Abstract

Nowadays, the search for a distribution capable of modeling the probability density function (PDF) of irradiance data under all conditions of atmospheric turbulence in the presence of aperture averaging still continues. Here, a family of PDFs alternative to the widely accepted Log-Normal and Gamma-Gamma distributions is proposed to model the PDF of the received optical power in free-space optical communications, namely, the Weibull and the exponentiated Weibull (EW) distribution. Particularly, it is shown how the proposed EW distribution offers an excellent fit to simulation and experimental data under all aperture averaging conditions, under weak and moderate turbulence conditions, as well as for point-like apertures. Another very attractive property of these distributions is the simple closed form expression of their respective PDF and cumulative distribution function.

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References

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  1. D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
    [CrossRef]
  2. 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]
  3. P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967).
  4. J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A4, 727–737 (1987).
    [CrossRef]
  5. L. C. Andrews and R. L. Phillips, “I – K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A2, 160–163 (1985).
    [CrossRef]
  6. E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546–550 (1978).
    [CrossRef]
  7. F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt.46, 3780–3790 (2007).
    [CrossRef] [PubMed]
  8. N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE5338, 233–242 (2004).
  9. B. Epple, “Simplified channel model for simulation of free-space optical communications,” J. Opt. Commun. Netw.2(5), 293–304 (2010).
    [CrossRef]
  10. N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.
  11. M.-S. Alouini and M. Simon, “Performance of generalized selection combining over Weibull fading channels,” in Vehicular Technology Conf., (Atlantic City, NJ, USA, 2001), pp. 1735–1739.
  12. M. Lupupa and M. Dlodlo, “Performance of MIMO system in Weibull fading channel—channel capacity analysis,” in EUROCON 2009, EUROCON ’09. IEEE, (St. Petersburg, Russia, 2009), pp. 1735–1740.
    [CrossRef] [PubMed]
  13. W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech.-Trans. ASME18, 293–297 (1951).
  14. J. V. Seguro and T. W. Lambert, “Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis,” J Wind. Eng. Ind. Aerod.85, 75–84 (2000).
    [CrossRef]
  15. Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
    [CrossRef]
  16. D. Schleher, “Radar detection in Weibull clutter,” IEEE Trans. Aerosp. Electron. Syst.AES-12, 736–743 (1976).
    [CrossRef]
  17. G. Mudholkar and D. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Trans. Reliab.42, 299–302 (1993).
    [CrossRef]
  18. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
    [CrossRef]
  19. R. R. Parenti and R. J. Sasiela, “Distribution models for optical scintillation due to atmospheric turbulence,” MIT Lincoln Laboratory Technical Report TR-1108, (2005).
  20. S. Nadarajah and A. K. Gupta, “On the moments of the exponentiated Weibull distribution,” Comm. Statist. Theory Methods34, 253–256 (2005).
  21. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt.38, 2161–2160 (1999).
    [CrossRef]
  22. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
    [CrossRef]
  23. J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 51–62 (2005).
  24. F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt.46, 2099–2109 (2007).
    [CrossRef] [PubMed]
  25. K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math.2, 164–168 (1944).
  26. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
    [CrossRef]
  27. R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
    [CrossRef]
  28. S. D. Lyke, D. G. Voelz, and M. C. Roggemann, “Probability density of aperture-averaged irradiance fluctuations for long range free space optical communication links,” Appl. Opt.48, 6511–6527 (2009).
    [CrossRef] [PubMed]
  29. R. J. Hill, R. G. Frehlich, and W. D. Otto, “The probability distribution of irradiance scintillation,” National Oceanic and Atmospheric Administration (NOAA) Technical Report ETL-270, (1997).

2010

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
[CrossRef]

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

B. Epple, “Simplified channel model for simulation of free-space optical communications,” J. Opt. Commun. Netw.2(5), 293–304 (2010).
[CrossRef]

2009

2007

2005

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 51–62 (2005).

S. Nadarajah and A. K. Gupta, “On the moments of the exponentiated Weibull distribution,” Comm. Statist. Theory Methods34, 253–256 (2005).

2004

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE5338, 233–242 (2004).

2001

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]

2000

J. V. Seguro and T. W. Lambert, “Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis,” J Wind. Eng. Ind. Aerod.85, 75–84 (2000).
[CrossRef]

1999

1993

G. Mudholkar and D. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Trans. Reliab.42, 299–302 (1993).
[CrossRef]

Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
[CrossRef]

1992

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
[CrossRef]

1987

1985

1978

E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546–550 (1978).
[CrossRef]

1976

D. Schleher, “Radar detection in Weibull clutter,” IEEE Trans. Aerosp. Electron. Syst.AES-12, 736–743 (1976).
[CrossRef]

1963

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

1951

W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech.-Trans. ASME18, 293–297 (1951).

1944

K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math.2, 164–168 (1944).

Al-Habash, M. A.

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]

Alouini, M.-S.

M.-S. Alouini and M. Simon, “Performance of generalized selection combining over Weibull fading channels,” in Vehicular Technology Conf., (Atlantic City, NJ, USA, 2001), pp. 1735–1739.

Andrews, L.

Andrews, L. C.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
[CrossRef]

F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt.46, 2099–2109 (2007).
[CrossRef] [PubMed]

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]

L. C. Andrews and R. L. Phillips, “I – K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A2, 160–163 (1985).
[CrossRef]

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

Barrios, R.

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

Beckmann, P.

P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967).

Chatzidiamantis, N.

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

Churnside, J. H.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
[CrossRef]

Dios, F.

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 51–62 (2005).

Dlodlo, M.

M. Lupupa and M. Dlodlo, “Performance of MIMO system in Weibull fading channel—channel capacity analysis,” in EUROCON 2009, EUROCON ’09. IEEE, (St. Petersburg, Russia, 2009), pp. 1735–1740.
[CrossRef] [PubMed]

Epple, B.

Fang, Z.

Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
[CrossRef]

Frehlich, R. G.

R. J. Hill, R. G. Frehlich, and W. D. Otto, “The probability distribution of irradiance scintillation,” National Oceanic and Atmospheric Administration (NOAA) Technical Report ETL-270, (1997).

Fritzsche, D.

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE5338, 233–242 (2004).

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
[CrossRef]

Gupta, A. K.

S. Nadarajah and A. K. Gupta, “On the moments of the exponentiated Weibull distribution,” Comm. Statist. Theory Methods34, 253–256 (2005).

Harding, C. M.

Hill, R. J.

J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A4, 727–737 (1987).
[CrossRef]

R. J. Hill, R. G. Frehlich, and W. D. Otto, “The probability distribution of irradiance scintillation,” National Oceanic and Atmospheric Administration (NOAA) Technical Report ETL-270, (1997).

Jakeman, E.

E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546–550 (1978).
[CrossRef]

Johnston, R. A.

Karagiannidis, G.

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

Kotsopoulos, S.

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

Lambert, T. W.

J. V. Seguro and T. W. Lambert, “Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis,” J Wind. Eng. Ind. Aerod.85, 75–84 (2000).
[CrossRef]

Lane, R. G.

C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt.38, 2161–2160 (1999).
[CrossRef]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math.2, 164–168 (1944).

Lupupa, M.

M. Lupupa and M. Dlodlo, “Performance of MIMO system in Weibull fading channel—channel capacity analysis,” in EUROCON 2009, EUROCON ’09. IEEE, (St. Petersburg, Russia, 2009), pp. 1735–1740.
[CrossRef] [PubMed]

Lyke, S. D.

Marquardt, D.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

Matthaiou, M.

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

Mudholkar, G.

G. Mudholkar and D. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Trans. Reliab.42, 299–302 (1993).
[CrossRef]

Nadarajah, S.

S. Nadarajah and A. K. Gupta, “On the moments of the exponentiated Weibull distribution,” Comm. Statist. Theory Methods34, 253–256 (2005).

Otto, W. D.

R. J. Hill, R. G. Frehlich, and W. D. Otto, “The probability distribution of irradiance scintillation,” National Oceanic and Atmospheric Administration (NOAA) Technical Report ETL-270, (1997).

Parenti, R. R.

R. R. Parenti and R. J. Sasiela, “Distribution models for optical scintillation due to atmospheric turbulence,” MIT Lincoln Laboratory Technical Report TR-1108, (2005).

Patterson, B. R.

Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
[CrossRef]

Perlot, N.

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE5338, 233–242 (2004).

Phillips, R. L.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
[CrossRef]

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]

L. C. Andrews and R. L. Phillips, “I – K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A2, 160–163 (1985).
[CrossRef]

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

Pusey, P.

E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546–550 (1978).
[CrossRef]

Recolons, J.

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt.46, 2099–2109 (2007).
[CrossRef] [PubMed]

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 51–62 (2005).

Rodríguez, A.

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

Roggemann, M. C.

Sandalidis, H.

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

Sasiela, R. J.

R. R. Parenti and R. J. Sasiela, “Distribution models for optical scintillation due to atmospheric turbulence,” MIT Lincoln Laboratory Technical Report TR-1108, (2005).

Schleher, D.

D. Schleher, “Radar detection in Weibull clutter,” IEEE Trans. Aerosp. Electron. Syst.AES-12, 736–743 (1976).
[CrossRef]

Seguro, J. V.

J. V. Seguro and T. W. Lambert, “Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis,” J Wind. Eng. Ind. Aerod.85, 75–84 (2000).
[CrossRef]

Simon, M.

M.-S. Alouini and M. Simon, “Performance of generalized selection combining over Weibull fading channels,” in Vehicular Technology Conf., (Atlantic City, NJ, USA, 2001), pp. 1735–1739.

Srivastava, D.

G. Mudholkar and D. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Trans. Reliab.42, 299–302 (1993).
[CrossRef]

Turner, M. E.

Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
[CrossRef]

Vetelino, F. S.

Voelz, D. G.

Wayne, D. T.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
[CrossRef]

Weibull, W.

W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech.-Trans. ASME18, 293–297 (1951).

Young, C.

Appl. Opt.

Comm. Statist. Theory Methods

S. Nadarajah and A. K. Gupta, “On the moments of the exponentiated Weibull distribution,” Comm. Statist. Theory Methods34, 253–256 (2005).

IEEE Trans. Aerosp. Electron. Syst.

D. Schleher, “Radar detection in Weibull clutter,” IEEE Trans. Aerosp. Electron. Syst.AES-12, 736–743 (1976).
[CrossRef]

IEEE Trans. Reliab.

G. Mudholkar and D. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Trans. Reliab.42, 299–302 (1993).
[CrossRef]

J Wind. Eng. Ind. Aerod.

J. V. Seguro and T. W. Lambert, “Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis,” J Wind. Eng. Ind. Aerod.85, 75–84 (2000).
[CrossRef]

J. Appl. Mech.-Trans. ASME

W. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech.-Trans. ASME18, 293–297 (1951).

J. Opt. Commun. Netw.

J. Opt. Soc. Am. A

Mater. Charact.

Z. Fang, B. R. Patterson, and M. E. Turner, “Modeling particle size distributions by the Weibull distribution function,” Mater. Charact.31, 177–182 (1993).
[CrossRef]

Opt. Eng.

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]

Phys. Rev. Lett.

E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett.40, 546–550 (1978).
[CrossRef]

Proc. SPIE

N. Perlot and D. Fritzsche, “Aperture-averaging: theory and measurements,” Proc. SPIE5338, 233–242 (2004).

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE7814, 78140K (2010).
[CrossRef]

R. Barrios, F. Dios, J. Recolons, and A. Rodríguez, “Aperture averaging in a laser gaussian beam: simulations and experiments,” Proc. SPIE7814, 78140C (2010).
[CrossRef]

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 51–62 (2005).

Quart. Appl. Math.

K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math.2, 164–168 (1944).

SIAM J. Appl. Math.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

Waves Random Media

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2, 209–224 (1992).
[CrossRef]

Other

R. J. Hill, R. G. Frehlich, and W. D. Otto, “The probability distribution of irradiance scintillation,” National Oceanic and Atmospheric Administration (NOAA) Technical Report ETL-270, (1997).

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

R. R. Parenti and R. J. Sasiela, “Distribution models for optical scintillation due to atmospheric turbulence,” MIT Lincoln Laboratory Technical Report TR-1108, (2005).

P. Beckmann, Probability in Communication Engineering, (Harcourt, Brace & World, 1967).

N. Chatzidiamantis, H. Sandalidis, G. Karagiannidis, S. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in Telecommunications (ICT), 2010 IEEE 17th International Conference on, (Doha, Qatar, 2010), pp. 487–492.

M.-S. Alouini and M. Simon, “Performance of generalized selection combining over Weibull fading channels,” in Vehicular Technology Conf., (Atlantic City, NJ, USA, 2001), pp. 1735–1739.

M. Lupupa and M. Dlodlo, “Performance of MIMO system in Weibull fading channel—channel capacity analysis,” in EUROCON 2009, EUROCON ’09. IEEE, (St. Petersburg, Russia, 2009), pp. 1735–1740.
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Weibull (WB) and exponentiated Weibull (EW) models fitted to simulation data for several aperture diameters D, link distance L = 375m, coherence radius ρ0 = 18.89 mm and σ R 2 = 0.15, under weak turbulence conditions. The Gamma-Gamma (GG) model is shown for comparison purposes.

Fig. 2
Fig. 2

Weibull (WB) and exponentiated Weibull (EW) models fitted to simulation data for several aperture diameters D, link distance L = 1225 m, coherence radius ρ0 = 9.27 mm and σ R 2 = 1.35, under moderate turbulence conditions. The Gamma-Gamma (GG) model is shown for comparison purposes.

Fig. 3
Fig. 3

Exponentiated Weibull fitted parameters (solid line) and estimated parameters (dashed line) for several aperture diameters D.

Fig. 4
Fig. 4

Weibull (WB) and exponentiated Weibull (EW) models fitted to experimental data for several aperture diameters D, link distance L = 1200m, coherence radius ρ0 = 9.40 mm and σ R 2 = 1.30, under moderate turbulence conditions. The Gamma-Gamma (GG) model is shown for comparison purposes.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

σ I 2 = I 2 I 2 1 ,
f W ( I ; β , η ) = β η ( I η ) β 1 exp   [ ( I η ) β ] ,
F W ( I ; β , η ) = 1 exp   [ ( I η ) β ] ,
I n = η n Γ ( 1 + n β ) ,
σ I 2 = Γ ( 1 + 2 / β ) Γ ( 1 + 1 / β ) 2 1 β 11 / 6 .
η = 1 Γ ( 1 + 1 / β ) .
f EW ( I ; β , η , α ) = α β η ( I η ) β 1 exp   [ ( I η ) β ] { 1 exp   [ ( I η ) β ] } α 1 ,
F EW ( I ; β , η , α ) = { 1 exp   [ ( I η ) β ] } α ,
I n = α η n Γ ( 1 + n β ) i = 0 ( 1 ) i ( i + 1 ) ( n + β ) / β Γ ( α ) i ! Γ ( α i ) .
α 3.931 ( D ρ 0 ) 0.519 ,
β ( α σ I 2 ) 6 / 11 ,
η = 1 α Γ ( 1 + 1 / β ) g ( α , β ) ,
g ( α , β ) = i = 0 ( 1 ) i ( i + 1 ) ( 1 + β ) / β Γ ( α ) i ! Γ ( α i ) .

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