# Time Series Irradiance Synthesizer for Optical GEO Satellite Downlinks in 5G Networks
## 3 Proposed Time Series Synthesizer  
### Important Metrics  
The strength of atmospheric turbulence in downlink can be estimated by the scintillation index (SI). The scintillation index is the normalized variance of the received irradiance and is derived from experimental data [5; 6; 11]:  
\[\sigma_{I}^{2}=\frac{\left\langle I^{2}\right\rangle}{\left\langle I\right\rangle ^{2}}-1 \tag{1}\]  
where \(I\) is the irradiance. The [PERSON] method indicates that the scintillation index can be expressed in terms of the variance of the field log-amplitude \(\sigma_{\chi}^{2}\) as:  
\[\sigma_{I}^{2}=\exp(4\sigma_{\chi}^{2})-1 \tag{2}\]  
and under weak turbulence (SI \(<1\)), the following approximation is valid [5; 6; 11]:  
\[\sigma_{I}^{2}\approx 4\sigma_{\chi}^{2} \tag{3}\]  
A theoretical formula for the evaluation of scintillation index for a plane wave is provided by [PERSON] [5; 6; 11]:  
\[\sigma_{R}^{2}=2.25k^{\frac{7}{6}}\sec^{\frac{11}{6}}(\zeta)\int\limits_{H_{GS }}^{H_{C}+H_{Turb}}C_{n}^{2}(h)(h-H_{GS})^{\frac{5}{6}}dh \tag{4}\]  
where \(k=2\pi/\lambda(rad/m)\) is the wavenumber, \(\lambda(m)\) is the wavelength, \(\zeta(rad)\) is the zenith angle, \(h(m)\) is the altitude, \(H_{GS}(m)\) is the ground station's altitude, \(H_{Turb}(m)\) is the turbulence altitude and is considered negligible for altitudes higher than 20 km [5], that is assuming \(H_{Turb}\) = 20,000 m.  
Finally, \(C_{n}^{2}(h)\) is the refractive index structure parameter as a function of the altitude \(h\). The most common model for \(C_{n}^{2}(h)\) is the Hufnagel-Valley Model (H-V) modified however to incorporate the ground station's altitude [16]:  
\[ \begin{array}{l}C_{n}^{2}(h)=A_{0}\exp(-\frac{H_{GS}}{700})\exp((H_{GS}-h)/1 00)+5.94\times 10^{-53}\times\left(\frac{u_{rms}}{27}\right)^{2}h^{10}\exp(- \frac{h}{1000})+\\ +2.7\times 10^{-16}\exp(-\frac{h}{1500})\end{array}  \tag{5}\]  
where \(A_{0}\)(\(m^{-2/3}\)) is the refractive index structure parameter at ground level and \(u_{rms}(m/s)\) is the rms wind speed at high altitude:  
\[u_{rms}=\sqrt{\frac{1}{15\times 10^{3}}\int\limits_{5\times 10^{3}}^{20\times 1 0^{3}}V^{2}(h)dh} \tag{6}\]  
where \(V(h)\) is the wind profile and can be estimated using the Bufton wind model [5; 6; 11]:  
\[V(h)=w_{s}h+V_{g}+30\exp\left[-\left(\frac{h-9400}{4800}\right)^{2}\right] \tag{7}\]  
where \(w_{s}(rad/sec)\) is the slew rate indicating the relative motion between the satellite and the ground station--minimal for GEO satellite--and \(V_{g}(m/s)\) is the wind speed on ground. To take into account the aperture averaging effect to (4), the Aperture Averaging Factor \(A(D_{r})=\sigma_{I}^{2}/\sigma_{R}^{2}<1\) is employed where \(D_{r}(m)\) is the receiver's aperture diameter. SI for a receiving telescope can be expressed as the following [6; 11; 12]:  
\[\sigma^{2}_{I}=A(D_{r})\sigma^{2}_{R} \tag{8}\]Aperture averaging factor can also be derived from experimental data:  
\[A(D_{r})=\left\langle\sigma_{I,Data}^{2}\right\rangle/\left\langle\sigma_{R}^{2}\right\rangle \tag{9}\]  
where \(\left\langle\sigma_{I,Data}^{2}\right\rangle\) is the mean value of the SIs calculated from the data. A notable theoretical approach for \(A(D_{r})\) is the following [17]:  
\[A(D_{r})=\left[1+1.1\left(\frac{{D_{r}}^{2}}{\lambda h_{s}\sec(\zeta)}\right) ^{\frac{7}{6}}\right]^{-1} \tag{10}\]  
where  
\[h_{s}=\left[ \begin{matrix}H_{GS}+H_{Turb}\\ \int\limits_{H_{GS}}^{H_{GS}+H_{Turb}}C_{n}^{2}(h)(h-H_{GS})^{2}dh\\ \int\limits_{H_{GS}}^{H_{Turb}}C_{n}^{2}(h)(h-H_{GS})^{5/6}dh\end{matrix} \right] \tag{11}\]  
For better understanding the aperture averaging, in Figure 2 the time series of irradiance values as measured by LUCE and ESA's OGS receivers are depicted. It is obvious that in the case of the LUCE telescope, there is more irradiance fluctuation than in ESA's bigger telescope, implying that large size terminals benefit more from the aperture averaging effect. The SI computed from LUCE data using (1) is 0.0130 while the SI from OGS data is 0.0015.