Abstract

We analyze the performance of a free-space optical (FSO) link affected by atmospheric turbulence and line-of-sight (LOS) blockage. For this purpose, the atmospheric turbulence induced fading is modeled by the distribution, which includes the Gamma-Gamma distribution as special case. We exploit the fact that the physical interpretation of the distribution allows to split the optical energy through the propagation link into three different components: two coherent components and one incoherent scatter component. Based on this separation, we derive novel analytical expressions for the probability density function (PDF), for the cumulative distribution function (CDF) and for the moment generating function (MGF) of the distribution under the temporary blockage of the coherent components, hereinafter referred to as LOS blockage. Further, a new closed-form expression for the outage probability (OP) under LOS blockage is derived in terms of the turbulence model parameters and the LOS blockage probability. By means of an asymptotic analysis, this expression is simplified in the high-SNR regime and the OP in terms of the diversity order and diversity gain is then deduced. Obtained results show that the impact of the LOS blockage on the OP strongly depends on the intensity of the turbulence and on the LOS blockage probability.

© 2017 Optical Society of America

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References

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    [Crossref]
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2016 (3)

2015 (4)

2012 (2)

2011 (1)

2010 (1)

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

2009 (2)

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

2006 (2)

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Z. Kolka, O. Wilfert, D. Biolek, and V. Biolkova, “Availability model of free-space optical data link,” Int. J. Microw. Opt. Technol. 1(2), 612–616 (2006).

2004 (1)

P. M. Shankar, “Error rates in generalized shadowed fading channels,” Wirel. Pers. Commun. 28(3), 233–238 (2004).
[Crossref]

2003 (2)

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

S. Bloom, E. Korevaar, J. Schuster, and H. Willebrand, “Understanding the performance of free-space optics,” J. Opt. Netw. 2(6), 178–200 (2003).

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

2001 (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), 1554–1562 (2001).
[Crossref]

1987 (2)

1986 (1)

1970 (1)

R. M. Lerner and A. E. Holland, “The optical scatter channel,” Proc. IEEE 58(10), 1547–1563, (1970).
[Crossref]

Abbas, T.

T. Abbas, K. Sjoberg, J. Karedal, and F. Tufvesson, “A Measurement Based Shadow Fading Model for Vehicle-to-Vehicle Network Simulations,” Int. J. Antennas Propag. 2015, 190607 (2015).
[Crossref]

Ahmadi, V.

Z. Ghassemlooy, W. O. Popoola, V. Ahmadi, and E. Leitgeb., “MIMO free-space optical communication employing subcarrier intensity modulation in atmospheric turbulence channels,” in International Conference on Communications Infrastructure. Systems and Applications in Europe, R. Mehmood, E. Cerqueira, R. Piesiewicz, and I. Chlamtac, eds. (Springer, 2009), 16, pp. 61–73.

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]

Alouini, M. S.

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (ℳ) Turbulence Channels With Pointing Errors,” IEEE Trans. Wirel. Comm. 15(1), 91–102 (2016).
[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 and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Anh, T. P.

V. P. Thanh, T. Cong-Thang, and T. P. Anh, “On the MGF-Based Approximation of the Sum of Independent Gamma-Gamma Random Variables,” in Proceedings of the IEEE 81st Vehicular Technology Conference, (Glasgow, 2015).

Ansari, I. S.

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (ℳ) Turbulence Channels With Pointing Errors,” IEEE Trans. Wirel. Comm. 15(1), 91–102 (2016).
[Crossref]

Antic, D. S.

Aoki, T.

Arimoto, Y.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Barakat, R.

Biolek, D.

Z. Kolka, O. Wilfert, D. Biolek, and V. Biolkova, “Availability model of free-space optical data link,” Int. J. Microw. Opt. Technol. 1(2), 612–616 (2006).

Biolkova, V.

Z. Kolka, O. Wilfert, D. Biolek, and V. Biolkova, “Availability model of free-space optical data link,” Int. J. Microw. Opt. Technol. 1(2), 612–616 (2006).

Bithas, P. S.

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Bloom, S.

Castillo-Vázquez, M.

Churnside, J. H.

Ciaramella, E.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Clifford, S. F.

Cong-Thang, T.

V. P. Thanh, T. Cong-Thang, and T. P. Anh, “On the MGF-Based Approximation of the Sum of Independent Gamma-Gamma Random Variables,” in Proceedings of the IEEE 81st Vehicular Technology Conference, (Glasgow, 2015).

Contestabile, G.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

D’Errico, A.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Djordjevic, G. T.

Endo, H.

Fujiwara, M.

Garrido-Balsells, J. M.

Ghassemlooy, Z.

Z. Ghassemlooy, W. O. Popoola, V. Ahmadi, and E. Leitgeb., “MIMO free-space optical communication employing subcarrier intensity modulation in atmospheric turbulence channels,” in International Conference on Communications Infrastructure. Systems and Applications in Europe, R. Mehmood, E. Cerqueira, R. Piesiewicz, and I. Chlamtac, eds. (Springer, 2009), 16, pp. 61–73.

Giannakis, G. B.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

Gomez, G.

F. J. Lopez-Martinez, G. Gomez, and J. M. Garrido-Balsells, “Physical-Layer Security in Free-Space Optical Communications,” IEEE Photon. J. 7(2), 1–14 (2015).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Guarino, V.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Higashino, T.

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

Holland, A. E.

R. M. Lerner and A. E. Holland, “The optical scatter channel,” Proc. IEEE 58(10), 1547–1563, (1970).
[Crossref]

Hranilovic, S.

S. Hranilovic, Wireless Optical Communication Systems (Springer, 2005).

Ito, T.

Jakerman, E.

Jurado-Navas, A.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Karagiannidis, G. K.

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Karedal, J.

T. Abbas, K. Sjoberg, J. Karedal, and F. Tufvesson, “A Measurement Based Shadow Fading Model for Vehicle-to-Vehicle Network Simulations,” Int. J. Antennas Propag. 2015, 190607 (2015).
[Crossref]

Kazaura, K.

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

Kennedy, R. S.

R. S. Kennedy and J. H. Shapiro, “Multipath dispersion in low visibility optical communication channels,” Res. Lab. Elect. Report RADC-TR-77-73 (MIT, 1977).
[Crossref]

Kitamura, M.

Kolka, Z.

Z. Kolka, O. Wilfert, D. Biolek, and V. Biolkova, “Availability model of free-space optical data link,” Int. J. Microw. Opt. Technol. 1(2), 612–616 (2006).

Komaki, S.

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

Korevaar, E.

Laurenti, N.

Leitgeb., E.

Z. Ghassemlooy, W. O. Popoola, V. Ahmadi, and E. Leitgeb., “MIMO free-space optical communication employing subcarrier intensity modulation in atmospheric turbulence channels,” in International Conference on Communications Infrastructure. Systems and Applications in Europe, R. Mehmood, E. Cerqueira, R. Piesiewicz, and I. Chlamtac, eds. (Springer, 2009), 16, pp. 61–73.

Lerner, R. M.

R. M. Lerner and A. E. Holland, “The optical scatter channel,” Proc. IEEE 58(10), 1547–1563, (1970).
[Crossref]

Lopez-Martinez, F. J.

F. J. Lopez-Martinez, G. Gomez, and J. M. Garrido-Balsells, “Physical-Layer Security in Free-Space Optical Communications,” IEEE Photon. J. 7(2), 1–14 (2015).
[Crossref]

Lu, X.

Majumdar, A. K.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances (Springer, 2010).

Mathiopoulos, P. T.

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Matsumoto, M.

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Monroy, I. T.

Nistazakis, H. E.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Paris, J. F.

Petkovic, M. I.

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 and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Popoola, W. O.

Z. Ghassemlooy, W. O. Popoola, V. Ahmadi, and E. Leitgeb., “MIMO free-space optical communication employing subcarrier intensity modulation in atmospheric turbulence channels,” in International Conference on Communications Infrastructure. Systems and Applications in Europe, R. Mehmood, E. Cerqueira, R. Piesiewicz, and I. Chlamtac, eds. (Springer, 2009), 16, pp. 61–73.

Presi, M.

E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
[Crossref]

Puerta-Notario, A.

Ricklin, J. C.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances (Springer, 2010).

Rontogiannis, A. A.

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Elsevier, 2007).

Sagias, N. C.

P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
[Crossref]

Sasaki, M.

Schuster, J.

Shankar, P. M.

P. M. Shankar, “Error rates in generalized shadowed fading channels,” Wirel. Pers. Commun. 28(3), 233–238 (2004).
[Crossref]

Shapiro, J. H.

R. S. Kennedy and J. H. Shapiro, “Multipath dispersion in low visibility optical communication channels,” Res. Lab. Elect. Report RADC-TR-77-73 (MIT, 1977).
[Crossref]

Shimizu, R.

Sjoberg, K.

T. Abbas, K. Sjoberg, J. Karedal, and F. Tufvesson, “A Measurement Based Shadow Fading Model for Vehicle-to-Vehicle Network Simulations,” Int. J. Antennas Propag. 2015, 190607 (2015).
[Crossref]

Spasic, M.

Takayama, Y.

Takenaka, H.

Tatarczak, A.

Thanh, V. P.

V. P. Thanh, T. Cong-Thang, and T. P. Anh, “On the MGF-Based Approximation of the Sum of Independent Gamma-Gamma Random Variables,” in Proceedings of the IEEE 81st Vehicular Technology Conference, (Glasgow, 2015).

Tombras, G. S.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Tough, R. J.

Toyoshima, M.

Tsiftsis, T. A.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Tsukamoto, K.

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
[Crossref]

Tufvesson, F.

T. Abbas, K. Sjoberg, J. Karedal, and F. Tufvesson, “A Measurement Based Shadow Fading Model for Vehicle-to-Vehicle Network Simulations,” Int. J. Antennas Propag. 2015, 190607 (2015).
[Crossref]

Vallone, G.

Vegas Olmos, J. J.

Villoresi, P.

Wakamori, K.

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Z. Kolka, O. Wilfert, D. Biolek, and V. Biolkova, “Availability model of free-space optical data link,” Int. J. Microw. Opt. Technol. 1(2), 612–616 (2006).

Willebrand, H.

Yilmaz, F.

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (ℳ) Turbulence Channels With Pointing Errors,” IEEE Trans. Wirel. Comm. 15(1), 91–102 (2016).
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IEEE Comm. Magazine (1)

K. Kazaura, K. Wakamori, M. Matsumoto, T. Higashino, K. Tsukamoto, and S. Komaki, “RoFSO: a universal platform for convergence of fiber and free-space optical communication networks,” IEEE Comm. Magazine 48(2), 130–137 (2010).
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P. S. Bithas, N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, “On the performance analysis of digital communications over generalized-K fading channels,” IEEE Commun. Lett. 10(5), 353–355 (2006).
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IEEE Photon. J. (1)

F. J. Lopez-Martinez, G. Gomez, and J. M. Garrido-Balsells, “Physical-Layer Security in Free-Space Optical Communications,” IEEE Photon. J. 7(2), 1–14 (2015).
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E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M. Matsumoto, “1.28-Tb/s (32×40 Gb/s) free-space optical WDM transmission system,” IEEE Photon. Technol. Lett. 21(16), 1121–1123 (2009).
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X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
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IEEE Trans. Wirel. Comm. (1)

I. S. Ansari, F. Yilmaz, and M. S. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (ℳ) Turbulence Channels With Pointing Errors,” IEEE Trans. Wirel. Comm. 15(1), 91–102 (2016).
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H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
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T. Abbas, K. Sjoberg, J. Karedal, and F. Tufvesson, “A Measurement Based Shadow Fading Model for Vehicle-to-Vehicle Network Simulations,” Int. J. Antennas Propag. 2015, 190607 (2015).
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A. Jurado-Navas, A. Tatarczak, X. Lu, J. J. Vegas Olmos, J. M. Garrido-Balsells, and I. T. Monroy, “850-nm hybrid fiber/free-space optical communications using orbital angular momentum modes,” Opt. Express 23(26), 33721–33732 (2015).
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J. M. Garrido-Balsells, A. Jurado-Navas, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “Novel formulation of the ℳ-model through the Generalized-K distribution for atmospheric optical channels,” Opt. Express 23(5), 6345–6358 (2015).
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Figures (6)

Fig. 1
Fig. 1 -model laser beam propagation scheme. The three components received are: the line-of-sight (LOS) term, UL, the coupled-to-LOS scattering term, U S C and the classic scattering term related to the off-axis eddies, U S G.
Fig. 2
Fig. 2 (a) Propagation model of a partially coherent Gaussian laser beam: W0 is the initial beam radius, Db = 2We is the beam diameter and Dc = 2ρ0 is the transverse coherence diameter, both at z = L. (b) Laser beam radius, We, and transverse coherence radius, ρ0, as a function of the propagation length, L, for moderate and strong turbulence conditions.
Fig. 3
Fig. 3 (a) PDF of the distribution under LOS blockage for different values of ρ assuming Pb = 0 and Pb = 1. (b) PDF of the distribution under LOS blockage for different values of blockage probability, Pb, assuming a very high coupling factor ρ = 0.99.
Fig. 4
Fig. 4 Outage probability of a FSO link over -turbulence channels and LOS blockage with different values of ρ and blockage probabilities Pb = 0 and Pb = 1.
Fig. 5
Fig. 5 (a) Power boost needed to achieve a Pout = 10−3 under LOS blockage as a function of the coupling factor, ρ, and the blockage probability Pb. (b) Outage probability of a FSO link over -turbulence channels with ρ = 0.99 and several values of the LOS blockage probabilities.
Fig. 6
Fig. 6 Impact of the coupling factor, ρ, on the outage probability for several normalized SNR, γn, and blockage probabilities, Pb.

Equations (43)

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y ( t ) = R I r x ( t ) + n ( t )
γ = ( R I 0 I ) 2 σ n 2 = γ 0 I 2
I = Y X = | U L + U S C + U S G | 2 exp ( 2 χ ) ,
f I ( I ) = k = 1 k ˜ m ˜ k K G ( I ; α , k , μ ˜ k ) ,
m ˜ k = { ( β 1 k 1 ) p k 1 ( 1 p ) β k β Γ ( k 1 + β ) Γ ( k ) Γ ( β ) p k 1 ( 1 p ) β β ,
p = [ 1 + ( 1 β Ω ξ g ) 1 ] 1 .
μ ˜ k = { k β ( ξ g β + Ω ) β k ξ g β ,
K G ( I ; α , k , k ) = 2 B ( α + k ) / 2 Γ ( α ) Γ ( k ) I ( α + k ) / 2 1 K α k ( 2 B I )
W ( z ) = W 0 ( 1 z F 0 ) 2 + ( 2 z k W 0 2 ) 2 ,
W e ( z ) = W ( z ) 1 + 1.625 σ R 12 / 5 Λ ,
ρ 0 ( z ) = ( 1.46 C n 2 k 2 z ) 3 / 5
I = | b ( U L + U S C ) + U S G | 2 exp ( 2 χ ) .
p = [ 1 + ( 1 β Ω ξ g ) 1 ] 1 = Ω Ω + β ξ g = 0 .
m ˜ k = { 1 k = 1 0 k 1 ,
f I , b ( I ) = P b f I ( I | b = 0 ) + ( 1 P b ) f I ( I | b = 1 ) ,
f I , b ( I ) = P b K G ( I ; α , 1 , ξ g ) + ( 1 P b ) k = 1 k ˜ m ˜ k K G ( I ; α , k , μ ˜ k ) .
M I ( s ) 0 exp ( s I ) f I ( I ) d I ,
M I ( s ) = k = 1 k ˜ m ˜ k M I , Kg ( k ) ( s ) ,
M I , Kg ( k ) ( s ) = ( α k μ k ˜ s ) α + k 1 2 exp ( α k 2 μ k ˜ s ) W α + k 1 2 , α k 2 ( α k μ k ˜ s ) ,
W v , μ ( z ) = z μ + 1 2 exp ( z 2 ) U ( μ v + 1 2 , 2 μ + 1 , z ) .
M I , Kg ( k ) ( s ) = ( α k μ k ˜ s ) α U ( α , α k + 1 , α k μ k ˜ s ) .
M I , Kg ( k ) ( s ) = Γ ( k α ) Γ ( k ) ( α k μ k ˜ s ) α F 1 1 ( α , α k + 1 , α k μ k ˜ s ) + + Γ ( α k ) Γ ( α ) ( α k μ k ˜ s ) k F 1 1 ( k , k α + 1 , α k μ k ˜ s ) .
M I , b ( s ) = P b M I ( s | b = 0 ) + ( 1 P b ) M I ( s | b = 1 ) ,
M I , b ( s | b = 1 ) = k = 1 k ˜ m ˜ k M I , Kg ( k ) ( s ) .
P out = Pr { I < γ t h γ 0 } = Pr { I < 1 γ n } ,
P out = 0 1 γ n f I , b ( I ) d I = F I , b ( 1 γ n ) ,
F I , b ( I ) = P b F I , Kg ( I ; α , 1 , ξ g ) + ( 1 P b ) k = 1 k ˜ m ˜ k F I , Kg ( I ; α , k , μ ˜ k ) ,
F I , Kg ( I ; α , k , k ) = 1 Γ ( α ) Γ ( k ) ( B I ) α + k 2 G 1 , 3 2 , 1 ( B I | 1 α + k 2 α k 2 , α k 2 , α + k 2 ) ,
P out = P b F I , Kg ( γ n 1 / 2 ; α , 1 , ξ g ) + ( 1 P b ) k = 1 k ˜ m ˜ k F I , Kg ( γ n 1 / 2 ; α , k , μ ˜ k ) .
P out = P b P out , Kg ( 1 ) + ( 1 P b ) k = 1 k ˜ m ˜ k P out , Kg ( k ) .
P out a D M ( γ th γ 0 ) D M / 2 = a D M γ n D M / 2 ,
a = f I ( t ) ( 0 ) t ! ,
lim s s D M M I , b ( s ) = b M = a Γ ( D M ) .
lim s s D k M I , Kg ( k ) ( s ) = Γ ( k α ) Γ ( k ) ( α k μ k ˜ s ) α lim s s D k s α + + Γ ( α k ) Γ ( α ) ( α k μ k ˜ s ) k lim s s D k s k ,
D k = min { α , k } .
b k = { Γ ( k α ) Γ ( k ) α k μ k ˜ for D k = α if α < k Γ ( α k ) Γ ( α ) α k μ k ˜ for D k = k if α > k .
b M = k = 1 k ˜ m ˜ k [ lim s s D M M I , Kg ( k ) ( s ) ] .
b M = m ˜ 1 b 1 = m ˜ 1 Γ ( α 1 ) Γ ( α ) α μ ˜ 1 = m ˜ 1 μ ˜ 1 α α 1 ,
P out b M γ n 1 / 2 .
P out α α 1 [ P b 1 ξ g + ( 1 P b ) m ˜ 1 μ ˜ 1 ] γ n 1 / 2
Δ ( dB ) 10 log 10 [ γ n ( P b ) γ n ( P b = 0 ) ] 20 log 10 [ 1 + P b ( m ˜ 1 ξ g μ ˜ 1 1 ) ] .
f I , b ( I ) = P b δ ( I ) + ( 1 P b ) f I , G G ( I ; α , β )
Δ max ( dB ) = 10 log 10 [ γ n ( P b = 1 ) γ n ( P b = 0 ) ] 20 log 10 [ μ ˜ 1 ξ g m ˜ 1 ] .

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