Abstract

This paper presents a modified channel likelihood model for optical communication systems with a photon-counting array receiver where photon-counting events are impaired by undesirable dead time and jitters. After the photon-counting detector detects a photon, the detector will go into a period of dead time under which it cannot detect any incident photons. In this context, the channel will have memory. We derive the channel likelihood in the presence of the detector dead time and the random jitter of the photon arrival. The impact of dead time and jitters on the performance of a pulse-position-modulated (PPM) optical communication system is also investigated. The simulation results indicate that the modified channel likelihood expressions can obtain a more obvious performance gain under the context of a stronger background noise, fewer detection elements, longer dead time and bigger jitter.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. M. Boroson and B. S. Robinson, “The lunar laser communication demonstration: NASA’s first step toward very high data rate support of science and exploration missions,” Space Sci. Rev. 185(1–4), 115–128 (2014).
    [Crossref]
  2. M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
    [Crossref]
  3. T. W. Murphy, “Lunar laser ranging: the millimeter challenge,” Rep. Prog. Phys. 76(7), 076901 (2013).
    [Crossref] [PubMed]
  4. L. Xue, M. Li, L. Zhang, D. Zhai, Z. Li, L. Kang, Y. Li, H. Fu, M. Ming, X. T. Sen Zhang, Y. Xiong, and P. Wu, “Long-range laser ranging using superconducting nanowire single-photon detectors,” Chin. Opt. Lett. 14(7), 071201 (2016).
    [Crossref]
  5. A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).
  6. S. Deng and A. P. Morrison, “Active quench and reset integrated circuit with novel hold-off time control logic for Geiger-mode avalanche photodiodes,” Opt. Lett. 37(18), 3876–3878 (2012).
    [Crossref] [PubMed]
  7. V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun. 50(7), 1091–1097 (2002).
    [Crossref]
  8. A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
    [Crossref]
  9. A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
    [Crossref]
  10. B. Moision, “Photon jitter mitigation for the optical channel,” IPN Progress Rep. 171, 1–13 (2007).
  11. B. Moision and W. Farr, “Communication limits due to photon detector jitter,” IEEE Photonics Technol. Lett. 20(9), 715–717 (2008).
    [Crossref]
  12. E. Sarbazi and H. Haas, “Detection statistics and error performance of SPAD-based optical receivers,” in Proceedings of IEEE Conference on Personal, Indoor, and Mobile Radio Communications (IEEE, 2015), pp. 830–834.
    [Crossref]
  13. Z. Li, J. Lai, C. Wang, W. Yan, and Z. Li, “Influence of dead-time on detection efficiency and range performance of photon-counting laser radar that uses a Geiger-mode avalanche photodiode,” Appl. Opt. 56(23), 6680–6687 (2017).
    [Crossref] [PubMed]
  14. B. Moision and J. Hamkin, “Coded modulation for the deep-space optical channel: serially concatenated pulse-position modulation,” Dev. Biol. 283(1), 113–127 (2005).
    [PubMed]

2017 (1)

2016 (1)

2014 (2)

D. M. Boroson and B. S. Robinson, “The lunar laser communication demonstration: NASA’s first step toward very high data rate support of science and exploration missions,” Space Sci. Rev. 185(1–4), 115–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

2013 (1)

T. W. Murphy, “Lunar laser ranging: the millimeter challenge,” Rep. Prog. Phys. 76(7), 076901 (2013).
[Crossref] [PubMed]

2012 (3)

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).

S. Deng and A. P. Morrison, “Active quench and reset integrated circuit with novel hold-off time control logic for Geiger-mode avalanche photodiodes,” Opt. Lett. 37(18), 3876–3878 (2012).
[Crossref] [PubMed]

2008 (1)

B. Moision and W. Farr, “Communication limits due to photon detector jitter,” IEEE Photonics Technol. Lett. 20(9), 715–717 (2008).
[Crossref]

2007 (1)

B. Moision, “Photon jitter mitigation for the optical channel,” IPN Progress Rep. 171, 1–13 (2007).

2005 (1)

B. Moision and J. Hamkin, “Coded modulation for the deep-space optical channel: serially concatenated pulse-position modulation,” Dev. Biol. 283(1), 113–127 (2005).
[PubMed]

2002 (1)

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

Adibi, A.

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

Amoozegar, F.

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

Bardin, J.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Beyer, A. D.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Boroson, D. M.

D. M. Boroson and B. S. Robinson, “The lunar laser communication demonstration: NASA’s first step toward very high data rate support of science and exploration missions,” Space Sci. Rev. 185(1–4), 115–128 (2014).
[Crossref]

Chang, S.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Cova, S.

A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).

Deng, S.

Eftekhar, A.

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

Farr, W.

B. Moision and W. Farr, “Communication limits due to photon detector jitter,” IEEE Photonics Technol. Lett. 20(9), 715–717 (2008).
[Crossref]

Farr, W. H.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Fu, H.

Gin, J. W.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Haas, H.

E. Sarbazi and H. Haas, “Detection statistics and error performance of SPAD-based optical receivers,” in Proceedings of IEEE Conference on Personal, Indoor, and Mobile Radio Communications (IEEE, 2015), pp. 830–834.
[Crossref]

Hamkin, J.

B. Moision and J. Hamkin, “Coded modulation for the deep-space optical channel: serially concatenated pulse-position modulation,” Dev. Biol. 283(1), 113–127 (2005).
[PubMed]

Hashmi, A. J.

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

Kang, L.

Lai, J.

Li, M.

Li, Y.

Li, Z.

Marsili, F.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Ming, M.

Mirin, R. P.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Moision, B.

B. Moision and W. Farr, “Communication limits due to photon detector jitter,” IEEE Photonics Technol. Lett. 20(9), 715–717 (2008).
[Crossref]

B. Moision, “Photon jitter mitigation for the optical channel,” IPN Progress Rep. 171, 1–13 (2007).

B. Moision and J. Hamkin, “Coded modulation for the deep-space optical channel: serially concatenated pulse-position modulation,” Dev. Biol. 283(1), 113–127 (2005).
[PubMed]

Morrison, A. P.

Murphy, T. W.

T. W. Murphy, “Lunar laser ranging: the millimeter challenge,” Rep. Prog. Phys. 76(7), 076901 (2013).
[Crossref] [PubMed]

Nam, S. W.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Patawaran, F. D.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Ravindran, P.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Resta, G. V.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Robinson, B. S.

D. M. Boroson and B. S. Robinson, “The lunar laser communication demonstration: NASA’s first step toward very high data rate support of science and exploration missions,” Space Sci. Rev. 185(1–4), 115–128 (2014).
[Crossref]

Russell, D. S.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Sarbazi, E.

E. Sarbazi and H. Haas, “Detection statistics and error performance of SPAD-based optical receivers,” in Proceedings of IEEE Conference on Personal, Indoor, and Mobile Radio Communications (IEEE, 2015), pp. 830–834.
[Crossref]

Sen Zhang, X. T.

Shaw, M. D.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Srinivasan, M.

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

Stern, J. A.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

Tosi, A.

A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).

Verma, V. B.

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[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]

Wang, C.

Wu, P.

Xiong, Y.

Xue, L.

Yan, W.

Zappa, F.

A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).

Zhai, D.

Zhang, L.

Appl. Opt. (1)

Chin. Opt. Lett. (1)

Dev. Biol. (1)

B. Moision and J. Hamkin, “Coded modulation for the deep-space optical channel: serially concatenated pulse-position modulation,” Dev. Biol. 283(1), 113–127 (2005).
[PubMed]

IEEE Photonics Technol. Lett. (1)

B. Moision and W. Farr, “Communication limits due to photon detector jitter,” IEEE Photonics Technol. Lett. 20(9), 715–717 (2008).
[Crossref]

IEEE Trans. Commun. (1)

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

IPN Progress Rep. (1)

B. Moision, “Photon jitter mitigation for the optical channel,” IPN Progress Rep. 171, 1–13 (2007).

Opt. Commun. (2)

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of adaptive optics-based telescope arrays in a deep-space inter-planetary optical communications link between Earth and Mars,” Opt. Commun. 333(4), 120–128 (2014).
[Crossref]

A. J. Hashmi, A. Eftekhar, A. Adibi, and F. Amoozegar, “A Kalman filter based synchronization scheme for telescope array receivers in deep-space optical communication links,” Opt. Commun. 285(24), 5037–5043 (2012).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

A. Tosi, F. Zappa, and S. Cova, “Single-photon detectors for practical quantum cryptography,” Proc. SPIE 8542, 854208 (2012).

Rep. Prog. Phys. (1)

T. W. Murphy, “Lunar laser ranging: the millimeter challenge,” Rep. Prog. Phys. 76(7), 076901 (2013).
[Crossref] [PubMed]

Space Sci. Rev. (1)

D. M. Boroson and B. S. Robinson, “The lunar laser communication demonstration: NASA’s first step toward very high data rate support of science and exploration missions,” Space Sci. Rev. 185(1–4), 115–128 (2014).
[Crossref]

Other (2)

M. D. Shaw, F. Marsili, A. D. Beyer, J. A. Stern, G. V. Resta, P. Ravindran, S. Chang, J. Bardin, D. S. Russell, J. W. Gin, F. D. Patawaran, V. B. Verma, R. P. Mirin, S. W. Nam, and W. H. Farr, “Arrays of WSi superconducting nanowire single photon detectors for deep-space optical communications,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2015), pp. 1–2.
[Crossref]

E. Sarbazi and H. Haas, “Detection statistics and error performance of SPAD-based optical receivers,” in Proceedings of IEEE Conference on Personal, Indoor, and Mobile Radio Communications (IEEE, 2015), pp. 830–834.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Left: The transmitted signal pulse P(t). Middle: The Gaussian jitter’s probability density function f δ (δ). Right: The probability density function f(t) of the received signal.
Fig. 2
Fig. 2 BER performance of photon-counting detector array receiver under different detector dead time, detector components and background noise.
Fig. 3
Fig. 3 BER performance of photon-counting detector array receiver under N = 8, nb = 1.0 and Td = 64ns.
Fig. 4
Fig. 4 BER performance of photon-counting detector array receiver for N = 8, nb = 1.0, Td = 64ns and the standard deviation σ δ : (a) σ δ =0.1 T s . (b)   σ δ =0.2 T s .

Equations (13)

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

{ p n,j i ( k n,j i |1)= [ ( n s + n b )/NH ] k n,j i k n,j i ! exp[ ( n s + n b )/NH ] p n,j i ( k n,j i |0)= ( n b /NH ) k n,j i k n,j i ! exp( n b /NH )
LL R i =ln n=1 N j=(i1)H+1 iH p n,j i ( k n,j i |1) n=1 N j=(i1)H+1 iH p n,j i ( k n,j i |0) = n=1 N j=(i1)H+1 iH k n,j i ln( 1+ n s n b ) n s
{ P 01 = p n,j i ( k n,j i =0|1)=exp[ ( n s + n b )/NH ] P 00 = p n,j i ( k n,j i =0|0)=exp( n b /NH)
{ P 11 =1 P 01 =1exp[ ( n s + n b )/NH ] P 10 =1 P 00 =1exp( n b /NH)
{ P 01 ' = p n,j i ( k n,j i =0|1)=1 P 00 ' = p n,j i ( k n,j i =0|0)=1
L R i = n=1 N j=(i1)H+1 iH p n,j i ( k n,j i |1) n=1 N j=(i1)H+1 iH p n,j i ( k n,j i |0) = p 1,(i1)H+1 i ( k 1,(i1)H+1 i |1) p 1,(i1)H+2 i ( k 1,(i1)H+2 i |1) p N,iH i ( k N,iH i |1) p 1,(i1)H+1 i ( k 1,(i1)H+1 i |0) p 1,(i1)H+2 i ( k 1,(i1)H+2 i |0) p N,iH i ( k N,iH i |0) = ( P 01 P 00 ) H 1 i ( P 11 P 10 ) H 2 i ( P 01 ' P 00 ' ) H 3 i
{ H 1 i + H 2 i + H 3 i =NH H 2 i = n=1 N j=(i1)H+1 iH k n,j i H 3 i = n=1 N j=(i1)H+1 iH t=j H d j-1 k n,t i H 1 i =NH( n=1 N j=(i1)H+1 iH k n,j i + n=1 N j=(i1)H+1 iH t=j H d j-1 k n,t i ) =NH n=1 N j=(i1)H+1 iH ( t=j H d j-1 k n,t i + k n,j i ) =NH n=1 N j=(i1)H+1 iH t=j H d j k n,t i
LL R i = H 1 i ln P 01 P 00 + H 2 i ln P 11 P 10 = n=1 N j=(i1)H+1 iH k n,j i ln 1exp[ ( n s + n b )/NH ] 1exp( n b /NH ) + n s NH n=1 N j=(i1)H+1 iH t=j H d j k n,t i n s
f(t)=(P f δ (δ))(t)
k n,j i = w n,j i n s /N+ n b /NH
w n,j i = (j1)Δti T s jΔti T s f(ti T s ) dt
L R i L R 1 i L R 0 i L R 1 i = n=1 N j=(i2)H+1 (i+1)H p n,j i ( k n,j i |1) n=1 N j=(i2)H+1 (i+1)H p n,j i ( k n,j i |0) = j=(i2)H+1 (i+1)H [ exp[ ( w n,j i n s /N+ n b /NH ) ] exp( n b /NH ) ] N n=1 N t=j H d j k n,t i ( 1exp[ ( w n,j i n s /N+ n b /NH ) ] 1exp( n b /NH ) ) n=1 N k n,j i
LL R i =lnL R i n=1 N j=(i2)H+1 (i+1)H k n,j i ln( 1exp[ ( w n,j i n s /N+ n b /NH ) ] 1exp( n b /NH ) )+ n s N n=1 N j=(i2)H+1 (i+1)H t=j H d j w n,j i k n,t i n s

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