Abstract

We propose a method to obtain the maximum likelihood (ML) parameter estimation of the Γ-Γ (\Γ-\Γ) distribution representing the free space optical (FSO) channel irradiance fluctuations. The proposed method is based on the expectation maximization (EM) algorithm and the generalized Newton method using a non-quadratic approximation. The numerical results show that, for all turbulence conditions, the proposed ML method is more accurate than the fractional moments (FMOM) method and the numerical ML method (two dimensional numerical maximization of log-likelihood function using the Nelder-Mead algorithm). Moreover, the proposed ML is a fast and stable iterative method, because the iterations always converge to the global optimum with high convergence rate.

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  1. L. Andrews, R. L. Philips, C. Y. Hopen, Laser Beam Scintillation With Applications (SPIE, 2001).
  2. A. C. Motlagh, V. Ahmadi, Z. Ghassemlooy, K. Abedi, "The effect of atmospheric turbulence on the performance of the free space optical communications," Proc. 6th Int. Symp. Commun. Syst., Netw. Digital Signal Process. (2008) pp. 540-543.
  3. L. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," J. Opt. Fiber Commun. Res. 2, 345-396 (2005).
  4. J. W. Strohbehn, "Laser beam propagation in the atmosphere," Optic ACTA 28, (1981).
  5. N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, M. Matthaiou, "Inverse gaussian modeling of turbulence-induced fading in free-space optical systems," J. Lightw. Technol. 29, (2011).
  6. E. Jakeman, P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antennas Propag. 24, 806-814 (1976).
  7. L. C. Andrews, R. L. Phillips, "I-K distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Amer. A 2, 160-163 (1985).
  8. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace and World, 1967).
  9. M. Al-Habash, L. Andrews, R. Phillips, "Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media," Opt. Eng. 40, 554-562 (2001).
  10. A. Komaee, "Channel estimation for free-space optical communication," Proc. IEEE Conf. Decision Contr. Eur. Contr. (CDC-ECC) (2011) pp. 7299-7304.
  11. N. D. Chatzidiamantis, G. K. Karagiannidis, M. Uysal, "Generalized maximum-likelihood sequence detection for photon-counting free space optical systems," IEEE Trans. Commun. 58, 3381-3385 (2010).
  12. N. D. Chatzidiamantis, M. Uysal, T. A. Tsiftsis, G. K. Karagiannidis, "Iterative near maximum-likelihood sequence detection for MIMO optical wireless systems," J. Lightw. Technol. 28, 1064-1070 (2010).
  13. Y. Ko, M. S. Alouini, "Estimation of Nakagami-m fading channel parameters with application to optimized transmitter diversity systems," IEEE Trans. Wireless Commun. 2, 250-259 (2003).
  14. H. E. Nistazakis, T. A. Tsiftsis, G. S. Tombras, "Performance analysis of free-space optical communication systems over atmospheric turbulence channels," IET Communications 3, 1402-1409 (2009).
  15. A. D. Luong, C. T. Truong, A. T. Pham, "Effect of APD and thermal noises on the performance of SC-BPSK/FSO systems over turbulence channels," Proc. Asia-Pacific Conf. Commun. (2012) pp. 344-349.
  16. M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, "On the capacity of hybrid FSO/RF links," Proc. Global Telecommun. Conf. (GLOBECOM) (2010) pp. 1-5.
  17. N. Wang, J. Cheng, "Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model," Opt. Exp. 18, 12824-12831 (2010).
  18. R. S. Raghavan, "A method for estimating parameters of K-distributed clutter," IEEE Trans. Aerosp. Electron. Syst. 27, 238-246 (1991).
  19. W. J. J. Roberts, S. Furui, "Maximum likelihood estimation of K-distribution parameters via the expectation-maximization algorithm," IEEE Trans. Signal Process. 48, 3303-3306 (2000).
  20. P. Chung, W. Roberts, J. Bohme, "Recursive K-distribution parameters estimation," IEEE Trans. Signal Process. 53, 397-402 (2005).
  21. I. R. Joughin, D. B. Percival, D. P. Winebrenner, "Maximum likelihood estimation of K-distribution parameters for SAR data," IEEE Trans. Geosci. Remote Sens. 32, 989-999 (1993).
  22. D. A. Abraham, A. P. Lyons, "Reliable methods for estimating the K-distribution shape parameter," IEEE J. Ocean. Eng. 35, 288-302 (2010).
  23. T. P. Minka, "Estimating a Γ distribution," http://research.mi-crosoft.com/en-us/um/people/minka/papers/minka-gamma.pdf (2002).
  24. T. P. Minka, "Beyond newton's method," http://research.mi-crosoft.com/~minka/papers/newton.html (2000).
  25. S. M. Kay, Fundamental of Statistical Signal Processing Estimation Theory (Printice Hall, 1993).
  26. M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (U.S. Dept. Commerce, Nat. Bur. Stand., 1972).
  27. G. Almkvist, D. Zeilberger, "The method of differentiating under the integral sign," J. Symb. Comp 10, 571-591 (1990).

2011

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, M. Matthaiou, "Inverse gaussian modeling of turbulence-induced fading in free-space optical systems," J. Lightw. Technol. 29, (2011).

2010

N. D. Chatzidiamantis, G. K. Karagiannidis, M. Uysal, "Generalized maximum-likelihood sequence detection for photon-counting free space optical systems," IEEE Trans. Commun. 58, 3381-3385 (2010).

N. D. Chatzidiamantis, M. Uysal, T. A. Tsiftsis, G. K. Karagiannidis, "Iterative near maximum-likelihood sequence detection for MIMO optical wireless systems," J. Lightw. Technol. 28, 1064-1070 (2010).

N. Wang, J. Cheng, "Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model," Opt. Exp. 18, 12824-12831 (2010).

D. A. Abraham, A. P. Lyons, "Reliable methods for estimating the K-distribution shape parameter," IEEE J. Ocean. Eng. 35, 288-302 (2010).

2009

H. E. Nistazakis, T. A. Tsiftsis, G. S. Tombras, "Performance analysis of free-space optical communication systems over atmospheric turbulence channels," IET Communications 3, 1402-1409 (2009).

2005

P. Chung, W. Roberts, J. Bohme, "Recursive K-distribution parameters estimation," IEEE Trans. Signal Process. 53, 397-402 (2005).

L. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," J. Opt. Fiber Commun. Res. 2, 345-396 (2005).

2003

Y. Ko, M. S. Alouini, "Estimation of Nakagami-m fading channel parameters with application to optimized transmitter diversity systems," IEEE Trans. Wireless Commun. 2, 250-259 (2003).

2001

M. Al-Habash, L. Andrews, R. Phillips, "Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media," Opt. Eng. 40, 554-562 (2001).

2000

W. J. J. Roberts, S. Furui, "Maximum likelihood estimation of K-distribution parameters via the expectation-maximization algorithm," IEEE Trans. Signal Process. 48, 3303-3306 (2000).

1993

I. R. Joughin, D. B. Percival, D. P. Winebrenner, "Maximum likelihood estimation of K-distribution parameters for SAR data," IEEE Trans. Geosci. Remote Sens. 32, 989-999 (1993).

1991

R. S. Raghavan, "A method for estimating parameters of K-distributed clutter," IEEE Trans. Aerosp. Electron. Syst. 27, 238-246 (1991).

1990

G. Almkvist, D. Zeilberger, "The method of differentiating under the integral sign," J. Symb. Comp 10, 571-591 (1990).

1985

L. C. Andrews, R. L. Phillips, "I-K distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Amer. A 2, 160-163 (1985).

1981

J. W. Strohbehn, "Laser beam propagation in the atmosphere," Optic ACTA 28, (1981).

1976

E. Jakeman, P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antennas Propag. 24, 806-814 (1976).

IEEE Trans. Geosci. Remote Sens.

I. R. Joughin, D. B. Percival, D. P. Winebrenner, "Maximum likelihood estimation of K-distribution parameters for SAR data," IEEE Trans. Geosci. Remote Sens. 32, 989-999 (1993).

IEEE Trans. Signal Process.

W. J. J. Roberts, S. Furui, "Maximum likelihood estimation of K-distribution parameters via the expectation-maximization algorithm," IEEE Trans. Signal Process. 48, 3303-3306 (2000).

IEEE J. Ocean. Eng.

D. A. Abraham, A. P. Lyons, "Reliable methods for estimating the K-distribution shape parameter," IEEE J. Ocean. Eng. 35, 288-302 (2010).

IEEE Trans. Aerosp. Electron. Syst.

R. S. Raghavan, "A method for estimating parameters of K-distributed clutter," IEEE Trans. Aerosp. Electron. Syst. 27, 238-246 (1991).

IEEE Trans. Antennas Propag.

E. Jakeman, P. N. Pusey, "A model for non-Rayleigh sea echo," IEEE Trans. Antennas Propag. 24, 806-814 (1976).

IEEE Trans. Commun.

N. D. Chatzidiamantis, G. K. Karagiannidis, M. Uysal, "Generalized maximum-likelihood sequence detection for photon-counting free space optical systems," IEEE Trans. Commun. 58, 3381-3385 (2010).

IEEE Trans. Signal Process.

P. Chung, W. Roberts, J. Bohme, "Recursive K-distribution parameters estimation," IEEE Trans. Signal Process. 53, 397-402 (2005).

IEEE Trans. Wireless Commun.

Y. Ko, M. S. Alouini, "Estimation of Nakagami-m fading channel parameters with application to optimized transmitter diversity systems," IEEE Trans. Wireless Commun. 2, 250-259 (2003).

IET Communications

H. E. Nistazakis, T. A. Tsiftsis, G. S. Tombras, "Performance analysis of free-space optical communication systems over atmospheric turbulence channels," IET Communications 3, 1402-1409 (2009).

J. Lightw. Technol.

N. D. Chatzidiamantis, M. Uysal, T. A. Tsiftsis, G. K. Karagiannidis, "Iterative near maximum-likelihood sequence detection for MIMO optical wireless systems," J. Lightw. Technol. 28, 1064-1070 (2010).

J. Opt. Fiber Commun. Res.

L. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," J. Opt. Fiber Commun. Res. 2, 345-396 (2005).

J. Lightw. Technol.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, M. Matthaiou, "Inverse gaussian modeling of turbulence-induced fading in free-space optical systems," J. Lightw. Technol. 29, (2011).

J. Opt. Soc. Amer. A

L. C. Andrews, R. L. Phillips, "I-K distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Amer. A 2, 160-163 (1985).

J. Symb. Comp

G. Almkvist, D. Zeilberger, "The method of differentiating under the integral sign," J. Symb. Comp 10, 571-591 (1990).

Opt. Exp.

N. Wang, J. Cheng, "Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model," Opt. Exp. 18, 12824-12831 (2010).

Opt. Eng.

M. Al-Habash, L. Andrews, R. Phillips, "Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media," Opt. Eng. 40, 554-562 (2001).

Optic ACTA

J. W. Strohbehn, "Laser beam propagation in the atmosphere," Optic ACTA 28, (1981).

Other

L. Andrews, R. L. Philips, C. Y. Hopen, Laser Beam Scintillation With Applications (SPIE, 2001).

A. C. Motlagh, V. Ahmadi, Z. Ghassemlooy, K. Abedi, "The effect of atmospheric turbulence on the performance of the free space optical communications," Proc. 6th Int. Symp. Commun. Syst., Netw. Digital Signal Process. (2008) pp. 540-543.

A. Komaee, "Channel estimation for free-space optical communication," Proc. IEEE Conf. Decision Contr. Eur. Contr. (CDC-ECC) (2011) pp. 7299-7304.

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

A. D. Luong, C. T. Truong, A. T. Pham, "Effect of APD and thermal noises on the performance of SC-BPSK/FSO systems over turbulence channels," Proc. Asia-Pacific Conf. Commun. (2012) pp. 344-349.

M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, "On the capacity of hybrid FSO/RF links," Proc. Global Telecommun. Conf. (GLOBECOM) (2010) pp. 1-5.

T. P. Minka, "Estimating a Γ distribution," http://research.mi-crosoft.com/en-us/um/people/minka/papers/minka-gamma.pdf (2002).

T. P. Minka, "Beyond newton's method," http://research.mi-crosoft.com/~minka/papers/newton.html (2000).

S. M. Kay, Fundamental of Statistical Signal Processing Estimation Theory (Printice Hall, 1993).

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (U.S. Dept. Commerce, Nat. Bur. Stand., 1972).

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