# An Accurate Computational Tool for Performance Estimation of FSO Communication Links over Weak to Strong Atmospheric Turbulent Channels

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Channel Model

_{0}/2, whereas I is the normalized instantaneous irradiance on the receiver’s side [20].

_{r}is the expected signal power at the receiver while N

_{0}is given as [32,38,39]:

_{B}being the Boltzmann constant, T

_{abs}being the absolute system temperature, F

_{n}is the photodiode noise figure, R

_{L}the load resistor [36], I

_{D}, is the dark current on photodetector [40], RIN the relative intensity noise process and a typical value is −130 dB/Hz, and ${q}_{e}$ is the electron charge, whereas ${I}_{ph}={P}_{r}\eta $ is the average receiver photocurrent [31].

_{f}, before the receiver’s lens are usually used [45]. Thus, assuming that the background noise is Gaussian with spectral density N

_{b}, the total power of the ambient noise, which arrives at the receiver’s input, is B

_{f}N

_{b}. The influence of the ambient noise at the total system’s noise is given through [45] and Equation (6) and has the mathematical form:

_{r}, taking into account the visibility at the area of the FSO link and losses due to the scintillation effect, using [2,33,46,47], we make a conclusion of the following expression:

_{R}and D

_{T}are the transmitter’s and receiver’s aperture, $\theta $ stands for the beam’s divergence, a

_{tot}, represents the atmospheric attenuation that includes the fog phenomenon, a

_{f}, the rain, a

_{r}, the snow a

_{sn}, and the scintillation effect, a

_{s}, which affect significantly the performance of the FSO links while ${L}_{m}$, stands for the miscellaneous losses of the system, as it is presented in [2,4,33,46,47]. More specifically, the attenuation parameter due to the fog is given through the Kim or the Kruse model depending on the effect’s strength, and the atmospheric visibility V with sky droplets are set to τ

_{TH}= 5% [47]. Furthermore, the scintillation losses parameter in dB is given as: ${a}_{s}=\sqrt{92.68\cdot {\left(2\pi {10}^{9}/\lambda \right)}^{7/6}{C}_{n}^{2}{L}^{11/6}}$ [47].

#### 2.1. Lognormal Turbulence Model

#### 2.2. Gamma–Gamma Turbulence Model

_{v}is the modified Bessel function of the second kind of order v, and a, b are parameters which can be defined from the link’s parameters and given through the expressions [35,39]:

## 3. Performance of the FSO System

#### 3.1. Outage Probability of the System

_{out}. More specifically, this quantity represents the probability of the instantaneous SNR to fall below a critical threshold, γ

_{th}, which is set by the receiver’s sensitivity limit [9,27,29,30]:

#### 3.2. Average Channel Capacity

_{k}parameters that are taken into account are given in [13,29].

## 4. Algorithm Structure for the Computational Tool

## 5. Numerical Results

^{−16}m

^{−2/3}, 6.0 × 10

^{−15}m

^{−2/3}, ${C}_{n}^{2}$ = 20.0 × 10

^{−15}m

^{−2/3}, for weak, moderate and strong turbulence conditions, respectively [36], for link lengths up to 5 km. For these parameter values and contiguous to these, by using the specific computational tool, the following figures for the presented above performance metrics are obtained.

_{out}≅ 6.5 × 10

^{−4}. In addition, the normalized average capacity also has the maximum efficiency of 18.63 (b/s/Hz) for weak turbulence conditions at 5 km. For longer link lengths, the scintillation effect strongly affects the system’s operation, and, thus, the lognormal distribution cannot give accurate results. For this reason, for moderate to strong turbulence conditions, i.e., ${\sigma}_{l}^{2}$ > 0.3, we use the gamma–gamma distribution [1,3,21,30]. Thus, from Figure 3 and Figure 4 and Table 2 and Table 3, it is clear that when the turbulence strength increases, the system’s performance decreases significantly.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

Algorithm A1. Outage Probability on Gamma—Gamma Distribution |

Require: Received signal Equation (6) P; parameters _{r}a, b from Equation (13); the receiver sensitivity thresh as clear value not in dBm; the number of lengths N that we simulate until achieving the final path; |

fgg=@(x,aa,bb,xm) %implement the Equation (14) as a function of x, aa, bb ,xm parameters |

for i1=1 to 3 do % i1 the number of 3 conditions that we study like weak, moderate, strong |

for i=1 to N do %Loop until achieve the final path |

X=logspace(−10,10,10000) %create the regions for numerical method trapezoidal |

Ftrapez = fgg(X,a(i,i1),b(i,i1),Pr(i,i1)) % the trapez function |

Pout_trapez(i,i1)=trapz(X,ftrapez) %the Outage probability with numerical method of trapezoidal |

Pout_integ(i,i1)=integral(@(x)fgg(x,a,b,Pr(i,i1)),0,thresh) %the integration method of outage probability with regions from 0 to thresh |

end |

find first element >=10^{−20} of Pout_integ that is j to avoid any chances of getting stuck in the Meijer implementation below and update i with j |

for i to N do %Loop until achieve the final path |

A=a(i1)+b(i1)/2, B= a(i1)+b(i1)/2,C=b(i)-a(i) |

K=((a*b)^A)/(gamma(a)*gamma(b)) |

K1(i)=(thresh / Pr(i,i1))^A/2 , z= a(i1)* b(i1) *(√ thresh / Pr(i,i1)) |

Gmeijer(i)= meijerG((2,1,[1-A],[B,C,-A],a*b),z) %the analytical method by meijer function in eq(18) |

Pout_mejer(i,i1)= K1(i)*K*Gmeijer(i) % Outage probability by analytical method |

end |

end |

Output: Pout_trapez, Pout_int,Pout_mejer . |

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**Figure 3.**Outage probability estimation, with the proposed computational tool for weak to strong turbulence conditions, (

**a**) for links up to 4 km with receiver threshold set as −30 dBm; (

**b**) for total link length up to 5 km with receiver threshold set as −40 dBm. These results use the lognormal or the gamma–gamma distribution as mentioned in the plot legend.

**Figure 4.**Average capacity performance metric for weak to strong atmospheric turbulence conditions (

**a**) for links up to 3 km with receiver threshold set as −40 dBm; (

**b**) for links up to 5 km with receiver threshold set as −30 dBm. These results use the suitable distribution model, i.e., lognormal or gamma–gamma depending on the value of the Rytov parameter, as presented in Equation (3).

Parameter | Symbol | Value |
---|---|---|

Transmitted Power | P_{T} | 400 mW (26 dBm) |

Wavelength | λ | 1550 nm |

Transmitted Aperture Diameter | D_{T} | 2 mm |

Beam Divergence | θ (theta) | 1 mrad |

Miscellaneous Losses | L_{m} | 1 dB |

Bandwidth | B | 0.5 Ghz |

Receiver Aperture | D_{R} | 180 mm |

Receiver Sensitivity | S (threshold) | −30 or −40 dBm |

Sky Droplets | τ_{TH} | 5% |

Detector Responsitivity | η | 0.8 A/W |

Boltzmann Constant | K_{B} | 1.38 × 10^{−23} |

Electron Charge Constant | q_{e} | 1.602 × 10^{−19} Cb |

Relative Intensity Noise | RIN | −130 dB/Hz |

Receiver Noise Figure | F_{n} | 1 (0 dB) |

Dark Current | I_{D} | 6 nA |

Load Resistor | R_{L} | 50 Ω |

Temperature | Tabs | 288 K |

Visibility | V | 20 km |

**Table 2.**Outage probability for an FSO link with the parameters of Table 1.

Threshold | Length | ${\mathit{C}}_{\mathit{n}}^{\mathbf{2}}$ (m^{−2/3}) | Rytov Variance ^{1}, ${\mathit{\sigma}}_{\mathit{l}}^{\mathbf{2}}$ | Outage Probability | SNR, μ (dB) | |
---|---|---|---|---|---|---|

Pr (<10^{−3}) | Drop System | |||||

1.0 × 10^{−15} | 0.253 | 0.00065 | 3450 m | 62.08 | ||

−30 dB | 4 km | 8.0 × 10^{−15} | 2.023 | 0.00046 | 2100 m | 47.88 |

2.0 × 10^{−14} | 5.057 | 0.00044 | 1700 m | 33.87 | ||

7.8 × 10^{−16} | 0.297 | 5.4 × 10^{−7} | 54.52 | |||

−40 dBm | 5 km | 6 × 10^{−1}^{5} | 2.284 | 0.00070 | 3050 | 38.53 |

2.0 × 10^{−14} | 7.613 | 0.00057 | 2250 | 17.00 |

^{1}${\sigma}_{l}^{2}$ ≤ 0.3 denotes weak turbulence, whereas 0.3 < ${\sigma}_{l}^{2}$ < 5 denotes moderate to strong (3).

**Table 3.**Average channel capacity for an FSO link with the parameters of Table 1.

Theshold | Length | ${\mathit{C}}_{\mathit{n}}^{\mathbf{2}}$ (m^{−2/3}) | Rytov Variance ^{1}, ${\mathit{\sigma}}_{\mathit{l}}^{\mathbf{2}}$ | Average Capacity (b/s/Hz) | SNR, μ (dB) | |
---|---|---|---|---|---|---|

Capacity | Distribution | |||||

2 × 10^{−1}^{5} | 0.298 | 22.91 | LN | 69.11 | ||

−40 dBm | 3 km | 6 × 10^{−1}^{5} | 0.895 | 21.22 | GG | 64.14 |

2.0 × 10^{−14} | 2.984 | 17.32 | GG | 52.60 | ||

5.0 × 10^{−16} | 0.190 | 18.63 | LN | 56.21 | ||

−30 dBm | 5 km | 4 × 10^{−1}^{5} | 1.523 | 14.18 | GG | 43.24 |

2.0 × 10^{−14} | 7.613 | 5.46 | GG | 17.00 |

^{1}${\sigma}_{l}^{2}$ ≤ 0.3 denotes weak turbulence, whereas 0.3 < ${\sigma}_{l}^{2}$ < 5 denotes moderate to strong (3).

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**MDPI and ACS Style**

Katsilieris, T.D.; Latsas, G.P.; Nistazakis, H.E.; Tombras, G.S.
An Accurate Computational Tool for Performance Estimation of FSO Communication Links over Weak to Strong Atmospheric Turbulent Channels. *Computation* **2017**, *5*, 18.
https://doi.org/10.3390/computation5010018

**AMA Style**

Katsilieris TD, Latsas GP, Nistazakis HE, Tombras GS.
An Accurate Computational Tool for Performance Estimation of FSO Communication Links over Weak to Strong Atmospheric Turbulent Channels. *Computation*. 2017; 5(1):18.
https://doi.org/10.3390/computation5010018

**Chicago/Turabian Style**

Katsilieris, Theodore D., George P. Latsas, Hector E. Nistazakis, and George S. Tombras.
2017. "An Accurate Computational Tool for Performance Estimation of FSO Communication Links over Weak to Strong Atmospheric Turbulent Channels" *Computation* 5, no. 1: 18.
https://doi.org/10.3390/computation5010018