## Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels |

Optics Express, Vol. 18, Issue 19, pp. 20445-20454 (2010)

http://dx.doi.org/10.1364/OE.18.020445

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### Abstract

A new upper bound on the capacity of power- and bandwidth-constrained optical wireless links using selection transmit diversity over exponential atmospheric turbulence channels with intensity modulation and direct detection is derived when non-uniform on-off keying (OOK) formats are used. In this strong turbulence free-space optical (FSO) scenario, average capacity is investigated subject to an average optical power constraint and not only to an average electrical power constraint when the transmit diversity technique assumed is based on the selection of the optical path with a greater value of irradiance. Simulation results for the mutual information are further demonstrated to confirm the analytical results for different diversity orders.

© 2010 Optical Society of America

## 1. Introduction

*last mile*”

*problem*, above all in densely populated urban areas. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as

*atmospheric scintillation*, severely degrading the link performance [1

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

2. 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]

3. 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]

6. H. G. Sandalidis and T. A. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. **44**(1), 46–47 (2008). [CrossRef]

7. 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,” J. Lightwave Technol. **27**(8), 974–979 (2009). [CrossRef]

## 2. System and channel model

*L*laser sources, assumed to be intensity-modulated only and all pointed towards a distant photodetector, and to be ideal noncoherent (direct-detection) receiver. The sources and the detector are physically situated so that all transmitters are simultaneously observed by the receiver. Additionaly, the fading experienced between source-detector pairs

*I*is assumed to be statistically independent. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [3

_{j}3. 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]

*I*, can be written as

_{m}*y*(

*t*), can be written as

*η*is the detector responsivity, assumed hereinafter to be the unity,

*X*≜

*x*(

*t*) represents the optical power supplied by the source,

*I*≜

_{m}*i*(

_{m}*t*) the equivalent real-valued fading gain (irradiance), and

*h*(

*t*) the impulse response of an ideal low-pass filter, which cuts out all frequencies greater than

*W*hertz, modelling the fact that these systems are intrinsically bandwidth limited due to the use of large inexpensive optoelectronic components;

*Z*≜

*z*(

*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance

*σ*

^{2}=

*N*

_{0}/2, i.e.

*Z*~

*N*(0,

*N*

_{0}/2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity,

*X*must satisfy ∀

*t*

*x*(

*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of

*X*is limited. The received electrical signal

*Y*≜

*y*(

*t*), however, can assume negative amplitude values. We use

*Y*,

*X*,

*I*and

_{m}*Z*to denote random variables and

*y*(

*t*),

*x*(

*t*),

*i*(

_{m}*t*) and

*z*(

*t*) their corresponding realizations.

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

4. B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER Expression for FSO Links with Transmit Laser Selection over Exponential Atmospheric Turbulence Channels,” Electron. Lett. **45**
(23), 1185–1187 (2009). [CrossRef]

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

*E*[

*I*] = 1 and the second moment is given by

_{j}*E*[

*I*

^{2}

_{j}] = 2, the scintillation index (

*SI*), a parameter of interest used to describe the strength of atmospheric fading

_{j}*I*experienced between source-detector pairs [1

_{j}1. L. Andrews, R. Phillips, and C. Hopen, *Laser Beam Scintillation with Applications* (Bellingham, WA: SPIE Press, 2001). [CrossRef]

*SI*=

_{j}*E*[

*I*

^{2}

_{j}]/(

*E*[

*I*])

_{j}^{2}− 1 = 1. According to [4], for i.i.d. random variables of {

*I*}

_{j}_{j=1,2,…L}, the PDF,

*I*, is

_{m}*ϕ*(

*t*) is defined as

*g*(

*t*) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤

*g*(

*t*) ≤ 1 in the bit period and 0 otherwise, and

*E*=

_{g}*∫*

^{∞}

_{−∞}

*g*

^{2}(

*t*)

*dt*is the electrical energy. In this way, an expression for the optical intensity can be written as

*G*(

*f*= 0) represents the Fourier transform of

*g*(

*t*) evaluated at frequency

*f*= 0, i.e. the area of the employed pulse shape, and

*T*parameter is the bit period. The random variable

_{b}*a*follows a Bernoulli distribution with parameter

_{k}*p*, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is

*P*

_{opt}. The constellation here defined for the OOK format using any pulse shape consists of two points (

*x*

_{0}= 0 and

*x*

_{1}=

*d*) in a one-dimensional space with an Euclidean distance of

*ξ*=

*T*/

_{b}E_{g}*G*

^{2}(

*f*= 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse. This parameter

*d*represents the distance between the two possible transmitted signals and, hence, in this case, the square root of the energy of the baseband signal waveform corresponding to the bit “1” (on pulse) [15, Chapter 3]. Assuming

*h*(

*t*) as the impulse response of an ideal low-pass filter, which cuts out all frequencies greater than

*W*hertz, and the use of a matched filter at the receiver, as in [15, Chapter 6] or [16, Chapter 9], the electrical power of

*X*

_{rx}, random variable corresponding to the signal at the detector output, conditioned to the irradiance, can be written as

*P*

_{el}=

*pd*

^{2}

*i*

^{2}

*= (1/*

_{m}θ*p*)

*P*

^{2}

_{opt}i^{2}

*where*

_{m}T_{b}ξθ*θ*is obtained from

*θ*< 1, representing the fact that the channel under study is constrained to

*κ*= 2

*WT*degrees of freedom. The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fast fading statistics. In spite of scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, this approach is valid because temporal correlation can in practice be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [5–7

_{b}5. 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]

## 3. Upper bound on channel capacity

17. A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory **43**(6), 1986–1992 (1997). [CrossRef]

*P*

_{opt}, of the conditional mutual information between the input and output,

*I*(

*X*;

*Y*∣

*i*), averaged over the PDF corresponding to the equivalent turbulence model. It must be noted that unlike the approach followed in [6

_{m}6. H. G. Sandalidis and T. A. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. **44**(1), 46–47 (2008). [CrossRef]

7. 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,” J. Lightwave Technol. **27**(8), 974–979 (2009). [CrossRef]

17. A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory **43**(6), 1986–1992 (1997). [CrossRef]

21. 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]

*E*[

*X*

^{2}

_{rx}], the second moment of

*X*

_{rx}, takes a value of up to

*P*

_{el}. Additionally, in our channel model, assuming an unidimensional space where the non-negativity constraint is satisfied and one of the two points of the constellation takes the value of 0, it is easy to deduce that an average electrical power constraint of

*P*

_{el}, and, hence,

*E*[

*X*

^{2}] ≤

*P*

_{el}/(

*i*

^{2}

*), implies an Euclidean distance as*

_{m}θ*P*

_{opt}. Thus, an average electrical power constraint of

*P*

_{el}is necessary and sufficient condition for satisfying an average optical power constraint of

*P*

_{opt}. This is only valid for OOK signaling, representing the basis of our work in order to achieve a tighter performance if compared with previously reported bounds. In relation to the equivalent discrete-time channel, it must be emphasized that the transmitted optical signal is represented by the random variable

*X*, the atmospheric turbulence-induced signal is represented by the product

*XI*and the corresponding signal performed in electrical domain is represented by

_{m}*X*

_{rx}, being the latter the signal to be finally considered in our analysis. As presented in [8], applying the fact that the Gaussian distribution maximizes the entropy over all distributions with the same variance [16, Theorem 8.6.5], we obtain

*N*/2 watts/Hz and average optical power constraint of

_{o}*P*

_{opt}watts. In this way, assuming that the channel is constrained to

*κ*dimensions and even without maximizing over the input distribution, the channel capacity

*C*(

*γ*,

*p*), channel capacity depending on SNR and input distribution

*p*, can be obtained by averaging over the PDF in (4) as follows

25. S. Hranilovic and F. R. Kschischang, “Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise,” IEEE Trans. Inf. Theory **50**(5), 784–795 (2004). [CrossRef]

6. H. G. Sandalidis and T. A. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. **44**(1), 46–47 (2008). [CrossRef]

7. 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,” J. Lightwave Technol. **27**(8), 974–979 (2009). [CrossRef]

20. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Communications **3**(8), 1402–1409 (2009). [CrossRef]

26. S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. A. Neifeld, “Information Theoretic Limits for Free-Space Optical Channels With and Without Memory,” J. Lightwave Technol. **26**(19), 3376–3384 (2008). [CrossRef]

*H*(

_{B}*p*) = −

*p*log

_{2}

*p*− (1 −

*p*) log

_{2}(1 −

*p*) represents the entropy of the Bernoulli random variable

*a*in (5), presenting the maximum value achievable because of OOK is the signaling technique considered in this analysis. After a simple transformation of the random variable

_{k}*I*as

_{m}*I*=

_{n}*I*(

_{m}h*p,γ*), being

*C*(

*γ, p*) as

*C*(

*γ, p*) is also upper bounded by the binary entropy

*H*(

_{B}*p*), the ergodic capacity in bits per channel use is obtained by maximizing

*C*(

*γ, p*) over the parameter

*p*as

## 4. Numerical results and conclusions

*Y*by

*Y*/

*σ*and now considering

*X*∈ {0,1}. In this way, our channel model can be rewritten as

*I*(

*X*;

*Y*∣

*i*) for this channel is derived as in [8] as follows

_{m}*P*(

_{X}*x*= 1) =

*p*,

*P*(

_{X}*x*= 0) = 1 −

*p*,

*I*(

*X*;

*Y*), function depending on SNR and input distribution

*p*, can be numerically obtained by averaging (15) over the PDF in (4) as follows

*p*as

*C*

_{1}(

*γ*), and mutual information, i.e.

*C*

_{2}(

*γ*), for the exponential atmospheric turbulent optical channel and non-turbulent optical channel are displayed for different diversity orders. It must be commented that the mutual information for the non-turbulent optical channel is numerically solved in a similar way as in (15) but not yet considering the impact of the atmospheric turbulence. In Fig. 1, a value of

*κ*= 20 has been considered and, hence, a value of

*θ*= 0.9898 when using a rectangular pulse of duration

*T*has been obtained from (7). For this rectangular pulse shape, the integral in (7) can be written as

_{b}*f*as

*q*=

*πT*and using integration by parts, it is easy to deduce that

_{b}f*θ*= (2

*πκ*si(

*πκ*) + 2cos(

*πκ*) +

*π*

^{2}

*κ*− 2)/(

*π*

^{2}

*κ*). As a result, a relevant improvement in average capacity for IM/DD exponential atmospheric turbulence FSO links is obtained when a transmit diversity technique based on the selection of the optical path with a greater value of irradiance is adopted, showing the fact that a non-uniform input signaling improves the channel capacity.

*p*for different values of SNR and diversity orders is displayed. From this figure it can be deduced the fact that a non-uniform input signaling improves the channel capacity, especially at low SNR [23], depending the maximizing input distribution on the SNR and the diversity order corresponding to the transmit laser selection scheme here analyzed.

*L*≥ 4 is assumed. This can be justified from the greater robustness provided by the transmit laser selection scheme here studied against fluctuations in the irradiance of the transmitted optical beam proper to the atmospheric turbulence. A better comprehension of this fact can be achieved by calculating the equivalent scintillation index (

*SI*), used to describe the strength of the equivalent fading gain

_{m}*I*and, hence, defined as

_{m}*SI*=

_{m}*E*[

*I*

^{2}

_{m}]/(

*E*[

*I*])

_{m}^{2}− 1, where

*E*[

*I*] and

_{m}*E*[

*I*

^{2}

_{m}] represent the mean value and the second moment of the equivalent turbulence model configured by the MISO system under study, respectively. The mean value of the equivalent irradiance,

*E*[

*I*], is obtained as follows

_{m}*E*[

*I*

^{2}

_{m}], is also obtained as follows

*E*[

*I*]) and scintillation index (

_{m}*SI*) of the equivalent turbulence model configured by the MISO system under study versus the number of laser sources

_{m}*L*are displayed.

*L*≥ 4 if compared with the non-turbulent case can be explained from the decreasing strength of the equivalent fading gain as

*L*is increased together with the fact that the mean value of the equivalent fading gain and, hence, the resulting SNR has been improved more than double. This can also explain that the parameter

*p*tends the value of 0.5 as

*L*is increased, as shown in Fig. 2. In this fashion, since a non-uniform input signaling improves the channel capacity especially at low SNR [8, 23], the increasing mean value of the equivalent fading gain implies a higher and higher equivalent SNR and, hence, the fact that the maximizing input distribution tends to be closer and closer to uniform signaling.

*ξ*in (6), expression corresponding to the Euclidean distance in the constellation here defined for the OOK format, it is shown that a relevant improvement in performance must be noted as a consequence of the pulse shape used [8], concluding that an increase of the PAOPR provides higher capacity values. From the relevant improvement in terms of capacity here obtained when using non-uniform signaling OOK and the transmit diversity technique based on the selection of the optical path with a greater value of irradiance is adopted over exponential atmospheric turbulence channels, investigating the performance in alternative FSO scenarios covering a wider range of atmospheric turbulence conditions as well as incorporating pointing error effects are interesting topics for future research.

## Acknowledgments

## References and links

1. | L. Andrews, R. Phillips, and C. Hopen, |

2. | 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. |

3. | 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. |

4. | B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER Expression for FSO Links with Transmit Laser Selection over Exponential Atmospheric Turbulence Channels,” Electron. Lett. |

5. | 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 |

6. | H. G. Sandalidis and T. A. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. |

7. | 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,” J. Lightwave Technol. |

8. | 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 Networking2010. Article ID 127657, 9 pages, 2010. doi:10.1155/2010/127657. |

9. | 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. |

10. | M. Simon and V. Vilnrotter, “Alamouti-Type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. |

11. | C. Abou-Rjeily and W. Fawaz, “Space-Time Codes for MIMO Ultra-Wideband Communications and MIMO Free-Space Optical Communications with PPM,” IEEE J. Sel. Areas Commun. |

12. | W. O. Popoola and Z. Ghassemlooy, “BPSK Subcarrier Intensity Modulated Free-Space Optical Communications in Atmospheric Turbulence,” J. Lightwave Technol. |

13. | N. Letzepis, K. Nguyen, A. Guillen i Fabregas, and W. Cowley, “Outage analysis of the hybrid free-space optical and radio-frequency channel,” IEEE J. Sel. Areas Commun. |

14. | K. Davaslioğlu, E. Çağiral, and M. Koca, “Free space optical ultra-wideband communications over atmospheric turbulence channels,” Opt. Express |

15. | U. Madhow, |

16. | T. M. Cover and J. A. Thomas, |

17. | A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory |

18. | J. Li and M. Uysal, “Optical wireless communications: system model, capacity and coding,” in |

19. | J. Li and M. Uysal, “Achievable information rate for outdoor free space optical communication with intensity modulation and direct detection,” in |

20. | H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Communications |

21. | J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. |

22. | A. A. Farid and S. Hranilovic, “Design of non-uniform capacity-approaching signaling for optical wireless intensity channels,” in |

23. | A. A. Farid and S. Hranilovic, “Outage capacity with non-uniform signaling for free-space optical channels,” in |

24. | A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun. |

25. | S. Hranilovic and F. R. Kschischang, “Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise,” IEEE Trans. Inf. Theory |

26. | S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. A. Neifeld, “Information Theoretic Limits for Free-Space Optical Channels With and Without Memory,” J. Lightwave Technol. |

27. | I. S. Gradshteyn and I. M. Ryzhik, |

28. |
Wolfram Research, Inc., |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

(060.2605) Fiber optics and optical communications : Free-space optical communication

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 6, 2010

Revised Manuscript: August 26, 2010

Manuscript Accepted: September 3, 2010

Published: September 10, 2010

**Citation**

Antonio García-Zambrana, Beatriz Castillo-Vázquez, and Carmen Castillo-Vázquez, "Average capacity of FSO links with transmit laser selection using non-uniform OOK
signaling over exponential atmospheric turbulence channels," Opt. Express **18**, 20445-20454 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20445

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### References

- L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [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. Wirel. Comm. 8(2), 951–957 (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]
- B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009). [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]
- H. G. Sandalidis, and T. A. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008). [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,” J. Lightwave Technol. 27(8), 974–979 (2009). [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 J. Wireless Commun. Networking 2010), Article ID 127657, 9 pages doi:10.1155/2010/127657.
- M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 8 (2001). [CrossRef]
- M. Simon, and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wirel. Comm. 4(1), 35–39 (2005). [CrossRef]
- C. Abou-Rjeily, and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Comm. 26(6), 938–947 (2008). [CrossRef]
- W. O. Popoola, and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol. 27(8), 967–973 (2009). [CrossRef]
- N. Letzepis, K. Nguyen, and A. Guillen i Fabregas, “andW. Cowley, “Outage analysis of the hybrid free-space optical and radio-frequency channel,” IEEE J. Sel. Areas Comm. 27(9), 1709–1719 (2009). [CrossRef]
- K. Davaslioglu, E. Cagiral, and M. Koca, “Free-space optical ultra-wideband communications over atmospheric turbulence channels,” Opt. Express 18(16), 618–16,627 (2010).
- U. Madhow, Fundamentals of Digital Communication (Cambridge Univ. Press, 2008).
- T. M. Cover, and J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley & Sons, 2006).
- A. J. Goldsmith, and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory 43(6), 1986–1992 (1997). [CrossRef]
- J. Li, and M. Uysal, “Optical wireless communications: system model, capacity and coding,” in Proc. VTC 2003-Fall Vehicular Technology Conference 2003 IEEE 58th, vol. 1, pp. 168–172 (2003).
- 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).
- H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]
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