Abstract

The receiver design for a high-speed free-space (wireless) optics (FSO) signal is necessarily highly complex when channel state information (CSI) is not available. Currently, although most approaches provide high detection performance in terms of bit error, receiver design is difficult to implement. This paper proposes two practical thresholding-based detection schemes, which offer significant improvement to receiver throughput on a computational load basis when CSI is not available. The first is based on a simple maximum likelihood (ML) function where the bit error rate (BER) is the same as conventional symbol-by-symbol detection. This method, however, causes a loss of BER performance. The second uses the aid of pilot-symbol-assisted modulation (PSAM) to modify the ML function when channel coefficients are temporally correlated. While numerical analysis based on this method shows that the BER performance in a lognormally distributed fading channel is very close to detection achieved with perfect CSI, the receiver suffers from increased complexity. If random processes for fading and noise are assumed as stationary and given that the detection threshold is quickly calculated and applied during a given period, such complexity of PSAM-based and symbol-by-symbol detection methods can be reduced.

© 2010 Optical Society of America

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References

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  1. H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.
  2. N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
    [Crossref]
  3. H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.
  4. W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
    [Crossref]
  5. X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
    [Crossref]
  6. S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
    [Crossref]
  7. M. L. B. Riediger, R. Schober, L. Lampe, “Decision-feedback detection for free-space optical communications,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1193–1197.
  8. M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
    [Crossref]
  9. X. Zhu, J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-space optical channels,” Proc. SPIE, vol. 4489, pp. 138–145, 2002.
    [Crossref]
  10. I. I. Kim, E. Korevaar, “Availability of free space optics (FSO) and hybrid FSO/RF systems,” Proc. SPIE, vol. 4530, pp. 84–95, Aug. 2001.
    [Crossref]
  11. A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
    [Crossref]
  12. R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed.Wiley, 1995.
  13. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
    [Crossref]
  14. S. M. Kay, Intuitive Probability and Random Processes Using MATLAB. Springer, 2006.
    [Crossref]
  15. A. Kaw, E. E. Kalu, Numerical Methods With Applications, 1st ed.2008.
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, 1964.
  17. G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed.Johns Hopkins U. Press, 1996.
  18. http://www.alglib.net/matrixops/symmetric/cholesky.php.
  19. A. Jurado-Navas, A. Puerta-Notario, “Generation of correlated scintillations on atmospheric optical communication,” J. Opt. Commun. Netw., vol. 1, no. 5, pp. 452–462, Oct. 2009.
    [Crossref]

2010 (2)

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

2009 (4)

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

A. Jurado-Navas, A. Puerta-Notario, “Generation of correlated scintillations on atmospheric optical communication,” J. Opt. Commun. Netw., vol. 1, no. 5, pp. 452–462, Oct. 2009.
[Crossref]

2008 (1)

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
[Crossref]

2006 (1)

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

2002 (2)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[Crossref]

X. Zhu, J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-space optical channels,” Proc. SPIE, vol. 4489, pp. 138–145, 2002.
[Crossref]

2001 (1)

I. I. Kim, E. Korevaar, “Availability of free space optics (FSO) and hybrid FSO/RF systems,” Proc. SPIE, vol. 4530, pp. 84–95, Aug. 2001.
[Crossref]

Abdullah, M. K.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, 1964.

Atiquzzaman, M.

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

Boucouvalas, A. C.

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

Cowley, W.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
[Crossref]

Falahpour, M.

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

Fauzi, M.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed.Wiley, 1995.

Ghassemlooy, Z.

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed.Johns Hopkins U. Press, 1996.

Harris, A.

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

Harun, H.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Hitam, S.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Holland, I.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
[Crossref]

Jurado-Navas, A.

Kahn, J. M.

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[Crossref]

X. Zhu, J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-space optical channels,” Proc. SPIE, vol. 4489, pp. 138–145, 2002.
[Crossref]

Kalu, E. E.

A. Kaw, E. E. Kalu, Numerical Methods With Applications, 1st ed.2008.

Karagiannidis, G. K.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed.Wiley, 1995.

Kaw, A.

A. Kaw, E. E. Kalu, Numerical Methods With Applications, 1st ed.2008.

Kay, S. M.

S. M. Kay, Intuitive Probability and Random Processes Using MATLAB. Springer, 2006.
[Crossref]

Kim, I. I.

I. I. Kim, E. Korevaar, “Availability of free space optics (FSO) and hybrid FSO/RF systems,” Proc. SPIE, vol. 4530, pp. 84–95, Aug. 2001.
[Crossref]

Korevaar, E.

I. I. Kim, E. Korevaar, “Availability of free space optics (FSO) and hybrid FSO/RF systems,” Proc. SPIE, vol. 4530, pp. 84–95, Aug. 2001.
[Crossref]

Lampe, L.

M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
[Crossref]

M. L. B. Riediger, R. Schober, L. Lampe, “Decision-feedback detection for free-space optical communications,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1193–1197.

Lee, C. G.

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

Letzepis, N.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
[Crossref]

LoPresti, P. G.

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

Mahdi, M. A.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Moradi, H.

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

Popoola, W. O.

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

Puerta-Notario, A.

Refai, H. H.

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

Riediger, M. L. B.

M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
[Crossref]

M. L. B. Riediger, R. Schober, L. Lampe, “Decision-feedback detection for free-space optical communications,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1193–1197.

Sali, A.

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Sandalidis, H. G.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

Schober, R.

M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
[Crossref]

M. L. B. Riediger, R. Schober, L. Lampe, “Decision-feedback detection for free-space optical communications,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1193–1197.

Sluss, J. J.

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, 1964.

Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

Uysal, M.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed.Johns Hopkins U. Press, 1996.

Zhu, X.

X. Zhu, J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-space optical channels,” Proc. SPIE, vol. 4489, pp. 138–145, 2002.
[Crossref]

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[Crossref]

IEEE Trans. Commun. (2)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[Crossref]

M. L. B. Riediger, R. Schober, L. Lampe, “Fast multiple-symbol detection for free-space optical communications,” IEEE Trans. Commun., vol. 57, no. 4, pp. 1119–1128, Apr. 2009.
[Crossref]

IEEE Trans. Wireless Commun. (2)

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, 2008.
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[Crossref]

J. Opt. Commun. Netw. (1)

J. Opt. Fiber Commun. Res. (1)

S. Hitam, M. K. Abdullah, M. A. Mahdi, H. Harun, A. Sali, M. Fauzi, “Impact of increasing threshold level on higher bit rate in free space optical communications,” J. Opt. Fiber Commun. Res., vol. 6, no. 1–6, pp. 22–34, Dec. 2009.
[Crossref]

Opt. Eng. (1)

A. Harris, J. J. Sluss, H. H. Refai, P. G. LoPresti, “Free-space optical wavelength diversity scheme for fog mitigation in a ground-to-unmanned aerial-vehicle communications link,” Opt. Eng., vol. 45, pp. 1–12, 2006.
[Crossref]

Opt. Laser Technol. (1)

W. O. Popoola, Z. Ghassemlooy, C. G. Lee, A. C. Boucouvalas, “Scintillation effect on intensity modulated laser communication systems—a laboratory demonstration,” Opt. Laser Technol., vol. 42, no. 4, pp. 682–692, June 2010.
[Crossref]

Proc. SPIE (3)

H. Moradi, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “A PSAM-based estimator of noise and fading statistics for optimum receivers of free space optics signals,” Proc. SPIE, vol. 7587, pp. 75870O1–75870O10, Jan. 2010.

X. Zhu, J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-space optical channels,” Proc. SPIE, vol. 4489, pp. 138–145, 2002.
[Crossref]

I. I. Kim, E. Korevaar, “Availability of free space optics (FSO) and hybrid FSO/RF systems,” Proc. SPIE, vol. 4530, pp. 84–95, Aug. 2001.
[Crossref]

Other (8)

M. L. B. Riediger, R. Schober, L. Lampe, “Decision-feedback detection for free-space optical communications,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1193–1197.

H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, M. Atiquzzaman, “BER analysis of optical wireless signals through lognormal fading channels with perfect CSI,” in IEEE 17th Int. Conf. on Telecommunications, Doha, Qatar, 2010, pp. 493–497.

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed.Wiley, 1995.

S. M. Kay, Intuitive Probability and Random Processes Using MATLAB. Springer, 2006.
[Crossref]

A. Kaw, E. E. Kalu, Numerical Methods With Applications, 1st ed.2008.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, 1964.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed.Johns Hopkins U. Press, 1996.

http://www.alglib.net/matrixops/symmetric/cholesky.php.

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Figures (7)

Fig. 1
Fig. 1

MSA estimator.

Fig. 2
Fig. 2

Typical threshold value I D for optimal detection based on a given Gaussian noise with I 0 = 0.6 , averaged SNR = 6 dB , and four different lognormal channels. The value of I D is highly dependent on the fading strength σ x . For example, I D 1.1 when σ x = 0.1 , whereas I D 0.9 when σ x = 0.4 .

Fig. 3
Fig. 3

Slot structure used for pilot-based optimal detection.

Fig. 4
Fig. 4

Threshold value of the LTD method with SNR, while I 0 = 0.2 and for different values of σ x . Such information can be used as a lookup table to find I D in practice.

Fig. 5
Fig. 5

Probability of error using the LTD detection method compared with that of detection with CSI for three different lognormal channels.

Fig. 6
Fig. 6

Comparing the probability of errors obtained for PTD detection, LTD detection, and detection with CSI methods, while σ χ = 0.1 .

Fig. 7
Fig. 7

BER performance versus processing load for different discussed detection schemes. LTD and PTD are suggested by this work. The complexities of CSI estimation for processing load of detection with CSI has not been included.

Tables (1)

Tables Icon

Table 1 Performance Loss in Decibels at σ x = 0.1 Based on Simulation Analysis in Fig. 6

Equations (64)

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

i d [ n ] = h [ n ] s [ n ] + i n [ n ] ,
I D , G = σ 0 I 1 + σ 1 I 0 σ 0 + σ 1 ,
γ G = 4 R 2 P t 2 ( σ 1 + σ 0 ) 2 .
h = I I m = e 2 X ,
μ I = E [ h ] = M x ( 2 ) = e ( 2 μ x + 2 σ x 2 ) .
σ I 2 = E [ h 2 ] E [ h ] 2 = M x ( 4 ) ( M x ( 2 ) ) 2 = e ( 4 μ x + 4 σ x 2 ) ( e 4 σ x 2 1 ) .
μ I = E [ h ] = 1 ,
σ I 2 = e 4 σ x 2 1.
f I ( h ) = 1 8 π h σ x exp ( [ ln ( h ) + 2 σ x 2 ] 2 8 σ x 2 ) ,
σ x = ln ( S.I. + 1 ) 2 .
γ L = 4 h 2 R 2 P t 2 ( σ 1 + σ 0 ) 2 .
γ ¯ 4 R 2 P t 2 ( σ 1 + σ 0 ) 2 ,
I D , With- CSI = σ 0 ( I 0 + 2 P t R h ) + σ 1 I 0 σ 0 + σ 1 ,
P e , L ( γ G , σ x ) = 1 2 1 π e σ x 2 k = 0 K ( 1 ) k γ G ( 2 k + 1 ) 2 2 ( 2 k + 1 ) 2 ( 2 k + 1 ) k ! exp ( ( 4 k + 1 ) σ x 2 ) 2 ,
i p = 2 P t R h + i n .
f p , 1 ( X ) = f I ( X ) f n ( X ) = 1 4 π σ x σ 1 0 1 h exp ( [ ln ( h ) + 2 σ x 2 ] 2 8 σ x 2 ) exp ( ( h I 0 X ) 2 2 σ 1 2 ) d h .
μ p , 1 = μ I + I 0 = 1 + I 0 ,
σ p , 1 2 = σ I 2 + σ 1 2 = e 4 σ x 2 + σ 1 2 1 .
σ 1 σ 0 .
σ ̂ x 1 2 ln [ σ ̂ p , 1 2 σ ̂ 0 2 + 1 ] ,
C x = [ σ x 2 C 1 , 2 C 1 , N h C 2 , 1 σ x 2 C 2 , N h C N h , 1 C N h , 2 σ x 2 ] N h × N h .
C n 1 , n 2 = σ x 2 exp [ ( T n 1 , n 2 τ c ) 5 3 ] ,
ρ n 1 , n 2 = E [ ( h [ n 1 ] μ I ) ( h [ n 2 ] μ I ) ] σ I 2 = C n 1 , n 2 σ I 2 ,
f I ( h n 1 , h n 2 ) = 1 8 π h n 1 h n 2 det ( Σ n 1 , n 2 ) × exp ( 1 8 [ ln ( h n 1 ) + 2 σ x 2 ln ( h n 2 ) + 2 σ x 2 ] T Σ n 1 , n 2 1 [ ln ( h n 1 ) + 2 σ x 2 ln ( h n 2 ) + 2 σ x 2 ] ) ,
Σ n 1 , n 2 = [ σ x 2 C n 1 , n 2 C n 1 , n 2 σ x 2 ] .
R h ( τ ) = E [ h n 1 h n 2 ] = exp [ 4 σ x 2 exp [ ( τ τ c ) 5 3 ] ] .
E [ h ] = μ ( a constant ) ,
E [ ( h n 1 μ I ) ( h n 2 μ I ) ] = f ( τ )
R p ( τ ) = E [ i p 1 i p 1 ] = E [ ( h n 1 + i 1 ) ( h n 2 + i 2 ) ] = exp [ 4 σ x 2 exp [ ( τ τ c ) 5 3 ] ] + I 0 2 + 2 I 0 .
R p ( τ ) exp [ 8 σ x 2 2 σ x 4 ] + I 0 2 + 2 I 0 ,
σ x = 2 4 0.5 ln [ R p ( τ ) I 0 2 2 I 0 ] .
Λ ( i d ) = p ( i d | 1 ) p ( i d | 0 ) 0 1 1
p ( i d | 0 ) = 1 2 π σ 0 exp [ ( i d I 0 ) 2 2 σ 0 2 ] ,
p ( i d | 1 ) = 1 2 π σ 1 0 f I ( h ) exp [ ( i d I o 2 R P t h ) 2 2 σ 1 2 ] d h .
Λ ( i d ) = σ 0 σ 1 0 f I ( h ) exp [ ( i d I o 2 R P t h ) 2 2 σ 1 2 + ( i d I o ) 2 2 σ 0 2 ] d h .
BER S - by- S = 1 2 Λ ( i d ) > 1 p ( i d | 0 ) d i d + 1 2 Λ ( i d ) < 1 p ( i d | 1 ) d i d .
p e ( 1 | 0 ) = 1 2 erfc ( I D I 0 2 σ 0 ) ,
p e ( 0 | 1 ) = 1 32 π σ x 0 1 h exp ( [ ln ( h ) 2 μ x ] 2 8 σ x 2 ) erfc ( I 0 I D + 2 R P t h 2 σ 1 ) d h .
BER LTD = 0.5 p e ( 0 | 1 ) + 0.5 p e ( 1 | 0 ) .
Λ ( I D ) = p e ( 0 | 1 ) p e ( 1 | 0 ) = 1.
exp ( [ X + σ x 2 ] 2 2 σ x 2 ) ( erfc ( I 0 I D + 2 R P t e 2 X 2 σ 1 ) erfc ( I D I 0 2 σ 0 ) ) d X = 0.
BER LTD = 1 2 erfc ( I D I 0 2 σ 0 ) .
p ( i k , n , m | 0 ) = 1 2 π σ 0 σ 1 exp [ ( i d ( n , m ) I 0 ) 2 2 σ 0 2 ] × 0 f I ( h ) exp [ ( i p , 1 ( n , k ) I 0 2 R P t h ) 2 2 σ 1 2 ] d h ,
p ( i k , n , m | 1 ) = 1 2 π σ 1 2 × 0 0 f I ( h p , h d ) exp [ ( i p , 1 ( k , m ) I 0 2 R P t h p ) 2 2 σ 1 2 ( i d ( n , m ) I 0 2 R P t h d ) 2 2 σ 1 2 ] d h p d h d ,
Λ ( i k , n , m ) = p ( i k , n , m | 1 ) p ( i k , n , m | 0 ) .
E [ Z ] = 0 z f Z ( z ) d z ,
E [ Z ] = E [ g ( h p , h d ) ] = 0 0 f I ( h p , h d ) g ( h p , h d ) d h p d h d .
g ( h p , h d ) 1 2 π σ 1 2 [ exp [ ( I ̂ [ n p ] 2 R P t h p ) 2 2 σ 1 2 ( I ̂ [ n d ] 2 R P t h d ) 2 2 σ 1 2 ] ] ,
E [ Z ] = p ( i k , n , m | 1 ) .
Ψ [ n ] 1 2 π σ 1 exp [ ( I ̂ [ n ] 2 R P t h [ n ] ) 2 2 σ 1 2 ] .
E [ Ψ [ n ] ] = 1 2 π σ 1 0 f I ( h ) exp [ ( I ̂ [ n ] 2 R P t h ) 2 2 σ 1 2 ] d h ,
E [ Ψ [ n ] ] C ,
R Ψ [ n 1 , n 2 ] = E [ Ψ [ n 1 ] Ψ [ n 2 ] ] = 1 2 π σ 1 2 0 0 f I ( h n 1 , h n 2 ) [ exp [ ( I ̂ [ n 1 ] 2 R P t h n 1 ) 2 2 σ 1 2 ( I ̂ [ n 2 ] 2 R P t h n 2 ) 2 2 σ 1 2 ] ] d h n 1 d h n 2 ,
R Ψ [ n 1 , n 2 ] = R Ψ ( τ ) C ; τ 0.
R Ψ ( τ ) R Ψ ( 0 ) ,
R Ψ ( τ ) = E [ Z ] = p ( i k , n , m | 1 ) ,
P ̃ PCD ( k , n , m ) = 0.5 i p , 1 ( k , m ) , i d ( n , m ) , Λ ( i d ( n , m ) ) < 1 p ( i k , n , m | 1 ) d i d ( n , m ) d i p , 1 ( k , m )
+ 0.5 i p , 1 ( k , m ) , i d ( n , m ) , Λ ( i d ( n , m ) ) > 1 p ( i k , n , m | 0 ) d i d ( n , m ) d i p , 1 ( k , m ) .
BER ¯ PCD = 1 ( N N p ) n = N p + 1 N P ̃ PCD ( k , n , m ) ,
p k , n , m ( 1 | 0 ) = I D p ( i k , n , m | 0 ) d i d ( n , m ) d i p , 1 ( k , m ) ,
p k , n , m ( 0 | 1 ) = I D p ( i k , n , m | 1 ) d i d ( n , m ) d i p , 1 ( k , m ) ,
p ¯ ( 1 | 0 ) = 1 N N p n = N p + 1 N p k , n , m ( 1 | 0 ) ,
p ¯ ( 0 | 1 ) = 1 N N p n = N p + 1 N p k , n , m ( 0 | 1 ) .
Λ k , n , m ( I D ) = p ¯ ( 0 | 1 ) p ¯ ( 1 | 0 ) = 1 .