Abstract

Atmospheric turbulence is a major limiting factor in an optical wireless communication (OWC) link. The turbulence distorts the phase of the propagating optical fields and limits the focusing capabilities of the telescope antennas. Hence, a detector array is required to capture the widespread signal energy in the focal-plane. This paper addresses the bit-error rate (BER) performance of optical wireless communication (OWC) systems employing a detector array in the presence of turbulence. Here, considering the gamma-gamma turbulence model, we propose a blind estimation scheme that provides the closed-form expression of the BER by exploiting the information of the data output of each pixel, which is based on the singular value decomposition of the sample matrix of the received signals after the code-matched filter. Instead of assuming spatially white additive noise, we consider the case where the noise spatial covariance matrix is unknown. The new method can be applied to either the single transmitter or the multi-transmitter cases. Simulation results for different Rytov variances are presented, which conform closely to the results of the proposed model.

© 2011 OSA

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References

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  1. V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
    [CrossRef]
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    [CrossRef]
  3. J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16(6), 1417–1429 (1999).
    [CrossRef]
  7. V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
    [CrossRef]
  8. J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
    [CrossRef]
  9. R. M. Gagliardi, and S. Karp, Optical Commun., Second Edition (John Wiley and Sons, New York, 1995).
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  11. H. Hemmati, ed., Deep space optical comunications, (John Wiley and Sons, New Jersey, 2006).
  12. W. Mao and J. M. Kahn, “Free-space heterochronous imaging reception of multiple optical signals,” IEEE Trans. Commun. 52(2), 269–279 (2004).
    [CrossRef]
  13. G. Kats and S. Arnon, “Analysis of optical coherence multiplexing networks for satellite communication,” IEEE Trans. Wirel. Comm. 3(5), 1444–1451 (2004).
    [CrossRef]
  14. R. M. Gagliardi, “Pulse-coded multiple access in space optical communications,” IEEE J. Sel. Areas Comm. 13(3), 603–608 (1995).
    [CrossRef]
  15. Y. Han, A. Dang, J. Tang, and H. Guo, “Weak beacon detection for air-to-ground optical wireless link establishment,” Opt. Express 18(3), 1841–1853 (2010).
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    [CrossRef]
  17. J. A. Salehi, “Code division multiple access techniques in optical fiber networks,” IEEE Trans. Commun. 37(8), 824–833 (1989).
    [CrossRef]
  18. S. Arnon, “Free-space optical communication: detector array aperture for optical communication through thin clouds,” Opt. Eng. 34(2), 518–522 (1995).
    [CrossRef]
  19. G. H. Golub, and C. F. V. Loan, Matrix Computation. (Baltimore, MD: Johns Hopkins Univ. Press, 1996).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. 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(8), 1554–1562 (2001).
    [CrossRef]
  24. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series, vol. 3: More Special Functions (Amsterdam: Gordon and Breach Science Publishers, 1986).
  25. N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22(9), 1382–1385 (1983).
    [CrossRef] [PubMed]
  26. K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
    [CrossRef]

2010

2009

2008

2007

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[CrossRef]

2006

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wirel. Comm. 5(6), 1229–1233 (2006).
[CrossRef]

2005

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

D. O'Brien and M. Katz, “Optical wireless communications within fourth-generation wireless systems [Invited],” J. Opt. Netw. 4(6), 312–322 (2005).
[CrossRef]

2004

W. Mao and J. M. Kahn, “Free-space heterochronous imaging reception of multiple optical signals,” IEEE Trans. Commun. 52(2), 269–279 (2004).
[CrossRef]

G. Kats and S. Arnon, “Analysis of optical coherence multiplexing networks for satellite communication,” IEEE Trans. Wirel. Comm. 3(5), 1444–1451 (2004).
[CrossRef]

2002

V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
[CrossRef]

2001

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(8), 1554–1562 (2001).
[CrossRef]

2000

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[CrossRef]

1999

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16(6), 1417–1429 (1999).
[CrossRef]

1995

R. M. Gagliardi, “Pulse-coded multiple access in space optical communications,” IEEE J. Sel. Areas Comm. 13(3), 603–608 (1995).
[CrossRef]

S. Arnon, “Free-space optical communication: detector array aperture for optical communication through thin clouds,” Opt. Eng. 34(2), 518–522 (1995).
[CrossRef]

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

1989

J. A. Salehi, “Code division multiple access techniques in optical fiber networks,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[CrossRef]

1986

P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. 4(5), 547–554 (1986).
[CrossRef]

1983

1966

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(8), 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. 16(6), 1417–1429 (1999).
[CrossRef]

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(8), 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. 16(6), 1417–1429 (1999).
[CrossRef]

Arnon, S.

G. Kats and S. Arnon, “Analysis of optical coherence multiplexing networks for satellite communication,” IEEE Trans. Wirel. Comm. 3(5), 1444–1451 (2004).
[CrossRef]

S. Arnon, “Free-space optical communication: detector array aperture for optical communication through thin clouds,” Opt. Eng. 34(2), 518–522 (1995).
[CrossRef]

Baba, H.

Bourennane, S.

Cardoso, J.

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

Causs, P.

Chan, V. W. S.

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[CrossRef]

Dang, A.

Duhamel, P.

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

Dwivedi, A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

Fan, T.

P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. 4(5), 547–554 (1986).
[CrossRef]

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, “Pulse-coded multiple access in space optical communications,” IEEE J. Sel. Areas Comm. 13(3), 603–608 (1995).
[CrossRef]

Guo, H.

Hammons, A. R.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

Han, Y.

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. 16(6), 1417–1429 (1999).
[CrossRef]

Horwath, J.

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[CrossRef]

Jones, S. D.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

Juarez, J. C.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

Kahn, J. M.

W. Mao and J. M. Kahn, “Free-space heterochronous imaging reception of multiple optical signals,” IEEE Trans. Commun. 52(2), 269–279 (2004).
[CrossRef]

Kats, G.

G. Kats and S. Arnon, “Analysis of optical coherence multiplexing networks for satellite communication,” IEEE Trans. Wirel. Comm. 3(5), 1444–1451 (2004).
[CrossRef]

Katz, M.

Khalighi, A.

Kiasaleh, K.

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

Knapek, M.

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[CrossRef]

Li, J.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wirel. Comm. 5(6), 1229–1233 (2006).
[CrossRef]

Louthain, J. A.

Mao, W.

W. Mao and J. M. Kahn, “Free-space heterochronous imaging reception of multiple optical signals,” IEEE Trans. Commun. 52(2), 269–279 (2004).
[CrossRef]

Mayrargue, S.

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

Moll, F.

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[CrossRef]

Moulines, E.

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

Nichols, R. A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

O'Brien, D.

Perlot, N.

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[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(8), 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. 16(6), 1417–1429 (1999).
[CrossRef]

Prucnal, P.

P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. 4(5), 547–554 (1986).
[CrossRef]

Sakurai, K.

Salehi, J. A.

J. A. Salehi, “Code division multiple access techniques in optical fiber networks,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[CrossRef]

Santoro, M.

P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. 4(5), 547–554 (1986).
[CrossRef]

Schmidt, J. D.

Srinivasan, M.

V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
[CrossRef]

Sugimoto, N.

Takeuchi, N.

Tang, J.

Uysal, M.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wirel. Comm. 5(6), 1229–1233 (2006).
[CrossRef]

Vilnrotter, V. A.

V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
[CrossRef]

Weerackody, V.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

Xu, F.

Yu, M.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wirel. Comm. 5(6), 1229–1233 (2006).
[CrossRef]

Appl. Opt.

IEEE Commun. Mag.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-Space Optical Communications for Next-generation Military Networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[CrossRef]

IEEE J. Sel. Areas Comm.

R. M. Gagliardi, “Pulse-coded multiple access in space optical communications,” IEEE J. Sel. Areas Comm. 13(3), 603–608 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[CrossRef]

IEEE Trans. Commun.

W. Mao and J. M. Kahn, “Free-space heterochronous imaging reception of multiple optical signals,” IEEE Trans. Commun. 52(2), 269–279 (2004).
[CrossRef]

V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
[CrossRef]

J. A. Salehi, “Code division multiple access techniques in optical fiber networks,” IEEE Trans. Commun. 37(8), 824–833 (1989).
[CrossRef]

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

IEEE Trans. Signal Process.

E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. Signal Process. 43(2), 516–525 (1995).
[CrossRef]

IEEE Trans. Wirel. Comm.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wirel. Comm. 5(6), 1229–1233 (2006).
[CrossRef]

G. Kats and S. Arnon, “Analysis of optical coherence multiplexing networks for satellite communication,” IEEE Trans. Wirel. Comm. 3(5), 1444–1451 (2004).
[CrossRef]

Int. J. Satell. Commun. Network

J. Horwath, N. Perlot, M. Knapek, and F. Moll, “Experimental verification of optical backhaul links for high-altitude platform networks: Atmospheric turbulence and downlink availability,” Int. J. Satell. Commun. Network 25(5), 501–528 (2007).
[CrossRef]

J. Lightwave Technol.

P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. 4(5), 547–554 (1986).
[CrossRef]

J. Opt. Netw.

J. Opt. Soc. Am.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16(6), 1417–1429 (1999).
[CrossRef]

D. L. Fried, “Optical Resolution through a Randomly Inhomogeneous Medium,” J. Opt. Soc. Am. 56(10), 1372–1379 (1966).
[CrossRef]

Opt. Eng.

S. Arnon, “Free-space optical communication: detector array aperture for optical communication through thin clouds,” Opt. Eng. 34(2), 518–522 (1995).
[CrossRef]

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(8), 1554–1562 (2001).
[CrossRef]

Opt. Express

Other

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series, vol. 3: More Special Functions (Amsterdam: Gordon and Breach Science Publishers, 1986).

G. H. Golub, and C. F. V. Loan, Matrix Computation. (Baltimore, MD: Johns Hopkins Univ. Press, 1996).

R. M. Gagliardi, and S. Karp, Optical Commun., Second Edition (John Wiley and Sons, New York, 1995).

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

H. Hemmati, ed., Deep space optical comunications, (John Wiley and Sons, New Jersey, 2006).

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Figures (4)

Fig. 1
Fig. 1

Multi-signal receiver for optical communication system

Fig. 2
Fig. 2

BER comparison of the theoretical and simulation results

Fig. 3
Fig. 3

Estimate error versus the received SNR over a chip interval

Fig. 4
Fig. 4

Estimated SNR versus the number of frames

Equations (16)

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

Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) 11 / 6
C n 2 ( h ) = 0.00594 ( v 27 ) 2 ( 10 5 h ) 10 e h 1000 + 2.7 × 10 16 e h 1500 + A e h 100
r 0 = [ 0.42 k 2 0 H C n 2 ( x ) d x ] 3 / 5
f ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 K α β ( 2 α β I ) , I > 0
K α β ( 2 α β I ) = 1 2 G 0 2 , 2 , 0 [ α β I | α β / 2 α β / 2 ]
α = ( exp [ 0.49 σ R 2 ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ] 1 ) 1 , β = ( exp [ 0.51 σ R 2 ( 1 + 0.69 σ R 12 / 5 ) 5 / 6 ] 1 ) 1
S ( R , t ) = l = 1 L η l S l ( t ) δ ( R R l )
a l ( t ) = j = 0 N 1 α l j E T c ( t j T c ) , 0 < t < T ,
R Γ l , Γ l ( q ) = { 1 f o r q = 0 1 / N f o r q 0 , R Λ l , Γ l ( q ) = { w / N f o r q = 0 0 f o r q 0
y m ( t ) = n = l = 1 L s l , n j = 1 N α l , j h l , m ( t j T c n T τ l ) + n m ( t )
Z l = N + 1 2 [ s l ( 1 ) , s l ( 2 ) , , s l ( P ) ] h l + s = 1 , s l L [ s s ( 1 ) R Λ s , Γ l ( 1 ) , s s ( 2 ) R Λ s , Γ l ( 2 ) , , s s ( P ) R Λ s , Γ l ( P ) ] h s + [ N ( 1 ) , N ( 2 ) , , N ( P ) ]
O = [ U s U n ] [ s n ] [ U s H U n H ] .
ζ i = { σ s i 2 + σ I 2 + σ n 2 i = 1 , , P σ I 2 + σ n 2 i = P + 1 , , L P σ n 2 i = L P + 1 , , N P .
σ ˜ n 2 = 1 N P L P j = L P + 1 N P ζ j σ ˜ I 2 = 1 ( L 1 ) P j = P + 1 L P ( ζ j σ ˜ n 2 ) . σ ˜ s 2 = 1 P ( j = 1 P ( ζ j ( σ ˜ I 2 + σ ˜ n 2 ) ) )
ξ ˜ l = σ ˜ s 2 σ ˜ I 2 + σ ˜ n 2 .
p e = 2 α + β 4 Γ ( α ) Γ ( β ) π 3 / 2 G 5 , 2 , 2 4 [ 8 σ ˜ s 2 ( α β ) 2 ( σ ˜ I 2 + σ ˜ n 2 ) | 1 α 2 2 α 2 1 β 2 2 β 2 1 0 1 2 ]

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