Abstract

In this paper, we investigate the error rate performance of a coherent free-space optical communication system with differential phase-shift keying over M-distributed turbulence channels. In our derivations, we express the probability density function of M distribution in terms of a series expansion. The coefficients of this series only include elementary and gamma functions. For numerical evaluation, this power series is truncated to a finite number of terms. An upper bound for the associated truncation error is provided and used for the convergence analysis of the series expansion. In the next step, using the moment generating function approach, we derive a bit error rate (BER) expression as a generalized infinite power series and demonstrate that existing BER results in the literature reported for gamma–gamma channels and K channels can obtained as special cases.

© 2013 Optical Society of America

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  1. H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks. Sams, 2002.
  2. L. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. SPIE, 2001.
  3. G. R. Osche, Optical Detection Theory for Laser Applications. Wiley, 2002.
  4. L. C. Andrews and R. L. Phillips, “Mathematical genesis of the I-K distribution for random optical fields,” J. Opt. Soc. Am. A, vol.  3, no. 11, pp. 1912–1919, Nov. 1986.
    [CrossRef]
  5. 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., vol.  40, no. 8, pp. 1554–1562, 2001.
    [CrossRef]
  6. N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.
  7. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .
  8. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.
  9. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett., vol.  36, no. 20, pp. 4095–4097, 2011.
    [CrossRef]
  10. H. Samimi, “Optical communication using subcarrier intensity modulation through generalized turbulence channels,” J. Opt. Commun. Netw., vol.  4, no. 5, pp. 378–381, 2012.
    [CrossRef]
  11. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express, vol.  20, no. 11, pp. 12550–12562, 2012.
    [CrossRef]
  12. S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels. Plenum, 1988.
  13. L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech House, 1996.
  14. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol.  54, no. 4, pp. 604–607, Apr. 2006.
    [CrossRef]
  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. Communications (ICC), New York, 2009.
  16. T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” Electron. Lett., vol.  44, pp. 373–375, Feb. 2008.
    [CrossRef]
  17. M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express, vol.  20, no. 6, pp. 6515–6520, 2012.
    [CrossRef]
  18. M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels, 1st ed. Wiley, 2001.
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. Academic, 2000.
  20. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, pp. 3415–3424, Nov. 2009.
    [CrossRef]
  21. E. Lee and V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol.  22, no. 9, pp. 1896–1906, Nov. 2004.
    [CrossRef]
  22. G. Xie, A. Dang, and H. Guo, “Effects of atmosphere dominated phase fluctuation and intensity scintillation to DPSK system,” IEEE Int. Conf. Communications (ICC), 2011.

2012 (3)

2011 (1)

2009 (1)

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, pp. 3415–3424, Nov. 2009.
[CrossRef]

2008 (1)

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” Electron. Lett., vol.  44, pp. 373–375, Feb. 2008.
[CrossRef]

2006 (1)

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol.  54, no. 4, pp. 604–607, Apr. 2006.
[CrossRef]

2004 (1)

E. Lee and V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol.  22, no. 9, pp. 1896–1906, Nov. 2004.
[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., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

1986 (1)

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., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Alouini, M.-S.

M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels, 1st ed. Wiley, 2001.

Andrews, L.

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

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., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mathematical genesis of the I-K distribution for random optical fields,” J. Opt. Soc. Am. A, vol.  3, no. 11, pp. 1912–1919, Nov. 1986.
[CrossRef]

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., vol.  57, pp. 3415–3424, Nov. 2009.
[CrossRef]

Benedetto, S.

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech House, 1996.

Chan, V.

E. Lee and V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol.  22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

Cheng, J.

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express, vol.  20, no. 6, pp. 6515–6520, 2012.
[CrossRef]

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. Communications (ICC), New York, 2009.

Dang, A.

G. Xie, A. Dang, and H. Guo, “Effects of atmosphere dominated phase fluctuation and intensity scintillation to DPSK system,” IEEE Int. Conf. Communications (ICC), 2011.

Gagliardi, R.

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

Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express, vol.  20, no. 11, pp. 12550–12562, 2012.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett., vol.  36, no. 20, pp. 4095–4097, 2011.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .

Ghuman, B. S.

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

Gradshteyn, I. S.

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

Guo, H.

G. Xie, A. Dang, and H. Guo, “Effects of atmosphere dominated phase fluctuation and intensity scintillation to DPSK system,” IEEE Int. Conf. Communications (ICC), 2011.

Holzman, J. F.

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express, vol.  20, no. 6, pp. 6515–6520, 2012.
[CrossRef]

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. Communications (ICC), New York, 2009.

Hopen, C. Y.

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

Jurado-Navas, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express, vol.  20, no. 11, pp. 12550–12562, 2012.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett., vol.  36, no. 20, pp. 4095–4097, 2011.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.

Karagiannidis, G. K.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

Karp, S.

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

Kazovsky, L.

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech House, 1996.

Kiasaleh, K.

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol.  54, no. 4, pp. 604–607, Apr. 2006.
[CrossRef]

Kotsopoulos, S. A.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

Lee, E.

E. Lee and V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol.  22, no. 9, pp. 1896–1906, Nov. 2004.
[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., vol.  57, pp. 3415–3424, Nov. 2009.
[CrossRef]

Matthaiou, M.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

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. Communications (ICC), New York, 2009.

Moran, S. E.

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

Niu, M.

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express, vol.  20, no. 6, pp. 6515–6520, 2012.
[CrossRef]

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. Communications (ICC), New York, 2009.

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications. Wiley, 2002.

Paris, J. F.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express, vol.  20, no. 11, pp. 12550–12562, 2012.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett., vol.  36, no. 20, pp. 4095–4097, 2011.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .

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., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mathematical genesis of the I-K distribution for random optical fields,” J. Opt. Soc. Am. A, vol.  3, no. 11, pp. 1912–1919, Nov. 1986.
[CrossRef]

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

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express, vol.  20, no. 11, pp. 12550–12562, 2012.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett., vol.  36, no. 20, pp. 4095–4097, 2011.
[CrossRef]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .

Ryzhik, I. M.

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

Samimi, H.

Sandalidis, H. G.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

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., vol.  57, pp. 3415–3424, Nov. 2009.
[CrossRef]

Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels, 1st ed. Wiley, 2001.

Song, X.

Stotts, L. B.

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

Tsiftsis, T. A.

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” Electron. Lett., vol.  44, pp. 373–375, Feb. 2008.
[CrossRef]

Willebrand, H.

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

Willner, A.

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech House, 1996.

Xie, G.

G. Xie, A. Dang, and H. Guo, “Effects of atmosphere dominated phase fluctuation and intensity scintillation to DPSK system,” IEEE Int. Conf. Communications (ICC), 2011.

Electron. Lett. (1)

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” Electron. Lett., vol.  44, pp. 373–375, Feb. 2008.
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

E. Lee and V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol.  22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

IEEE Trans. Commun. (2)

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, pp. 3415–3424, Nov. 2009.
[CrossRef]

K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun., vol.  54, no. 4, pp. 604–607, Apr. 2006.
[CrossRef]

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. A (1)

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., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Other (12)

G. Xie, A. Dang, and H. Guo, “Effects of atmosphere dominated phase fluctuation and intensity scintillation to DPSK system,” IEEE Int. Conf. Communications (ICC), 2011.

M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels, 1st ed. Wiley, 2001.

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

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. Communications (ICC), New York, 2009.

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

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

G. R. Osche, Optical Detection Theory for Laser Applications. Wiley, 2002.

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

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech House, 1996.

N. D. Chatzidiamantis, H. G. Sandalidis, G. K. Karagiannidis, S. A. Kotsopoulos, and M. Matthaiou, “New results on turbulence modeling for free-space optical systems,” in IEEE 17th Int. Conf. Telecommunications (ICT), 2010, pp. 487–492.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv:1102.1915v1, 2011 [Online]. Available: http://arxiv.org/abs/1102.1915v1 .

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes. Intech, 2011, pp. 181–206.

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Figures (5)

Fig. 1.
Fig. 1.

Truncation error with respect to the number of terms used in the series representation of the M distribution.

Fig. 2.
Fig. 2.

Exact pdf of the irradiance fluctuations [given by Eq. (6)] compared with the truncated pdf [given by Eq. (8)].

Fig. 3.
Fig. 3.

BER performance for two values of ρ assuming α=8.1, β=4.

Fig. 4.
Fig. 4.

BER performance for different values of α, β, ρ.

Fig. 5.
Fig. 5.

Comparison of exact and asympotic BER expressions.

Tables (1)

Tables Icon

Table I Measurement Data [11]

Equations (37)

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

fI(I)=Ak=1βakIα+k21Kαk(2αβIγβ+Ω),
A=2αα2γ1+α2Γ(α)(γβγβ+Ω)β+α2,
ak=(β1k1)(γβ+Ω)1k2(k1)!(Ωγ)k1(αβ)k2.
Ξ=2(γξ)βγΓ(α),
ωk=(β1k1)β1kξ(k1)!(Ωγ)k1.
fI(I)=Ξk=1βωk(αξ)α+k2Iα+k21Kαk(2αξI).
Kv(x)=π2sin(πv)p=0[(x/2)2pvΓ(pv+1)p!(x/2)2p+vΓ(p+v+1)p!],
fI(I)=Ξk=1βπωk2sin[π(αk)]{p=0[dp(α,k)Ip+k1dp(k,α)Ip+α1]},
dp(x,y)=(αξ)p+yΓ(px+y+1)p!.
f˜I(I)=Ξk=1βπωk2sin[π(αk)]{p=0P[dp(α,k)Ip+k1dp(k,α)Ip+α1]}.
E(P)<Ξexp(αξI)maxp>P{cp(α,k)}k=1βπωk2sin[π(αk)].
ΦI(s)=E[esI]=Ξk=1βπωk2sin[π(αk)]{p=0[dp(α,k)0esIIp+k1dIdp(k,α)0esIIp+α1dI]}.
ΦI(s)=Ξk=1βπωk2sin[π(αk)]{p=0[gp(α,k)s(p+k)gp(k,α)s(p+α)]},
gp(x,y)=dp(x,y)Γ(p+y)=(αξ)p+yΓ(p+y)Γ(px+y+1)p!.
Pe(I)=12eSNR0I,
Pe=0Pe(I)fI(I)dI=012eSNR0IfI(I)dI=12E[eSNR0I]=12ΦI(SNR0),
Pe=Ξk=1βπωk4sin[π(αk)]{p=0[gp(α,k)SNR0(p+k)gp(k,α)SNR0(p+α)]}.
Pe,asym=Ξk=1βπωk4sin[π(αk)]g0(α,k)SNR0k,
Pe=12Γ(α)πsin[π(α1)]{p=0[gp(α,1)SNR0(p+1)gp(1,α)SNR0(p+α)]}.
Pe=Γ(α1)Γ(2α)2Γ(α){p=0[gp(α,1)SNR0(p+1)gp(1,α)SNR0(p+α)]}.
p=0gp(x,y)SNR0(p+y)=Γ(y)Γ(yx+1)F11(y,yx+1,αξSNR0).
Pe=Γ(α1)2Γ(α)(αγSNR0)F11(1,2α,αγSNR0)Γ(α1)Γ(2α)2Γ(α)(αγSNR0)αF11(α,α,αγSNR0).
Pe=Γ(α1)2Γ(α)(αγSNR0)F11(1,2α,αγSNR0)+Γ(1α)2(αγSNR0)αF11(α,α,αγSNR0),
Pe=2Γ(α)k=1β(β1k1)ββkγβk(k1)!π4sin[π(αk)]{p=0[gp(α,k)SNR0(p+k)gp(k,α)SNR0(p+α)]}.
Pe=12(β1)!Γ(α)πsin[π(αβ)]{p=0[gp(α,β)SNR0(p+k)gp(β,α)SNR0(p+α)]}.
Pe=Γ(αβ)Γ(1α+β)2Γ(β)Γ(α){p=0[gp(α,β)SNR0(p+β)gp(β,α)SNR0(p+α)]},
Pe,asym=Ξπωβ4sin[π(αβ)]g0(α,β)SNR0β=Γ(αβ)2Γ(α)(αβSNR0)β,
Pe=Γ(αβ)Γ(1α+β)2Γ(β)Γ(α)(αβSNR0)βΓ(β)Γ(βα+1)F11(β,βα+1,αβSNR0)Γ(αβ)Γ(1α+β)2Γ(β)Γ(α)(αβSNR0)αΓ(α)Γ(αβ+1)F11(α,αβ+1,αβSNR0).
Pe=Γ(αβ)2Γ(α)(αβSNR0)βF11(β,βα+1,αβSNR0)+Γ(βα)2Γ(β)(αβSNR0)αF11(α,αβ+1,αβSNR0).
E(P)=Ξk=1βπωk2sin[π(αk)]{p=P+1[dp(α,k)Ip+k1dp(k,α)Ip+α1]}.
E(P)=Ξk=1βπωk2sin[π(αk)]{p=P+1(αξI)pp!cp(α,k)},
bp(x,y)=(αξ)yIy1Γ(px+y+1),
cp(x,y)=bp(x,y)bp(y,x).
E(P)<Ξmaxp>Pcp(α,k)k=1βπωk2sin[π(αk)]{p=P+1(αξI)pp!}.
E(P)<Ξexp(αξI)maxp>P{cp(α,k)}k=1βπωk2sin[π(αk)].
p=0gp(x,y)SNR0(p+y)=p=0(αξ)p+yΓ(p+y)Γ(px+y+1)p!1SNR0(p+y)=p=0Γ(y)Γ(yx+1)(y)pp!(yx+1)p(αξSNR0)p+y=Γ(y)Γ(yx+1)(αξSNR0)yp=0(y)pp!(yx+1)p(αξSNR0)p.
p=0gp(x,y)SNR0(p+y)=Γ(y)Γ(yx+1)F11(y,yx+1,αξSNR0).