Transparencia y composicionalidad semánticas

In order to describe from a statistical point of view the effect of atmospheric turbulence, different statistical models have been proposed in the literature. On the one hand, these statistical models must be capable of validating the practical measurements. On the other hand, an easy mathematical treatment is required in order to obtain simple analytical ex-pressions.

Early probability density function (PDF) models developed for the irradiance were the modified Rician distribution, which is obtained from the Born approximation, and the log-normal (LN) model, which is based on the first Rytov approximation and was proposed several decades ago in [12,13]. Both of them are suitable for weak turbulence. However, the modified Rician distribution is used under extremely weak fluctuations [27]. The LN turbulence model is widely used to study the performance under weak turbulence conditions.

However, it is well known that the LN turbulence model is not appropriate for moderate-to-strong turbulence conditions in agreement with experimental data [12,13,28]. For that reason, a number of statistical models were developed to address this problem in strong turbulence regime.

Most statistical models are based on heuristic arguments and observed experimental data and, hence, they show good agreement with experimental data under certain conditions.

One of the early models that gained wide acceptance for strong turbulence was the K distribution [29,30], which was originally proposed as a model for non-Rayleigh sea clutter.

One extension of the K distribution was the I-K distribution, which was presented in [31] to also cover weak turbulence. Later, other statistical models were arising such as lognormally modulated exponential distribution [32], and the more general lognormal-Rician distribution [33], also known as the Beckmann distribution. The Beckmann distribution was presented for the first time in [34]. These models, like K and I-K distributions, present the inconvenience of not being able to relate their statistical parameters to atmospheric conditions. Another atmospheric turbulence model is the gamma-gamma (GG) distribution [12,13,28], which has gained a wide acceptance by the research community since the parameters involved in this distribution can be measured directly from the channel. This density function has been extensively utilized in the literature to evaluate the FSO system performance since it provides a close agreement with measurement data. In addition, this distribution can be used to study the performance in a wide range of turbulence conditions, i.e., from moderate-to-strong. However, the GG turbulence model does not provide a good fit to simulation data

2.2. OPTICAL TURBULENCE THEORY 21 in moderate to strong turbulence regimes when D≥ ρc, i.e., when aperture averaging takes place [35,36]. In Subsection 2.2.4, a brief description of this atmospheric turbulence model is given since GG density function is the most used statistical model in this thesis.

Over the years, many attempts have been conducted to propose atmospheric turbulence statistical models that can be used under all turbulence conditions, i.e, from weak to strong.

A universal model able to characterize different turbulence strengths is one of the main concerns. A number of authors have proposed universal statistical channel models, such as the M´alaga (M)-distributed atmospheric turbulence [37], the double GG generalized fading channels [38,39], and the exponentiated Weibull (EW) turbulence model [40–43]. The EW model provides a good fit between simulation and experimental data under moderate to strong aperture averaging conditions [40–43]. It should be noted that the EW distribution offers an excellent fit to simulation and experimental data under all aperture averaging conditions D ≥ ρc, from weak to strong [40,41]. Indeed, this atmospheric turbulence model has been used in a significant number of research articles in order to study the performance of FSO communication systems [44–51].

Log-Normal (LN) Model

The LN turbulence model is the most widely accepted model under weak turbulence con-ditions, which was proposed several decades ago in [12,13]. The corresponding PDF of LN model is given by

f_Ia(i) = 1 i

q 8πσ²_X

exp − ln(i) + 2σ_X²
2

8σ_X²

!

, i≥ 0 (2.16)

where σ_X² is the log-amplitude variance given by σ_X² ≈ σ²_R/4. The SI is related to the log-amplitude variance according to

σ_I²_a = exp 4σ²_X

− 1. (2.17)

Gamma-Gamma (GG) Model

In the GG statistical model, the irradiance Ia is expressed as a product of two independent gamma RVs, i.e., Ia= IL· IS. These RVs represent the irradiance fluctuations from large (I_L) and small (I_S) scale turbulence, whose PDF is given by

fIa(i) = 2(αβ)^(α+β)/2

Γ(α)Γ(β) i((α+β)/2)−1Kα−β

 2p

αβi

, i≥ 0 (2.18)

where Γ(·) is the well-known Gamma function (See Appendix A.1), Kν(·) is the νth-order modified Bessel function of the second kind (See AppendixA.2), and the parameters α and

β represent the effective numbers of large and small scale turbulence cells according to

The parameters α and β can be selected to achieve a good agreement between Eq. (2.18) and measurement data [28]. Expressions for α and β for plane, spherical and Gaussian-beam wave are given in [13], which are all expressed in terms of Rytov variance. In the case of plane wave propagation, α and β can be expressed as follows

α =

It must be emphasized that parameters α and β cannot arbitrarily be chosen in FSO ap-plications since both parameters are related to Rytov variance. The SI can be computed as

σ_I²_a = 1 α + 1

β + 1

αβ. (2.21)

It must be noted that the PDF in Eq. (2.18) contains other statistical atmospheric turbu-lence models adopted in strong turbuturbu-lence such as the K distribution (β = 1 and α > 0) and the negative exponential distribution (β = 1 and α→ ∞). From the scintillation index point of view, it is easy to deduce the fact that the strength of atmospheric turbulence represented by the GG turbulence model with channel parameters β = 1 and increasing α tends to be closer and closer to 1, i.e., the corresponding SI of the negative exponential atmospheric turbulence model.

Exponentiated Weibull (EW) Model

Atmospheric turbulence is also modeled using the EW distribution in order to consider a wide range of turbulence conditions (weak-to-strong) as well as aperture averaging conditions i.e., when the condition D≥ ρcholds. The corresponding PDF was derived in [40, eqn. (7)]

as follows

2.3. FSO CHANNEL MODELING 23