Abstract

A number of existing spatial diversity schemes have been shown to improve the performance of optical wireless communication systems in diversity-rich environments. Among all, switched diversity has low complexity and is simple to implement. In this paper, an innovative spatial diversity scheme based on switched diversity is proposed. The scheme, namely switch-to-dominant combining, contributes to a higher bit error rate (BER) performance when compared to conventional switched diversity schemes, including switch-and-stay and switch-and-examine diversity. The optical multireceiver wireless system operates in a spatially correlated and lognormally distributed fading channel. Analytical analyses are conducted to demonstrate BER and processing load performance offered by the new scheme and compare them to available schemes, i.e., conventional switched combining and selection combining.

© 2011 Optical Society of America

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References

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  1. M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, 2nd ed. (Wiley, 2005).
  2. A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems on generalized-fading channels,” IEEE Trans. Commun. 42, 2959–2966 (1994).
    [CrossRef]
  3. H. Nam and M. S. Alouini, “Optimization of multi-branch switched diversity systems,” IEEE Trans. Commun. 57, 2960–2970 (2009).
    [CrossRef]
  4. Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
    [CrossRef]
  5. G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
    [CrossRef]
  6. H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high speed FSO links,” IET Optoelectron.(to be published).
  7. A. A. Abu-Dayya and N. C. Beaulieu, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Technol. 43, 970–976 (1994).
    [CrossRef]
  8. H. C. Yang and M. S. Alouini, “Performance analysis of multi-branch switched diversity systems,” IEEE Trans. Commun. 51, 782–794 (2003).
    [CrossRef]
  9. R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (Wiley, 1995).
  10. H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
    [CrossRef]
  11. H. Moradi, H. H. Refai, and P. G. LoPresti, “Selection diversity for wireless optical communications with non-CSI non-coherent optimal detection,” in 2010 IEEE GLOBECOM Workshops (IEEE, 2010), pp. 1010–1014.
    [CrossRef]
  12. A. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  13. J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).
  14. A. C. Andrews and R. L. Philips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
    [CrossRef]
  15. X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 509–516 (2003).
    [CrossRef]
  16. N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
    [CrossRef]
  17. M.-A. Khalighi, N. Schwartz, N. Aitamer, and S. Bourennane, “Fading reduction by aperture averaging and spatial diversity in optical wireless systems,” J. Opt. Commun. Netw. 1, 580–593 (2009).
    [CrossRef]
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    [CrossRef]
  20. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (3)

G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
[CrossRef]

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Netw. 2, 689–700 (2010).
[CrossRef]

2009 (2)

2008 (1)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
[CrossRef]

2007 (1)

2004 (1)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[CrossRef]

2003 (2)

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 509–516 (2003).
[CrossRef]

H. C. Yang and M. S. Alouini, “Performance analysis of multi-branch switched diversity systems,” IEEE Trans. Commun. 51, 782–794 (2003).
[CrossRef]

2002 (1)

M. S. Alouini and M. K. Simon, “Dual diversity over correlated log-normal fading channels,” IEEE Trans. Commun. 50, 1946–1959 (2002).
[CrossRef]

2000 (1)

Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
[CrossRef]

1994 (2)

A. A. Abu-Dayya and N. C. Beaulieu, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Technol. 43, 970–976 (1994).
[CrossRef]

A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems on generalized-fading channels,” IEEE Trans. Commun. 42, 2959–2966 (1994).
[CrossRef]

Abu-Dayya, A. A.

A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems on generalized-fading channels,” IEEE Trans. Commun. 42, 2959–2966 (1994).
[CrossRef]

A. A. Abu-Dayya and N. C. Beaulieu, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Technol. 43, 970–976 (1994).
[CrossRef]

Ahmadi, V.

W. O. Popoola, Z. Ghassemlooy, E. Leitgeb, and V. Ahmadi, “Terrestrial free-space optical links with temporal diversity,” in CSNDSP’10 (IEEE/IET, 2010), pp. 598–603.

Aitamer, N.

Alexandropoulos, G. C.

G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
[CrossRef]

Alouini, M. S.

H. Nam and M. S. Alouini, “Optimization of multi-branch switched diversity systems,” IEEE Trans. Commun. 57, 2960–2970 (2009).
[CrossRef]

H. C. Yang and M. S. Alouini, “Performance analysis of multi-branch switched diversity systems,” IEEE Trans. Commun. 51, 782–794 (2003).
[CrossRef]

M. S. Alouini and M. K. Simon, “Dual diversity over correlated log-normal fading channels,” IEEE Trans. Commun. 50, 1946–1959 (2002).
[CrossRef]

Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
[CrossRef]

M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, 2nd ed. (Wiley, 2005).

Andrews, A. C.

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

A. C. Andrews and R. L. Philips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Anguita, J. A.

Atiquzzaman, M.

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

Beaulieu, N. C.

A. A. Abu-Dayya and N. C. Beaulieu, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Technol. 43, 970–976 (1994).
[CrossRef]

A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems on generalized-fading channels,” IEEE Trans. Commun. 42, 2959–2966 (1994).
[CrossRef]

Bourennane, S.

Brandt-Pearce, M.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
[CrossRef]

Chan, V. W. S.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[CrossRef]

Cvijetic, N.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (Wiley, 1995).

Ghassemlooy, Z.

W. O. Popoola, Z. Ghassemlooy, E. Leitgeb, and V. Ahmadi, “Terrestrial free-space optical links with temporal diversity,” in CSNDSP’10 (IEEE/IET, 2010), pp. 598–603.

Haas, S. M.

S. M. Haas, “Capacity of and coding for multiple-aperture wireless optical communications,” Ph.D. dissertation (Massachusetts Institute of Technology, 2003).

Hopen, C. Y.

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

Kahn, J. M.

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 509–516 (2003).
[CrossRef]

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (Wiley, 1995).

Khalighi, M.-A.

Ko, Y. C.

Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
[CrossRef]

Lee, E. J.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[CrossRef]

Leitgeb, E.

W. O. Popoola, Z. Ghassemlooy, E. Leitgeb, and V. Ahmadi, “Terrestrial free-space optical links with temporal diversity,” in CSNDSP’10 (IEEE/IET, 2010), pp. 598–603.

LoPresti, P. G.

H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Netw. 2, 689–700 (2010).
[CrossRef]

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high speed FSO links,” IET Optoelectron.(to be published).

H. Moradi, H. H. Refai, and P. G. LoPresti, “Selection diversity for wireless optical communications with non-CSI non-coherent optimal detection,” in 2010 IEEE GLOBECOM Workshops (IEEE, 2010), pp. 1010–1014.
[CrossRef]

Mathiopoulos, P. T.

G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
[CrossRef]

Moradi, H.

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Netw. 2, 689–700 (2010).
[CrossRef]

H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high speed FSO links,” IET Optoelectron.(to be published).

H. Moradi, H. H. Refai, and P. G. LoPresti, “Selection diversity for wireless optical communications with non-CSI non-coherent optimal detection,” in 2010 IEEE GLOBECOM Workshops (IEEE, 2010), pp. 1010–1014.
[CrossRef]

Nam, H.

H. Nam and M. S. Alouini, “Optimization of multi-branch switched diversity systems,” IEEE Trans. Commun. 57, 2960–2970 (2009).
[CrossRef]

Neifeld, M. A.

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

Philips, R. L.

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

A. C. Andrews and R. L. Philips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Popoola, W. O.

W. O. Popoola, Z. Ghassemlooy, E. Leitgeb, and V. Ahmadi, “Terrestrial free-space optical links with temporal diversity,” in CSNDSP’10 (IEEE/IET, 2010), pp. 598–603.

Refai, H. H.

H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimal detection of wireless optical signals,” J. Opt. Commun. Netw. 2, 689–700 (2010).
[CrossRef]

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high speed FSO links,” IET Optoelectron.(to be published).

H. Moradi, H. H. Refai, and P. G. LoPresti, “Selection diversity for wireless optical communications with non-CSI non-coherent optimal detection,” in 2010 IEEE GLOBECOM Workshops (IEEE, 2010), pp. 1010–1014.
[CrossRef]

Sagias, N. C.

G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
[CrossRef]

Schwartz, N.

Simon, M. K.

M. S. Alouini and M. K. Simon, “Dual diversity over correlated log-normal fading channels,” IEEE Trans. Commun. 50, 1946–1959 (2002).
[CrossRef]

Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
[CrossRef]

M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, 2nd ed. (Wiley, 2005).

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).

Vasic, B. V.

Wilson, S. G.

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
[CrossRef]

Yang, H. C.

H. C. Yang and M. S. Alouini, “Performance analysis of multi-branch switched diversity systems,” IEEE Trans. Commun. 51, 782–794 (2003).
[CrossRef]

Zhu, X.

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 509–516 (2003).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Areas Commun. (2)

N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, “Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE J. Sel. Areas Commun. 26, 3–12 (2008).
[CrossRef]

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[CrossRef]

IEEE Trans. Commun. (5)

A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems on generalized-fading channels,” IEEE Trans. Commun. 42, 2959–2966 (1994).
[CrossRef]

H. Nam and M. S. Alouini, “Optimization of multi-branch switched diversity systems,” IEEE Trans. Commun. 57, 2960–2970 (2009).
[CrossRef]

H. C. Yang and M. S. Alouini, “Performance analysis of multi-branch switched diversity systems,” IEEE Trans. Commun. 51, 782–794 (2003).
[CrossRef]

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 509–516 (2003).
[CrossRef]

M. S. Alouini and M. K. Simon, “Dual diversity over correlated log-normal fading channels,” IEEE Trans. Commun. 50, 1946–1959 (2002).
[CrossRef]

IEEE Trans. Veh. Technol. (3)

A. A. Abu-Dayya and N. C. Beaulieu, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Technol. 43, 970–976 (1994).
[CrossRef]

Y. C. Ko, M. S. Alouini, and M. K. Simon, “Analysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol. 49, 1813–1831 (2000).
[CrossRef]

G. C. Alexandropoulos, P. T. Mathiopoulos, and N. C. Sagias, “Switch-and-examine diversity over arbitrarily correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol. 59, 2080–2087 (2010).
[CrossRef]

J. Opt. Commun. Netw. (2)

Proc. SPIE (1)

H. Moradi, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE 7587, 75870O (2010).
[CrossRef]

Other (10)

H. Moradi, H. H. Refai, and P. G. LoPresti, “Selection diversity for wireless optical communications with non-CSI non-coherent optimal detection,” in 2010 IEEE GLOBECOM Workshops (IEEE, 2010), pp. 1010–1014.
[CrossRef]

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

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, 1978).

A. C. Andrews and R. L. Philips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, E. Leitgeb, and V. Ahmadi, “Terrestrial free-space optical links with temporal diversity,” in CSNDSP’10 (IEEE/IET, 2010), pp. 598–603.

S. M. Haas, “Capacity of and coding for multiple-aperture wireless optical communications,” Ph.D. dissertation (Massachusetts Institute of Technology, 2003).

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high speed FSO links,” IET Optoelectron.(to be published).

M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, 2nd ed. (Wiley, 2005).

R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (Wiley, 1995).

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Figures (6)

Fig. 1
Fig. 1

Setup configuration of a multireceiving FSO system model.

Fig. 2
Fig. 2

BER versus switching threshold γ T for several different values of γ ¯ using an N = 6 SDC combiner. σ χ = 0.2 , ρ = 0 .

Fig. 3
Fig. 3

BER versus switching threshold γ T for several different values of correlation coefficients ρ for an N = 6 SDC. σ χ = 0.2 , γ ¯ = 17 dB .

Fig. 4
Fig. 4

Comparison of BER versus average SNR γ ¯ when optimum switching threshold is applied for SSC and γ T = γ ¯ is applied for SDC. σ χ = 0.2 .

Fig. 5
Fig. 5

Outage probability versus normalized switching threshold γ T and outage threshold γ out for dual-branch SSC and SDC schemes ( N = 2 ).

Fig. 6
Fig. 6

APL due to diversity combining. σ χ = 0.2 , ρ = 0 .

Equations (29)

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

r i [ k ] = 2 R P t h i [ k ] s [ k ] + v i [ k ] , i = 1 , , N ,
γ i R 2 h i 2 P t 2 σ v 2 , γ i 0 , i = 1 , ... , N .
γ ¯ R 2 P t 2 σ v 2 .
f h i ( h i ) = 1 8 π h i σ χ exp [ ( ln ( h i ) 2 μ χ ) 2 8 σ χ 2 ] ,
F h i ( h i ) = Q [ ln ( h i ) + 2 σ χ 2 2 σ χ ] = 1 Q [ ln ( h i ) + 2 σ χ 2 2 σ χ ] ,
f h i ( γ i ) = 1 32 π γ i σ χ exp [ ( ln ( γ i / γ ¯ ) 4 σ χ 2 ) 2 32 σ χ 2 ]
F γ i ( γ i ) = 1 Q [ ln ( γ i / γ ¯ ) + 4 σ χ 2 4 σ χ ] ,
f Γ ( γ 1 , γ 2 , , γ N ) = exp ( 1 32 ( ln [ Γ ] ln [ Γ ¯ ] 4 Ψ ¯ ) Σ χ 1 ( ln [ Γ ] ln [ Γ ¯ ] 4 Ψ ¯ ) T ) 4 N ( 2 π ) N 2 ( det [ Σ χ ] ) 1 2 P [ Γ ] ,
χ = [ σ x 2 C 1 , 2 C 1 , N C 2 , 1 σ x 2 C 2 , N C N , 1 C N , 2 σ x 2 ] N × N ,
t comb = k τ ,
F SDC ( γ ) = N P ( γ [ k ] = γ n [ k ] γ n [ k ] γ ) .
γ [ k ] = γ n [ k ] i f f { γ [ k 1 ] = γ 1 [ k 1 ] γ 1 [ k ] < γ T γ n [ k ] = max { γ 1 , γ 2 , , γ N } or γ [ k 1 ] = γ 2 [ k 1 ] γ 2 [ k ] < γ T γ n [ k ] = max { γ 1 , γ 2 , , γ N } or γ [ k 1 ] = γ n [ k 1 ] γ n [ k ] < γ T γ n [ k ] = max { γ 1 , γ 2 , , γ N } or γ [ k 1 ] = γ 1 [ k 1 ] γ 1 [ k ] γ T or γ [ k 1 ] = γ N [ k 1 ] γ N [ k ] < γ T γ n [ k ] = max { γ 1 , γ 2 , , γ N } .
F SDC ( γ ) = i = 1 N P { γ [ k 1 ] = γ i [ k 1 ] γ i [ k ] < γ T γ n [ k ] = max { γ 1 [ k ] , γ 2 [ k ] , , γ N [ k ] } γ n [ k ] γ } + N P { γ [ k 1 ] = γ n [ k 1 ] γ n [ k ] γ T γ n [ k ] γ } .
P { γ [ k 1 ] = γ 1 [ k 1 ] γ 1 [ k ] γ } = P { γ [ k 1 ] = γ 2 [ k 1 ] γ 2 [ k ] γ } = = P { γ [ k 1 ] = γ N [ k 1 ] γ N [ k ] γ }
F SDC ( γ ) = N P { γ [ k 1 ] = γ m [ k 1 ] γ m [ k ] < γ T γ n [ k ] = max { γ 1 [ k ] , γ 2 [ k ] , , γ N [ k ] } γ n [ k ] γ } + N P { γ [ k 1 ] = γ n [ k 1 ] γ T γ n [ k ] γ } .
P SDC e = 0 f SDC ( γ ) Q ( γ ) d γ ,
F SDC ( γ ) = N P { γ [ k 1 ] = γ m [ k 1 ] } P { γ m [ k ] < γ T γ n [ k ] = max { γ 1 [ k ] , γ 2 [ k ] , , γ N [ k ] } γ n [ k ] γ } + N P { γ [ k 1 ] = γ n [ k 1 ] } P { γ T γ n [ k ] γ } .
F SDC ( γ ) = P { i = 1 , i n N γ i < γ n γ m < γ T γ n γ } + P { γ T γ n γ } .
F SDC γ < γ T ( γ ) = 0 γ 0 γ 0 γ n ... 0 γ n N -fold f Γ ( γ 1 , ... , γ N ) d γ N ... d γ j ... d γ 1 ( N 2 ) -fold ; j m , n d γ m d γ n ,
F SDC γ < γ T ( γ ) = 0 γ 0 γ ... 0 γ N -fold f Γ ( γ 1 , γ 2 , ... , γ N ) d γ N ... d γ 2 d γ 1 ,
f SDC γ < γ T ( γ ) = N 0 γ 0 γ ... 0 γ ( N 1 ) -fold f Γ ( γ , γ 2 , ... , γ N ) d γ N ... d γ 2 , γ < γ T .
F SDC γ γ T ( γ ) = 0 γ 0 γ T 0 γ ... 0 γ N -fold f Γ ( γ 1 , ... , γ N ) d γ N ... d γ j ... d γ 1 ( N 2 ) -fold ; j m , n d γ m d γ n + γ T γ f γ ( γ ) d γ ,
F SDC γ γ T ( γ ) = 0 γ 0 γ T 0 γ ... 0 γ N -fold f Γ ( γ , γ 2 , ... , γ N ) d γ N ... d γ 2 d γ 1 N -fold + γ T γ f γ ( γ ) d γ .
f SDC γ < γ T ( γ ) = ( N 1 ) 0 γ 0 γ 0 γ ( N 1 ) -fold f Γ ( γ , γ 2 , ... , γ N ) d γ N ... d γ 2 ( N 1 ) -fold + f γ ( γ ) , γ γ T .
P SDC e = 0 γ T f SDC γ < γ T ( γ ) Q ( γ ) d γ + γ T f SDC γ γ T ( γ ) Q ( γ ) d γ .
P SDC out = P { i = 1 N γ i [ k ] γ T } ,
f SDC ( γ ) f SC ( γ ) if     γ T .
f SDC ( γ ) f γ ( γ ) if     γ T 0.
P SC e = 0 d d γ P { i = 1 N γ i γ } P b ( γ ) d γ = 1 8 π 0 1 γ P { i = 1 N γ i γ } exp ( γ 2 ) d γ .

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