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Performance Analysis of Free–Space Optical

Systems in Gamma–Gamma Fading

Ehsan Bayaki†, Robert Schober†, and Ranjan K. Mallik††

†University of British Columbia,††Indian Institute of Technology Delhi

Abstract—Atmospheric turbulence induced fading is one of the

main impairments affecting free–space optics (FSO). In recent

years, Gamma–Gamma fading has become the dominant fading

model for FSO links because of its excellent agreement with

measurement data for a wide range of turbulence conditions. In

this paper, we express the bit error rate of intensity modulated

FSO with direct detection in single–input single–output and

multiple–input multiple–output Gamma–Gamma fading channels

as generalized infinite power series with respect to the signal–to–

noise ratio. For fast numerical evaluation these power series are

truncated to a finite number of terms and an upper bound for the

associated approximation error is provided. Another contribution

of this paper is the extension of the well–known RF concepts

of diversity and combining gain to FSO and Gamma–Gamma

fading.

I. INTRODUCTION

Free–space optics (FSO) has received considerable attention

recently as attractive solution for high–rate last–mile terrestrial

communications [1]. Attractive features compared to more

traditional RF solutions include ease of deployment, license–

free operation, high security, and high data rates. On the other

hand, FSO systems are susceptible to pointing errors, severe

attenuation under adverse weather conditions (e.g. fog), and

atmospheric turbulence [1]. Viable solutions to overcome these

problems have to be found before widespread deployment of

FSO systems will be possible. In this paper, we concentrate on

the effects of atmospheric turbulence on intensity–modulated

FSO systems with direct detection (IM/DD).

Atmospheric turbulence caused by variations in the refrac-

tive index due to inhomogeneities in temperature, pressure

fluctuations, humidity variations, and motion of the air along

the propagation path of the laser beam introduces irradiance

fluctuations in the received signal. The resulting signal fading

causes severe performance degradations. Similar to RF com-

munications, the effect of fading in FSO can be substantially

reduced by creating a multiple–input multiple output (MIMO)

FSO system with multiple lasers at the transmitter and multiple

photodetectors at the receiver [2]–[6]. In order to evaluate the

impact of atmospheric turbulence and the effectiveness of cor-

responding countermeasures, accurate models for the fading

distribution are important. While the lognormal distribution

is often used to model weak turbulence conditions, recently

the Gamma–Gamma distribution has received considerable

attention because of its excellent fit with measurement data

for a wide range of turbulence conditions (weak to strong)

[7], [8]. However, despite the popularity of the Gamma–

Gamma distribution in the FSO literature [9]–[15], a basic

understanding of the effects of Gamma–Gamma fading on the

performance of FSO systems is not available.

In this paper, we analyze the performance of uncoded on–

off keying (OOK) over single–input single–output (SISO) and

MIMO FSO channels subject to Gamma–Gamma fading. For

MIMO FSO we assume repetition coding across antennas

at the transmitter and equal gain combining (EGC) at the

receiver [5]. The difficulties that arise in the analysis of

Gamma–Gamma fading channels have their origin in the fact

that the Gamma–Gamma probability density function (pdf)

contains a modified Bessel function of the second kind. This

Bessel function precludes simple closed–form expressions for

the bit error rate (BER). Therefore, existing performance

analysis techniques for IM/DD FSO systems resort to nu-

merical integration techniques [8], [12], [14]. However, nu-

merical integration obscures the impact of the basic system

and channel parameters on performance and may become

numerically instable at high signal–to–noise ratios (SNRs)

[14]. Recently, the BER of a SISO FSO link impaired by K–

fading (a special case of Gamma–Gamma fading) and pointing

errors was expressed in terms of Meijer’s G–function in [15].

Unfortunately, this closed–form result does not provide much

insight either because of the presence of Meijer’s G–function.

The presented novel approach to performance analysis is

based on a generalized infinite power series representation of

the modified Bessel function of the second kind. With this

representation, we can express the BER of SISO and MIMO

IM/DD FSO systems as power series with respect to the SNR.

The coefficients of this series expansion only include elemen-

tary and Gamma functions and are easy to compute. We prove

that these series converge for any finite SNR and we provide

an upper bound for the approximation error that is caused

by truncating the infinite series to a finite number of terms,

which is necessary for numerical evaluation. Interestingly,

unlike numerical integration techniques, the accuracy of the

truncated series BER approximation improves with increasing

SNR. Furthermore, we extend the concepts of diversity and

combining gain, which are well known from the RF commu-

nication literature [16], to FSO and Gamma–Gamma fading.

Diversity and combining gain allow us to provide simple,

insightful, and accurate closed–form approximations for the

BER of SISO and MIMO FSO at high SNR.

The remainder of this paper is organized as follows. In

Section II, the considered system model and the generalized

power series representation of the Gamma–Gamma pdf are

introduced. The performance of IM/DD systems in SISO and

MIMO FSO channels is analyzed in Sections III and IV,

respectively. Performance results are presented in Section V,

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and some conclusions are drawn in Section VI.

Notation: In this paper, [·]T, ∗, and E{·} denote transpo-

sition, convolution, and statistical expectation, respectively.

Q(·), Γ(·), and Kν(·) are the Gaussian Q–function, the

Gamma function, and the νth order modified Bessel function

of the second kind, respectively.

II. PRELIMINARIES

In this section, we present the equivalent discrete–time

model for the received signal in a MIMO FSO system with

IM/DD, review the Gamma–Gamma distribution for the fading

gain, and introduce a novel series representation for this

distribution.

A. Received Signal Model

We consider an FSO system with M lasers, which are simul-

taneously intensity modulated with identical signals (repetition

coding across lasers, cf. e.g. [6]), and N photodetectors [8].

The photocurrentgenerated by each photodetectoris integrated

over one pulse interval and the constant bias due to background

radiation is removed. Under these conditions, for high–energy

FSO systems the sufficient statistic in the kth symbol interval

can be modeled as [2], [12]

rn[k] =

√γ

Ms[k]

M

?

m=1

Imn+ zn[k],1 ≤ n ≤ N

(1)

where s ∈ {0,1}1represents the data–carrying OOK symbol,

Imn> 0 with E{Imn} = 1 is the fading gain (irradiance) be-

tween laser m and photodetector n, znis real–valued additive

white Gaussian noise (AWGN) with variance σ2

1/4, and γ denotes the SNR. Imnis assumed to be constant

for the duration of one symbol because of the high data rates

(hundreds to thousands of Mbps) typical for FSO systems.

Furthermore, we assume that the Imn are independent and

identically distributed (i.i.d.) random variables (RVs), which

is justified if the transmit and receive apertures are separated

by more than a few centimeters, respectively, cf. [3], [6].

z? E{z2

n} =

B. Gamma–Gamma Distribution

For a wide range of turbulence conditions (weak to strong)

the fading gain Imn in FSO systems can be modeled by a

Gamma–Gamma distribution [7]–[9], [11], [12]

fI(Imn) =2(αβ)(α+β)/2

Γ(α)Γ(β)

I(α+β)/2−1

mn

Kα−β

?

2

?

αβImn

?

(2)

where parameters α > 0 and β > 0 are linked to the so–called

scintillation index S.I. ? 1/α+1/β+1/(αβ). α and β can be

adjusted to achieve a good agreement between fI(Imn) and

measurement data [7], [8]. Alternatively, assuming spherical

wave propagation, α and β can be directly linked to physical

1In the following, for simplicity of notation we drop the symbol index “k”.

parameters via [7], [12]

α=

?

?

exp

?

?

0.49χ2

(1 + 0.18d2+ 0.56χ12/5)7/6

?

− 1

?−1

?−1

(3)

β=exp

0.51χ2(1 + 0.69χ12/5)−5/6

(1 + 0.9d2+ 0.62d2χ12/5)5/6

?

− 1

(4)

where χ2= 0.5C2

2π/λ. Here, λ, D, C2

the diameter of the receiver’s aperture in meters, the index of

refraction structure parameter, and the link distance in meters,

respectively. We note that (2) contains the K–distribution (α >

0 and β = 1) and the negative exponential distribution (α →

∞ and β = 1) as special cases. The K–distribution is typically

used to model strong turbulence conditions [8], [9], [11], [15],

while the negative exponential distribution can be seen as a

limit distribution for extremely strong turbulence [7] and has

been used in the literature mostly because of its mathematical

tractability [6], [17].

nk7/6L11/6, d = (kD2/4L)1/2, and k =

n, and L are the wavelength in meters,

C. Series Representation of Gamma–Gamma Distribution

Performance analysis of MIMO FSO systems in Gamma–

Gamma fading is difficult because of the modified Bessel

function of the second kind in (2). To gain more insight and

to avoid having to deal with the modified Bessel function, we

base our analysis on the generalized power series representa-

tion [18]

Kν(x)=

π

2sin(πν)

?

∞

?

j=0

1

Γ(j − ν + 1)j!

1

Γ(j + ν + 1)j!

?x

2

?2j−ν

−

∞

?

j=0

?x

2

?2j+ν?

(5)

which is valid for ν ?∈ Z Z and |x| < ∞. We note that for

the case of integer ν a simple power series representation of

Kν(x) does not seem to exist. Combining (2) and (5) leads

to

fI(Imn) =

∞

?

j=0

?aj(α,β)Ij+β−1

mn

+ aj(β,α)Ij+α−1

mn

?,

(6)

(α − β) ?∈ Z Z, where

aj(α,β) ?

π(αβ)j+β

sin[π(α − β)]Γ(α)Γ(β)Γ(j − α + β + 1)j!.

We point out that the condition (α − β) ?∈ Z Z is not a severe

restriction since (α − β) ?∈ Z Z holds for typical parameters λ,

D, C2

and β such that (α − β) + ? ∈ Z Z with some small constant ?

to approximate the case (α − β) ∈ Z Z.

(7)

n, and L, cf. (3), (4), and if necessary, we can choose α

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D. EGC and BER

Assuming popular EGC at the receiver, the combined re-

ceived signal can be represented as [6]2

r = M

N

?

n=1

rn=√γIs + z

(8)

where I ??M

(8) is valid with M = 1 and N = 1. We assume that the

channel state I has been accurately estimated [9], [12], [17].

Thus, the optimum detector computes the symbol estimate

m=1

?N

n=1Imnand z ? M?N

n=1znwith vari-

ance σ2? M2Nσ2

z. For the special case of SISO transmission

ˆ s = argmin

s∈{0,1}

?|r −√γIs|2?.

(9)

Based on (9), the BER conditioned on I can be obtained as

Pb(I) = Q(?γI2/(4σ2)). Consequently, the BER averaged

∞

?

0

over the fading gain is

Pb=

Q

??

γ

4σ2I2

?

f(I)dI

(10)

where f(I) denotes the pdf of I.

III. PERFORMANCE ANALYSIS FOR SISO CHANNELS

We first consider SISO channels (i.e., M = 1, N = 1) since

they are analytically more tractable than MIMO channels. We

also use the SISO case to establish some basic understanding

of the convergence of the derived BER formulas.

A. BER in SISO Channels

For SISO transmission f(I) = fI(I) and σ2= 1/4 hold.

Thus, applying the series representation (6) of fI(I) in (10)

and expoiting key result (27) established in Appendix A, we

obtain for the average BER in SISO Gamma–Gamma fading

Pb

=

∞

?

j=0

(aj(α,β)X(√γ,j + β)

+aj(β,α)X(√γ,j + α))

=

∞

?

j=0

?

ξj(α,β)γ−j+β

2 + ξj(β,α)γ−j+α

2

?

, (11)

α − β ?∈ Z Z, where X(·,·) is defined in (27) and

ξj(α,β)

?

√π(√2αβ)j+βΓ

2sin[π(α − β)]Γ(α)Γ(β)Γ(j − α + β + 1)(j + β)j!(12)

Since (11) is an infinite series, the question of convergence

arises. For this purpose, we calculate the convergence radius

?

j+β+1

2

?

2We note that the multiplication of the combined signal in (8) with M is

not necessary in practice but facilitates our exposition here.

R1of the first sub–series in (11) as

????

= lim

j→∞

R1 = lim

j→∞

ξj(α,β)

ξj+1(α,β)

????

(j + 1)(j + β + 1)(j − α + β + 1)Γ

√2αβ(j + β)Γ

?

j+β+1

2

?

?

j+β+2

2

?

→ ∞.

(13)

Similarly, we obtain for the convergence radius R2 of the

second sub–series in (11) R2 → ∞. Thus, (11) converges

for all γ < ∞.

In practice, some finite value J has to be used for the upper

limit in (11) and we denote the resulting BER approximation

by˜Pb(J). In Appendix B, it is shown that the approximation

error ε(J) ? |Pb−˜Pb(J)| is bounded by

√πsmaxxJ+1

2|sin[π(α − β)]|Γ(α)Γ(β)(J + 1)!

?????

−

Γ(j − β + α + 1)(j + α)

with x0=√2αβ/√γ. This bound illustrates that the approx-

imation error can be made arbitrarily small by increasing J

or/and γ.

ε(J)<

0

ex0

(14)

smax

?

max

j>J

xβ

0Γ

?

j+β+1

2

?

Γ(j − α + β + 1)(j + β)

xα

0Γ?j+α+1

2

?

?????

(15)

B. Diversity and Combining Gain

In the RF communication literature, it is customary to

characterize fading channels in terms of their diversity gain

Gd and combining gain Gc, cf. e.g. [16]. In particular, for

high SNR the BER can be approximated as Pb≈ (Gcγ)−Gd,

i.e., on a log–log scale Gc and Gd specify, respectively, a

relative horizontal shift and the slope of the BER curves in the

asymptotic regime of γ → ∞. Considering (11) we obtain for

the diversity and combining gains of OOK in SISO Gamma–

Gamma fading

Gd

= min{α/2,β/2}

1

2α2β2

(16)

Gc

=

?

2√πΓ(max{α,β})Γ(2Gd+ 1)

Γ(|α − β|)Γ?Gd+1

2

?

? 1

Gd

(17)

where we have used the identity π/sin(πx) = Γ(x)Γ(1 − x)

[21, Eq. (6.1.17)]. We note that typically α > β is considered

in the literature since it provides the best fit with measurements

for most scenarios. However, α < β provides a better match

to measurements in some cases, cf. e.g. [7, Figs. 9–11, 13].

The results in this section are significant since they show that,

despite the complicated nature of the Gamma–Gamma pdf, for

high SNR the performance of FSO in Gamma–Gamma fading

can be characterized by only two parameters. Interestingly, we

observe from (16) that the diversity gain of SISO FSO only

depends on the minimum of α and β. On the other hand, (17)

shows that at high SNR for a given Gdperformance improves

with increasing max{α,β}.

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IV. PERFORMANCE OF MIMO FSO WITH EGC

In this section, we extend the analysis from the previous

section to MIMO FSO with EGC which is the dominant

combining scheme in FSO, e.g. [6].

A. BER for EGC

For application of (10), we have to determine the pdf

f(I) = fEGC(I) of the combined channel. For this purpose,

we introduce the moment generating function (MGF) ΦI(s) ?

E{e−sImn} of a single link, which can be calculated to

∞

?

cf. (6), where bj(α,β) ? Γ(j + β)aj(α,β). Since the Imn

are i.i.d. RVs, the MGF of I =?M

MN

?

where the coefficients cj(MN − k,k) can be efficiently

calculated as [20, Eq. (0.316)]

ΦI(s) =

j=0

?

bj(α,β)s−(j+β)+ bj(β,α)s−(j+α)?

(18)

m=1

?N

n=1Imnis given by

ΦEGC(s) = (ΦI(s))MNand can be expressed as

ΦEGC(s)=

k=0

?MN

k

?∞

j=0

?

cj(MN − k,k)s−j−(MN−k)β−kα

(19)

cj(µ,ν) = b(µ)

j(α,β) ∗ b(ν)

means that yjis convolved x−1

= yj∗ yj, y(1)

j(β,α)

(20)

where superscript (x) in y(x)

times with itself, e.g. y(2)

Based on (19) we can express the fading pdf for EGC as

j

jj

= yj, and y(0)

j

= 1.

fEGC(I)=

MN

?

Γ(j + (MN − k)β + kα)Ij+(MN−k)β+kα−1

Applying f(I) = fEGC(I) in (10) yields

k=0

?MN

cj(MN − k,k)

k

?

(21)

×

∞

?

j=0

Pb=

MN

?

k=0

?MN

k

?∞

j=0

?

ξEGC

j

(MN − k,k)γ−j+(MN−k)β+kα

2

,

(22)

α−β ?∈ Z Z, where we have exploited (27) with σ2= M2N/4,

and the SNR independent coefficients ξEGC

given by

(√2NM)j+µβ+ναΓ

2√πΓ(j + µβ + να + 1)

j

(MN − k,k) are

ξEGC

j

(µ,ν) =

?

j+µβ+να

2

+1

2

?

cj(µ,ν)

.

(23)

To arrive at (22) we basically added and multiplied convergent

series having an infinite convergence radius. Thus, the series

in (22) also converges for all γ < ∞ [20].

In practice, the upper limit of the inner infinite series in (22)

has to be truncated to some finite value J, and the resulting

BER approximation is again referred to as˜Pb(J). While it is

difficult to obtain a simple upper bound for the approximation

error in this case, we expect that the approximation error for

EGC behaves qualitatively similar to the approximation error

in the SISO case, cf. (14). We point out that even for large

J the coefficients ξEGC

j

(µ,ν) required for evaluation of (22)

can be easily computed. The main complexity for computation

of ξEGC

j

(µ,ν) stems from the convolution in (20), which can

be performed very efficiently with standard software such as

MATLABTM.

B. Diversity and Combining Gain

Similar to the SISO case, more light can be shed onto the

performance of MIMO FSO with EGC by considering the

high SNR regime, where the BER is fully characterized by

the combining gain and the diversity gain. In case of EGC,

we obtain for these quantities from (22)

GEGC

d

= MNGd

(24)

GEGC

c

=

1

2NM2α2β2

2√πΓ(2MNGd+ 1)

Γ?MNGd+1

?Γ(max{α,β})

Γ(|α − β|)

? 1

1

MNGd

Gd

×

?

2

?

?

,

(25)

where Gd denotes the diversity gain in the SISO case as

specified in (16). Eq. (24) shows that repetition coding across

lasers and EGC at the receiver can exploit the full diversity

gain offered by the FSO MIMO channel, which formally

proves the conjecture in [6].

V. PERFORMANCE RESULTS

In this section, we verify the analytical results from Sections

III and IV by computer simulations and use them to study the

performance of SISO and MIMO FSO systems in Gamma–

Gamma fading.

In Fig. 1, we consider the BER of OOK over a SISO FSO

link with β = 2. Analytical results˜Pb(J) based on (11),

asymptotic results Pb≈ (Gcγ)−Gdbased on (16), (17), and

simulation results are compared. From Fig. 1 we observe that

for a given J the theoretical BER˜Pb(J) converges faster to

the true BER for higher SNR and for smaller α. For J = 10

and J = 20 the theoretical BER is practically identical to

the simulated BER even for SNR = 0 dB for α = 2.1 and

α = 4.1, respectively. As expected from (16), (17) both α

values yield the same diversity gain but α = 4.1 has a larger

combininggain. The asymptotic BER approaches the true BER

faster for α = 4.1. For α = 2.1 the convergence of the

asymptotic BER is quite slow since in this case√γ−βand

√γ−αare similar for small to medium SNRs. Consequently,

the first term in the series in (11), which is considered for the

asymptotic BER, becomes dominant only at very high SNRs.

In Fig. 2, the approximation error ε(J) = |Pb−˜Pb(J)| is

compared with the corresponding upper bound in (14). Both

the error and the bound are normalized by Pb which was

obtained by simulation. The same link parameters as for Fig. 1

are valid. Fig. 2 shows that the upper bound is tight, and thus,

is a useful tool for predicting the accuracy of the approximate

BER˜Pb(J). As expected, the approximation error rapidly

decreases with increasing J, increasing SNR, and decreasing

α.

Page 5

The impact of receive diversity on the accuracy of the

proposed BER approximation is studied in Fig. 3 for α = 3.1

and β = 2. As one would expect, the number of terms J

required in the truncated series to get an accurate BER estimate

increases with the diversity gain. While, in the considered SNR

range, J = 20 is sufficient to approach the simulated BER

for the SISO case (M = N = 1), J = 40 is required for

N = 3 photodetectors. The asymptotic BER also converges

considerably faster for the SISO channel than for N = 2 and

N = 3.

Having established the accuracy of the proposed perfor-

mance analysis technique, in Fig. 4, we choose J sufficiently

large to guarantee that the approximation error is negligi-

ble. Fig. 4 shows the BER of OOK as a function of the

link distance L between transmitter and receiver assuming

spherical wave propagation. α and β were calculated from

(3), (4), where we followed [12] and adopted λ = 1550 nm,

C2

be equal to 20 dB independent of L in order to separate the

effect of attenuation from the distance dependence of α and β,

cf. (3), (4). Fig. 4 shows that even in the absence of attenuation

the BER of SISO and MIMO FSO systems degrades with

increasing distance since α and β decrease. Transmit and

receive diversity are effective means to improve performance.

For all considered cases the BER obtained from the presented

analysis accurately predicts the simulated performance.

n= 1.7·10−14, and D/L → 0. The link SNR is assumed to

VI. CONCLUSIONS AND EXTENSIONS

In this paper, we have presented a novel approach to

performance analysis of SISO and MIMO FSO systems in

Gamma–Gamma fading. The proposed technique is based on a

generalized power series representation of the modified Bessel

function of the second kind and enables fast and accurate

performance evaluation. This is especially true for high SNR

where the first few terms of the series suffice to get accurate

results. This is in contrast to existing techniques which are

mostly based on numerical integration and may suffer from

stability problems at high SNR. Furthermore, the proposed

approach has allowed us to extend the concepts of combining

gain and diversity gain to FSO and Gamma–Gamma fading.

Closed–form expressions for both gains have been provided

for SISO FSO and MIMO FSO with EGC. These expressions

provide valuable insight into the impact of various system and

channel parameters on performance.

While we have focused on OOK and EGC in this paper, the

proposed general approach can be straightforwardly extended

to other modulation schemes (e.g. pulse position modulation)

and maximum ratio combining.

APPENDIX

A. A Useful Result

In this appendix, we establish a key result which is useful for

the performance analysis in Sections III and IV. In particular,

in the proposed analysis the integral

X(c,y) =

∞

?

0

Q

?√

c2I2?

Iy−1dI

(26)

with c2> 0, y > 0 has to be evaluated. Using the alter-

native representation of the Gaussian Q–function Q(x) =

1

π

0

exp−

ten as

?π/2

?

x2

2sin2θ

?

π/2

?

0

dθ [19, Eq. (4.2)], (26) can be rewrit-

X(c,y)=

1

π

∞

?

0

exp

?

−

c2I2

2sin2θ

?

Iy−1dI dθ

=

2y/2−1Γ(y/2)

πcy

π/2

?

0

sinyθdθ

=

2y/2−1Γ((y + 1)/2)

√πcyy

,

(27)

where we have used the identities?∞

?π/2

B. Bound on Approximation Error

In this appendix, we develop an upper bound for the

approximation error ε(J) ? |Pb −˜Pb(J)| which can be

expressed as

0xν−1exp(−µx2)dx =

0 [20, Eq. (3.4782)],

µ−ν/2Γ(ν/2)/2, ν

sinµ−1xdx = Γ(1/2)Γ(µ/2)/[2Γ((µ + 1)/2)] [20,

Eq. (3.6211)], Γ(x + 1) = xΓ(x), and Γ(1/2) =√π.

>0, µ>

0

ε(J) =

√π

2|sin[π(α − β)]|Γ(α)Γ(β)

??????

∞

?

j=J+1

sjxj

0

j!

??????

(28)

where sj ? µj(α,β) − µj(β,α), µj(α,β) ? xβ

1)/2)/[Γ(j −α+β +1)(j +β)], and x0?√2αβ/√γ. From

(28) we observe that ε(J) can be bounded as

√πsmax

2|sin[π(α − β)]|Γ(α)Γ(β)

√πsmaxxJ+1

2|sin[π(α − β)]|Γ(α)Γ(β)(J + 1)!

where we have used the definition smax? maxj>J|sj|. smax

can be easily computed since it can be shown that there is a

j0≥ J such that sj monotonically decreases if j > j0. For

sufficiently large J (J ≥ 10 is sufficient for typical values of

α and β), j0= J holds.

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