Abstract

We present an analytical framework for the performance evaluation of laser satellite uplinks over the major probabilistic impairments, i.e., atmospheric turbulence and beam wander. Specifically, we consider a ground-station-to-space laser uplink with a Gaussian beam wave model, and we focus on the particular regime assuming untracked beams where beam wandering takes place. In that regime, the modulated gamma–gamma distribution has been proposed as an effective irradiance model to characterize the combined effect of turbulence and beam wander. First we provide a closed-form expression of the probability density function and deduce the fundamental statistics of the new model. Then we evaluate the performance of the laser system assuming coherent detection for several modulation schemes. An appropriate set of numerical results is presented to verify the accuracy of the derived expressions.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. J. Ma, Y. Jiang, L. Tan, S. Yu, and W. Du, “Influence of beam wander on bit-error rate in a ground-to-satellite laser uplink communication system,” Opt. Lett. 33, 2611–2613 (2008).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
    [CrossRef]
  27. V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
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  28. 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, 951–957 (2009).
    [CrossRef]
  29. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
    [CrossRef]
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    [CrossRef]
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2010

2009

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, 951–957 (2009).
[CrossRef]

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
[CrossRef]

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

2008

A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008).
[CrossRef] [PubMed]

J. Ma, Y. Jiang, L. Tan, S. Yu, and W. Du, “Influence of beam wander on bit-error rate in a ground-to-satellite laser uplink communication system,” Opt. Lett. 33, 2611–2613 (2008).
[CrossRef] [PubMed]

V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
[CrossRef]

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791(2008).
[CrossRef] [PubMed]

L. C. Andrews and R. L. Philips, “Recent results on optical scintillation in the presence of beam wander,” Proc. SPIE 6878, 687802 (2008).
[CrossRef]

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

2007

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

2006

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

2005

R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

A. Rodríguez-Gomez, F. Dios, J. A. Rubio, and A. Comerón, “Temporal statistics of the beam-wander contribution to scintillation in ground-to-satellite optical links: an analytical approach,” Appl. Opt. 44, 4574–4581 (2005).
[CrossRef] [PubMed]

2004

2003

2002

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

1999

Y. E. Yenice and B. G. Evans, “Adaptive beam-size control scheme for ground-to-satellite optical communications,” Opt. Eng. 38, 1889–1895 (1999).
[CrossRef]

1973

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

Adamchik, V. S.

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

Alouini, M.-S.

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

Andrews, L.

L. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Philips, “Recent results on optical scintillation in the presence of beam wander,” Proc. SPIE 6878, 687802 (2008).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Arai, K.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Barros, D. J. F.

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
[CrossRef]

Belmonte, A.

Canning, D. E.

R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Chan, V. W. S.

Comerón, A.

Dhang, A.

Dios, F.

Du, W.

Evans, B. G.

Y. E. Yenice and B. G. Evans, “Adaptive beam-size control scheme for ground-to-satellite optical communications,” Opt. Eng. 38, 1889–1895 (1999).
[CrossRef]

Fried, D. L.

Gradshteyn, I. S.

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

Guo, H.

Ho, K.-P.

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer-Verlag, 2005).

Hopen, C. Y.

L. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Ip, E.

Jiang, Y.

Jono, T.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Kahn, J. M.

Karagiannidis, G. K.

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, 951–957 (2009).
[CrossRef]

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

Khandekar, R. M.

V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
[CrossRef]

Kimura, S.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Klaus, W.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Kunimori, H.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Kuri, T.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Lau, A. P. T.

Luo, B.

Ma, J.

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
[CrossRef]

Marichev, O. I.

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

Nikulin, V. V.

V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
[CrossRef]

Parenti, R.

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

Parenti, R. R.

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Philips, R. L.

L. C. Andrews and R. L. Philips, “Recent results on optical scintillation in the presence of beam wander,” Proc. SPIE 6878, 687802 (2008).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

L. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001).

Ren, Y.

Rodríguez, A.

Rodríguez-Gomez, A.

Rubio, J. A.

Ryzhik, I. M.

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

Sandalidis, H. G.

H. G. Sandalidis, “Performance analysis of a laser ground station-to-satellite link with modulated gamma distributed irradiance fluctuations,” J. Opt. Commun. Netw. 2, 938–943(2010).
[CrossRef]

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

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, 951–957 (2009).
[CrossRef]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. (to be published).
[CrossRef]

Sasiela, R. J.

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
[CrossRef]

Simon, M. K.

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

Sofka, J.

V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
[CrossRef]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

Suzuki, K.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Takahashi, T.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Takayama, Y.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Takizawa, K.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Tan, L.

Tharp, J. S.

R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Titterton, P. J.

Toyoda, M.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Toyoshima, M.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

Tsiftsis, T. A.

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, 951–957 (2009).
[CrossRef]

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

Tyson, R. K.

R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Uysal, M.

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, 951–957 (2009).
[CrossRef]

Wolfram,

Wolfram, “The Wolfram functions site,” http://functions.wolfram.com.

Yenice, Y. E.

Y. E. Yenice and B. G. Evans, “Adaptive beam-size control scheme for ground-to-satellite optical communications,” Opt. Eng. 38, 1889–1895 (1999).
[CrossRef]

Yu, S.

Zhao, S.

Zhu, X.

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

Appl. Opt.

IEEE Aerosp. Electron. Syst. Mag.

M. Toyoshima, Y. Takayama, T. Takahashi, K. Suzuki, S. Kimura, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, and K. Arai, “Ground-to-satellite laser communication experiments,” IEEE Aerosp. Electron. Syst. Mag. 23, 10–18 (2008).
[CrossRef]

IEEE Trans. Commun.

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

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57, 3415–3424 (2009).
[CrossRef]

IEEE Trans. Wireless Commun.

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, 951–957 (2009).
[CrossRef]

J. Lightwave Technol.

J. Opt. Commun. Netw.

Opt. Eng.

Y. E. Yenice and B. G. Evans, “Adaptive beam-size control scheme for ground-to-satellite optical communications,” Opt. Eng. 38, 1889–1895 (1999).
[CrossRef]

R. K. Tyson, D. E. Canning, and J. S. Tharp “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

V. V. Nikulin, J. Sofka, and R. M. Khandekar, “Effect of the sampling rate of the tracking system on free-space laser communications,” Opt. Eng. 47, 036003 (2008).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

L. C. Andrews and R. L. Philips, “Recent results on optical scintillation in the presence of beam wander,” Proc. SPIE 6878, 687802 (2008).
[CrossRef]

L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. Parenti, “PDF models for uplink to space in the presence of beam wander,” Proc. SPIE 6551, 655109 (2007).
[CrossRef]

Other

L. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer-Verlag, 2005).

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

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

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

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001).

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. (to be published).
[CrossRef]

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

Wolfram, “The Wolfram functions site,” http://functions.wolfram.com.

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Figures (7)

Fig. 1
Fig. 1

SI in terms of γ and α / β ( β = 10 ).

Fig. 2
Fig. 2

Probability of fade versus F T .

Fig. 3
Fig. 3

DPSK BER versus average SNR.

Fig. 4
Fig. 4

128-PSK BER versus average SNR.

Fig. 5
Fig. 5

M-PSK BER versus average SNR for W 0 / r 0 = 0.5 and different constellation sizes.

Fig. 6
Fig. 6

512-QAM BER versus average SNR.

Fig. 7
Fig. 7

QAM BER versus average SNR for W 0 / r 0 = 0.5 and different constellation sizes.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (48)

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Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 .
Θ = Θ 0 Θ 0 2 + Λ 0 2 = 1 + L F , Λ = Λ 0 Θ 0 2 + Λ 0 2 = 2 L k W 2 , Θ ¯ = 1 Θ .
σ I 2 ( L ) = 34.29 ( Λ L k r 0 2 ) 5 / 6 ( σ pe W ) 2 + exp ( σ ln X 2 + σ ln Y 2 ) 1 ,
σ ln X 2 = 0.49 σ Bu 2 ( 1 + ( 1 + Θ ) 0.56 σ Bu 12 / 5 ) 7 / 6 , σ ln Y 2 = 0.51 σ Bu 2 ( 1 + 0.69 σ Bu 12 / 5 ) 5 / 6 ,
r 0 [ 0.423 k 2 sec ζ h 0 H C n 2 ( h ) d h ] 3 5 , H 20 km ,
σ Bu 2 = 8.7 μ u k 7 6 ( H h 0 ) 5 6 sec 11 6 ( ζ ) ,
μ u = Re [ h 0 H C n 2 ( h ) { ξ 5 6 [ Λ ξ + j ( 1 Θ ¯ ξ ) ] 5 6 Λ 5 6 ξ 5 3 } d h ] ,
σ pe 2 = r c 2 [ 1 ( π 2 W 0 2 / r 0 2 1 + π 2 W 0 2 / r 0 2 ) 1 6 ] ,
r c 2 0.54 L 2 ( λ 2 W 0 ) 2 ( 2 W 0 r 0 ) 5 3 , H 20 km .
C n 2 ( h ) = 0.00594 ( u 27 ) 2 ( 10 5 h ) 10 exp ( h 1000 ) + 2.7 × 10 16 exp ( h 1500 ) + A exp ( h 100 ) ,
f I ( I ) = 2 Γ ( α ) Γ ( β ) I ( α β I I ¯ ) α + β 2 K α β ( 2 α β I I ¯ ) , I > 0 .
α = [ exp ( σ ln X 2 ) 1 ] 1 , β = [ exp ( σ ln Y 2 ) 1 ] 1 .
I ¯ = { Θ 2 + Λ 2 1 + 5.66 ( W 0 / r 0 ) 5 3 , W 0 r 0 < 1 Θ 2 + Λ 2 [ 1 + 5.66 ( W 0 / r 0 ) 5 3 ] 6 5 , W 0 r 0 1 .
f 2 ( u ) = γ u γ 1 , 0 u 1 ,
γ = 1 + exp ( σ ln X 2 + σ ln Y 2 ) 34.29 ( Λ L / k r 0 2 ) 5 / 6 ( σ pe / W ) 2 1 .
f I ( I ) = 0 1 f 2 ( u ) f 1 ( I | u ) d u u ,
f I ( I ) = γ Γ ( α ) Γ ( β ) I G 1 , 3 3 , 0 [ α β γ I ( γ + 1 ) I ¯ | 1 + γ α , β , γ ] .
f I ( I ) = π 2 γ csc ( π ( α β ) ) csc ( π ( α γ ) ) csc ( π ( β γ ) ) I Γ ( a ) Γ ( b ) Γ ( α + γ + 1 ) Γ ( β + γ + 1 ) × { sin ( π ( β γ ) ) Γ ( β + γ + 1 ) ( α β γ I I ¯ ( γ + 1 ) ) α F ˜ 1 2 ( α γ ; α β + 1 , α γ + 1 ; α β γ I I ¯ ( γ + 1 ) ) sin ( π ( α γ ) ) Γ ( α + γ + 1 ) ( α β γ I I ¯ ( γ + 1 ) ) β F ˜ 1 2 ( β γ ; α + β + 1 , β γ + 1 ; α β γ I I ¯ ( γ + 1 ) ) + sin ( π ( α β ) ) ( α β γ I I ¯ ( γ + 1 ) ) γ } .
F I ( I ) 0 I f I ( I ) d I = 0 I γ Γ ( α ) Γ ( β ) I G 1 , 3 3 , 0 [ α β γ I ( γ + 1 ) I ¯ | 1 + γ α , β , γ ] .
F I ( I ) = γ Γ ( α ) Γ ( β ) G 2 , 4 3 , 1 [ α β γ I ( γ + 1 ) I ¯ | 1 , γ + 1 α , β , γ , 0 ] .
F I ( I ) = π Γ ( α ) Γ ( β ) Γ ( α + γ + 1 ) Γ ( β + γ + 1 ) { csc ( π ( α β ) ) Γ ( α + γ + 1 ) Γ ( β + γ + 1 ) × [ ( I α β γ I ¯ ( γ + 1 ) ) α { Γ ( α γ ) F ˜ 1 2 ( α γ ; α β + 1 , α γ + 1 ; I α β γ I ¯ ( γ + 1 ) ) Γ ( α ) F ˜ 1 2 ( α ; α + 1 , α β + 1 ; I α β γ I ¯ ( γ + 1 ) ) } + ( I α β γ I ¯ ( γ + 1 ) ) β { Γ ( β ) F ˜ 1 2 ( β ; β + 1 , α + β + 1 ; I α β γ I ¯ ( γ + 1 ) ) Γ ( β γ ) F ˜ 1 2 β γ ; α + β + 1 , β γ + 1 ; I α β γ I ¯ ( γ + 1 ) } ] + π csc ( π ( α γ ) ) csc ( π ( β γ ) ) ( I α β γ I ¯ ( γ + 1 ) ) γ } .
F I ( I ) = 1 Γ ( α ) Γ ( β ) G 1 , 3 2 , 1 [ α β I | 1 α , β , 0 ] .
μ I ( ν ) 0 I ν f I ( I ) d I .
μ I ( ν ) = ( α β γ I ¯ ( γ + 1 ) ) ν γ Γ ( α + ν ) Γ ( β + ν ) Γ ( α ) Γ ( β ) ( γ + ν ) .
SI = E [ I 2 ] E 2 [ I ] E 2 [ I ] = ( α + 1 ) ( β + 1 ) ( γ + 1 ) 2 α β γ ( γ + 2 ) 1 .
M I ( σ ) = γ Γ ( α ) Γ ( β ) G 2 , 2 2 , 1 [ α β γ σ I ¯ ( 1 + γ ) | 1 , γ + 1 α , β , γ ] .
y k = η I x k + n k .
f μ ( μ ) = γ 2 Γ ( α ) Γ ( β ) μ G 1 , 3 3 , 0 [ α β γ μ ( γ + 1 ) μ ¯ | 1 + γ α , β , γ ] .
F μ ( μ ) = 1 Γ ( α ) Γ ( β ) G 1 , 3 2 , 1 [ α β μ μ ¯ | 1 α , β , 0 ] .
M μ ( σ ) = 0 μ 1 e σ μ γ 2 Γ ( α ) Γ ( β ) G 1 , 3 3 , 0 [ α β γ μ ( γ + 1 ) μ ¯ | 1 + γ α , β , γ ] d μ ,
M μ ( σ ) = 2 α + β γ 8 π Γ ( α ) Γ ( β ) G 3 , 6 6 , 1 [ α 2 β 2 γ 2 16 σ ( γ + 1 ) 2 μ ¯ | 1 , γ + 1 2 , γ + 2 2 α 2 , α + 1 2 , β 2 , β + 1 2 , γ 2 , γ + 1 2 ] .
F T = 10 log 10 ( E [ I ] I T ) [ dB ] .
P b DPS K = 1 2 e μ .
P ¯ b DPSK = 0 P ( e ) f μ ( μ ) d μ .
P ¯ b DPS K = 1 2 M μ ( 1 ) = 2 α + β γ 16 π Γ ( α ) Γ ( β ) G 3 , 6 6 , 1 [ α 2 β 2 γ 2 16 ( γ + 1 ) 2 μ ¯ | 1 , γ + 1 2 , γ + 2 2 α 2 , α + 1 2 , β 2 , β + 1 2 , γ 2 , γ + 1 2 ] .
P b PSK ( M ) 1 log 2 M erfc ( log 2 ( M ) μ sin ( π μ ) ) ,
P b PSK ( 2 ) = P b PSK ( 4 ) = 1 2 erfc ( μ ) .
P ¯ b PSK ( M ) = γ 2 π log 2 ( M ) Γ ( α ) Γ ( β ) × 0 μ 1 G 1 , 2 2 , 0 [ log 2 ( M ) sin 2 ( π M ) μ | 1 0 , 1 2 ] G 1 , 3 3 , 0 [ α β γ γ + 1 μ μ ¯ | γ + 1 α , β , γ ] d μ .
P ¯ b PSK ( M ) = 2 α + β γ 8 π π log 2 ( M ) Γ ( α ) Γ ( β ) G 4 , 7 6 , 2 [ α 2 β 2 γ 2 csc 2 ( π M ) 16 ( γ + 1 ) 2 log 2 ( M ) μ ¯ | 1 2 , 1 , γ + 1 2 , γ + 2 2 α 2 , α + 1 2 , β 2 , β + 1 2 , γ 2 , γ + 1 2 , 0 ] .
P ¯ b PSK ( 2 ) = γ 4 π Γ ( α ) Γ ( β ) × 0 μ 1 G 1 , 2 2 , 0 [ μ | 1 0 , 1 2 ] G 1 , 3 3 , 0 [ α β γ γ + 1 μ μ ¯ | γ + 1 α , β , γ ] d μ ,
P ¯ b PSK ( 2 ) = 2 α + β γ 16 π π Γ ( α ) Γ ( β ) G 4 , 7 6 , 2 [ α 2 β 2 γ 2 16 ( γ + 1 ) 2 μ ¯ | 1 2 , 1 , γ + 1 2 , γ + 2 2 α 2 , α + 1 2 , β 2 , β + 1 2 , γ 2 , γ + 1 2 , 0 ] .
P b QAM ( M ) 2 log 2 ( M ) M 1 M erfc ( 3 log 2 ( M ) μ 2 ( M 1 ) ) .
P ¯ b QAM ( M ) = ( M 1 ) γ π M log 2 ( M ) Γ ( α ) Γ ( β ) × 0 μ 1 G 1 , 2 2 , 0 [ 3 log 2 ( M ) μ 2 ( M 1 ) | 1 0 , 1 2 ] G 1 , 3 3 , 0 [ α β γ γ + 1 μ μ ¯ | γ + 1 α , β , γ ] d μ .
P ¯ b QAM ( M ) = ( M 1 ) 2 α + β γ 4 π π log 2 ( M ) M Γ ( α ) Γ ( β ) G 4 , 7 6 , 2 [ ( M 1 ) α 2 β 2 γ 2 24 log 2 ( M ) ( γ + 1 ) 2 μ ¯ | 1 2 , 1 , γ + 1 2 , γ + 2 2 α 2 , α + 1 2 , β 2 , β + 1 2 , γ 2 , γ + 1 2 , 0 ] .
f I ( I ) = 2 γ Γ ( α ) Γ ( β ) I ( α β γ I ( γ + 1 ) I ¯ ) ( α + β ) / 2 0 1 u γ α + β 2 1 K α β ( 2 α β γ I ( γ + 1 ) u I ¯ ) d u .
f I ( I ) = γ Γ ( α ) Γ ( β ) I ( α β γ I ( γ + 1 ) I ¯ ) 0 1 u γ α + β 2 1 G 2 , 0 0 , 2 [ ( γ + 1 ) I ¯ u α β γ I | 1 α β 2 , 1 β α 2 ] d u .
f I ( I ) = γ Γ ( α ) Γ ( β ) I ( α β γ I ( γ + 1 ) I ¯ ) G 1 , 3 3 , 0 [ α β γ I ( γ + 1 ) I ¯ | 1 α β 2 + γ α β 2 , β α 2 , γ α + β 2 ] d u .
f I ( I ) = γ Γ ( α ) Γ ( β ) I G 1 , 3 3 , 0 [ α β γ I ( γ + 1 ) I ¯ | 1 + γ α , β , γ ] .

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