4.2 K-distribution

The K-distribution turbulence model is often used to describe strong atmospheric turbulence conditions (non-Rayleigh sea clutter). For K-distribution atmospheric turbulence model, the probability distribution function p Ið Þ is expressed as [14]

$$p(I) = \frac{2a}{\Gamma(a)} (aI)^{\frac{a-1}{2}} K\_{a-1} \Big( 2\sqrt{aI} \Big), I > 0, a > 0 \tag{2}$$

x0, x1, x2, …, xt, …, where each element xt of this sequence is the space-time symbol

Mitigating Turbulence-Induced Fading in Coherent FSO Links: An Adaptive Space-Time Code…

Space-time code (STC) leverages on the features of both time diversity and space diversity to combat turbulence-induced fading in wireless communication systems. RF wireless systems in particular have witness an explosion of interest in the use of space-time coding to improve communication system performance in terms of error control and turbulence mitigation, and FSO communication systems are also witnessing a lot of interest in using this same tool for similar purpose. In this chapter, we present an adaptive four-state space-time trellis coded coherent FSO system with two transmit lasers, as illustrated in Figure 3. Firstly, the error correction performance of the system is complemented by the interleaver, a mechanism put in place to distribute the burst errors—an effect of deep fade, onto

Denoting the average SNR as γ, we take the received signal matrix for each

where H and Z are the channels and noise matrices, respectively, and H is modeled in terms of the uniformly distributed channel gain phase ϕμ<sup>v</sup> and the

R ¼ ffiffi

xi <sup>t</sup> ¼ ∑ m k¼1 ∑ vk j¼0 gk j,i c k t�j

DOI: http://dx.doi.org/10.5772/intechopen.84911

<sup>t</sup> of the encoder for the ith transmitter at time t is expressed as [19]

Mod 2, i ¼ 1, 2 (7)

<sup>γ</sup> <sup>p</sup> CH <sup>þ</sup> <sup>Z</sup> (8)

at a given time t. The output xi

different codeword lengths.

channel gain amplitude aμ<sup>v</sup> as [20]

codeword C as [20]

Figure 2. STTC encoder [18].

Figure 3.

109

Space-time trellis coded FSO communication system.

KmðÞ¼ ∙ modified Bessel function of second kind and order m:

#### 4.3 Negative exponential distribution

Negative exponential turbulence model is employed for saturated turbulence cases where the probability distribution function of the received irradiance value is expressed as [15]

$$p(I) = \frac{1}{I\_0} \exp\left(\frac{-I}{I\_0}\right), I\_0 > 0\tag{3}$$

where I<sup>0</sup> denotes the mean irradiance.

#### 4.4 Gamma-gamma distribution

The gamma-gamma model is very commonly used in FSO communication literatures because it is applicable for a wider range of turbulence conditions. In comparison with measured data, gamma-gamma distribution is effective in describing weak to strong atmospheric turbulence conditions. The PDF is expressed as [16]

$$f\widetilde{H}^{GG}(h) = \frac{2(a\beta)^{\frac{a+\beta}{2}}}{\Gamma(a)\Gamma(\beta)} h^{\frac{a+\beta}{2}-1} K\_{a-\beta} \left(2\sqrt{a\beta h}\right) \tag{4}$$

where Kvð Þ x is the modified Bessel function of the second kind and α and β are the turbulence parameter.
