## Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels |

Optics Express, Vol. 19, Issue 14, pp. 13480-13496 (2011)

http://dx.doi.org/10.1364/OE.19.013480

Acrobat PDF (931 KB)

### Abstract

Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the performance over free-space optical (FSO) links. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. In this paper, the outage probability as a performance measure for multiple-input/multiple-output (MIMO) FSO communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels with pointing errors is analyzed. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results show that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver. Simulation results are further demonstrated to confirm the analytical results. Additionally, since proper FSO transmission requires transmitters with accurate control of their beamwidth, asymptotic expressions here obtained for different diversity techniques are used to find the optimum beamwidth that minimizes the outage performance.

© 2011 OSA

## 1. Introduction

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

*“last mile” problem*, 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]

*atmospheric scintillation*, severely degrading the link performance [4

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

5. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. **50**(8), 1293–1300 (2002). [CrossRef]

6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. **22**(9), 1896–1906 (2004). [CrossRef]

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

12. 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]

13. 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]

14. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. **2**(4), 626–629 (2003). [CrossRef]

15. S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. **28**(2), 129–131 (2003). [CrossRef] [PubMed]

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**(7), 1702–1710 (2007). [CrossRef]

17. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. **12**(1), 44–46 (2008). [CrossRef]

18. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. **59**(3), 669–674 (2011). [CrossRef]

19. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. **27**(18), 3965–3973 (2009). [CrossRef]

20. W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. **14**(5), 468–470 (2010). [CrossRef]

21. A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in *Proc. IEEE GLOBECOM Workshops* (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

21. A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in *Proc. IEEE GLOBECOM Workshops* (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**(7), 1702–1710 (2007). [CrossRef]

18. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. **59**(3), 669–674 (2011). [CrossRef]

21. A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in *Proc. IEEE GLOBECOM Workshops* (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

*Proc. IEEE GLOBECOM Workshops* (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

## 2. System and channel model

*L*laser sources, assumed to be intensity-modulated only and all pointed towards a distant array of

*M*photodetectors, assumed to be ideal noncoherent (direct-detection) receivers. The transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. The use of infrared technologies based on IM/DD links is considered, where the instantaneous current

*y*(

_{lm}*t*) in the

*m*th receiving photodetector corresponding to the information signal transmitted from the

*l*th laser can be written as where

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

*X*≜

*x*(

*t*) represents the optical power supplied by the

*l*th source and

*I*≜

_{lm}*i*(

_{lm}*t*) the equivalent real-valued fading gain (irradiance) through the optical channel between the

*l*th transmit and the

*m*th receive aperture. Additionaly, the fading experienced between source-detector pairs

*I*is assumed to be statistically independent.

_{lm}*Z*≜

_{m}*z*(

_{m}*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 the

*m*th detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance

*Z*∼

_{m}*N*(0,

*N*

_{0}/2

*M*), independent of the on/off state of the received bit. The factor

*M*is consequence of the fact that the sum of the

*M*receive aperture areas is the same as the aperture area of a system with no receive diversity, wherein the noise is usually modelled as AWGN with zero mean and variance

*σ*

^{2}=

*N*

_{0}/2. This allows the systems to be compared fairly [6

6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. **22**(9), 1896–1906 (2004). [CrossRef]

*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*≜

_{lm}*y*(

_{lm}*t*), however, can assume negative amplitude values. We use

*Y*,

_{lm}*X*,

*I*and

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

_{m}*y*(

_{lm}*t*),

*x*(

*t*),

*i*(

_{lm}*t*) and

*z*(

_{m}*t*) their corresponding realizations.

*l*th transmitter and

*m*th receiver. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [19

19. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. **27**(18), 3965–3973 (2009). [CrossRef]

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

22. 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**, 1554–1562 (2001). [CrossRef]

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express **18**(24), 25422–25440 (2010). [CrossRef] [PubMed]

8. 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]

9. 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. **26**(6), 938–947 (2008). [CrossRef]

24. 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]

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**(7), 1702–1710 (2007). [CrossRef]

*ω*, on the receiver plane at distance

_{z}*z*from the transmitter and a circular receive aperture of radius

*r*, the PDF of

*φ*=

*ω*/2

_{zeq}*σ*is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver,

_{s}*A*

_{0}= [erf(

*v*)]

^{2}and erf(·) is the error function [25, eqn. (7.1.1)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being

*I*was derived in [18

_{lm}18. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. **59**(3), 669–674 (2011). [CrossRef]

*M*> 1) since it depends on locations of the transmitter and receiver, requiring to take into account the relative position for each receive aperture regarding initial pointing in undisturbed position. This represents a fixed pointing error called boresight. As previously indicated, we adopt a MIMO array wherein the transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. For the sake of simplicity, it is here assumed that the receivers are placed very closely together and beam radius is large enough so that the relative displacement for each receiver regarding the initial pointing can be considered negligible, and hence the pointing error model in Eq. (3) can be approximately applied. This assumption can be reinforced by the fact that strong turbulence channels are more robust to boresight [19

19. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. **27**(18), 3965–3973 (2009). [CrossRef]

12. 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]

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express **18**(24), 25422–25440 (2010). [CrossRef] [PubMed]

## 3. Outage performance analysis

*P*

_{out}, is another standard performance criterion of communications systems operating over fading channels [26]. It is defined as the probability that the instantaneous combined SNR,

*γ*, falls below a certain specified threshold,

_{T}*γ*, which represents a protection value of the SNR above which the quality of the channel is satisfactory, i.e., where

_{th}*f*(

_{γ}_{T}*γ*) and

_{T}*F*(

_{γ}_{T}*γ*) are the PDF and the cumulative distribution function (CDF) of

_{T}*γ*, respectively. Next, the outage performance of a SISO FSO system is firstly evaluated in order to be considered as a benchmark in the analysis of the remaining MIMO configurations, corresponding to a multiple-input/single-output (MISO) system with a

_{T}*L*× 1 array, a single-input/multiple-output (SIMO) system with a 1 ×

*M*array and, finally, a MIMO FSO system with a

*L*×

*M*array.

### 3.1. SISO FSO system

*L*=

*M*= 1 is evaluated. According to Eq. (1) and the OOK signaling described in the appendix, a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of

*d*, the statistical channel model corresponding to the SISO configuration can be written as The received electrical SNR can be defined, as in [16

**25**(7), 1702–1710 (2007). [CrossRef]

*γ*represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats,

*T*parameter is the bit period,

_{b}*P*

_{opt}is the average optical power transmitted and

*ξ*represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in the appendix. Using Eq. (5), the outage probability for a SISO FSO system can be written as where the CDF of

*I*,

*F*(

_{I}*i*), can be easily derived from Eq. (4) as by applying the differential relation in [25, eqn. (6.5.26)] and the fact that the series expansion corresponding to the upper incomplete Gamma function can be simplified by

*ξ*= 1 are used, assuming different values of normalized beamwidth,

*ω*/

_{z}*r*, and normalized jitter,

*σ*/

_{s}*r*(i.e.

*ω*/

_{z}*r*= 5 with

*σ*/

_{s}*r*= {1,3,4,5} and

*ω*/

_{z}*r*= 10 with

*σ*/

_{s}*r*= {1,4,7,11}). Outage simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (8) for different misalignment fading conditions.

*F*

_{I(a)}(

*i*) = 1 − exp(−

*i*) and the fact that the series expansion for the exponential function can be simplified by exp(

*z*) ∝ 1 +

*z*+

*O*(

*z*

^{2}), an asymptotic expression can be easily derived as

*φ*is a key parameter in order to optimize the outage diversity. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e.

*φ*> 1. Once this condition is satisfied, taking into account the coding gain in Eq. (11

*a*), the impact of the pointing error effects translates into a coding gain disadvantage,

*D*[

*dB*], relative to strong atmospheric turbulence without misalignment fading given by According to this expression, it can be observed in Fig. 1 that coding gain disadvantages of 23.7, 34.4 and 42.7 decibels are achieved for values of (

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (5, 1), (

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (10,1) and (

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (10,4), respectively.

### 3.2. MISO FSO system with a L ×1 array

*L*apertures and received by only one aperture (i.e.

*M*= 1) is evaluated. Here, two different transmit diversity strategies are considered: repetition coding (RC) and transmit laser selection (TLS).

#### 3.2.2. Transmit diversity using laser selection

12. 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

_{max}*I*= max

_{max}

_{l}_{=1,2,…}

_{L}*I*. In this way, having a SISO system as a reference, the statistical channel model corresponding to this MISO configuration can be written as where, in order to maintain the average optical power transmitted at the same constant level

_{l}*P*

_{out}, the division by

*L*is not included in Eq. (19) because of not more than one laser is simultaneously used. According to order statistics [30

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

*F*(

_{Imax}*i*), of the resulting channel irradiance,

*I*, is given by

_{max}*F*(

_{Imax}*i*) = [

*F*(

_{I}*i*)]

*and, hence, the outage probability can be expressed as In the same way, its corresponding asympotic expressions can be easily derived from Eq. (11) as Results for the exact outage probability in Eq. (20) together with the corresponding asymptotic expressions in Eq. (21) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (20). Comparing both transmit diversity techniques and their corresponding asymptotic expressions in Eq. (18) and Eq. (21) , respectively, it can be stated that the superiority of the TLS scheme translates into a coding gain advantage,*

^{L}*A*[

_{TLS}*dB*], relative to the RC scheme given by As observed in Fig. 2, it can be deduced from expressions in Eq. (22) that the greater the number of transmit apertures, the better performance in terms of coding gain advantage. Additionally, this superiority is even more significant when

*φ*< 1 and the value of

*φ*is decreasing.

### 3.3. SIMO FSO system with a 1 × M array

*L*= 1) and received by

*M*apertures. Here, two different receive diversity strategies are considered: equal gain combining (EGC) and selection combining (SC).

#### 3.3.1. Receive diversity using equal gain combining

*M*photodetectors and, hence, according to Eq. (1), the statistical channel model for this SIMO configuration can be written as where

*I*represents the equivalent irradiance through the optical channel between the transmit aperture and the

_{m}*m*th photodetector. Here, the division by

*M*is considered to ensure that the area of the receive aperture in SISO links has the same size as in the sum of

*M*receive aperture areas of SIMO links [6

6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. **22**(9), 1896–1906 (2004). [CrossRef]

10. 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*by

*M*in Eq. (15) and Eq. (18) to obtain the outage probability and its corresponding asympotic expressions for SIMO FSO links using equal gain combining.

#### 3.3.2. Receive diversity using selection combining

**22**(9), 1896–1906 (2004). [CrossRef]

10. 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]

*I*, can be written as

_{max}*I*= max

_{max}

_{m}_{=1,2,···}

_{M}*I*and, hence, the statistical channel model can be written as where

_{m}*M*represents the same role as previously explained when EGC is adopted. It can be observed that the noise variance is given by

*N*

_{0}/2

*M*since only one of the

*M*receive apertures is used. Taking into account Eq. (7), the resulting received electrical SNR can be defined as and, hence, the outage probability for a SIMO FSO system with selection combining can be written as In the same way, its corresponding asympotic expressions, depending on the value of

*φ*, can be easily derived from Eq. (11) as Results for the exact outage probability in Eq. (26) together with the corresponding asymptotic expressions in Eq. (27) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (26). Comparing both receive diversity techniques and their corresponding asymptotic expressions, a different relative performance is deduced depending on the value of

*φ*, corroborating the superiority of EGC scheme when

*φ*> 1 and the superiority of both SC scheme or EGC scheme when

*φ*< 1, depending on the values of

*M*and

*φ*. This improvement in performance of the EGC scheme when

*φ*> 1 translates into a coding gain advantage,

*A*[

_{EGC}*dB*], relative to the SC scheme as follows corroborating the fact that the greater the number of apertures, the higher coding gain advantage. Nonetheless, it can be noted that there is no difference between both diversity techniques for a value of

*M*= 2. In a similar way, the superiority of SC scheme when

*φ*< 1 translates into a coding gain advantage,

*A*[

_{SC}*dB*], relative to the EGC scheme as follows Nonetheless, it can be concluded from this expression that the superiority of the SC scheme is only valid for low values of

*φ*and when a small number of receive apertures is considered, being only justified the adoption of this receive diversity technique in the context of very severe pointing errors. As depicted in Fig. 3 for a normalized beamwidth of

*ω*/

_{z}*r*= 10, better performance for the EGC scheme is again achieved as the number of receive apertures increases and the value of the normalized jitter,

*σ*/

_{s}*r*, is lower and lower. So, the value of

*φ*becomes higher and higher, and hence it is closer to the unity.

## 4. Outage optimization

*φ*> 1, being

*φ*a main parameter to consider in order to optimize the outage performance. Once this condition is satisfied, comparing different diversity techniques and their corresponding asymptotic expressions, it can be noted that better performance is achieved when increasing the number of transmit apertures, i.e.

*L*, instead of the number of receive apertures, i.e.

*M*, in order to guarantee a same diversity order, as shown in Fig. 4 when assuming a normalized beamwidth of

*ω*/

_{z}*r*= 5 with a normalized jitter of

*σ*/

_{s}*r*= 1 and a diversity order of 8 with values of {

*L*,

*M*} = {4,2} and {

*L*,

*M*} = {2,4}. Taking into account expressions in Eq. (35

*a*) and Eq. (21

*a*), this can be quantified as a coding gain disadvantage,

*D*[

_{MIMO}*dB*], for the MIMO FSO system relative to the MISO FSO system using laser selection as follows where

*O*and

_{d}*M*denote outage diversity and the number of receive apertures, respectively. In agreement with this expression, it can be seen in Fig. 4 that a difference of 2.13 decibels is achieved for a diversity order of 8 and values of

*M*= 2 and

*M*= 4 when a normalized beamwidth of

*ω*/

_{z}*r*= 5 with a normalized jitter of

*σ*/

_{s}*r*= 1 is assumed. Lastly, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth,

*ω*/

_{z}*r*, that gives the minimum outage [16

**25**(7), 1702–1710 (2007). [CrossRef]

17. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. **12**(1), 44–46 (2008). [CrossRef]

31. H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. **12**(5), 395–397 (2008). [CrossRef]

*a*), it can be observed that this is equivalent to minimize the expression in Eq. (12) corresponding to the SISO configuration, being independent of the MIMO configuration adopted. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter,

*σ*/

_{s}*r*[32]. Numerical results for the optimum beamwidth are illustrated in Fig. 5 for a range of values in the normalized jitter from

*σ*/

_{s}*r*= 1 to

*σ*/

_{s}*r*= 10 in discrete steps of 0.5 when the stochastic function minimizer simulated annealing is used. From this figure, it can be deduced that the outage optimization provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows considering the values of optimum beamwidth corresponding to normalized jitters of

*σ*/

_{s}*r*= 1 and

*σ*/

_{s}*r*= 10, respectively. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is shown in Fig. 4, where results assuming the optimum beamwidth corresponding to values of jitter of

*σ*/

_{s}*r*= 1 and

*σ*/

_{s}*r*= 7 are also included for

*L*= 4 and

*M*= 2. As expected, a greater improvement in performance is achieved when

*σ*/

_{s}*r*= 7 compared to the improvement in performance corresponding to the normalized jitter of

*σ*/

_{s}*r*= 1 since not only coding gain but also diversity gain is obtained, making outage diversity performance independent of the impact of misalignment fading.

## 5. Conclusions

## Appendix

**21**(14), 1017–1019 (2009). [CrossRef]

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express **18**(24), 25422–25440 (2010). [CrossRef] [PubMed]

*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

*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 (RV)

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

_{k}*P*

_{opt}, defining a constellation of two equiprobable 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.

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

5. | X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. |

6. | E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. |

7. | 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. |

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

9. | 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. |

10. | 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. |

11. | E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. |

12. | 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. |

13. | 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 |

14. | S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. |

15. | S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. |

16. | A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. |

17. | H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. |

18. | H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. |

19. | D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. |

20. | W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. |

21. | A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in |

22. | 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. |

23. | A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express |

24. | 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. |

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

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

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

28. | Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. |

29. | S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. |

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

31. | H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. |

32. | 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:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: April 19, 2011

Revised Manuscript: June 6, 2011

Manuscript Accepted: June 10, 2011

Published: June 28, 2011

**Citation**

Antonio García-Zambrana, Carmen Castillo-Vázquez, and Beatriz Castillo-Vázquez, "Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels," Opt. Express **19**, 13480-13496 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13480

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