**1. Introduction**

22 Will-be-set-by-IN-TECH

[16] Kahn, J. M., You, R., Djahani, P., Weisbin, A. G., Teik, B. K. & Tang, A. [1998]. Imaging diversity receivers for high-speed infrared wireless communication, *IEEE*

[17] Komine, T., Lee, J. H., Haruyama, S. & Nakagawa, M. [2009]. Adaptive equalization system for visible light wireless communication utilizing multiple white LED lighting

[18] Komine, T. & Nakagawa, M. [2003]. Integrated system of white LED visible-light communication and power-line communication, *IEEE Transactions on Consumer*

[19] Komine, T. & Nakagawa, M. [2004]. Fundamental analysis for visible-light communication system using LED lights, *IEEE Transactions on Consumer Electronics*

[20] Li, X., Vuˇci´c, J., Jungnickel, V. & Armstrong, J. [2012]. On the capacity of intensity-modulated direct-detection systems and the information rate of ACO-OFDM for indoor optical wireless applications, *IEEE Transactions on Communications*

[21] Lomba, C. R., Valadas, R. T. & de Oliveira Duarte, A. M. [1998]. Experimental characterisation and modelling of the reflection of infrared signals on indoor surfaces,

[22] Mesleh, R., Elgala, H. & Haas, H. [2011]. Optical spatial modulation, *IEEE/OSA Journal*

[23] Minh, H. L., O'Brien, D. C., Faulkner, G. E., Zeng, L., Lee, K., Jung, D., Oh, J. & Won, E. T. [2009]. 100-Mb/s NRZ visible light communications using a postequalized white LED,

[24] O'Brien, D. C., Minh, H. L., Faulkner, G. E., Zeng, L., Lee, K., Jung, D. & Oh, J. [2008]. High-speed visible light communications using multiple-resonant equalization, *IEEE*

[25] O'Brien, D. C., Zeng, L., Minh, H. L., Faulkner, G. E., Walewski, J. W. & Randel, S. [2008]. Visible light communications: challenges and possibilities, *Proceedings of IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC*

[26] Proakis, J. G. [2001]. *Digital Communications*, McGraw-Hill International Edition,

[27] Vuˇci´c, J., Kottke, C., Habel, K. & Langer, K.-D. [2011]. 803 Mbit/s visible light WDM link based on DMT modulation of a single RGB LED luminary, *Proceedings of Optical Fiber Communication Conference and Exposition, and the National Fiber Optic Engineers Conference*

[28] Vuˇci´c, J., Kottke, C., Nerreter, S., Langer, K.-D. & Walewski, J. W. [2010]. 513 Mbit/s visible light communications link based on DMT-modulation of a white LED, *Journal of*

[29] Zeng, L., O'Brien, D. C., Minh, H. L., Faulkner, G. E., Lee, K., Jung, D., Oh, J. & Won, E. T. [2009]. High data rate multiple iinput multiple output (MIMO) optical wireless communications using white LED lighting, *IEEE Journal on Selected Areas in*

equipment, *IEEE Transactions on Wireless Communications* 8(6): 2892–2900.

*Communications Magazine* 36(12): 88–94.

*IEE Proceedings-Optoelectronics* 145(3): 191–197.

*of Optical Communications and Networking* 3(3): 234–244.

*IEEE Photonics Technology Letters* 21(15): 1063–1065.

*Photonics Technology Letters* 20(15): 1243–1245.

*Electronics* 49(1): 71–79.

50(1): 100–107.

60(3): 799–809.

*2008*, pp. 1–5.

*(OFC/NFOEC) 2011*, pp. 1–3.

*Lightwave Technology* 28(4): 3512–3518.

*Communications* 27(9): 1654–1662.

Singapore.

Free-space optical (FSO) communications using intensity modulation and direct detection (IM/DD), is a cost-effective and high bandwidth access technique, which has recently received significant attention and commercial interest for a variety of applications [30, 44]. Optical wireless communication systems are rapidly gaining popularity as effective means of transferring data at high rates over short distances due to the necessity of a cost-effective, license-free, and high-bandwidth access communication technique [12, 30, 36, 44]. These systems facilitate rapidly deployable, lightweight, high-capacity communication without licensing fees and tariffs. Terrestrial FSO is not free of challenges though. A major impairment over FSO links is the atmospheric turbulence, caused by the variations in the refractive index because of inhomogeneities in temperature and pressure changes. In clear weather conditions, the atmospheric turbulence results in fluctuations at the intensity of the received signal, i.e., signal fading, also known as scintillation in optical communication terminology [6, 11]. Turbulence is caused by inhomogeneities of both temperature and pressure in the atmosphere and can severely degrade the link performance, particularly over link distances of 1 km or longer. The performance of this technology depends strongly on the atmospheric conditions between the transmitter and the receiver and the parameters of the link such as the length and the operation wavelength. Effects of fog, rain, atmospheric gases, and aerosols also result in beam attenuation due to photon absorption and scattering [25].

The performance of FSO systems over turbulence channels has been addressed in many previous works. Representative examples can be found in [47, 66, 70, 81] and the references therein. The results presented in these papers have demonstrated that the performance of single-input single-output (SISO) FSO links is severely degraded from turbulence. More specifically, the average bit error probability of such systems is far away from satisfying the typical targets for FSO applications within practical ranges of signal-to-noise ratio. To circumvent this problem, powerful fading mitigation techniques have to be deployed. In the open technical literature on FSO communication, the two most popular existing techniques

©2012 Peppas et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Peppas et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 416 Optical Communication

for mitigation of the degrading effects of atmospheric turbulence are error control coding in conjunction with interleaving [70], [83] and maximum likelihood sequence detection (MLSD) [82]. However, for most scenarios the first one requires large-size interleavers to achieve the promised coding gains. On the other hand, MLSD requires complicated multidimensional integrations and suffers from excessive computational complexity. Some sub-optimal temporal-domain fading mitigation techniques are further explored in [83] and [84].

Another promising solution is the use of diversity techniques and the most popular scheme is the spatial diversity, i.e., the employment of multiple transmit/receive apertures, a well known diversity technique in Radio-Frequency (RF) systems [46, 48, 49, 61, 69, 72, 78, 80]. By using multiple apertures at the transmitter and/or the receiver, the inherent redundancy of spatial diversity has the potential to significantly enhance the performance. Moreover, the possibility for temporal blockage of the laser beams by obstructions is further reduced and longer distances can be covered through heavier weather conditions [46]. Concerning the performance analysis of FSO systems employing spatial diversity, the technical literature is rather rich. Representative past examples can be found in [14, 19, 21, 26, 27, 29, 46, 54–56, 62, 63, 69, 73–75].

On the other hand, various statistical models, e.g. the log normal, the gamma gamma (G − G), the I-K, the K, the negative exponential, and the Rician log normal distribution, have been used in order to describe the optical channel characteristics with respect to the atmospheric turbulence strength [4, 9, 10, 12, 23, 30, 32, 35, 38, 44, 47, 51–53, 59, 65, 71]. Recently, Al-Habash et al. proposed the G-G distribution [4] as a tractable mathematical model for atmospheric turbulence. This model is a two parameter distribution which is based on a doubly stochastic theory of scintillation and assumes that small-scale irradiance fluctuations are modulated by large-scale irradiance fluctuations of the propagating wave, both governed by independent gamma distributions. This distribution has become the dominant fading channel model for FSO links due to its excellent agreement with measurement data for a wide range of turbulence conditions [4].

For many practical FSO applications, however, irradiance is temporally correlated. Thus the derivation of a correlated G − G model is of significant theoretical and practical interest. It is noted that multivariate distributions have recently attracted the interest within the research community due to their importance in studying the performance of diversity systems operating over a multipath fading channels, see, e.g., [1, 2, 5, 15, 33, 45, 49, 58, 64, 68] and references therein. These distributions are particularly useful in the performance analysis of practical systems configurations where antenna branches are closely spaced and the correlation between diversity signals cannot be ignored [68, Chapt. 9]. In the past, several spatial correlation models have been proposed [5, 68] and used for the performance evaluation of wireless communication systems over correlated fading channels. Among them, the exponential correlation model has gained particular interest [2, 33, 45, 64]. This model corresponds to the scenario of multichannel reception by equispaced diversity antennas, in which the correlation between pairs of combined signals decays as the spacing between antenna branches increases [68].

In this chapter, we investigate the performance of multiple-input-multiple output (MIMO) FSO links over both independent and identically distributed (i.i.d.) G − G turbulence channels as well as for exponentially correlated G − G turbulent channels. A closed-form expression for the average bit error probability (ABEP) of SISO links is first derived. This result serves as benchmark for the performance analysis of FSO links employing multiple apertures and IM/DD. Rapidly convergent infinite series representations for the joint G − G probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) with exponential correlation are also derived. Based on these statistical results, the outage probability (OP) of selection diversity (SD) receivers as well as the ABEP of single-input-multiple-output (SIMO) FSO systems over exponentially correlated turbulent channels is investigated. Finally, we propose a simple yet highly accurate closed form approximation to the sum of arbitrary i.i.d. G − G variates. In the context of this chapter, the *α*-*μ* distribution [79] has been chosen as the convenient approximation, for which the parameters are adequately estimated from the sum of the G − G variates. Based on this result, simple accurate approximations for the OP and ABEP of MIMO FSO systems operating over i.i.d. G − G channels and employing equal gain combining (EGC) at the receiver are provided. Various numerically evaluated and computer simulation results demonstrate the accuracy of the proposed analysis. The validity of the presented analysis is testified by comparing numerically evaluated with equivalent computer simulations performance results.
