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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 18238–18245
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Asymptotic BER analysis of FSO with multiple receive apertures over -distributed turbulence channels with pointing errors

Liang Yang, Mazen Omar Hasna, and Xiqi Gao  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 18238-18245 (2014)
http://dx.doi.org/10.1364/OE.22.018238


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Abstract

In this paper, we consider a free-space optical (FSO) communication with multiple receive apertures over -distributed turbulence channels with pointing errors. In particular, we consider two different combining schemes at the receiver: optimal combining (OC) and selection combining (SC). With these setups, the statistic characters of the instantaneous electrical signal-to-noise ratio (SNR) are derived. Then, using the cumulative density function (CDF)-based method, we analyze the asymptotic bit-error rate (BER) performance. The derived results help quantifying the diversity order of our considered systems.

© 2014 Optical Society of America

1. Introduction

Recently, free-space optical (FSO) communication systems have received considerable attention due to their high capacity ability. Operating over the unlicensed optical spectrum, FSO techniques can provide high speed line-of-sight wireless transmission with low cost and good security [1

1. G. Yang, M. A. Khalighi, S. Bourennane, and Z. Ghassemlooy, “Fading correlation and analytical performance evaluation of the space-diversity free-space optical communications system,” J. Opt. 16, 035403 (2014). [CrossRef]

3

3. S. Zvanovec, J. Perez, Z. Ghassemlooy, S. Rajbhandari, and J. Libich, “Route diversity analyses for free-space optical wireless links within turbulent scenarios,” Opt. Express 21(6), 7641–7650 (2013). [CrossRef] [PubMed]

].

Over the years, there have been many models used to describe the atmospheric turbulence for different degrees of strength, such as log-normal distribution, K distribution, gamma-gamma distribution and generalized -distribution [13

13. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, and A. P.- Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engneering Processed, J. Awrejcewicz, ed. (Intech, 2011), ch.8. [CrossRef]

].

2. System and channel models

A. System model

Consider a MIMO FSO system with one transmit and M receive apertures as shown in Fig. 1. We assume that the signals are transmitted through the turbulence-induced fading channels and are corrupted at the receiver by the additive white Gaussian noise (AWGN). Assuming a SIM modulation, the transmitted optical power at the laser transmitter can be written as [21

21. X. Song and J. Cheng, “Optical communication using subcarrier intensity modulation in strong atmospheric turbulence,” J. Lightwave Technol. 30(22), 3484–3492 (2012). [CrossRef]

]
P(t)=PS(1+ηSx(t))
(1)
where PS is the total transmit optical power at the transmitter, x(t) is the transmitted RF base-band signal with unit energy and which is used to modulate the irradiance of a continuous optical wave beam, and ηS is the electrical-to-optical conversion coefficient. Therefore, the received signal at the m receive aperture is given by [21

21. X. Song and J. Cheng, “Optical communication using subcarrier intensity modulation in strong atmospheric turbulence,” J. Lightwave Technol. 30(22), 3484–3492 (2012). [CrossRef]

]
ym=RA(1+ηSx(t))Im+nm
(2)
where nm is the AWGN at the mth receive aperture, R is the photodetector responsivity, A′ is the photodetector area, and Im is the normalized channel fading coefficient between the mth receive aperture and the transmit aperture and represents the intensity fluctuations due to the joint effects of atmospheric turbulence and misalignment. As shown in [17

17. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

], the channel state is assumed to be a product of two random components, i.e., Im=ImaImp, where Ima is the attenuation due to atmospheric turbulence (modeled as a -distributed random variable) and Imp is the attenuation due to pointing errors.

Fig. 1 System model of a FSO system with multiple receive apertures.

At the receiver, a photodetector converts its received optical signal into an electrical signal. Then, an electrical demodulator can be used to recover the transmitted data. If optimal combining is used, the resulting instantaneous SNR at the input of the electrical demodulator can be expressed as
γOC=γ¯m=1MIm2
(3)
where γ̄ is defined as the average received SNR. If we apply the SC scheme, the selection policy is m* = arg max Im and the corresponding SNR is γSC=γ¯Im*2.

B. Channel statistical model

When taking the combined effects of turbulence-induced scintillation and misalignment-induced fading into consideration, the combined unconditional probability density function (PDF) for generalized atmospheric optical channels has been derived in [17

17. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

]. However, this PDF involves the Meijer G-function and makes the closed-form analysis untractable. Therefore, we intend to rewrite the PDF expression in terms of the generalized power series representation.

As derived in [17

17. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

], the combined PDF of the irradiance Im is given by
fIm(I)=g2A2I1k=1βak(αβμβ+Ω)α+k2G1,33,0(αβμβ+ΩIA0|g2,α,kg2+1)
(4)
where, μ=E[|USC|2]=2b0(1ρ), α is a positive parameter depending on the effective number of large-scale cells of the scattering process, β denotes the amount of fading parameter and is a natural number, while Ω=Ω+2ρb0+22ρb0Ωcos(φAφB) represents the average power coming from the coherent contribution. Notice that the parameter ρ (0 ≤ ρ ≤ 1) denotes the amount of scattering power coupled to the line of sight (LOS) component, the parameters φA and φB are the deterministic phases of the LOS and the coupled-to-LOS component. In Eq. (4), Gp,qm,n() is the Meijer G-function, and the parameters A and ak are defined as
A=2αα2μ1+α2Γ(α)(μβμβ+Ω)β+α2
(5)
ak=(β1k1)(μβ+Ω)10.5k(k1)!(Ωμ)k1(αβ)0.5k
(6)
Furthermore, in Eq. (4), g = wzeq/(2σs) is the ratio between the equivalent beam radius and the pointing error displacement standard deviation at the receiver σs, and A0 is the fraction of the collected power at r = 0 (r radial distance) and the parameter wzeq can be calculated by using the relations, v=(πa)/(2wz) and wzeq=wz2πerf(v)/(2vev2, where wz is the beam waist at distance z, a is the radius of a circular detection aperture, and erf() is the error function (See [16

16. H. Samimi and M. Uysal, “End-to-end performance of mixed RF/FSO transmission systems,” J. Opt. Commun. Netw. 5(11), 1139–1144 (2013). [CrossRef]

,17

17. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

] for more details). As discussed in [13

13. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, and A. P.- Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engneering Processed, J. Awrejcewicz, ed. (Intech, 2011), ch.8. [CrossRef]

], the gamma-gamma and K distributions are special cases of the -distribution model, for example, (ρ = 1, Ω′ = 1), (ρ = 0, Ω = 0 or β = 1), and (ρ = 0, Ω = 0 or α → ∞) corresponding to gamma-gamma, K distribution, and negative exponential model, respectively.

From [22

22. “The Wolfram Functions Site.”, [Online]. Available: http://functions.wolfram.com.

, Eq. (07.34.06.0002.01)], the Meijer G-function can be expressed in terms of the generalized power series representation as
Gp,qm,n(z|b1,,bm,bqa1,,an,,ap)=t=1mΠj=1,jtmΓ(bjbt)Πj=n+1pΓ(ajbt)zbti=0Πj=n+1p(1aj+bt)iΠj=1nΓ(1aj+bt+i)Πj=1m(1bj+bt)iΠj=m+1qΓ(1bj+bj+i)((1)pmnz)i
(7)
where q > p, and (a)n is the Pochhamer number defined as (a)n = a(a + 1)· · · (a + n−1), with (a)0 = 1. Using Eq. (7), we can rewrite Eq. (4) as
fIm(I)=g2A2k=1βakt=13Πj=1,jt3Γ(bjbt)Γ(g2+1bt)i=0(btg2)iδbt+iα+k2Πj=13(1bj+bt)iA0bt+iIbt+i1
(8)
where δ = αβ/(μβ + Ω′), b1 = g2, b2 = α, and b3 = k. From Eq. (8), the PDF of Im2 can be obtained as
fIm2(I)=12I12fIm(I)
(9)

3. Performance analysis

In this section, we analyze the BER performance of the above two FSO systems over generalized turbulence fading channels with pointing errors. More specifically, assuming BPSK modulation, we use the CDF-based method to evaluate the BER performance [23

23. Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-and-forward relay systems,” IEEE Commun. Lett. 10(11), 757–759 (2006). [CrossRef]

], namely,
Pe=EX[F(X22)]
(10)
where X is a standard normal-distributed random variable and F(·) is the CDF expression.

A. BER analysis for OC

From Eq. (10), to compute the BER, we need to find the CDF of γOC. Combining Eq. (8) and Eq. (9), we express the moment generating function (MGF) of Im2 as
MIm2(s)=g2A4k=1βakt=13Πj=1,jt3Γ(bjbt)Γ(g2+1bt)i=0(btg2)iδbt+iα+k2Πj=13(1bj+bt)iA0bt+iΓ(bt+i2)sbt+i2
(11)
Let Y=Σm=1MIm2. Since Y is the sum of M independent random variables, the MGF of Y is readily given by
MY(s)=[MIm2(s)]M
(12)
If we use Eq. (11), we note that the expansion in Eq. (12) will be very complicated. Therefore, in this paper, we focus on the asymptotic analysis. Let i = 0, Eq. (11) simplifies to
MIm2(s)g2A4[A(1,g2)sg22+A(2,α)sα2+A(3,k)sk2]
(13)
where the constants A(1, g2), A(2, α), and A(3, k) are given by
A(t,x)=k=1βakΠj=1,jt3Γ(bjx)Γ(g2+1x)δxα+k2A0xΓ(x2)

By substituting Eq. (13) into Eq. (12), the asymptotic MGF of Y can be expressed as
MY(s)(g2A)M4Mr=0M(Mr)t=0Mr(A(1,g2))Mrt(A(2,α))tv=rrβλvs(Mrt)g2+tα+v2
(14)
where λv is the coefficient of sv2 in the expansion of (A(3,k)sk2)r. Using the inverse Laplace transform, we can obtain the CDF of γOC as
FγOC(γ)(g2A)M4Mr=0M(Mr)t=0Mr(A(1,g2))Mrt(A(2,α))tv=rrβλvΓ(Mrt)g2+tα+v2+1)(γγ¯)(Mrt)g2+tα+v2
(15)
Substituting Eq. (15) into Eq. (10) yields
PeOC(g2A)M22M+1πr=0M(Mr)t=0Mr(A(1,g2))Mrt(A(2,α))tv=rrβλvΓ((Mrt)g2+tα+v+12)Γ((Mrt)g2+tα+v2+1)γ¯(Mrt)g2+tα+v2
(16)
For large SNR, the diversity order is determined by the smallest exponent of γ̄. Therefore, we can conclude that the diversity order of a FSO system with multiple receive apertures over generalized fading channels with pointing errors and OC is M(min{g22,α2,β2}).

B. BER analysis for SC

From Eq. (13), the CDF of γm=γ¯Im2 can be obtained as
Fγm(γ)g2A4[A(1,g2)Γ(g22+1)(γγ¯)g22+A(2,α)Γ(α2+1)(γγ¯)α2+A(3,k)Γ(k2+1)(γγ¯)k2]
(17)
For independent and identically distributed γm, the CDF of γSC is given by
FγSC(γ)=[Fγm(γ)]M(g2A)M4Mr=0M(Mr)t=0Mr(A(1,g2)Γ(g22+1))Mrt(A(2,α)Γ(α2+1))tv=rrβχv(γγ¯)(Mrt)g2+tα+v2
(18)
where χv is the coefficient of (γγ¯)v2 in the expansion of (A(k)Γ(k2+1)(γγ¯)k2)r. Then, combining Eq. (10) and Eq. (18), we have
PeSC(γ)(g2A)M22M+1πr=0M(Mr)t=0Mr(A(1,g2)Γ(g22+1))Mrt(A(2,α)Γ(α2+1))tv=rrβχvΓ((Mrt)g2+tα+v+12)γ¯(Mrt)g2+tα+v2
(19)
We can see that the diversity order for SC scheme is also M(min{g22,α2,β2}).

4. Numerical results

In this section, we illustrate our analysis with some numerical results for the channel model under consideration. For the FSO networks, we note that the current references seldom present the simulation results. In this work, we provide the simulation results to verify our analysis. For the BER performance, we assume that BPSK is used.

The BER performance for different values of g, α, and β is plotted in Fig. 2, where we set M = 3. It should be noted that α = 3.6, β = 1, and ρ = 0, and α = 4.2, β = 2, and ρ = 1 correspond to the K and gamma-gamma distributions, respectively. It is clearly shown that the asymptotic results approach the exact values at high SNR regions. As expected, we can observe that increasing the channel parameters α, β, and g can improve the system performance. Furthermore, we can see that the optimal combining has a better performance than the selection scheme. However, for given α, β, and g, OC and SC have the same diversity order since the slopes of BER curves for SC and OC are the same. In Fig. 3, we plot the curves for different pairs of M and g for α = 4.2, β = 2, and ρ = 1. It is clearly shown that increasing M can improve the system performance. Also, the increasing of g can improve the BER performance significantly because g → ∞ means no pointing errors.

Fig. 2 BER of a FSO network with multiple receive apertures over -distributed channels with pointing errors. M = 3
Fig. 3 Simulation of BER of a FSO network with multiple receive apertures over -distributed channels with pointing errors. α = 4.2, β = 2, and ρ = 1.

5. Conclusions

In this letter, we analyzed the asymptotic BER performance of a FSO system with multiple receive apertures over -distributed channels with pointing errors. Results show that the diversity order for both OC and SC are Mmin{g22,α2,β2}

Acknowledgments

The authors would like to acknowledge the support of Ooredoo under the project QUEX-Qtel-09/10-10, the National Natural Science Foundation of China (NSFC) under Grants 61372096 and 61320106003, and the Open Research Fund of the State Key Laboratory of Integrated Services Networks ( ISN15-07).

References and links

1.

G. Yang, M. A. Khalighi, S. Bourennane, and Z. Ghassemlooy, “Fading correlation and analytical performance evaluation of the space-diversity free-space optical communications system,” J. Opt. 16, 035403 (2014). [CrossRef]

2.

J. Perez, S. Zvanovec, Z. Ghassemlooy, and W. Popoola, “Experimental characterization and mitigation of turbulence induced signal fades within an ad hoc FSO network,” Opt. Express 22(3), 3208–3218 (2014). [CrossRef] [PubMed]

3.

S. Zvanovec, J. Perez, Z. Ghassemlooy, S. Rajbhandari, and J. Libich, “Route diversity analyses for free-space optical wireless links within turbulent scenarios,” Opt. Express 21(6), 7641–7650 (2013). [CrossRef] [PubMed]

4.

S. M. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun. 21(8), 1346–1357 (2003). [CrossRef]

5.

E. J. Shin and V. W. S. Chan, “Optical communication over the turbulent atmospheric channel using spatial diversity,” IEEE Globecom, 2055–2060 (2002).

6.

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]

7.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009). [CrossRef]

8.

M. Safari and M. Uysal, “Do we really need OSTBCS for free-space optical communication with direct detection? ” IEEE Trans. Wireless Commun. 7(11), 4445–4448 (2008). [CrossRef]

9.

N. D. Chatzidiamantis, M. Uysal, T. A. Tsiftsis, and G. K. Karagiannidis, “Iterative near maximum-likelihood sequence detection for MIMO optical wireless systems,” J. Lightwave Technol. 28(7), 1064–1070 (2010). [CrossRef]

10.

S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010). [CrossRef]

11.

H. Moradi, H. H. Refai, and P. G. LoPresti, “Circular MIMO FSO nodes with transmit selection and receive generalized selection diversity,” IEEE Trans. Veh. Technol. 61(3), 1174–1181 (2012). [CrossRef]

12.

M. Liu, J. Cheng, and J. F. Holzman, “Error rate performance comparison of coherent and subcarrier intensity modulated optical wireless communications,” J. Opt. Commun. Netw. 5(6), 554–564 (2013). [CrossRef]

13.

A. J.- Navas, J. M. G.- Balsells, J. F. Paris, and A. P.- Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engneering Processed, J. Awrejcewicz, ed. (Intech, 2011), ch.8. [CrossRef]

14.

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]

15.

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]

16.

H. Samimi and M. Uysal, “End-to-end performance of mixed RF/FSO transmission systems,” J. Opt. Commun. Netw. 5(11), 1139–1144 (2013). [CrossRef]

17.

A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

18.

I. E. Lee, Z. Ghassemlooy, W. P. Ng, and Mo.-A. Khalighi, “Joint optimization of a partially coherent Gaussian beam for free-space optical communication over turbulent channels with pointing errors,” Opt. Lett. 38(3), 350–352 (2013). [CrossRef] [PubMed]

19.

H. Samimi, “Optical communication using subcarrier intensity modulation through generalized turbulence channels,” J. Opt. Commun. Netw , 4(5), 378–381 (2012). [CrossRef]

20.

H. Samimi and M. Uysal, “Performance of coherent differential phase-shift keying free-space optical communication systems in M-distributed turbulence,” J. Opt. Commun. Netw , 5(7), 704–710 (2013). [CrossRef]

21.

X. Song and J. Cheng, “Optical communication using subcarrier intensity modulation in strong atmospheric turbulence,” J. Lightwave Technol. 30(22), 3484–3492 (2012). [CrossRef]

22.

“The Wolfram Functions Site.”, [Online]. Available: http://functions.wolfram.com.

23.

Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-and-forward relay systems,” IEEE Commun. Lett. 10(11), 757–759 (2006). [CrossRef]

OCIS Codes
(010.7060) Atmospheric and oceanic optics : Turbulence
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Optical Communications

History
Original Manuscript: May 26, 2014
Revised Manuscript: July 1, 2014
Manuscript Accepted: July 2, 2014
Published: July 21, 2014

Citation
Liang Yang, Mazen Omar Hasna, and Xiqi Gao, "Asymptotic BER analysis of FSO with multiple receive apertures over ℳ -distributed turbulence channels with pointing errors," Opt. Express 22, 18238-18245 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18238


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References

  1. G. Yang, M. A. Khalighi, S. Bourennane, and Z. Ghassemlooy, “Fading correlation and analytical performance evaluation of the space-diversity free-space optical communications system,” J. Opt.16, 035403 (2014). [CrossRef]
  2. J. Perez, S. Zvanovec, Z. Ghassemlooy, and W. Popoola, “Experimental characterization and mitigation of turbulence induced signal fades within an ad hoc FSO network,” Opt. Express22(3), 3208–3218 (2014). [CrossRef] [PubMed]
  3. S. Zvanovec, J. Perez, Z. Ghassemlooy, S. Rajbhandari, and J. Libich, “Route diversity analyses for free-space optical wireless links within turbulent scenarios,” Opt. Express21(6), 7641–7650 (2013). [CrossRef] [PubMed]
  4. S. M. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun.21(8), 1346–1357 (2003). [CrossRef]
  5. E. J. Shin and V. W. S. Chan, “Optical communication over the turbulent atmospheric channel using spatial diversity,” IEEE Globecom, 2055–2060 (2002).
  6. 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]
  7. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun.8(2), 951–957 (2009). [CrossRef]
  8. M. Safari and M. Uysal, “Do we really need OSTBCS for free-space optical communication with direct detection? ” IEEE Trans. Wireless Commun.7(11), 4445–4448 (2008). [CrossRef]
  9. N. D. Chatzidiamantis, M. Uysal, T. A. Tsiftsis, and G. K. Karagiannidis, “Iterative near maximum-likelihood sequence detection for MIMO optical wireless systems,” J. Lightwave Technol.28(7), 1064–1070 (2010). [CrossRef]
  10. S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw.2(12), 1087–1094 (2010). [CrossRef]
  11. H. Moradi, H. H. Refai, and P. G. LoPresti, “Circular MIMO FSO nodes with transmit selection and receive generalized selection diversity,” IEEE Trans. Veh. Technol.61(3), 1174–1181 (2012). [CrossRef]
  12. M. Liu, J. Cheng, and J. F. Holzman, “Error rate performance comparison of coherent and subcarrier intensity modulated optical wireless communications,” J. Opt. Commun. Netw.5(6), 554–564 (2013). [CrossRef]
  13. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, and A. P.- Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engneering Processed, J. Awrejcewicz, ed. (Intech, 2011), ch.8. [CrossRef]
  14. 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]
  15. 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]
  16. H. Samimi and M. Uysal, “End-to-end performance of mixed RF/FSO transmission systems,” J. Opt. Commun. Netw.5(11), 1139–1144 (2013). [CrossRef]
  17. A. J.- Navas, J. M. G.- Balsells, J. F. Paris, M. C.- Vázquez, and A. P.- Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express20(11), 12550–12562 (2012). [CrossRef]
  18. I. E. Lee, Z. Ghassemlooy, W. P. Ng, and Mo.-A. Khalighi, “Joint optimization of a partially coherent Gaussian beam for free-space optical communication over turbulent channels with pointing errors,” Opt. Lett.38(3), 350–352 (2013). [CrossRef] [PubMed]
  19. H. Samimi, “Optical communication using subcarrier intensity modulation through generalized turbulence channels,” J. Opt. Commun. Netw, 4(5), 378–381 (2012). [CrossRef]
  20. H. Samimi and M. Uysal, “Performance of coherent differential phase-shift keying free-space optical communication systems in M-distributed turbulence,” J. Opt. Commun. Netw, 5(7), 704–710 (2013). [CrossRef]
  21. X. Song and J. Cheng, “Optical communication using subcarrier intensity modulation in strong atmospheric turbulence,” J. Lightwave Technol.30(22), 3484–3492 (2012). [CrossRef]
  22. “The Wolfram Functions Site.”, [Online]. Available: http://functions.wolfram.com .
  23. Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-and-forward relay systems,” IEEE Commun. Lett.10(11), 757–759 (2006). [CrossRef]

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