On the Asymptotic Capacity of Dual-Aperture FSO...

1536-1276 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2016.2585486, IEEETransactions on Wireless Communications

1

On the Asymptotic Capacity of Dual-Aperture FSOSystems with Generalized Pointing Error ModelHessa AlQuwaiee, Student Member, IEEE, Hong-Chuan Yang, Senior Member, IEEE, and Mohamed-Slim

Alouini, Fellow, IEEE

Abstract—Free-space optical (FSO) communication systemsare negatively affected by two physical phenomenon, namely,scintillation due to atmospheric turbulence and pointing errors.To quantify the effect of these two factors on FSO systemperformance, we need an effective mathematical model for them.In this paper, we propose and study a generalized pointing errormodel based on the Beckmann distribution. We then derive ageneric expression of the asymptotic capacity of FSO systemsunder the joint impact of turbulence and generalized pointingerror impairments. Finally, the asymptotic channel capacityformula are extended to quantify the FSO systems performancewith selection and switched-and-stay diversity.

Index Terms—Free-space optical, log-normal fading, Gamma-Gamma fading, Ergodic capacity, asymptotic, high SNR, pointingerrors, Beckmann distribution, correlated branches.

I. INTRODUCTION

One of the potential solutions to the spectrum scarcityproblem is optical wireless communications (OWC), whichutilizes the unlicensed optical spectrum. Long-range outdoorOWC are usually referred to as free-space optical (FSO)communications in the literature. Unlike radio frequency (RF)communications, FSO is immune to interference and multi-path fading. Also, the deployment of FSO systems is flexibleand much faster than optical fibers. These attractive featuresmake FSO applicable for broadband wireless transmissionsuch as optical fiber backup, metropolitan area network, andlast mile access.

Although FSO communications is a promising technology,it still faces challenges that prompted intensive research inthe last decade. One of the primary concern is that weatherconditions such as fog and snow can cause some attenuationin the intensity of the laser beam. Moreover, FSO is prone toatmospheric turbulence in which different intensity air layersformed locally by temperature differences vary the refractiveindex leading to scintillation of the laser beam [1]. In addition,the misalignment between the transmitter and the receiverleads to pointing error and additional performance degradation.The misalignment originates from either mechanical error inthe tracking system or mechanical vibrations in the systemdue to winds or/and building sway [2]. More specifically,pointing error results from the displacement of the laser beamalong vertical (elevation) and horizontal (azimuth) directionsthat are typically assumed to be independent Gaussian randomvariables (RVs). A pointing error has two main components:the boresight and the jitter. The boresight is caused by thermalexpansion of the laser beam and defined as the fixed displace-ment between the beam footprint center and the center of

detection plane. On the other hand, the jitter is the randomoffset of the beam center at the detector plane, typicallycaused by building sway, weak earthquakes, and dynamic windloads [3].

As widely adopted in RF, spatial diversity, in which multipletransmitter and/or multiple receivers are employed, is usuallyutilized to suppress the effect of channel fading. Similarly, thistechnique can efficiently overcome scintillation according tosome reported work in the literature [4], [5]. However, whenconsidering diversity links and due to system design, it iscrucial to take into account the correlation of the underly-ing channels since the spacing between beams or aperturescan not always ensure uncorrelated signals [6]. Furthermore,identifying the statistical model of the received irradiance thatcounts for both scintillation and pointing error facilities theperformance analysis of FSO systems especially in diversitylinks. To illustrate, the scintillation effect is widely modeled asLog-Normal (LN) for weak turbulence conditions and Gamma-Gamma (ΓΓ) for strong turbulence [7]–[13]. In terms of thepointing error, it was first modeled in [14] by a Rayleighdistribution, in [15] by a Hoyt distribution, in [3] by a Riciandistribution and more generally by a Beckmann distribution1 inthe conference version of this paper [17]. Moreover, combinedstatistics of turbulence and pointing errors were too investi-gated for Gamma-Gamma\Rayleigh channel model in [18],for double generalized Gamma\Rayleigh channel in [19], forLog-Normal\Rician in [3], and for Málaga (M)\Rayleighchannel model in [20]. Finding the combined effect of pointingerrors and turbulence becomes a harder task when consideringgeneral models of each (i.e. considering Beckmann model forthe pointing error effect) and even more complicated for spatialdiversity systems [21]–[24]. Hence, obtaining closed-formexpression of any of the performance measures might not befeasible especially in the case of the irradiance being a mixtureof two independent processes. This has raised the interest toinvestigate the asymptotic limit at high signal-to-noise ratio(SNR). In this work, our interest is one of the fundamentalinformation-theoretic measures namely channel capacity. Athigh SNR regime, it can be easily derived by utilizing thenth moment of the effective receive SNR. Interestingly, thisapproach can be utilized as well to find the asymptotic capacityof diversity links.

Several research work have been reported in the literature

1The Beckmann distribution [16] is a four-parameter distribution corre-sponding to the envelope of two independent Gaussian random variables, eachwith their own mean and variance. It is different than the Log-Normal Ricandistribution which can be also called as Beckmann distribution [8].



2

Table I: Summary of research work on ergodic capacity.

Detection Technique Channel Model Turbulence Only Turbulence and Pointing ErrorIM/DD ΓΓ [11], [25], [26] [27], [18]

LN [11], [25] *Other [28] [29] [19]

Heterodyne ΓΓ * [30], [18]Detection LN * *

Other * [19]

aiming to find closed-form expressions of the ergodic capacityof FSO systems as summarized in Table. I. It is noticed thatthe studies focusing on the joint effect of turbulence andpointing errors assume zero boresight and same jitter variancefor horizontal and vertical misalignment. In our work, we try togeneralize the pointing error model. Thus, our contribution inthis paper is twofold. First, we derive closed-form asymptoticresults of the ergodic capacity of single link FSO systemunder weak and strong turbulence conditions and under theimpact of generalized pointing error impairments. Our resultis general for any turbulence channel model and for anyscenario of the pointing error. Second, we extend our resultto be utilized in the derivation of the channel capacity ofFSO systems with diversity reception. In particular, we obtainclosed-form asymptotic expressions of the capacity of dual-branch correlated FSO channels considering general model ofthe pointing error.

The remainder of the paper is organized as follows. In Sec-tion II, system and channel model of turbulence and pointingerror are described. In Section III, analytical expressions of thechannel capacity for single and dual-aperture FSO system areobtained. Section IV highlights some numerical examples ofthe results and finally concluding remarks are given in SectionV. Φ

II. CHANNEL AND SYSTEM MODEL

A. System Model

In this work, we consider single-input single-output (SISO)and dual-branch single-input multiple-output (SIMO) config-urations of the FSO system with two types of detection tech-niques, heterodyne and intensity modulation/direct detection(IM/DD). Data transmission is affected by path loss, atmo-spheric turbulence conditions, pointing errors, and additivewhite Gaussian noise (AWGN)2. As such, the received vectory = [y1 y2]T is given by

y = ηIaIpx+ ω , (1)

where η is the effective photoelectric conversion ratio, Ia =diag(I1 , I2) is a 2×2 diagonal matrix reflects the turbulence-induced fading such that Ii represents the irradiance receivedat the ith aperture, Ip = [Ip1 Ip2 ]T , with (.)T is the transpose

2Most optical wireless systems operate in shot-noise limited regime and inthat case the ambient light shot noise component dominants the shot noisegenerated from signal and the circuit. Thus, the resulting noise of the channelbecomes white shot noise and can be distributed as Poisson random variables.By the central limit theorem, as the number of random variables approachesinfinity, the cumulative distribution function is approximated by Gaussiandistribution.

operator, is the 2×1 pointing error matrix consists of indepen-dent random variables where each component represents themisalignment between the center of the beam footprint andthe center of the ith aperture, and x ∈ {0, 2Pt} is the on-off keying (OOK) modulated transmitted signal with Pt beingaverage transmitted optical power. The vector ω = [ω1 ω2]T

is a noise vector of independent components modeled as whiteand Gaussian distributed RVs. It is important to note that Ii’sare not necessarily independent random variables. The spatialmatrix R can be of the form

R =

[1 ρρ 1

], (2)

where ρ is the correlation between I1 and I2. The electricalSNR of the ith branch can be defined as

γi =(ηIiIpi)

r

N0, (3)

where r depends on the detection techniques: r = 1 forheterodyne and r = 2 for IM/DD. Hence, the average electricalSNR, µ, is then expressed as

µ =ηrE[Ii]

rE[Ipi ]r

N0, (4)

where E[.] denotes the expectation operator. Then, the nthmoment of the electrical SNR can be written as

E[γni ] =E[Irni ]E[Irnpi ]

E[Ii]rnE[Ipi ]

rnµn. (5)

B. Channel Model

1) Atmospheric Turbulence: The atmospheric turbulencecan be classified into two categories: large-scale (diffractive)and small-scale fluctuations (refractive) (i.e. if the turbulencecells are larger than the beam diameter and vise versa).Moreover, the resulting irradiance can be modeled as

Ii = IxIy , (6)

where Ix and Iy are statistically independent unit mean RVsrepresenting large-scale and small-scale fluctuations, respec-tively. For instant, assuming plane wave, turbulence conditionscan be categorized into three regimes according to the Rytovvariance, σ2

Ri[31]: a weak fluctuations regime (σ2

Ri< 0.3),

a moderate-fluctuations regime (0.3 ≤ σ2Ri< 5), and a strong

fluctuations regime (σ2Ri≥ 5).

For weak turbulence conditions, in which large-scale fluc-tuations dominate, Ii is modeled as [7].

Ii = exp(2Xi) , (7)



3

where Xi ∼ N (µXi , σ2Xi

) is the log-amplitude of the opticalintensity such that σ2

Xi≈ σ2

Ri/4 = 0.30545 k7/6 C2

niz11/6

where k = 2π/λ is the optical wavenumber with λ beingthe wavelength, z being the transmission distance, and C2

niis the index of refraction structure parameter of atmosphere3.The refractive index is considered as one of the most criticalparameters to measure the strength of turbulence [7]. It is analtitude dependent parameter defined as

C2n(h) = 0.00594

(t

27

)2

(10−5h)10 exp

[− h

1000

]+ 2.7× 10−16 exp

[− h

1500

]+A exp

[− h

100

], (8)

such that h is the height, t is the wind speed in metersper second and C0 is the nominal value of the refractiveindex at the ground which is estimated to be 1.7 × 10−14.Typical C2

n values can vary from 10−17 m23 to 10−12 m

23

indicating weak atmospheric turbulence and strong conditions,respectively [25]. Then the probability density function (PDF)of Ii can be given as

fIi(I) =1

IσIi√

2πexp

{− (ln(I)− µIi)2

2σ2Ii

}, (9)

where µIi = 2µXi and σ2Ii

= 4σ2Xi

are the mean andstandard deviation of Ii. To ensure that the average poweris not amplified by fading, the irradiance is normalized (i.eE[Ii] = 1 and µIi = −2σ2

Xi) [21]. The nth moment for a

log-normal RV Ii can then be calculated as

E[Ini ] = exp

(nµIi +

n2σ2Ii

2

)= exp

(2nσ2

Xi(n− 1)).

(10)

On the other hand, moderate-to-strong turbulence conditionsresulted from combined effect of the large-scale and small-scale fluctuations, leading to Ii modeled as ΓΓ RV with aPDF given by [8]

fIi(I) =2(αiβiI)

αi+βi2

Γ(αi)Γ(βi)IKαi−βi

(2√αiβiI

), (11)

where Γ(.) is the Gamma function, Kj(.) is the modifiedBessel function of order j, αi and βi are the fading parametersof large-scale and small-scale fluctuations, defined as [8, Eq.14] in the case of plane wave

αi =

[exp

(0.49σ2

Ri

(1 + 1.11σ12/5Ri

)7/6

)− 1

]−1

,

βi =

[exp

(0.51σ2

Ri

(1 + 0.69σ12/5Ri

)5/6

)− 1

]−1

. (12)

The PDF in Eq. (10) can be rewritten in terms of the Meijer’s

3For plane wave propagation, the Rytov variance is given as σ2Ri

=

1.23k7/6 C2niz11/6 and σ2

Xi= 0.30545k7/6 C2

niz11/6. However, for

spherical wave propagation, the Rytov variance is equal to σ2Ri

= 0.5k7/6

C2niz11/6 and therefore, σ2

Xi= 0.1250k7/6 C2

niz11/6 [7, Eq. (97)].

G-function as

fIi(I) =(αiβiI)

αi+βi2

Γ(αi)Γ(βi)IG2,0

0,2

[αiβiI

∣∣∣∣ −αi−βi

2 ,βi−αi

2

], (13)

where G.,..,.[.] is the Meijer G-function. Next, the nth momentof Ii can be derived by utilizing [32, Eq. 07.34.21.0009.01]as

E[Ini ] =Γ(α+ n)Γ(β + n)

(αβ)nΓ(α)Γ(β). (14)

(a) Zero boresight: Nomisalignment.

(b) Bi-Directional misalign-ment: zero boresight andidentical jitters [14].

(c) Uni-Directionalmisalignment: zeroboresight [22].

(d) Bi-Directional misalign-ment: zero boresight andnon-identical jitters.

(e) Bi-Directional misalignment: non-zeroboresight and identical jitters.

Figure 1: Beam footprint on the detector plane.

2) Pointing Error Generalized Model: Assuming Gaussianbeam with initial beamwaist, w0, and radius of curvature, F0,propagating through atmospheric turbulence of distance z, thebeam waist at the receiver in long term, wz , can be defined[33, Eq. 45, p.238].

For the SISO setup as shown in Fig. 1a, the beam is initiallypointing at A = [q p]t in the detection plane and it is detectedby the ith aperture of radius a placed at Bi = [ui vi]

t

(i.e. i = 1 for the SISO case). Moreover, due to importantinitialization or other effect, A may not collocated with Bi.Also, the beam may experience random displacements intwo directions namely, horizontal, x and vertical, y as theresult of building sway. It is commonly assumed that bothdisplacements are modeled as independent Gaussian randomvariable i.e. x ∼ N (µx, σx) and y ∼ N (µy, σy). Then thedistance between the center of the beam footprint and the



4

center of the ith aperture can be expressed as

ri =

∥∥∥∥A− Bi +

[xy

] ∥∥∥∥ =

∥∥∥∥ [xy] ∥∥∥∥. (15)

It follows that the attenuation due to geometric spread andpointing errors can be approximated by [14]

Ipi(ri : z) ≈ A0 exp

(− 2r2

i

w2zeq

), (16)

where ri =√x2 + y2 such that x ∼ N (µx, σ

2x) and

y ∼ N (µy, σ2y), µx = µx+q−ui, µy = µy+p−vi, and w2

zeq

is the equivalent beamwidth defined as w2zeq = w2

z

√A0π

2g exp{−g2}such that A0 = [erf(g)]2 is the maximum fraction of thecollected power (i.e. the fraction of power at ri = 0) , andg =

√a2π2w2

zis the ratio between the aperture radius and the

beamwidth. It is important to note that the approximation in(16) is valid when wz > 6a [14]. Moreover, the distribution ofri depends on x = x+ q−ui and y = y+p−vi. Since x andy are independent Gaussian RVs, then ri can be distributedaccording to the following distributions.

a) Beckmann Distribution: The Beckmann distribution isa versatile model that includes many distributions as specialcases. It is a four-parameter distribution modeling the enve-lope of two independent Gaussian RVs. In our case, if bothdisplacements are nonzero mean Gaussian RVs with differentjitters, i.e. x ∼ N (µx, σ

2x) and y ∼ N (µy, σ

2y), then ri follows

the Beckmann distribution [34, Eq. 2.37] with probabilitydensity function (PDF) given by

fr(r) =r

2πσxσy

×∫ 2π

0

exp

(− (r cos θ − µx)2

2σ2x

− (r sin θ − µy)2

2σ2y

)dθ.

(17)

With the PDF of r, we can calculate the nth moment of Ip as

E[Inp ] = E

[An0 exp

(−2nr2

w2zeq

)]= An0 Mr2

(− 2n

w2zeq

),

(18)

where Mr2(.) is the moment-generating function (MGF) ofthe random variable r2 and given by [34, Eq. (2.38)]

Mr2(ν) =1√

(1− σ2xν)(1− σ2

yν)

× exp

(µ2xν

1− 2σ2xν

+µ2yν

1− 2σ2yν

). (19)

Therefore, the nth moment of Ip becomes in this case

E[Inp ] =An0 ξxξy√

(n+ ξ2x)(n+ ξ2

y)

× f exp

(− 2n

w2zeq

[µ2x

1 + nξ2x

+µ2y

1 + nξ2y

]), (20)

where ξx =wzeq2σx

and ξy =wzeq2σy

, are the ratio betweenthe equivalent beam width and the jitter variance for eachdirection. From this relation, we can state that for high jittervariance, ξi gets closer to zero and vice versa. Similarly, forwide beams, the effect of the pointing error is trivial and thusξi →∞.

b) Rayleigh Distribution: When both displacement havezero mean and common variance as shown in Fig. 1b (i.e.µx = µy = 0 and σx = σy = σ), r is a Rayleigh distributedRV whose PDF is given by

fr(r) =r

σ2exp

(− r2

2σ2

). (21)

The PDF of Ip reduces in this case to [14]

fIp(Ip) =ξ2

Aξ2

0

Iξ2−1p , (22)

where ξ =wzeq2σ . The nth moment can be deduced from (20)

asE[Inp ] =

An0 ξ2

n+ ξ2. (23)

c) Hoyt Distribution: Here, zero mean but differentvariances are assumed for the two displacements directionsas shown in Fig. 1d (i.e. µx = µy = 0 and σx 6= σy). In thiscase, r becomes a Hoyt distributed RV with PDF given by

fr(r) =r

qσ2y

exp

(−r

2(1 + q2)

4q2σ2y

)I0

(r2(1− q2)

4q2σ2y

), (24)

where q = σxσy

=ξyξx

. This special case was studied in [15]and the PDF of Ip was found to be given by

fIp(Ip) =ξxξyA0

(IpA0

) ξ2x(1+q2)

2 −1

× I0(ξ2x(1− q2)

2lnIpA0

), 0 ≤ Ip ≤ A0. (25)

The nth moment can be deduced from (20) as

E[Inp ] =An0 ξxξy√

(ξ2x + n)(ξ2

y + n). (26)

d) Rician Distribution: If both displacements have dis-tinct non-zero mean and common variance as shown in Fig.1e (i.e. µx + µy 6= 0 and σx = σy = σ) then r is a Riciandistributed RV with PDF given by

fr(r) =r

σ2exp

(−(r2 + s2)

2σ2

)I0

( rsσ2

), (27)

where s =√µ2x + µ2

y and Ij(.) is the modified Besselfunction of the first kind with order j. This case has beenvisited in [3] and the PDF of the pointing error has beenderived as

fIp(Ip) =ξ2 exp

(−s22σ2

)Aξ

2

0

Iξ2−1p I0

(s√2σ2

√wzeq ln

(A0

Ip

)).

(28)



5

The nth moment can be deduced also from (20) as

E[Inp ] =An0 ξ

2

n+ ξ2exp

(− 2n ξ2 s

w2zeq (n+ ξ2)

), (29)

which matches [3, Eq. 6].e) Zero-mean Single-sided Gaussian Distribution: In this

scenario, the displacement occur in only one direction eitherparallel or orthogonal to the detection plane as shown in Fig.1c (i.e. µx = µy = 0, σx = σ, and σy = 0). This model wasfirst introduced in [22] and the PDF of Ip can be derived inthis case by simple RV transformation of (16), yielding

fIp(Ip) =ξ Iξ

2−1p

Aξ2

0

√π ln

(A0

Ip

) , (30)

and the resulting nth moment can be expressed as

E[Inp ] =ξAn0√n+ ξ2

. (31)

f) Non-zero Mean Single-sided Gaussian Distribution:In this case, assuming µx = µy = µp, σx = σ, and σy = 0and we can obtain the PDF of Ip as

fIp(Ip) = Iξ2−1p ξ2

√√√√ µp√

2

wzeq

√ln A0

Ip

exp

(−

2µ2pξ

2

w2zeq

)

× I− 12

(2µpξ

2

wzeq

√2 ln

A0

Ip

), (32)

and then the nth moment can as a result be expressed as

E[Inp ] =An0 ξ√n+ ξ2

exp

(−

2nµ2p(n+ 2ξ2)

w2zeq (n+ ξ2)

). (33)

III. PERFORMANCE ANALYSIS: ASYMPTOTIC ERGODICCAPACITY

Complete performance analysis of the FSO systems requiresthe statistics of the irradiance, which might not be math-ematical tractable. For the purpose of this work, to studythe asymptotic channel capacity of FSO links with generalpointing error model, the nth moment of the irradiance is theonly requirement.

At high SNR and considering that perfect channel-stateinformation (CSI) is available at both the receiver and thetransmitter of an FSO communication system, the ergodic ca-pacity C , E

γend[log2(1+δγend)]

4 can be obtained through anasymptotic analysis by utilizing the moments of instantaneousend-to-end SNR, γend [38, Eq. (8) and (9)] [18, Eq. (22)] as

C uµ�1

∂

∂nE[γnend]

∣∣∣∣n=0

. (34)

4Although optical channel is well described as slow varying channel wherecoherence time is much greater than transmission time, ergodic capacity is stilla widely adopted metric. The suitability of ergodic capacity as performancemetric improves in this work as we consider pointing errors which typicallyincrease the rate of channel fluctuations. In the case of heterodyne detection(r = 1) the expression C = E[log2(1 + δγend)] represents the exact ergodiccapacity where δ = 1. However, in the case of IM/DD (r = 2) the expressionC = E[log2(1 + δγ)] where δ = e/2π represents a tight lower bound forcapacity [35] [36, Eq. (26)] [37, Eq. (7.43)].

A. Single Link Asymptotic Capacity:

Assuming a single link FSO system. Since Ia and Ipare statistically independent processes, the moments of theirradiance and the electrical SNR can be written as

E[In] = E[Ina ]E[Inp ] = An0E[Ina ]Mr2

(− 2n

w2zeq

), (35)

E[γn] =

(ηr

N0

)nE[Irn]. (36)

Taking the derivative of (36), the asymptotic capacity of asingle link at high SNR can be written as

C uµ�1W − r log(E[Ia]) +

∂

∂nE[Irna ]

∣∣∣∣n=0

, (37)

such that

W = log

(δµAr0E[Ip]

r

)− 2r

wzeqM′r2(0) , (38)

whereMr2(.) is given in (19). Therefore, W can take severalforms as listed in Table II according to the specific pointingerror model adopted. For example, for the most general case,W becomes

W = log

[ (1 + ξ2x)(1 + ξ2

y)

ξ2xξ

2y

] r2

δµ

− r

2ξ2x

− r

2ξ2y

− 2r

w2zeq

(µ2x

1 + ξ2x

+µ2y

1 + ξ2y

). (39)

The result in Eq. (37) is generic that it can be applied toany turbulence channel model and only requires finding themoment of Ia and its derivative. In this work, we considerweak and strong atmospheric turbulence modeled as Log-Normal and Gamma-Gamma turbulences, respectively. Hence,the capacity of the channel can be asymptotically be found bytaking the first derivatives of (10), (14) and substituting it in(37), we reach after some manipulations and simplificationsthe following results

C|ΓΓ uµ�1W − r log(αβ) + rψ(α) + rψ(β), (40)

C|LN uµ�1W − 2rσ2

X , (41)

for the Gamma-Gamma and the Log-Normal cases, respec-tively. This result can specialize to different cases accordingto the pointing error impairments model based on the choiceof W .

B. Dual-Branch Diversity Link:

In this section, we investigate the dual-aperture FSO systemover correlated Log-Normal channels in which the footprintof the beam is shown in Fig. 2. The beam is originallypointing at A = [0 0]T while the apertures are placed inB1 = [d2 0]T and B2 = [−d2 0]T such that d is the separationdistance between the centers of the apertures. Note that dueto building sway or other factors, the center of the beammoved to a random location A + [x y]T . Then, the distanceto the center of the ith aperture can be given as Eq. (15) (i.e.



6

Table II: Different forms of W depends mainly on the pointing error model considered.

Pointing Errors Model W

Beckmann (µx 6= µy and ξx 6= ξy) log

([(1+ξ2x)(1+ξ2y)

ξ2xξ2y

] r2

δµ

)− r

2ξ2x− r

2ξ2y− 2r

w2zeq

(µ2x

1+ξ2x+

µ2y

1+ξ2y

)Rayleigh (µx = µy = 0 and ξx = ξy = ξ) log

([1+ξ2

ξ2

]rδµ)− r

ξ2

Rician (µx + µy 6= 0 and ξx = ξy = ξ) log([

1+ξ2

ξ2

]rδµ)− r

ξ2 −2rs2

w2zeq

+ 2rs2ξ2

w2zeq

(1+ξ2)

Hoyt (µx = µy = 0 and ξx 6= ξy) log

([1+ξ2xξ2x

] r2[

1+ξ2yξ2y

] r2

δµ

)− r

2ξ2x− r

2ξ2y

Gaussian (µx = 0, ξx = ξ, and ξy =∞) log

([1+ξ2

ξ2

] r2

δµ

)− r

2ξ2

Shifted Gaussian (µx = µp, ξx = ξ, and ξy =∞) log

([1+ξ2

ξ2

] r2

δµ

)− r

2ξ2 −4rµ2

p

w2zeq

+2rµ2

p(1+2ξ2)

w2zeq

(1+ξ2)

No pointing errors log(δµ)

Figure 2: Beam foot print of the dual-aperture FSO system.

r1 =√

(d2 − x)2 + y2 and r2 =√

(d2 + x)2 + y2), whichaffect the pointing error. The received signals captured by theoptical apertures will be combined together. More specifically,two combining techniques namely selection combining (SC)and switched and stay combining (SSC) are examined:

1) Selection-Combining (SC): In this technique, the branchwith the larger instantaneous SNR is selected and therefore theoutput SNR can be written as

γSC = max(γ1, γ2), (42)

where γi is defined in Eq. (3). Thus, Eq. (42) can be rewrittenas

γSC = max

(ηIr1I

rp1

N0,ηIr2I

rp2

N0

)=

η

N0max (I1Ip1 , I2Ip2)

r.

(43)

Note that I1 and I2 are two correlated Log-Normal variateswith power correlation coefficient ρ that is a function of theseparation distance and the coherence length [39], [40], givenby

ρ = exp

[−(

d

ρo(z)

)−35

], (44)

where ρo(z) = (1.46C2nk

2z)(−3/5) is the coherence length ofa plane wave propagation5. It is clear from the expression thatthe links are correlated when the spacing between apertures isless than the coherence length of the beam.

The pointing error components Ip1 and Ip2 mainly dependon the distance ri. Noting that r2

2 = r21 + 2dx, we can rewrite

Eq. (43) as

γSC =η

N0max

(I1A0 exp

(−2r21w2

zeq

), I2A0 exp

(−2r22w2

zeq

))r

=η

N0max

(I1A0 exp

(−2r21w2

zeq

), I2A0 exp

(−2(r21 + 2dx)

w2zeq

))r

=η

N0

(A0 exp

(−2r21w2

zeq

))r

max

(I1, I2 exp

(−4dx

w2zeq

))r

=η

N0Ip1 max

(I1, I2

)r

=η

N0Irp1I

rSC , (45)

where I2 = exp(

2X2 − 4dw2zeq

x)

leading to I2 ∼lnN (−2σ2

x2− λ1, 4σ

2x2

+ λ22) such that λ1 = 4dµx

w2zeq

and

λ2 = 4dσxw2zeq

. Furthermore, the nth moment of γSC can beexpressed as

E[γnSC ] =E[Irnp1 ]E[IrnSC ]

E[Ip1 ]rnE[ISC ]

rnµnSC , (46)

where E[Inp1 ] can be derived using Eq. (20), and E[InSC ] canbe expressed shown in Eq. (47), where ζ = λ1−2σ2

x1+2σ2

x2,

κ = λ22 + 4σ2

x2, φ =

2σx1√κ

, and P21 = f(2σx1

,√κ,−ρ) such

that f(a, b, c) = a2 + b2 + 2abc.Based on Eq. (37), the asymptotic capacity of dual branch

FSO system with SC can be written as shown in Eq. (48)where P2 = f(σx1

, σx2,−ρ) .

2) Switched and Stay Combining (SSC): The SNR γSSCat the output of a dual-branch SSC receiver is given by [34],

5Physically, the coherence parameter is the measure of light coherenceacross each transverse plane along the propagation path [41]. The co-herence length of a spherical wave propagation is equal to ρo(z) =(0.55C2

nk2z)(−3/5) [7, Eq. (65)].



7

E[InSC ] = exp

(1

2n2κ− n

(λ1 + 2σ2

x2

))Q

(ζ − nκ (1− ρφ)

P1

)+ exp

(2n2σ2

x1− 2nσ2

x1

)Q

(−ζ − 4nσ2

x1

(1− ρφ−1

)P1

).

(47)

C|SC uµ�1W +

r√2π

exp

(−ζ2

2P21

)√λ2

2 + 4λ2(σx2− ρσx1

) + 4P2 − 2rσ2x1− rζQ

(ζ

P1

)− r log

[Q

(−ζ − 4σ2x1

(1− ρφ−1)

P1

)+ exp

(−(ζ + 2σ2

x1) +

1

2κ2

)Q

(ζ − κ(1− ρφ)

P1

)]. (48)

[42]

γSSC =

{γ1 , γ1 ≥ γtγ2 , γ1 < γt

=

{ηIr1 I

rp1

N0I1Ip1 ≥ It

ηIr2 Irp2

N0I1Ip1 < It

(49)

where γt is the switching threshold below which the receiverswitches to the other diversity branch and It = r

√N0γtη .

Following similar analysis in Sec. III-B1, we have γSSC =ηN0Irp1I

rSSC , where IrSSC is given by .

ISSC =

{I1 , Ip1I1 ≥ ItI2 , Ip1I1 < It

(50)

Then the nth moment of the γSSC can be expressed as

E[γnSSC ] =E[Irnp1 ]E[InSSC ]

E[Ip1 ]rnE[ISSC ]

nµn, (51)

where the moments of ISSC is given as in Eq. (56) utilizingearlier results in [43] such that

K1 =2σ2

x1+ log(γt)

2σx1

, (52)

K2 = λ1 + 2σ2x2

+ log(γt), (53)

ε1 =Q (−K1)

Q (−K1) +Q(−K2√

κ

) , (54)

ε2 =Q(−K2√

κ

)Q (−K1) +Q

(−K2√

κ

) , . (55)

Since the moment of the irradiance ISSC is available, wecan directly apply Eq. (37) to get the asymptotic capacity forSSC case as in Eq. (57).

IV. NUMERICAL SIMULATIONS

In this section, we validate and evaluate our analyticalexpressions of the channel capacity for two systems, singlelink FSO and dual-aperture FSO system over correlated Log-Normal channels. It is important to mention that Monte-Carlocomputer based simulations are utilized to obtain all exact re-sults. In each plot, we specify all parameters considered in thesimulation i.e. the beamwaist, w0, phase front F0, the distancez, aperture radius a, refractive index C2

n, jitter variances σ2x

σ2y and boresight mean µx and µy . First, we compare between

the single link and dual-aperture FSO systems for different

Average Electrical Signal-to-Noise Ratio (SNR), µ (dB).0 10 20 30 40 50 60 70 80

Ergodic

Cap

acity,

C,(N

ats/Sec/H

z).

0

2

4

6

8

10

12

14

16

Exact, SC.Exact, no diversity.Asymptotic.

a = 2.5 cm, σx = 0.01, σy = 0.01,

d = 2a , µx =µy = 0,

Cn2 = 2*10-13, λ = 1550 nm,

w0 = 1.66 cm, F0 = -10, ρ = 0.

z = 2 KM

z = 1 KM

Figure 3: Comparison between the channel capacity of a single aperture anddual-aperture FSO systems.

distances, z, as shown in Fig. 3. It is expected that the diversitylink can enhance the capacity when the turbulence conditionsget severe since the distance is directly proportional to theRytov variance (i.e. as distance increases, more turbulenceeddies are added). Also, our asymptotic results are tight athigh SNR. However, as the channel gets more turbulent, theconvergence of the asymptotic results to the exact ones happenat larger SNR. In Fig. 4 we plot the ergodic capacity ofa single link versus the average electrical SNR over weakturbulence modeled as Log-normal fading channel. Assumingno-boresight, we consider different scenarios of jitter variancei.e. high pointing error effect is presented by high values of σ2

x

and σ2y . Clearly, highly varying beam due to winds or other

factor results in performance degradation. Interestingly, ourasymptotic analytical results show an excellent match withthe exact ones generated by Monte-Carlo computer basedsimulation. Next, we show the effect of beam waist at thetransmitter on the capacity of the link in Fig. 5. First thing toobserve is that our asymptotic results converge well to the onesgenerated by Monte-Carlo simulation. Wide beam delivers thebest result as it resists the pointing errors in contrast with thenarrow beam. Lastly for the single link, we compare betweenunidirectional and bidirectional misalignments over Gamma-



8

E(InSSC) = ε1 exp(2n2σ2

x1− 2nσ2

x1

)(Q (2nρσx1 −K1) +Q(K1 − 2nσx1))

+ γnt ε2 exp

(κn2

2− nK2

)(Q

(κnρ−K2√

κ

)+Q

(−κn+K2√

κ

)). (56)

C|SSC uµ�1W +

r(1− ρ)√π

(√2ε1σx1

exp

(−K2

1

2

)+ε2√κ√

2πexp

(−K2

2

2κ

))− r

(ε1σ

2x1

+ ε2σ2x2

+1

2ε2λ1

)− r log

[ε1Q (−2σx1 + σx1K1) + ε1Q

(2ρσ2

x1−K1

σx1

)+ ε2 exp

(λ2

2 − 2λ1

2

)(Q

(ρκ−K2√

κ

)+Q

(−κ+K2√

κ

))].

(57)

Average electrical Signal-to-Noise Ratio (SNR), µr (dB)0 5 10 15 20 25 30 35 40 45 50

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

1

2

3

4

5

6

7

8

9

10

Exact results, σx2 = σ

y2 = 0.1


y2 = 0.5


y2 = 0.9

Asymptotic results.

a = 10 cm, z = 1KM,

Cn2 = 10-15, F0 = -10, w0=

1.66 cm, µx = µy = 0

Figure 4: The effect of jitter variance on the capacity of a single link overLog-Normal turbulence.

Average electrical Signal-to-Noise Ratio (SNR), µr (dB)0 10 20 30 40 50 60

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

2

4

6

8

10

12

14

Exact result, w0 = 1.66 cm

Exact results, w0 = 1 cm

Exact result, w0 = 0.8 cm.

Asymptotic results

a = 10 cm, z = 1KM,

Cn2 = 10-15, F0 = -10,

w0= 1.66 cm, µx = 0.5,

µy = 0.1, σx2 = σy

2 = 0.1

Figure 5: The effect of beam waist at the transmitter on the capacity overLog-Normal turbulence under boresight error.

Gamma turbulence in Fig. 6. Clearly, for low jitter variancei.e. high values of ξ, both directions of misalignment have

Average electrical Signal-to-Noise Ratio (SNR), µr (dB)0 50 100

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

2

4

6

8

10

12

14

16

18

20ξ = 2.5

Exact results, 1D

Exact results, 2D

Asymptotic results

0 50 1000

2

4

6

8

10

12

14

16

18

20ξ = 0.6

a = 10 cm, z = 1KM,

Cn2 = 10 -15, F

0 = -10,

w0= 1.66 cm, µ

x = µ

y

= 0

Figure 6: Comparison between unidirectional and bidirectional misalignmentover Gamma-Gamma turbulence.

the same effect of the channel capacity. In contrast with highjitter variance case, channel capacity is less affected by theunidirectional misalignment.

With regard to diversity link, first we show the effect ofboresight on the channel capacity of selection combined FSOlink with two scenarios of jitter variance and beam waistin Fig. 7 and Fig. 8, respectively. We can conclude thatlow varying beam and wide beam are more resistant to theboresight.

The main purpose of diversity is to overcome scintillation.It can also help suppressing the effect of beam boresight error.From our observation in Fig. 9, with small boresight error, thedistance does not make any difference to the channel capacity.On the other hand, larger boresight, spacing distance can behelpful in diminishing the effect of boresight. Finally, we showin Fig. 10 the effect of the pointing error on the channel capac-ity if SSC are employed to combine the signal at the receiveside. Here, we consider only unidirectional misalignment, sowe can obtain the optimal threshold γt = exp(−2σ2

X). It isclear that even with the misalignment being in one direction,performance degradation is expected.



9

Average electrical Signal-to-Noise Ratio (SNR), µSC (dB)0 50 100

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

5

10

15

20

25σ

x2 = σ

y2 = 0.01

Exact results, µx = µ

y = 0

Exact result, µx = 2, µ

y = 3

Asymptotic results.

0 50 1000

5

10

15

20

25σ

x2 = σ

y2 = 0.3

a = 2.5 cm,

L = 1 kM, Cn2

= 10-15, r =2, d/a = 5,w0 = 1.66 cm,

F0 = -10

Figure 7: The effect of boresight and jitter variance on the channel capacityof dual-branch FSO link.

Average electrical Signal-to-Noise Ratio (SNR), µSC (dB)0 50 100

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

5

10

15

20

25w

0 = 1.2 cm

0 50 1000

5

10

15

20

25w

0 = 1.66 cm

Exact results.Asymptotic results.

b

a

a

b

a) µx = µy = 0.

b) µx = 2, µy = 3.

a = 2.5 cm,

L = 1 kM, Cn2

= 10-15, r =2, d/a = 5,

σx2 = σy

2 = 0.1,

F0 = -10

Figure 8: The effect of boresight and beam waist on the channel capacity ofdual-branch FSO link.

V. CONCLUSION

In this work, we developed a general model for the point-ing errors affecting FSO systems based on the generalizedBeckmann distribution. Our result is generic and includespreviously published models as special cases. Furthermore,due to the interest in the high SNR regime, we derived ageneral expression for the asymptotic ergodic capacity of FSOsystems subject to generalized pointing error impairments.In this work, we have provided closed-form expressions ofthe Log-Normal and Gamma-Gamma channels. However, ourformula can be applied to any turbulence channel such as K-distributed, double Weibull, double Generalized Gamma andM channels. Actually our approach can be extended as well todiversity links. In particular, we have derived the asymptoticcapacity of dual-aperture FSO system over correlated Log-Normal channels. Numerical results, validated by computersimulations, show that our asymptotic results can accurately

Average electrical Signal-to-Noise Ratio (SNR), µSC (dB)0 10 20 30 40 50

Ergodic

Capacity,C

(Nats/Sec/Hz))

0

2

4

6

8

10

12µ

x = µ

y = 0.1

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10µ

x = 2, µ

y = 3.

Exact results. d/a = 2.Exact results, d/a = 30.Asymptotic results.

a = 2.5 cm,

L = 1 kM, Cn2= 10-15,

w0 = 1.66 cm, r =2, σx2

= σy2 = 0.1, F0 = -10

Figure 9: The effect of distance and boresight on the channel capacity ofdual-branch FSO link.

Average Electrical Signal-to-Noise Ratio (SNR), µSSC (dB).0 10 20 30 40 50 60 70 80 90 100

Ergodic

Capacity,C

SSC,(N

ats/Sec/Hz).

0

2

4

6

8

10

12

14

16

18

20

Exact, ξy = 10.

Exact, ξy = 0.7.

Exact, ξy = 0.5.

Asymptotic.

a = 2.5 cm, z = 1 KM, µx = µy= 0, d = 2a,

ξx → ∞, Cn2 = 10-14, λ = 1550 nm,

θ = 1 mrad, ρ = 0.54.

Figure 10: Unidirectional misalignment and corresponding channel capacityfor the dual-aperture FSO system.

predict the performance of FSO systems in the high SNRregime.

REFERENCES

[1] S. Karp, R. Gagliardi, S. Moran, and L. Stotts, Optical Channels:Fibers, Clouds, Water, and the Atmosphere, ser. Applications ofCommunications Theory. Springer US, 1988.

[2] D. K. Borah and D. G. Voelz, “Pointing error effects on free-spaceoptical communication links in the presence of atmospheric turbulence,”Journal of Lightwave Technology, vol. 27, no. 18, pp. 3965–3973, Sep.2009.

[3] F. Yang, J. Cheng, and T. Tsiftsis, “Free-space optical communicationwith nonzero boresight pointing errors,” IEEE Transactions on Commu-nications, vol. 62, no. 2, pp. 713–725, Feb. 2014.

[4] E. Lee and V. Chan, “Part 1: Optical communication over the clearturbulent atmospheric channel using diversity,” IEEE Journal on SelectedAreas in Communications, vol. 22, no. 9, pp. 1896–1906, Nov. 2004.

[5] T. Tsiftsis, H. Sandalidis, G. Karagiannidis, and M. Uysal, “Opticalwireless links with spatial diversity over strong atmospheric turbulencechannels,” IEEE Transactions on Communications, vol. 8, no. 2, pp.951–957, Feb. 2009.



10

[6] G. Yang, M. A. Khalighi, S. Bourennane, and Z. Ghassemlooy, “Fadingcorrelation and analytical performance evaluation of the space-diversityfree-space optical communications system,” Journal of Optics, vol. 16,no. 3, p. 035403, Feb. 2014.

[7] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillationwith Applications. SPIE press, 2001, vol. 99.

[8] M. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematicalmodel for the irradiance probability density function of a laser beampropagating through turbulent media,” Optical Engineering, vol. 40,no. 8, pp. 1554–1562, Feb. 2001.

[9] X. Zhu and J. Kahn, “Free-space optical communication through atmo-spheric turbulence channels,” IEEE Transactions on Communications,vol. 50, no. 8, pp. 1293–1300, Aug. 2002.

[10] T. Tsiftsis, “Performance of heterodyne wireless optical communicationsystems over Gamma-Gamma atmospheric turbulence channels,” Elec-tronics Letters, vol. 44, no. 5, pp. 372–373, Feb. 2008.

[11] H. Nistazakis, T. Tsiftsis, and G. Tombras, “Performance analysis offree-space optical communication systems over atmospheric turbulencechannels,” IET Communications, vol. 3, no. 8, pp. 1402–1409, Aug.2009.

[12] A. Prokeš, “Modeling of atmospheric turbulence effect on terrestrialFSO link,” Radio Eng, vol. 18, no. 1, pp. 42–47, Jan. 2009.

[13] X. Tang, S. Rajbhandari, W. Popoola, Z. Ghassemlooy, E. Leitgeb,S. Muhammad, and G. Kandus, “Performance of BPSK subcarrier inten-sity modulation free-space optical communications using a Log-Normalatmospheric turbulence model,” in 2010 Symposium on Photonics andOptoelectronic (SOPO), Chengdu, China, June 2010, pp. 1–4.

[14] A. Farid and S. Hranilovic, “Outage capacity optimization for free-spaceoptical links with pointing errors,” IEEE/OSA Journal of LightwaveTechnology, vol. 25, no. 7, pp. 1702–1710, July 2007.

[15] W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for ter-restrial FSO links in turbulent atmosphere with pointing errors modeledby Hoyt distributions,” IEEE Communications Letters, vol. 15, no. 8,pp. 875–877, July 2011.

[16] P. Beckmann, “Statistical distribution of the amplitude and phase of amultiply scattered field,” Journal of research of the National Bureau ofStandards-D.Radio propagation, vol. 66D, no. 3, June 1962.

[17] H. AlQuwaiee, H.-C. Yang, and M.-S. Alouini, “On the asymptoticergodic capacity of FSO links with generalized pointing error model,”in 2015 IEEE International Conference on Communications, (ICC2015), London, UK, June 2015, pp. 5072–5077.

[18] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “A unified performance offree-space optical links over Gamma-Gamma turbulence channels withpointing errors,” in Proceedings of IEEE 81st Vehicular TechnologyConference (VTC Spring’ 2015), Glasgow, Scotland, May 2015.

[19] H. AlQuwaiee, I. Ansari, and M.-S. Alouini, “On the performanceof free-space optical communication systems over double generalizedGamma channel,” IEEE Journal on Selected Areas in Communications,vol. 33, no. 9, pp. 1829–1840, May. 2015.

[20] I. Ansari, F. Yilmaz, and M.-S. Alouini, “Performance analysis of free-space optical links over Malaga (M) turbulence channels with pointingerrors,” IEEE Transactions on Wireless Communications, vol. PP, no. 99,pp. 1–1, Aug. 2015.

[21] S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance offree-space optical transmission with spatial diversity,” IEEE Transac-tions on Wireless Communications, vol. 6, no. 8, pp. 2813–2819, Aug.2007.

[22] A. Farid and S. Hranilovic, “Diversity gain and outage probability forMIMO free-space optical links with misalignment,” IEEE Transactionson Communications, vol. 60, no. 2, pp. 479 – 487, Feb. 2012.

[23] H. Moradi, H. Refai, and P. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IETOptoelectronics, vol. 6, no. 1, pp. 34–42, Feb. 2012.

[24] M. Abaza, R. Mesleh, A. Mansour, and E.-H. M. Aggoune, “Diversitytechniques for a free-space optical communication system in correlatedLog-Normal channels,” Optical Engineering, vol. 53, no. 1, pp. 016 102–016 102, Jan. 2014.

[25] H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios,and G. S. Tombras, “Average capacity of optical wireless communicationsystems over atmospheric turbulence channels,” Journal of LightwaveTechnology, vol. 27, no. 8, pp. 974–979, April 2009.

[26] H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacityof free-space optical links over strong turbulence,” Electronics Letters,vol. 44, no. 1, pp. 46–47, Jan. 2008.

[27] W. Gappmair, “Further results on the capacity of free-space opticalchannels in turbulent atmosphere,” IET Communications, vol. 5, no. 9,pp. 1262–1267, June 2011.

[28] K. P. Peppas, A. N. Stassinakis, G. K. Topalis, H. E. Nistazakis, andG. S. Tombras, “Average capacity of optical wireless communicationsystems over IK atmospheric turbulence channels,” Journal of OpticalCommunications and Networking, vol. 4, no. 12, pp. 1026–1032, Dec.2012.

[29] “Performance estimation of free space optical links over negative expo-nential atmospheric turbulence channels,” Optik - International Journalfor Light and Electron Optics, vol. 122, no. 24, pp. 2191 – 2194, Dec.2011.

[30] C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyneFSO communication systems over gamma-gamma turbulence channelswith pointing errors,” Electronics letters, vol. 46, no. 12, pp. 851–853,June 2010.

[31] N. Perlot, Characterization of Signal Fluctuations in Optical Commu-nications with Intensity Modulation and Direct Detection Through theTurbulent Atmospheric Channel, ser. Berichte aus der Kommunikation-stechnik. Shaker, 2006.

[32] I. Wolfram, Mathematica Edition: Version 8.0. Champaign, Illinois:Wolfram Research Inc., 2010.

[33] L. C. Andrews and R. L. Phillips, Laser Beam Propagation throughRandom Media. SPIE press, 2005, vol. 152.

[34] M. K. Simon and M.-S. Alouini, Digital Communication over FadingChannels. New York: Wiley, 2000.

[35] A. Chaaban, J. M. Morvan, and M. S. Alouini, “Free-space opticalcommunications: Capacity bounds, approximations, and a new sphere-packing perspective,” IEEE Transactions on Communications, vol. 64,no. 3, pp. 1176–1191, March 2016.

[36] A. Lapidoth, S. Moser, and M. Wigger, “On the capacity of free-spaceoptical intensity channels,” IEEE Transactions on Information Theory,vol. 55, no. 10, pp. 4449–4461, Oct. 2009.

[37] S. Arnon, J. Barry, G. Karagiannidis, R. Schober, and M. Uysal,Advanced Optical Wireless Communication Systems. Cambridge Uni-versity Press, 2012.

[38] F. Yilmaz and M.-S. Alouini, “Novel asymptotic results on the high-order statistics of the channel capacity over generalized fading channels,”in Proceedings of IEEE 13th International Workshop on Signal Pro-cessing Advances in Wireless Communications (SPAWC’ 2012), Cesme,Turkey, June 2012, pp. 389–393.

[39] X. Zhu and J. M. Kahn, “Maximum-likelihood spatial-diversity receptionon correlated turbulent free-space optical channels,” in IEEE GlobalTelecommunications Conference (GLOBECOM’00), vol. 2, San Fran-cisco, CA, Dec. 2000, pp. 1237–1241.

[40] G. R. Osche, “Optical detection theory for laser applications,” OpticalDetection Theory for Laser Applications, by Gregory R. Osche, pp. 424.ISBN 0-471-22411-1. Wiley-VCH, July 2002., vol. 1, 2002.

[41] J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects ona partially coherent Gaussian beam: Implications for free-space lasercommunication,” J. Opt. Soc. Am. A, vol. 19, no. 9, pp. 1794–1802,Sep. 2002.

[42] M. Blanco and K. Zdunek, “Performance and optimization of switcheddiversity systems for the detection of signals with rayleigh fading,” IEEETransactions on Communications, vol. 27, no. 12, pp. 1887–1895, Dec.1979.

[43] M.-S. Alouini and M. K. Simon, “Dual diversity over correlated Log-Normal fading channels,” IEEE Transactions on Wireless Communica-tions, vol. 50, no. 12, pp. 1946–1959, Dec. 2002.

Hessa AlQuwaiee (S’09) received the B.S degreein Computer Engineering from Prince MohamedUniversity (PMU), Alkhobar, Saudi Arabia in 2011.In 2008, she was awarded the discovery scholar-ship to complete graduate studies in King AbdullahUniversity of Science and Technology (KAUST).In 2013, she earned her M.S. degree in ElectricalEngineering. She is now toward her PhD degree atKAUST. Her current research interests include, butnot limited to, channel characterization and perfor-mance analysis of optical wireless communications,

body-to-body communications, diversity combining techniques and MIMOsystems.



11

Hong-Chuan Yang (S’00, M’03, SM’07) receivedhis Ph.D. degree in Electrical Engineering fromthe University of Minnesota, Minneapolis, USA, in2003. Since September 2003, Dr. Yang has beenwith the Department of Electrical and ComputerEngineering at the University of Victoria, Victoria,B.C., Canada, where he is now a professor. From1995 to 1998, Dr. Yang was a Research Associateat the Science and Technology Information Center(STIC) of Ministry of Posts & Telecomm. (MPT),Beijing, China. His research interest include wireless

channel modeling, diversity techniques, system performance evaluation, cross-layer design and energy efficient communications. He is an Editor for IEEETransactions on Communications

Mohamed-Slim Alouini (S’94, M’98, SM’03, F’09)was born in Tunis, Tunisia. He received the Ph.D.degree in Electrical Engineering from the CaliforniaInstitute of Technology (Caltech), Pasadena, CA,USA, in 1998. He served as a faculty memberin the University of Minnesota, Minneapolis, MN,USA, then in the Texas A&M University at Qatar,Education City, Doha, Qatar before joining KingAbdullah University of Science and Technology(KAUST), Thuwal, Makkah Province, Saudi Arabiaas a Professor of Electrical Engineering in 2009.

His current research interests include the modeling, design, and performanceanalysis of wireless communication systems.

On the Asymptotic Capacity of Dual-Aperture FSO...

Documents

Transcript of On the Asymptotic Capacity of Dual-Aperture FSO...