Abstract

In this paper, a new and simple rate-adaptive transmission scheme for free-space optical (FSO) communication systems with intensity modulation and direct detection (IM/DD) over atmospheric turbulence channels is analyzed. This scheme is based on the joint use of repetition coding and variable silence periods, exploiting the potential time-diversity order (TDO) available in the turbulent channel as well as allowing the increase of the peak-to-average optical power ratio (PAOPR). Here, repetition coding is firstly used in order to accomodate the transmission rate to the channel conditions until the whole time diversity order available in the turbulent channel by interleaving is exploited. Then, once no more diversity gain is available, the rate reduction can be increased by using variable silence periods in order to increase the PAOPR. Novel closed-form expressions for the average bit-error rate (BER) as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam follows negative exponential and gamma-gamma distributions, covering a wide range of atmospheric turbulence conditions. Obtained results show a diversity order as in the corresponding rate-adaptive transmission scheme only based on repetition codes but providing a relevant improvement in coding gain. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also OOK formats with any pulse shape, corroborating the advantage of using pulses with high PAOPR, such as Gaussian or squared hyperbolic secant pulses. We also determine the achievable information rate for the rate-adaptive transmission schemes here analized.

© 2010 OSA

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2010 (5)

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

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]

I. B. Djordjevic, “Adaptive Modulation and Coding for Free-Space Optical Channels,” IEEE/OSA Journal of Optical Communications and Networking 2(5), 221–229 (2010).
[Crossref]

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “An efficient rate-adaptive transmission technique using shortened pulses for atmospheric optical communications,” Opt. Express 18( 16), 17,346–17,363 (2010).
[Crossref]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18( 19), 20,445–20,454 (2010).
[Crossref]

2009 (9)

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the Distribution of the Sum of Gamma-Gamma Variates and Application in MIMO Optical Wireless Systems,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM 2009, pp. 1–6 (2009).
[Crossref]

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
[Crossref]

I. B. Djordjevic and G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17(20), 18,250–18,262 (2009).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

C. Abou-Rjeily, “Orthogonal Space-Time Block Codes for Binary Pulse Position Modulation,” IEEE Trans. Commun. 57(3), 602–605 (2009).
[Crossref]

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

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

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

F. Xu, A. Khalighi, P. Caussé, and S. Bourennane, “Channel coding and time-diversity for optical wireless links,” Opt. Express 17(2), 872–887 (2009).
[Crossref] [PubMed]

2008 (2)

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
[Crossref]

C. H. Kwok, R. V. Penty, and I. H. White, “Link Reliability Improvement for Optical Wireless Communication Systems with Temporal-Domain Diversity Reception,” IEEE Photon. Technol. Lett. 20(9), 700–702 (2008).
[Crossref]

2007 (4)

A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient lognormal channel model for turbulent FSO communications,” IEE Electronics Letters 43(3), 178–179 (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(8), 2813–2819 (2007).
[Crossref]

A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett. 11(5), 390–392 (2007).
[Crossref]

I. B. Djordjevic, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation,” Opt. Express 15(16), 10,026–10,032 (2007).
[Crossref]

2006 (1)

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(6), 1229–1233 (2006).
[Crossref]

2005 (4)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[Crossref]

M. Simon and V. Vilnrotter, “Alamouti-Type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Characterization of delayed diversity optical wireless system to mitigate atmospheric turbulence induced fading,” in Proc. SPIE, pp. 589,215.1–589,215.10 (2005).

2004 (1)

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Delayed diversity for fade resistance in optical wireless communication system through simulated turbulence,” in Proc. SPIE, pp. 385–394 (2004).
[Crossref]

2003 (3)

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Inf. Theory 49(6), 1385–1399 (2003).
[Crossref]

J. Li and M. Uysal, “Achievable information rate for outdoor free space optical communication with intensity modulation and direct detection,” in Proc. IEEE Global Telecommunications Conference GLOBECOM ’03, vol.  5, pp. 2654–2658 (2003).

A. Garcia-Zambrana and A. Puerta-Notario, “Novel approach for increasing the peak-to-average optical power ratio in rate-adaptive optical wireless communication systems,” IEE Proceedings -Optoelectronics 150(5), 439–444 (2003).
[Crossref]

2002 (1)

X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

2001 (2)

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,” Optical Engineering 40, 8 (2001).
[Crossref]

A. García-Zambrana and A. Puerta-Notario, “Large change rate-adaptive indoor wireless infrared links using variable silence periods,” IEE Electronics Letters 37(8), 524–525 (2001).
[Crossref]

1999 (1)

A. García-Zambrana and A. Puerta-Notario, “RZ-Gaussian pulses reduce the receiver complexity in wireless infrared links at high bit rates,” IEE Electronics Letters 35(13), 1059–1061 (1999).
[Crossref]

1998 (1)

1997 (1)

J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

1989 (1)

A. A. Ali and I. A. Al-Kadi, “On the use of repetition coding with binary digital modulations on mobile channels,” IEEE Trans. Veh. Technol. 38(1), 14–18 (1989).
[Crossref]

1975 (1)

R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc. IEEE 63(12), 1669–1692 (1975).
[Crossref]

Abou-Rjeily, C.

C. Abou-Rjeily, “Orthogonal Space-Time Block Codes for Binary Pulse Position Modulation,” IEEE Trans. Commun. 57(3), 602–605 (2009).
[Crossref]

Adamchik, V. S.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. on Symbolic and Algebraic Computation, pp. 212–224 (Tokyo, Japan, 1990).

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Optical Engineering 40, 8 (2001).
[Crossref]

Ali, A. A.

A. A. Ali and I. A. Al-Kadi, “On the use of repetition coding with binary digital modulations on mobile channels,” IEEE Trans. Veh. Technol. 38(1), 14–18 (1989).
[Crossref]

Al-Kadi, I. A.

A. A. Ali and I. A. Al-Kadi, “On the use of repetition coding with binary digital modulations on mobile channels,” IEEE Trans. Veh. Technol. 38(1), 14–18 (1989).
[Crossref]

Alouini, M.-S.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, WA: SPIE Press, 2001).
[Crossref]

Andrews, L. C.

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

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Optical Engineering 40, 8 (2001).
[Crossref]

C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-Arrival Fluctuations of a Space–Time Gaussian Pulse in Weak Optical Turbulence: an Analytic Solution,” Appl. Opt. 37(33), 7655–7660 (1998).
[Crossref]

Anguita, J.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
[Crossref]

J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[Crossref]

Balakrishnan, N.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd ed. (Wiley Series in Probability and Statistics, 1994).

Barry, J. R.

J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

Bayaki, E.

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

Bourennane, S.

Brandt-Pearce, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Castillo-Vazquez, B.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

Castillo-Vazquez, C.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

Castillo-Vázquez, B.

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18( 19), 20,445–20,454 (2010).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “On the Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels Using OOK Signaling,” EURASIP Journal on Wireless Communications and Networking 2010. Article ID 127657, 9 pages, 2010. .
[Crossref]

Castillo-Vázquez, C.

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18( 19), 20,445–20,454 (2010).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “On the Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels Using OOK Signaling,” EURASIP Journal on Wireless Communications and Networking 2010. Article ID 127657, 9 pages, 2010. .
[Crossref]

Castillo-Vázquez, M.

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “An efficient rate-adaptive transmission technique using shortened pulses for atmospheric optical communications,” Opt. Express 18( 16), 17,346–17,363 (2010).
[Crossref]

Caussé, P.

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the Distribution of the Sum of Gamma-Gamma Variates and Application in MIMO Optical Wireless Systems,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM 2009, pp. 1–6 (2009).
[Crossref]

Cherry, P. C.

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

Davis, C. C.

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Characterization of delayed diversity optical wireless system to mitigate atmospheric turbulence induced fading,” in Proc. SPIE, pp. 589,215.1–589,215.10 (2005).

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Delayed diversity for fade resistance in optical wireless communication system through simulated turbulence,” in Proc. SPIE, pp. 385–394 (2004).
[Crossref]

Denic, S.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
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Diels, J. C.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Optics and Photonics Series, 2nd ed. (Academic Press, 2006).

Djordjevic, G. T.

I. B. Djordjevic and G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17(20), 18,250–18,262 (2009).
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Djordjevic, I.

Djordjevic, I. B.

I. B. Djordjevic, “Adaptive Modulation and Coding for Free-Space Optical Channels,” IEEE/OSA Journal of Optical Communications and Networking 2(5), 221–229 (2010).
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I. B. Djordjevic and G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17(20), 18,250–18,262 (2009).
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I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
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I. B. Djordjevic, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation,” Opt. Express 15(16), 10,026–10,032 (2007).
[Crossref]

Fafalios, M. E.

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
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R. L. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc. IEEE 63(12), 1669–1692 (1975).
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Foshee, J. J.

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

Garcia-Zambrana, A.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient lognormal channel model for turbulent FSO communications,” IEE Electronics Letters 43(3), 178–179 (2007).
[Crossref]

A. Garcia-Zambrana and A. Puerta-Notario, “Novel approach for increasing the peak-to-average optical power ratio in rate-adaptive optical wireless communication systems,” IEE Proceedings -Optoelectronics 150(5), 439–444 (2003).
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García-Zambrana, A.

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18( 19), 20,445–20,454 (2010).
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A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
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A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett. 11(5), 390–392 (2007).
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A. García-Zambrana and A. Puerta-Notario, “Large change rate-adaptive indoor wireless infrared links using variable silence periods,” IEE Electronics Letters 37(8), 524–525 (2001).
[Crossref]

A. García-Zambrana and A. Puerta-Notario, “RZ-Gaussian pulses reduce the receiver complexity in wireless infrared links at high bit rates,” IEE Electronics Letters 35(13), 1059–1061 (1999).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “On the Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels Using OOK Signaling,” EURASIP Journal on Wireless Communications and Networking 2010. Article ID 127657, 9 pages, 2010. .
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Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “An efficient rate-adaptive transmission technique using shortened pulses for atmospheric optical communications,” Opt. Express 18( 16), 17,346–17,363 (2010).
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Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Hiniesta-Gomez, A.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
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Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, WA: SPIE Press, 2001).
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Hranilovic, S.

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Inf. Theory 49(6), 1385–1399 (2003).
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Ishimaru, A.

J. H.,

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
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Johnson, N. L.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd ed. (Wiley Series in Probability and Statistics, 1994).

Jurado-Navas, A.

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “An efficient rate-adaptive transmission technique using shortened pulses for atmospheric optical communications,” Opt. Express 18( 16), 17,346–17,363 (2010).
[Crossref]

A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient lognormal channel model for turbulent FSO communications,” IEE Electronics Letters 43(3), 178–179 (2007).
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Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
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J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” Proc. IEEE 85, 265–298 (1997).
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Karagianni, E. A.

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
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Karagiannidis, G. K.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the Distribution of the Sum of Gamma-Gamma Variates and Application in MIMO Optical Wireless Systems,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM 2009, pp. 1–6 (2009).
[Crossref]

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

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(8), 2813–2819 (2007).
[Crossref]

Khalighi, A.

Kim, K.

Kolodzy, P. J.

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

Kotz, S.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd ed. (Wiley Series in Probability and Statistics, 1994).

Kschischang, F. R.

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Inf. Theory 49(6), 1385–1399 (2003).
[Crossref]

Kwok, C. H.

C. H. Kwok, R. V. Penty, and I. H. White, “Link Reliability Improvement for Optical Wireless Communication Systems with Temporal-Domain Diversity Reception,” IEEE Photon. Technol. Lett. 20(9), 700–702 (2008).
[Crossref]

Leveque, I.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Li, J.

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(6), 1229–1233 (2006).
[Crossref]

J. Li and M. Uysal, “Achievable information rate for outdoor free space optical communication with intensity modulation and direct detection,” in Proc. IEEE Global Telecommunications Conference GLOBECOM ’03, vol.  5, pp. 2654–2658 (2003).

Lim, W.

Marichev, O. I.

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. on Symbolic and Algebraic Computation, pp. 212–224 (Tokyo, Japan, 1990).

McIntire, W. K.

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

Michalopoulos, D. S.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the Distribution of the Sum of Gamma-Gamma Variates and Application in MIMO Optical Wireless Systems,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM 2009, pp. 1–6 (2009).
[Crossref]

Milner, S. D.

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Characterization of delayed diversity optical wireless system to mitigate atmospheric turbulence induced fading,” in Proc. SPIE, pp. 589,215.1–589,215.10 (2005).

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Delayed diversity for fade resistance in optical wireless communication system through simulated turbulence,” in Proc. SPIE, pp. 385–394 (2004).
[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(8), 2813–2819 (2007).
[Crossref]

Neifeld, M.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
[Crossref]

J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[Crossref]

Nistazakis, H. E.

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
[Crossref]

Northcott, M.

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

Penty, R. V.

C. H. Kwok, R. V. Penty, and I. H. White, “Link Reliability Improvement for Optical Wireless Communication Systems with Temporal-Domain Diversity Reception,” IEEE Photon. Technol. Lett. 20(9), 700–702 (2008).
[Crossref]

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, WA: SPIE Press, 2001).
[Crossref]

Phillips, R. L.

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

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Optical Engineering 40, 8 (2001).
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Pike, H. A.

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

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “An efficient rate-adaptive transmission technique using shortened pulses for atmospheric optical communications,” Opt. Express 18( 16), 17,346–17,363 (2010).
[Crossref]

A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient lognormal channel model for turbulent FSO communications,” IEE Electronics Letters 43(3), 178–179 (2007).
[Crossref]

A. Garcia-Zambrana and A. Puerta-Notario, “Novel approach for increasing the peak-to-average optical power ratio in rate-adaptive optical wireless communication systems,” IEE Proceedings -Optoelectronics 150(5), 439–444 (2003).
[Crossref]

A. García-Zambrana and A. Puerta-Notario, “Large change rate-adaptive indoor wireless infrared links using variable silence periods,” IEE Electronics Letters 37(8), 524–525 (2001).
[Crossref]

A. García-Zambrana and A. Puerta-Notario, “RZ-Gaussian pulses reduce the receiver complexity in wireless infrared links at high bit rates,” IEE Electronics Letters 35(13), 1059–1061 (1999).
[Crossref]

Rudolph, W.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Optics and Photonics Series, 2nd ed. (Academic Press, 2006).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Sandalidis, H. G.

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

Schober, R.

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

Simon, M.

M. Simon and V. Vilnrotter, “Alamouti-Type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

Smolyaninov, I. I.

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Characterization of delayed diversity optical wireless system to mitigate atmospheric turbulence induced fading,” in Proc. SPIE, pp. 589,215.1–589,215.10 (2005).

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Delayed diversity for fade resistance in optical wireless communication system through simulated turbulence,” in Proc. SPIE, pp. 385–394 (2004).
[Crossref]

Stadler, B.

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

Stotts, L. B.

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

Tombras, G. S.

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
[Crossref]

Trisno, S.

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Characterization of delayed diversity optical wireless system to mitigate atmospheric turbulence induced fading,” in Proc. SPIE, pp. 589,215.1–589,215.10 (2005).

S. Trisno, I. I. Smolyaninov, S. D. Milner, and C. C. Davis, “Delayed diversity for fade resistance in optical wireless communication system through simulated turbulence,” in Proc. SPIE, pp. 385–394 (2004).
[Crossref]

Tsiftsis, T. A.

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

Tsigopoulos, A. D.

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” IEEE/OSA Journal of Lightwave Technology 27(8), 974–979 (2009).
[Crossref]

Uysal, M.

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

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER Performance of Free-Space Optical Transmission with Spatial Diversity,” IEEE Trans. Wireless Commun. 6(8), 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(6), 1229–1233 (2006).
[Crossref]

J. Li and M. Uysal, “Achievable information rate for outdoor free space optical communication with intensity modulation and direct detection,” in Proc. IEEE Global Telecommunications Conference GLOBECOM ’03, vol.  5, pp. 2654–2658 (2003).

Vasic, B.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
[Crossref]

J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005).
[Crossref]

Vilnrotter, V.

M. Simon and V. Vilnrotter, “Alamouti-Type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

White, I. H.

C. H. Kwok, R. V. Penty, and I. H. White, “Link Reliability Improvement for Optical Wireless Communication Systems with Temporal-Domain Diversity Reception,” IEEE Photon. Technol. Lett. 20(9), 700–702 (2008).
[Crossref]

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Xu, F.

Young, C. Y.

Young, D. W.

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

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(6), 1229–1233 (2006).
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Yun, C.

Zhu, X.

X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
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Appl. Opt. (1)

IEE Electronics Letters (3)

A. García-Zambrana and A. Puerta-Notario, “Large change rate-adaptive indoor wireless infrared links using variable silence periods,” IEE Electronics Letters 37(8), 524–525 (2001).
[Crossref]

A. García-Zambrana and A. Puerta-Notario, “RZ-Gaussian pulses reduce the receiver complexity in wireless infrared links at high bit rates,” IEE Electronics Letters 35(13), 1059–1061 (1999).
[Crossref]

A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient lognormal channel model for turbulent FSO communications,” IEE Electronics Letters 43(3), 178–179 (2007).
[Crossref]

IEE Proceedings -Optoelectronics (1)

A. Garcia-Zambrana and A. Puerta-Notario, “Novel approach for increasing the peak-to-average optical power ratio in rate-adaptive optical wireless communication systems,” IEE Proceedings -Optoelectronics 150(5), 439–444 (2003).
[Crossref]

IEEE Commun. Lett. (1)

A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett. 11(5), 390–392 (2007).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

C. H. Kwok, R. V. Penty, and I. H. White, “Link Reliability Improvement for Optical Wireless Communication Systems with Temporal-Domain Diversity Reception,” IEEE Photon. Technol. Lett. 20(9), 700–702 (2008).
[Crossref]

IEEE Trans. Commun. (4)

C. Abou-Rjeily, “Orthogonal Space-Time Block Codes for Binary Pulse Position Modulation,” IEEE Trans. Commun. 57(3), 602–605 (2009).
[Crossref]

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

S. G. Wilson, M. Brandt-Pearce, Q. Cao, I. Leveque, and J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

IEEE Trans. Inf. Theory (1)

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Inf. Theory 49(6), 1385–1399 (2003).
[Crossref]

IEEE Trans. Veh. Technol. (1)

A. A. Ali and I. A. Al-Kadi, “On the use of repetition coding with binary digital modulations on mobile channels,” IEEE Trans. Veh. Technol. 38(1), 14–18 (1989).
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IEEE Trans. Wireless Commun. (4)

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(6), 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(8), 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(2), 951–957 (2009).
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IEEE/OSA Journal of Lightwave Technology (2)

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE/OSA Journal of Lightwave Technology 26(5), 478–487 (2008).
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IEEE/OSA Journal of Optical Communications and Networking (1)

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Figures (4)

Fig. 1
Fig. 1

Performance comparison of different rate-adaptive transmission schemes in FSO IM/DD links over the gamma-gamma atmospheric turbulence channel when different levels of turbulence (a) (α,β) = (4, 3) and (b) (α,β) = (4, 1) are assumed, corresponding to values of scintillation index of SI = 0.66 and SI = 1.5, respectively.

Fig. 2
Fig. 2

Performance comparison of different rate-adaptive transmission schemes in FSO IM/DD links over the exponential atmospheric turbulence channel, corresponding to a value of scintillation index of SI = 1.

Fig. 3
Fig. 3

(a) Performance of the sech2 pulse shape with κ = 0.25 for the rate-adaptive scheme (Rep&Sil) with TDO = 2 and RR = {1, 2, 4, 8} in FSO IM/DD links over the exponential atmospheric turbulence channel; (b) Performance of the rate-adaptive scheme (Rep&Sil) here proposed with TDO = 2, rectangular pulse shape with κ = 1 and RR = {1, 2, 4, 8} in FSO IM/DD links over the gamma-gamma atmospheric turbulence channel with analytical results in Eq. (20) when different levels of turbulence are assumed, together with the analytical results in Eq. (24), corresponding to the negative exponential distributed turbulence model.

Fig. 4
Fig. 4

Achievable information rate corresponding to the rate-adaptive transmission schemes Sil, Rep and Rep&Sil for two different target BER requirements, (a) Pb = 10−4 and (b) Pb = 10−8, and time-diversity orders of TDO={2, 4}, together with the ergodic capacity for the exponential atmospheric turbulent optical channel.

Equations (31)

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y ( t ) = η i ( t ) x ( t ) + z ( t )
f I ( i ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) μ ( α + β ) ) / 2 i ( ( α + β ) / 2 ) 1 K α β ( 2 α β i μ ) , i 0
α = [ exp ( 0.49 χ 2 ( 1 + 0.18 d 2 + 0.56 χ 12 / 5 ) 7 / 6 ) 1 ] 1
β = [ exp ( 0.51 χ 2 ( 1 + 0.69 χ 12 / 5 ) 5 / 6 ( 1 + 0.9 d 2 + 0.62 d 2 χ 12 / 5 ) 7 / 6 ) 1 ] 1
SI = E [ I 2 ] ( E [ I ] ) 2 1 = 1 α + 1 β + 1 α β .
f I ( i ) = exp ( i ) , i 0
x ( t ) = k = a k 2 T b P G ( f = 0 ) g ( t k T b )
x R R s ( t ) = k = a k R R s 2 T b P G ( f = 0 ) g ( t kR R s T b )
x R R rc ( t ) = k = a k 2 T b P G ( f = 0 ) l = 0 R R rc 1 g ( t l T b kR R rc T b )
x R R rcs ( t ) = k = a k R R s 2 T b P G ( f = 0 ) l = 0 R R rc 1 g ( t lR R s T b kRR T b )
P b s ( E | I ) = Q ( R R s 2 d 2 i 2 / 2 N 0 )
P b s ( E ) = 0 Q ( 2 γ ξ Ω c s i ) f I ( i ) d i .
P b r c ( E | { I k } k = 1 R R r c ) = Q ( d 2 2 R R r c N 0 k = 1 R R r c i k ) = Q ( 2 γ ξ R R r c k = 1 R R r c i k ) .
P b rc ( E | { I k } k = 1 TDO ) = Q ( 2 γ ξ R R r c R R r c TDO k = 1 TDO i k ) .
P b r c ( E | { I k } k = 1 Ω d r c ) = Q ( 2 γ ξ Ω c r c k = 1 Ω d r c i k )
P b r c ( E ) = 0 0 0 Ω d r c - fold Q ( 2 γ ξ Ω c r c k = 1 Ω d r c i k ) k = 1 Ω d r c f I k ( i k ) d i 1 d i 2 d i Ω d r c .
P b rcs ( E | { I k } k = 1 Ω d rcs ) = Q ( 2 γ ξ Ω c rcs k = 1 Ω d rcs i k )
P b r c s ( E ) = 0 0 0 Ω d rcs - fold Q ( 2 γ ξ Ω c r c s k = 1 Ω d r c s i k ) k = 1 Ω d r c s f I k ( i k ) d i 1 d i 2 d i Ω d rcs .
P b ( E ) = 0 Q ( 2 γ ξ Ω c i ) f I T ( i ) d i
P b ( E ) = ( α T β T ) α T + β T 2 Ω d α T + β T 2 ( Ω c 2 γ ξ ) α T + β T 4 8 π 3 / 2 Γ ( α T ) Γ ( β T ) × G 2 , 5 4 , 2 ( α T 2 β T 2 16 Ω c 2 Ω d 2 γ ξ | α T β T + 2 4 , α T β T + 4 4 α T β T 4 , α T β T + 2 4 , β T α T 4 , α T + β T + 2 4 , α T + β T 4 ) .
P b ( E ) ( ( Γ ( ( Ω d β + 1 ) α β + 1 Ω d β ) Ω d β Γ ( Ω d β 2 ) Γ ( ( Ω d β + 1 ) α β + 1 ) ) 1 / ( Ω d β ) 2 ( β + 1 ) Ω c α β ( Ω d β + 1 ) γ ξ ) Ω d β .
A [ d B ] G c rcs [ d B ] G c r c [ d B ] = 10 log 10 ( Ω c r c s / Ω c r c ) = 5 log 10 ( R R / TDO )
f I T ( i ) = i Ω d 1 Γ ( Ω d ) exp ( i ) , i 0 .
P b ( E ) = 2 Ω d 3 Ω c Ω d 1 ( γ ξ ) Ω d 2 1 2 × ( 4 Ω c Γ ( Ω d + 2 2 ) γ ξ 2 F 2 ( Ω d + 1 2 , Ω d 2 ; 1 2 , Ω d + 2 2 ; 1 4 Ω c 2 γ ξ ) 2 Ω d Γ ( Ω d + 3 2 ) 2 F 2 ( Ω d + 1 2 , Ω d + 2 2 ; 3 2 , Ω d + 3 2 ; 1 4 Ω c 2 γ ξ ) )
P b ( E ) ( ( Ω d Γ ( Ω d 2 ) ) 1 / Ω d 2 Ω c γ ξ ) Ω d .
R Sil ( γ ) = 2 π P b γ ξ
R Rep ( γ ) = { ( TDO P b 2 TDO Γ ( TDO / 2 ) ( γ ξ ) TDO / 2 ) 2 / TDO , if R R TDO ; TDO 2 ( TDO P b 2 TDO Γ ( TDO / 2 ) ( γ ξ ) TDO / 2 ) 2 / TDO , if R R > TDO ;
R Rep & Sil ( γ ) = { ( TDO P b 2 TDO Γ ( TDO / 2 ) ( γ ξ ) TDO / 2 ) 2 / TDO , if R R TDO ; TDO 3 / 2 ( TDO P b 2 TDO Γ ( TDO / 2 ) ( γ ξ ) TDO / 2 ) 1 / TDO , if R R > TDO ;
Y = AXI + Z , X { 0 , 1 } , Z N ( 0 , 1 )
I ( X ; Y | i ) = x = 0 1 P X ( x ) f Y ( y | x , i ) log 2 ( f Y ( y | x , i ) Σ r = 0 , 1 P X ( r ) f Y ( y | x = r , i ) d y
C ( γ ) = 0 I ( X ; Y | i ) f I ( i ) d i .

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