multiple transmitter performance with appropriate amplitude modulation for free-space optical...

Multiple transmitter performance with appropriateamplitude modulation for free-space

optical communication

Jason A. Tellez and Jason D. Schmidt*Department of Electrical and Computer Engineering, Air Force Institute of Technology,

2950 Hobson Way, Wright-Patterson Air Force Base, Ohio 45433-7765, USA

*Corresponding author: [email protected]

Received 6 June 2011; accepted 1 July 2011;posted 11 July 2011 (Doc. ID 148814); published 11 August 2011

The propagation of a free-space optical communications signal through atmospheric turbulence experi-ences random fluctuations in intensity, including signal fades, which negatively impact the performanceof the communications link. The gamma–gamma probability density function is commonly used to modelthe scintillation of a single beam. One proposed method to reduce the occurrence of scintillation-inducedfades at the receiver plane involves the use of multiple beams propagating through independent paths,resulting in a sum of independent gamma–gamma random variables. Recently an analytical model forthe probability distribution of irradiance from the sum of multiple independent beams was developed.Because truly independent beams are practically impossible to create, we present here a more generalbut approximate model for the distribution of beams traveling through partially correlated paths. Thismodel compares favorably with wave-optics simulations and highlights the reduced scintillation as thenumber of transmitted beams is increased. Additionally, a pulse-position modulation scheme is used toreduce the impact of signal fades when they occur. Analytical and simulated results showed significantlyimproved performance when compared to fixed threshold on/off keying.OCIS codes: 010.1330, 060.2605, 290.5930.

1. Introduction

When laser beams are transmitted over long, turbu-lent paths, the variation in the index of refractionalong the path causes the beam to randomly wanderand scintillate, resulting in the fluctuation of powerat the receiver end of the propagation path. Whenused in free-space optical communications (FSOC),the fluctuations can result in signal fades of varyingdurations, decreasing the received irradiance and in-creasing the bit error rate (BER). More dramatically,for a unipolar modulation scheme, such as on/off key-ing (OOK), when a fade causes the received intensityto drop below the detection threshold, the BER staysat the maximum value for the duration of the fade.Adaptive optics can compensate for the accumulationof optical phase distortions occurring along the path.However, certain engagement scenarios, such as forhigh-altitude air-to-air links, result in a large atmo-spheric coherence radius r0, while still experiencing

strong scintillation. In this turbulence regime, adap-tive optics provides little benefit. Another method toreduce the probability of a fade is to use multiplebeams that are incoherent with respect to each other.This averages the intensity fluctuations due to scin-tillation, reducing the probability of signal fades andtheir duration [1,2]. Recent literature has derivedanisoplanatic separation distances for multiplebeams and exploited this knowledge to reduce fadesin FSOC systems over long, horizontal paths, withparticular benefits for air-to-air links [3].

The probability density function (PDF) of the re-ceived intensity from a single beam propagatingthrough moderate-to-strong turbulence is usuallymodeled as a gamma–gamma probability density;however, a closed-form solution for multiple, spa-tially separated beams has been elusive. Thisresearch extends a recently developed approxima-tion of the PDF resulting from the sum of multiple

20 August 2011 / Vol. 50, No. 24 / APPLIED OPTICS 4737

independent gamma–gamma random variables toaccount for partially dependent propagating beams.The resulting PDFs are then compared to resultsobtained from a wave-optics simulation of a typicalair-to-air FSOC link. Since atmospherically inducedsignal fades cannot be completely eliminated, differ-ent modulation schemes are studied to determinewhich is least sensitive to the remaining intensityfluctuations, thus further mitigating the impact ofsignal intensity fades [2,4].

2. Review of N Independent Gaussian Beams

This section reviews the aperture-averaged irradi-ance results for multiple, identically distributed, in-dependent beams propagating through turbulence[5]. Once the independent case is established, themodel is modified to account for partially correlatedscintillation between the beams.

A. Single-Beam Irradiance Statistics

The normalized irradiance resulting from a single-beam propagation through atmospheric turbulencecan be described as the product of two independentgamma-distributed random variables, I ¼ xy, wherex and y arise from large-scale and small-scale atmo-spheric effects, respectively [6]. The resulting PDF ofthe normalized intensity is then

pðIÞ ¼ 2ðαβÞðαþβÞ=2

ΓðαÞΓðβÞ IðαþβÞ=2−1 × Kα−β

�2

ffiffiffiffiffiffiffiffiαβI

p �;

I > 0; ð1Þ

where α ¼ 1=σ2x and β ¼ 1=σ2y represent the inverse ofthe large- and small-scale variances, respectively,Γð·Þ is the gamma function, and Kpð·Þ is the modifiedBessel function of the second kind. More often, thecumulative distribution function (CDF) is used to de-termine fade statistics since it describes the probabil-ity of receiving less than a given threshold. The CDFfor the gamma–gamma function can be determinedanalytically to be

PðI ≤ ITÞ ¼Z

IT

0pðIÞdI

¼ πsin½πðα − βÞ�ΓðαÞΓðβÞ

×�ðαβITÞββΓðβ1Þ

× 1F2ðβ; β þ 1; β1; αβITÞ

−ðαβITÞααΓðα1Þ

× 1F2ðα; αþ 1; α1; αβITÞ�; ð2Þ

where IT is a threshold, β1 ¼ β − αþ 1, α1 ¼ α − β þ 1,and 1F2 is a generalized hypergeometric function.

The large- and small-scale variances, σ2x and σ2y , aredetermined based on the initial beam characteristicsand the turbulence along the propagation path. Forpractical applications, such as for FSOC, the irradi-ance is averaged across the receiver aperture ratherthan determined at a single point, and the light is

focused onto a receiver that detects the total powerin the aperture. This allows the receiver to capturemore energy from the propagating beam and alsoaverages out the spatially varying scintillation ef-fects. For a Gaussian beam, assuming Kolmogorovturbulence and the receiver is centered on the beamcenter, the aperture-averaged log variance of x is [7]

σ2ln xðDÞ ≅ 0:49σ21�ΩG −Λ1

ΩG þΛ1

�2�13−12�Θ1 þ

15�Θ21

�

×� ηx1þ 0:40ηxð2 −

�Θ1Þ=ðΛ1 þ ΩGÞ

�7=6

; ð3Þ

where

ηx ¼

�13 −

12�Θ1 þ 1

5�Θ21

�−6=7

ðσB=σ1Þ12=7

1þ 0:56σ12=5B

: ð4Þ

The aperture-averaged log variance of y is

σ2ln yðDÞ ≅ 1:27σ21η−5=6y

1þ 0:40ηy=ðΛ1 þ ΩGÞ; ηy ≪ 1; ð5Þ

where

ηy ¼ 3

�σ1σB

�12=5

ð1þ 0:69σ12=5B Þ: ð6Þ

The variance is determined via σ2x;y ¼ expðσ2ln x;yÞ − 1.The parameter σ2B is the Rytov variance for a beamwave, which is approximated as

σ2B ≅ 3:86σ21�0:40½ð1þ 2Θ1Þ2 þ 4Λ2

1�5=12

× cos�56tan−1

�1þ 2Θ1

2Λ1

��−1116

Λ5=61

�; ð7Þ

where σ21 is the plane-wave Rytov variance given asσ21 ¼ 1:23C2

nk7=6L11=6. The values Θ1 and Λ1 are thecurvature parameter and Fresnel ratio at the receiveplane for vacuum propagation, respectively, which,given in terms of their respective values at the sourceplane, are [8]

Θ1 ¼ Θ0

Θ20 þΛ2

0

; ð8Þ

Λ1 ¼ Λ0

Θ20 þΛ2

0

: ð9Þ

The curvature parameter and Fresnel ratio at thesource plane are

Θ0 ¼ 1 −LF0

; Λ0 ¼ 2L

kW20

; ð10Þ

4738 APPLIED OPTICS / Vol. 50, No. 24 / 20 August 2011

respectively, where L is the propagation distance, F0is the initial radius of curvature, W0 is the initialGaussian beam radius, and k ¼ 2π=λ is the wave-number. The parameter �Θ0 is simply defined as�Θ0 ¼ 1 −Θ0. Finally, the value ΩG characterizes thefinite size of a Gaussian lens and is given to beΩG ¼ 2L=kW2

G, where WG ¼ ðD2=8Þ1=2 with D repre-senting the receive aperture diameter.

A comparison between the aperture-averagedgamma–gamma CDF from Eq. (3) using α and β re-sulting from Eqs. (2) and (5) with a wave-optics simu-lation for a single-beam system is shown in Fig. 1.This nominal representative FSOC engagement usesa 1550nm wavelength initially collimated Gaussianbeam with a beam waist of W0 ¼ 5 cm propagatingalong a L ¼ 100km horizontal path with a constantturbulence strength at C2

n ¼ 1 × 10−17 m−2=3 [1,2].This engagement results in a plane-wave atmo-spheric coherence diameter of r0;pw ¼ 31:2 cm and aplane-wave Rytov variance of σ21 ¼ 0:924. The totalpower transmitted is P ¼ 1W, and the direct-detection receiver has an aperture diameter of D ¼10 cm. Figure 1 shows good agreement betweenthe analytical model from Eq. (2) and the wave-opticssimulation.

B. Multiple Independent-Beam Irradiance Statistics

The development of the theoretical approximationfor the sum of multiple, independent, identicallydistributed gamma—gamma random variables de-veloped by Chatzidiamantis et al. begins with thesum of the product of two independent gamma ran-dom variables [9]:

IN ¼XNi¼1

xiyi; ð11Þ

where N is the number of independent beams. Thiscan be rewritten as

IN ¼ 1N

�XNi¼1

xi

��XNi¼1

yi

�þ 1N

XN−1

i¼1

XNj¼iþ1

ðxi − xjÞðyi − yjÞ:

ð12ÞIn this form, the equation is seen to be simply thescaled product of the sum of two gamma randomvariables plus an error term, ϵ, given by

ϵ ¼ 1N

XN−1

i¼1

XNj¼iþ1

ðxi − xjÞðyi − yjÞ: ð13Þ

Since the gamma distribution is infinitely divisible,the two sums in Eq. (12) are gamma distributedthemselves. The two resulting gamma random vari-ables then have variances of σ2x=N and σ2y=N, respec-tively, resulting in αN ¼ Nα and βN ¼ Nβ, which,using Eq. (1), produces the new multibeam PDF ap-proximation. Chatzidiamantis et al. accounted forthe error term in Eq. (13) by adding a correctionfactor to αN , such that

αN ¼ Nαþ ϵN ; ð14Þwhere ϵN can be approximated byminimizing the dif-ference between the first four moments generatedusing the correction factor with a single gamma–gamma random variable and the moments generatedwithout the correction factor using multinomial ex-pansion for N gamma–gamma random variables.Using nonlinear regression, the result can be closelyapproximated to be

ϵN ¼ ðN − 1Þ−0:127 − 0:95α − 0:0058β1þ 0:00124αþ 0:98β : ð15Þ

C. Comparison with Simulation

As a basis for comparison with the analytic model, awave-optics simulation of a nominal FSOC systemwas performed using the angular-spectrum methodto evaluate the Fresnel diffraction integral [10]. Thesampling requirements were determined using themethods outlined by Coy to avoid aliasing while stilladequately sampling the field in the telescope pupil[11]. For the independent-beam case, each beam wastransmitted through a unique random realization ofthe atmospheric phase screens with the irradianceaveraged across the receive aperture. For multiple-beam results, irradiances were combined in uniquegroups of N beams. Once the results were obtained,each irradiance value was given a marginal probabil-ity of 1=n, where n is the total number of values. Bysorting the resulting received irradiances and plot-ting against a cumulative sum of the marginal prob-abilities, a CDF was constructed.

The simulation and theoretical curves were gener-ated using the same engagement scenario describedin Subsection 2.A. The total power transmitted,regardless of the number of beams propagated is

Fig. 1. (Color online) Analytical and wave-optics generated CDFfor an aperture-averaged, single-beam irradiance.


PT ¼ 1W. The theoretical average power received iscalculated by integrating the received irradianceacross the aperture via

PR ¼ PT

Z2π

0

ZD=2

0

2πWe

exp�−2r2

W2e

�rdrdθ

¼ PT

�1 − exp

�−D2

2W2e

��; ð16Þ

where We is the effective beam radius given by

We ≅ W1ð1þ 1:63σ12=51 Λ1Þ1=2 ð17Þ

andW1 is the vacuum-propagated beam radius givenby

W1 ¼ W0ðΘ20 þΛ2

0Þ1=2: ð18Þ

The results shown in Fig. 2 for one to seven beamsdemonstrate a decrease in fade probability as thenumber of transmitters is increased. One potentialsource of differences arises from transmitting thedisplaced, collimated beams in parallel, resulting inthe receiver detecting off-center regions of the result-ing Gaussian beams. The model assumes the statis-tics of the aperture-averaged beam are the sameregardless of which portion of the beam is received.From the results this appears to be a reasonablevalid assumption for this scenario. Additional turbu-lence strengths using the same engagement condi-tions but with constant C2

n values ranging from5 × 10−18 to 1 × 10−16 were evaluated with similarlevels of agreement.

3. Spatially Separated Beams

The case of spatially separated beams requires thedevelopment of a solution for the sum of partially cor-related gamma–gamma random variables. The pro-posed solution modifies the independent case toadjust for the partial correlation. Prior to developingthe model for spatially separated beams, the separa-tion distance and the beam placement geometry areaddressed.

A. Anisoplanatic Separations

In previous work, Louthain determined the analyticlog-amplitude and phase structure functions for ahorizontal path starting with the von Kármán powerspectral density:

Φnðκ; zÞ ¼0:033C2

nðzÞðκ2 þ κ20Þ11=6

; ð19Þ

where κ is the three-dimensional radial spatialfrequency and κ0 ¼ 2π=L0 is the outer scale roll-offfrequency for outer scale L0 [8,12]. The derived struc-ture functions as a function of separation d are

DχðdÞ ¼ 3:089

�L0

r0

�Z∞

0

�1 − J0

�κdL0

��

×�1 −

2πL20

λLκ2 sin�λLκ22πL2

0

�� κdκðκ2 þ 4π2Þ11=6 ð20Þ

for the log-amplitude structure function and

DψðdÞ ¼ 3:089

�L0

r0

�Z∞

0

�1 − J0

�κdL0

��

×�1þ 2πL2

0

λLκ2 sin�λLκ22πL2

0

�� κdκðκ2 þ 4π2Þ11=6 ð21Þ

for the phase structure function, where J0 is thezeroth-order Bessel function of the first kind [1,13].Normalized versions of the structure functions areplotted together in Fig. 3. Also shown are the separa-tion angles required for phase anisoplanatism andtilt anisoplanatism, which are

θ0 ¼ 0:949ðk2C2nL8=3Þ−3=5; ð22Þ

θTA ¼ 0:319λD1=6

CnL3=2; ð23Þ

Fig. 2. (Color online) Analytical CDF (solid lines) and wave-opticsresults (dashed lines) plotted for one through seven beams.

Fig. 3. (Color online) Normalized amplitude and phase structurefunctions. When the structure function no longer increases withincreased separation, d, then points separated by that distanceare uncorrelated. The vertical lines show the separations dueto the isoplanatic angle θ0, the tilt isoplanatic angle θTA, andthe chosen separation of twice the Fresnel zone size 2ρc.


respectively [1,14]. The plot shows that two pointsseparated such that they have log-amplitude inde-pendence, at a distance of twice the Fresnel zone sizeof ρc ¼ ðL=kÞ1=2, where k ¼ 2π=λ [8], can still tilt glob-ally in a similar manner. This allows a single track-ing mechanism to be used for all of the beamswithout sacrificing too much wander between beams(however, in this research no tracking device isused). For this scenario, twice the Fresnel zone is2ρc ¼ 31:4 cm.

B. Beam Array Patterns

To minimize the area occupied by the multiple trans-mitters, a hexagonal close-pack spacing is used [15].Figure 4 shows the arrangement and relative inten-sities of each of the patterns, assuming the power isdistributed equally among the transmitters. Eachpattern used is centered in relation to the singlebeam. The closest any two beams can be is the chosenseparation distance, in this case 2ρc. For the largernumber of transmitters, particularly five or more,the maximum separation between the furthestspaced beams is twice the chosen separation dis-tance, or 4ρc. Those beams begin to wander inde-pendently, since 4ρc > LθTA, which would limit atracking system’s ability to stabilize pointing of thebeam array. One solution would be to track eachbeam individually at the cost of increased systemcomplexity. Additionally, the total power remains thesame regardless of the number of transmitters used.

C. Model Modification for Partially Correlated Beams

To account for the finite beam separation, a methodneeds to be developed to account for the partially cor-

related scintillation resulting from several beams.The independent, identically distributed solution ef-fectively sums the identical shaping parameters. The

partially correlated solution can be thought of in thesame way, where the number of shaping parametersto be summed is varied between one, for full correla-tion, and N, for full independence. By rewritingEq. (14) and similarly modifying βN, a proposed mod-ification to the independent-beam solution can bewritten as

αN ¼ α�Nαþ ϵ

α

�f ðρðdÞÞ

; ð24Þ

βN ¼ βNf ðρðdÞÞ; ð25Þ

where ρðdÞ is the correlation between the result-ing integrated irradiance from the multiple beams,which is a function of separation distance d. Thismodification necessitates the lower limiting case suchthat, when the beams are uncorrelated, i.e., ρðdÞ ¼ 0,αN and βN match the N independent-beam solution.Theupper limiting case,when thebeamsareperfectlycorrelated, i.e., ρðdÞ ¼ 1, results in αN and βN ap-proaching the single-beam solution. In this methodthe effective strength of the multiplicative factor isvaried as a function of the correlation coefficient.

The correlation coefficient was determined usingwave-optics simulations of Gaussian beam pairs ofvarious separation distances propagated over thesimulated distance and integrated over a commonaperture located between the propagated beams atthe receive plane. The resulting relationship be-tween separation distance and correlation coefficientwas determined via

ρðdÞ ¼ hðIAð−d=2; 0Þ − μIAð−d=2; 0ÞÞðIAðd=2; 0Þ − μIAðd=2; 0ÞÞiσIAð−d=2; 0ÞσIAðd=2; 0Þ

; ð26Þ

where h·i is the expectation operator and IAðx; yÞ isthe received irradiance integrated over the receiveaperture for a Gaussian beam centered on x, y andμIAðx; yÞ and σIAðx; yÞ are the mean and variance ofIAðx; yÞ, respectively. The integrated irradiance foreach beam is determined via

IAðx; yÞ ¼Z Z

AIðξ − x; η − yÞdηdξ ð27Þ

¼Z

D=2

−D=2

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD=2Þ2−ðξ−xÞ2

p

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD=2Þ2−ðξ−xÞ2

p Iðξ − x; η − yÞdηdξ; ð28Þ

where D represents the receive aperture diameterand I is the irradiance at the receiver plane. Theresults are shown in Fig. 5 along with a model ofthe results using a first-order, single-parameter func-tional fit of

Fig. 4. Configuration used for multiple transmitters. Each config-uration is a subset of a hexagonal close-pack grid. The top rowshows one, two, three, and four transmitters from left to right,while the bottom row shows five, six, and seven transmitters. Inall cases, the total transmitted power is held constant at P ¼ 1W.


ρðdÞ ≈ expð−0:6875d=ρcÞ: ð29ÞThe value of −0:6875 in the model was determinedusing a least-squares fit to the simulation data. Thisfunctional form provides the appropriate limitingconditions as d approaches zero and infinity. Thesame simulation data were then used to determinethe functional form of f ðρÞ by first determining theleast-squares fit value for f ðρÞ for the correspondingCDF. The resulting relationship between the correla-tion coefficient and f ðρÞ was modeled using a first-order, single-parameter function resulting in

f ðρÞ ≈ ð1 − ρÞ1:4894: ð30ÞThis functional form provides the appropriate limit-ing values of f ð1Þ ¼ 0 and f ð0Þ ¼ 1 with the value of1.4894 determined via a least-squares fit to the simu-lation data. Combining Eq. (29) and (30) results in

f ðρðdÞÞ ≈ ½1 − expð−0:6875d=ρcÞ�1:4894: ð31ÞA comparison of Eq. (31) with results from simula-tion data is shown in Fig. 6. The simulation resultsshow that, at a separation distance greater than 3ρc,the correlation seems to increase. The aperture-averaged model for the gamma–gamma shapingparameter assumes the receive aperture is centeredon the beam center. As the beam separation in-creases, this assumption is no longer valid. To inves-tigate the impact of the displacement on the CDF,several normalized CDFs were created from simula-tion data with varying receiver displacements. Theresult shown in Fig. 7 indicates that there is a slightshift in the CDFof around 0:1dBm at 10−2. This errorsource is small when compared to the CDF shift as-sociated with increasing the number of transmitters,which is of the order of 1:0dBm.

Another potential source of the decreasing value off ðρÞ beyond 3ρc in Fig. 6 is the variation in mean re-ceived irradiance as a function of receiver displace-ment from beam center. Figure 8 demonstrates that,as the receiver displacement increases, the meanirradiance decreases. This, coupled with a corre-

sponding decrease in the variance as displacementis increased, provides a partial explanation for thedownward trend evident in Fig. 6. However, the pri-mary region of interest lies between 1ρc and 3ρcwhere there is still reasonable agreement betweenthe simulation data and model shown in Fig. 6.

D. Comparison with Simulation

Wave-optics simulations were done to test the CDFmodel and explore the transmitter configurations.As part of this, the beam power levels were adjusted.An issue associated with the placement of multiplebeams resulting from the necessarily nonuniform se-paration distance for four or more beams is the var-iation in relative intensity over the receive aperturefrom each of the beams. When the beams are trans-mitted parallel to each other, beams placed furtherfrom the center clearly result in a smaller contribu-tion to the aperture-averaged irradiance since theaperture intercepts the beam edge, not its center.As a result, fades from beams near the center couldhave a greater impact on the frequency of fades thanthose farther away. To compensate for this effect, thetransmitted power for each beam is adjusted to en-sure that the mean power received from each beam,using Eq. (27), is the same.

Fig. 5. (Color online) Correlation coefficient versus beam separa-tion for two Gaussian beams propagated over 100km path withσ21 ¼ 1:0.

Fig. 6. (Color online) Exponent argument versus beam separa-tion for two Gaussian beams propagated over 100km path withσ21 ¼ 1:0.

Fig. 7. (Color online) Normalized CDFs for integrated irradiancewith receivers placed at various off-axis distances.


Using a combination of multiple beams and adjust-ing the power among the beams such that the meanpower received from each is the same, a wave-opticssimulation was done showing the marginal improve-ment as the number of beams is increased. The CDFresults are shown in Fig. 9. Additionally, the theore-tical estimates determined using Eqs. (1), (24), and(30) were compared to the CDFs from simulation.Figure 9 shows that the model provides a good de-scription of wave-optics results up to five transmit-ters. As the number of transmitters increases pastfive, the approximation becomes less valid. The as-sumption of gamma–gamma statistics for off-centerbeams is likely failing again. Referring back to Fig. 4,one can see that, as the number of transmitters in-creases, separations between the center of the beamconfigurations, where the receiver is located, and themost distant beams increases. This may account forthe disagreement between the model and wave-optics simulations when the number of transmittersis greater than four.

4. Mean BER Results

To evaluate the end-to-end performance of usingmultiple beams, the BER was studied. The evalua-tion used fixed threshold OOK and binary pulse-position modulation (PPM) for the modulationschemes. This comparison highlights the differences

between using a fixed threshold modulation schemethat is highly sensitive to deep signal fades versus ascheme that is not significantly impacted by deepfades. In the case of OOK, a deep fade due to scintil-lation causes the received signal to drop below thethreshold, resulting in every received bit being esti-mated as a signal low. Assuming the likelihood of atransmitted low or high signal is equal, this amountsto a 50% BER. Binary PPM places all of the energyeither in the first half of the transmitted symbol orthe second half. The receiver then subtracts the re-ceived energy for the first half of the symbol fromthe second half, resulting in either a positive or ne-gative value. With a threshold of zero, the negativevalues correspond with a signal low and positivevalues with a signal high. As it turns out, when con-sidering additive white Gaussian noise (AWGN),Binary PPM is equivalent in performance to an idealvariable threshold where the threshold is adjusted tohalf the received intensity instantaneously. However,PPM does not require the constant detection and up-date of the threshold value, which draws irradiancefrom the communications detector [2]. Additionally,PPM always has either a rising or falling edge in thecenter of the pulse, which could be used to aid in tim-ing and synchronization of the demodulator [16].

First, an estimated BER from simulated intensi-ties was determined using a time domain wave-optics simulation. At each time step, a theoreticalBER was produced, and these BERs were then aver-aged together to produce an overall BER. The overallBER, PB, can be determined via

PB ¼ P½s1�P½H2js1� þ P½s2�P½H1js2�; ð32Þ

where s1;2 represents the transmitted symbols forthe binary signaling scheme. P½s1;2� is the probabilitythat s1 or s2 was transmitted, respectively, andP½H2;1js1;2� is the probability that the receiver hy-pothesized s2 when s1 was transmitted and viceversa. Using OOK and making the simplifying as-sumption of AWGN, the probability of false positiveand missed detection becomes

P½H2js1� ¼Z

∞

γTpðzjs1Þdz

¼Z

∞

γT

1

σ0ffiffiffiffiffiffi2π

p exp�−12

�z − a1

σ0

�2�dz

¼ Q

�γT − μ1σ0

�; ð33Þ

P½H1js2� ¼ Q

�μ2 − γTσ0

�; ð34Þ

where γT is the detection threshold, μ1;2 are the sym-bol mean values, σ0 is the standard deviation of thenoise, and Qð·Þ is the Q function defined as

Fig. 8. (Color online) Mean integrated irradiance plotted as afunction of receiver distance from beam center.

Fig. 9. (Color online) Comparison of wave-optics results withanalytical CDF for one to seven spatially separated beams.


QðxÞ ¼ 1ffiffiffiffiffiffi2π

pZ

∞

xexp

�−u2

2

�du: ð35Þ

In this case μ1 ¼ 0 and μ2 are intensity over the aper-ture and are constant for each wave-optics time step.Themodulation schemes used were OOKwith a fixedthreshold throughout the wave-optics simulationand PPM. The previously derived OOK fixed thresh-old is determined using

γTh¼μ1σ22−μ2σ21σ22−σ21

þ σ1σ2σ22−σ21

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðμ2−μ1Þ2þ2ðσ22−σ21Þ ln

�σ2σ1

�s; ð36Þ

where σ22 ¼ σ2I þ σ20 and σ21 ¼ σ20 [17]. This representsthe best possible threshold assuming all of themeansand variances of the signal and noise are known,which in practice would likely be determined by sam-pling the irradiance statistics. The signal values usedwere μ1 ¼ 0W and μ2 ¼ 1W. For PPM, the thresholdbecomes γT ¼ 0W, while the signal values are μ1 ¼−0:5W and μ2 ¼ 0:5W.

The analytical BER using the probability distribu-tion of intensity is computed via

PB ¼Z

∞

0P½γ�pðγÞdγ; ð37Þ

where γ represents the integrated received intensity.Additionally a Monte Carlo simulation was per-formed at each time step with various noise levelsto validate the analytical model. The Monte Carlosimulation generated random bit patterns, wasmodulated using both OOK and PPM, and was thenmultiplied by the received power from the wave-optics simulation. After the addition of a randomGaussian noise, the signals were demodulated to ob-tain a bit estimate. The estimate was then comparedto the original random bit pattern, with differencescounted as bit errors. The results for both the analy-tical solutions and the Monte Carlo simulations areshown in Figs. 10 and 11. The calculated BER isshown as a function of the signal-to-noise ratio(SNR) where the signal is defined as the full trans-mitted power and the noise is the AWGN at the re-ceiver. While this definition of the SNR limits theinterpretation of the results to the aperture sizedused in the simulation, D ¼ 10 cm, it allows for ameaningful and fair comparison between the varioustransmitter configurations, since different transmit-ter configurations result in different mean receivedirradiance.

The use of additional transmitters clearly im-proves the BER seen in Figs. 10 and 11. However,the improvement is more dramatic for the fixedthreshold than the improvement for PPM. By way ofcomparison, at a BER of 10−3, the increase in codegain between one to two transmitters is 3:4dB for

fixed threshold, while it is 1:6dB for PPM. Similarly,the gain for going from two to four transmitters is2:4dB for fixed threshold and 1:3dB for PPM. Whilethe improvement is greater when fixed threshold isused, the performance still does not surpass that ofPPM, which, at a BER of 10−3, is 15:7dB, better evenwhen comparing seven transmitters using fixedthreshold against a single transmitter using PPM.Figure 10 also shows the more dramatic impact themodeling errors have in estimating the BER for fixedthreshold as the number of transmitters is increased.While the BER curve for one or two transmitters ismodeled well, the assumptions used for determiningthe PDF for additional transmitters more dramati-cally impacts BER estimates. This is less evidentfor the PPM case shown in Fig. 11, where there ismuch better agreement between themodel and simu-lation results.

Fig. 10. (Color online) BER for multiple beams using OOK withfixed threshold. The markers indicate results from the MonteCarlo simulation of the communications link. The dashed linesare the “analytical” results using the data from the wave-opticssimulation. The solid line is the analytical result obtained fromthe newly derived model of the received intensity. Results areshown for one, two, four, and seven transmitters.

Fig. 11. (Color online) BER for multiple beams using PPM. Themarkers indicate results from the Monte Carlo simulation of thecommunications link. The dashed lines are the “analytical” resultsusing the data from the wave-optics simulation. The solid line isthe analytical result obtained from the newly derived model of thereceived intensity. Results are shown for one, two, four, and seventransmitters.


5. Conclusion

The original analytical approximation for the sumof multiple gamma–gamma random variables pro-vides a basis for a good approximation for multiple,independent Gaussian beams propagating throughatmospheric turbulence. While that approximationassumes complete independence, the newly derivedapproximation considers partially independentbeams, with correlation acting as a function of spatialseparation. The results provide a useful approxima-tion for fade characteristics that could then be usedto provide the expected number of fades and themean fade duration for a FSOC system propagatingthrough moderate-to-strong turbulence. The result-ing probability distribution can also be used to predictBERperformance. Finally, theBERperformance for amodulation scheme less sensitive to scintillation,such as PPM, can have significant performance im-provement over traditional OOK.

Future expansion of this work includes analyzingthe results for a wider set of turbulence conditions.While the correlation of scintillation at differentpoints is primarily driven by Fresnel zone size [8],changes in the turbulence conditions would likely im-pact the constants appearing in Eqs. (29) and (30).Additionally, a higher-order form of each of theseequations would provide a better fit between themodel and simulation results. Finally, the model as-sumes gamma–gamma statistics are valid for off-center portions of the Gaussian beam, an assumptionthat does not appear to hold as the receive aperturesget farther from the center of the beam. As such, themodel loses fidelity as the number of transmittersincreases beyond five.

This research is sponsored by the Physics andElectronics Directorate of the United States AirForce Office of Scientific Research (USAFOSR) underfederal grant F1ATA0035J001.

The technical results and conclusions expressed inthis article are those of the authors and do not neces-sarily reflect the official policy or position of thesponsors, the U.S. Air Force, the U.S. Departmentof Defense, or the U.S. government.

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