Abstract

Wireless optical communication systems with coherent detection are analyzed for Gamma–Gamma distributed turbulence channels. In addition to the shot noise, we consider the impacts of both turbulence amplitude fluctuations and phase fluctuations on the error performance. Error rate analyses of predetection and postdetection equal gain combining (EGC) are carried out. We derive the exact error rate expressions for predetection and postdetection EGC using a characteristic function method. In the case of predetection EGC, we also study the impact of phase noise compensation error on the error rate performance. It is shown that the error rate performance of predetection EGC is sensitive to phase noise compensation errors for both weak and strong turbulence conditions. In order to alleviate the impact of phase noise, postdetection EGC with differential phase-shift keying is introduced and analyzed. In addition, postdetection EGC is compared with predetection EGC in the presence of phase noise compensation errors, and it is found to be an effective alternative to predetection EGC with low complexity implementation.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. W. S. Chan, "Free-space optical communications," J. Lightwave Technol. 24, 4750‒4762 (2006).
    [CrossRef]
  2. S. Hranilovic, Wireless Optical Communication Systems, Springer, New York, 2004.
  3. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications, SPIE Press, Bellingham, WA, 2001.
  4. H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, Indianapolis, IN, 2002.
  5. 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, 1896‒1906 (2004).
    [CrossRef]
  6. S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
    [CrossRef]
  7. 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, 951‒957 (2009).
    [CrossRef]
  8. J. H. Shapiro and R. C. Harney, "Burst-mode atmospheric optical communication," Nat. Telecommunication Conf. (NTC’80), 1980, Houston, TX, pp. 27.5.1‒27.5.7.
  9. J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
    [CrossRef]
  10. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
    [CrossRef]
  11. S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.
  12. K. Kiasaleh, "Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence," IEEE Trans. Commun. 54, 604‒607 (2006).
    [CrossRef]
  13. E. J. Lee and V. W. S. Chan, "Diversity coherent and incoherent receivers for free-space optical communication in the presence and absence of interference," J. Opt. Commun. Netw. 1, 463‒483 (2009).
    [CrossRef]
  14. 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, 4440‒4445 (2009).
    [CrossRef]
  15. A. Belmonte and J. M. Kahn, "Performance of synchronous optical receivers using atmospheric compensation techniques," Opt. Express 16, 14151‒14162 (2008).
    [CrossRef] [PubMed]
  16. A. Belmonte and J. M. Kahn, "Capacity of coherent free-space optical links using atmospheric compensation techniques," Opt. Express 17, 2763‒2773 (2009).
    [CrossRef] [PubMed]
  17. A. Belmonte and J. M. Kahn, "Capacity of coherent free-space optical links using diversity combining techniques," Opt. Express 17, 12601‒12611 (2009).
    [CrossRef] [PubMed]
  18. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 1998.
  19. X. Zhu and J. M. Kahn, "Free-space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 1293‒1300 (2002).
    [CrossRef]
  20. X. Zhu and J. M. Kahn, "Performance bounds for coded free-space optical communications through atmospheric turbulence channels," IEEE Trans. Commun. 51, 1233‒1239 (2003).
    [CrossRef]
  21. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
    [CrossRef]
  22. M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
    [CrossRef]
  23. R. L. Phillips and L. C. Andrews, "Measured statistics for laser light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440‒1445 (1981).
    [CrossRef]
  24. 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, 1554‒1562 (2001).
    [CrossRef]
  25. 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, 1229‒1233 (2006).
    [CrossRef]
  26. T. A. Tsiftsis, "Performance of heterodyne wireless optical communication systems over Gamma–Gamma atmospheric turbulence channels," Electron. Lett. 44, 373‒375 (2008).
    [CrossRef]
  27. E. Bayaki, R. Schober, and R. K. Mallik, "Performance analysis of MIMO free-space optical systems in Gamma–Gamma fading," IEEE Trans. Commun. 57, 3415‒3424 (2009).
    [CrossRef]
  28. M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.
  29. M. Niu, J. Cheng, and J. F. Holzman, "Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels," Opt. Express 18, 13915‒13926 (2010).
    [CrossRef] [PubMed]
  30. J. A. Anguita and J. E. Cisternas, "Influence of turbulence strength on temporal correlation of scintillation," Opt. Lett. 36, 1725‒1727 (2011).
    [CrossRef] [PubMed]
  31. J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
    [CrossRef]
  32. G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed., Wiley, New York, 2002.
  33. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
    [CrossRef]
  34. J. W. Goodman, Statistical Optics, 1st ed., Wiley-Interscience, 1985.
  35. N. Wang and J. Cheng, "Moment-based estimation for the shape parameters of the Gamma–Gamma atmospheric turbulence model," Opt. Express 18, 12824‒12831 (2010).
    [CrossRef] [PubMed]
  36. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.
  37. N. Letzepis and A. Guillén i Fàbregas, "Outage analysis in MIMO free-space optical channels with pulse-position modulation," Tech. Rep. CUED/FINFENG/TR 597, Department of Engineering, University of Cambridge, Feb. 2008.
  38. G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
    [CrossRef]
  39. M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.
  40. J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
    [CrossRef]
  41. D. Banerjee, PLL Performance, Simulation and Design Handbook, 4th ed., National Semiconductor, 2006.
  42. M. A. Najib and V. K. Prabhu, "Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol. 49, 783‒791 (2000).
    [CrossRef]
  43. J. G. Proakis, Digital Communications, 4th ed., McGraw-Hill, New York, 2000.
  44. A. J. Viterbi, Principle of Coherent Communication, McGraw-Hill, New York, 1966.
  45. A. C. Bordonalli, C. Walton, and A. J. Seeds, "High-performance phase locking of wide linewidth semiconductor lasers by combined use of optical injection locking and optical phase-lock loop," J. Lightwave Technol. 17, 328‒342 (1999).
    [CrossRef]
  46. R. Noe, "Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery," J. Lightwave Technol. 11, 1226‒1233 (2005).
  47. B. A. Khawaja and M. J. Cryan, "Wireless hybrid mode locked lasers for next generation radio-over-fiber systems," J. Lightwave Technol. 28, 2268‒2276 (2010).
    [CrossRef]
  48. J. Gil-Pelaez, "Note on the inversion theorem," Biometrika 38, 481‒482 (1951).
  49. Q. T. Zhang, "Outage probability in cellular mobile ratio due to Nakagami signal and interferers with arbitrary parameters," IEEE Trans. Veh. Technol. 45, 364‒372 (1996).
    [CrossRef]

2011 (1)

2010 (3)

2009 (6)

2008 (3)

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[CrossRef]

T. A. Tsiftsis, "Performance of heterodyne wireless optical communication systems over Gamma–Gamma atmospheric turbulence channels," Electron. Lett. 44, 373‒375 (2008).
[CrossRef]

A. Belmonte and J. M. Kahn, "Performance of synchronous optical receivers using atmospheric compensation techniques," Opt. Express 16, 14151‒14162 (2008).
[CrossRef] [PubMed]

2007 (2)

J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

2006 (4)

K. Kiasaleh, "Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence," IEEE Trans. Commun. 54, 604‒607 (2006).
[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, 1229‒1233 (2006).
[CrossRef]

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
[CrossRef]

V. W. S. Chan, "Free-space optical communications," J. Lightwave Technol. 24, 4750‒4762 (2006).
[CrossRef]

2005 (3)

R. Noe, "Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery," J. Lightwave Technol. 11, 1226‒1233 (2005).

J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
[CrossRef]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

2004 (2)

M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
[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, 1896‒1906 (2004).
[CrossRef]

2003 (2)

X. Zhu and J. M. Kahn, "Performance bounds for coded free-space optical communications through atmospheric turbulence channels," IEEE Trans. Commun. 51, 1233‒1239 (2003).
[CrossRef]

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

2002 (1)

X. Zhu and J. M. Kahn, "Free-space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 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, 1554‒1562 (2001).
[CrossRef]

2000 (1)

M. A. Najib and V. K. Prabhu, "Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol. 49, 783‒791 (2000).
[CrossRef]

1999 (2)

1996 (1)

Q. T. Zhang, "Outage probability in cellular mobile ratio due to Nakagami signal and interferers with arbitrary parameters," IEEE Trans. Veh. Technol. 45, 364‒372 (1996).
[CrossRef]

1981 (1)

1951 (1)

J. Gil-Pelaez, "Note on the inversion theorem," Biometrika 38, 481‒482 (1951).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed., Wiley, New York, 2002.

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, 1554‒1562 (2001).
[CrossRef]

Al-Habash, M. A. 

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
[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, 1554‒1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
[CrossRef]

R. L. Phillips and L. C. Andrews, "Measured statistics for laser light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440‒1445 (1981).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 1998.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications, SPIE Press, Bellingham, WA, 2001.

Anguita, J. A.

Baedke, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Banerjee, D.

D. Banerjee, PLL Performance, Simulation and Design Handbook, 4th ed., National Semiconductor, 2006.

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, 3415‒3424 (2009).
[CrossRef]

Belmonte, A.

Bordonalli, A. C.

Brandt-Pearce, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

Chan, V. W. S.

Cheng, J.

N. Wang and J. Cheng, "Moment-based estimation for the shape parameters of the Gamma–Gamma atmospheric turbulence model," Opt. Express 18, 12824‒12831 (2010).
[CrossRef] [PubMed]

M. Niu, J. Cheng, and J. F. Holzman, "Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels," Opt. Express 18, 13915‒13926 (2010).
[CrossRef] [PubMed]

M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.

M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.

Cisternas, J. E.

Cryan, M. J.

Edwards, D. J.

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[CrossRef]

Gagliardi, R.

S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.

Geng, J.

J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
[CrossRef]

Ghuman, B. S.

H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, Indianapolis, IN, 2002.

Gil-Pelaez, J.

J. Gil-Pelaez, "Note on the inversion theorem," Biometrika 38, 481‒482 (1951).

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed., Wiley-Interscience, 1985.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.

Guillén i Fàbregas, A.

N. Letzepis and A. Guillén i Fàbregas, "Outage analysis in MIMO free-space optical channels with pulse-position modulation," Tech. Rep. CUED/FINFENG/TR 597, Department of Engineering, University of Cambridge, Feb. 2008.

Harney, R. C.

J. H. Shapiro and R. C. Harney, "Burst-mode atmospheric optical communication," Nat. Telecommunication Conf. (NTC’80), 1980, Houston, TX, pp. 27.5.1‒27.5.7.

Holzman, J. F.

M. Niu, J. Cheng, and J. F. Holzman, "Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels," Opt. Express 18, 13915‒13926 (2010).
[CrossRef] [PubMed]

M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.

M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications, SPIE Press, Bellingham, WA, 2001.

Hranilovic, S.

S. Hranilovic, Wireless Optical Communication Systems, Springer, New York, 2004.

Jafar, M.

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[CrossRef]

Jiang, S.

J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
[CrossRef]

Kahn, J. M.

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. Lightwave Technol. 27, 4440‒4445 (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, 951‒957 (2009).
[CrossRef]

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
[CrossRef]

Karp, S.

S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.

Kavehrad, M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

Khawaja, B. A.

Kiasaleh, K.

K. Kiasaleh, "Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence," IEEE Trans. Commun. 54, 604‒607 (2006).
[CrossRef]

Kim, J.

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

Lee, E. J.

E. J. Lee and V. W. S. Chan, "Diversity coherent and incoherent receivers for free-space optical communication in the presence and absence of interference," J. Opt. Commun. Netw. 1, 463‒483 (2009).
[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, 1896‒1906 (2004).
[CrossRef]

Letzepis, N.

N. Letzepis and A. Guillén i Fàbregas, "Outage analysis in MIMO free-space optical channels with pulse-position modulation," Tech. Rep. CUED/FINFENG/TR 597, Department of Engineering, University of Cambridge, Feb. 2008.

Li, J.

J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
[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, 1229‒1233 (2006).
[CrossRef]

M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
[CrossRef]

Liu, J. Q.

J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
[CrossRef]

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, 3415‒3424 (2009).
[CrossRef]

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
[CrossRef]

Maneatis, J. G.

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

Maxey, J.

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

McClatchie, I.

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

McPhail, L.

M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.

Moran, S. E.

S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.

Najib, M. A.

M. A. Najib and V. K. Prabhu, "Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol. 49, 783‒791 (2000).
[CrossRef]

Navidpour, S. M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[CrossRef]

M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
[CrossRef]

Niu, M.

M. Niu, J. Cheng, and J. F. Holzman, "Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels," Opt. Express 18, 13915‒13926 (2010).
[CrossRef] [PubMed]

M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.

M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.

Noe, R.

R. Noe, "Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery," J. Lightwave Technol. 11, 1226‒1233 (2005).

O’Brien, D. C.

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[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, 1554‒1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
[CrossRef]

R. L. Phillips and L. C. Andrews, "Measured statistics for laser light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440‒1445 (1981).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications, SPIE Press, Bellingham, WA, 2001.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 1998.

Prabhu, V. K.

M. A. Najib and V. K. Prabhu, "Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol. 49, 783‒791 (2000).
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications, 4th ed., McGraw-Hill, New York, 2000.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.

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. Lightwave Technol. 27, 4440‒4445 (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, 951‒957 (2009).
[CrossRef]

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, 3415‒3424 (2009).
[CrossRef]

M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.

Seeds, A. J.

Shankaradas, M.

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro and R. C. Harney, "Burst-mode atmospheric optical communication," Nat. Telecommunication Conf. (NTC’80), 1980, Houston, TX, pp. 27.5.1‒27.5.7.

Spiegelberg, C.

J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
[CrossRef]

Stevens, C. J.

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[CrossRef]

Stotts, L. B.

S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.

Taylor, D. P.

J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
[CrossRef]

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. Lightwave Technol. 27, 4440‒4445 (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, 951‒957 (2009).
[CrossRef]

T. A. Tsiftsis, "Performance of heterodyne wireless optical communication systems over Gamma–Gamma atmospheric turbulence channels," Electron. Lett. 44, 373‒375 (2008).
[CrossRef]

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
[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, 951‒957 (2009).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[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, 1229‒1233 (2006).
[CrossRef]

M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
[CrossRef]

Viterbi, A. J.

A. J. Viterbi, Principle of Coherent Communication, McGraw-Hill, New York, 1966.

Walton, C.

Wang, N.

Willebrand, H.

H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, Indianapolis, IN, 2002.

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[CrossRef]

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, 1229‒1233 (2006).
[CrossRef]

Zhang, Q. T.

Q. T. Zhang, "Outage probability in cellular mobile ratio due to Nakagami signal and interferers with arbitrary parameters," IEEE Trans. Veh. Technol. 45, 364‒372 (1996).
[CrossRef]

Zhu, X.

X. Zhu and J. M. Kahn, "Performance bounds for coded free-space optical communications through atmospheric turbulence channels," IEEE Trans. Commun. 51, 1233‒1239 (2003).
[CrossRef]

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

Biometrika (1)

J. Gil-Pelaez, "Note on the inversion theorem," Biometrika 38, 481‒482 (1951).

Electron. Lett. (1)

T. A. Tsiftsis, "Performance of heterodyne wireless optical communication systems over Gamma–Gamma atmospheric turbulence channels," Electron. Lett. 44, 373‒375 (2008).
[CrossRef]

IEEE Commun. Lett. (1)

M. Uysal, S. M. Navidpour, and J. Li, "Error rate performance of coded free-space optical links over strong turbulence channels," IEEE Commun. Lett. 8, 635‒637 (2004).
[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, 1896‒1906 (2004).
[CrossRef]

IEEE J. Solid-State Circuits (1)

J. G. Maneatis, J. Kim, I. McClatchie, J. Maxey, and M. Shankaradas, "Self-biased high-bandwidth low-jitter 1-to-4096 multiplier clock generator PLL," IEEE J. Solid-State Circuits 38, 1795‒1803 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Geng, C. Spiegelberg, and S. Jiang, "Narrow linewidth fiber laser for 100-km optical frequency domain reflectometry," IEEE Photon. Technol. Lett. 17, 1827‒1829 (2005).
[CrossRef]

IEEE Trans. Commun. (7)

G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, "Bounds of multihop relayed communications in Nakagami-m fading," IEEE Trans. Commun. 54, 18‒22 (2006).
[CrossRef]

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

X. Zhu and J. M. Kahn, "Performance bounds for coded free-space optical communications through atmospheric turbulence channels," IEEE Trans. Commun. 51, 1233‒1239 (2003).
[CrossRef]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, "Free-space optical MIMO transmission with Q-ary PPM," IEEE Trans. Commun. 53, 1402‒1412 (2005).
[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, 3415‒3424 (2009).
[CrossRef]

J. Li, J. Q. Liu, and D. P. Taylor, "Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels," IEEE Trans. Commun. 55, 1598‒1606 (2007).
[CrossRef]

K. Kiasaleh, "Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence," IEEE Trans. Commun. 54, 604‒607 (2006).
[CrossRef]

IEEE Trans. Veh. Technol. (2)

M. A. Najib and V. K. Prabhu, "Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol. 49, 783‒791 (2000).
[CrossRef]

Q. T. Zhang, "Outage probability in cellular mobile ratio due to Nakagami signal and interferers with arbitrary parameters," IEEE Trans. Veh. Technol. 45, 364‒372 (1996).
[CrossRef]

IEEE Trans. Wireless Commun. (3)

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, 1229‒1233 (2006).
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6, 2813‒2819 (2007).
[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, 951‒957 (2009).
[CrossRef]

IET Commun. (1)

M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, "Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system employing coherent detection," IET Commun. 2, 18‒26 (2008).
[CrossRef]

J. Lightwave Technol. (5)

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. (2)

R. L. Phillips and L. C. Andrews, "Measured statistics for laser light scattering in atmospheric turbulence," J. Opt. Soc. Am. 71, 1440‒1445 (1981).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A.  Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417‒1429 (1999).
[CrossRef]

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, 1554‒1562 (2001).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Other (15)

M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, "Performance analysis of coherent free space optical communication systems with K-distributed turbulence," IEEE Int. Conf. on Communications (ICC’09), June 2009, Dresden, Germany, pp. 1‒5.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.

N. Letzepis and A. Guillén i Fàbregas, "Outage analysis in MIMO free-space optical channels with pulse-position modulation," Tech. Rep. CUED/FINFENG/TR 597, Department of Engineering, University of Cambridge, Feb. 2008.

M. Niu, J. Cheng, J. F. Holzman, and R. Schober, "Coherent free-space optical transmission with diversity combining for Gamma–Gamma atmospheric turbulence," 25th Biennial Symp. on Communications, May 2010, Kingston, Canada, pp. 217‒220.

J. W. Goodman, Statistical Optics, 1st ed., Wiley-Interscience, 1985.

G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed., Wiley, New York, 2002.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, SPIE Press, Bellingham, WA, 1998.

S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, Plenum, New York, 1988.

S. Hranilovic, Wireless Optical Communication Systems, Springer, New York, 2004.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications, SPIE Press, Bellingham, WA, 2001.

H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, Indianapolis, IN, 2002.

J. H. Shapiro and R. C. Harney, "Burst-mode atmospheric optical communication," Nat. Telecommunication Conf. (NTC’80), 1980, Houston, TX, pp. 27.5.1‒27.5.7.

J. G. Proakis, Digital Communications, 4th ed., McGraw-Hill, New York, 2000.

A. J. Viterbi, Principle of Coherent Communication, McGraw-Hill, New York, 1966.

D. Banerjee, PLL Performance, Simulation and Design Handbook, 4th ed., National Semiconductor, 2006.

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 (8)

Fig. 1
Fig. 1

Block diagram for coherent wireless optical receiver with predetection EGC reception.

Fig. 2
Fig. 2

Block diagram for coherent wireless optical receiver with postdetection EGC reception.

Fig. 3
Fig. 3

BER of BPSK for MRC and predetection EGC reception (assuming perfect channel state information) operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2 . 23 , β = 1 . 70 .

Fig. 4
Fig. 4

BER of BPSK for predetection EGC reception with phase noise compensation error operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2 . 23 , β = 1 . 70 .

Fig. 5
Fig. 5

BER of BPSK for predetection EGC reception with phase noise compensation error operating over L weakly turbulent Gamma–Gamma channels with channel parameters α = 6 . 52 , β = 6 . 92 .

Fig. 6
Fig. 6

BER of DPSK for postdetection EGC reception operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2 . 23 , β = 1 . 70 .

Fig. 7
Fig. 7

BER of BPSK with predetection EGC reception and DPSK with postdetection EGC reception operating over dual-branch strongly turbulent Gamma–Gamma channels with channel parameters α = 2 . 23 , β = 1 . 70 .

Fig. 8
Fig. 8

BER versus standard deviation of phase noise compensation error for BPSK with predetection EGC reception and DPSK with postdetection EGC reception operating over dual-branch strongly turbulent Gamma–Gamma channels with channel parameters α = 2 . 23 , β = 1 . 70 .

Equations (48)

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

P l ( t ) = P s , l + P L O + 2 P s , l P L O g ( t ) × cos ( ω I F t + ϕ s + ϕ n , l ) , l = 1 , 2 , , L ,
g ( t ) = 1 T , 0 t T 0 , else,
i l ( t ) = R P l ( t ) = i d c , l + i a c , l ( t ) + n l ( t ) , l = 1 , 2 , , L ,
i a c , l ( t ) = 2 R P s , l P L O g ( t ) × cos ( ω I F t + ϕ s + ϕ n , l ) , l = 1 , 2 , , L
σ 2 = 2 q R P L O Δ f ,
i a c , l ( t ) = c l g ( t ) cos ϕ s cos ( ω I F t + ϕ n , l ) sin ϕ s sin ( ω I F t + ϕ n , l ) .
y c , l ( t ) = 2 c l g ( t ) [ cos ϕ s cos ( ω I F t + ϕ n , l ) sin ϕ s sin ( ω I F t + ϕ n , l ) ] cos ( ω I F t )
y s , l ( t ) = 2 c l g ( t ) [ cos ϕ s cos ( ω I F t + ϕ n , l ) sin ϕ s sin ( ω I F t + ϕ n , l ) ] sin ( ω I F t ) .
i ̃ l ( t ) = i ̃ a c , l ( t ) + n ̃ l ( t ) = 2 R A P L O g ( t ) I s , l e j ϕ s e j ϕ n , l + n ̃ l ( t ) , l = 1 , 2 , , L ,
i ̃ l = 0 T 2 A R I s , l P L O g ( t ) e j ϕ n , l e j ϕ s g ( t ) d t + 0 T n ̃ l ( t ) g ( t ) d t = 2 A R I s , l P L O e j ϕ n , l e j ϕ s + n ̃ l , l = 1 , 2 , , L ,
f I s ( I s , l ) = 2 Γ ( α ) Γ ( β ) ( α β ) α + β 2 I s , l α + β 2 1 × K α β 2 α β I s , l , I s , l > 0 ,
σ s i 2 E [ I s , l 2 ] ( E [ I s , l ] ) 2 1 = 1 α + 1 β + 1 α β .
σ R 2 = 1 . 23 C n 2 k 7 6 L p 11 6 ,
M I s ( s ) = 0 e s I s f I s ( I s ) d I s = ( α β ) α + β 1 2 e α β 2 s ( s ) α + β 1 2 Γ ( β α ) Γ ( β ) M k 1 , k 2 α β s + Γ ( α β ) Γ ( α ) M k 1 , k 2 α β s ,
M k 1 , k 2 α β s = α β s α β + 1 2 e α β 2 s × 1 F 1 α , α β + 1 , α β s
M k 1 , k 2 α β s = α β s β α + 1 2 e α β 2 s × 1 F 1 β , β α + 1 , α β s .
f z ( z ) = 4 Γ ( α ) Γ ( β ) ( α β ) α + β 2 z α + β 1 × K α β 2 α β z , z > 0 .
f I x ( I x ) = α ( α I x ) α 1 e α I x Γ ( α ) , I x > 0
f I y ( I y ) = β ( β I y ) β 1 e β I y Γ ( β ) , I y > 0 .
f x ( x ) = 2 α α x 2 α 1 e α x 2 Γ ( α ) , x > 0
f y ( y ) = 2 β β y 2 β 1 e β y 2 Γ ( β ) , y > 0 ,
Φ z ( ω ) = 0 e j ω z f z ( z ) d z = E [ e j ω z ] , ω R .
ϕ z x ( ω ) = E z x [ e j ω z ] = 2 1 β Γ ( 2 β ) Γ ( β ) exp ω 2 x 2 8 β D 2 β j ω x 2 β ,
ϕ z x ( ω ) = 1 F 1 β , 1 2 , ω 2 x 2 4 β + j Γ β + 1 2 Γ ( β ) ω x β × 1 F 1 β + 1 2 , 3 2 , ω 2 x 2 β .
R { Φ z ( ω ) } = 2 F 1 β , α , 1 2 , ω 2 4 α β
I { Φ z ( ω ) } = Γ α + 1 2 Γ β + 1 2 ω Γ ( α ) Γ ( β ) α β 1 2 × 2 F 1 β + 1 2 , α + 1 2 , 3 2 , ω 2 4 α β .
i ̃ = l = 1 L e j ϕ ˆ n , l 2 R P s , l P L O e j ϕ n , l e j ϕ s + l = 1 L e j ϕ ˆ n , l n ̃ l = l = 1 L e j Δ ϕ l 2 R I s , l A P L O e j ϕ s + ν ,
f Δ ϕ ( Δ ϕ l ) = exp cos ( Δ ϕ l ) σ Δ ϕ 2 2 π I 0 1 σ Δ ϕ 2 , | Δ ϕ | π ,
D = l = 1 L 2 R A P L O cos ϕ s I s , l cos Δ ϕ l + R { ν } = cos ϕ s l = 1 L S l + ν R ,
γ ̃ E G C = 2 R 2 A P L O L σ 2 l = 1 L I s , l cos Δ ϕ l 2 = γ ¯ L l = 1 L I s , l cos Δ ϕ l 2 ,
γ E G C = γ ¯ l = 1 L I s , l 2 L .
F D ( ξ | ϕ s = 0 ) = Pr { D < ξ | ϕ s = 0 } .
Φ D ( ω | ϕ s = 0 ) = Φ ν R ( ω ) l = 1 L Φ S l ( ω ) = Φ ν R ( ω ) [ Φ S 1 ( ω ) ] L
Φ S 1 | Δ ϕ 1 ( ω ) = Φ z ω 2 A P L O R cos Δ ϕ 1 .
Φ D ( ω | ϕ s = 0 ) = [ Φ S 1 ( ω ) ] L Φ ν R ( ω ) = E Δ ϕ 1 Φ z ω 2 A P L O R cos Δ ϕ 1 L × Φ ν R ( ω ) .
F D ( ξ | ϕ s = 0 ) = 1 2 1 π 0 I { Φ D ( ω | ϕ s = 0 ) e j ω ξ } ω d ω .
P e = 1 2 1 π 0 I { Φ D ( ω | ϕ s = 0 ) } ω d ω .
P e = 1 2 π π 2 π 2 R sec 2 θ Φ z * γ ¯ tan θ 2 L L × Ϝ ( tan θ ) 2 d θ ,
Ϝ ( ω ) = 1 π 1 F 1 1 , 3 2 , ω 2 4 + j ω 1 e ω 2 4
i ̃ k , l ( t ) = i ̃ a c , k , l ( t ) + n ̃ k , l ( t ) = 2 R A P L O g ( t ) I s , l e j ϕ s , k e j ϕ n , l + n ̃ k , l ( t ) ,
i ̃ k 1 , l ( t ) = i ̃ a c , k 1 , l ( t ) + n ̃ k 1 , l ( t ) = 2 R A P L O g ( t ) × I s , l e j ϕ s , k 1 e j ϕ n , l + n ̃ k 1 , l ( t ) .
V ̃ k , l = R A q R Δ f I s , l e j ϕ s , k e j ϕ n , l + μ ̃ k = γ ̄ I s , l e j ϕ s , k e j ϕ n , l + μ ̃ k
V ̃ k 1 , l = R A q R Δ f I s , l e j ϕ s , k 1 e j ϕ n , l + μ ̃ k 1 = γ ̄ I s , l e j ϕ s , k 1 e j ϕ n , l + μ ̃ k 1 ,
D ̃ = l = 1 L U ̃ l = l = 1 L R { V ̃ k 1 , l * V ̃ k , l } ,
Φ U ̃ | I s , l ( ω | Δ ϕ s , k = 0 ) = 1 ω 2 + 1 exp γ ̄ ω 2 j ω ω 2 + 1 I s , l .
Φ U ̃ ( ω | Δ ϕ s , k = 0 ) = 1 ω 2 + 1 M I s γ ̄ ω 2 j ω ω 2 + 1 ,
Φ D ̃ ( ω | Δ ϕ s , k = 0 ) = Φ U ̃ L ( ω | Δ ϕ s , k = 0 ) = 1 ( ω 2 + 1 ) L M I s γ ̄ ω 2 j ω ω 2 + 1 L .
P e = Pr { D ̃ < 0 | Δ ϕ s , k = 0 } = 1 2 1 π 0 I M I s γ ̄ ω 2 j ω ω 2 + 1 L ω ( ω 2 + 1 ) L d ω ,