JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 20, OCTOBER 15, 2013 3221

Coherent Polarization Modulated Transmissionthrough MIMO Atmospheric Optical

Turbulence ChannelXuan Tang, Zhengyuan Xu, Senior Member, IEEE, and Zabih Ghassemlooy, Senior Member, IEEE

Abstract—An optical signal suffers from irradiance and phasefluctuations when propagating through the free space optical (FSO)turbulence channel, thus resulting in the degradation of the biterror rate (BER) performance. The BER performance can be im-proved by adopting the multiple-input multiple-output (MIMO)scheme. In this paper, we propose a coherent binary polarizationshift keying (BPOLSK) modulation scheme with MIMO employ-ing maximum ratio combining and equal gain combining diversitytechniques to mitigate the turbulence effect. The gamma–gammastatistical channel model is adopted for all the turbulence regimes.The BER performance for the proposed BPOLSK-MIMO FSOlink is compared with the single-input single-output and ON–OFF-keying systems by means of computer simulation. The op-tical power gain is investigated and demonstrated under differentturbulence regimes for a number of transmitters/receivers.

Index Terms—Atmospheric turbulence, bit error rate (BER), bi-nary polarization shift keying (BPOLSK), free space optical (FSO),gamma–gamma (GG), multiple-input multiple-output (MIMO),single-input single-output (SISO).

I. INTRODUCTION

FREE space optical (FSO) communication, considered asa cost-effective and high bandwidth access technique, has

received increasing attention following its recent commercial-ization successes [1], [2]. It is an attractive solution for the “lastmile” access networks to bridge the bandwidth gap betweenthe end user and the fiber optic-based back-bone network al-ready in place. Its unique properties make it also appealing fora number of other applications, including fiber backup, enter-prise/local area network connectivity, metropolitan area networkextensions, redundant link, and back-haul for wireless cellularnetworks.

Manuscript received March 10, 2013; revised June 13, 2013 and July 29,2013; accepted September 4, 2013. Date of publication September 8, 2013; dateof current version September 23, 2013. This work was supported in part byNational 973 Program of China under Grant 2013CB329201, National NaturalScience Foundation of China under Grant 61171066, Tsinghua National Labo-ratory for Information Science and Technology under Grant 2011Z02289, andthe EU COST ACTIONS IC0802 and IC1101.

X. Tang and Z. Xu are with the Department of Electronic Engineering,Tsinghua University, Beijing 100084, China (e-mail: xtang2012@gmail.com;xuzy@tsinghua.edu.cn).

Z. Ghassemlooy is with the Optical Communications Research Group, Fac-ulty of Engineering and Environment, Northumbria University, Newcastle, NE18ST, U.K. (e-mail: z.ghassemlooy@northumbria.ac.uk).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2013.2281216

However, the FSO link reliability and availability are sus-ceptible to the atmospheric conditions [1]–[3]. Apart from at-tenuation, the major impairment is the atmospheric turbulence,which is due to the variations in the refractive index as the resultof inhomogeneities in the pressure and temperature fluctuations.The atmospheric turbulence leads to signal fading, also known asscintillation, which severely degrades the link performance es-pecially for link length >1 km [4].The deep fading could last for1–100 μs, thus resulting in a loss of up to 105 consecutive bits for1 Gb/s data link [5]. The theoretical models for statistical distri-bution of the random fading irradiance signals have already beendeveloped, comprising the lognormal, gamma–gamma, and neg-ative exponential corresponding to weak, weak-to-strong, andsaturation regimes, respectively [1], [6].

There are a number of schemes to mitigate the turbulence in-duced fading in FSO links. The maximum-likelihood sequencedetection scheme is not feasible for most practical applicationsdue to the high complexity in determining the metric [7]. Thedeployment of large receiving aperture can improve the systemperformance by increasing the total received signal power. How-ever, incoherency of the received field caused by the atmosphericturbulence results in significant performance degradation partic-ularly for large apertures [8]. The spatial diversity (SD) utilizingthe multiple-input multiple-output (MIMO) technique createsa large aperture at the receiver by deploying multiple smallerapertures, which provides a striking approach to compensate forthe turbulence induced fading [1]. When the diameter of eachaperture becomes smaller, the received wavefront is more co-herent over each aperture compared to the system with a largeraperture. The outage probabilities of MIMO FSO systems overlognormal turbulence channels have been investigated in [9].In [10] and [11], the results for MIMO FSO systems employingpulse-position-modulation (PPM) and Q-ary PPM in lognormaland Rayleigh fading channels have been studied. The bit er-ror rate (BER) performance of MIMO FSO links for both theindependent and correlated lognormal atmospheric turbulencechannels have been fully covered in [12], whereas the coherentFSO with MIMO in Rician statistic channel has been reportedin [8]. The analysis for MIMO FSO systems in gamma–gamma(GG) channel is intractable due to the involvement of the modi-fied Bessel function of the second kind [3]. A limited number ofresearchers have published results of the distribution of the sumof independent GG variables. The pairwise error probabilities ofSISO and MIMO FSO systems with intensity modulation/directdetection (IM/DD) have been derived in [1], and the BER per-formance of IM/DD MIMO FSO systems over independent and

0733-8724 © 2013 IEEE

3222 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 20, OCTOBER 15, 2013

not necessarily identically distributed K atmospheric turbulencechannels have been reported in [2].

The simplest modulation technique widely used in the FSOsystems is the IM/DD ON-OFF-keying (OOK). However, OOKsuffers highly from the atmospheric turbulence, thus requir-ing the adaptive detection schemes at the receiver [13], [14].The adaptive threshold detection schemes have been developedwhere both the means and variances are precisely tracked us-ing a Kalman filter algorithm, which cannot be realized in thepractical FSO systems. Alternatively, the experimental resultshave shown that the polarization states of a propagating opticalbeam are the most stable parameters compared to the intensityand phase [15] and can be maintained over the entire FSO linkspan [16]. The FSO links employing polarization shift keyingare significantly insensitive to the phase noise of the laser localoscillator (LO) at the receiver, provided the intermediate fre-quency (IF) filter bandwidth is large enough [17]. The circlepolarization shift keying with DD offers a 3 dB lower signal-to-noise ratio (SNR) to achieve the same BER compared toOOK [15]. In [18], it was shown that polarization multiplexingtransmission with coherent detection in the lognormal fadingchannel offers a power gain of 5–14 dB along with the doubleof data rate compared to IM/DD links.

The main goal of this paper is to systematically investigatethe heterodyne binary polarization shift keying (BPOLSK)-FSOsystem by employing the MIMO technology over the GG tur-bulence channel. The system noise (the background radiation,thermal noise, and shot noise) is modeled as an additive whiteGaussian noise (AWGN) process. Maximum ratio combining(MRC) and equal gain combining (EGC) combining techniquesare considered to further improve the BER performance. Thedetailed unconditional BER analysis and the outage probabili-ties are also carried out. The error probabilities are compared toOOK to illustrate the advantages of the proposed scheme. TheSD gains for a range of transmitter/receiver configurations arealso demonstrated under different turbulence regimes.

The remainder of this paper is organized as follows: the GGstatistic channel model is introduced in Section II; the principlesof SISO BPOLSK-FSO system are outlined in Section III; thein-depth analytical and numerical results for MIMO BPOLSK-FSO are presented in Section IV. The results are discussed andcompared in Section V, and the conclusion is given in Section VI.

II. GAMMA–GAMMA TURBULENCE CHANNEL

The GG model is first proposed by Andrews et al. [6], [19].This distribution is the product of two independent GG randomvariables. Both are statistically independent random processesand governed by GG distributions [6]. The PDF of a three-parameter GG random variable is derived as [3], [6]

fγ (γ;α, β, μ) =(αβ)

α + β2

Γ (α) Γ (β)√

μγ

(√γ

μ

) α + β2 −1

× Kα−β

(2

√αβ

√γ

μ

)(1)

TABLE ITURBULENCE PARAMETERS FOR ALL TURBULENCE REGIMES

Fig. 1. Block diagram of the BPOLSK-FSO system: (a) the transmitter, and(b) the receiver.

where Kv (·) denotes the modified Bessel function of the secondkind (see [20, eq. (8.432.2)]) and Γ (·) is the Gamma function(see [20, eq. (8.310.1)]). The electrical SNR and the averageSNR are defined as γ and μ = E [γ], respectively. Assumingspherical wave propagation, α and β can be directly related toatmospheric conditions [3], [6]

α =

⎡⎢⎣exp

⎛⎜⎝ 0.49σ2

l(1 + 0.18d2 + 0.56σ

12/5l

)7/6

⎞⎟⎠− 1

⎤⎥⎦−1

, (2a)

β =

⎡⎢⎣exp

⎛⎜⎝ 0.51σ2

l

(1 + 0.69σ

12/5l

)−5/6

(1 + 0.9d2 + 0.62d2σ

12/5l

)5/6

⎞⎟⎠− 1

⎤⎥⎦−1

. (2b)

where Rytov variance σ2l = 0.5C2

nk7/6L11/6 , d = (kD2/4L)1/2 , L is the link span, k = 2π/λ is the wave number, λ

is the wavelength, and D is the diameter of the receiver collect-ing lens aperture. The index of refraction structure parameterC2

n varies from 10−13 to 10−17 m−2/3 for the strong and weakturbulence regimes, respectively, with a typical average value of10−15 m−2/3 [21]. Values of the channel parameters are givenin Table I.

III. SYSTEM CONFIGURATION

A. BPOLSK-FSO System Model

Fig. 1 shows the block diagram of the coherent BPOLSK-FSO system. The transmitter consists of a transmitting laser(TL), a polarization beam splitter (PBS), and two external

TANG et al.: COHERENT POLARIZATION MODULATED TRANSMISSION THROUGH MIMO ATMOSPHERIC OPTICAL TURBULENCE CHANNEL 3223

LiNbO3-Mach-Zehnder Interferometers (MZIs) based ampli-tude modulators. The optical carrier �E0 (t) is linearly polarizedalong π/4 regarding the reference axis of PBS. �E0 (t) is splitequally into two light beams with �x and �y polarizations, respec-tively, which are modulated by MZIs and then combined by apolarization beam combiner to form the BPOLSK signal. MZIsintroduce constructive and destructive interferences accordingto the bits “0”and “1,” respectively.

The transmitted optical signal viewed as a combination oftwo orthogonal amplitude modulated signals is given as [22]

�Es (t) = a (t) {[1 − m (t)] · �x + m (t) · �y} (3)

where a (t) = Aei(ωt+ϕ(t)) is the radio frequency (RF) carrier,A,ω, and ϕ (t) representing the amplitude, angular frequency,and the phase noise of the optical carrier, respectively.

BPOLSK-FSO using the heterodyne receiver is shown inFig. 1(b). The large-aperture lens can focus the incoming lightsignal and project it onto highly sensitive photo-detectors (PDs)located at its focal point. An optical bandpass filter (OBPF)(bandwidth typically 1 nm) can help reduce the potential im-pact of the background light interference on the FSO link per-formance. OBPF’s bandwidth depends on the linewidth of thelaser, which is <1 nm. The working principle of automaticfrequency control circuit is same as a phase-locked-loop cir-cuit for system synchronization with control signals derivedfrom the intermediate components. It is used to compensate forslow-frequency fluctuations from LO [23]. The received sig-nal is expressed as �Er (t) = ar (t){[1-m(t)]�x+m(t)�y} wherear (t) = Are

i(ωt+ϕr (t)) with Ar and ϕr (t) are the time-variantstatistics due to the turbulence fluctuation. The optical LO sig-nal �Elo (t) = alo (t) {�x + �y} is linearly polarized at π/4, wherealo (t) = Aloe

i(ωl o t+ϕl o (t)) , Alo , ωlo and ϕlo (t) are the ampli-tude, angular frequency and phase noise, respectively. �Er (t) issplit by PBS, which is then mixed with �Elo (t) thus resulting in�Ex (t) and �Ey (t).

At the receiver, the optical fields are detected by two identi-cal PDs with a unit area. As the received irradiance is equallydivided between two channels, the electrical signals cx,y (t) arepassed through ideal electric bandpass filter (BPF) with a band-width and a center frequency of Bbp = 2 (Rs + kF BL ) andωIF , respectively. Rs and BL are the symbol rate and linewidthof the laser sources, respectively. kF is chosen to pass the signalthrough the filter with a minimum distortion. The signals at theoutput of BPF are given as

cxb (t) = �ArAlocos (ωIF t + ϕIF (t)) [1 − m (t)] + nx (t)

(4a)

cyb (t) = �ArAlocos (ωIF t + ϕIF (t)) m (t) + ny (t) (4b)

where � is the PD responsivity, ωIF = ω − ωlo and ϕIF (t) =ϕr (t) − ϕlo (t) are the frequency and phase noise of the IFsignal, respectively. The noise terms nx,y (t) ∼ N

(0, σ2

n

)are

assumed to be AWGN. Both noise terms are uncorrelated suchthat nx (t) · ny (t) = 0. An ideal square-law demodulator con-sisting of an electrical mixer, a sampler, and a threshold detectoris used to recover the information signal. Note that the IF phase

noise disappears because of the square-law demodulation, thusillustrating BPOLSK insensitivity to the phase noise. The outputVj (t) from matched filter with a bandwidth of Rs is sampled atthe symbol period t = T to obtain the decision variable Vj . Vj isthen compared with a zero threshold level yielding the signal V̂j ,thus leading to detection of the transition between two adjacentsymbols, by which information is encoded. Assuming indepen-dent and identically distributed (i.i.d.) data transmission, theconditional error probability Pec is given by [24], [25]

Pec =12

exp (−γ/2) (5)

where γ = �2A2rA

2lo/σ2

n is the electrical SNR at the input ofthe coherent demodulator. The unconditional error probabilityfor BPOLSK-FSO using SISO is obtained by averaging (5) overGG distribution (1)

Pe =∫ ∞

0Pec (γ) fγ (γ;α, β, μ) dγ. (6)

A direct use of (6) yields an expression that unfortunately doesnot have a closed form solution. To further simplify it, exp [·]and Kv [·] using (see [26, eqs. (11) and (14)]) are rewritten interms of Meijer’s G-function Gm,n

p,q [·] (see [20, eq. (9.301)]).A closed-form error probability for BPOLSK-FSO using SISObecomes

Pe =1

8πΓ (α) Γ (β)

(√2αβ√

μ

)(α+β )/2

× G4,11,4

⎡⎢⎢⎣ (αβ)2

8μ

∣∣∣∣∣∣∣∣1 − α + β

4

α−β

4,α−β+2

4,β−α

4,β−α+2

4

⎤⎥⎥⎦ .

(7)

Using (7), the BER performance of the SISO BPOLSK-FSOsystem against the normalized electrical SNR (μ) for all theturbulence regimes (represented by L) is plotted as shown inFig. 2. For fair comparison with OOK, the average transmit-ted power for both modulation schemes is made equal. In anatmospheric turbulence channel, the threshold level for OOKwill vary with the irradiance fluctuation and the noise level.Therefore, the threshold is no longer fixed at half of the re-ceived irradiance. The unconditional BER for OOK system inGG turbulence channel is given as [27]

POOK = p (0)∫ ∞

it h

1√2πσ2

n

exp(−γ

2

)dγ + p (1)

×∫ ∞

0

∫ it h

0

{1√

2πσ2n

exp

[−(x−√

γ)2

2

](αβ)

α + β2

Γ (α) Γ (β)√

μγ

×(√

γμ−1) α + β

2 −1Kα−β

(2√

αβ√

γμ−1

)}dxdγ. (8)

Fig. 2 illustrates the BER performances of BPOLSK along-side that of OOK system employing adaptive and fixed thresh-old (of 0.5) values in GG turbulence induced fading. OOK with

3224 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 20, OCTOBER 15, 2013

Fig. 2. BER against the normalized electrical SNR across the whole turbulenceregimes using the BPOLSK and OOK with fixed (Fix) and adaptive (Adp)threshold.

adaptive detection shows performance relatively insensitive toSNR at the BER of 10−3 , and much worse than BPOLSK inweak to medium turbulence channels. It does require an accu-rate knowledge of both additive noise and fading levels, whichis not always practical [27]. OOK with a fixed threshold levelof 0.5 displays the worst system performance, and also createsthe BER floor, see Fig. 2 [27]. Another observation is that BERincreases drastically as the turbulence effect gets stronger forboth modulation schemes. This motivates the use of MIMOtechnology, as discussed later.

B. Outage Probability

The outage probability Pout is an alternative performancemetric for quantifying the performance of communication sys-tems in the fading channels, which is defined as Pe > P ∗

e and P ∗e

is a predetermined BER threshold. The probability can be trans-lated into the probability of SNR γ falling below the specifiedthreshold γ∗. That is Pout = P (Pe > P ∗

e ) ≡ P (mγ < γ∗),where the power margin m is the extra power needed to mitigateturbulence induced signal fading. By using (see [26, eq. (26)]),a closed-form solution for the outage probability is derived as

Pout =∫ μ

m

0

(αβ)α + β

2

Γ (α) Γ (β)√

μγ

(√γμ−1

) α + β2 −1

× Kα−β

(2√

αβ√

γμ−1

)dγ

=(αβ)

α + β2

2Γ (α) Γ (β) (m)α + β

4

× G2,11,3

⎡⎢⎢⎣ αβ√

m

∣∣∣∣∣∣∣∣1 − α + β

4

α − β

2,β − α

2,−α + β

4

⎤⎥⎥⎦ . (9)

IV. MULTIPLE-INPUT-MULTIPLE-OUTPUT BPOLSK-FSO

BPOLSK-FSO with MIMO employing M-transmitter andN -PD is depicted in Fig. 3. To avoid any correlation in thereceived irradiance, the spacing between detectors is assumedto be greater than the transverse correlation size ρ0 of the laserradiation in atmospheric turbulence channel, where ρ0 is inthe order of a few centimeters [27]. We have assumed thatthe propagation delay across the receiver array is negligible.Since the terrestrial FSO uses a line-of-sight with a negligibledelay spread, the intersymbol interference is not considered.The received optical power γij is assumed to be constant andtime invariant during one symbol duration T � τ0 , where thecoherence time τ0 of the atmospheric fluctuation is in the orderof milliseconds [27].

The received signal from each branch is scaled by the gainfactor {Gi}Ni=1 . The output of the combiner is the sum of theweighted and cophased signals. Each receiver aperture sizeof N -PD is (1/N ) th of the aperture area of the single re-ceiver [2], [27]. Accordingly, the background noise varianceon each branch is proportional to the receiver aperture area,which is reduced by a factor of N , whereas the thermal noiseon each branch is not affected. Assuming the backgroundnoise being the dominant source, the system noise becomesσ2

T ≈∑N

i=1 G2i σ

2n/N , where i = 1, 2, 3, . . . ,N [2], [27]. The

electric currents are scaled by {Gi}Ni=1 before being cophasedand coherently combined. The total SNR is obtained as

γT =(�Alo)

2

M2N

(∑Ni=1 Gi

∑Mj=1 Arij

)2

∑Ni=1 G2

i σ2n

(10)

where Arij denotes the received signal amplitude through theoptical turbulence channel between the ith transmitter and thejth receive aperture. M in the denominator ensures that a con-stant transmitted optical power is maintained to ensure a faircomparison. The unconditional BER for MIMO MRC is ob-tained by averaging the conditional error probability over thestatistics of GG distribution (1)

PMIMO =∫ ∞

0

12

exp(−γT

2

)fγ (�γ;α, β, μ) d�γ. (11)

A. Maximum Ratio Combining

On the receiver side using the MRC linear combiner scheme,{Gi}Ni=1 is proportional to γij . As MRC linear combinerresults in a maximum-likelihood receiver structure [2], itis optimal regardless of the fading statistics. However, γij

level and the phase on each branch have to be estimatedprior to coherent combining. In this section, we will com-pare the exact BER with the approximate BER for MIMOMRC in i.i.d. turbulence channels. For MIMO using MRC(11) cannot be derived directly, applying Cauchy inequality[20], (

∑Ni=1 Gi

∑Mj=1 Arij )2≤(

∑Ni=1 G2

i )(∑N

i=1∑M

j=1 A2rij ),

where equality holds with the optimal gain values proportionalto signal power of the corresponding branch, the optimal SNRfor MIMO with MRC γMRC is derived as the upper bound of

TANG et al.: COHERENT POLARIZATION MODULATED TRANSMISSION THROUGH MIMO ATMOSPHERIC OPTICAL TURBULENCE CHANNEL 3225

Fig. 3. Block diagram of BPOLSK-FSO-based MIMO with M-transmitter and N -PD.

the following:

γMRC (�γ) ≤ (�Alo)2

M2N

(∑Ni=1 G2

i

)(∑Ni=1∑M

j=1 A2rij

)σ2

n

∑Ni=1 G2

i

=

∑Ni=1∑M

j=1 γij

M2N . (12)

By substituting (12) into (11) and replacing exp [·] and Kv [·]using (see [26, eq. (11) and (14)]) Meijer’s G-function Gm,n

p,q [·](see [20, eq. (9.301)]), the closed-form expression for MIMOMRC is derived

PapMRC =N∏

i=1

M∏j=1

(√2M2Nαβ

) α + β2

8πΓ (α) Γ (β) (μij )α + β

4

× G4,11,4

×

⎡⎢⎢⎣M

2N (αβ)2

8μij

∣∣∣∣∣∣∣∣1 − α + β

4

α − β

4,α − β + 2

4,β − α

4,β − α + 2

4

⎤⎥⎥⎦

(13)

where μij denotes the average electrical SNR of each FSO link.To gain further insight into the performance of FSO links withSD, we only investigate the receiver diversity as special cases,i.e., M = 1, (11) can be calculated as [2]

PexMRC (�γ) =∫ ∞

0

12

exp(−γ

2

)fγ (�γ;α, β, μ) d�γ. (14)

By applying (see [26, eq. (11), (14), and (21)]) into (14), theclosed form for the exact BER of SIMO MRC can be derived

PexMRC =12

⎛⎜⎜⎝(√

2Nαβ) α + β

2

2πΓ (α) Γ (β) μα + β

4

⎞⎟⎟⎠

N

×(

G4,11,4

×

⎡⎢⎢⎣ α2β2N

8μ

∣∣∣∣1 − α + β

4

α − β

4,α − β + 2

4,β − α

4,β − α + 2

4

⎤⎥⎥⎦)N

.

(15)

Fig. 4. Comparison of exact (Ex.) and approximate (Ap.) BER for the SIMOMRC with N=3 in GG channel.

In Fig. 4 we plot the approximate BER given by (13) andinclude the exact BER (15) as a reference for M = 1 and N =3 using different values of channel parameters α and β. It isobserved that the derived BER provides a good approximationand coincides with the exact BER for higher SNRs.

B. Equal Gain Combining

In EGC, the knowledge of the phase on each branch is es-sential. Here, all the received signals of �x- and �y-channels arecombined coherently with equal weights of {Gci}N

i=1 = 1 [27].The SNR for EGC,γEGC (�γ), is given as

γEGC (�γ) =(�Alo

MN

)2(∑N

i=1∑M

j=1 Aij

)2

σ2n

=

(∑Ni=1∑M

j=1√

γij

)2

M2N 2 . (16)

Following the approach detailed in [28], γT =(∑N

i=1∑M

j=1√

γij )2 denotes the sum of MN i.i.d. GGrandom variables with mean μγT

= MNμ and parametersαγT

= MNα + εγ and βγT= MNβ, with εγ representing

the adjustment parameter in order to improve the accuracy ofproposed approximation [28]

εγ

(MN , α, β

)=(MN − 1

)−0.127 − 0.95α − 0.0058β

1 + 0.00124α + 0.98β.

(17)

3226 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 20, OCTOBER 15, 2013

The PDF for MN received uncorrelated GG signals is givenby [3]

fγ (γT ;αγT, βγT

, μγT) =

(αγTβγT

)α γ T + β γ T

2

Γ (αγT) Γ (βγT

)√μγTγT

×(√

γT

μγT

) α γ T + β γ T2 −1

Kαγ T−βγ T

(2

√αγT

βγT

√γT

μγT

).

(18)

The unconditional BER for MIMO-BPOLSK using EGC ina GG FSO link is obtained as

PeEGC=∫ ∞

0

12

exp(−γEGC (�γ)

2

)fγ (γT ;αγT

, βγT, μγT

) dγT

(19)

It is simplified as (20) by replacing exp [·] and Kv [·] using(see [26, eq. (11), and (14)]) Meijer’s G-function Gm,n

p,q [·] (see[20, eq. (9.301) and eq. (9.31.5)])

PeEGC =2αγ T

+βγ T−3

πΓ (αγT) Γ (βγT

)

× G4,11,4

[(MN αγT

βγT

)28μγT

∣∣∣∣∣1

αγT

2,αγT

+ 12

,βγT

2,βγT

+ 12

].

(20)

C. Outage Probability

The probability that SNR of the combined signal at the re-ceiver γT < γ∗

T , is an important parameter for the design of theFSO link as a data network. When MIMO FSO link is i.i.d.distributed, the distribution of γT can be approximated by thePDF of a single GG variable with parameters (μγT

, αγT, βγT

)as defined previously [28]. Following the approach in [28] andusing (see [26, eq. (26)]), a closed-form solution for Pout ofBPOLSK-MIMO system is defined as [28]

Pout =∫ μγ T

/mT

0fγT

(γT ;αγT, βγT

, μγT) dγT (21)

where the power margin mT is the additional power requirementas explained earlier. The closed form of (21) is written as (22)using Meijer’s G-function

Pout =(αγT

βγT)

α γ T + β γ T2

2Γ (αγT) Γ (βγT

) (mT )α γ T + β γ T

4

× G2,11,3

×

⎡⎢⎢⎣αγT

+ βγT√mT

∣∣∣∣∣∣∣∣1 − αγT

+ βγT

2

αγT− βγT

2,βγT

− αγT

2,−αγT

+ βγT

2

⎤⎥⎥⎦ .

(22)

V. RESULTS AND DISCUSSION

Using (7), (13), and (20) the BER against the normalized SNRfor a range of N and M over the i.i.d. atmospheric turbulence

Fig. 5. BPOLSK with EGC and MRC against the normalized SNR for differentN and M at a link range of 4 km.

channel with L = 4 km is shown in Fig. 5. For SISO systemsto achieve a lower BER, the SNR needs to be increased. For ex-ample, reducing the BER from 10−5 to 10−6 , an extra ∼10 dBof SNR is needed. However, increasing the power margin inthe link budget to mitigate the turbulence induced fading isnot practical and even not feasible for many applications. Thepowerful scintillation-mitigation techniques, such as the diver-sity techniques, are therefore required. A parameter called the“power gain” (SNRσl,M N – SNRσl,SISO ) is introduced to ac-count for the amount of power reduced to achieve the targeterror probabilities at the same turbulence level compared withthe benchmark SISO scheme. It is noted that the power gainis significantly improved as M and N increase, which is alsodepicted. To achieve a BER of 10−5 , using 1×2, 1×3, and 2×2MIMO with MRC the power gains obtained are∼26 dB,∼36 dBand ∼38 dB for the moderate turbulence regime, respectively.To achieve a BER of 10−6 under the same channel conditions,power gains increase to ∼31, ∼42 and ∼45 dB, respectively.Increasing M and N can result in additional power gains. How-ever, achieving a sufficient spacing between receivers to captureuncorrelated signals is far more challenging. It is also noted thatthe performance of MRC outperforms that of EGC under thesame channel conditions as well as being more complex to im-plement. For instance, to achieve a BER of 10−6 , power gainsachieved by using 1×2, 1×3, and 2×2 MIMO with EGC are∼25, ∼33, and ∼37 dB, respectively. Further observation fromFig. 5 shows that the power gain is more than 3 dB for N = 2in a weak turbulence regime. Since the atmospheric turbulencechannel is modeled as GG distribution, a diversity gain of 3 dB(N = 2) is achievable in a Gaussian channel.

In Fig. 6, the power gain of different MIMO FSO systemsemploying MRC and EGC over i.i.d. atmospheric turbulencechannels is further plotted against a range of link spans. MRCoffers improved system performance compared to EGC withthe tradeoff of the increased system complexity. The power gainreaches up to ∼46 dB when 2×2 MRC MIMO is used at L =4 km. The power gain is higher for worse channel conditionssince using higher M and N will efficiently reduce the chanceof a catastrophic fading.

The outage probabilities of SISO and MIMO systems ((9)and (22)) are plotted against the normalized SNR for the weak

TANG et al.: COHERENT POLARIZATION MODULATED TRANSMISSION THROUGH MIMO ATMOSPHERIC OPTICAL TURBULENCE CHANNEL 3227

Fig. 6. Power gain of BPOLSK with EGC and MRC against the link lengthL at a BER of 10−6 .

Fig. 7. Outage probabilities against the normalized SNR for SISO and MIMOin weak turbulence channel.

turbulence regime as depicted in Fig. 7 for a link range of 2.8 km.To achieve a Pout of 10−5 , the power gain over SISO at the linkspan of 2.8 km is ∼20 dB for the 1×2 MIMO structure. Thisvalue raises to ∼27 dB by adding an extra receiver as 1×3MIMO. Further power gains will be obtained using higher Mand N . The power margin of ∼30.5 dB is obtained for M = 2and N = 2. The power gain increases as the outage probabil-ity decreases. For instance, to achieve a Pout of 10−6 under thesame turbulence level, the power gains become∼24,∼31.5, and∼36 dB for 1×2, 1×3, and 2×2 MIMO systems, respectively.The increasing power gain indicates that for a single communi-cation link to achieve lower Pout is extremely challenging due tothe turbulence induced fading, but the situation can be improvedby using higher values of M and N .

VI. CONCLUSION

The error rate performance of the coherent BPOLSK-FSOcommunication system using MIMO over GG-distributed at-mospheric turbulence channel was studied in this paper. MRCand EGC techniques were applied to circumvent the turbulenceinduced fading. The approximated closed-form expressions forthe average BER and Pout of SISO and MIMO FSO systems interms of the Meijer G-function were obtained analytically. Theperformances of BPOLSK- and OOK-based FSO systems werecompared across all turbulence regimes. According to the sim-ulation, BPOLSK offers improved performance. Meanwhile,

the performance of FSO systems can be enhanced by usingmultiple apertures at the transmitter and/or receiver. Significantpower gains were achieved under different turbulence regimes,especially when employing higher number of transmitters andreceivers. Moreover, it was shown that MRC outperforms EGCunder the same channel conditions with a tradeoff in complexityincrement.

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Xuan Tang received the diploma (with hons) in electrical engineering fromNanyang Polytechnic, Singapore in 2007 and the BEng (first class hons) degreein electric and communication engineering from Northumbria University, New-castle upon Tyne, U.K, in 2008. She was awarded her PhD degree in free-spaceoptical communications in Optical Communications Research Lab (OCRG) atthe same university in 2012. Her research interests include optical communi-cation (outdoor wireless), digital communication and digital signal processing.She also worked as a Research/Teaching Assistant in the Electrical EngineeringDepartment at Northumbria University between 2008 and 2009. She is cur-rently a postdoc with Optical Wireless Information Systems Lab, Departmentof Electronic Engineering, Tsinghua University, China on optical wireless com-munications.

Zhengyuan Xu received his B.S. and M.S. degrees in electronic engineeringfrom Tsinghua University, Beijing, China, in 1989 and 1991, respectively, andPh.D. degree in electrical engineering from Stevens Institute of Technology,Hoboken, NJ, in 1999. From 1991 to 1996, he was with Tsinghua UnisplendourGroup Corporation, Tsinghua University, as system engineer and departmentmanager. In 1999, he joined University of California, Riverside, first as AssistantProfessor and then tenured Associate Professor and Professor. He was FoundingDirector of the multi-campus Center for Ubiquitous Communication by Light(UC-Light), University of California. In 2010, he was selected by the “Thou-sand Talents Program” of China, and appointed as Professor in Departmentof Electronic Engineering and Tsinghua National Laboratory for InformationScience and Technology, Tsinghua University, where he has established OpticalWireless Information Systems (OWiSys) Laboratory. His research focuses onwireless communications, networking, optical wireless communications, geolo-cation, and intelligent transportation systems. He has published over 160 journaland conference papers. He has served as an associate editor and guest editor fordifferent IEEE journals, and now serves as an associate editor for a new OSAjournal Photonics Research. He was Founding Chair of IEEE GLOBECOMWorkshop on Optical Wireless Communications.

Zabih Ghassemlooy CEng, Fellow of IET, Senior Member of IEEE: Receivedhis BSc (Hons) degree in Electrical and Electronics Engineering from theManchester Metropolitan University in 1981, and his MSc and PhD in Op-tical Communications from the University of Manchester Institute of Scienceand Technology (UMIST), in 1984 and 1987, respectively with Scholarshipsfrom the Engineering and Physical Science Research Council, UK. From 1986–87 worked in UMIST and from 1987 to 1988 was a Post-doctoral ResearchFellow at the City University, London. In 1988 he joined Sheffield Hallam Uni-versity as a Lecturer, becoming a Reader in 1995 and a Professor in OpticalCommunications in 1997. From 2004 until 2012 was an Associate Dean forResearch in the School of Computing, Engineering and in 2012 he becameAssociate Dean for Research and Innovation in the Faculty of Engineeringand Environment, at Northumbria University at Newcastle, UK. He also headsthe Northumbria Communications Research Laboratories within the Faculty.In 2001 he was a recipient of the Tan Chin Tuan Fellowship in Engineer-ing from the Nanyang Technological University in Singapore to work on thephotonic technology. He is the Editor-in-Chief of the International Journal ofOptics and Applications The Mediterranean Journal Electronics and Communi-cations. He currently serves on the Editorial Committees of number internationaljournals. He is the founder and the Chairman of the IEEE, IET InternationalSymposium on Communication Systems, Network and Digital Signal Process-ing. His researches interests are on photonics switching, optical wireless andwired communications, visible light communications and mobile communica-tions. He has supervised a large number of PhD students (more than 40) andhas published over 450 papers (160 in journals +11 books/book chapters) andpresented several keynote and invited talks. He is a co-author of a CRC bookon “Optical Wireless Communications—Systems and Channel Modelling withMatlab (2012); a co-editor of an IET book on “Analogue Optical Fibre Commu-nications”. From 2004–06 he was the IEEE UK/IR Communications ChapterSecretary, the Vice-Chairman (2004–2008), the Chairman (2008–2011), andChairman of the IET Northumbria Network (Oct 2011-). Personal Web site:http://soe.northumbria.ac.uk/ocr/people/ghassemlooy/