Abstract

This paper proposes and analyzes the performance of the multihop free-space optical (FSO) communication links using a heterodyne differential phase-shift keying modulation scheme operating over a turbulence induced fading channel. A novel statistical fading channel model for multihop FSO systems using channel-state-information-assisted and fixed-gain relays is developed incorporating the atmospheric turbulence, pointing errors, and path-loss effects. The closed-form expressions for the moment generating function, probability density function, and cumulative distribution function of the multihop FSO channel are derived using Meijer's G-function. They are then used to derive the fundamental limits of the outage probability and average symbol error rate. Results confirm the performance loss as a function of the number of hops. Effects of the turbulence strength varying from weak-to-moderate and moderate-to-strong turbulence, geometric loss, and pointing errors are studied. The pointing errors can be mitigated by widening the beam at the expense of the received power level, whereas narrowing the beam can reduce the geometric loss at the cost of increased misalignment effects.

© 2014 IEEE

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  1. X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.
  2. X. Tang, Z. Xu, and Z. Ghassemlooy, “Outage probability of multihop free space optical communications over Nakagami fading channels,” in Proc. 8th Conf. Netw. Opt. Commun., Graz, Austria, Jul. 2013, pp. 199–202.
  3. M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.
  4. F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.
  5. L. Andrews, “Free-space laser propagation: atmospheric effects,” in Proc. Dig. LEOS Summer Top. Meet., 2005, pp. 3–4.
  6. W. Gappmair, “Novel results on pulse-position modulation performance for terrestrial free-space optical links impaired by turbulent atmosphere and pointing errors,” IET Commun., vol. 6, pp. 1300–1305, 2012.
  7. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.
  8. C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.
  9. G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.
  10. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, 2008.
  11. M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.
  12. E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.
  13. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightw. Technol., vol. 25, no. 7, pp. 1702–1710, 2007.
  14. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA, USA: SPIE Press, 2005.
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York, NY, USA: Academic Press, Inc., 2000.
  16. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.
  17. C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.
  18. V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. Symbolic Algebraic Comput., Minsk, Belarus1990, pp. 212–224.
  19. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Inverse Laplace Transforms, 5th ed. London, U.K.: Gordon and Breach, 1992, vol. 5.
  20. M. O. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, 2003.
  21. J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” in Proc. IEEE Wireless Commun. Netw. Conf., Chicago, IL, USA, Sep. 2000, pp. 7–12.
  22. G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

2014 (1)

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

2013 (3)

F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.

M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.

2012 (2)

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.

W. Gappmair, “Novel results on pulse-position modulation performance for terrestrial free-space optical links impaired by turbulent atmosphere and pointing errors,” IET Commun., vol. 6, pp. 1300–1305, 2012.

2010 (2)

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

2009 (1)

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.

2008 (2)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, 2008.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

2007 (2)

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightw. Technol., vol. 25, no. 7, pp. 1702–1710, 2007.

G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.

2006 (1)

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

2003 (1)

M. O. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, 2003.

Adamchik, V. S.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. Symbolic Algebraic Comput., Minsk, Belarus1990, pp. 212–224.

Alouini, M.-S.

F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.

M. O. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, 2003.

Andrews, L.

L. Andrews, “Free-space laser propagation: atmospheric effects,” in Proc. Dig. LEOS Summer Top. Meet., 2005, pp. 3–4.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA, USA: SPIE Press, 2005.

Bayaki, E.

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.

Benkhelifa, F.

F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Inverse Laplace Transforms, 5th ed. London, U.K.: Gordon and Breach, 1992, vol. 5.

Datsikas, C. K.

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

Farid, A. A.

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightw. Technol., vol. 25, no. 7, pp. 1702–1710, 2007.

Gappmair, W.

W. Gappmair, “Novel results on pulse-position modulation performance for terrestrial free-space optical links impaired by turbulent atmosphere and pointing errors,” IET Commun., vol. 6, pp. 1300–1305, 2012.

Ghassemlooy, Z.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Outage probability of multihop free space optical communications over Nakagami fading channels,” in Proc. 8th Conf. Netw. Opt. Commun., Graz, Austria, Jul. 2013, pp. 199–202.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York, NY, USA: Academic Press, Inc., 2000.

Hasna, M. O.

M. O. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, 2003.

Hranilovic, S.

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightw. Technol., vol. 25, no. 7, pp. 1702–1710, 2007.

Hulea, M.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

Karagiannidis, G. K.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

Kashani, M. A.

M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.

Laneman, J. N.

J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” in Proc. IEEE Wireless Commun. Netw. Conf., Chicago, IL, USA, Sep. 2000, pp. 7–12.

Liu, C.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

Mallik, R. K.

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

Marichev, O. I.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. Symbolic Algebraic Comput., Minsk, Belarus1990, pp. 212–224.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Inverse Laplace Transforms, 5th ed. London, U.K.: Gordon and Breach, 1992, vol. 5.

Mathiopoulos, T. T.

G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.

Michalopoulos, D. S.

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.

Peppas, K. P.

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA, USA: SPIE Press, 2005.

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Inverse Laplace Transforms, 5th ed. London, U.K.: Gordon and Breach, 1992, vol. 5.

Rajbhandari, S.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

Rezki, Z.

F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York, NY, USA: Academic Press, Inc., 2000.

Safari, M.

M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, 2008.

Sagias, N. C.

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.

Sandalidis, H. G.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

Schober, R.

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.

Sun, Y.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

Tang, X.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Outage probability of multihop free space optical communications over Nakagami fading channels,” in Proc. 8th Conf. Netw. Opt. Commun., Graz, Austria, Jul. 2013, pp. 199–202.

Tombras, G. S.

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

Tsiftsis, T. A.

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

Uysal, M.

M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, 2008.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

Wornell, G. W.

J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” in Proc. IEEE Wireless Commun. Netw. Conf., Chicago, IL, USA, Sep. 2000, pp. 7–12.

Xu, Z.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Outage probability of multihop free space optical communications over Nakagami fading channels,” in Proc. 8th Conf. Netw. Opt. Commun., Graz, Austria, Jul. 2013, pp. 199–202.

Yao, Y.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

Zhao, X.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

Electron. Lett. (1)

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett., vol. 46, pp. 851–853, 2010.

IEEE Commun. Lett. (3)

F. Benkhelifa, Z. Rezki, and M.-S. Alouini, “Low SNR capacity of FSO links over gamma-gamma atmospheric turbulence channels,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1264–1267, 2013.

M. O. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over Nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, 2003.

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, 2008.

IEEE Trans. Commun. (3)

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds for multihop relayed communications in Nakagami-m fading,” IEEE Trans. Commun., vol. 54, no. 1, pp. 18–22, 2006.

G. K. Karagiannidis, N. C. Sagias, and T. T. Mathiopoulos, “N*Nakagami: A novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453–1458, 2007.

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3797–3807, 2012.

IEEE Trans. Wireless Commun. (1)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, 2008.

IEEE/Opt. Soc. Am. J. Opt. Commun. Netw. (2)

M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 5, no. 1, pp. 37–47, 2013.

C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” IEEE/Opt. Soc. Am. J. Opt. Commun. Netw., vol. 2, no. 8, pp. 576–586, 2010.

IET Commun. (1)

W. Gappmair, “Novel results on pulse-position modulation performance for terrestrial free-space optical links impaired by turbulent atmosphere and pointing errors,” IET Commun., vol. 6, pp. 1300–1305, 2012.

J. Lightw. Technol. (4)

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightw. Technol., vol. 27, no. 20, pp. 4440–4445, 2009.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightw. Technol., vol. 32, no. 7, pp. 1323–1328, 2014.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightw. Technol., vol. 31, no. 20, pp. 3221–3228, 2013.

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free space optical links with pointing errors,” J. Lightw. Technol., vol. 25, no. 7, pp. 1702–1710, 2007.

Other (7)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA, USA: SPIE Press, 2005.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. New York, NY, USA: Academic Press, Inc., 2000.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. Symbolic Algebraic Comput., Minsk, Belarus1990, pp. 212–224.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Inverse Laplace Transforms, 5th ed. London, U.K.: Gordon and Breach, 1992, vol. 5.

X. Tang, Z. Xu, and Z. Ghassemlooy, “Outage probability of multihop free space optical communications over Nakagami fading channels,” in Proc. 8th Conf. Netw. Opt. Commun., Graz, Austria, Jul. 2013, pp. 199–202.

L. Andrews, “Free-space laser propagation: atmospheric effects,” in Proc. Dig. LEOS Summer Top. Meet., 2005, pp. 3–4.

J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” in Proc. IEEE Wireless Commun. Netw. Conf., Chicago, IL, USA, Sep. 2000, pp. 7–12.

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