Abstract

Analytic expression of the receiver-aperture-averaged scintillation index (SI) was derived for Gaussian-beam waves propagating through non-Kolmogorov maritime atmospheric environment by establishing a generalized maritime atmospheric spectrum model. The error performance of an intensity-modulated and direct-detection (IM/DD) free-space optical (FSO) system was investigated using the derived SI and log-normal distribution. The combined effects of non-Kolmogorov power-law exponent, turbulence inner scale, structure parameter, propagation distance, receiver aperture, and wavelength were also evaluated. Results show that inner scale and power-law exponent obviously affect SI. Large wavelength and receiver aperture can mitigate the effects of turbulence. The proposed model can be evaluated ship-to-ship/shore FSO system performance.

© 2015 Optical Society of America

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Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence

Linyan Cui, Bindang Xue, Lei Cao, Shiling Zheng, Wenfang Xue, Xiangzhi Bai, Xiaoguang Cao, and Fugen Zhou
Opt. Express 19(18) 16872-16884 (2011)

References

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  1. A. K. Majumdar, Advanced Free Space Optics (FSO): A Systems Approach, (Springer, 2014).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).
  3. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
    [Crossref]
  4. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
    [Crossref]
  5. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
    [Crossref]
  6. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010).
    [Crossref] [PubMed]
  7. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  9. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
    [Crossref] [PubMed]
  10. K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
    [Crossref]
  11. F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
    [Crossref] [PubMed]
  12. I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
    [Crossref]
  13. O. Korotkova, S. Avramov-Zamurovic, R. Malek-Madani, and C. Nelson, “Probability density function of the intensity of a laser beam propagating in the maritime environment,” Opt. Express 19(21), 20322–20331 (2011).
    [Crossref] [PubMed]
  14. L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
    [Crossref]
  15. L. Cui, “Temporal power spectra of irradiance scintillation for infrared optical waves’ propagation through marine atmospheric turbulence,” J. Opt. Soc. Am. A 31(9), 2030–2037 (2014).
    [Crossref] [PubMed]
  16. L. Cui, “Analysis of marine atmospheric turbulence effects on infrared imaging system by angle of arrival fluctuations,” Infrared Phys. Technol. 68, 28–34 (2015).
    [Crossref]
  17. F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).
  18. M. Li and M. Cvijetic, “Coherent free space optics communications over the maritime atmosphere with use of adaptive optics for beam wavefront correction,” Appl. Opt. 54(6), 1453–1462 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  22. 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(33), 6511–6527 (2009).
    [Crossref] [PubMed]
  23. W. H. Press, S. A. Teukolsky, W. A. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge Univ., 1992).

2015 (3)

2014 (3)

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[Crossref]

L. Cui, “Temporal power spectra of irradiance scintillation for infrared optical waves’ propagation through marine atmospheric turbulence,” J. Opt. Soc. Am. A 31(9), 2030–2037 (2014).
[Crossref] [PubMed]

2011 (4)

2010 (2)

L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[Crossref]

2009 (2)

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(33), 6511–6527 (2009).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[Crossref]

2008 (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

2007 (1)

1999 (1)

1994 (1)

Agrawal, B.

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[Crossref]

Al-Habash, M. A.

Andrews, L. C.

Avramov-Zamurovic, S.

Bai, X.

Belafhal, A.

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

Boustimi, M.

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

Cao, L.

Cao, X.

Cui, L.

Cvijetic, M.

Du, W.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[Crossref]

Grayshan, K.

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Han, Q.

Hopen, C. Y.

Khannous, F.

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

Korotkova, O.

Li, M.

Lu, G.

Lyke, S. D.

Ma, J.

Malek-Madani, R.

Miller, W. B.

Nebdi, H.

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

Nelson, C.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[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(6), 1417–1429 (1999).
[Crossref]

Restaino, S.

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[Crossref]

Ricklin, J. C.

Roggemann, M. C.

Tan, L.

Toselli, I.

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[Crossref]

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Voelz, D. G.

Xue, B.

Xue, W.

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Yu, S.

Zhai, C.

Zheng, S.

Zhou, F.

Appl. Opt. (2)

IEEE Trans. Antenn. Propag. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[Crossref]

Infrared Phys. Technol. (2)

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[Crossref]

L. Cui, “Analysis of marine atmospheric turbulence effects on infrared imaging system by angle of arrival fluctuations,” Infrared Phys. Technol. 68, 28–34 (2015).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Express (4)

Phys. Chem. News (1)

F. Khannous, M. Boustimi, H. Nebdi, and A. Belafhal, “Li’s flattened Gaussian beams propagation in maritime atmospheric turbulence,” Phys. Chem. News 73, 73–82 (2014).

Proc. SPIE (1)

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[Crossref]

Waves Random Complex Media (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Other (5)

A. K. Majumdar, Advanced Free Space Optics (FSO): A Systems Approach, (Springer, 2014).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

F. W. J. Olver, NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).

W. H. Press, S. A. Teukolsky, W. A. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge Univ., 1992).

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Figures (9)

Fig. 1
Fig. 1 Scintillation index of Gaussian-beam waves as a function of L with different α values, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.
Fig. 2
Fig. 2 Scintillation index of Gaussian-beam waves as a function of L with different l 0 value, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.
Fig. 3
Fig. 3 Scintillation index of Gaussian-beam waves as a function of L with different D, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.
Fig. 4
Fig. 4 Scintillation index of Gaussian-beam waves as a function of L with different C ˜ n 2 , in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.
Fig. 5
Fig. 5 Scintillation index of Gaussian-beam waves as a function of L with different λ, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.
Fig. 6
Fig. 6 BER of FSO links against normalized SNR 0 for different α values in weak non-Kolmogorov maritime atmospheric environment.
Fig. 7
Fig. 7 BER of FSO links against normalized SNR 0 for different l 0 in weak non-Kolmogorov maritime atmospheric environment.
Fig. 8
Fig. 8 BER of FSO links against normalized SNR 0 for different D in weak non-Kolmogorov maritime atmospheric environment.
Fig. 9
Fig. 9 BER of FSO links against SNR 0 for different λ in weak non-Kolmogorov maritime atmospheric environment

Equations (42)

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Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) 11 / 6 [ 1 + a 1 ( κ / κ l ) + a 2 ( κ / κ l ) 7 / 6 ] ,
Φ n ( κ , α ) = A ( α ) C ˜ n 2 exp ( κ 2 / κ H 2 ) ( κ 2 + κ 0 2 ) α / 2 [ 1 + a 1 ( κ κ H ) + a 2 ( κ κ H ) 3 α / 2 ] , 3 < α < 5 ,
A ( α ) = Γ ( α 1 ) 4 π 2 sin [ ( α 3 ) π 2 ] ,
c 0 ( α ) = { π A ( α ) [ Γ ( 3 2 α 2 ) 3 α 3 + a 1 Γ ( 2 α 2 ) 4 α 3 + a 2 Γ ( 3 3 α 4 ) 4 α 2 ] } 1 α 5 .
μ [ ( κ / κ l ) γ ] = μ 1 [ ( κ / κ l ) γ 1 ] + μ 2 [ ( κ / κ l ) γ 2 ] + μ 3 [ ( κ / κ l ) γ 3 ] .
Φ n ( κ , α ) = A ( α ) C ˜ n 2 μ exp ( κ 2 / κ H 2 ) ( κ 2 + κ 0 2 ) α / 2 [ ( κ / κ l ) γ ] , 3 < α < 5.
δ IG 2 ( D ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ , α ) exp ( Λ L κ 2 ξ 2 k ) exp ( D 2 κ 2 ξ 2 16 ) × [ 1 cos ( L κ 2 k ξ ( 1 Θ ¯ ξ ) ) ] d κ d ξ ,
δ IG 2 ( D ) = 8 π 2 k 2 L A ( α ) C ˜ n 2 μ 0 1 0 κ exp ( κ 2 / κ H 2 ) ( κ 2 + κ 0 2 ) α / 2 ( κ κ H ) γ exp ( Λ L κ 2 ξ 2 k D 2 κ 2 ξ 2 16 ) × [ 1 cos ( L κ 2 k ξ ( 1 Θ ¯ ξ ) ) ] d κ d ξ .
δ IG 2 ( D ) = 8 π 2 k 2 L A ( α ) C ˜ n 2 μ κ H γ Re 0 1 0 κ γ + 1 ( κ 2 + κ 0 2 ) α / 2 { exp [ ( 1 κ H 2 + Λ L ξ 2 k + D 2 ξ 2 16 ) κ 2 ] exp [ ( 1 κ H 2 + Λ L ξ 2 k + D 2 ξ 2 16 + i L k ξ ( 1 Θ ¯ ξ ) ) κ 2 ] } d κ d ξ .
U ( a ; c ; x ) = 1 Γ ( a ) 0 exp ( x t ) t a 1 ( 1 + t ) c a 1 d t ,
δ IG 2 ( D ) = 8 π 2 k 2 L A ( α ) C ˜ n 2 μ κ H γ κ 0 2 + γ α Re 0 1 Γ ( γ + 2 2 ) { U ( γ + 2 2 , 2 + ( γ α ) / 2 ; ( 1 + Q L ξ 2 ) κ 0 2 κ H 2 ) U ( γ + 2 2 , 2 + ( γ α ) / 2 ; [ 1 + Q L ξ 2 + i Q l ξ ( 1 Θ ¯ ξ ) ] κ 0 2 κ H 2 ) } d ξ .
U ( a ; c ; x ) Γ ( 1 c ) Γ ( 1 + a c ) + Γ ( c 1 ) Γ ( a ) x 1 c , | x | < < 1.
δ IG 2 ( D ) = 8 π 2 k 2 L A ( α ) C ˜ n 2 μ κ H 2 α Γ ( 1 ( α γ ) 2 ) Re 0 1 { ( 1 + Q L ξ 2 ) ( α γ ) / 2 1 [ 1 + Q L ξ 2 + i Q H ξ ( 1 Θ ¯ ξ ) ] ( α γ ) / 2 1 } d ξ .
0 1 ( 1 + Q L ξ 2 ) ( α γ ) / 2 1 d ξ = 2 F 1 ( 1 ( α γ ) 2 , 1 2 ; 3 2 ; Q L ) .
0 1 [ 1 + Q L ξ 2 + i Q H ξ ( 1 Θ ¯ ξ ) ] ( α γ ) / 2 1 d ξ = n = 0 ( ( α γ ) / 2 1 n ) ( i Q H ) n 0 1 ξ n ( 1 ( Θ ¯ Q L i Q H ) ξ ) n d ξ .
0 1 ξ n ( 1 ( Θ ¯ Q L i Q H ) ξ ) n d ξ = 1 n + 1 F 2 1 ( n , n + 1 ; n + 2 ; Θ ¯ Q L i Q H ) .
0 1 [ 1 + Q L ξ 2 + i Q H ξ ( 1 Θ ¯ ξ ) ] ( α γ ) / 2 1 d ξ = F 2 1 ( 1 + γ α 2 , 1 ; 2 ; i Q H 2 3 ( i Q H Θ ¯ Q L ) ) .
Re F 2 1 ( 1 + γ α 2 , 1 ; 2 ; i Q H 2 3 ( i Q H Θ ¯ Q L ) ) = Re [ ( 3 + 2 Q L ) / 3 + i ( 1 + 2 Θ ) Q H / 3 ] ( α γ ) / 2 1 [ 2 Q L / 3 + i ( 1 + 2 Θ ) Q H / 3 ] ( α γ ) / 2 .
Re { [ ( 3 + 2 Q L ) / 3 + i ( 1 + 2 Θ ) Q H / 3 ] ( α γ ) / 2 1 [ 2 Q L / 3 + i ( 1 + 2 Θ ) Q H / 3 ] ( α γ ) / 2 } = [ ( α γ ) Τ ] 1 { Ζ ( α γ ) / 2 cos [ ( α γ ) 2 φ 1 φ 2 ] cos ( φ 2 ) } ,
δ IG 2 ( D ) = 8 π 2 k 2 L A ( α ) C ˜ n 2 μ κ H 2 α Γ ( 1 + γ α 2 ) { 2 F 1 ( 1 + γ α 2 , 1 2 ; 3 2 ; Q L ) + [ ( α γ ) Τ ] 1 [ Ζ ( α γ ) / 2 cos [ ( α γ ) 2 φ 1 φ 2 ] cos ( φ 2 ) ] } .
δ IG,Mar 2 ( D ) = α Q H 1 α / 2 δ R 2 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) { Γ ( 1 α / 2 ) [ 2 F 1 ( 1 α 2 , 1 2 ; 3 2 ; Q L ) ( α Τ ) 1 [ Ζ α / 2 cos ( α φ 1 2 φ 2 ) cos ( φ 2 ) ] ] 0.061 Γ ( 3 / 2 α / 2 ) × [ 2 F 1 ( 3 α 2 , 1 2 ; 3 2 ; Q L ) + [ ( α 1 ) Τ ] 1 × [ Ζ ( α 1 ) / 2 cos [ ( α 1 ) φ 1 / 2 φ 2 ] cos ( φ 2 ) ] ] + 5.672 Γ ( 5 / 2 3 α / 4 ) [ 2 F 1 ( 10 3 α 4 , 1 2 ; 3 2 ; Q L ) [ ( 3 α 6 ) Τ / 2 ] 1 × [ Ζ ( 3 α 6 ) / 4 cos [ ( 3 α 6 ) φ 1 / 4 φ 2 ] cos ( φ 2 ) ] ] } ,
δ R 2 = 8 π 2 Γ ( 1 α / 2 ) α sin ( α π / 4 ) π 2 A ( α ) C ˜ n 2 k 3 α / 2 L α / 2 .
δ IG,Ter 2 ( D ) = α Q H 1 α / 2 δ R 2 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) { Γ ( 1 α / 2 ) [ 2 F 1 ( 1 α 2 , 1 2 ; 3 2 ; Q L ) ( α Τ ) 1 ( Ζ α / 2 cos ( α φ 1 2 φ 2 ) cos ( φ 2 ) ) ] + 1.802 Γ ( 3 / 2 α / 2 ) × [ 2 F 1 ( 3 α 2 , 1 2 ; 3 2 ; Q L ) + ( ( α 1 ) Τ ) 1 × ( Ζ ( α 1 ) / 2 cos [ ( α 1 ) φ 1 / 2 φ 2 ] cos ( φ 2 ) ) ] 0.508 Γ ( 5 / 2 3 α / 4 ) [ 2 F 1 ( 10 3 α 4 , 1 2 ; 3 2 ; Q L ) [ ( 3 α 6 ) Τ / 2 ] 1 × [ Ζ ( 3 α 6 ) / 4 cos [ ( 3 α 6 ) φ 1 / 4 φ 2 ] cos ( φ 2 ) ] ] } .
y = I x + n ,
f I ( I ) = [ 2 π δ I 2 ( α , L ) I ] 1 exp { [ ln ( I ) + δ IG,Mar 2 ( D ) / 2 ] 2 / [ 2 δ IG,Mar 2 ( D ) ] } .
BER = 0 f I ( I ) Q ( S N R 0 I ) d I ,
Q ( x ) = 1 π 0 2 π exp ( x 2 2 sin 2 θ ) d θ , x > 0 ,
BER = 1 π 0 2 π 1 π w i i = 1 n exp { S N R 0 2 exp [ 2 ( 2 δ I ( α , L ) x i δ I 2 ( α , L ) / 2 ) ] 2 sin 2 θ } d θ = 1 π i = 1 n Q { S N R 0 exp [ 2 δ I ( α , L ) x i δ I 2 ( α , L ) / 2 ] }
D n ( R , α ) = 8 π 0 κ 2 Φ n ( κ , α ) ( 1 sin ( κ R ) κ R ) d κ .
D n ( R , α ) = 8 π 0 κ 2 α A ( α ) C ˜ n 2 exp ( κ 2 / κ H 2 ) × [ 1 + a 2 ( κ κ H ) + a 2 ( κ κ H ) 3 α / 2 ] ( 1 sin ( κ R ) κ R ) d κ .
( 1 sin ( κ R ) κ R ) = n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! R 2 n κ 2 n ,
D n ( R , α ) = 8 π A ( α ) C ˜ n 2 n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! R 2 n 0 κ 2 + 2 n α exp ( κ 2 / κ H 2 ) × [ 1 + a 1 ( κ κ H ) + a 2 ( κ κ H ) 3 α / 2 ] d κ .
Γ ( x ) = 0 κ x 1 e κ d κ ( κ > 0 , x > 0 ) ,
F 1 1 ( a , b ; x ) = n = 0 ( a ) n x n ( b ) n n ! ,
D n ( R , α ) = 4 π A ( α ) C ˜ n 2 κ H 3 α { Γ ( 3 2 α 2 ) [ 1 F 1 1 ( 3 2 α 2 ; 3 2 ; R 2 κ H 2 4 ) ] + a 1 Γ ( 2 α 2 ) [ 1 F 1 1 ( 2 α 2 ; 3 2 ; R 2 κ H 2 4 ) ] + a 2 Γ ( 3 3 α 4 ) [ 1 F 1 1 ( 3 3 α 4 ; 3 2 ; R 2 κ H 2 4 ) ] } .
D ( R , α ) = { C ˜ n 2 R α 3 , l 0 < < R < < L 0 C ˜ n 2 l 0 α 5 R 2 , 0 R < < l 0 .
F 1 1 ( a ; b ; x ) Γ ( b ) Γ ( b a ) x α , ( x > > 1 ) .
D n ( R , α ) = 4 π A ( α ) C ˜ n 2 R α 3 Γ ( 3 2 α 2 ) Γ ( 3 / 2 ) Γ ( α / 2 ) ( 1 2 ) α 3 .
A ( α ) = Γ ( α 1 ) 4 π 2 sin [ ( α 3 ) π 2 ] .
F 1 1 ( a ; b ; x ) 1 + a x / b ( x < < 1 ) .
D n ( R , α ) = 4 π A ( α ) C ˜ n 2 κ H 5 α R 2 [ Γ ( 3 2 α 2 ) 3 α 12 + a 1 Γ ( 2 α 2 ) 4 α 12 + a 2 Γ ( 3 3 α 4 ) 4 α 8 ] .
c 0 ( α ) = { π A ( α ) [ Γ ( 3 2 α 2 ) 3 α 3 + a 1 Γ ( 2 α 2 ) 4 α 3 + a 2 Γ ( 3 3 α 4 ) 4 α 2 ] } 1 α 5 .

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