͳ
ǡ ǡ
ஶ
ଶ
‫ێ‬
‫ۑ‬
ߙଵ ߙଶ ݃ସ ʹఈభିଶ ʹఈమିଶ
ߙ
ʹ
ʹ
ʹ
ଵ
ሺ௉ାଵሻିଵ ଺ǡଵ
‫ێ‬െ
න
ሺെ‫ߛݍ‬ሻ
ߛ
‫ܩ‬
൮
ߛ
ተ
൲
ଶ
ଶ
௧௛
௧௛
ଷǡ଻
ଵ ଵ
െͳ ൅ ݃ ݃ ߙଵ െ ͳ ߙଵ
ͳ െͳ ‫ۑ‬
ͳ͸‫ܣ‬଴ ଶ ߤଵ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
‫ ێ‬Ͷߨ ଶ ‫ܣ‬଴ ଶ Ȟሺߙଵ ሻȞሺߙଶ ሻߤଵ ଶ ߤଶ ଶ ଴
ʹ ʹ ‫ۑ‬
ʹ
ʹ
ʹ
ʹ
‫ێ‬
‫ۑ‬
െͳ ݃ଶ ݃ଶ ൅ ͳ
‫ێ‬
‫ۑ‬
ǡ
ǡ
ଶ
ߙଶ
ʹ ʹ
ʹ
‫ێ‬
‫ۑ‬
଺ǡଵ
‫ܩ‬ଷǡ଻ ൮
ߛ௧௛ ተ െͳ ൅ ݃ଶ ݃ଶ ߙ െ ͳ ߙ
ͳ െͳ൲ ݀ߛ
‫ێ‬
‫ۑ‬
ଶ
ଶ
ͳ͸‫ܣ‬଴ ଶ ߤଶ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
‫ۏ‬
‫ے‬

(16)
Let,

020011-4

‫ܫ‬ଵ ൌ 

ఈభ ௚మ ଶഀభ షమ

భ

ஶ

షభ ೒మ ೒మ శభ

ǡ

ఈ మ

ǡ

ቀ௉ା ቁିଵ ଺ǡଵ
మ
‫ܩ‬ଷǡ଻ ൭ భమ ߛ௧௛ อ షభశ೒మ ೒మమ ഀమభషభమഀభ
భ ‫׬‬଴ ሺെ‫ߛݍ‬ሻ ߛ௧௛
ଵ଺஺ ఓ
బ

ଶగ஺బ ୻ሺఈభ ሻఓభ మ

భ

ǡ

మ

మ

ǡ

ǡ

మ

మ

൱ ݀ߛ

భ షభ
మ మ

ǡ଴ǡ ǡ

(17)

By using [14, eq. (19)],
‫ܫ‬ଵ ൌ 

ఈభ ௚మ ଶഀభ షమ

భ
ஶ
ቀ௉ା ቁିଵ
మ
భ ‫׬‬଴ ߛ௧௛

ଶగ஺బ ୻ሺఈభ ሻఓభ మ

మ మ

ଵǡ଴
‫ܩ‬଴ǡଵ

షభ ೒ ೒ శభ
ǡ ǡ
െ ଺ǡଵ
ఈ మ
ቀ‫ߛݍ‬ቚ Ͳ ቁ ‫ܩ‬ଷǡ଻
൭ భమ ߛ௧௛ อ షభశ೒మ ೒మమ ഀమభషభమഀభ భ షభ൱ ݀ߛ
ଵ଺஺బ ఓభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ

మ

మ

మ

(18)

మ మ

By solving eq.(18),

‫ܫ‬ଵ ൌ 

ఈభ ௚మ ଶഀభ షమ

భ
ଶగ஺బ ୻ሺఈభ ሻఓభ మ

‫ݍ‬

ିሺ௉ାଵȀଶሻ

଺ǡଶ
‫ܩ‬ସǡ଻

ቌ

ିଵ ௚మ ௚మ ାଵ ଵ

ǡ

ఈభ మ

ǡ

ǡ െܲ

ଶ ଶ
ଶ
ଶ
ቮ
ቍ
ଵ ିଵ
ଵ଺஺బ మ ఓభ ௤ ିଵା௚మ ௚మ ఈభ ିଵ ఈభ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
ଶ
ଶ
ଶ
ଶ
ଶ ଶ

Let,
‫ܫ‬ଶ ൌ 

ఈమ ௚మ ଶഀమ షమ

ǡ

ǡ

ሺ௉ାଵȀଶሻିଵ ଺ǡଵ
‫ܩ‬ଷǡ଻ ൭ మమ ߛ௧௛ อ షభశ೒మ ೒మమ ഀమమషభమഀమ
భ ‫׬‬଴ ሺെ‫ߛݍ‬ሻ ߛ௧௛
ଵ଺஺ ఓ
బ

ଶగ஺బ ୻ሺఈమ ሻఓమ మ

By using [14, eq. (44)] and solving,
‫ܫ‬ଶ ൌ 

షభ ೒మ ೒మశభ

ఈ మ

ஶ

ఈమ ௚మ ଶഀమ షమ

భ
ଶగ஺బ ୻ሺఈమ ሻఓమ మ

଺ǡଶ
‫ି ݍ‬ሺ௉ାଵȀଶሻ ‫ܩ‬ସǡ଻
ቌ

(19)

మ

మ

ǡ

మ

ǡ

మ

ǡ

మ

൱ ݀ߛ

భ షభ
మ మ

ǡ଴ǡ ǡ

(20)

ିଵ ௚మ ௚మ ାଵ ଵ

ఈమ మ

ǡ

ଶ

ቮ

ǡ

ଶ

ǡ െܲ

ଶ

ଶ

ଵ଺஺బ మ ఓమ ௤ ିଵା௚మ ௚మ ఈమ ିଵ ఈమ

ǡ

ଶ

Let,

ଶ

ǡ

ଶ

ǡ

ଶ

ଵ ିଵ

ǡ Ͳǡ ǡ
ଶ

ቍ

(21)

ଶ

െͳ ݃ଶ ݃ଶ ൅ ͳ
ǡ ǡ
ʹ
ʹ ʹ
ሺ௉ାଵሻିଵ ଺ǡଵ
න
ሺെ‫ߛݍ‬ሻ
ߛ
‫ܩ‬
൮
ߛ
ተ
‫ܫ‬ଷ ൌ
௧௛
௧௛ െͳ ൅ ݃ଶ ݃ଶ ߙ െ ͳ ߙ
ଷǡ଻
ଵ ଵ
ଶ
ͳ െͳ൲
ଵ
ଶ
ͳ͸‫ܣ‬
ߤ
ଶ
଴ ଵ
ǡ ǡ
ǡ ଵ ǡ Ͳǡ ǡ
Ͷߨ ‫ܣ‬଴ Ȟሺߙଵ ሻȞሺߙଶ ሻߤଵ ଶ ߤଶ ଶ ଴
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
ߙଵ ߙଶ ݃ସ ʹఈభିଶ ʹఈమିଶ

ൈ

ߙଵ ଶ

ஶ

଺ǡଵ
‫ܩ‬ଷǡ଻

൭

ఈమ మ

ଵ଺஺బ మ ఓమ

షభ ೒మ ೒మ శభ
ǡ ǡ
మ మ
మ
௧௛ షభశ೒మ ೒మ ഀమ షభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

൱ ݀ߛ

(22)

By using [Adamchik,1990, eq. (24)],
‫ܫ‬ଷ ൌ
ൈ

଺ǡଵ
‫ܩ‬ଷǡ଻

൭

ఈభ మ

ଵ଺஺బ మ ఓభ

ߙଵ ߙଶ ݃ସ ʹఈభିଶ ʹఈమିଶ

ஶ

ଵ ଵන
Ͷߨ ଶ ‫ܣ‬଴ ଶ Ȟሺߙଵ ሻȞሺߙଶ ሻߤଵ ଶ ߤଶ ଶ ଴

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀభషభ ഀభ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

଺ǡଵ
൱ ‫ܩ‬ଷǡ଻

ߛ อ

By solving eq. (23),
‫ܫ‬ଷ ൌ

൭

െ
ଵǡ଴
ߛ௧௛ ሺ௉ାଵሻିଵ ‫ܩ‬଴ǡଵ
ቀ‫ߛݍ‬ቚ Ͳ ቁ

ఈమ మ

ଵ଺஺బ మ ఓమ

షభ ೒మ ೒మ శభ
ǡ ǡ మ
మ మ
௧௛ షభశ೒మ ೒మ ഀమషభ ഀమ భ షభ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
మ
మ
మ
మ
మ మ

ߛ อ

ߙଵ ߙଶ ݃ସ ʹఈభିଶ ʹఈమିଶ
ଵ ଵ
Ͷߨ ଶ ‫ܣ‬଴ ଶ Ȟሺߙଵ ሻȞሺߙଶ ሻߤଵ ଶ ߤଶ ଶ

൱ ݀ߛ

(23)

‫ି ݍ‬ሺ௉ାଵሻ

െͳ ݃ଶ ݃ଶ ൅ ͳ
െͳ ݃ଶ ݃ଶ ൅ ͳ
ǡ ǡ
ǡ ǡ
ߙଵ ଶ
ߙଶ ଶ
ʹ ʹ
ʹ ʹ
ʹ
ʹ
଴ǡଵǣ଺ǡଵǣ଺ǡଵ െܲ
ተെͳ ൅ ݃ଶ ݃ଶ ߙ െ ͳ ߙ
ൈ ‫ܩ‬ଵǡ଴ǣଷǡ଻ǣଷǡ଻
ቀ
ተ
ተ
ǡ
ͳ െͳ െͳ ൅ ݃ଶ ݃ଶ ߙଶ െ ͳ ߙଶ
ͳ െͳ ͳ͸‫ ܣ‬ଶ ߤ ‫ͳ ݍ‬͸‫ ܣ‬ଶ ߤ ‫ ݍ‬ቇ
െ
଴ ଵ
଴
ଶ
ǡ ǡ ଵ
ǡ ଵ ǡ Ͳǡ ǡ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
ʹ ʹ
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
ʹ
ʹ
ʹ
ʹ
(24)
By using the values of eq. (21), (22) and (24) in eq. (16),

020011-5

ିଵ ௚మ ௚మ ାଵ ଵ

ǡ ǡ
ǡ െܲ
‫ۍ‬
‫ ۍ‬ఈభ௚మ ଶഀభషమ ଺ǡଶ
‫ې ې‬
ఈభ మ
ଶ ଶ
ଶ
ଶ
ቍ ‫ۑ ۑ‬
భ ‫ܩ‬ସǡ଻ ቌ
మ ఓ ௤ ቮିଵା௚మ ௚మ ఈ ିଵ ఈ
‫ێ‬
‫ێ‬
ଵ
ିଵ
ଵ଺஺
బ భ
ǡ ǡ భ ǡ భ ǡ Ͳǡ ǡ
భ ‫ ێ‬ଶగ஺బ ୻ሺఈభ ሻఓభ మ
‫ۑ ۑ‬
‫ێ‬
ଶ
ଶ
ଶ
ଶ
ଶ ଶ
‫ି ݍ‬ቀ௉ାమቁ ‫ێ‬
‫ۑ ۑ‬
ିଵ ௚మ ௚మ ାଵ ଵ
‫ێ‬
ǡ ǡ
ǡ െܲ
‫ێ‬
‫ۑ ۑ‬
ఈమ ௚మ ଶഀమ షమ
ఈమ మ
଺ǡଶ
ଶ ଶ
ଶ
ଶ
‫ێ‬
ቮ
ቍ
భ ‫ܩ‬ସǡ଻ ቌ
‫ێ‬൅
௤ು
ଵ ିଵ ‫ۑ‬
ଵ଺஺బ మ ఓమ ௤ ିଵା௚మ ௚మ ఈమ ିଵ ఈమ
‫ێ‬
‫ۑ‬
మ
ሻఓ
ܲ௘ ൌ 
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
‫ ۏ‬ଶగ஺బ ୻ሺఈమ మ
ଶ୻ሺ௉ሻ
ଶ
ଶ
ଶ
ଶ
ଶ ଶ ‫ے‬
‫ێ‬
‫ۑ‬
ఈభ ఈమ ௚ర ଶഀభ షమ ଶഀమ షమ
ିሺ௉ାଵሻ
‫ێ‬
‫ۑ‬
െ
‫ݍ‬
భ భ
‫ێ‬
‫ۑ‬
ସగమ ஺బ మ ୻ሺఈభ ሻ୻ሺఈమ ሻఓభ మ ఓమ మ
‫ێ‬
‫ۑ‬
షభ ೒మ ೒మశభ
షభ ೒మ ೒మ శభ
ǡ ǡ
ǡ ǡ
ఈభ మ
ఈమ మ
଴ǡଵǣ଺ǡଵǣ଺ǡଵ െܲ
‫ܩێ‬ଵǡ଴ǣଷǡ଻ǣଷǡ଻
‫ۑ‬
ቀ
อషభశ೒మ ೒మమ ഀమభషభమഀభ భ షభ อషభశ೒మ ೒మమ ഀమమషభమഀమ భ షభอ
ǡ
ቁ
ଵ଺஺బ మ ఓభ ௤ ଵ଺஺బ మ ఓమ ௤
െ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
ǡ ǡ
ǡ ǡ଴ǡ ǡ
‫ۏ‬
‫ے‬
మ
మ
మ
మ
మ మ
మ
మ
మ
మ
మ మ
(25)
By Considering the special case with ߙଵ ൌ ߙଶ ൌ ߙǡ ߤଵ ൌ ߤଶ ൌ ߤǡ ߛଵ ൌ ߛଶ ൌ ߛ,
ܲ௘
െͳ ݃ଶ ݃ଶ ൅ ͳ ͳ
‫ۍ‬
‫ې‬
ଶ ఈିଶ
ଶ
ǡ ǡ
ǡ െܲ
ଵ
ߙ ଶ݃ସ ʹଶሺఈିଶሻ
ߙ݃ ʹ
ߙ
ିቀ௉ା ቁ ଺ǡଶ ‫ۇ‬
ʹ
ʹ ʹ
ʹ
ିሺ௉ାଵሻ ‫ۑ‬
‫ێ‬
‫ ۊ‬െ
ଶ ‫ܩ‬
ተ
‫ݍ‬
‫ݍ‬
ସǡ଻
ଵ
ͳ െͳ
‫ێ‬
‫ۑ‬
ͳ͸‫ܣ‬଴ ଶ ߤ‫ ݍ‬െͳ ൅ ݃ଶ ݃ଶ ߙ െ ͳ ߙ
Ͷߨ ଶ ‫ܣ‬଴ ଶ ሾȞሺߙሻሿଶ ߤ
ଶ
ǡ ǡ
ǡ ǡ Ͳǡ ǡ
‫ ݍ‬௉ ‫ܣߨ ێ‬଴ Ȟሺߙሻߤ
‫ۑ‬
‫ۉ‬
‫ی‬
ʹ
ʹ
ʹ
ʹ
ʹ
ʹ
ൌ
‫ۑ‬
ଶ
ଶ
ଶ
ଶ
ʹȞሺܲሻ ‫ێ‬
െͳ ݃ ݃ ൅ ͳ
െͳ ݃ ݃ ൅ ͳ
‫ێ‬
‫ۑ‬
ଶ
ଶ
ǡ ǡ
ǡ ǡ
ߙ
ߙ
ʹ
ʹ
ʹ ʹ
ʹ ʹ
‫ ܩێ‬଴ǡଵǣ଺ǡଵǣ଺ǡଵ ቀെܲ ተ
‫ۑ‬
ተ
ተ
ǡ
ቇ
‫ ێ‬ଵǡ଴ǣଷǡ଻ǣଷǡ଻ െ െͳ ൅ ݃ଶ ǡ ݃ଶ ǡ ߙ െ ͳ ǡ ߙ ǡ Ͳǡ ͳ ǡ െͳ െͳ ൅ ݃ଶ ǡ ݃ଶ ǡ ߙ െ ͳ ǡ ߙ ǡ Ͳǡ ͳ ǡ െͳ ͳ͸‫ܣ‬଴ ଶ ߤ‫ͳ ݍ‬͸‫ܣ‬଴ ଶ ߤ‫ۑ ݍ‬
ʹ ʹ
ʹ ʹ
ʹ ʹ
ʹ ʹ
ʹ
ʹ
ʹ
ʹ
‫ۏ‬
‫ے‬
(26)
Equation (26) represents final equation of BER for FSO system over K distribution turbulence channel with
pointing errors.

NUMERICAL RESULTS
An FSO link with turbulence fading parameterߙ ൌ ʹ . Fig.2 shows the outage probability in terms of
the average SNR (dB) for various values of the g. Where ‘g represents’ the ratio of equivalent beam radius at the
receiver and the pointing error displacement. As the value of g increases outage probability decreases and gives
better system performance. Lower value of g represents high misalignment error, as the value of g increases
misalignment error decreases. Fig.3 shows the average BER in terms of the average SNR (dB) for various values of
the g together with the various modulation schemes. For g = 1.2 and g = 4, system gives better performance for g = 4
which represents lower values of misalignment error. It is clear from the figure that coherent binary phase shift
keying (CBPSK) gives better performance than non-coherent binary frequency shift keying (NBFSK). For
improving the system performance multiple relay can be used.
Modulation Scheme

p

q

Coherent binary phase shift keying

0.5

1

Non-coherent binary frequency shift keying

1

0.5

TABLE 1. BER Parameters of Binary Modulation.

020011-6

.
FIGURE 2. Outage probability vs. Average SNR (dB) for
different value of g for ߙ ൌ ʹ

FIGURE 3. BER vs. Average SNR (dB) for different
modulation schemes.

020011-7

CONCLUSION
In the presence of misalignment fading, the outage and BER performance of a FSO system operating over kdistributed turbulence channel, is studied and the closed form expression of outage probability and BER is obtained.
We analyze that with an increase in misalignment errors, BER and Outage probability increases and degrade the
system performance. Further, we compared the results of different modulation scheme and found out with the BPSK
modulation scheme system gives the best performance. It is concluded that for getting excellent system performance
even with the strong turbulence conditions analysis can be done for multiple relay system.

REFERENCES
1.
2.
3.
4.
5.

6.
7.

8.
9.

10.

11.

12.

13.

14.

15.

Willebrand, Heinz, and Baksheesh S. Ghuman. Free space optics: enabling optical connectivity in today's
networks. SAMS publishing, 2002.
Borah, Deva K., David Voelz, and Santasri Basu. "Maximum-likelihood estimation of a laser system
pointing parameters by use of return photon counts." Applied optics 45.11 (2006): 2504-2509.
Sandalidis, Harilaos G., et al. "BER performance of FSO links over strong atmospheric turbulence channels
with pointing errors." IEEE Communications Letters 12.1 (2008): 44-46.
Borah, Deva K., and David G. Voelz. "Pointing error effects on free-space optical communication links in
the presence of atmospheric turbulence." Journal of Lightwave Technology 27.18 (2009): 3965-3973.
Ansari, Imran Shafique, Ferkan Yilmaz, and Mohamed-Slim Alouini. "Impact of pointing errors on the
performance of mixed RF/FSO dual-hop transmission systems." IEEE Wireless Communications Letters
2.3 (2013): 351-354.
Aggarwal, Mona, Parul Garg, and Parul Puri. "Dual-hop optical wireless relaying over turbulence channels
with pointing error impairments." Journal of Lightwave Technology 32.9 (2014): 1821-1828.
Prabu, K., and D. Sriram Kumar. "BER Analysis for BPSK Based SIM–FSO Communication System Over
Strong Atmospheric Turbulence with Spatial Diversity and Pointing Errors." Wireless Personal
Communications 81.3 (2015): pp. 1143-1157.
Prabu, K., and D. Sriram Kumar. "BER analysis of DPSK–SIM over MIMO free space optical links with
misalignment." Optik-International Journal for Light and Electron Optics 125.18 (2014): pp. 5176-5180.
Prabu, K., and D. Sriram Kumar. "Outage Analysis of Relay-Assisted BPSK-SIM Based FSO Systems over
Strong Atmospheric Turbulence with Pointing Errors." International Journal of Computer and
Communication Engineering 3.5 (2014): 317.
T. Ismail and E. Leitgeb, "Performance analysis of SIM-DPSK FSO system over lognormal fading with
pointing errors," 2016 18th International Conference on Transparent Optical Networks (ICTON), Trento,
Italy, 2016, pp. 1-4.
A. N. Stassinakis, H. E. Nistazakis, G. K. Varotsos, G. S. Tombras, A. D. Tsigopoulos and V.
Christofilakis, "Outage capacity estimation of FSO links with pointing errors over gamma turbulence
channels," 2016 5th International Conference on Modern Circuits and Systems Technologies (MOCAST),
Thessaloniki, 2016, pp. 1-4.
Aggarwal, M.; Garg, P.; Puri, P.; Sharma, P.K., "Performance analysis of optical wireless communication
system with a decode and forward relay," in Signal Processing and Integrated Networks (SPIN), 2014
International Conference on, vol., no., pp.333-337, 20-21 Feb. 2014.
A. K. M. N. Islam and S. P. Majumder, "Performance analysis of an FSO link in presence of pointing error
using multiple PIN photodetectors with equal gain combiner," 2015 18th International Conference on
Computer and Information Technology (ICCIT), Dhaka, 2015, pp. 341-346.
Adamchik, V. S., and O. I. Marichev. "The algorithm for calculating integrals of hypergeometric type
functions and its realization in REDUCE system." Proceedings of the international symposium on Symbolic
and algebraic computation. ACM, 1990.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic,
2000.

020011-8

