Long Term Irradiance Statistics for Optical GEO Satellite Feeder Links: Validation Against Experimental Data

Abstract

In this paper, optical feeder links to support high throuput satellite communication systems are studied. A methodology for the generation of the received irradiance/power time series for GEO optical feeder links (ground-to-GEO), that reproduces the long term irradiance/power statistics, is presented. In the proposed synthesizer, the atmospheric phenomena under clear sky that degrade the optical signal and the weak turbulence effects, i.e. scintillation, beam wandering and beam spreading, are considered. The proposed methodology takes advantage of the use of stochastic differential equations driven by fractional Brownian motion, for the time series synthesis. The methodology refers to the transmission of a single collimated beam. For the modelling of turbulence effects, Kolmogorov turbulence is assumed and the model is based on Rytov theory under weak fluctuations. Additionally, the proposed model is validated in terms of first order statistics with experimental data from the bi-directional optical GEO ARTEMIS satellite link experiment showing good agreement. Finally, the synthesizer is employed for the generation of fist order statistics for different transmission scenarios.

Appendix

More information about the use of Stochastic Differential Equations (SDE) driven by fractional Brownian Motion for the generation of a zero-mean unity-variance Gaussian process with a power spectral density of low-pass shape are reported in this “Appendix”. The underlined Gaussian process is generated through [26]:

$$\begin{aligned} d{\chi _t} = f\left( {t,{\chi _t}} \right) dt + g\left( {t,{\chi _t}} \right) d{B_H} \end{aligned}$$

(19)

Since for a given variance, ${\chi _{t}}$ can be considered as a Gaussian process with zero mean value, the Langevin equation with fBm (fractional Langevin equation) is used for modeling the time series of log-amplitude, given that the variance equals to 1, ${\chi _{t,1}}$:

$$\begin{aligned} {d{\chi _{t,1}} = - {\lambda _s}{\chi _{t,1}}dt + \sigma d{B_H}} \end{aligned}$$

(20)

The solution of the above equation with zero initial value is:

$$\begin{aligned} {\chi _{t,1}} = {e^{ - {\lambda _s}t}}\sigma \int \limits _0^t {{e^{{\lambda _s}u}}dB_u^H} \end{aligned}$$

(21)

The above process is called fractional Ornstein-Uhlenbeck process and it is a Gaussian process with zero mean value and the co-variance of ${\chi _{t}}$, ${\chi _{t,s}}$ is [26]:

$$\begin{aligned} Cov\left( {{\chi _{t,1}},{\chi _{t + s,1}}} \right) = {\sigma ^2}\frac{{\Gamma \left( {2H + 1} \right) \sin \left( {\pi H} \right) }}{{2\pi }}\int \limits _{ - \infty }^{ + \infty } {{e^{isy}}\frac{{{{\left| y \right| }^{1 - 2H}}}}{{{\lambda _s}^2 + {y^2}}}dy} \end{aligned}$$

(22)

H is Hurst index.

Therefore, the variance of the Ornstein-Uhlenbeck process is:

$$\begin{aligned} \sigma _{{\chi _{t,1}}}^2 = {\sigma ^2}\frac{{\Gamma \left( {2H + 1} \right) \sin \left( {\pi H} \right) }}{{2\pi }}\int \limits _{ - \infty }^{ + \infty } {\frac{{{{\left| y \right| }^{1 - 2H}}}}{{{\lambda _s}^2 + {y^2}}}dy} \end{aligned}$$

(23)

Since, firstly we want to have a unitary variance process we set:

$$\begin{aligned} \sigma = \frac{1}{{\sqrt{\frac{{\Gamma \left( {2H + 1} \right) \sin \left( {\pi H} \right) }}{{2\pi }}\int \nolimits _{ - \infty }^{ + \infty } {\frac{{{{\left| y \right| }^{1 - 2H}}}}{{{\lambda _s}^2 + {y^2}}}dy} } }} \end{aligned}$$

(24)

For the energy spectrum of fractional Ornstein-Uhlenbeck process holds that [26]:

$$\begin{aligned} {F_{{\chi _{t,1}}}}(f) = {\sigma ^2}\frac{{{F_{{\omega _b}}}(f)}}{{\lambda _s^2 + {{\left( {2\pi f} \right) }^2}}} \end{aligned}$$

(25)

where ${{F_{{\omega _b}}}(f)}$ is the energy spectrum of fractional Gaussian noise and it holds that:

$$\begin{aligned} {F_{{\omega _b}}}(f)\backsim {f^{ - \left( {2H - 1} \right) }} \end{aligned}$$

(26)

From (16) and (17), it holds for high frequencies that:

$$\begin{aligned} {F_{{\chi _{t,1}}}}(f)\backsim {f^{ - \left( {2H + 1} \right) }} \end{aligned}$$

(27)

with corner frequency:

$$\begin{aligned} {\omega _c} = {\lambda _s}^{\frac{1}{{H + 1/2}}} \end{aligned}$$

(28)

The corner frequency can be modeled from the temporal spectrum of turbulence.

To sum up, the main steps of the methodology are summarized. To begin with, taking into account the the slope of the power spectral density, the Hurst index is calculated using Eq. (27). Then, $\lambda _s$ parameter is computed from the corner frequency according to Eq. (28) and the parameter $\sigma$ is computed from Eq. (24). Finally, employing Eq. (21) time series are generated.