Abstract

Recently, a new and generalized statistical model, called ℳ or Málaga distribution, was proposed to model the irradiance fluctuations of an unbounded optical wavefront (plane and spherical waves) propagating through a turbulent medium under all irradiance fluctuation conditions in homogeneous, isotropic turbulence. Málaga distribution was demonstrated to have the advantage of unifying most of the proposed statistical models derived until now in the bibliography in a closed-form expression providing, in addition, an excellent agreement with published plane wave and spherical wave simulation data over a wide range of turbulence conditions (weak to strong). Now, such a model is completed by including the adverse effect of pointing error losses due to misalignment. In this respect, the well-known effects of aperture size, beam width and jitter variance are taken into account. Accordingly, after presenting the analytical expressions for the combined distribution of scintillation and pointing errors, we derive its centered moments of the overall probability distribution. Finally, we obtain the analytical expressions for the average bit error rate performance for the ℳ distribution affected by pointing errors. Numerical results show the impact of misalignment on link performance.

© 2012 OSA

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).
  2. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
    [CrossRef]
  3. 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]
  4. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  5. 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).
    [CrossRef]
  6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
    [CrossRef]
  7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  8. S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997).
    [CrossRef]
  9. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626–629 (2003).
    [CrossRef]
  10. K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
    [CrossRef]
  11. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25, 1702–1710 (2007).
    [CrossRef]
  12. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
    [CrossRef]
  13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011).
    [CrossRef] [PubMed]
  14. M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004).
    [CrossRef]
  15. ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010.
  16. R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
    [CrossRef]
  17. T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).
  18. J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970).
    [CrossRef]
  19. Wolfram, http://functions.wolfram.com/
  20. C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005)
    [CrossRef]
  21. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).
  22. P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)
  23. M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).
  24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  25. L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003).
    [CrossRef]

2011 (1)

2008 (1)

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

2007 (1)

2004 (1)

M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004).
[CrossRef]

2003 (3)

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626–629 (2003).
[CrossRef]

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003).
[CrossRef]

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[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]

1997 (1)

S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997).
[CrossRef]

1994 (1)

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

1987 (1)

1980 (1)

E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

1970 (2)

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[CrossRef]

J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970).
[CrossRef]

Abdi, A.

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

Al Naboulsi, M.

M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004).
[CrossRef]

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]

Alouini, M. S.

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

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, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Arnon, S.

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626–629 (2003).
[CrossRef]

S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997).
[CrossRef]

Billingsley, P.

P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)

Castillo-Vázquez, M.

Charalambides, C. A.

C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005)
[CrossRef]

Churnside, J. H.

Clark, J. R.

J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970).
[CrossRef]

Clifford, S. F.

Farid, A. A.

Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Hranilovic, S.

Jakerman, E.

E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

Jurado-Navas, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Karagiannidis, G. K.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

Karp, S.

J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970).
[CrossRef]

Kaveh, M. A.

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

Kennedy, R. S.

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[CrossRef]

Kiasaleh, K.

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

Kopeika, N. S.

S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997).
[CrossRef]

Lau, W. C.

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

Paris, J. F.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

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, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36, 4095–4097 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

Rappaport, T.

T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).

Sandalidis, H. G.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

Simon, M. K.

M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

Sizun, H.

M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004).
[CrossRef]

Tse, D. N. C.

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003).
[CrossRef]

Tsiftsis, T. A.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

Uysal, M.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

Zheng, L.

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003).
[CrossRef]

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

IEEE Commun. Lett. (1)

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12, 44–46 (2008).
[CrossRef]

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

IEEE Trans. Inf. Theory (1)

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory 49, 1073–1096 (2003).
[CrossRef]

IEEE Trans. Wireless Commun. (2)

A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. 2, 519–528 (2003).
[CrossRef]

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626–629 (2003).
[CrossRef]

J. Lightwave Technol. (1)

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

J. Phys. A (1)

E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

Opt. Eng. (3)

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]

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43, 319–329 (2004).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (3)

S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE 85, 1646–1661 (1997).
[CrossRef]

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[CrossRef]

J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE 58, 1964–1965 (1970).
[CrossRef]

Other (10)

Wolfram, http://functions.wolfram.com/

C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005)
[CrossRef]

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

P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)

M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).

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).
[CrossRef]

ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010.

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Figures (3)

Fig. 1
Fig. 1

Proposed propagation geometry for a laser beam in a Málaga model to form the small-scale fluctuations [5].

Fig. 2
Fig. 2

Average optical BER against average SNR for different values of ρ and wz/a. In all curves, α = 10, β = 5 and the transmitted power is normalized, i.e., Ω + 2b0 = 1. The case of ρ = 1 corresponds to the Gamma-Gamma distribution.

Fig. 3
Fig. 3

Average optical BER against average SNR for different values of α, β, ρ assuming σ R 2 = 0.36 and wz/a = 20. In all curves, the transmitted power is normalized, i.e., Ω + 2b0 = 1.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

P R ( t ) = h ( t ) R P T ( t ) + N ( t ) ,
U = ( U L + U S C + U S G ) exp ( χ + j S )
h a = | U L + U S C + U S G | 2 exp ( 2 χ ) = Y X , { Y | U L + U S C + U S G | 2 ( small - scale fluctuations ) X exp ( 2 χ ) ( large - scale fluctuations ) ;
R ( t ) = ( U L + U S C + U S G ) = G [ Ω exp ( j ϕ A ) + ρ 2 b 0 exp ( j ϕ B ) ] + ( 1 ρ ) U S ,
f Y ( y ) = 1 γ [ γ β γ β + Ω ] β exp [ y γ ] 1 F 1 ( β ; 1 ; 1 γ Ω ( γ β + Ω ) y )
f X ( x ) = α α Γ ( α ) x α 1 exp ( α x ) ,
f h a ( h a ) = A k = 1 β a k h a α + k 2 1 K α k ( 2 α β h a γ β + Ω )
{ A 2 α α 2 γ 1 + α 2 Γ ( α ) ( γ β γ β + Ω ) β + α 2 ; a k ( β 1 k 1 ) ( γ β + Ω ) 1 k 2 ( k 1 ) ! ( Ω γ ) k 1 ( α β ) k 2 .
f h a ( h a ) = A ( G ) k = 1 a k ( G ) h a α + k 2 1 K α k ( 2 α h a γ ) ,
{ A ( G ) 2 α α 2 γ 1 + α 2 Γ ( α ) ( γ β γ β + Ω ) β ; a k ( G ) ( β ) k 1 ( α γ ) k 2 [ ( k 1 ) ! ] 2 γ k 1 ( Ω + γ β ) k 1 .
I beam ( ρ ; z ) = 2 π w z 2 exp ( 2 ρ 2 w z 2 ) ,
h p ( r ; z ) A 0 exp ( 2 r 2 w z eq 2 ) ,
w z eq 2 = w z 2 π erf ( v ) 2 v exp ( v 2 ) .
f r ( r ) = r σ s 2 exp ( r 2 2 σ s 2 ) , r > 0 ;
f h p ( h p ) = g 2 A 0 g 2 h p g 2 1 , 0 h p A 0 ;
f h ( h ) = f h | h a ( h | h a ) f h a ( h a ) d h a ,
f h | h a ( h | h a ) = 1 h a f h p ( h h a ) = g 2 A 0 g 2 h a ( h h a ) g 2 1 , 0 h A 0 h a ;
f h ( h ) = g 2 A A 0 g 2 h g 2 1 k = 1 β a k h / A 0 h a α + k 2 1 g 2 K α k ( 2 α β h a γ β + Ω ) d h a
G 0 , 2 2 , 0 ( x | a , b ) = 2 x ( a + b ) / 2 K a b ( 2 x ) ,
f h ( h ) = g 2 A 2 A 0 g 2 h g 2 1 k = 1 β a k h | A 0 h a α + k 2 1 g 2 G 0 , 2 2 , 0 ( α β h a γ β + Ω | α k 2 , α k 2 ) d h a .
f h ( h ) = g 2 A 2 h 1 k = 1 β a k ( α β γ β + Ω ) α + k 2 G 1 , 3 3 , 0 ( α β γ β + Ω h A 0 | g 2 + 1 g 2 , α , k ) ,
x r G p , q m , n ( x | a p c q ) = G p , q m , n ( x | a p + r c q + r )
f h ( h ) = g 2 A ( G ) 2 h 1 k = 1 a k ( G ) ( α γ ) α + k 2 G 1 , 3 3 , 0 ( α γ h A 0 | g 2 + 1 g 2 , α , k ) .
m k ( h a ) E [ h a k ] = E [ X k ] E [ Y k ] = m k ( X ) m k ( Y ) ,
m k ( X ) = Γ ( α + k ) Γ ( α ) α k ,
m k ( Y ) = 1 γ ( γ β γ β + Ω ) β r = 0 β 1 ( β 1 r ) 1 r ! ( Ω γ ( γ β + Ω ) ) r Γ ( k + r + 1 ) ( β γ β + Ω ) k + r + 1 .
m k ( h a ) = Γ ( α + k ) Γ ( α ) α k γ ( γ β γ β + Ω ) β r = 0 β 1 ( β 1 r ) 1 r ! ( Ω γ ( γ β + Ω ) ) r Γ ( k + r + 1 ) ( β γ β + Ω ) k + r + 1 ,
m k ( Y ) = ( γ β γ β + Ω ) β γ k Γ ( k + 1 ) 2 F 1 ( k + 1 , β ; 1 ; Ω γ β + Ω ) ,
m k ( G ) ( h a ) = Γ ( α + k ) Γ ( α ) α k ( γ β γ β + Ω ) β γ k Γ ( k + 1 ) 2 F 1 ( k + 1 , β ; 1 ; Ω γ β + Ω ) .
m k ( h ) E [ h k ] = E [ h a k ] E [ h p k ] = m k ( h a ) m k ( h p ) .
m k ( h p ) = E [ h p k ] = h p k f h p ( h p ) d h p .
m k ( h p ) = g 2 A 0 g 2 0 A 0 h p g 2 + k 1 d h p = g 2 g 2 + k A 0 k .
m k ( h ) = Γ ( α + k ) Γ ( α ) α k γ ( γ β γ β + Ω ) β g 2 g 2 + k A 0 k × × r = 0 β 1 ( β 1 r ) 1 r ! ( Ω γ ( γ β + Ω ) ) r Γ ( k + r + 1 ) ( β γ β + Ω ) k + r + 1 .
m k ( h ) = g 2 A 0 k g 2 + k Γ ( α + k ) Γ ( α ) α k ( γ β γ β + Ω ) β γ k Γ ( k + 1 ) 2 F 1 ( k + 1 , β ; 1 ; Ω γ β + Ω ) .
P b ( e ) = p 0 p ( e | 0 ) + p 1 p ( e | 1 ) ,
P b ( e | h ) = p ( e | 0 , h ) = p ( e | 1 , h ) = Q ( P R h σ N )
P b ( e ) = 0 P b ( e | h ) f h ( h ) d h ,
G 1 , 2 2 , 0 ( z | a a 1 , a 1 2 ) = π z a 1 erfc ( z ) .
P b ( e ) = 2 α g 2 A B α 2 32 π π k = 1 β 2 k B k 2 a k G 7 , 4 2 , 6 ( 8 R 2 P 2 A 0 2 σ N 2 B 2 | 1 g 2 2 , 2 g 2 2 , 1 α 2 , 2 α 2 , 1 k 2 , 2 k 2 , 1 0 , 1 2 , g 2 2 , 1 g 2 2 )
P b ( e ) = 2 α g 2 A ( G ) 32 π π ( γ α ) α 2 k = 1 2 k ( γ α ) k 2 a k ( G ) × × G 7 , 4 2 , 6 ( 8 R 2 P 2 A 0 2 σ N 2 γ 2 α 2 | 1 g 2 2 , 2 g 2 2 , 1 α 2 , 2 α 2 , 1 k 2 , 2 k 2 , 1 0 , 1 2 , g 2 2 , 1 g 2 2 ) .
σ R 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 .

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