Abstract

The average symbol error rate is studied for subcarrier intensity modulated wireless optical communication systems employing general order rectangular quadrature amplitude modulation. We consider three different turbulence channel models, i.e., the Gamma–Gamma channel, the K-distributed channel, and the negative exponential channel with different levels of turbulence. Closed-form error rate expressions are derived using a series expansion of the modified Bessel function. In addition, detailed truncation error analysis and asymptotic error rate analysis are also presented. Numerical results demonstrate that our series solutions are highly accurate and efficient.

© 2012 OSA

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  1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol., vol. 24, pp. 4750–4762, Dec.2006.
    [CrossRef]
  2. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, pp. 1293–1300, Aug.2002.
    [CrossRef]
  3. W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.
  4. J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, pp. 1598–1606, Aug.2007.
    [CrossRef]
  5. W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
    [CrossRef]
  6. W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, pp. 967–973, Apr.2009.
    [CrossRef]
  7. N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
    [CrossRef]
  8. J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
    [CrossRef]
  9. H. Samimi and P. Azmi, “Subcarrier intensity modulated free-space optical communications in K-distributed turbulence channels,” J. Opt. Commun. Netw., vol. 2, pp. 625–632, Aug.2010.
    [CrossRef]
  10. X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
    [CrossRef]
  11. M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
    [CrossRef]
  12. N. Cvijetic and T. Wang, “WiMAX over free-space optics evaluating OFDM multi-subcarrier modulation in optical wireless channels,” in IEEE Sarnoff Symp., 27–28 Mar. 2006, pp. 1–4.
  13. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, pp. 6336–6350, May2007.
    [CrossRef] [PubMed]
  14. K. P. Peppas and C. K. Datsikas, “Average symbol error probability of general-order rectangular quadrature amplitude modulation of optical wireless communication systems over atmospheric turbulence channels,” J. Opt. Commun. Netw., vol. 2, pp. 102–110, Feb.2010.
    [CrossRef]
  15. G. K. Karagiannidis, “On the symbol error probability of general order rectangular QAM in Nakagami-m fading,” IEEE Commun. Lett., vol. 10, pp. 745–747, Nov.2006.
    [CrossRef]
  16. G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed.Wiley, New York, 2002.
  17. 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., vol. 40, pp. 1554–1562, Aug.2001.
    [CrossRef]
  18. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. SPIE Press, Bellingham, WA, 2001.
  19. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma–Gamma atmospheric turbulence model,” Opt. Express, vol. 18, pp. 12824–12831, June2010.
    [CrossRef] [PubMed]
  20. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol. 54, pp. 604–607, Apr.2006.
    [CrossRef]
  21. M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
    [CrossRef]
  22. H. A. Suraweera and J. Armstrong, “A simple and accurate approximation to the SEP of rectangular QAM in arbitrary Nakagami-m fading channels,” IEEE Commun. Lett., vol. 11, pp. 426–428, May2007.
    [CrossRef]
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed.Academic Press, San Diego, 2000.
  24. M. K. Simon, Digital Communication Over Fading Channel: A Unified Approach to Performance Analysis. John Wiley & Sons, Inc., Hoboken, NJ, 2004.
  25. The Wolfram Function Site, 2012 [Online]. Available: http://functions.wolfram.com/GammaBetaErf/Erfc/21/02/01/.
  26. M. Geller and E. W. Ng, “A table of integrals of the error function. II. Additions and corrections,” J. Res. Natl. Bur. Stand., vol. 75B, pp. 149–163, July–Dec.1971.

2012

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
[CrossRef]

2011

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[CrossRef]

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
[CrossRef]

M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
[CrossRef]

2010

2009

2008

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

2007

J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, pp. 1598–1606, Aug.2007.
[CrossRef]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, pp. 6336–6350, May2007.
[CrossRef] [PubMed]

H. A. Suraweera and J. Armstrong, “A simple and accurate approximation to the SEP of rectangular QAM in arbitrary Nakagami-m fading channels,” IEEE Commun. Lett., vol. 11, pp. 426–428, May2007.
[CrossRef]

2006

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol. 54, pp. 604–607, Apr.2006.
[CrossRef]

G. K. Karagiannidis, “On the symbol error probability of general order rectangular QAM in Nakagami-m fading,” IEEE Commun. Lett., vol. 10, pp. 745–747, Nov.2006.
[CrossRef]

V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol., vol. 24, pp. 4750–4762, Dec.2006.
[CrossRef]

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

2002

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

2001

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., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

1993

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

1971

M. Geller and E. W. Ng, “A table of integrals of the error function. II. Additions and corrections,” J. Res. Natl. Bur. Stand., vol. 75B, pp. 149–163, July–Dec.1971.

Agrawal, G. P.

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

Al-Habash, A.

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., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

Allen, J. I. H.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

Andrews, L. C.

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., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

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

Armstrong, J.

H. A. Suraweera and J. Armstrong, “A simple and accurate approximation to the SEP of rectangular QAM in arbitrary Nakagami-m fading channels,” IEEE Commun. Lett., vol. 11, pp. 426–428, May2007.
[CrossRef]

Arnon, S.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[CrossRef]

Azmi, P.

Chan, V. W. S.

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[CrossRef]

Cheng, J.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
[CrossRef]

M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
[CrossRef]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma–Gamma atmospheric turbulence model,” Opt. Express, vol. 18, pp. 12824–12831, June2010.
[CrossRef] [PubMed]

Cvijetic, N.

N. Cvijetic and T. Wang, “WiMAX over free-space optics evaluating OFDM multi-subcarrier modulation in optical wireless channels,” in IEEE Sarnoff Symp., 27–28 Mar. 2006, pp. 1–4.

Datsikas, C. K.

Djordjevic, I. B.

Gao, S.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

Geller, M.

M. Geller and E. W. Ng, “A table of integrals of the error function. II. Additions and corrections,” J. Res. Natl. Bur. Stand., vol. 75B, pp. 149–163, July–Dec.1971.

Ghassemlooy, Z.

W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, pp. 967–973, Apr.2009.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

Gradshteyn, I. S.

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

Holtzman, J. F.

M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
[CrossRef]

Hongou, J.

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

Hopen, C. Y.

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

Huang, W.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

Kahn, J. M.

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

Karagiannidis, G. K.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[CrossRef]

G. K. Karagiannidis, “On the symbol error probability of general order rectangular QAM in Nakagami-m fading,” IEEE Commun. Lett., vol. 10, pp. 745–747, Nov.2006.
[CrossRef]

Kasai, K.

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

Kiasaleh, K.

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol. 54, pp. 604–607, Apr.2006.
[CrossRef]

Lee, E.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
[CrossRef]

Leitgeb, E.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

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., vol. 55, pp. 1598–1606, Aug.2007.
[CrossRef]

Lioumpas, A. S.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[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., vol. 55, pp. 1598–1606, Aug.2007.
[CrossRef]

Nakagawa, M.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

Nakazawa, M.

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

Neifeld, M. A.

Ng, E. W.

M. Geller and E. W. Ng, “A table of integrals of the error function. II. Additions and corrections,” J. Res. Natl. Bur. Stand., vol. 75B, pp. 149–163, July–Dec.1971.

Niu, M.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
[CrossRef]

M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
[CrossRef]

Park, J.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
[CrossRef]

Peppas, K. P.

Phillips, R. L.

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., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

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

Popoola, W. O.

W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, pp. 967–973, Apr.2009.
[CrossRef]

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

Ryzhik, I. M.

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

Sakanaka, T.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

Samimi, H.

Simon, M. K.

M. K. Simon, Digital Communication Over Fading Channel: A Unified Approach to Performance Analysis. John Wiley & Sons, Inc., Hoboken, NJ, 2004.

Song, X.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
[CrossRef]

Suraweera, H. A.

H. A. Suraweera and J. Armstrong, “A simple and accurate approximation to the SEP of rectangular QAM in arbitrary Nakagami-m fading channels,” IEEE Commun. Lett., vol. 11, pp. 426–428, May2007.
[CrossRef]

Takayanagi, J.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

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., vol. 55, pp. 1598–1606, Aug.2007.
[CrossRef]

Vasic, B.

Wang, N.

Wang, T.

N. Cvijetic and T. Wang, “WiMAX over free-space optics evaluating OFDM multi-subcarrier modulation in optical wireless channels,” in IEEE Sarnoff Symp., 27–28 Mar. 2006, pp. 1–4.

Yoon, G.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
[CrossRef]

Yoshida, M.

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

Zhu, X.

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

Electron. Lett.

M. Nakazawa, M. Yoshida, K. Kasai, and J. Hongou, “20 Msymbol/s, 64 and 128 QAM coherent optical transmission over 525 km using heterodyne detection with frequency-stabilised laser,” Electron. Lett., vol. 42, pp. 710–712, June2006.
[CrossRef]

IEEE Commun. Lett.

X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr.2012.
[CrossRef]

G. K. Karagiannidis, “On the symbol error probability of general order rectangular QAM in Nakagami-m fading,” IEEE Commun. Lett., vol. 10, pp. 745–747, Nov.2006.
[CrossRef]

H. A. Suraweera and J. Armstrong, “A simple and accurate approximation to the SEP of rectangular QAM in arbitrary Nakagami-m fading channels,” IEEE Commun. Lett., vol. 11, pp. 426–428, May2007.
[CrossRef]

IEEE Photon. Technol. Lett.

J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb.2011.
[CrossRef]

IEEE Trans. Commun.

N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May2011.
[CrossRef]

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

J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, pp. 1598–1606, Aug.2007.
[CrossRef]

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol. 54, pp. 604–607, Apr.2006.
[CrossRef]

M. Niu, J. Cheng, and J. F. Holtzman, “Error rate analysis of M-ary coherent free space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun., vol. 59, pp. 664–668, Mar.2011.
[CrossRef]

IEICE Trans. Commun.

W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept.1993.

IET Optoelectron.

W. O. Popoola, Z. Ghassemlooy, J. I. H. Allen, E. Leitgeb, and S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, pp. 16–23, Feb.2008.
[CrossRef]

J. Lightwave Technol.

J. Opt. Commun. Netw.

J. Res. Natl. Bur. Stand.

M. Geller and E. W. Ng, “A table of integrals of the error function. II. Additions and corrections,” J. Res. Natl. Bur. Stand., vol. 75B, pp. 149–163, July–Dec.1971.

Opt. Eng.

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., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

Opt. Express

Other

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

M. K. Simon, Digital Communication Over Fading Channel: A Unified Approach to Performance Analysis. John Wiley & Sons, Inc., Hoboken, NJ, 2004.

The Wolfram Function Site, 2012 [Online]. Available: http://functions.wolfram.com/GammaBetaErf/Erfc/21/02/01/.

N. Cvijetic and T. Wang, “WiMAX over free-space optics evaluating OFDM multi-subcarrier modulation in optical wireless channels,” in IEEE Sarnoff Symp., 27–28 Mar. 2006, pp. 1–4.

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

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

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Figures (8)

Fig. 1
Fig. 1

ASER of subcarrier 4 × 4 QAM over the Gamma–Gamma channels with different levels of turbulence.

Fig. 2
Fig. 2

ASER of subcarrier 8 × 8 QAM over the Gamma–Gamma channels with different levels of turbulence.

Fig. 3
Fig. 3

ASER of subcarrier 4 × 4 QAM over the K-distributed turbulence channels.

Fig. 4
Fig. 4

ASER of subcarrier 8 × 8 QAM over the K-distributed turbulence channels.

Fig. 5
Fig. 5

ASER of subcarrier 4 × 4 and 8 × 8 QAM over negative exponential turbulence channel.

Fig. 6
Fig. 6

Absolute truncation error of subcarrier 4 × 4 QAM over a moderate Gamma–Gamma turbulence channel (α=2.50, β=2.06) using different values of J.

Fig. 7
Fig. 7

Absolute truncation error of subcarrier 8 × 8 QAM over a K-distributed fading channel (α=1.99) using different values of J.

Fig. 8
Fig. 8

Comparison of the ASER of subcarrier 4 × 4 QAM over the Gamma–Gamma turbulence channel using Eq. (28) and Eq. (24) in [14].

Equations (70)

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it(t)=P[1+ξs(t)],
s(t)=si(t)cos(2πfct)sq(t)sin(2πfct).
ir(t)=RI(t)A[1+ξs(t)]+n(t),
μ=(RAξ)22f(qRIb+2KbTk/RL)I2=CI2,
fI(I)=2(αβ)α+β2Γ(α)Γ(β)Iα+β21Kαβ(2αβI),
α=exp0.49σI2(1+1.11σI12/5)7/611,
β=exp0.51σI2(1+0.69σI12/5)5/611,
σSI2=E[I2](E[I])21=1α+1β+1αβ.
fI(I)=2αα+12Γ(α)Iα12Kα1(2αI).
fI(I)=λexp(λI),
Pe=0Pe(μ)fμ(μ)dμ,
Pe(μ)=211MIQAIμ+211MQQAQμ411MI11MQQAIμQAQμ.
AI=6/[(MI21)+r2(MQ21)],
AQ=6r2/(MI21)+r2(MQ21).
Kν(x)=π2sin(πν)p=0(x/2)2pνΓ(pν+1)p!(x/2)2p+νΓ(p+ν+1)p!,
Mμ(s)Eexpsμ=B(αβ,1α+β)2Γ(α)Γ(β)p=0ap(α,β)sp+β2μ¯p+β2ap(β,α)sp+α2μ¯p+α2.
0xmexp(bxn)dx=Γ(m+1n)nbm+1n
ap(x,y)=(xy)p+yΓ(px+y+1)p!Γp+y2.
Q(x)=1π0π2expx22sin2θdθ
Q(x)Q(y)=12π0π2tan1(yx)expx22sin2θdθ+12π0tan1(yx)expy22sin2θdθ.
Pe=2π11MI0π2MμAI22sin2θdθe1+2π11MQ0π2MμAQ22sin2θdθe22π11MI11MQ0π2tan1(AIAQ)MμAQ22sin2θdθe32π11MI11MQ0tan1(AIAQ)MμAI22sin2θdθe4
gp12,x=0π2sinp+xθdθ=2p+x1Bp+x+12,p+x+12,
e1=B(αβ,1α+β)πΓ(α)Γ(β)11MI×p=0ap(α,β)gp12,βAI(p+β)μ¯2p+β2ap(β,α)gp12,αAI(p+α)μ¯2p+α2.
e2=B(αβ,1α+β)πΓ(α)Γ(β)11MQ×p=0ap(α,β)gp12,βAQ(p+β)μ¯2p+β2ap(β,α)gp12,αAQ(p+α)μ¯2p+α2.
gp(η,x)=0ηπsinp+xθdθ=πΓ1+p+x22Γ1+p+x2cos(ηπ)F12,1px2;32;cos2(ηπ),
e3=B(αβ,1α+β)πΓ(α)Γ(β)11MI11MQ×p=0ap(α,β)gp(m,β)AQ(p+β)μ¯2p+β2ap(β,α)gp(m,α)AQ(p+α)μ¯2p+α2.
e4=B(αβ,1α+β)πΓ(α)Γ(β)11MI11MQ×p=0ap(α,β)gp(n,β)AI(p+β)μ¯2p+β2ap(β,α)gp(n,α)AI(p+α)μ¯2p+α2.
Pe,γγ=e1+e2e3e4.
Pe,K=Pe,γγ|β=1=e1|β=1+e2|β=1e3|β=1e4|β=1,
e1|β=1=Γ(2α)π(α1)11MI×p=0ap(α,1)gp12,1AI(p+1)μ¯2p+12ap(1,α)gp12,αAI(p+α)μ¯2p+α2,
e2|β=1=Γ(2α)π(α1)11MQ×p=0ap(α,1)gp12,1AQ(p+1)μ¯2p+12ap(1,α)gp12,αAQ(p+α)μ¯2p+α2,
e3|β=1=Γ(2α)π(α1)11MI11MQ×p=0ap(α,1)gp(m,1)AQ(p+1)μ¯2p+12ap(1,α)gp(m,α)AQ(p+α)μ¯2p+α2,
e4|β=1=Γ(2α)π(α1)11MI11MQ×p=0ap(α,1)gp(n,1)AI(p+1)μ¯2p+12ap(1,α)gp(n,α)AI(p+α)μ¯2p+α2.
Pe,NE=limαPe,K=limαe1|β=1+limαe2|β=1limαe3|β=1limαe4|β=1.
limαe1|β=1=limαΓ(2α)π(α1)11MI×p=0αp+1Γp+12Γ(pα+2)p!gp12,1AI(p+1)μ¯2p+12=1π11MIlimαp=0(1)pαp+1Γp+12k=0p(kα+1)p!×gp12,1AI(p+1)μ¯2p+12=1π11MIp=0(1)pΓp+12p!×gp12,1AI(p+1)μ¯2p+12.
limαe2|β=1=1π11MQp=0(1)pΓp+12p!×gp12,1AQ(p+1)μ¯2p+12,
limαe3|β=1=1π11MQ11MQ×p=0(1)pΓp+12p!gp(m,1)AQ(p+1)μ¯2p+12,
limαe4|β=1=1π11MQ11MQ×p=0(1)pΓp+12p!gp(n,1)AI(p+1)μ¯2p+12.
ϵe1,J=B(αβ,1α+β)πΓ(α)Γ(β)11MI×p=J+1ap(α,β)gp12,βAI(p+β)μ¯2p+β2ap(β,α)gp12,αAI(p+α)μ¯2p+α2.
ϵe1,J=B(αβ,1α+β)πΓ(α)Γ(β)11MIp=J+11p!2αβAIμ¯p×upα,β,AI,12upβ,α,AI,12,
up(x,y,z,η)=Γp+y2Γ(px+y+1)gp(η,y)2xyzμ¯y.
ϵe1,Jup=B(αβ,1α+β)πΓ(α)Γ(β)11MIexp2αβAIμ¯×maxp>Jupα,β,AI,12upβ,α,AI,12.
ϵe2,Jup=B(αβ,1α+β)πΓ(α)Γ(β)11MQexp2αβAQμ¯×maxp>Jupα,β,AQ,12upβ,α,AQ,12,
ϵe3,Jup=B(αβ,1α+β)πΓ(α)Γ(β)11MI11MQ×exp2αβAQμ¯maxp>J[up(α,β,AQ,m)up(β,α,AQ,m)],
ϵe4,Jup=B(αβ,1α+β)πΓ(α)Γ(β)11MI11MQ×exp2αβAIμ¯maxp>Jup(α,β,AI,n)up(β,α,AI,n).
ϵJ,γγϵe1,Jup+ϵe2,Jupϵe3,Jupϵe4,Jup.
e1,asym=B(αβ,1α+β)πΓ(α)Γ(β)(αβ)βΓβ2Γ(βα+1)×πΓ1+β22Γ1+β211MIAIβμ¯2β2,
e2,asym=B(αβ,1α+β)πΓ(α)Γ(β)(αβ)βΓβ2Γ(βα+1)×πΓ1+β22Γ1+β211MQAQβμ¯2β2,
e3,asym=B(αβ,1α+β)πΓ(α)Γ(β)(αβ)βΓβ2Γ(βα+1)×11MI11MQg0(m,β)AQβμ¯2β2,
e4,asym=B(αβ,1α+β)πΓ(α)Γ(β)(αβ)βΓβ2Γ(βα+1)×11MI11MQg0(n,β)AIβμ¯2β2.
Pe,γγ,asym=e1,asym+e2,asyme3,asyme4,asym.
Pe(I)=11MIerfcAIμ¯2I+1MI11MQerfcAQμ¯2I+11MI×11MQerfcAQμ¯2IerfAIμ¯2I.
e1A=11MI0erfcAIμ¯2IfI(I)dI,
e2A=1MI11MQ0erfcAQμ¯2IfI(I)dI,
e3A=11MI11MQ×0erfcAQμ¯2IerfAIμ¯2IfI(I)dI.
e1A=11MIB(αβ,1α+β)πΓ(α)Γ(β)×p=0bp(α,β)AI(p+β)μ¯2p+β2bp(β,α)AI(p+α)μ¯2p+α2,
bp(x,y)=(xy)p+yΓ(px+y+1)p!Γp+y+12p+y.
e2A=1MI11MQB(αβ,1α+β)πΓ(α)Γ(β)×p=0bp(α,β)AQ(p+β)μ¯2p+β2bp(β,α)AQ(p+α)μ¯2p+α2.
0ta1erfc(t)dt=1πaΓa+12.
e3A=11MI11MQB(αβ,1α+β)Γ(α)Γ(β)×p=0wp(α,β)μ¯2p+β2wp(β,α)μ¯2p+α2,
0xp1erfc(ax)erf(bx)dx=2bap+1Γ1+p2π(p+1)×F12,p+12,p+22;32,p+32;b2a2,a>b,p>1,
wp(x,y)=(xy)p+yΓ(px+y+1)p!2AIAQp+y+1Γ1+p+y2π(p+y+1)×F12,p+y+12,p+y+22;32,p+y+32;AI2AQ2.
Pe(I)=221MI1MQQAIμ¯I411MI11MQQ2AIμ¯I.
e4A=221MI1MQ0QAIμ¯IfI(I)dI,
e5A=411MI11MQ0Q2AIμ¯IfI(I)dI.
e4A=21MI1MQB(αβ,1α+β)πΓ(α)Γ(β)×p=0bp(α,β)AI(p+β)μ¯2p+β2bp(β,α)AI(p+α)μ¯2p+α2.
Q2(x)=1π0π4expx22   sin2θdθ.
e5A=11MI11MQ2B(αβ,1α+β)πΓ(α)Γ(β)×p=0vp(α,β)AI(p+β)μ¯2p+β2vp(β,α)AI(p+α)μ¯2p+α2,
vp(x,y)=(xy)p+yΓ(px+y+1)p!Γp+y2×F12,p+y+12;p+y+32;122p+y+12(p+y+1).
0π4sinxθdθ=F12,x+12;x+32;122x+12(x+1).