SPIE Proceedings [SPIE SPIE Defense, Security, and Sensing - Orlando, Florida (Monday 25 April 2011)] Atmospheric Propagation VIII - Monte-Carlo-based multiple-scattering channel modeling

Monte-Carlo-based multiple-scattering channel modeling fornon-line-of-sight ultraviolet communications

Robert J. Drost∗, Terrence J. Moore, and Brian M. Sadler

U.S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD, USA 20783-1138

ABSTRACT

Although the concept of non-line-of-sight (NLOS) ultraviolet (UV) communications has been studied for decades,recent advances in the design and manufacturing of light-emitting diodes, filters, and sensors have ignited newinterest. In this paper, we discuss a Monte Carlo channel model for NLOS UV communications that accountsfor the possibility that a transmitted photon experiences multiple scattering events before being received. Bysimulating the propagation of many photons based on probabilistic rules derived from physics considerations, acomputationally efficient algorithm is obtained that allows for the study of the contribution of various orders ofscattering to the received signal and to the system impulse response function. We then demonstrate the use ofthis channel model in the exploration of several system configurations. In particular, we examine the effect ofthe transmitter beam shape and receiver sensitivity function on the faithfulness of a well-known linear modelof path loss versus distance for short-range NLOS UV systems, and we explore geometry design for interferencereduction in a full-duplex link. The use of the model to study such diverse system implementations demonstratesits general applicability.

Keywords: Ultraviolet communications, channel modeling, atmospheric propagation, multiple scattering

1. INTRODUCTION

The employment of ultraviolet (UV) light for communications is an intriguing notion that has been studiedfor decades. In addition to exploiting untapped bandwidth to augment or replace traditional systems whenconventional spectrum usage is unavailable or undesirable, a UV communication system experiences a uniquechannel that provides opportunities to overcome some specific challenges. For example, while atmosphericabsorption can limit the range over which UV communications is possible, it also blocks nearly all solar radiationbelow 300 nm. The resulting low-noise channel enables the use of extremely sensitive photon-counting receivers(e.g., photomultiplier tubes). Meanwhile, scattering of UV light by the atmosphere makes non-line-of-sight(NLOS) operation possible.

Practical issues–such as the size, cost, and efficiency of transmitters, filters, and sensors–have limited thedevelopment of UV NLOS communication systems. However, recent advances in component design and man-ufacturing suggest that such barriers may soon be overcome. For example, commercially available UV light-emitting diodes (LEDs) have recently been used in short-range experimental communication systems.1 As thistechnology matures, improvements in both the efficiency and cost of such devices could enable real-world systemimplementations, and this possibility has renewed research interest.

While many early studies2 of UV NLOS communications focused on understanding atmospheric effects,3–6

recent studies have explored system design issues and tradeoffs. For example, the effect of pointing angles onpath loss has been examined for both coplanar geometries1 and noncoplanar geometries,7 and the performanceof various modulation and coding schemes has been analyzed.8 Novel hardware configurations have also beenexamined theoretically and experimentally. For example, the use of an imaging receiver array has been proposedin order to enhance the signal-to-dark-count ratio,9 while a nonimaging optical concentrator–designed to simul-taneously increase the receiver field-of-view and effective collection area–has been fielded in an experimentalsystem.10

∗[email protected]

Invited Paper

Atmospheric Propagation VIII, edited by Linda M. Wasiczko Thomas, Earl J. Spillar, Proc. of SPIE Vol. 8038803802 · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.888049

Proc. of SPIE Vol. 8038 803802-1

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System design and analysis is greatly aided by accurate channel models that can predict the effect of designchoices on communication performance. However, the NLOS channel is particularly challenging to model giventhe complex interaction of photons with the atmosphere. Nevertheless, steady progress has been made in this area.So-called single-scattering models–models that consider only a single scattering interaction for each transmittedphoton–are attractive for their relative ease of evaluation.11, 12 Indeed, such a model has been used to analyticallyderive a linear relationship between path loss and the distance between the transmitter and receiver in short-range NLOS communications (in contrast to a quadratic relationship found in line-of-sight systems).13 On theother hand, multiple-scattering models attempt to achieve improved accuracy by accounting for energy receivedby photons that experienced multiple scattering interactions. Such models have been developed using analyticaltechniques,14 applying multiple forward scatter theory,15 and employing Monte Carlo methods.16 Efforts toformalize the Monte Carlo approach have been recently undertaken,17, 18 and it is this model that we address.

In this paper, we promote the use of a Monte Carlo channel model as an intuitive, accurate, flexible, andefficient approach to UV communications channel modeling. By simulating the propagation of transmitted pho-tons through the atmosphere, the probability that such a photon will be detected at the receiver is determined,yielding an estimate of the overall path loss that accounts for returns from multiple-scattered photons. Further-more, this method can naturally incorporate a variety of system characteristics, such as arbitrary transmitterbeam patterns and receiver sensitivity functions and nonzero transmitted spectral linewidths. We provide anoverview of this approach in Section 2, highlighting the above characteristics. Then in Section 3, we demonstratethe power of this method with analysis of particular system configurations. Finally, we provide some concludingremarks in Section 4.

2. MONTE CARLO CHANNEL MODEL

In the Monte Carlo approach to multiple scattering UV channel modeling, the propagation of a large number ofphotons is simulated to determine the probability that a transmitted photon will be detected at the receiver; thepath loss of the channel is given by the reciprocal of this probability. In addition, by computing the total distancetraveled by each detected photon, one can obtain the impulse response function (IRF) of the system. Finally,the contributions of different orders of scattering to the path loss and IRF can be determined by classifying thereceived energy according to the number of scattering events experienced by the associated photon. Althoughthis framework is quite intuitive, there are several modeling and algorithmic challenges that must be addressed.We highlight several of these issues here; a more-detailed treatment is given in Ref. 18.

To simulate the propagation of a photon, we first require probabilistic models for its direction of transmission,its interaction with the atmosphere, and its detection by the receiver. The transmitter and receiver are oftenmodeled by cones with apex angles βT and βR determined from the beam width of the transmitter and thefield-of-view of the receiver, respectively. The initial direction of a transmitted photon is then taken to beuniformly distributed within the transmitter cone, while a photon impinging upon the receiver is assumed to bereceived if and only if it arrives from within the receiver cone. While such models can be convenient in single-scattering methods, where the complexity of computing the path loss may depend on the size of the intersectionof the two cones, the Monte Carlo simulation does not particularly benefit from this approximation and, inany case, can easily accommodate more-general transmitter radiant intensity functions and receiver angularsensitivity functions. Although these functions will depend on the devices used in a particular system, insightmay be gained by using reasonably representative functions. For example, one might model the transmitterbeam pattern with a Gaussian function or the receiver sensitivity as having a cosine roll-off.

To simulate the effect of the atmosphere on the propagation of a photon, the following model is oftenadopted. The direction a photon travels is assumed to be constant in between atmospheric interactions, and thedistance between interactions is taken to be exponentially distributed with parameter ke, called the extinctioncoefficient. When an interaction occurs, the photon is either absorbed, with probability ka/ke, or scattered, withprobability ks/ke, where ka and ks are known as the absorption and scattering coefficients, respectively, andsatisfy ke = ka + ks. If the photon is absorbed, it is no longer considered. On the other hand, if the photon isscattered, its trajectory experiences a deflection that depends on the nature of the atmospheric particle involvedin the interaction. If this particle is much smaller than the wavelength of the incident light, the deflection angle



α can be modeled by the Rayleigh phase function

pray(α) =3[1 + 3γ + (1− γ) cos2 α]

16π(1 + 2γ), (1)

where γ is a parameter of the function.17 If the size of the atmospheric particle is similar to the wavelength ofthe photon, the interaction is better modeled by Mie scattering. A generalized Henyey-Greenstein function canbe used for the phase function modeling such a scattering event:

pmie(α) =1− g2

4π

[1

(1 + g2 − 2g cosα)3/2+ f

3 cos2 α− 1

2(1 + g2)3/2

], (2)

where f and g are parameters of the function.14 Given the heterogeneous nature of the atmosphere, a linearcombination of the two functions provides a reasonable overall model of the phase function. That is, we assumeptotal(α) = (kr/ks)pray(α) + (km/ks)pmie(α), where kr and km are parameters such that ks = kr + km.

Note that the above atmospheric model contains several (independent) parameters: ka, kr, km, γ, f , andg. The values of these parameters depend on a variety of factors, such as the wavelength of the transmittedphoton, the atmospheric composition, and weather conditions. Unfortunately, the precise determination of theparameters corresponding to a particular set of conditions is challenging, and, as such, representative values areusually adopted. For example, Ref. 17 suggests values for ka, kr, and km as a function of wavelength, and assumesthat γ = 0.17, f = 0.5, and g = 0.72. Developing a deeper understanding of the effect of atmospheric conditionson these parameters, and on the atmospheric model in general, is clearly an important research avenue.

Having described the probabilistic modeling of photon propagation, we now turn our attention to severalalgorithmic details of the Monte Carlo simulation. An important observation is that path loss values on the orderof 100 dB are not uncommon in NLOS systems. As such, the straightforward simulation approach of launchingphotons and counting the number that arrive at the detector could require an average of 1010 simulated photonsto obtain a single detected photon. Since many received photons are needed for a reasonable probability estimate,it is clear that this method may be intractable.

To achieve a Monte Carlo simulation with reasonable complexity, we analytically account for the last legof the propagation of the photon from a scattering center to the receiver. (To avoid cumbersome languagedealing with the initial transmission of the photon, we call the transmitter a “scattering center” for purposesof this discussion.) In essence, we simulate only those paths in which the photon is not detected, but computethe probability that the photon could have scattered directly toward the receiver and been detected from eachscattering center instead of continuing along the simulated path. This has the added benefit that the possiblecontributions to all orders of scattering can be evaluated for each photon. Also, since absorption of a photonterminates the simulation, thus providing no additional information, we similarly only simulate propagationpaths in which the photon is not absorbed, again accounting for this possibility analytically.

Figure 1 depicts the simulation of a single photon; the overall Monte Carlo simulation requires many appli-cations of this algorithm. The location r of the photon is initialized at the transmitter, which, for convenience,can be interpreted as the 0th-order scattering center. As such, the current scattering order i is set to 0. Asurvival probability Ps, initially equal to 1, is assigned to the photon that tracks the probability that the photoncontinues along the simulated propagation path rather than being detected or absorbed. The photon wavelengthis then drawn at random from the emission spectrum of the source. (The standard inversion method19 is usedto obtain realizations of the random variables in the simulation, with numerical methods20 being used when acumulative distribution function cannot be analytically inverted.) Note that if the source is modeled as havingzero linewidth, then the wavelength is deterministic. In either case, the photon’s wavelength can subsequentlybe used to determine the atmospheric parameters to be applied during the simulation. The following procedureis then applied.

We first compute the probability that the photon would scatter towards the receiver and, without any furtheratmospheric interaction, be detected. A formula for this probability PD,i is given in the figure that accounts forthe photon’s current survival probability, the probability Pα(r, R) that the photon will leave the scattering centerin the direction of the receiver (as determined by the radiant intensity function or scattering phase function), the



Yes

OR

i > Nscat

reception probability:

PD,i ← PsPα(r,R) exp{−kedR(r)}PR(r)

Compute ith-order conditional

Initialization

Photon location r← Tx

Scattering order i← 0

Survival probability Ps ← 1

Draw photon wavelength

Abandon

photon

Draw transmission/scattering angles,

Compute ith scattering location

Update photon direction,

Draw propagation distance,

Update r

Does

photon pass through

receiver?

Increment iUpdate survival probability:

Ps ← (ks/ke)(Ps − PD,i−1)

No

Yes

No

Ps < Pmin

Figure 1. Computation of detection probability of a simulated photon.

probability that the photon will travel the distance dR(r) to the receiver without being scattered or absorbed, andthe receiver sensitivity function PR(r). After incrementing i, we update the survival probability of the photonaccording to the formula in the figure. This formula considers the survival probability at the (i− 1)th scatteringcenter and subtracts off the probability that the photon would have been detected from that scattering centeror absorbed at the ith scattering center. We then determine whether we wish to simulate additional orders ofscattering or if the survival probability has diminished sufficiently that the photon’s subsequent contributionto the received energy is negligible. In either case, the simulation of the photon terminates. Otherwise, wecompute the location of the ith scattering center. Since, as previously discussed, we are only simulating thosepaths in which the photon does not arrive at the receiver, we test for this condition and repeatedly draw newpath segments for the photon until one is obtained that does not pass through the receiver. Given an allowedpath extension, we then repeat the above procedure, starting with the computation of the detection probabilityfrom the current scattering center.

Resulting from the photon simulation are the detection probabilities PD,i for each order of scattering. Foreach i, we average the PD,i from many simulated photons to obtained an estimate of the contribution of ith-orderscattering to the received energy. Summing these estimates over i then yields the overall detection probabilityestimate.

3. RESULTS

3.1 Path-Loss/Range Relationship

In this section, we examine how the path loss of a short-range UV communication system varies with thedistance between the transmitter and receiver, assuming other system parameters, such as pointing angles, areheld constant. (By short-range, we mean distances less than approximately 1/ka–typically on the order of akilometer–so that absorption effects can be neglected.) This relationship has been previously explored in theliterature, but different conclusions have been drawn. For example, an analytical result indicates that pathloss should vary proportionally with range.13 However, experimental results detailed in Ref. 21 suggest a morecomplicated relationship in which path loss is proportional to rν , where r is the range from the transmitter tothe receiver and 1 ≤ ν ≤ 2, depending on the elevation of the transmitter and receiver. Figure 2, reproducedfrom Ref. 21, describes this relationship. (Note that the receiver elevation angle is alternatively called the “Rxapex angle” in this figure.) It has been hypothesized that the single-scattering and small-volume-intersectionapproximations applied in the analytical derivation could be the source of this apparent discrepancy. That theMonte Carlo channel model is not encumbered by these approximations suggests that it could be invaluable ininvestigating this relationship.



Figure 2. Experimentally obtained path loss exponents.21

0 10 20 30 40 50 60 70 80 900.30.50.70.91.11.31.51.71.92.12.32.5

Rx elevation angle (degrees)

Path

loss

exp

onen

t

Tx 20°

Tx 40°

Tx 60°

Tx 80°

Figure 3. Path loss exponents predicted by Monte Carlo channel model with conic transmitter and receiver models.

We begin by using the Monte Carlo channel model to reproduce the experimental conditions reported inRef. 21, to which we refer for details. Note that, based on the experimental description, this includes adoptingthe conic approximation to the transmitter radiant intensity and receiver angular sensitivity. We consider avariety of transmission and receiver elevation angles and use the Monte Carlo channel model to predict thepath loss for a set of ranges. Then, for each transmitter/receiver elevation configuration, we use the proceduredescribed in Ref. 21 to fit the model ξrν to the predicted path loss values. The resulting estimates of the path lossexponent ν are shown in Fig. 3. Clearly, the simulation indicates a linear path loss relationship independent ofthe system geometry, failing to support the hypothesis that a large volume-intersection and/or the considerationof multiple scattering would yield the more complex relationship exhibited by the experimental data.

Noting that line-of-sight (LOS) returns exhibit a quadratic path-loss/range relationship, we propose an alter-native explanation for the experimentally observed behavior that supports the linear NLOS channel hypothesis,namely that the complex experimental behavior was a result of a combination of LOS and NLOS paths. Thatis, if instead of approximating the transmitter radiant intensity and receiver sensitivity functions with step func-tions, we assume that they have small but nonzero values at large offset angles (as has been experimentallyobserved), then the received energy could always contain LOS and NLOS components as the elevation angles areincreased. The ultimate behavior of the path loss as a function of range would then depend on which componentdominates in a particular geometry.

To test the feasibility of our hypothesis, we heuristically chose radiant intensity and sensitivity functions thatresult in path loss values that are similar to those reported in Ref. 1. The chosen functions, shown in Fig. 4, area combination of a Gaussian shaped main lobe and either a cosine or squared-cosine roll-off. Figure 5 depictsthe resulting path loss exponents. The more-general model has clearly captured the quadratic path-loss/rangerelationship at low receiver elevation and the linear relationship at high elevation, suggesting the validity of our



−90 −60 −30 0 30 60 9010

−3

10−2

10−1

100

101

102

Nor

mal

ized

rad

iant

inte

nsity

(Sr

−1 )

Angle from normal (degrees)(a)

−90 −60 −30 0 30 60 90

10−4

10−2

100

Angle of incidence (degrees)(b)

Sens

itivi

ty

Figure 4. Heuristic model for (a) the transmitter and (b) the receiver.

0 10 20 30 40 50 60 70 80 900.30.50.70.91.11.31.51.71.92.12.32.5

Path

loss

exp

onen

t

Rx elevation angle (degrees)

Tx 20°

Tx 40°

Tx 60°

Tx 80°

Figure 5. Path loss exponents predicted by Monte Carlo channel model with heuristic transmitter and receiver models.

hypothesis. Precise measurements of the radiant intensity and sensitivity functions of the experimental systemmight allow for improved accuracy in the channel model, and further study is ongoing.

3.2 Duplex Link Geometry

We now turn our attention to the geometry design of a spatially multiplexed duplex communication system.In this scenario, two NLOS systems are in communication, both simultaneously transmitting and receiving.Focusing on one of these systems, we note that its transmitted energy has the potential to backscatter into itsreceiver, thus washing out returns from the second system. To avoid this possibility, the transmitter and receiverare to be pointed in different directions; provided that the second system appropriately adjusts its pointingdirection, a duplex link can be established. We examine the effect of these pointing angles on the performanceof such a system.

First, we assume that the transmitter and receiver can be modeled by cones with appropriate apex anglesto avoid stray photons from limiting the ability of the system to spatially multiplex the transmission. (Thiscan be enforced in a practical system by the addition of mechanical apertures to the transmitter and receiver.)Next, we assume that the two systems will be configured symmetrically, thus achieving identical performance.We employ on-off keying and maximum-likelihood receivers for both systems. As such, bit error rates (BERs)can be computed as an appropriate sum over Poisson probability distributions; we refer to Ref. 21 for details.

Given a particular system geometry, the Monte Carlo channel model could be applied in a straightforwardmanner to determine the probability that a receiver detects a desired photon from the second system or aninterfering photon from its own transmitter. An estimated BER can then be computed and the geometryoptimized. However, because we are interested in configurations where little backscattered energy is received,this approach could be computationally intensive, requiring many simulated photons to obtain a reasonableestimate. Noting that the probability of receiving a backscattered photon depends only on the angle between



0 30 60 90 120 150 18010

−13

10−11

10−9

10−7

10−5

Tx−Rx offset angle (degrees)

Prob

abili

ty o

f ph

oton

det

ectio

n

Figure 6. Backscattered photon detection probability.

−100

10−10

0

10

20

30

40

−5

0

5

(a)

−10

0

10 −10

0

10

20

30

40

0

5

10

(b)

−10

0

10

−100

1020

30400

5

10

15

(c)

Figure 7. Optimized duplex geometries: (a) unconstrained, (b) constrained, and (c) a local minimum.

a system’s transmitter and receiver pointing directions, we precompute a set of these probabilities and fit aparametric model. In particular, based on theoretical and empirical considerations, we choose the parametricmodel for the detection probability PD(θ) in dB, where θ, 0 ≤ θ ≤ π, is the angle between the transmitter andreceiver pointing directions, to have the form

PD(θ) =

{a2θ

2+a0

{[(βT+βR)/2]2−θ2}ae + b6(x− π)6 + b0, if θ ≤ 12 (βT + βR)

b6(x − π)6 + b0, otherwise,(3)

where a0, a2, ae, b0, and b6 are model parameters selected to minimize the mean squared regression error. Theresult of this procedure is depicted in Fig. 6. Note the steep drop as the transmitter and receiver cones transitionfrom overlapping to nonintersecting (i.e., from configurations where single-scattering returns are possible tothose where only multiple-scattering energy is collected.) Also, there is relatively little variation as the conesare directed further away from each other. This suggests the intuitive notion that the key to configuring aduplex NLOS UV link is ensuring that the geometry does not admit a single-scattering backscatter path, andthat further precise orientation of the devices will have a much smaller (though possibly significant) effect onthe interference.

Now, using the Monte Carlo algorithm to determine the probability of detecting the desired transmittedphotons and Fig. 6 to determine the probability of detecting interfering photons, we can compute the BERprobability for any given system geometry. Applying Powell’s optimization algorithm20 to minimize the BERover the pointing angles yielded the system geometries depicted in Fig. 7. Each image depicts the transmitterand receiver cones for a particular optimization; the transmitter cones have smaller apex angles (17◦) than the



receiver cones (30◦).

Figure 7(a) shows the result of an unconstrained optimization, where the transmitter and receiver from onesystem appear to point as closely as possible to the second system but with the previously discussed requirementthat a system’s own transmitter and receiver cones do not overlap. Noting that this configuration requires(near) LOS, we next constrain the elevation angle of the transmitter to be 30◦. The result of the constrainedoptimization is shown in Fig. 7(b), where similar conclusions can be drawn. Finally, further investigation revealeda local BER minimum in the configuration depicted in Fig. 7(c). Although the BER is significantly worse in thisconfiguration than in the prior two, it is worth noting that this configuration does not suffer from the possibilityof small perturbations causing a system’s transmitter and receiver cone to overlap, a condition that would leadto severe degradation in performance.

4. CONCLUSIONS

We have demonstrated the application of the Monte Carlo framework to UVmultiple-scattering channel modeling.After providing an overview of some of the probabilistic modeling and algorithmic details associated with thisapproach, we investigated its use in two applications. The first scenario demonstrated the flexibility of theMonte Carlo model with regards to system characteristics such as the transmitter beam pattern and the receiversensitivity function. Meanwhile, the second scenario involved a duplex link in which multiple-scattering returnscan be the dominant noise source, thus requiring a multiple-scattering channel model for analysis. In both cases,the Monte Carlo framework provided an intuitive method to investigate and predict system performance, andwe believe that future research will increasingly exploit the power of this model.

ACKNOWLEDGMENTS

This research was supported in part by an appointment to the U.S. Army Research Laboratory PostdoctoralFellowship Program administered by the Oak Ridge Associated Universities through a contract with the U.S.Army Research Laboratory.

REFERENCES

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[2] Harvey, G. L., “A survey of ultraviolet communication systems,” Tech. Rep. NRL-6037, U.S. Naval ResearchLaboratory, Washington, D.C. (1964).

[3] Reilly, D. M., “Atmospheric optical communications in the middle ultraviolet,” M.S. thesis, MassachusettsInstitute of Technology, Cambridge, MA (1976).

[4] Fishburne, E. S., Neer, M. E., and Sandri, G., “Voice communication via scattered ultraviolet radiation,”Tech. Rep. 274, Aeronautical Research Associates of Princeton, Inc., Princeton, NJ (1976).

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[7] Wang, L., Li, Y., Xu, Z., and Sadler, B. M., “Wireless ultraviolet network models and performance innoncoplanar geometry,” in [IEEE Globecom Workshop on Optical Wireless Communications ], 1037–1041(2010).

[8] He, Q., Sadler, B. M., and Xu, Z., “Modulation and coding tradeoffs for non-line-of-sight ultraviolet com-munications,” in [Proceeding of SPIE ], 7464, 74640H (2009).

[9] Shaw, G. A., Siegel, A. M., and Model, J., “Extending the range and performance of non-line-of-sightultraviolet communication links,” in [Proceedings of SPIE ], 6231, 93–104 (2006).

[10] Moriarty, D. and Hombs, B., “System design of tactical communications with solar blind ultraviolet nonline-of-sight systems,” in [IEEE Military Communications Conference ], (2009).

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[12] Luettgen, M. R. and Shapiro, J. H., “Non-line-of-sight single-scatter propagation model,” Journal of theOptical Society of America A 8, 1964–1972 (December 1991).

[13] Xu, Z., “Approximate performance analysis of wireless ultraviolet links,” in [IEEE International Conferenceon Acoustics, Speech, and Signal Processing ], 577–580 (2007).

[14] Zachor, A. S., “Aureole radiance field about a source in a scattering-absorbing medium,” Applied Optics 17,1911–1922 (June 1978).

[15] Ross, W. S. and Kennedy, R. S., “An investigation of atmospheric optically scattered non-line-of-sightcommunication links,” Tech. Rep. ARO-15365.2-A.EL, U.S. Army Research Office, Research Triangle Park,NJ (1980).

[16] Junge, D. M., “Non-line-of-sight electro-optic laser communications in the middle ultraviolet,” M.S. thesis,Naval Postgraduate School, Monterey, CA (1977).

[17] Ding, H., Chen, G., Majumdar, A. K., Sadler, B. M., and Xu, Z., “Modeling of non-line-of-sight ultravioletscattering channels for communication,” IEEE Journal on Selected Areas in Communications 27, 1535–1544(December 2009).

[18] Drost, R. J., Moore, T. J., and Sadler, B. M., “Ultraviolet communications channel modeling incorporatingmultiple scattering interactions,” Journal of the Optical Society of America A, to appear (2011).

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[20] Mathews, J. H. and Fink, K. D., [Numerical methods using MATLAB ], Prentice Hall, Upper Saddle River,NJ, 4th ed. (2004).

[21] Chen, G., Xu, Z., Ding, H., and Sadler, B. M., “Path loss modeling and performance trade-off study forshort-range non-line-of-sight ultraviolet communications,” Optics Express 17, 3929–3940 (March 2009).



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SPIE Proceedings [SPIE SPIE Defense, Security, and Sensing - Orlando, Florida (Monday 25 April 2011)] Atmospheric Propagation VIII - Monte-Carlo-based multiple-scattering channel modeling