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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13480–13496
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Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels

Antonio García-Zambrana, Carmen Castillo-Vázquez, and Beatriz Castillo-Vázquez  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13480-13496 (2011)
http://dx.doi.org/10.1364/OE.19.013480


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Abstract

Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the performance over free-space optical (FSO) links. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. In this paper, the outage probability as a performance measure for multiple-input/multiple-output (MIMO) FSO communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels with pointing errors is analyzed. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results show that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver. Simulation results are further demonstrated to confirm the analytical results. Additionally, since proper FSO transmission requires transmitters with accurate control of their beamwidth, asymptotic expressions here obtained for different diversity techniques are used to find the optimum beamwidth that minimizes the outage performance.

© 2011 OSA

1. Introduction

2. System and channel model

3. Outage performance analysis

In this section, the outage probability as a performance measure is evaluated for different MIMO FSO configurations over strong atmospheric turbulence channels with pointing errors. Together with the average error rate, outage probability, P out, is another standard performance criterion of communications systems operating over fading channels [26

26. M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

]. It is defined as the probability that the instantaneous combined SNR, γT, falls below a certain specified threshold, γth, which represents a protection value of the SNR above which the quality of the channel is satisfactory, i.e.,
Pout=Prob(γTγth)=0γthfγT(γT)dγT=FγT(γth)
(5)
where fγT (γT) and FγT (γT) are the PDF and the cumulative distribution function (CDF) of γT, respectively. Next, the outage performance of a SISO FSO system is firstly evaluated in order to be considered as a benchmark in the analysis of the remaining MIMO configurations, corresponding to a multiple-input/single-output (MISO) system with a L × 1 array, a single-input/multiple-output (SIMO) system with a 1 × M array and, finally, a MIMO FSO system with a L × M array.

3.1. SISO FSO system

In this subsection, the outage probability for the FSO system model previously presented with L = M = 1 is evaluated. According to Eq. (1) and the OOK signaling described in the appendix, a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of d, the statistical channel model corresponding to the SISO configuration can be written as
Y=XI+Z,X{0,d},ZN(0,N0/2).
(6)
The received electrical SNR can be defined, as in [16

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

], as
γTSISO(i)=12d2N0/2i2=4Popt2TbξN0i2=γξi2
(7)
where γ represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats, Tb parameter is the bit period, P opt is the average optical power transmitted and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in the appendix. Using Eq. (5), the outage probability for a SISO FSO system can be written as
Pout=Prob(γξi2γth)=Prob(i2γthξγ)=FI(1ξγthγ)
(8)
where the CDF of I, FI (i), can be easily derived from Eq. (4) as
FI(i)=1(iA0)φ2φ2Γ(φ2,iA0),i0
(9)
by applying the differential relation in [25

25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

, eqn. (6.5.26)] and the fact that the series expansion corresponding to the upper incomplete Gamma function can be simplified by Γ(a,z)Γ(a)(za/a)(1az1+a+O(z2)) [27

27. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

, eqn. (8.354.2)], where Γ(·) is the well-known Gamma function. The results corresponding to this FSO scenario are illustrated in Fig. 1, where rectangular pulse shapes with ξ = 1 are used, assuming different values of normalized beamwidth, ωz/r, and normalized jitter, σs/r (i.e. ωz/r = 5 with σs/r = {1,3,4,5} and ωz/r = 10 with σs/r = {1,4,7,11}). Outage simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (8) for different misalignment fading conditions.

Fig. 1 Probability of outage versus normalized average SNR in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming different values of normalized beamwidth, ωz/r, and normalized jitter, σs/r.

Additionally, we can take advantage of this series expansion in order to quantify the outage probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. Therefore, for large enough SNR (γ → ∞), the asymptotic outage performance can be expressed after some algebraic manipulations as
Pout(φ4(1φ2)2A02ξγthγ)1/2+(Γ(1φ2)2φ2A02ξγthγ)φ2/2.
(10)
Here, an even greater simplification can be deduced, leading to different asympotic expressions depending on the value of φ, as follows
Pout(φ4(1φ2)2A02ξγthγ)1/2,φ>1
(11a)
Pout(Γ(1φ2)2/φ2A02ξγthγ)φ2/2,φ<1
(11b)
Results corresponding to this asymptotic analysis are also illustrated in the Fig. 1 for different cases in Eq. (11) . It is straightforward to show that the outage probability behaves asymptotically as (Oc(γ/γth))Od, where Od and Oc denote outage diversity and coding gain, respectively [26

26. M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

, 28

28. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

]. At high SNR, if asymptotically the outage probability behaves as (Oc(γ/γth))Od, the outage diversity Od determines the slope of the outage performance versus average SNR curve in a log-log scale and the coding gain Oc (in decibels) determines the shift of the curve in SNR. Additionally, as previously reported by the authors [12

12. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009). [CrossRef]

, 13

13. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010). [CrossRef] [PubMed]

, 23

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24), 25422–25440 (2010). [CrossRef] [PubMed]

], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10 log10 ξ decibels.

At this point, it can be convenient to compare with the outage performance obtained in a similar context when misalignment fading is not present. In this case, knowing that the CDF corresponding to the negative exponential distribution in Eq. (2) is given by F I(a) (i) = 1 − exp(−i) and the fact that the series expansion for the exponential function can be simplified by exp(z) ∝ 1 + z + O(z 2), an asymptotic expression can be easily derived as Poutexp(γth/γξ))1/2. From these results, it can be deduced that φ is a key parameter in order to optimize the outage diversity. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1. Once this condition is satisfied, taking into account the coding gain in Eq. (11a), the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to strong atmospheric turbulence without misalignment fading given by
D[dB]20log10(φ2A0(φ21)).
(12)
According to this expression, it can be observed in Fig. 1 that coding gain disadvantages of 23.7, 34.4 and 42.7 decibels are achieved for values of (ωz/r,σs/r) = (5, 1), (ωz/r,σs/r) = (10,1) and (ωz/r,σs/r) = (10,4), respectively.

3.2. MISO FSO system with a L ×1 array

In this subsection, the outage probability for the FSO system model previously presented where the information signal is transmitted via L apertures and received by only one aperture (i.e. M = 1) is evaluated. Here, two different transmit diversity strategies are considered: repetition coding (RC) and transmit laser selection (TLS).

3.2.1. Transmit diversity using repetition coding

Fig. 2 Probability of outage versus normalized average SNR in MISO and SIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz/r = 5 and values of normalized jitter of (a) σs/r = 1 and (b) σs/r = 4.

3.2.2. Transmit diversity using laser selection

When channel side information is not only available at the receiver but also on the transmitter, an alternative transmit diversity scheme can be adopted. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [12

12. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009). [CrossRef]

], our MISO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the TLS scheme, Imax, can be written as Imax = maxl =1,2,… L Il. In this way, having a SISO system as a reference, the statistical channel model corresponding to this MISO configuration can be written as
Y=XImax+Z,X{0,d},ZN(0,N0/2)
(19)
where, in order to maintain the average optical power transmitted at the same constant level P out, the division by L is not included in Eq. (19) because of not more than one laser is simultaneously used. According to order statistics [30

30. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003). [CrossRef]

], the CDF, FImax (i), of the resulting channel irradiance, Imax, is given by FImax (i) = [FI (i)]L and, hence, the outage probability can be expressed as
Pout=FImax(1ξγthγ)=(FI(1ξγthγ))L
(20)
In the same way, its corresponding asympotic expressions can be easily derived from Eq. (11) as
Pout(φ4(1φ2)2A02ξγthγ)L/2,φ>1
(21a)
Pout(Γ(1φ2)2/φ2A02ξγthγ)Lφ2/2,φ<1
(21b)
Results for the exact outage probability in Eq. (20) together with the corresponding asymptotic expressions in Eq. (21) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (20). Comparing both transmit diversity techniques and their corresponding asymptotic expressions in Eq. (18) and Eq. (21) , respectively, it can be stated that the superiority of the TLS scheme translates into a coding gain advantage, ATLS[dB], relative to the RC scheme given by
ATLS[dB]10log10(L2Γ(L+1)2/L),φ>1
(22a)
ATLS[dB]10log10(L2Γ(1+φ2)2/φ2Γ(1+Lφ2)2/Lφ2),φ<1
(22b)
As observed in Fig. 2, it can be deduced from expressions in Eq. (22) that the greater the number of transmit apertures, the better performance in terms of coding gain advantage. Additionally, this superiority is even more significant when φ < 1 and the value of φ is decreasing.

3.3. SIMO FSO system with a 1 × M array

In this subsection, the outage probability for the FSO system model previously presented is evaluated when the information signal is transmitted via only one aperture (i.e. L = 1) and received by M apertures. Here, two different receive diversity strategies are considered: equal gain combining (EGC) and selection combining (SC).

3.3.1. Receive diversity using equal gain combining

3.3.2. Receive diversity using selection combining

Fig. 3 Coding gain advantage for the SC scheme relative to the EGC scheme, assuming a normalized beamwidth of ωz/r = 10 and values of normalized jitter of 6, 7, 8 and 9, corresponding to values of φ of 0.83, 0.71, 0.62 and 0.55, respectively.

3.4. MIMO FSO system with a L ×M array

Fig. 4 Probability of outage versus normalized average SNR in MIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 and a normalized beamwidth of ωz/r = 10 with a normalized jitter of σs/r = 7, corresponding to values of φ of 2.55 and 0.71, respectively. Additionally, results assuming the optimum beamwidth corresponding to values of jitter of σs/r = 1 and σs/r = 7 are also included for L = 4 and M = 2.

4. Outage optimization

From obtained results, it can be concluded that the outage diversity is independent of the pointing error effects when the equivalent beam radius value at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1, being φ a main parameter to consider in order to optimize the outage performance. Once this condition is satisfied, comparing different diversity techniques and their corresponding asymptotic expressions, it can be noted that better performance is achieved when increasing the number of transmit apertures, i.e. L, instead of the number of receive apertures, i.e. M, in order to guarantee a same diversity order, as shown in Fig. 4 when assuming a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 and a diversity order of 8 with values of {L,M} = {4,2} and {L,M} = {2,4}. Taking into account expressions in Eq. (35a) and Eq. (21a), this can be quantified as a coding gain disadvantage, DMIMO[dB], for the MIMO FSO system relative to the MISO FSO system using laser selection as follows
DMIMO[dB]20log10(MΓ(Od/M+1)M/OdΓ(Od+1)1/Od).
(36)
where Od and M denote outage diversity and the number of receive apertures, respectively. In agreement with this expression, it can be seen in Fig. 4 that a difference of 2.13 decibels is achieved for a diversity order of 8 and values of M = 2 and M = 4 when a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 is assumed. Lastly, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth, ωz/r, that gives the minimum outage [16

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

, 17

17. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008). [CrossRef]

, 31

31. H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008). [CrossRef]

]. From Eq. (35a), it can be observed that this is equivalent to minimize the expression in Eq. (12) corresponding to the SISO configuration, being independent of the MIMO configuration adopted. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter, σs/r [32

32. Wolfram Research Inc., Mathematica, version 8.0.1. ed. (Wolfram Research, Inc., Champaign, Illinois, 2011).

]. Numerical results for the optimum beamwidth are illustrated in Fig. 5 for a range of values in the normalized jitter from σs/r = 1 to σs/r = 10 in discrete steps of 0.5 when the stochastic function minimizer simulated annealing is used. From this figure, it can be deduced that the outage optimization provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows
ωz/roptimum2.85(σs/r1)+2.6
(37)
considering the values of optimum beamwidth corresponding to normalized jitters of σs/r = 1 and σs/r = 10, respectively. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is shown in Fig. 4, where results assuming the optimum beamwidth corresponding to values of jitter of σs/r = 1 and σs/r = 7 are also included for L = 4 and M = 2. As expected, a greater improvement in performance is achieved when σs/r = 7 compared to the improvement in performance corresponding to the normalized jitter of σs/r = 1 since not only coding gain but also diversity gain is obtained, making outage diversity performance independent of the impact of misalignment fading.

Fig. 5 Optimum normalized beamwidth versus normalized jitter, σs/r in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors.

5. Conclusions

Appendix

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12

12. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009). [CrossRef]

, 23

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24), 25422–25440 (2010). [CrossRef] [PubMed]

]. A new basis function ϕ (t) is defined as ϕ(t)=g(t)/Eg where g(t) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ g(t) ≤ 1 in the bit period and 0 otherwise, and Eg=g2(t)dt is the electrical energy. In this way, an expression for the optical intensity can be written as
x(t)=k=ak2TbPoptG(f=0)g(tkTb)
(38)
where G(f = 0) represents the Fourier transform of g(t) evaluated at frequency f = 0, i.e. the area of the employed pulse shape, and Tb parameter is the bit period. The random variable (RV) ak follows a Bernoulli distribution with parameter 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is P opt, defining a constellation of two equiprobable points (x 0 = 0 and x 1 = d) in a one-dimensional space with an Euclidean distance of d=2PoptTbξ where ξ = Tb Eg/G 2(f = 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse.

Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper. The authors are grateful for financial support from the Junta de Andalucía (research group “ Communications Engineering ( TIC-0102)”).

References and links

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2.

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W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009). [CrossRef] [PubMed]

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L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

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X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). [CrossRef]

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13.

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010). [CrossRef] [PubMed]

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S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003). [CrossRef]

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A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

17.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008). [CrossRef]

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H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011). [CrossRef]

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A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

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M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001). [CrossRef]

23.

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24), 25422–25440 (2010). [CrossRef] [PubMed]

24.

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009). [CrossRef]

25.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

26.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

27.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

28.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

29.

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007). [CrossRef]

30.

H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003). [CrossRef]

31.

H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008). [CrossRef]

32.

Wolfram Research Inc., Mathematica, version 8.0.1. ed. (Wolfram Research, Inc., Champaign, Illinois, 2011).

OCIS Codes
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.4510) Fiber optics and optical communications : Optical communications
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: April 19, 2011
Revised Manuscript: June 6, 2011
Manuscript Accepted: June 10, 2011
Published: June 28, 2011

Citation
Antonio García-Zambrana, Carmen Castillo-Vázquez, and Beatriz Castillo-Vázquez, "Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels," Opt. Express 19, 13480-13496 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13480


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References

  1. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997). [CrossRef]
  2. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009). [CrossRef]
  3. W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009). [CrossRef] [PubMed]
  4. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]
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