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Free Space Optical Multi-Carrier Code division Multiple Access with Receiver Spatial Diversity and

Turbo Coding: An Optimal Solution to Combat Strong Atmospheric Turbulence and Timing jitter

Md. Zoheb Hassan1, Tanveer Ahmed Bhuiyan

2, S.M. Shahrear Tanzil

3, S.P. Majumder

4

Electrical & Electronic Engineering, Bangladesh University of Engineering & TechnologyDhaka-1000, Bangladesh

Email: [1dip024.eee,

2casillas082,

3tanzil.eee]@gmail.com

4spmajumder@eee.buet.ac.bd

ABSTRACT

In this paper an analytical model of a free space multi carrier optical CDMA communication system with photo

detector spatial diversity in presence of strong atmospheric turbulence, timing jitter and multipleaccessinterference (MAI) is presented. The analysis presents a novel approach to the development of expression of the

output signal to interference plus noise ratio (SINR) of the single input single output (SISO) system considering

turbulence induced fading and timing jitter. The analysis is also extended to single input multiple output (SIMO)

with maximal ratio combining (MRC) system. A probability density function of the SINR at the output of MRC

receiver is developed representing joint effect of turbulence induced fading, jitter and photo detector spatial

diversity. The system performance is evaluated in terms of average bit error rate (BER) of the MRC receiver byintegrating conditional BER over the proposed probability density function (pdf). Here subcarrier intensity

modulation (SIM) based technique BPSK is used for modulation scheme and we have also analytically evaluated

system performance for different higher order M-ary PSK modulation schemes. The present analysis is also

extended to the application with forward error correction code (FEC) which is the most popular turbo code. Bit

error rate performance between with and without turbo encoding is compared. It is seen from numerical analysis

that applying multiple carrier with receiver spatial diversity improves bit error rate performance several times in

presence of strong turbulence and jitter. It is also reported that application of spatial diversity can make some

higher order PSK as a better choice than traditional BPSK or QPSK in terms of required received power so that

switching to higher order PSK is a better choice to improves system spectral efficiency. Finally application of

turbo encoding with optimally selected parameter makes bit error rate several times lower and can achieve a

satisfactory level of BER with less amount of received power or less number of receiving photo detectors, thuscan reduce receiver complexity.

Keywords: FSO communication, Atmospheric turbulence, Gamma-gamma

pdf, Diversity, MRC, Bit error rate, Timing jitter, Turbo encoding, Bit interleaving.

I. INTRODUCTION

Free space optical communication (FSO) is a wireless

optical communication technique in which point to point

optical transceivers communicate with light propagating

through air. FSO is a promising technology that replete the

gap between end user and fiber optic infrastructure [1]. The

main advantage of FSO is high bit rate transmission, low biterror rate, low cost and power requirement, unlicensed

wireless transmission (UWT) and security [2]. In fiber optic

communication optical code division multiple access has

gained popularity over other multiple access scheme due to

its ability to use the full available time, frequency and

wavelength slot and self routing by the code sequence. The

potential integration of FSO and OCDMA is the interest of

recent years research. FSO CDMA is reported in recent

article [1] where performance of FSO multicarrier CDMA is

evaluated in another recent article [3].

The main challenge of building FSO communication is theeffect of atmosphere on the propagating light. The

atmosphere contains aerosol and suspended water particle.The propagating light get scattered and deflected by these

particle and optical pulse attenuation take place [4]. In dense

atmosphere this attenuation is as high as 270db/km where inclear atmosphere it is only 0.043db/km [5]. Although FSO

communication suffers very low attenuation in clear

atmosphere the most severe challenge is atmospheric

turbulence induced irradiance fluctuation. Atmospheric

turbulence is caused by small change in temperature which

causes change in refractive index of air. The change in

refractive index causes phase and amplitude fluctuationwhich results beam scintillation, beam spread and beam

wander [6]. The irradiance fluctuation due to atmospheric

turbulence is a statistical process and therefore a tractable

model is necessary to describe it. Although On-Off keying

(OOK) is a widely used modulation scheme in FSO, the

turbulence induced irradiance fluctuation requires adaptive

threshold for OOK which is difficult to acquire. Hence

subcarrier intensity modulation (SIM) is proposed and it is

shown that SIM exhibits better performance than OOK [7]. In

order to mitigate the effect of turbulence diversity combiningis a very effective option. Using spatial diversity the BER

performance of FSO communication link in presence of log-

normal irradiance fluctuation is evaluated [8]. Channelcoding for optical wireless link along with diversity

combining is also proposed to alleviate the effect of

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turbulence [9]. However no closed from analysis is presented

yet on the performance of free space optical CDMA with

diversity combining and forward error correction coding.

In this paper we have formulated an analytical model on bit

error rate (BER) performance of free space optical multi

carrier code division multiple access using subcarrier

intensity modulation based technique with photo detector

spatial diversity in receiver and turbo encoding at transmitterside. We have also analysed effect of timing jitter onreceiving side. This paper is organized as follow: Section II

represents BER performance with spatial diversity and with

timing error. Section IV represents performance improvement

with turbo encoding. Finally section V and VI presents

numerical evaluation and conclusion respectively.

II. Model Of Atmospheric Turbulence

Temperature variation and wind flow give rise to the change

of refractive index of different layer of atmosphere. This

refractive index change causes phase perturbation of the

wavefront (isophase plane) of propagating light [10]. As aresult, secondary waves generated from different point of

wavefront have different phase and their interference with

each other gives amplitude variation. So atmospheric

turbulence causes both phase and amplitude fluctuation

which results intensity fading. Several statistical models are

proposed to represent turbulence induced intensity fading i.e.

log-normal pdf, K distribution and gamma-gamma pdf. The

weak turbulence is modelled by log- normal pdf while strong

turbulence is approximated by exponential model. Both

strong and weak turbulence are modelled by gamma-gamma

distribution which represents received optical intensityI=IxIyWhere Ix is strong eddies induced turbulence and Iy is weak

eddies induced turbulence. Gamma-gamma pdf forturbulence induced irradiance is following [11]

( ))(2)(2)( )(1)

2(

2/)(

IKIIf

++

=

(1)

Where,Iis the signal intensity (.) is the gamma function,

andK-is the modified Bessel function of the second kind of

order -. Here and are the effective number of small and

large scale eddies of the scattering environment. These

parameters can be directly related to atmospheric conditions

according to [11].1

)6/7(5/12

2

11)11.11(

49.0

exp

+=

x

x

(2)

1

67

512

2

1)69.1(

51.0exp

+

=

x

x

(3)

Here 611

6/722 23.1 LkCnx = is the rytov variance where

Cn2 is the strength of atmospheric turbulence, k is the wave

number and L is the link length. In week turbulence region

this pdf approaches to the log normal distribution while forstrong turbulence this approaches to exponential pdf.

III. Bit error rate performance evaluation

a.. Transmitter and channel model

In fig. 1(a) and 1(b) block diagram of transmitter and

receiver are showed. Here [ ]1,1kb is the input bitsequence and g(t) is unit amplitude rectangular pulse i.e

rec(t/Tb). Now input bit sequence is first converted into M

number of parallel bit sequence. Each parallel bit stream isthen spread by a chip and BPSK modulated by orthogonal

subcarrier having fundamental frequency fc and spaced apart

by integer multiple of F/Tb where F is integer.

Fig1(a) block diagram of transmitter

Fig .1(b):Block diagram of Receiver

If F=1 the system is similar to OFDM and a FFT block is

required instead of BPSK modulator. However to analyse in

continuous time domain we assume 2F . The spreadingsequence hasMnumber of chip where M is also the number

of orthogonal subcarrier. The DC bias is applied to drive the

laser diode. Let Pt is the power of transmitted optical signal

and is the modulation index. So transmitted optical signal[3] for k-th user is following:

[ ] [ ]=

++=

M

i b

ckbkt tT

FticiTtgibPtx

1

))2cos()(1()(

(4)

Next, consider the channel response for this optical signal.

Here channel responses limited by the effect of atmospheric

turbulence. Due to presence of atmospheric turbulence beamscintillation i.e. random fluctuation of beam intensity takes

place. This random intensity fluctuation can be modelled as

gamma-gamma pdf as equation (1). Beside this, atmospheric

turbulence also cause phase shift and distortion of point

spread function (PSF) of beam wave [10]. The distortion of

PSF occurs when beam coherence length, ro gets so small

because of strong turbulence and receiver aperture diameter

Do> > ro. As a result incoming optical power gets no more

concentrated around a focal point but gets spread. This

phenomenon is very similar to the multipath propagation in

RF wireless communication where several replica of RFwave is received with different delays. Again, due to multi-

carrier communication symbol duration on each parallel path

is increased which makes channel as slowly time varying and

frequency flat fading channel. So the channel response for m-

th subcarrier of n-th receiving photo detector will be

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nmj

nmnmnm etth,)()( ,,,

= (5)

here

2

,nm intensity fluctuation which is gamma- gamma

distributed with parameter and . mm, is the delay for the

m-th subcarrier of n-th receiving photo detector and nm, is

the phase shift of the m-th subcarrier of n-th receiving photodetector.

b.. Performance analysis

Now we consider K number of users simultaneously access

the network and assume each user has some transmitted

power level. If we consider total received power is Pr and

responsivity of photo detector is R then the photocurrent

generated by n-th photo detector will be:

= =

++=

K

k

j

b

c

M

i

kbknmrnmet

T

FticiTtgibRPtr

1 1

,,)2cos(][)(][1)(

)(tn+ (6)

Here n(t) is the noise current generated consist of shot noiseand thermal noise current. Now at first we consider n-th

receiving detector and decision variable for n-th photo

detector of u-th user after demodulation, despreading and low

pass filtering will be following. Here we assume

orthogonality among subcarriers, which is logical for high

frequency orthogonal carrier, so we can neglect inter carrier

interference.

dtt

T

Ftmctr

TZ

bT

t b

c

M

i

u

b

u

n = =

+=

0 1

)2cos(][)(1

)][][(2 ,1 1

nn j

k

K

ukk

n

M

i

uk

j

unr ebicicebMRP

= =

+=

dttT

Ftictn

T

bT

b

c

M

i

u

b

++=0 1

)2cos(][)(1

(7)

)][][(2 ,1 1

nn j

k

K

ukk

n

M

i

uk

j

unr ebicicebM

RP = =

+=

)(1 tn+ (8)

Here Tbis the bit period. Now let assume we have an array of

N photo detectors. Each photo detector can independently

receive signal and output of each photo detector is assumedindependent of another.

Fig1(c) : Block diagram of receiver diversity with maximal

ratio combining

Next we combine signals from all receiving photo detector

with MRC scheme. Having perfect estimation of channel

fading parameter is assumed. So after MRC decision variable

for u-th user is following:

nj

n

N

n

u

n

u eZZ

=

=1

= = ==

+=N

n

n

K

ukk

k

M

i

ukr

N

n

nur bicic

RPb

MRP

1

2

,1 11

2 ][][22

nj

n

N

n

etn

=

+1

1 )( (9)

Here in above equation first term is signal component, second

term is multiple access component while last term is noise

term representing photo detector noise after at the output of

MRC receiver

Therefore we find following term after maximal ration

combining for u-th user

Signal power, U2=

2

1

2

2

2

=

N

n

nr

MRP

MAIvariance,

][]][][[2

2

1 ,1 1

222

2

2

n

N

n

k

ukk

M

i

uknr

MAI EicicERP

= = =

=

( ) =

=

N

n

nhukr M

RPK

1

222

,

2

)0(2

1

Noise variance, No= Nsh + NthHereNsh=shot noise of the photo detector

BIRP

e Br

+= 2

2

AndNth =thermal noise of the photo detector

BR

KT

L

4=

Where

uk= cross correlation of spreading code between u-th user

and k-th user

=

=

N

n

nh E1

22 is the average power of the channel

B= bandwidth of the signal

K=Boltzmann constante= charge of electron

IB= background noise currentT=temperature

RL= Load resistance of photo detectorSo output SINR (signal to noise and interference ratio) willbe following:

( ) ohukr

M

i

nr

NMKRP

MRP

+

=

=

22

,

2

1

2

2

)0(12

2

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( )M

NK

RP

MRP

hukr

M

i

nr

022

,

2

1

2

2

)0(12

2

+

=

=

=

=

n

n

nC1

2

Where C

( )M

NK

RP

MRP

hukr

r

022

,

2

2

)0(12

2

+

=

is the average SINR of the system only with photo detectornoise and without turbulence fading

Now let, =

=

N

n

nrI1

2

Here2

n is gamma -gamma distributed with parameter

and . Each photo detector output is assumed independent aswe assumed uncorrelated fading of signal soIris the sum ofN number of independent random variable having gamma -gamma distribution. So pdf of Ir will be another gamma-gamma distribution having parameter t andtas follow[12]

++

+=

98.00124.1

0058.95.127.)1(NNt

Nt =

=

+

+

+

)(2)(2

)( )(1)

2(

2

2/)(

IN

KI

N

If tt

tt

ttrI tt

tt

tt

tt

r

Now pdf of output SINR will be following

)(1

)(C

fC

frI

= (10)

Now above analysis is done considering no timing error i.e.ideal sampling circuit. But in practical non-ideal sampling

circuit amplitude of sample is affected by a random timingjitter which is defined as variation of sample instant fromideal position over a time slot. In any communication systemthe effect of timing jitter is twofold. First signal power isdegraded by the factor of (1- ) and second is an additionalnoise proportional to jitter variable and input power isadded to the photo detector shot noise and thermal noise [13].

Here is normalized over symbol period i.e = /Tb whereis total timing error over a bit period. Therefore in presenceof jitter expression of SINR is modified as follow:

( )( )

( )( )

2

022

,

2

1

2

2

,

2)0(11

2

12

++

=

=

rhuk

r

M

i

nr

n

RP

M

NK

RP

MRP

jitter variance follows Gaussian distribution with zero mean

and variance 2as follow[13]:

( )2

2

2

22

1

= ef

Therefore mean SINR without turbulence induced fading is

calculated in following way

( )

( )( )

dfRP

M

NK

RP

MRP

C

rhuk

r

r

avg

++

=

0

2

022

,

2

2

)(

2)0(11

2

12

So by replacing C in former equation of the pdf of SINR byCavgwe find the probability density function of output SINRrepresenting combined effect of turbulence induced fadingand timing error with receiver spatial diversity

)(1

)(avg

Iavg C

fC

fr

= (11)

Now for BPSK modulation bit error rate

( )

=

2

2

1 SINRerfcyPe

(12)

For M ary PSK expression of BER is following [14]

( )

= )(

2

1

MSin

SINRKerfc

kyPe

(13)

Here k=log2M

For all these cases average BER can be computed as follow

( ) ( ) ( )dyyfyPavgP yee

=

0

(14)

It is difficult to obtain a close from expression of bit error ratefrom integration of above equation. Rather we willnumerically evaluate bit error rate which will be discussed insection V.

IV.SYSTEMWITH RECEIVER DIVERSITY AND TURBOENCODING-DECODING

Error control code is a powerful tool to handle error inreceived signal through wireless propagation. In recent papers[15] use of error control coding for wireless opticalcommunication gained attention. Although error control codecan correct error in received signal, however they alonecannot solve signal degradation by fading. Diversity is must

to handle fading. However a joint integration of error controland correcting code with diversity becomes an excellentexample to beat fading and other impairments. Among manyavailable forward error corrections code (FEC) turbo codehas gained most popularity. Fig.2(a) shows block diagram ofturbo encoder-decoder and fig 2(b) shows block diagram of

turbo coded system with diversity.

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Fig. 2(a)Block diagram of turbo encoder- decoder

Fig. 2 (b)Turbo encoded decoded communication system

A turbo encoders is a combination of two simpleRSC(recursive systematic convolutional) encoder For a block

of k information bit sequence each RSC encoder generates aset of parity bit sequence and so the code rate of the turboencoder is1/3. The important block is interleaver (P) whichpermutes the original bit sequence before second RSC

encoder. The iterative decoder employs A-PosterioriProbability (APP) decoder for each constitutes codes. Log

likelihood ratio (LLR) information of systematic sequenceand parity sequence is given to APP decoder along with a -piori information. Each APP decoder generates extrinsicinformation which is essentially independent of incoming

systematic sequence [16]. The resulting code achieves nearperformance of Shannons random code [17]

Now it is found that bit error probability of a BPSK

modulated system with turbo encoding is following [18]

( )

=

2

21 SINRCderfcByP iide

(15) Where,Bd ,= average number of ones on the minimum free distancepath in the overall turbo code trails (known as errorcoefficients)di = code minimum free distanceand Ci = code rateFor M-ary PSK above BER with turbo encoding will be as

follow

( )

= )(

2

2

1

MSin

SINRCdKerfcB

kyP iide

(16)

So average bit error rate (BER) for turbo encoding can becomputed by integrating equation a(15) and (16) over the pdfof the SINR found in equation (11).

V.NUMERICAL RESULTS AND DISCUSSION

Following theoretical analysis in Sec. III and IV, BER andother system performance results are numerically evaluatedin this section. The common parameter used in this section is

listed below:

parameter Strong

turbulence

Medium

turbulence

Weak

turbulencex

23.5 1.6 0.2

4.2 4.0 11.2

1.4 1.6 10

Parameter relevant to spreading and jitter ar listed below

Parameter Value

Spreading sequence used GOLD sequence

Number of chips of goldsequence

255

Timing jitter variance 0.2

-20 -15 -10 -5 0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

100

recived SINR in db

bit

error

rate

weak tubulence without jitter

weak turbulence with jit ter varriance 0.2

medium turbulence with jitt er varriance 0.2

strong turbulence with jit ter varriance 0.2

Fig. 3 SINR vs bit error rate

In Fig 3 BER vs. received SINR is profiled for differentturbulence conditions and timing error. This is clear fromfigure that increasing turbulence with timing error results

increased power penalty. Also it is also clear the effect ofturbulence strength is more pronounced than effect of jitter.

In Fig 4 effect of the increase of number of user on bit errorrate is presented for fixed received power and 255 chip lengthgold sequence. Here we see applying receiver diversityimproves BER several times in presence of both strong

turbulence and timing jitter..

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50 100 150 200 25010

-10

10-9

10-8

10-7

10

-6

10-5

10-4

10-3

10-2

number of user

biterro

rrate

withut timing error and spatial diversity

with jitter varriance =0.2 and single photodetector

with jitter varriance=0. and 2 receiving photodetector

Fig. 4number of user vs. Bit error rate for -20dbm power and255 chip length gold code in presence of strong turbulence

In Fig 5 the effect of increasing number of photo detector on

BER is profiled. Here it is clear that increasing number ofphoto detector improves bit error rate performance. Againthis performance improvement is limited by turbulence

strength and jitter variance.

Fig. 5BER vs. number of photo detector for SINR=10db

Fig.6 represents bit error rate vs. received SINR for differenthigher order PSK based subcarrier intensity modulation

(SIM). Higher order PSK system has more spectral efficiencyand consequently has more information carrying capability.But this results with sacrifice of transmitted power. Here we

see without receiver diversity for a fixed BER higher orderPSK requires more power. But with diversity we see above afixed threshold power 8 -ary PSK with 2 photo detectoroutperforms 4- ary PSK with single photo detector and 32 -

ary PSK with 3 photo detector outperforms all but it has morethreshold point. The significance of this result is higher orderPSK transmission is more preferable with spatial diversity tosave transmitted power along with increase of spectralefficiency above a threshold point.

-20 -15 -10 -5 0 5 10 15 20 25 3010

-10

10-8

10-6

10-4

10-2

100

received SINR in db

biterrr

orate

4-ary with 1 photodetector

8-ary with 1 photodetector

8-ary with 2 photodetector

32-ary with 2 photodetecto

32-ary with 3 phpdetector

Fig.6:BER vs. SINR for different M-ary PSK inpresence of

strong turbulence and timing jitter

-20 -15 -10 -5 0 5 10 15 20 25 3010

-10

10-8

10-6

10-4

10-2

10

0

received SINR in db

biterrorrate

without turbo coding

with turbo coding:P=20,d=7,BD=0.22

with turbo coding:P=50,d=9,BD=0.04

Fig. 7(a)BER vs. SINR for turbo coding without diversitywith storng turbulence and jitter and 1/3 code rate with

memory order 2

1 2 3 4 5 6 7 810

-25

10-20

10-15

10-10

10-5

100

number of receiver photo detector

biterrorrate

without turbo coding

turbo coding:P=50,d=7,Bd=7.05e-03

turbo coding:P=50,d=12,Bd=0.18

turbo coding:P=50,d=28,Bd=3.18e2

Fig. 7(b)BER vs. number of photodetector for SINR=10dbwith strong turbulence and jitter and 1/3 code rate with

memory order=2

In Fig .7(a) bit error rate performance with turbo encoding

and decoding is profiled and in Fig. 7(b) effect of differentparameter of turbo coding on different parameter is profiled.

With turbo encoding it is seen in Fig.7(a) that BERperformance gets several times improved than without turbo

encoding. It is also seen increasing interleaver size is a betterway to save transmitted power and number of receiver

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antenna for a fixed BER. In Fig.7(b) we see for fixed

interleaver size (P=50) the code free distance d and errorcoefficient Bd are two important parameters need to beoptimized. For small number of photo detectors turbo code

having lower code free distance preferable where for photodetectors number greater than 5 turbo coding having highererror coefficient is preferable

0.5 1 1.5 2 2.5 3 3.5 410

-25

10-20

10-15

10-10

10-5

100

scintillation index varriance

biterrorrate

without jitter,uncoded,with single photodetector

turbocoded with jitter varriance 0.2&single photodetector

turbocoded with jitter varriance 0.2& 2 photodetectors

Fig.8(a) BER vs. .variance of x for 10db SINR with turbo

encoder having P=50,d=7,Bd=7.05e-03andC=1/3

0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

scintillation index variance

SINRs

avingindb

turbo encoder with P=50,d=9,bd=0.04

turbo encoder with P=20,d=9,bd=0.22

Fig 8(b) SINR saving vs. variance of x for turbo coding

with 3 receiving photodetector for bit error rate 10-10

Figure 8(a) and 8(b) represents the benefit of the

commingling of receiver diversity and turbo encoding in

presence of strong atmospheric turbulence and timing jitter.Fig 8(a) shows a comparison of MC-OCDMA withoutdiversity and turbo coding and MC-OCDMA with using bothof them together with variation of turbulence strength. FromFig. 8(a) we see by using both technique together it ispossible to lower bit error rate(BER) very effectively without

high power transmission in case of strong turbulence withtiming jitter. Fig 8(b) shows the effective SINR saving forapplying turbo coding with diversity against variance of

scintillation index. Here we see about 16-25 db SINR issaved. At the same time we see turbo coding with higherinterleaver size outperforms turbo coding with lower

interleaver size .

VI.CONCLUSION

In this paper we have analyzed performance of free spacemulti carrier optical CDMA communication system inpresence of strong turbulence fading and timing jitter. A

closed form expression of the probability density function ofSINR representing joint effect of fading and timing jitter is

developed and average bit error rate is numerically computed.Performance improvement is proposed by means of spatialphoto detector diversity with maximal ratio combining atreceiver side and with turbo encoding decoding. It is seen

from numerical results that system performance is moredegraded by turbulence fading than jitter. It has been shownthat by applying receiver diversity it is possible to increasesystem spectral efficiency by using higher order PSK withoutincreasing transmitted power. It is also seen to combat jointeffect of strong turbulence and jitter combination of diversityand turbo coding becomes a very intelligent choice. Finally it

is seen along with diversity, turbo coding with optimallyselected parameter is an efficient way to reduce transmitted

power or to reduce number of receiving photo detector whichobviously reduce receiver complexity. From application pointof view this type analysis can be used to select optimumnumber of photo detectors in receiving side and optimumparameter of turbo coding with a fixed amount of transmitted

power for required level of BER. In future we expect toextend current analysis with transmit diversity along withreceiver diversity and channel coding .

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