Abstract

The performance of lasercom systems operating in the atmosphere is reduced by optical turbulence, which causes irradiance fluctuations in the received signal. The result is a randomly fading signal. Fade statistics obtained from experimental data were compared to theoretical predictions based on the lognormal and gamma–gamma distributions. The probability of fade, the expected number of fades per second, and the mean fade time were calculated from the irradiance fluctuations of a Gaussian beam wave propagating through the atmosphere along a horizontal path, near ground, in the moderate-to-strong optical turbulence regime. Irradiance data were collected simultaneously at three receiving apertures, each with a different size. Atmospheric propagation parameters were inferred from the measurements and were used in calculations for the theoretical distributions. It was found that fade predictions made by the gamma–gamma and lognormal distributions provide an upper and lower bound, respectively, for the probability of fade and the number of fades per second for the irradiance data collected in the moderate-to-strong fluctuation regime. What is believed to be a new integral expression for the expected number of fades based on the gamma–gamma distribution was developed. This new expression tracked the gamma–gamma distributed data more closely than the existing approximation and resulted in a higher number of fades.

© 2007 Optical Society of America

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References

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  1. F. Strömqvist Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, "Characterizing the propagation path in moderate-to-strong optical turbulence," Appl. Opt. 45, 3534-3543 (2006).
    [CrossRef] [PubMed]
  2. F. Strömqvist 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] [PubMed]
  3. H. T. Yura and W. G. McKinley, "Optical scintillation statistics for IR ground-to-space laser communication systems," Appl. Opt. 22, 3353-3358 (1983).
    [CrossRef] [PubMed]
  4. L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system," Appl. Opt. 34, 7742-7751 (1995).
    [CrossRef] [PubMed]
  5. L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system: errata," Appl. Opt. 36, 6068 (1997).
    [CrossRef]
  6. L. C. Andrews and R. L. Phillips, "Pointing errors and fade statistics associated with a laser satellite communication system," Proc. SPIE 2956, 166-178 (1997).
    [CrossRef]
  7. L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
    [CrossRef]
  8. A. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," Journal of Optical and Fiber Communications Reports 2, 345-396 (2005).
    [CrossRef]
  9. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
    [CrossRef]
  10. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
    [CrossRef]
  11. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001), Chap. 8.4.
  12. A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
    [CrossRef]
  13. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001).
  14. 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]
  15. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
    [CrossRef]
  16. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
    [CrossRef]
  20. J. H. Churnside and R. J. Hill, "Probability density of irradiance scintillations for strong path-integrated refractive turbulence," J. Opt. Soc. Am. A 4, 727-733 (1987).
    [CrossRef]
  21. Z. Azar, H. M. Loebenstein, G. Appelbaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur, "Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence," Appl. Opt. 24, 2401-2407 (1985).
    [CrossRef] [PubMed]
  22. Equations (82)-(85) in Section 6.6 of Ref. 13.

2007 (1)

2006 (1)

2005 (1)

A. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," Journal of Optical and Fiber Communications Reports 2, 345-396 (2005).
[CrossRef]

2002 (1)

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

2001 (3)

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

2000 (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
[CrossRef]

1999 (2)

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[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]

1997 (2)

L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system: errata," Appl. Opt. 36, 6068 (1997).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Pointing errors and fade statistics associated with a laser satellite communication system," Proc. SPIE 2956, 166-178 (1997).
[CrossRef]

1995 (1)

1989 (1)

1987 (1)

1985 (1)

1983 (2)

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

H. T. Yura and W. G. McKinley, "Optical scintillation statistics for IR ground-to-space laser communication systems," Appl. Opt. 22, 3353-3358 (1983).
[CrossRef] [PubMed]

Al-habash, A.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
[CrossRef]

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[CrossRef]

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF 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.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
[CrossRef]

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[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, "Pointing errors and fade statistics associated with a laser satellite communication system," Proc. SPIE 2956, 166-178 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system: errata," Appl. Opt. 36, 6068 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system," Appl. Opt. 34, 7742-7751 (1995).
[CrossRef] [PubMed]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001), Chap. 8.4.

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

Appelbaum, G.

Azar, Z.

Azoulay, E.

Churnside, J. H.

Clare, B.

Corbett, K.

Cornish, C. S.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

Desmet, K. N.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

Diament, P.

Fischer, K. W.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

Grant, K.

Halavee, U.

Hill, R. J.

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
[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, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001), Chap. 8.4.

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

Lewinski, D. J.

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

Loebenstein, H. M.

Majumdar, A. K.

A. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," Journal of Optical and Fiber Communications Reports 2, 345-396 (2005).
[CrossRef]

McKinley, W. G.

Nash, J.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

Phillips, R. L.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
[CrossRef]

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[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, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system: errata," Appl. Opt. 36, 6068 (1997).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Pointing errors and fade statistics associated with a laser satellite communication system," Proc. SPIE 2956, 166-178 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillations and fade statistics for a satellite-communication system," Appl. Opt. 34, 7742-7751 (1995).
[CrossRef] [PubMed]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001), Chap. 8.4.

Recolons, J.

Tamir, M.

Teich, M. C.

Tjin-Tham-Sjin, D. E.

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[CrossRef]

Tur, M.

Vetelino, F. Strömqvist

Young, C.

Young, C. Y.

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[CrossRef]

Yu, P. T.

Yura, H. T.

Appl. Opt. (6)

IEEE Trans. Antennas Propag. (1)

D. J. Lewinski, "Nonstationary probabilistic target and clutter scattering models," IEEE Trans. Antennas Propag. AP-31, 490-498 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

Journal of Optical and Fiber Communications Reports (1)

A. K. Majumdar, "Free-space laser communication performance in the atmospheric channel," Journal of Optical and Fiber Communications Reports 2, 345-396 (2005).
[CrossRef]

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001).
[CrossRef]

Proc. SPIE (5)

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," in Proc. SPIE 4873, 79-89 (2002).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mean fade time of an optical communication channel under moderate-to-strong atmospheric turbulence," in Proc. SPIE 3927, 240-248 (2000).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Double-pass fade statistics of a laser beam under moderate to strong atmospheric turbulence," in Proc. SPIE 4272, 260-267 (2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, "Pointing errors and fade statistics associated with a laser satellite communication system," Proc. SPIE 2956, 166-178 (1997).
[CrossRef]

L. C. Andrews, C. Y. Young, A. Al-Habash, R. L. Phillips, and D. E. Tjin-Tham-Sjin, "Fade statistics associated with a space/ground laser communication link at large zenith angles," in Proc. SPIE 3763, 268-277 (1999).
[CrossRef]

Waves Random Media (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, "Aperture averaging of optical scintillations: power fluctuations and temporal spectrum," Waves Random Media 10, 53-70 (2000).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001).
[CrossRef]

Other (3)

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

Equations (82)-(85) in Section 6.6 of Ref. 13.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, 2001), Chap. 8.4.

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Figures (3)

Fig. 1
Fig. 1

Probability of fade as a function of the fade threshold parameter, F T , for the receiving apertures of diameters: (a) (b), 1   mm ; (c) (d) 5   mm ; and (e) (f) 13   mm . Comparisons are made between the experimental data (Data), the GG distribution and the LN for C n 2 = 6.5 × 10 14 (a), (c), (e), and C n 2 = 4.6 × 10 13 (b), (d), (f).

Fig. 2
Fig. 2

Expected number of fades per second as a function of the fade threshold parameter, F T , for the receiving apertures of diameters: (a) (b) 1 mm; (c) (d) 5   mm ; and (e) (f) 13   mm . Comparisons are made between the experimental data (Data), the LN distribution, the approximation of the GG-a distribution, and the integral expression for the GG-b for C n 2 = 6.5 × 10 14 (a), (c), (e), and C n 2 = 4.6 × 10 13 (b), (d), (f).

Fig. 3
Fig. 3

Mean fade time as a function of the fade threshold parameter, F T , for the receiving apertures of diameters: (a) (b) 1 mm; (c) (d) 5   mm ; and (e) (f) 13   mm . Comparisons are made between the experimental data (Data), the LN, the approximation of the GG-a distribution and the integral expression for the GG-b distribution for C n 2 = 6.5 × 10 14 (a), (c), (e), and C n 2 = 4.6 × 10 13 (b), (d), (f).

Tables (1)

Tables Icon

Table 1 Atmospheric Parameters Inferred From the Experimental Data and the PDF Parameters Used to Calculate the Lognormal and Gamma–Gamma Distributions When Compared to the Experimental Data

Equations (26)

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P f a = Pr ( I I T ) = 0 I T p I ( I ) d I ,
F T = 10 log 10 ( I ( 0 , L ) I T ) [ dB ] ,
p I ( I ) = 1 I 2 π σ ln I 2 exp [ [ ln ( I ) + 1 2 σ ln I 2 ] 2 2 σ ln I 2 ] ,   I > 0 ,
P f a = 1 2 [ 1 + erf ( 1 2 σ ln I 2 0.23 F T 2 σ ln I ) ] ,
σ ln I 2 = ln ( σ I 2 + 1 ) ,
I = x y ,
p I ( I ) = 2 Γ ( α ) Γ ( β ) I ( α β I ) ( α + β ) / 2 K α β ( 2 α β I ) I > 0 ,
P f a = π Γ ( α ) Γ ( β ) sin [ π ( α β ) ] × { ( α β ) β β Γ ( β α 1 ) exp ( 0.23 β F T ) × 1 F 2 [ β ; β + 1 , β α 1 ; α β exp ( 0.23 F T ) ] ( α β ) α α Γ ( α β 1 ) exp ( 0.23 α F T ) × 1 F 2 [ α ; α + 1 , α β 1 ; α β exp ( 0.23 F T ) ] } ,
α = 1 σ x 2 = 1 exp ( σ ln x 2 ) 1 ,
β = 1 σ y 2 = 1 exp ( σ ln y 2 ) 1 ,
n ( I T ) = 1 2 | I ˙ | p I , I ˙ ( I T , I ˙ ) d I ˙ ,
p I , I ˙ ( I , I ˙ ) = p I ( I ) p I ˙ ( I ˙ | I ) ,
t ( I T ) = Pr ( I I T ) n ( I T ) .
p I , I ˙ ( I , I ˙ ) = 1 I σ ln I 2 π exp { [ ln ( I ) + 1 2 σ ln I 2 ] 2 2 σ ln I 2 } × 1 2 b I 2 π exp ( I ˙ 2 8 b 2 I 2 ) , I > 0 ,
n ( F T ) = ν 0 exp [ ( 1 2 σ ln I 2 0.23 F T ) 2 2 σ ln I 2 ] ,
ν 0 = b π σ ln I = 1 2 π B ln I ( 0 ) B ln I ( 0 ) [ Hz ] ,
p I , I ˙ ( I , I ˙ ) = p I ( I ) p I ˙ ( I ˙ | I ) = 2 Γ ( α ) Γ ( β ) I ( α β I ) ( α + β ) / 2 K α β ( 2 α β I ) × 1 8 π b 2 I exp [ I ˙ 2 8 I b 2 ] , I > 0 .
n ( F T ) = 2 ν 0 σ ln I 2 π α β Γ ( α ) Γ ( β ) [ α β exp ( 0.23 F T ) ] ( α + β 1 ) 2 × K α β [ 2 α β exp ( 0.23 F T ) ] ,
I = x y , I ˙ = x ˙ y + x y ˙ ,
p x , x ˙ ( x , x ˙ ) = 1 8 π b x 2 α α Γ ( α ) x α 3 / 2 e α x e x ˙ 2 / 8 x b x 2 , x > 0 ,
p y , y ˙ ( y , y ˙ ) = 1 8 π b y 2 β β Γ ( β ) y β 3 / 2 e β y e y ˙ 2 / 8 y b y 2 , y > 0 ,
Φ I , I ˙ ( u , v ) = e i ( u I + v I ˙ ) p I , I ˙ ( I , I ˙ ) d I d I ˙ = exp { i [ u x y + v ( x ˙ y + x y ˙ ) ] } p x , x ˙ ( x , x ˙ ) × p y , y ˙ ( y , y ˙ ) d x d x ˙ d y d y ˙ = 1 4 π b 2 α α β β Γ ( α ) Γ ( β ) 0 x α 3 / 2 e α x 0 y β 3 / 2 e β y e i u x y × e x ˙ 2 / 4 b 2 x e i v y x ˙ e y ˙ 2 / 4 b 2 y e i v x y ˙ d y ˙ d x ˙ d y d x .
Φ I , I ˙ ( u , v ) = α α β β Γ ( α ) Γ ( β ) 0 x α 1 e α x 0 y β 1 × exp [ β y + i u x y b 2 v 2 ( x 2 y + y 2 x ) ] d y d x .
p I , I ˙ ( I , I ˙ ) = 1 ( 2 π ) 2 e i ( I u + I ˙ v ) Φ I , I ˙ ( u , v ) d u d v = α α β β Γ ( α ) Γ ( β ) 0 x α 1 e α x 0 y β 1 e β y × { 1 2 π e i I ˙ v exp [ b 2 ( x 2 y + y 2 x ) v 2 ] d v } × { 1 2 π e i I u e i x y u d u } d y d x .
p I , I ˙ ( I , I ˙ ) = 1 4 π b 2 α α β β Γ ( α ) Γ ( β ) I β 3 / 2 0 x α β 1 / 2 I + x 2 × exp [ α x β I x I ˙ 2 x 4 b 2 I ( I + x 2 ) ] d x .
n ( I T ) = π σ ln I ν 0 α α β β Γ ( α ) Γ ( β ) I T β 1 / 2 0 x α β 3 / 2 I T + x 2 × exp [ α x β I T x ] d x .

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