Abstract

Free-space optical communication can experience severe fading due to optical scintillation in long-range links. Channel estimation is also corrupted by background and electrical noise. Accurate estimation of channel parameters and scintillation index (SI) depends on perfect removal of background irradiance. In this paper, we propose three different methods, the minimum-value (MV), mean-power (MP), and maximum-likelihood (ML) based methods, to remove the background irradiance from channel samples. The MV and MP methods do not require knowledge of the scintillation distribution. While the ML-based method assumes gamma–gamma scintillation, it can be easily modified to accommodate other distributions. Each estimator’s performance is compared using simulation data as well as experimental measurements. The estimators’ performance are evaluated from low- to high-SI areas using simulation data as well as experimental trials. The MV and MP methods have much lower complexity than the ML-based method. However, the ML-based method shows better SI and background-irradiance estimation performance.

© 2013 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  2. A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications (Springer, 2008).
  3. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering, 2001).
  4. J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978), vol. 25.
  5. P. Beckmann, Probability in Communication Engineering (Harcourt, Brace & World, 1967).
  6. E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
    [CrossRef]
  7. 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]
  8. 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), pp. 181–206.
  9. E. Jakeman and P. N. Pusey, “Significance of k distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550(1978).
    [CrossRef]
  10. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099–2108 (2007).
    [CrossRef]
  11. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma–gamma atmospheric turbulence model,” Opt. Express 18, 12824–12831 (2010).
    [CrossRef]
  12. J. Anguita, M. Neifeld, B. Hildner, and B. Vasic, “Rateless coding on experimental temporally correlated FSO channels,” J. Lightwave Technol. 28, 990–1002 (2010).
    [CrossRef]
  13. 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]
  14. D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. 51, 3145–3152 (2012).
    [CrossRef]
  15. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  16. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  17. H. A. David, Order Statistics (Wiley, 2003).
  18. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  19. S. S. Rao, Engineering Optimization Theory and Practice(Wiley, 1996).
  20. A. Dogandzic and J. Jin, “Maximum likelihood estimation of statistical properties of composite gamma-lognormal fading channels,” IEEE Trans. Signal Process. 52, 2940–2945(2004).
    [CrossRef]
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  22. A. Khatoon, W. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in Australian Communications Theory Workshop (AusCTW), 2011 (IEEE, 2011), pp. 112–117.

2012 (1)

2010 (2)

2009 (1)

2007 (1)

2004 (1)

A. Dogandzic and J. Jin, “Maximum likelihood estimation of statistical properties of composite gamma-lognormal fading channels,” IEEE Trans. Signal Process. 52, 2940–2945(2004).
[CrossRef]

2001 (1)

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]

1999 (1)

1978 (1)

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

1976 (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

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]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Andrews, L.

Andrews, L. C.

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering, 2001).

Anguita, J.

Beckmann, P.

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

Cheng, J.

Cowley, W.

D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. 51, 3145–3152 (2012).
[CrossRef]

A. Khatoon, W. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in Australian Communications Theory Workshop (AusCTW), 2011 (IEEE, 2011), pp. 112–117.

David, H. A.

H. A. David, Order Statistics (Wiley, 2003).

Dogandzic, A.

A. Dogandzic and J. Jin, “Maximum likelihood estimation of statistical properties of composite gamma-lognormal fading channels,” IEEE Trans. Signal Process. 52, 2940–2945(2004).
[CrossRef]

Garrido-Balsells, J. M.

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), pp. 181–206.

Giggenbach, D.

Grant, K.

Hildner, B.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering, 2001).

Jakeman, E.

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

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

Jin, J.

A. Dogandzic and J. Jin, “Maximum likelihood estimation of statistical properties of composite gamma-lognormal fading channels,” IEEE Trans. Signal Process. 52, 2940–2945(2004).
[CrossRef]

Jurado-Navas, A.

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), pp. 181–206.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Khatoon, A.

A. Khatoon, W. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in Australian Communications Theory Workshop (AusCTW), 2011 (IEEE, 2011), pp. 112–117.

Letzepis, N.

A. Khatoon, W. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in Australian Communications Theory Workshop (AusCTW), 2011 (IEEE, 2011), pp. 112–117.

Lyke, S. D.

Majumdar, A. K.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications (Springer, 2008).

Neifeld, M.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Paris, J. F.

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), pp. 181–206.

Perlot, N.

Phillips, R. L.

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering, 2001).

Puerta-Notario, A.

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), pp. 181–206.

Pusey, P. N.

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

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

Rao, S. S.

S. S. Rao, Engineering Optimization Theory and Practice(Wiley, 1996).

Recolons, J.

Ricklin, J. C.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications (Springer, 2008).

Roggemann, M. C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978), vol. 25.

Vasic, B.

Vetelino, F. S.

Voelz, D. G.

Wang, N.

Young, C.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. 24, 806–814 (1976).
[CrossRef]

IEEE Trans. Signal Process. (1)

A. Dogandzic and J. Jin, “Maximum likelihood estimation of statistical properties of composite gamma-lognormal fading channels,” IEEE Trans. Signal Process. 52, 2940–2945(2004).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Eng. (1)

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]

Opt. Express (1)

Phys. Rev. Lett. (1)

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

Other (12)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

H. A. David, Order Statistics (Wiley, 2003).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

S. S. Rao, Engineering Optimization Theory and Practice(Wiley, 1996).

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), pp. 181–206.

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

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications (Springer, 2008).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering, 2001).

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978), vol. 25.

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

A. Khatoon, W. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in Australian Communications Theory Workshop (AusCTW), 2011 (IEEE, 2011), pp. 112–117.

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Figures (4)

Fig. 1.
Fig. 1.

Estimator RMS error performance comparison with P=1, z=1, where the solid curve with circle markers shows the CRB, dashed curve shows the ML-based method, the solid curve with diamond markers shows the ML-GG method [22], and the solid curve with square markers shows the MOM-GG method [11]. (a) RMS error for α with β=2 and (b) RMS error for β with α=6.

Fig. 2.
Fig. 2.

Comparison of estimation methods for simulated time-varying sinusoidal background irradiance: solid curve, ML-based method; dotted–dashed curve, MV method; and dashed curve, MP method. (a) mean bias error, (b) RMS error, and (c) mean SI estimation error.

Fig. 3.
Fig. 3.

Estimation of background irradiance from experimental data.

Fig. 4.
Fig. 4.

Estimation of background irradiance from frequently blocked experimental data.

Tables (1)

Tables Icon

Table 1. Estimation Error Statistics with Experimental Data

Equations (32)

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I(t)=Px(t)+z(t),
y[n]=Px[n]+z[n].
y[n]=Px[n]+z,
fY(y)=1PfX(yzP;θ).
SIXσX2μX2=E[X2]E[X]21.
SIY=SIX(1+1γ)2,
z^=minn=1,,Ny[n].
fZ^(u)=N[1FY(u)]N1fY(u).
E[z^z]=P0(1FX(x))Ndx>0,
z^off=1Noffn=1Noffyoff[n],
P^=1Nonn=1Nonyon[n]z^off.
z^on[n]=g[n]*yon[n]P^,
L(y;P,z,θ)=n=1NlogfY(y[n])=NlogP+n=1NlogfX(y[n]zP;θ).
(P^ML,z^ML,θ^ML)=argmaxP,z,θL(y;P,z,θ),
fX(x;α,β)=2(αβ)(α+β)2Γ(α)Γ(β)xα+β21Kαβ(2αβx),
z^=argmaxzL(y;P,z,θ),
zL(y;P,z,θ)=0.
zL(y;P,z,α,β)=(α+β21)n=1N1y[n]zαβPn=1N1y[n]zKαβ(2αβP(y[n]z))Kαβ(2αβP(y[n]z)),
Kν(a)=ddaKν(a)=12(Kν1(a)+Kν+1(a)).
zL(y;P^(i1),z,α^(i1),β^(i1))=0,
θ^=argmaxθL(y;P,z,θ),
αL(y;P,z,α,β)=N2(1+βα+log(αβ))Nψ(α)+12n=1Nlog(y[n]zP)+n=1NαKαβ(2αβP(y[n]z))Kαβ(2αβP(y[n]z)),
ddνKν(s(ν)x)=s(ν)xKν(s(ν)x)+ην(s(ν)x),
ην(a)=0ueacosh(u)sinh(νu)du.
z^(0)=(1ρ)min(y[1],,y[N]),
P^(0)=1Nn=1Ny[n]z^(0),
S^I=Nn=1N(y[n]z^(0))2[n=1N(y[n]z^(0))]21.
SI=1α+1β+1αβ,
β=1+ααS^I1
αL(y;P^(0),z^(0),α,1+ααS^I1)=0,
crb(α)=(E[(αL(y;P,z,α,β))2])1
z[n]=z˜+bsin(2πFFsn),

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