the mean value of the irradiance, and m1 > 0 is an extra shape parameter that is strongly dependent on the receiver aperture size. By fitting the EW turbulence model to simulated or experimental PDF data, several specific values of the parameters m1, m2 and m3 as well as some expressions for evaluating these parameters have been obtained in [40,41].
In this thesis, expressions obtained in [41, eqs. (20)-(22)] are used for moderate-to-strong turbulence conditions. The corresponding expressions of the EW parameters are given as a function of the SI as follows
m1 = 7.220σ2/3I
a
Γ
2.487σI2/6
a
− 0.104
, (2.23a)
m2 = 1.012(m1σ2Ia)−13/25+ 0.142, (2.23b)
m3 = 1
m1Γ (1 + 1/m2) g1(m1, m2), (2.23c) where gn(m1, m2) is defined as follows
gn(m1, m2) = X∞ k=0
(−1)kΓ(m1)
k!(k + 1)1+n/m2Γ(m1− k). (2.24) It must be noted that the above expressions are valid when AA < 0.9 [41]. At the same time, we can express the SI as a function of the EW parameters as
σ2Ia = Γ
1 +m2
2
g2(m1, m2) a
Γ 1 +m1
2
g1(m1, m2)2 − 1. (2.25)
2.3 FSO Channel Modeling
2.3.1 Atmospheric Attenuation
The atmospheric turbulence along with absorption and scattering are the three basic pro-cesses that affect optical wave propagation. The two phenomena that produce optical power attenuation in optical wave propagation are absorption and scattering. Both of them are wavelength and weather dependent. During the absorption process, atmospheric molecules absorb energy from incident photons. Scattering is the result of photons colliding with atmo-spheric particles [12,13]. Unlike the absorption process, there is no loss of energy during the scattering process, only a directional redistribution of energy that may result in a significant reduction in beam intensity for longer distances. There are two kinds of scattering according to the physical size: Rayleigh scattering and Mie scattering. The first one is produced by haze and air molecules that are smaller than wavelength, and the second one is produced by particles that present a size quite similar to wavelength. Note that the atmospheric effects
Table 2.1: Values of q for different visibility conditions.
Visibility q
V > 50 km 1.6
6 km < V < 50 km 1.3 V < 6 km 0.585V−1/3
are a consequence of using shorter wavelengths. The atmospheric attenuation has been well studied in the literature.
The attenuation of laser power through atmosphere is determined by the exponential Beers-Lambert law as follows
L(dm) = P (dm)
P (0) = exp (−Φdm) , (2.26)
where L(dm) is the loss over a propagation link of length dm, P (0) is the laser power at the transmitter, P (dm) is the laser power at a distance dm, and Φ is the atmospheric attenuation coefficient described in [52]. The path loss is considered as a deterministic factor that depends on size and distribution of the atmosphere particles and wavelength. Furthermore, this factor is expressed in terms of the visibility, which can be measured directly from the atmosphere. A formula to compute the attenuation coefficient is given in [52]:
Φ = 3.91 V
λ
550× 10−9
−q
, (2.27)
where V is the visibility, and q is the size distribution of the particles related to visibility as can be seen in Table 2.1.
2.3.2 Dynamic Misalignment Statistical Model
FSO communication links are strongly affected by pointing errors, resulting in serious mis-alignment of fixed-position laser communication systems. An accurate mis-alignment between transmitter and receiver is required [10,22,53]. Pointing accuracy is a critical issue in deter-mining link performance and reliability and they can be arisen due to many different factors such as building sway and mechanical errors. Firstly, building sway is due to wind loads, differential heating and cooling, or ground motion over time that can result in an impor-tant misalignment error [24,54]; secondly, mechanical errors are due to errors in tracking systems or mechanical vibrations present in the FSO system [55]. Due to the narrowness of the optical beam and the fact that aperture receivers have a limited field of view (FOV), building sway can even lead to link outages [22]. Hence, pointing errors play an important role in channels fading characteristics.
2.3. FSO CHANNEL MODELING 25 Statistical modeling of the pointing errors have been studied in the literature. In this thesis, a general misalignment fading model given in [53] by Farid and Hranilovic is used as the cornerstone of pointing error models, where the effect of beam width, detector size and jitter variance is considered. In this way, the attenuation due to geometric spread and pointing errors can be approximated, as in [53], by
Ip(r; z)≈ A0exp −2r2 ω2zeq
!
, r≥ 0, (2.28)
where v =√ πa/√
2ωz, A0 = [erf(v)]2 is the fraction of the collected power at r = 0, a = D/2 is the radius of a circular detection aperture, and ωz2eq = ωz2√
πerf(v)/2v exp(−v2) is the equivalent beam width. The beam width ωz can be approximated by ωz = θz, where θ is the transmit divergence angle describing the increase in beam radius with distance from the transmitter. The approximation in Eq. (2.28) is in good agreement with the exact value when the beam width ωz > 6a, as shown in [53, appendix]. The approximate expression for Ip can even be used when ωz< 6a but obtaining a normalized mean-squared error (NMSE) NMSE > 10−3.
The radial displacement r at the receiver plane can be expressed as r2 = x2+ y2, where x and y represent the horizontal displacement and the elevation, respectively. Moreover, the radial displacement is distributed according to a Rayleigh distribution when x and y are modeled as independent Gaussian RVs with zero means and same jitters for the horizontal displacement and the elevation, i.e. x∼ N(0, σs) and y∼ N(0, σs), whose PDF is given by
fr(r) = r σ2s exp
− r2 2σs2
. r ≥ 0. (2.29)
Thus, combining Eq. (2.28) and Eq. (4.28), the corresponding PDF of the irradiance Ip is obtained as follows
fIp(i) = ϕ2
Aϕ02iϕ2−1, 0≤ i ≤ A0 (2.30) where ϕ = ωzeq/2σs is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver. This pointing error model and its corresponding derivation can be seen in greater detail in [53].
Note that this pointing error model does not take into account the effect of nonzero boresight, which will be addressed later. In that model, the radial displacement r at the receiver follows a lognornal-Rice distribution, and it was presented in [56]. This effect and other ones such as different jitters for the horizontal displacement and the elevation and correlated sways will take into consideration in Chapter 4.
2.3.3 Composite Fading Channel
Before computing the probability distribution of the channel I, some comments about con-sidering atmospheric turbulence and pointing errors statistically independent are required.
As we know well, pointing errors are due to building sway and, hence, the correlation time is on the order of a few seconds [57], which is bigger than correlation time of the atmo-spheric turbulence (10-100 ms). Hence, both effects can be considered to be statistically independent.
The probability distribution of the channel I = L· Ia· Ip can be expressed as follows fI(i) =
Z ∞ i/A0L
fI|Ia(i|ia)fIa(ia)dia, (2.31) where fI|Ia(i|ia) is the conditional probability given an atmospheric turbulence state, fIa(ia) is the corresponding PDF of atmospheric turbulence, and L acts as a scaling factor. In this way, the conditional probability is expressed as
fI|Ia(i|ia) = 1
The integral in Eq. (2.32) can be computed by substituting the corresponding statisti-cal model of atmospheric turbulence. In GG atmospheric turbulence, we can substitute Eq. (2.18) and Eq. (2.32) into Eq. (2.31), resulting in
fI(i) = ϕ22(αβ)α+β2 In order to solve the above integral, we can express the function Kν(·) in terms of the Meijer’s G-function (See Appendix A.3.1) and, then, we can use [58, eqn. (07.34.21.0085.01)] (See AppendixA.3.2) to obtain the corresponding closed-form expression of the combined PDF of I as follows
where Gm,np,q [·] is the Meijer’s G-function (See AppendixA.3). The corresponding cumulative distribution function (CDF) is given by
FI(i) = Prob(I ≤ i) =
The above integral can be derived by using [59, eqn. (1.16.2.1)] (See Appendix A.3.2) as follows
It must be noted that the above equations appear to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of FSO communication systems, resulting in
2.4. PERFORMANCE OF FSO COMMUNICATION SYSTEMS 27