Abstract

We study the parameter estimation problem for the Gamma-Gamma turbulence model for free-space optical communication. An estimation scheme for the shape parameters of the Gamma-Gamma distribution is proposed based on the concept of fractional moments and convex optimization. To improve the estimation performance, we further propose a modified scheme which exploits the relationship between the Gamma-Gamma shape parameters in free-space optical communication. Simulation results reveal that the modified estimation scheme can achieve satisfactory performance for a wide range of turbulence conditions.

© 2010 OSA

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  1. G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta (Lond.) 28, 715–728 (1981).
    [CrossRef]
  2. R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71(12), 1440–1445 (1981).
    [CrossRef]
  3. J. H. Churnside and R. G. Frehlich, “Experimental evaluation of lognormally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6(11), 1760–1766 (1989).
    [CrossRef]
  4. E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40(9), 546–550 (1978).
    [CrossRef]
  5. M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
    [CrossRef]
  6. A. C. Cohen., “Estimating parameters of logarithmic-normal distributions by maximum likelihood,” J. Am. Stat. Assoc. 46(254), 206–212 (1951).
    [CrossRef]
  7. A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small samples from a three-parameter lognormal distribution,” J. Am. Stat. Assoc. 65(329), 212–225 (1970).
    [CrossRef]
  8. I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
    [CrossRef]
  9. D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
    [CrossRef]
  10. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications (SPIE Press, 2001).
  11. A. Prokeš, “Modeling of atmospheric turbulence effect on terrestrial FSO link,” Radio Eng. 18(1), 42–47 (2009).
  12. D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), 036001–036007 (2009).
    [CrossRef]
  13. W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with gamma-gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
    [CrossRef]
  14. A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laser scintillation,” Pure Appl. Opt. 7(5), 1013–1032 (1998).
    [CrossRef]

2009 (2)

A. Prokeš, “Modeling of atmospheric turbulence effect on terrestrial FSO link,” Radio Eng. 18(1), 42–47 (2009).

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), 036001–036007 (2009).
[CrossRef]

2007 (1)

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with gamma-gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

1999 (1)

D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
[CrossRef]

1998 (1)

A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laser scintillation,” Pure Appl. Opt. 7(5), 1013–1032 (1998).
[CrossRef]

1993 (1)

I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
[CrossRef]

1989 (1)

1981 (2)

R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71(12), 1440–1445 (1981).
[CrossRef]

G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta (Lond.) 28, 715–728 (1981).
[CrossRef]

1978 (1)

E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40(9), 546–550 (1978).
[CrossRef]

1970 (1)

A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small samples from a three-parameter lognormal distribution,” J. Am. Stat. Assoc. 65(329), 212–225 (1970).
[CrossRef]

1951 (1)

A. C. Cohen., “Estimating parameters of logarithmic-normal distributions by maximum likelihood,” J. Am. Stat. Assoc. 46(254), 206–212 (1951).
[CrossRef]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71(12), 1440–1445 (1981).
[CrossRef]

Boashash, B.

D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
[CrossRef]

Churnside, J. H.

Cohen, A. C.

A. C. Cohen., “Estimating parameters of logarithmic-normal distributions by maximum likelihood,” J. Am. Stat. Assoc. 46(254), 206–212 (1951).
[CrossRef]

Consortini, A.

A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laser scintillation,” Pure Appl. Opt. 7(5), 1013–1032 (1998).
[CrossRef]

Frehlich, R. G.

Gappmair, W.

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with gamma-gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[CrossRef]

Iskander, D. R.

D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
[CrossRef]

Jakeman, E.

E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40(9), 546–550 (1978).
[CrossRef]

Joughin, I. R.

I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
[CrossRef]

Muhammad, S. S.

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with gamma-gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[CrossRef]

Munro, A. H.

A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small samples from a three-parameter lognormal distribution,” J. Am. Stat. Assoc. 65(329), 212–225 (1970).
[CrossRef]

Parry, G.

G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta (Lond.) 28, 715–728 (1981).
[CrossRef]

Percival, D. B.

I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
[CrossRef]

Philips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Phillips, R. L.

Prokeš, A.

A. Prokeš, “Modeling of atmospheric turbulence effect on terrestrial FSO link,” Radio Eng. 18(1), 42–47 (2009).

Pusey, P. N.

E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40(9), 546–550 (1978).
[CrossRef]

Rigal, F.

A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laser scintillation,” Pure Appl. Opt. 7(5), 1013–1032 (1998).
[CrossRef]

Voelz, D. G.

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), 036001–036007 (2009).
[CrossRef]

Winebrenner, D. P.

I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
[CrossRef]

Wixley, R. A. J.

A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small samples from a three-parameter lognormal distribution,” J. Am. Stat. Assoc. 65(329), 212–225 (1970).
[CrossRef]

Xiao, X.

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), 036001–036007 (2009).
[CrossRef]

Zoubir, A. M.

D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
[CrossRef]

Electron. Lett. (1)

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with gamma-gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

I. R. Joughin, D. B. Percival, and D. P. Winebrenner, “Maximum likelihood estimation of K distribution parameters for SAR data,” IEEE Trans. Geosci. Rem. Sens. 31(5), 989–999 (1993).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. R. Iskander, A. M. Zoubir, and B. Boashash, “A method for estimating the parameters of the K distribution,” IEEE Trans. Signal Process. 47(4), 1147–1151 (1999).
[CrossRef]

J. Am. Stat. Assoc. (2)

A. C. Cohen., “Estimating parameters of logarithmic-normal distributions by maximum likelihood,” J. Am. Stat. Assoc. 46(254), 206–212 (1951).
[CrossRef]

A. H. Munro and R. A. J. Wixley, “Estimators based on order statistics of small samples from a three-parameter lognormal distribution,” J. Am. Stat. Assoc. 65(329), 212–225 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (Lond.) (1)

G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta (Lond.) 28, 715–728 (1981).
[CrossRef]

Opt. Eng. (2)

M. A. Al-Habash, L. C. Andrews, and R. L. Philips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), 036001–036007 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

E. Jakeman and P. N. Pusey, “The significance of K-distributions in scattering experiments,” Phys. Rev. Lett. 40(9), 546–550 (1978).
[CrossRef]

Pure Appl. Opt. (1)

A. Consortini and F. Rigal, “Fractional moments and their usefulness in atmospheric laser scintillation,” Pure Appl. Opt. 7(5), 1013–1032 (1998).
[CrossRef]

Radio Eng. (1)

A. Prokeš, “Modeling of atmospheric turbulence effect on terrestrial FSO link,” Radio Eng. 18(1), 42–47 (2009).

Other (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications (SPIE Press, 2001).

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Figures (4)

Fig. 1
Fig. 1

Gamma-Gamma shape parameters α and β as functions of σR.

Fig. 2
Fig. 2

Gamma-Gamma PDF’s with Rytov variance σ R 2 = 0.25, 2, and 11.

Fig. 3
Fig. 3

MSE performance of the MoM/CVX estimator, the modified MoM/CVX estimator, as well as the estimator for the root of Rytov variance with k = 0.5 and sample size N = 100,000.

Fig. 4
Fig. 4

Flow chart of the modified MoM/CVX Gamma-Gamma shape parameter estimator.

Equations (18)

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f G ( I ) = 2 ( α β ) ( α + β ) / 2 λ Γ ( α ) Γ ( β ) ( I λ ) α + β 2 1 K α β ( 2 α β I / λ ) ,     α > 0 , β > 0 , λ > 0
α = g ( σ R ) = [ exp ( 0.49 σ R 2 ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ) 1 ] 1
β = h ( σ R ) = [ exp ( 0.51 σ R 2 ( 1 + 0.69 σ R 12 / 5 ) 5 / 6 ) 1 ] 1
μ k + 1 μ k = 1 + k α + k β + k 2 α β .
μ 2 = 1 + 1 α + 1 β + 1 α β .
{ 1 α + 1 β = c 1 α 1 β = d
c = k 2 μ 2 μ k + 1 μ k ( k 2 1 ) k 2 k ,
d = k μ 2 μ k + 1 μ k ( k 1 ) k k 2 .
x 2 c d x + 1 d = 0.
α ^ = c ^ 2 d ^ + 1 2 c ^ 2 d ^ 2 4 d ^ ,
β ^ = c ^ 2 d ^ 1 2 c ^ 2 d ^ 2 4 d ^
lim k 0 c = lim k 0 k 2 μ 2 μ k + 1 μ k ( k 2 1 ) k 2 k = lim k 0 2 k μ 2 ( 1 α + 1 β + 2 k α β ) 2 k 2 k 1 = 1 α + 1 β ,
lim k 0 d = lim k 0 k μ 2 μ k + 1 μ k ( k 1 ) k k 2 = lim k 0 μ 2 ( 1 α + 1 β + 2 k α β ) 1 1 2 k = 1 α 1 β .
Δ = ( c ^ d ^ ) 2 4 d ^ > 0.
minimize α , β         [ f ( α ) 0 ] 2 + [ f ( β ) 0 ] 2 subject   to       α > 0 , β > 0 , α β .
α ^ = β ^ = c ^ 2 d ^ .
σ ^ R = h 1 ( β ^ )
α ^ i m p v = g ( h 1 ( β ^ ) ) .

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