## Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels

Optics Express, Vol. 18, Issue 6, pp. 5356-5366 (2010)

http://dx.doi.org/10.1364/OE.18.005356

Acrobat PDF (171 KB)

### Abstract

Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance. In this paper, a scheme combining transmit laser selection (TLS) and space-time trellis code (STTC) for multiple-input-single-output (MISO) free-space optical (FSO) communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels is analyzed. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available *L* lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas. Based on a pairwise error probability (PEP) analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2*L*-1 when *L* transmit lasers are available and a simple two-state STTC with rate 1 *bit*/(*s∙Hz*) is used. Simulation results are further demonstrated to confirm the analytical results.

© 2010 Optical Society of America

## 1. Introduction

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

*“last mile” problem*, above all in densely populated urban areas, as well as a supplement to radio-frequency (RF) links [2

2. 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]

3. 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]

4. K. Tsukamoto, A. Hashimoto, Y. Aburakawa, and M. Matsumoto, “The case for free space,” IEEE Microwave Mag. **10**(5), 84–92 (2009). [CrossRef]

*atmospheric scintillation*, severely degrading the link performance [5

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

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

7. X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. **51**(8), 1233–1239 (2003). [CrossRef]

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

17. 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]

20. M. Safari and M. Uysal, “Do We Really Need OSTBCs for Free-Space Optical Communication with Direct Detection?” IEEE Trans. Wireless Commun. **7**(11), 4445–4448 (2008). [CrossRef]

21. V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory **44**(2), 744–765 (1998). [CrossRef]

23. Z. Chen, B. Vucetic, and J. Yuan, “Space-time trellis codes with transmit antenna selection,” Electron. Lett. **39**(11), 854–855 (2003). [CrossRef]

26. 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]

27. 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]

29. 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]

## 2. Atmospheric turbulence channel model

*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, and

*I*≜

*i*(

*t*) the scintillation at the optical path;

*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

*N*

_{0}/2, i.e.

*Z*~

*N*(0,

*N*

_{0}/2), independent of the on/off state of the received bit [1

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

*X*must satisfy ∀

*tx*(

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

*X*is limited. Although limits are placed on both the average and peak optical power transmitted, in the case of most practical modulated optical sources, it is the average optical power constraint that dominates [31

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

*Y*≜

*y*(

*t*), however, can assume negative amplitude values. In this fashion, the atmospheric turbulence channel model consists of a multiplicative noise model, where the optical signal is multiplied by the channel irradiance. Considering strong turbulence conditions [5

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

16. 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]

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

18. 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]

*α*→ ∞.

18. 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]

26. 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]

*t*) is defined as ϕ(

*t*) =

*g*(

*t*)/√

*E*where

_{g}*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. The random variable (RV)

*a*follows a Bernoulli distribution with parameter

_{k}*p*= 1/2, 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*, defining a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of

*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. 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 [8

8. 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]

9. 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]

11. 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]

12. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” J. Lightwave Technol. **26**(5), 478–487 (2008). [CrossRef]

## 3. Proposed transmit diversity scheme

*L*laser sources, assumed to be intensity-modulated only and all pointed towards a distant photodetector, assumed 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, and the separation distance between the lasers is large enough so that the fading experienced between source-detector pairs

*I*(

_{j}*t*) is assumed to be statistically independent. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available

*L*lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas [21

21. V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory **44**(2), 744–765 (1998). [CrossRef]

*bit*/(

*s∙Hz*) using OOK is displayed [33

33. H. Jafarkhani, *Space-Time Coding: Theory and Practice* (Cambridge University Press, New York, 2005). [CrossRef]

*L*lasers corresponding to greater values of scintillation,

*I*

_{(L)}(

*t*) and

*I*

_{(L-1)}(

*t*), where

*I*

_{(1)}(

*t*),

*I*

_{(2)}(

*t*),…,

*I*

_{(L)}(

*t*) is a new sequence of

*L*auxiliary random variables obtained by arranging the random sequence

*I*

_{1}(

*t*),

*I*

_{2}(

*t*),…,

*I*(

_{L}*t*) in an increasing order of magnitude. The labeling

*i/kk*along each branch of the trellis refers to the input bit (

*i*) and the corresponding pair of output symbols (

*kk*) that result from the transition between the states at the beginning and end of the branch.

## 4. Performance analysis

*L*= {1,2,4,8} laser sources, all pointed towards a distant photodetector, is considered. We present aproximate closed-form expressions for the bit error rate (BER) using a pairwise error probability analysis when the scintillation follows negative exponential and

**K**distributions, which cover a wide range of strong atmospheric turbulence conditions. The PEP represents the probability of choosing the space-time sequence

**X**̂ when in fact the sequence

**X**was transmitted [30, Chapter 16]. Assuming that the correct path is the all-zeros sequence, then for the shortest error event path of length

*N*= 2 illustrated by shading in Fig. 1, we have

**X**and

**X**̂ is associated with the two symbols transmitted from the two lasers in a given symbol interval (time slot) and

*d*is the Euclidean distance in (6) corresponding to the OOK signaling (i.e., the OOK symbols are the elements of the

**X**and

**X**̂ matrices associated with the trellis). In the proposed scheme, for example, we associate the first and second rows with the (L-1)th and Lth order statistics corresponding to the scintillation. Under the assumption of perfect CSI, the conditional PEP with respect to scintillation coefficients of greater value,

*I*

_{(L)}and

*I*

_{(L-1)}, is given as [30, Chapter 16]

*Q*(∙) is the Gaussian-

*Q*function. Similar expressions to evaluate the pairwise error probability of coded FSO IM/DD links using OOK signaling can be found in [7

7. X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. **51**(8), 1233–1239 (2003). [CrossRef]

9. 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]

*d*by 2 is considered so as to maintain the average optical power in the air at a constant level of

*P*, being transmitted by each laser an average optical power of

*P*/2. Substituting the value of

*d*obtained in (6) gives

*P*

^{2}

*T*/

_{b}*N*

_{0}is the average receiver electrical signal-to-noise spectral density ratio (SNR) in the presence of the turbulence [6

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

*P*(

**X**→

**X**̂), by averaging (9) as follows

34. H. A. David and H. N. Nagaraja, *Order Statistics*, 3rd ed. (John Wiley and Sons Inc., 2003). [CrossRef]

*I*}

_{j}_{j=1,2,⋯L}, the PDF corresponding to

*I*

_{(L)}and

*I*

_{(L-1)}can be written as

*F*(

_{I}*i*) the cumulative density function (CDF) corresponding to the turbulence model. An union bound on the average BER can be found as [30, eq. (13.44)]

*P*(

**X**) is the probability that the coded sequence

**X**is transmitted,

*n*(

**X**,

**X**̂) is the number of information bit errors in choosing another coded sequence

**X**̂ instead of

**X**and

*n*is the number of information bits per transmission. Next, if we were to choose to approximate the average BER by considering only error event paths of minimum length (i.e.,

_{c}*N*= 2) [30, Section 14.6.4], we can use (10) to obtain

*P*(

_{b}*E*) ≃

*P*(

**X**→

**X**̂). To simplify the expression in (10), we use the approximation for the

*Q*-function presented in [35

35. M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. **2**(4), 840–845 (2003). [CrossRef]

*Q*(

*x*) ≃ (1/12)exp(-

*x*

^{2}/2)+(1/4)exp(-2

*x*

^{2}/3)), finally obtaining

26. 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]

### 4.1. K atmospheric turbulence channel

36. Wolfram Research, Inc., “The Wolfram functions site,” URL http://functions.wolfram.com.

*K*(∙) is a even function with respect to its parameter, the derived integral for the CDF of the K channel can be written as

_{v}*α*= 1 and

*α*= 4 are assumed, corresponding to values of scintillation index of

*SI*= 3 and

*SI*= 1.5, respectively, and where rectangular pulse shapes with ξ = 1 are used. Additionally, a relevant improvement in performance must be noted as a consequence of pulse shape used, providing an increment in the average SNR of 10log

_{10}ξ

*dB*. So, for instance, when a rectangular pulse shape of duration

*κT*, with 0 <

_{b}*κ*≤ 1, is adopted, a value of ξ = 1/

*κ*can be easily shown. Nonetheless, a significantly higher value of ξ = 4/

*κ*√π is obtained when a Gaussian pulse of duration

*κT*as

_{b}*g*(

*t*) = exp (-

*t*

^{2}/2σ

^{2}) ∀|

*t*| <

*κT*/2 is adopted, where σ =

_{b}*κT*/8 and the reduction of duty cycle is also here controlled by the parameter

_{b}*κ*. In this fashion, 99.99% of the average optical power of a Gaussian pulse shape is being considered. Then, a Gaussian pulse shape with

*κ*= 0.25 is also adopted when

*L*= 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Numerical results for TLS/STTC in (14), TLS in (15) and STTC without laser selection in (16) are computed using a symbolic mathematics package [37]. BER simulation results are furthermore included as a reference. Due to the long simulation time involved, simulation results only up to BER=10

^{-6}are included. Simulation results demonstrate an excellent agreement with the analytical results for

*L*= {2,4,8}, as well as the greater diversity order for the transmit diversity technique here proposed if compared with TLS and STTC, being superior to the number of available transmit lasers

*L*.

### 4.2. Exponential atmospheric turbulence channel

*Laser Beam Scintillation with Applications* (SPIE Press, 2001). [CrossRef]

18. 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]

*F*(

_{I}*i*) = 1 - exp (-

*i*). Using the binomial theorem in (11) and (12), we obtain

*κ*= 1 are used.

*κ*= 0.25 is also adopted when

*L*= 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Here, results for TLS/STTC from evaluating the expression in (20) are displayed together with numerical results for the TLS and STTC schemes computed as in previous subsection. BER simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical results for

*L*= {2,4,8}, as well as a better performance in terms of diversity gain, as previously concluded for the K channel. With the purpose of analyzing the diversity order achieved for the TLS/STTC scheme here proposed when

*L*transmit lasers are available, we can use in (20) the series expansions corresponding to the exponential function [38, eqn. (4.2.1)] (i.e., exp (

*x*) = ∑

^{∞}

_{k=0}

*x*/

^{k}*k*!) and the error function [38, eqn. (7.1.5)] (i.e., erfc(

*x*) = 1 - (2/√π)∑

^{∞}

_{k=0}(-1)

^{k}

*x*

^{2k+1}/((2

*k*+ 1)

*k*!)). In this way, it is straightforward to show that the average BER behaves asymptotically as 1/γ

^{(2L-1)/2}, corroborating a diversity gain of 2

*L*- 1 in relation to the absence of space-time trellis coding with laser selection, wherein the average BER varies as 1/γ

^{1/2}[17

17. 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]

**11**(5), 390–392 (2007). [CrossRef]

27. 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]

*L*can be demonstrated in a similar way as before, the additional use of the simple two-state STTC in Fig. 1 provides a performance improvement of 20 dB at a target BER rate of 10

^{-9}with only two transmit lasers.

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” Proc. IEEE |

2. | 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 |

3. | 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 |

4. | K. Tsukamoto, A. Hashimoto, Y. Aburakawa, and M. Matsumoto, “The case for free space,” IEEE Microwave Mag. |

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

6. | X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. |

7. | X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. |

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

9. | 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. |

10. | E. J. Shin and V. W. S. Chan, “Optical communication over the turbulent atmospheric channel using spatial diversity,” in Proc. IEEE GLOBECOM , pp. 2055–2060 (2002). |

11. | I. B. Djordjevic, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation,” Opt. Express |

12. | I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” J. Lightwave Technol. |

13. | F. Xu, A. Khalighi, P. Caussé, and S. Bourennane, “Channel coding and time-diversity for optical wireless links,” Opt. Express |

14. | S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-Space Optical MIMO Transmission With |

15. | S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER Performance of Free-Space Optical Transmission with Spatial Diversity,” IEEE Trans. Wireless Commun. |

16. | 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. |

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

18. | A. García-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett. |

19. | C. Abou-Rjeily, “Orthogonal Space-Time Block Codes for Binary Pulse Position Modulation,” IEEE Trans. Commun. |

20. | M. Safari and M. Uysal, “Do We Really Need OSTBCs for Free-Space Optical Communication with Direct Detection?” IEEE Trans. Wireless Commun. |

21. | V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory |

22. | E. Bayaki and R. Schober, “On Space-Time Coding for Free-Space Optical Systems,” (2009). Accepted for future publication in IEEE Trans. Commun. |

23. | Z. Chen, B. Vucetic, and J. Yuan, “Space-time trellis codes with transmit antenna selection,” Electron. Lett. |

24. | D. A. Gore and A. J. Paulraj, “MIMO antenna subset selection with space-time coding,” IEEE Trans. Sig. Process. |

25. | A. F. Molisch and M. Z. Win, “MIMO systems with antenna selection,” IEEE Microwave Magazine |

26. | 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. |

27. | 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. |

28. | N. Letzepis and A. G. Fabregas, “Outage probability of the MIMO Gaussian free-space optical channel with PPM,” in |

29. | 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. |

30. | M. K. Simon and M.-S. Alouini, |

31. | S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Information Theory |

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

33. | H. Jafarkhani, |

34. | H. A. David and H. N. Nagaraja, |

35. | M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. |

36. | Wolfram Research, Inc., “The Wolfram functions site,” URL http://functions.wolfram.com. |

37. |
Wolfram Research, Inc., |

38. | M. Abramowitz and I. A. Stegun, |

**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:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: November 16, 2009

Revised Manuscript: January 9, 2010

Manuscript Accepted: February 21, 2010

Published: March 1, 2010

**Citation**

Antonio García-Zambrana, Carmen Castillo-Vázquez, and Beatriz Castillo-Vázquez, "Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels," Opt. Express **18**, 5356-5366 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5356

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### References

- J. M. Kahn and J. R. Barry, "Wireless Infrared Communications," Proc. IEEE 85, 265-298 (1997). [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]
- K. Tsukamoto, A. Hashimoto, Y. Aburakawa, and M. Matsumoto, "The case for free space," IEEE Microwave Mag. 10(5), 84-92 (2009). [CrossRef]
- L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]
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