Abstract

In this paper, we present analytical expressions for the performance of urban free-space optical (FSO) communication systems under the combined influence of atmospheric turbulence- and misalignment-induced fading (pointing errors). The atmospheric turbulence channel is modeled by the exponentiated Weibull (EW) distribution that can accurately describe the probability density function (PDF) of the irradiance fluctuations associated with a transmitted Gaussian-beam wave and a finite-sized receiving aperture. The nonzero boresight pointing error PDF model, which is recently proposed for considering the effects of both boresight and jitter, is adopted in analysis. We derive a novel expression for the composite PDF in terms of a convergent double series involving a Meijer’s G-function. Based on the statistical results mentioned above, exact expressions for the average bit error rate of on-off keying modulation scheme and the outage probability are developed. To provide more insight, we also perform an asymptotic error rate analysis at high average signal-to-noise ratio. Our analytical results indicate that the diversity gain for the zero boresight case is determined only by the ratio between the equivalent beamwidth at the receiver and the jitter standard deviation, while for the nonzero boresight case, the diversity gain is related to the ratio of the equivalent beamwidth to the jitter variance as well as the parameter of the EW distribution.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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  22. J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).
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2014 (4)

P. Wang, L. Zhang, L. X. Guo, F. Huang, T. Shang, R. Wang, and Y. T. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22, 20828–20841 (2014).
[Crossref] [PubMed]

M. J. Cheng, Y. X. Zhang, J. Gao, F. Wang, and F. S. Zhao, “Average capacity for optical wireless communication systems over exponentiated Weibull distribution non-Kolmogorov turbulent channels,” Appl. Opt. 53, 4011–4017 (2014).
[Crossref] [PubMed]

F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62, 713–725 (2014).
[Crossref]

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

2013 (2)

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

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

2012 (5)

2010 (4)

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14, 468–470 (2010).
[Crossref]

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

C. Liu, Y. Yao, Y. X. Sun, J. J. Xiao, and X. H. Zhao, “Average capacity optimization in free-space optical communication system over atmospheric turbulence channels with pointing errors,” Opt. Lett. 35, 3171–3173 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (1)

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

2007 (2)

2003 (2)

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

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

Adamchik, V. S.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometic type functions and its realization in reduce system,” Proceedings of the International Symposium on Symbolic and Algebraic Computation (ACM, 1990), pp. 212–224.

Alexandridis, A.

Andrews, L.

Andrews, L. C.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed., (SPIE, 2005).
[Crossref]

Arnon, S.

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

Barrios, R.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20, 13055–13064 (2012).
[Crossref] [PubMed]

Borah, D. K.

Castillo-Vazquez, M.

Chen, M.

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

Cheng, J.

F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62, 713–725 (2014).
[Crossref]

Cheng, M. J.

Dios, F.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20, 13055–13064 (2012).
[Crossref] [PubMed]

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

Dongakis, K.

Farid, A.

Gao, J.

Gappmair, W.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14, 468–470 (2010).
[Crossref]

Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vazquez, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20, 12550–12562 (2012).
[Crossref] [PubMed]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011).
[Crossref]

Ghassemlooy, Z.

Giannakis, G.

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Guo, L. X.

Hranilovic, S.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14, 468–470 (2010).
[Crossref]

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

Huang, F.

Jurado-Navas, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vazquez, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20, 12550–12562 (2012).
[Crossref] [PubMed]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011).
[Crossref]

Karagiannidis, G.

H. Sandalidis, T. Tsiftsis, and G. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27, 4440–4445 (2009).
[Crossref]

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

Khalighi, M.

Kilbas, A.

A. Kilbas, H-Transforms: Theory and Applications. Analytical Methods and Special Functions. (Taylor and Francis, 2004).
[Crossref]

Lazarakis, F.

Leclerc, T.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

Lee, I. E.

Leitgeb, E.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14, 468–470 (2010).
[Crossref]

Liu, C.

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

C. Liu, Y. Yao, Y. X. Sun, J. J. Xiao, and X. H. Zhao, “Average capacity optimization in free-space optical communication system over atmospheric turbulence channels with pointing errors,” Opt. Lett. 35, 3171–3173 (2010).
[Crossref] [PubMed]

Liu, Z. J.

Marichev, O. I.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometic type functions and its realization in reduce system,” Proceedings of the International Symposium on Symbolic and Algebraic Computation (ACM, 1990), pp. 212–224.

Ng, W. P.

Paris, J. F.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vazquez, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20, 12550–12562 (2012).
[Crossref] [PubMed]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011).
[Crossref]

Peppas, K. P.

Phillips, R. L.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed., (SPIE, 2005).
[Crossref]

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vazquez, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20, 12550–12562 (2012).
[Crossref] [PubMed]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011).
[Crossref]

Recolons, J.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Sandalidis, H.

H. Sandalidis, T. Tsiftsis, and G. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27, 4440–4445 (2009).
[Crossref]

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

Sauer, P.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

Shang, T.

Stryjewski, J.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

Sun, Y. X.

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

C. Liu, Y. Yao, Y. X. Sun, J. J. Xiao, and X. H. Zhao, “Average capacity optimization in free-space optical communication system over atmospheric turbulence channels with pointing errors,” Opt. Lett. 35, 3171–3173 (2010).
[Crossref] [PubMed]

Tang, Y.

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

Tsiftsis, T.

H. Sandalidis, T. Tsiftsis, and G. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27, 4440–4445 (2009).
[Crossref]

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

Tsiftsis, T. A.

F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62, 713–725 (2014).
[Crossref]

Uysal, M.

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

Vetelino, F. S.

Voelz, D. G.

Wang, F.

Wang, J. B.

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

Wang, J. Y.

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

Wang, P.

Wang, R.

Wang, Z.

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

Wayne, D. T.

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

Xiao, J. J.

Yang, F.

F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62, 713–725 (2014).
[Crossref]

Yang, Y. T.

Yao, Y.

C. Liu, Y. Yao, Y. X. Sun, J. J. Xiao, and X. H. Zhao, “Average capacity optimization in free-space optical communication system over atmospheric turbulence channels with pointing errors,” Opt. Lett. 35, 3171–3173 (2010).
[Crossref] [PubMed]

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

Yi, X.

Young, C.

Yue, P.

Zhang, L.

Zhang, Y.

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

Zhang, Y. X.

Zhao, F. S.

Zhao, X. H.

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

C. Liu, Y. Yao, Y. X. Sun, J. J. Xiao, and X. H. Zhao, “Average capacity optimization in free-space optical communication system over atmospheric turbulence channels with pointing errors,” Opt. Lett. 35, 3171–3173 (2010).
[Crossref] [PubMed]

Appl. Opt. (2)

Electron. Lett. (1)

C. Liu, Y. Yao, Y. X. Sun, and X. H. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 851–853 (2010).
[Crossref]

IEEE Commun. Lett. (2)

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14, 468–470 (2010).
[Crossref]

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

IEEE Photon. J. (1)

J. Y. Wang, J. B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photon. J. 6, 7901815 (2014).

IEEE Trans. Commun. (2)

F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62, 713–725 (2014).
[Crossref]

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

IEEE Trans. Wireless. Commun. (1)

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

J. Lightwave Technol. (3)

Opt. Express (3)

Opt. Laser Technol. (1)

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[Crossref]

Opt. Lett. (4)

Proc. SPIE (2)

R. Barrios and F. Dios, “Probability of fade and BER performance of FSO links over the exponentiated Weibull fading channel under aperture averaging,” Proc. SPIE 8540, 85400D (2012).
[Crossref]

D. T. Wayne, R. L. Phillips, L. C. Andrews, T. Leclerc, P. Sauer, and J. Stryjewski, “Comparing the Log-Normal and Gamma-Gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[Crossref]

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A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed., (SPIE, 2005).
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometic type functions and its realization in reduce system,” Proceedings of the International Symposium on Symbolic and Algebraic Computation (ACM, 1990), pp. 212–224.

Wolfram, http://functions.wolfram.com .

A. Kilbas, H-Transforms: Theory and Applications. Analytical Methods and Special Functions. (Taylor and Francis, 2004).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Probability of outage versus the transmitted optical power for weak turbulence conditions σ R 2 = 0.15 and R0 = 0.5 bits/chan. use. D = 3mm and D = 25mm represent point receiver and aperture-averaged receiver, respectively. s = 0 and s > 0 represent zero bore-sight and nonzero boresight cases, respectively.
Fig. 2
Fig. 2 Probability of outage versus the transmitted optical power for moderate turbulence conditions σ R 2 = 1.3 and R0 = 0.5 bits/chan. use. D = 3mm and D = 25mm represent point receiver and aperture-averaged receiver, respectively. s = 0 and s > 0 represent zero boresight and nonzero boresight cases, respectively.
Fig. 3
Fig. 3 Average BER versus the transmitted optical power for weak turbulence conditions σ R 2 = 0.15. D = 3mm and D = 25mm represent point receiver and aperture-averaged receiver, respectively. s = 0 and s > 0 represent zero boresight and nonzero boresight cases, respectively.
Fig. 4
Fig. 4 Average BER versus the transmitted optical power for moderate turbulence conditions σ R 2 = 1.3. D = 3mm and D = 25mm represent point receiver and aperture-averaged receiver, respectively. s = 0 and s > 0 represent zero boresight and nonzero boresight cases, respectively.

Tables (1)

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Table 1 Link parameters

Equations (28)

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y = R h x + n ,
f h a ( h a ) = α β η ( h a η ) β 1 exp [ ( h a η ) β ] { 1 exp [ ( h a η ) β ] } α 1 , h a > 0 ,
f h p ( h p ) = γ 2 A 0 γ 2 exp ( s 2 2 σ s 2 ) h p γ 2 1 I 0 ( s σ s 2 ω z eq 2 2 ln h p A 0 ) , 0 h p A 0 ,
f h ( h ) = f h | h a ( h | h a ) f h a ( h a ) d h a .
f h | h a ( h | h a ) = 1 h l h a f h p ( h h l h a ) = γ 2 A 0 γ 2 h l h a exp ( s 2 2 σ s 2 ) ( h h l h a ) γ 2 1 I 0 ( s σ s 2 ω z eq 2 2 ln h A 0 h l h a ) .
f h ( h ) = α β γ 2 η β ( A 0 h l ) γ 2 exp ( s 2 2 σ s 2 ) h γ 2 1 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) × h / A 0 h l h a β γ 2 1 I 0 ( s σ s 2 ω z eq 2 2 ln h A 0 h l h a ) exp [ ( 1 + j ) ( h a η ) β ] d h a ,
f h ( h ) = α γ 2 ( η A 0 h l ) β exp ( s 2 2 σ s 2 ) j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) h β 1 × 0 exp [ ( 1 + j ) h β ( η A 0 h l ) β e y ( γ 2 β 1 ) y ] I 0 ( s σ s 2 ( ω z eq 2 2 β ) 1 / 2 y 1 / 2 ) d y .
f h ( h ) = α γ 2 ( η A 0 h l ) β exp ( s 2 2 σ s 2 ) j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) h β 1 × k = 0 1 ( k ! ) 2 ( s 2 σ s 2 ) k ( γ 2 β ) k 0 y k exp [ ( 1 + j ) h β ( η A 0 h l ) β e y ( γ 2 β 1 ) y ] d y .
f h ( h ) = α γ 2 ( η A 0 h l ) β exp ( s 2 2 σ s 2 ) j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) h β 1 × k = 0 1 k ! ( s 2 σ s 2 ) k ( γ 2 β ) k G k + 1 , k + 2 k + 2 , 0 [ ( 1 + j ) h β ( η A 0 h l ) β | γ 2 β , , γ 2 β ( k + 1 ) terms 0 , γ 2 β 1 , , γ 2 β 1 ( k + 1 ) terms ] ,
f h ( h ) = α γ 2 ( η A 0 h l ) β h β 1 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) × 0 exp [ ( 1 + j ) h β ( η A 0 h l ) β e y ( γ 2 β 1 ) y ] d y .
f h ( h ) = α γ 2 ( η A 0 h l ) γ 2 h γ 2 1 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 / β × Γ ( 1 γ 2 β , ( 1 + j ) ( h η A 0 h l ) β ) ,
C ( h ^ ) = x f y | x ( y | x ) p x ( x ) log 2 f y | x ( y | x ) f y ( y ) d y ,
P out ( R 0 ) = Pr ( h < C 1 ( R 0 ) ) .
P out ( R 0 ) = 0 h 0 f h ( h ) d h .
P out ( R 0 ) = α γ 2 β ( h 0 η A 0 h l ) β exp ( s 2 2 σ s 2 ) j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) k = 0 1 k ! ( s 2 σ s 2 ) k ( γ 2 β ) k × G k + 2 , k + 3 k + 2 , 1 [ ( 1 + j ) ( h 0 η A 0 h l ) β | 0 , γ 2 β , , γ 2 β ( k + 1 ) terms 0 , γ 2 β 1 , , γ 2 β 1 , 1 ( k + 1 ) terms ] .
P out ( R 0 ) = α γ 2 β ( h 0 η A 0 h l ) γ 2 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 / β × G 2 , 3 2 , 1 [ ( 1 + j ) ( h 0 η A 0 h l ) β | 1 γ 2 β , 1 0 , 1 γ 2 β , γ 2 β ] .
P e ( e | h ) = 1 2 erfc ( R P t h 2 σ n ) ,
P e = 0 P e ( e | h ) f h ( h ) d h .
P e = α γ 2 2 π ( η A 0 h l ) γ 2 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β × 0 h γ 2 1 G 1 , 2 2 , 0 [ R 2 P t 2 2 σ n 2 h 2 | 1 0 , 1 2 ] G 1 , 2 2 , 0 [ ( 1 + j ) ( h η A 0 h l ) β | 1 0 , 1 γ 2 β ] d h ,
P e = α γ 2 4 π ( η A 0 h l ) γ 2 ( R P t 2 σ n ) γ 2 j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β × H 3 , 3 2 , 2 [ ( 1 + j ) ( η A 0 h l ) β ( R P t 2 σ n ) β | ( 1 γ 2 2 , β 2 ) , ( 1 γ 2 2 , β 2 ) , ( 1 , 1 ) ( 0 , 1 ) , ( 1 γ 2 β , 1 ) , ( γ 2 2 , β 2 ) ] ,
P e = α γ 2 4 π ( η A 0 h l ) β exp ( s 2 2 σ s 2 ) j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) × k = 0 1 k ! ( s 2 2 σ s 2 ) k ( γ 2 β ) k ( R P t 2 σ n ) β × H k + 3 , k + 3 k + 2 , 2 [ ( 1 + j ) ( η A 0 h l ) β ( R P t 2 σ n ) β | ( 1 β 2 , β 2 ) , ( 1 β 2 , β 2 ) , ( γ 2 β , 1 ) , , ( γ 2 β , 1 ) ( k + 1 ) terms ( 0 , 1 ) , ( γ 2 β 1 , 1 ) , , ( γ 2 β 1 , 1 ) ( k + 1 ) terms , ( β 2 , β 2 ) ] .
P e ( G c SNR ¯ ) G d , SNR ¯ ,
H 3 , 3 2 , 2 [ ( η A 0 h l ) β ( 1 + j ) ( R P t 2 σ n ) β | ( 1 , 1 ) , ( γ 2 β , 1 ) , ( 1 + γ 2 2 , β 2 ) ( γ 2 2 , β 2 ) , ( 1 + γ 2 2 , β 2 ) , ( 0 , 1 ) ] 2 γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) + 2 γ 2 β Γ ( 1 + β 2 ) [ ( 1 + j ) ( η A 0 h l / 2 ) β ] 1 γ 2 β ( 2 R 2 P t 2 σ n 2 ) β γ 2 2 .
P e α 2 π ( 2 η A 0 h l ) γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β ( 2 R 2 P t 2 σ n 2 ) γ 2 2 + α γ 2 2 π ( γ 2 β ) ( 2 η A 0 h l ) β Γ ( 1 + β 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 2 R 2 P t 2 σ n 2 ) β 2 .
P e α 2 π ( 2 η A 0 h l ) γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β ( 2 R 2 P t 2 σ n 2 ) γ 2 2 .
H k + 3 , k + 3 k + 2 , 2 [ ( 1 + j ) ( η A 0 h l ) β ( R P t 2 σ n ) β | ( 1 β 2 , β 2 ) , ( 1 β 2 , β 2 ) , ( γ 2 β , 1 ) , , ( γ 2 β , 1 ) ( k + 1 ) terms ( 0 , 1 ) , ( γ 2 β 1 , 1 ) , , ( γ 2 β 1 , 1 ) ( k + 1 ) terms , ( β 2 , β 2 ) ] 2 [ Γ ( γ 2 β 1 ) ] k + 1 Γ ( 1 + β 2 ) β [ Γ ( γ 2 β ) ] k + 1 + 2 γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) × ( 1 ) k k ! { log [ ( 1 + j ) ( η A 0 h l ) β ( R P t 2 σ n ) β ] } k [ ( 1 + j ) ( η A 0 h l ) β ( R P t 2 σ n ) β ] γ 2 β 1 .
P e α 2 π ( 2 η A 0 h l ) γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) exp ( s 2 2 σ s 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β k = 0 ( 1 ) k ( k ! ) 2 ( s 2 2 σ s 2 ) k ( γ 2 β ) k × { log [ ( 1 + j ) ( η A 0 h l / 2 ) β ( 2 R 2 P t 2 σ n 2 ) β 2 ] } k ( 2 R 2 P t 2 σ n 2 ) γ 2 2 + α 2 π ( 2 η A 0 h l ) β Γ ( 1 + β 2 ) exp ( s 2 2 σ s 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) k = 0 1 k ! ( s 2 2 σ s 2 ) k ( γ 2 γ 2 β ) k + 1 ( 2 R 2 P t 2 σ n 2 ) β 2 .
P e α 2 π ( 2 η A 0 h l ) γ 2 Γ ( 1 γ 2 β ) Γ ( 1 + γ 2 2 ) exp ( s 2 2 σ s 2 ) × j = 0 ( 1 ) j Γ ( α ) j ! Γ ( α j ) ( 1 + j ) 1 γ 2 β k = 0 ( 1 ) k ( k ! ) 2 ( s 2 2 σ s 2 ) k ( γ 2 β ) k × { log [ ( 1 + j ) ( η A 0 h l / 2 ) β ( 2 R 2 P t 2 σ n 2 ) β 2 ] } k ( 2 R 2 P t 2 σ n 2 ) γ 2 2 .

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