Abstract

In this paper, the deployment of novel space-time trellis codes (STTCs) with transmit laser selection (TLS) for free-space optical (FSO) communication systems using intensity modulation and direct detection (IM/DD) over atmospheric turbulence and misalignment fading channels is presented. Combining TLS and STTC with rate 1 bit/(s · Hz), a new code design criterion based on the use of the largest order statistics is here proposed for multiple-input/single-output (MISO) FSO systems in order to improve the diversity order gain by properly chosing the transmit lasers out of the available L lasers. Based on a pairwise error probability (PEP) analysis, closed-form asymptotic bit error-rate (BER) expressions in the range from low to high signal-to-noise ratio (SNR) are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results show diversity orders of 2L and 3L when simple two-state and four-state STTCs are considered, respectively. Simulation results are further demonstrated to confirm the analytical results.

© 2015 Optical Society of America

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References

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2015 (2)

M. Bhatnagar, “A One Bit Feedback Based Beamforming Scheme for FSO MISO System Over Gamma-Gamma Fading,” IEEE Trans. Commun. 63(4), 1306–1318 (2015).
[Crossref]

C. Abou-Rjeily, “Performance Analysis of FSO Communications with Diversity Methods: Add More Relays or More Apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

2014 (2)

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23861–23874 (2014).
[Crossref] [PubMed]

M. Khalighi and M. Uysal, “Survey on Free Space Optical Communication: A Communication Theory Perspective,” IEEE Commun. Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

2013 (1)

2012 (2)

2011 (2)

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13,480–13,496 (2011).
[Crossref]

C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (2011).
[Crossref]

2010 (4)

2009 (6)

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” J. Lightwave Technol. 27(8), 974–979 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009).
[Crossref]

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

F. Xu, A. Khalighi, P. Caussé, and S. Bourennane, “Channel coding and time-diversity for optical wireless links,” Opt. Express 17(2), 872–887 (2009).
[Crossref] [PubMed]

W. Gappmair and M. Flohberger, “Error performance of coded FSO links in turbulent atmosphere modeled by gamma-gamma distributions,” IEEE Trans. Wireless Commun. 8(5), 2209–2213 (2009).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

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. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

2005 (2)

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[Crossref]

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(9), 1896–1906 (2004).
[Crossref]

2003 (1)

M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003).
[Crossref]

2001 (2)

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 (2001).
[Crossref]

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 1-12 (2001).

Abou-Rjeily, C.

C. Abou-Rjeily, “Performance Analysis of FSO Communications with Diversity Methods: Add More Relays or More Apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (2011).
[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, 8 (2001).
[Crossref]

Alouini, M.-S.

M. K. Simon and M.-S. Alouini, Digital communications over fading channels, 2nd ed. (Wiley-IEEE, 2005).

Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, 2001).
[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 (2001).
[Crossref]

Anguita, J.

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

Bhatnagar, M.

M. Bhatnagar, “A One Bit Feedback Based Beamforming Scheme for FSO MISO System Over Gamma-Gamma Fading,” IEEE Trans. Commun. 63(4), 1306–1318 (2015).
[Crossref]

Boluda-Ruiz, R.

Bourennane, S.

Castillo-Vazquez, B.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

Castillo-Vazquez, C.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Caussé, P.

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(9), 1896–1906 (2004).
[Crossref]

Cheng, J.

Chiani, M.

M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003).
[Crossref]

Dang, A.

Dardari, D.

M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003).
[Crossref]

David, H. A.

H. A. David and H. N. Nagaraja, Order statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

Deng, P.

Denic, S.

Dhungana, Y.

Y. Dhungana and C. Tellambura, “New Simple Approximations for Error Probability and Outage in Fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

Djordjevic, I.

Djordjevic, I. B.

Fafalios, M. E.

Farid, A. A.

Flohberger, M.

W. Gappmair and M. Flohberger, “Error performance of coded FSO links in turbulent atmosphere modeled by gamma-gamma distributions,” IEEE Trans. Wireless Commun. 8(5), 2209–2213 (2009).
[Crossref]

Gappmair, W.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

W. Gappmair and M. Flohberger, “Error performance of coded FSO links in turbulent atmosphere modeled by gamma-gamma distributions,” IEEE Trans. Wireless Commun. 8(5), 2209–2213 (2009).
[Crossref]

Garcia-Zambrana, A.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

García-Zambrana, A.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, 7th ed. (Academic Inc., 2007).

Guo, H.

Han, Y.

Hiniesta-Gomez, A.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, 2001).
[Crossref]

Hranilovic, S.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007).
[Crossref]

Jafarkhani, H.

H. Jafarkhani, Space-Time Coding: Theory and Practice (Cambridge University, 2005).
[Crossref]

Karagianni, E. A.

Karagiannidis, G. K.

Kavehrad, M.

Khalighi, A.

Khalighi, M.

M. Khalighi and M. Uysal, “Survey on Free Space Optical Communication: A Communication Theory Perspective,” IEEE Commun. Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

Kim, I. I.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 1-12 (2001).

Korevaar, E. J.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 1-12 (2001).

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(9), 1896–1906 (2004).
[Crossref]

Leitgeb, E.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[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. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Liu, Z.

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

McArthur, B.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 1-12 (2001).

Nagaraja, H. N.

H. A. David and H. N. Nagaraja, Order statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

Neifeld, M.

Nistazakis, H. E.

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, 2001).
[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 (2001).
[Crossref]

Ren, Y.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, 7th ed. (Academic Inc., 2007).

Sandalidis, H. G.

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

Simon, M.

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

Simon, M. K.

M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003).
[Crossref]

M. K. Simon and M.-S. Alouini, Digital communications over fading channels, 2nd ed. (Wiley-IEEE, 2005).

Tang, J.

Tellambura, C.

Y. Dhungana and C. Tellambura, “New Simple Approximations for Error Probability and Outage in Fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

Tombras, G. S.

Tsiftsis, T. A.

Tsigopoulos, A. D.

Uysal, M.

M. Khalighi and M. Uysal, “Survey on Free Space Optical Communication: A Communication Theory Perspective,” IEEE Commun. Surveys Tutorials 16(4), 2231–2258 (2014).
[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. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Vasic, B.

Vilnrotter, V.

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

Wang, N.

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. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Yuan, X.

Zhou, Z.

IEEE Commun. Lett. (3)

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (2011).
[Crossref]

Y. Dhungana and C. Tellambura, “New Simple Approximations for Error Probability and Outage in Fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

IEEE Commun. Surveys Tutorials (1)

M. Khalighi and M. Uysal, “Survey on Free Space Optical Communication: A Communication Theory Perspective,” IEEE Commun. Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

IEEE J. Sel. Areas Commun. (2)

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(9), 1896–1906 (2004).
[Crossref]

C. Abou-Rjeily, “Performance Analysis of FSO Communications with Diversity Methods: Add More Relays or More Apertures?” IEEE J. Sel. Areas Commun. 33(9), 1890–1902 (2015).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

IEEE Trans. Commun. (2)

M. Bhatnagar, “A One Bit Feedback Based Beamforming Scheme for FSO MISO System Over Gamma-Gamma Fading,” IEEE Trans. Commun. 63(4), 1306–1318 (2015).
[Crossref]

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

IEEE Trans. Wireless Commun. (4)

W. Gappmair and M. Flohberger, “Error performance of coded FSO links in turbulent atmosphere modeled by gamma-gamma distributions,” IEEE Trans. Wireless Commun. 8(5), 2209–2213 (2009).
[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. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003).
[Crossref]

J. Lightwave Technol. (4)

J. Opt. Netw. (1)

Opt. Eng. (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, 8 (2001).
[Crossref]

Opt. Express (8)

F. Xu, A. Khalighi, P. Caussé, and S. Bourennane, “Channel coding and time-diversity for optical wireless links,” Opt. Express 17(2), 872–887 (2009).
[Crossref] [PubMed]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13,480–13,496 (2011).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010).
[Crossref]

Y. Han, A. Dang, Y. Ren, J. Tang, and H. Guo, “Theoretical and experimental studies of turbo product code with time diversity in free space optical communication,” Opt. Express 18(26), 26978–26988 (2010).
[Crossref]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20(3), 2096–2109 (2012).
[Crossref] [PubMed]

P. Deng, M. Kavehrad, Z. Liu, Z. Zhou, and X. Yuan, “Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence,” Opt. Express 21(13), 15213–15229 (2013).
[Crossref] [PubMed]

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23861–23874 (2014).
[Crossref] [PubMed]

Proc. SPIE (1)

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 1-12 (2001).

Other (6)

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, 7th ed. (Academic Inc., 2007).

H. Jafarkhani, Space-Time Coding: Theory and Practice (Cambridge University, 2005).
[Crossref]

M. K. Simon and M.-S. Alouini, Digital communications over fading channels, 2nd ed. (Wiley-IEEE, 2005).

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, 2001).
[Crossref]

Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com .

H. A. David and H. N. Nagaraja, Order statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 MISO FSO system using STTC with transmit laser selection.
Fig. 2
Fig. 2 Trellis diagram of two-state STTC, 1 bit/(s · Hz).
Fig. 3
Fig. 3 Trellis diagram of two-state STTC modified.
Fig. 4
Fig. 4 Trellis diagram of four-state STTC modified.
Fig. 5
Fig. 5 Error event paths for the two-state STTC scheme proposed.
Fig. 6
Fig. 6 Error event paths of length (a) three and (b) four for the four-state STTC scheme proposed.
Fig. 7
Fig. 7 BER performance using STTC with TLS over atmospheric turbulence and misalignment fading channels, when different weather conditions (a) C n 2 = 1.7 × 10 14 m 2 / 3 and (b) C n 2 = 8 × 10 14 m 2 / 3 are assumed for a link distance of d = 3 km and values of normalized beamwidth and jitter of (ωz/r, σs/r) = (5, 1).
Fig. 8
Fig. 8 Error event path for the STTC scheme in Fig. 2

Equations (37)

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y m ( t ) = η i m ( t ) x ( t ) + z ( t ) ,
f I m ( i ) = α m β m φ m 2 A 0 ζ m Γ ( α m ) Γ ( β m ) G 1 , 3 3 , 0 ( α m β m A 0 ζ m i | φ m 2 φ m 2 1 , α m 1 , β m 1 ) , i 0
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 ,
f I m ( i ) φ m 2 ( α m β m ) β m Γ ( α m β m ) ( A 0 ζ m ) β m Γ ( α m ) Γ ( β m ) ( φ m 2 β m ) i β m 1 e i α m β m ( φ m 2 β m ) A 0 ζ ( α m β m 1 ) ( β m φ m 2 + 1 ) , φ m 2 > β m
f I m ( i ) φ m 2 ( α m β m ) α m 2 Γ ( α m φ m 2 ) Γ ( β m φ m 2 ) ( A 0 ζ m ) φ m 2 Γ ( α m ) Γ ( β m ) i φ m 2 1 , φ m 2 < β m
P b 1 n c X P ( X ) X X ^ n ( X , X ^ ) P ( X X ^ ) ,
0 x τ 1 1 exp ( τ 2 x τ 3 x 2 ) d x = τ 3 τ 1 2 Γ ( τ 1 ) H τ 1 ( τ 2 2 τ 3 ) .
X ^ 1 = X S 1 S 2 S 1 = [ 0 0 0 0 ] , X ^ 2 = X S 2 S 1 S 2 = [ 0 0 0 0 ] ,
X 1 = X S 1 S 1 S 1 = [ 1 1 1 1 ] , X 2 = X S 2 S 2 S 2 = [ 1 1 0 0 ] ,
P ( X X ^ ) = 1 2 P ( X 1 X ^ 1 ) + 1 2 P ( X 2 X ^ 2 ) .
P ( X 1 X ^ 1 ) = 0 0 i 11 0 0 i 21 Q ( γ ξ 3 / 2 ( ( i 11 + i 12 ) 2 + ( i 21 + i 22 ) 2 ) ) × f I ( L 1 ) ( L ) ( i 2 , i 11 ) f I ( L 1 ) ( L ) ( i 22 , i 21 ) d i 11 d i 12 d i 21 d i 22 ,
P ( X 2 X ^ 2 ) = 0 0 Q ( γ ξ 3 / 2 ( i 11 2 + i 21 2 ) ) f I ( L ) ( i 11 ) f I ( L ) ( i 21 ) d i 11 d i 21 ,
P b L a L b 1 L 6 ( 0 i b L 1 exp ( c ( 1 + b L ) a ( 1 + b ) i 4 γ ξ 9 i 2 ) d i ) 2 + L a L b 1 L 2 ( 0 i b L 1 exp ( c ( 1 + b L ) a ( 1 + b ) i 16 γ ξ 27 i 2 ) d i ) 2 .
P b a 2 L 2 4 b L 1 3 2 b L 1 b 2 L Γ ( b L + 1 ) 2 ( γ ξ ) b L × ( 4 b L H b L ( 3 ( b c L + c ) 4 a ( b + 1 ) γ ξ ) 2 + 3 b L + 1 H b L ( 3 3 ( b c L + c ) 8 a ( b + 1 ) γ ξ ) 2 ) .
( X ^ X ^ ) | continuous = [ 1 1 1 0 0 1 1 0 0 ] , ( X X ^ ) | dashed = [ 1 1 1 1 1 0 0 1 1 ] .
P ( X X ^ ) | length 3 = 0 0 i 11 0 i 12 0 0 0 i 31 Q ( γ ξ 2 ( ( i 11 + i 13 ) 2 + i 21 2 + ( i 31 + i 32 ) 2 ) ) × f I ( L 2 ) ( L ) ( i 13 , i 11 ) f I ( L ) ( i 21 ) f I ( L 1 ) ( L ) ( i 32 , i 31 ) d i 11 d i 12 d i 13 d i 21 d i 31 d i 32 ,
X X ^ = [ 1 0 0 1 1 1 0 0 0 0 1 1 ] ,
P ( X X ^ ) | length 4 = 0 0 i 11 0 0 0 0 i 41 0 i 42 Q ( γ ξ 2 ( ( i 11 + i 12 ) 2 + i 22 2 + i 33 2 + ( i 41 + i 43 ) 2 ) ) × f I ( L 1 ) ( L ) ( i 12 , i 11 ) f I ( L 1 ) ( i 22 ) f I ( L 2 ) ( i 33 ) × f I ( L 2 ) ( L ) ( i 43 , i 41 ) d i 11 d i 12 d i 22 d i 33 d i 41 d i 42 d i 43 .
P b 3 P ( X X ^ ) | length 3 + 2 P ( X X ^ ) | length 4 .
f I Σ L , L 1 ( x ) = L ( L 1 ) 0 x 2 F I L 2 ( y ) f I ( y ) f I ( x y ) d y ,
f I Σ L , L 2 ( x ) = L ( L 1 ) ( L 2 ) 0 x 2 F I L 3 ( y ) f I ( y ) f I ( x y ) ( F I ( x y ) F I ( y ) ) d y .
f I Σ L , L 1 ( x ) L ( L 1 ) a L exp ( c x / a ) B 1 2 ( b ( L 1 ) , b ) x b L 1 ,
f I Σ L , L 2 ( x ) L ( L 1 ) ( L 2 ) a L b L 2 exp ( c x / a ) ( B 1 2 ( b ( L 2 ) , 2 b ) B 1 2 ( b ( L 1 ) , b ) ) x b L 1 .
P b 1 4 T L , L 2 ( γ ξ / 4 ) T L ( γ ξ / 4 ) T L , L 1 ( γ ξ / 4 ) + 3 4 T L , L 2 ( γ ξ / 3 ) T L ( γ ξ / 3 ) T L , L 1 ( γ ξ / 3 ) + 1 6 T L , L 1 ( γ ξ / 4 ) T L 1 ( γ ξ / 4 ) T L 2 ( γ ξ / 4 ) T L , L 2 ( γ ξ / 4 ) + 1 2 T L , L 1 ( γ ξ / 3 ) T L 1 ( γ ξ / 3 ) T L 2 ( γ ξ / 3 ) T L , L 2 ( γ ξ / 3 ) ,
P b Ψ a 3 L b 5 3 L 8 ( L 2 ) Γ ( b L ) ( γ ξ ) b 2 3 L ( 2 3 b H b L ( c a γ ξ ) 2 H b L ( c ζ 1 a γ ξ ) + 3 3 b L 2 + 1 H b L ( 3 c 2 a γ ξ ) 2 H b L ( 3 c ζ 1 2 a γ ξ ) ) + Ψ a 4 L 3 b 8 4 L 2 3 b + 2 3 3 b 2 + 1 ( L 1 ) L ( γ ξ ) b 2 ( 4 L 3 ) × Γ ( ( L 2 ) b + 1 ) Γ ( b ( L 1 ) ) 1 ( 3 3 b 2 16 b L H b L ( c a γ ξ ) 2 H b b L ( c ζ 2 a γ ξ ) H 2 b b L ( c ζ 3 a γ ξ ) + 2 3 b 3 3 2 b L H b L ( 3 c 2 a γ ξ ) 2 H b b L ( 3 c ζ 2 2 a γ ξ ) H 2 b b L ( 3 c ζ 3 2 a γ ξ ) ) ,
X = X S 1 S 1 S 1 = [ 0 0 0 0 ] , X ^ = X S 1 S 2 S 1 = [ 0 1 1 0 ] ,
P ( X X ^ | { I 1 , I 2 } ) = Q ( ( d E / 2 ) 2 2 N 0 ( i 1 2 + i 2 2 ) ) = Q ( γ ξ 2 ( i 1 2 + i 2 2 ) ) ,
P ( X X ^ ) = 0 0 Q ( γ ξ 2 ( i 1 2 + i 2 2 ) ) f I 1 ( i 1 ) f I 2 ( i 2 ) d i 1 d i 2 .
P b a 2 12 ( 0 i b 1 exp ( c a i γ ξ 2 i 2 ) d i ) 2 + a 2 4 ( 0 i b 1 exp ( c a i γ ξ 3 i 2 ) d i ) 2 ,
P b 1 12 a 2 ( γ ξ ) b Γ ( b ) 2 ( 4 b H b ( c a γ ξ ) 2 + 3 b + 1 H b ( 3 c 2 a γ ξ ) 2 ) .
P ( X X ^ | + { I ( L ) , I ( L 1 ) } ) = Q ( γ ξ 2 ( i 1 2 + i 2 2 ) ) .
P ( X X ^ ) = 0 0 Q ( γ ξ 2 ( i 1 2 + i 2 2 ) ) f I ( L ) ( i 1 ) f I ( L 1 ) ( i 2 ) d i 1 d i 2 .
f I ( r ) ( i ) = 1 B ( r , L r + 1 ) f I ( i ) ( 1 F I ( i ) ) L r ( F I ( i ) ) r 1 ,
f I ( r ) ( i ) f I ( i ) ( F I ( i ) ) r 1 B ( r , L r + 1 ) = a r b 1 r B ( r , L r + 1 ) exp ( c ( b r + 1 ) a ( b + 1 ) i ) i b r 1 .
P b 1 12 0 exp ( γ ξ i 1 2 4 ) f I ( L ) ( i 1 ) d i 1 0 exp ( γ ξ i 2 2 4 ) f I ( L 1 ) ( i 2 ) d i 2 + 1 4 0 exp ( γ ξ i 1 2 3 ) f I ( L ) ( i 1 ) d i 1 0 exp ( γ ξ i 2 2 3 ) f I ( L 1 ) ( i 2 ) d i 2 .
P b 2 b 2 3 b 2 1 a 2 L 1 b 1 2 L L Γ ( ( L 1 ) b + 1 ) Γ ( b L + 1 ) ( γ ξ ) b 2 ( 2 L 1 ) ( 3 b / 2 4 b L H b L ( c ( b L + 1 ) a ( b + 1 ) γ ξ ) H b b L ( c ( b ( L ) + b 1 ) a ( b + 1 ) γ ξ ) + 2 b 3 b L + 1 H b L ( 3 c ( b L + 1 ) 2 a ( b + 1 ) γ ξ ) H b b L ( 3 c ( b L 1 ) + 1 2 a ( b + 1 ) γ ξ ) ) ,

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