Abstract

An unsuitable alignment between transmitter and receiver together with fluctuations in the irradiance of the transmitted optical beam due to the atmospheric turbulence can severely degrade the performance of free-space optical (FSO) systems. In this paper, cooperative FSO communications with decode-and-forward (DF) relaying and equal gain combining (EGC) reception over atmospheric turbulence and misalignment fading channels is analyzed in order to mitigate these impairments. Novel closed-form asymptotic bit error-rate (BER) expressions are derived for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2β1/2 times the value of the pointing error displacement standard deviation at the receiver. It is contrasted that the available diversity order is strongly dependent on the relay location, achieving greater diversity gains when the diversity order is determined by βAC + βBC, where βAC and βBC are parameters corresponding to the turbulence of the source-destination and relay-destination links. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
    [CrossRef]
  2. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
    [CrossRef]
  3. W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009).
    [CrossRef] [PubMed]
  4. L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001).
    [CrossRef]
  5. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
    [CrossRef]
  6. 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]
  7. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol. 26(5), 478–487 (2008).
    [CrossRef]
  8. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
    [CrossRef]
  9. 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]
  10. E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
    [CrossRef]
  11. 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]
  12. A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010).
    [CrossRef]
  13. S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003).
    [CrossRef] [PubMed]
  14. 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]
  15. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
    [CrossRef]
  16. 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]
  17. 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]
  18. 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),480–496 (2011).
    [CrossRef]
  19. 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]
  20. A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
    [CrossRef]
  21. A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
    [CrossRef]
  22. J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
    [CrossRef]
  23. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
    [CrossRef]
  24. M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009).
    [CrossRef]
  25. M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010).
    [CrossRef]
  26. C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011).
    [CrossRef]
  27. C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol. 29(7), 1058 –1065 (2011).
    [CrossRef]
  28. M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012).
    [CrossRef]
  29. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
    [CrossRef]
  30. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (2009).
    [CrossRef]
  31. 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]
  32. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, 7th ed. (Academic Press Inc., 2007).
  33. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010).
    [CrossRef]
  34. 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]
  35. Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com .
  36. 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. on Symbolic and Algebraic Computation, 212–224 (Tokyo, Japan, 1990).

2012 (2)

2011 (4)

C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol. 29(7), 1058 –1065 (2011).
[CrossRef]

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011).
[CrossRef]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[CrossRef]

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),480–496 (2011).
[CrossRef]

2010 (7)

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010).
[CrossRef]

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. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010).
[CrossRef]

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[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]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
[CrossRef]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010).
[CrossRef]

2009 (7)

W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009).
[CrossRef] [PubMed]

D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (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]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (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]

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009).
[CrossRef]

2008 (2)

2007 (1)

2006 (1)

2004 (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]

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
[CrossRef]

2003 (4)

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
[CrossRef]

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. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003).
[CrossRef] [PubMed]

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 (2001).
[CrossRef]

Aazhang, B.

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
[CrossRef]

Abou-Rjeily, C.

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011).
[CrossRef]

C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol. 29(7), 1058 –1065 (2011).
[CrossRef]

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. on Symbolic and Algebraic Computation, 212–224 (Tokyo, Japan, 1990).

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]

Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001).
[CrossRef]

Andrews, L. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

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.

Arnon, S.

Bayaki, E.

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[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]

Bhatnagar, M.

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012).
[CrossRef]

Borah, D. K.

Castillo-Vázquez, B.

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]

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),480–496 (2011).
[CrossRef]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
[CrossRef]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010).
[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]

Castillo-Vázquez, C.

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]

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),480–496 (2011).
[CrossRef]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
[CrossRef]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010).
[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]

Chan, V. W. S.

V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
[CrossRef]

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[CrossRef]

Cheng, J.

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010).
[CrossRef]

Cherry, P. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Denic, S.

Djordjevic, I. B.

Erkip, E.

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
[CrossRef]

Farid, A. A.

Foshee, J. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (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]

García-Zambrana, A.

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]

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),480–496 (2011).
[CrossRef]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
[CrossRef]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010).
[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]

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]

Gradshteyn, I. S.

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

Haddad, S.

Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 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]

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.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (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]

Karimi, M.

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010).
[CrossRef]

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009).
[CrossRef]

Kim, K.

Kolodzy, P. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Laneman, J.

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
[CrossRef]

Lee, E. J.

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

Lim, W.

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]

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. on Symbolic and Algebraic Computation, 212–224 (Tokyo, Japan, 1990).

McIntire, W. K.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Nasiri-Kenari, M.

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010).
[CrossRef]

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009).
[CrossRef]

Neifeld, M.

Northcott, M.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001).
[CrossRef]

Phillips, R. L.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

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]

Pike, H. A.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Ryzhik, I. M.

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

Safari, M.

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[CrossRef]

Sandalidis, H. G.

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[CrossRef]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (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]

Schober, R.

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[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]

Sendonaris, A.

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
[CrossRef]

Slim, A.

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011).
[CrossRef]

Stadler, B.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Stotts, L. B.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Tse, D.

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
[CrossRef]

Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (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]

Uysal, M.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[CrossRef]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[CrossRef]

Vasic, B.

Voelz, D. G.

Wang, N.

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010).
[CrossRef]

Wang, Z.

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]

Wornell, G.

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
[CrossRef]

Young, D. W.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Yun, C.

Zhu, X.

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

IEEE Commun. Lett. (1)

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]

IEEE J. Sel. Areas Commun. (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]

IEEE Photon. Technol. Lett. (1)

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012).
[CrossRef]

IEEE Trans. Commun. (8)

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]

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

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[CrossRef]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[CrossRef]

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003).
[CrossRef]

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003).
[CrossRef]

IEEE Trans. Inf. Theory (1)

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004).
[CrossRef]

IEEE Trans. Wireless Commun. (2)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[CrossRef]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[CrossRef]

IET Communications (1)

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010).
[CrossRef]

J. Light-wave Technol. (1)

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009).
[CrossRef]

J. Lightwave Technol. (6)

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 (7)

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),480–496 (2011).
[CrossRef]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (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]

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]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010).
[CrossRef]

W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009).
[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), 824–831 (2010).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[CrossRef]

Other (4)

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001).
[CrossRef]

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

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

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. on Symbolic and Algebraic Computation, 212–224 (Tokyo, Japan, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Block diagram of the considered 3-way FSO communication system, where LAC is the A–C link distance and (xB, yB) represents the location of the node B.

Fig. 2
Fig. 2

Diversity order gain Gd for a 3-way FSO communication setup with BDF relaying and EGC reception for a source-destination link distance of (a) LAC = 3 km and (b) LAC = 6 km when different relay locations of yB={0.5 km, 1 km, 1.5 km, 2 km, 2.5 km} are assumed, once the condition φ2 > β is satisfied for each link.

Fig. 3
Fig. 3

BER performance for a 3-way FSO communication setup with BDF relaying and EGC reception over atmospheric turbulence and misalignment fading channels, when different relay locations for source-destination link distances of (a) LAC = 3 km and (b) LAC = 6 km are assumed together with values of normalized beamwidth and normalized jitter of (ωz/r, σs/r) = (5, 1) and (ωz/r, σs/r) = (10, 2).

Fig. 4
Fig. 4

(a) Diversity order gain Gd for a source-destination link distance of LAC = 2 km and vertical displacement of the relay node of yB={0.2 km} when values of normalized beamwidth of ωz/r = 7 and normalized jitter of σs/r = {1, 1.5, 1.75, 2, 3} are assumed. (b) BER performance is depicted for the same source-destination link distance and a relay location of (xB=0.8 km; yB=0.2 km) when values of normalized beamwidth of ωz/r = 7 and normalized jitter of σs/r = {1, 2, 3} are assumed as well as when no pointing errors are considered.

Equations (32)

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

y m ( t ) = η i m ( t ) x ( t ) + z m ( t )
f I m ( a ) ( i ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) i ( ( α + β ) / 2 ) 1 K α β ( 2 α β i ) , 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 ) 7 / 6 ) 1 ] 1
f I m ( p ) ( i ) = φ 2 A 0 φ 2 i φ 2 1 , 0 i A 0
f I l m ( i ) = α β φ 2 A 0 Γ ( α ) Γ ( β ) G 1 , 3 3 , 0 ( α β A 0 i | φ 2 φ 2 1 , α 1 , β 1 ) , i 0
f I m ( i ) a m i b m ,
a m = { a m 0 = φ 2 ( α β ) β Γ ( α β ) A 0 β Γ ( α ) Γ ( β ) ( φ 2 β ) , φ 2 > β a m 1 = φ 2 ( α β ) φ 2 Γ ( α φ 2 ) Γ ( β φ 2 ) A 0 φ 2 Γ ( α ) Γ ( β ) , φ 2 < β
b m = { b m 0 = β 1 , φ 2 > β b m 1 = φ 2 1 , φ 2 < β
Y A B = 1 2 X I A B + Z A B , X { 0 , d } , Z A B ~ N ( 0 , N 0 / 2 )
P b A B ( E | I A B ) = Q ( ( d / 2 ) 2 i 2 / 2 N 0 )
P b A B ( E ) = 0 Q ( ( γ / 2 ) ξ i ) f I A B ( i ) d i .
P b A B ( E ) a A B 2 b A B Γ ( b A B 2 + 1 ) π ( b A B + 1 ) ( γ ξ ) 1 2 ( 1 + b A B )
Y B D F = 1 2 X I A C + Z A C + X * I B C + Z B C , X { 0 , d } , Z A C , A B C ~ N ( 0 , N 0 / 2 )
Y B D F 0 = X ( 1 2 I A C + I B C ) + Z E G C , X { 0 , d } , Z E G C ~ N ( 0 , N 0 )
P b B D F 0 ( E ) = 0 0 Q ( ( γ / 4 ) ξ ( i 1 + 2 i 2 ) ) f I A C ( i 1 ) f I B C ( i 2 ) d i 1 d i 2 .
f I T ( i ) a A C a B C 2 b B C 1 Γ ( b A C + 1 ) Γ ( b B C + 1 ) Γ ( b A C + b B C + 2 ) i b A C + b B C + 1 .
P b B D F 0 ( E ) = 0 Q ( ( γ / 4 ) ξ i ) f I T ( i ) d i .
P b B D F 0 ( E ) a A C a B C 2 1 2 ( b A C b B C 2 ) Γ ( b A C + 1 ) Γ ( b B C + 1 ) Γ ( 1 2 ( b A C + b B C + 4 ) ) ( γ ξ ) 1 2 ( b A C + b B C + 2 )
Y B D F 1 = X ( 1 2 I A C I B C ) + d I B C + Z E G C , X { 0 , d } , Z E G C ~ N ( 0 , N 0 )
P b B D F 1 ( E ) = 0 0 Q ( ( γ / 4 ) ξ ( i 1 2 i 2 ) ) f I A C ( i 1 ) f I B C ( i 2 ) d i 1 d i 2 .
P b B D F 1 ( E ) 0 0 2 i 2 f I A C ( i 1 ) f I B C ( i 2 ) d i 1 d i 2 .
P b B D F 1 ( E ) ( φ A C 2 φ B C 2 ) G 5 , 5 3 , 4 ( 2 α A C β A C A 0 B C α B C β B C A 0 A C | 1 , 1 α B C , 1 β B C , 1 φ B C 2 , φ A C 2 + 1 φ A C 2 , α A C , β A C , φ B C 2 , 0 ) Γ ( α A C ) Γ ( α B C ) Γ ( β A C ) Γ ( β B C ) .
P b B D F ( E ) = P b B D F 0 ( E ) ( 1 P b A B ( E ) ) + P b B D F 1 ( E ) P b A B ( E ) .
P b B D F ( E ) a A C a B C Γ ( b A C + 1 ) Γ ( b B C + 1 ) ( γ ξ ) 1 2 ( 2 + b A C + b B C ) 2 1 2 ( b A C + b B C ) 2 Γ ( 1 2 ( b A C + b B C + 4 ) ) , b A C + b B C + 1 < b A B
P b B D F ( E ) P b B D F 1 ( E ) a A B 2 b A B Γ ( b A B 2 + 1 ) ( γ ξ ) 1 2 ( 1 + b A B ) π ( b A B + 1 ) , b A C + b B C + 1 > b A B
G d = min ( 2 + b A C + b B C , 1 + b A B ) / ( 1 + b A C )
P b T L S ( E ) ( a A C b A C + 1 ) M Γ ( 1 2 ( b A C M + M + 1 ) ) 2 π γ 1 2 ( b A C + 1 ) M .
P b B D F ( E ) a A C n p e a B C n p e Γ ( β A C ) Γ ( β B C ) ( γ ξ ) 1 2 ( β A C + β B C ) 2 1 2 ( β A C + β B C ) 2 Γ ( 1 2 ( β A C + β B C + 2 ) ) ,
a m n p e = ( α m β m ) m β Γ ( α m β m ) Γ ( α m ) Γ ( β m )
D p e [ d B ] 20 β A C + β B C log 10 ( φ A C 2 φ B C 2 A 0 β A C + β B C ( φ A C 2 β A C ) ( φ B C 2 β B C ) ) .
D sym [ d B ] 20 β A C + β B C log 10 ( 2 β A C + β B C 1 + 1 2 ) .

Metrics