Abstract

In this paper, the bit error rate (BER) performance of a subcarrier intensity-modulated (SIM) free-space optical (FSO) communications system using binary phase shift keying (BPSK) is investigated over a K-distributed turbulence channel. First, the performance is analyzed employing a negative exponential turbulence model, and an exact closed-form expression is derived for the BER. Then, it is shown that the probability density function (PDF) of the K distribution can be approximated accurately by a finite sum of weighted negative exponential PDFs. Based on this interesting result and by using the closed-form expression, which is derived for the case of a negative exponential model, an approximate, closed-form expression for the BER of the BPSK-based SIM FSO over a K channel is derived. Moreover, to improve the BER performance, spatial diversity using selection combining (SC) is considered. It is shown that the PDF of the resulting channel irradiance corresponding to the SC diversity scheme over a K channel can be approximated accurately by a finite linear combination of negative exponential functions. The derived approximate PDF accurately estimates the PDF of the channel irradiance for arbitrary values of diversity order and is valid for a wide range of channel parameters. Then, an approximate, closed-form expression is derived for the average BER of the BPSK-based SIM FSO system employing the SC diversity technique over a K channel. Numerical results presented in this paper show that the derived approximate expressions are very accurate and can be used as efficient tools for performance analysis of the system.

© 2010 Optical Society of America

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  1. J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
    [CrossRef]
  2. W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
    [CrossRef]
  3. W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
    [CrossRef]
  4. W. O. Popoola, Z. Ghassemlooy, “BPSK subcarrier modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, no. 8, pp. 967–973, Apr. 2009.
    [CrossRef]
  5. H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.
  6. R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. New York: Morgan Kaufmann, 2002.
  7. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag., vol. 24, pp. 806–814, Nov. 1976.
    [CrossRef]
  8. E. J. Lee, V. W. S. Chan, “Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
    [CrossRef]
  9. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 1965.
  10. A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett., vol. 11, no. 5, pp. 390–392, May 2007.
    [CrossRef]
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover, 1972.
  12. B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
    [CrossRef]
  13. J. M. Holtzman, “On using perturbation analysis to do sensitivity analysis: derivatives versus differences,” IEEE Trans. Autom. Control, vol. 37, no. 2, pp. 243–247, Feb. 1992.
    [CrossRef]
  14. J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles With Applications. Academic: New York, 1992.
  15. H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed.Wiley, 2003.
    [CrossRef]

2009

W. O. Popoola, Z. Ghassemlooy, “BPSK subcarrier modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, no. 8, pp. 967–973, Apr. 2009.
[CrossRef]

H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.

B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
[CrossRef]

2008

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
[CrossRef]

2007

J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
[CrossRef]

A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett., vol. 11, no. 5, pp. 390–392, May 2007.
[CrossRef]

2004

E. J. Lee, V. W. S. Chan, “Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

1992

J. M. Holtzman, “On using perturbation analysis to do sensitivity analysis: derivatives versus differences,” IEEE Trans. Autom. Control, vol. 37, no. 2, pp. 243–247, Feb. 1992.
[CrossRef]

1976

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover, 1972.

Ahmadi, V.

W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
[CrossRef]

Allen, J. I. H.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Castillo-Vazquez, B.

B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
[CrossRef]

Castillo-Vazquez, C.

B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
[CrossRef]

Chan, V. W. S.

E. J. Lee, V. W. S. Chan, “Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

David, H. A.

H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed.Wiley, 2003.
[CrossRef]

Gao, S.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Garcia-Zambrana, A.

B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
[CrossRef]

García-Zambrana, A.

A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett., vol. 11, no. 5, pp. 390–392, May 2007.
[CrossRef]

Ghassemlooy, Z.

W. O. Popoola, Z. Ghassemlooy, “BPSK subcarrier modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, no. 8, pp. 967–973, Apr. 2009.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 1965.

Holtzman, J. M.

J. M. Holtzman, “On using perturbation analysis to do sensitivity analysis: derivatives versus differences,” IEEE Trans. Autom. Control, vol. 37, no. 2, pp. 243–247, Feb. 1992.
[CrossRef]

Jakeman, E.

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

Kapur, J. N.

J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles With Applications. Academic: New York, 1992.

Kesavan, H. K.

J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles With Applications. Academic: New York, 1992.

Lee, E. J.

E. J. Lee, V. W. S. Chan, “Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

Leitgeb, E.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Li, J.

J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
[CrossRef]

Li, X.

H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.

Liu, J. Q.

J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
[CrossRef]

Nagaraja, H. N.

H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed.Wiley, 2003.
[CrossRef]

Popoola, W. O.

W. O. Popoola, Z. Ghassemlooy, “BPSK subcarrier modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, no. 8, pp. 967–973, Apr. 2009.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
[CrossRef]

Pusey, P. N.

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

Ramaswami, R.

R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. New York: Morgan Kaufmann, 2002.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 1965.

Sivarajan, K.

R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. New York: Morgan Kaufmann, 2002.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover, 1972.

Taylor, D. P.

J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
[CrossRef]

Wu, H.

H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.

Yan, H.

H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.

Electron Lett.

B. Castillo-Vazquez, A. Garcia-Zambrana, C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron Lett., vol. 45, no. 23, pp. 1185–1187, 2009.
[CrossRef]

IEEE Commun. Lett.

A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett., vol. 11, no. 5, pp. 390–392, May 2007.
[CrossRef]

IEEE J. Sel. Areas Commun.

E. J. Lee, V. W. S. Chan, “Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

IEEE Trans. Antennas Propag.

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

IEEE Trans. Autom. Control

J. M. Holtzman, “On using perturbation analysis to do sensitivity analysis: derivatives versus differences,” IEEE Trans. Autom. Control, vol. 37, no. 2, pp. 243–247, Feb. 1992.
[CrossRef]

IEEE Trans. Commun.

J. Li, J. Q. Liu, D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.
[CrossRef]

IET Optoelectron.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Int. J. Auton. Adapt. Commun. Syst.

W. O. Popoola, Z. Ghassemlooy, V. Ahmadi, “Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment,” Int. J. Auton. Adapt. Commun. Syst., vol. 1, no. 3, pp. 342–355, 2008.
[CrossRef]

J. Lightwave Technol.

Opt. Appl.

H. Wu, H. Yan, X. Li, “Performance analysis of bit error rate for free space optical communication with tip-tilt compensation based on gamma-gamma distribution,” Opt. Appl., vol. 39, no. 3, pp. 533–545, 2009.

Other

R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. New York: Morgan Kaufmann, 2002.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 1965.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover, 1972.

J. N. Kapur, H. K. Kesavan, Entropy Optimization Principles With Applications. Academic: New York, 1992.

H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed.Wiley, 2003.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Block diagram of BPSK-based SIM FSO: (a) transmitter, (b) receiver.

Fig. 2
Fig. 2

Approximate PDF [Eq. (16)] compared with the exact PDF [Eq. (14)] for K distribution: (a) α = 3.1 , (b) α = 10 .

Fig. 3
Fig. 3

Kullback and Leibler (KL) measure between the exact PDF [Eq. (13)] and the proposed approximate PDF [Eq. (20)].

Fig. 4
Fig. 4

Approximate BER [Eq. (25)] compared with the exact BER [which is derived by numerical integration of Eq. (14)] for various turbulence levels.

Fig. 5
Fig. 5

Approximate PDF [Eq. (33)] of the resulting channel irradiance corresponding to the SC diversity scheme over a K channel compared with the exact PDF [Eq. (29)].

Fig. 6
Fig. 6

Approximate BER [Eq. (37)] compared with the exact BER for a K channel for different diversity order ( L ) and turbulence conditions.

Equations (37)

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

i r ( t ) = R I ( 1 + ξ m ( t ) ) + n ( t ) ,
i r ( t ) = ± R I g ( t ) cos ( ω c t ) + n ( t ) .
γ = ( I R ) 2 2 σ 2 .
P e | I = Q ( γ ) = Q ( I R 2 σ ) ,
f Neg - Exp ( I ; β ) = β exp ( β I ) , β > 0 ,
P e ( Neg - Exp ) = 0 Q ( I R 2 σ ) f Neg - Exp ( I ; β ) d I = β 0 Q ( I R 2 σ ) exp ( β I ) d I ,
ϑ ( n , m ) 0 Q ( n I ) exp ( m I ) d I .
ϑ ( n , m ) = 1 2 m { 1 exp ( m 2 2 n 2 ) erfc ( m 2 n ) } ,
P e ( Neg - Exp ) = β ϑ ( R 2 σ , β ) .
f y ( y ) = exp ( y ) , y 0 ,
f z ( z ) = α α z α 1 Γ ( α ) exp ( α z ) , z 0 ,
f I | z ( I | z ) = ( 1 z ) f y ( I z ) = ( 1 z ) exp ( I z ) ,
f I ( I ) = E z [ f I | z ( I | z ) ] = 2 Γ ( α ) α α + 1 2 I α 1 2 K α 1 ( 2 α I ) , I > 0 ,
P e ( K ) = 0 Q ( I R 2 σ ) f I ( I ) d I = 2 Γ ( α ) α α + 1 2 0 Q ( I R 2 σ ) I α 1 2 K α 1 ( 2 α I ) d I ,
f I ( I ) 2 3 f Neg - Exp ( I ; 1 ) + 1 6 f Neg - Exp ( I ; β 1 ) + 1 6 f Neg - Exp ( I ; β 2 ) ,
f I | z ( I | z ) f I | z ( I | z = μ z ) + ( z μ z ) f I | z | ( I | z ) | z = μ z + 1 2 ( z μ z ) 2 f I | z | ( I | z ) | z = μ z ,
E z [ f I | z ( I | z ) ] f I | z ( I | z = μ z ) + 1 2 σ z 2 f I | z | ( I | z ) | z = μ z .
E z [ f I | z ( I | z ) ] f I | z ( I | z = μ z ) + 1 2 σ z 2 f I | z ( I | z = μ z + h ) 2 f I | z ( I | z = μ z ) + f I | z ( I | z = μ z h ) h 2 ,
E z [ f I | z ( I | z ) ] 2 3 f I | z ( I | z = μ z ) + 1 6 f I | z ( I | z = μ z + 3 σ z ) + 1 6 f I | z ( I | z = μ z 3 σ z ) .
f I ( I ) 2 3 exp ( I ) + 1 6 1 1 + 3 α exp ( I 1 + 3 α ) + 1 6 1 1 3 α exp ( I 1 3 α ) , I > 0 .
D ( h g ) = 0 h ( x ) ln ( h ( x ) g ( x ) ) d x .
F I ( I ) = 1 2 α α 2 Γ ( α ) I α 2 K α ( 2 I α ) .
F I ( I ) = 0 I f I ( x ) d x 1 2 3 exp ( I ) 1 6 exp ( β 1 I ) 1 6 exp ( β 2 I ) .
P e ( K ) = 0 Q ( I R 2 σ ) f I ( I ) d I 0 Q ( I R 2 σ ) { 2 3 f Neg - Exp ( I ; 1 ) + 1 6 f Neg - Exp ( I ; β 1 ) + 1 6 f Neg - Exp ( I ; β 2 ) } d I = 2 3 0 Q ( I R 2 σ ) f Neg - Exp ( I ; 1 ) d I + 1 6 0 Q ( I R 2 σ ) f Neg - Exp ( I ; β 1 ) d I + 1 6 0 Q ( I R 2 σ ) f Neg - Exp ( I ; β 2 ) d I ,
P e ( K ) 2 3 ϑ ( R 2 σ , 1 ) + 1 6 β 1 ϑ ( R 2 σ , β 1 ) + 1 6 β 2 ϑ ( R 2 σ , β 2 ) .
I sc = max j = 1 , 2 , , L I j .
f I sc ( I sc ) = L f I ( I ) [ F I ( I ) ] L 1 .
f I sc ( I sc ) = L ( 2 Γ ( α ) α α + 1 2 I α 1 2 K α 1 ( 2 α I ) ) × ( 1 2 α α 2 Γ ( α ) I α 2 K α ( 2 I α ) ) L 1 .
f I m ( I m ) = L ( 2 3 exp ( I ) + 1 6 A exp ( A I ) + 1 6 B exp ( B I ) ) × [ 1 2 3 exp ( I ) 1 6 exp ( A I ) 1 6 exp ( B I ) ] L 1 .
f I sc ( I sc ) L ( 2 3 exp ( I ) + 1 6 β 1 exp ( β 1 I ) + 1 6 β 2 exp ( β 2 I ) ) × k 1 , k 2 , k 3 , k 4 k 1 + k 2 + k 3 + k 4 = L 1 ( L 1 k 1 , k 2 , k 3 , k 4 ) ( 2 3 ) k 2 ( 1 6 ) k 3 + k 4 exp ( ( k 2 + A k 3 + B k 4 ) I ) .
H ( I ; β ) k 1 , k 2 , k 3 , k 4 k 1 + k 2 + k 3 + k 4 = L 1 ( L 1 k 1 , k 2 , k 3 , k 4 ) ( 2 3 ) k 2 ( 1 6 ) k 3 + k 4 exp ( ( k 2 + A k 3 + B k 4 + β ) I ) ,
f I sc ( I sc ) L { 2 3 H ( I ; 1 ) + β 1 6 H ( I ; β 1 ) + β 2 6 H ( I ; β 2 ) } .
γ SC = ( R I L ) 2 2 ( 1 L ) σ 2 = ( R I ) 2 2 L σ 2 .
P e | SC = Q ( γ SC ) = Q ( I R A 2 L σ ) .
P e ( SC ) = 0 Q ( I R 2 L σ ) f I sc ( I sc ) .
P e ( SC ) ( E ) 0 Q ( I R 2 L σ ) { L { 2 3 H ( I ; 1 ) + β 1 6 H ( I ; β 1 ) + β 2 6 H ( I ; β 2 ) } } = L { 2 3 0 Q ( I R 2 L σ ) H ( I ; 1 ) d I + β 1 3 0 Q ( I R 2 L σ ) H ( I ; β 1 ) d I + β 2 3 0 Q ( I R 2 L σ ) H ( I ; β 2 ) d I } .
P e ( SC ) L k 1 , k 2 , k 3 , k 4 k 1 + k 2 + k 3 + k 4 = L 1 ( L 1 k 1 , k 2 , k 3 , k 4 ) ( 2 3 ) k 2 ( 1 6 ) k 3 + k 4 × { 2 3 ϑ ( I R 2 L σ , k 2 + β 1 k 3 + β 2 k 4 + 1 ) + β 1 6 ϑ ( I R 2 L σ , k 2 + β 1 k 3 + β 2 k 4 + β 1 ) + β 2 6 ϑ ( I R 2 L σ , k 2 + β 1 k 3 + β 2 k 4 + β 2 ) } .