# Time Series Irradiance Synthesizer for Optical GEO Satellite Downlinks in 5G Networks
## 3 Proposed Time Series Synthesizer
### Time Series Generator  
The analysis begins examining the case where no turbulence is present and the optical signal, a single collimated Gaussian beam, is only affected by free-space diffraction. Therefore, for a free space  
Figure 2: Received irradiance time series as measured by optical ground station (OGS) and LUCE terminals.  
optical GEO downlink communication system the received irradiance can be given according to the formula [6; 11; 15]:  
\[I_{no\;turb}(r,L)=\frac{2}{\pi}P_{T}\eta_{T}\eta_{R}\eta_{atm}\frac{1}{W^{2}(L)} \exp\!\left(\frac{-2r^{2}}{W^{2}(L)}\right) \tag{13}\]  
where \(P_{T}(Watts)\) is the transmitted power, \(\eta_{T},\eta_{R}\) are the efficiencies of transmitter and receiver respectively, \(\eta_{atm}\) is the atmospheric transmittance, \(L(m)\) is the link distance (slant path), \(r\) is the radial distance from the beam center and \(W(L)(m)\) is the beam waist after propagation of distance \(L\) given according to the next formula [6; 11; 15]:  
\[W^{2}(L)=W_{0}^{2}\!\left[1+\left(\frac{\lambda L}{\pi W_{0}^{2}}\right)^{2}\right] \tag{14}\]  
where \(W_{0}(m)\) is the initial beam waist of the source at which the irradiance falls to \(1/e^{2}\) (spot size). In (13) the transmitter and receiver gains as well as the free space losses are incorporated. The received power in Watts can be extracted from (13) as \(P_{r}=I\pi D_{r}^{2}/4\).  
In the presence of clear air turbulence, the signal propagation suffers from scintillation effects and beam effects, that is, beam wandering and beam spreading. However, in downlink only the scintillation effects are most significant, and based on the assumptions about Rytov theory and the Kolmogorov spatial spectrum, the received irradiance time series are given by the next expression [6; 11; 15]:  
\[I(r,L,t)=\frac{2}{\pi}P_{T}\eta_{T}\eta_{R}\eta_{atm}\frac{1}{W^{2}(L)}\exp\! \left(\frac{-2r^{2}}{W^{2}(L)}\right)e^{2X_{s}(t)} \tag{15}\]  
The scintillation effects are integrated in the term \(e^{2X_{s}(t)}\) where \(X_{s}\) is the fluctuations log-amplitude. Assuming weak turbulence conditions the log-amplitude is a normally distributed Gaussian process with \(-\sigma_{\chi}^{2}\) mean value, \(\sigma_{\chi}^{2}\) variance [19] and low pass PSD with \(-80/3\) dB/decade slope. In Figure 3 the PSD of downlink irradiance as received by Luce is displayed along with the graph of the function \((cf)^{-\frac{8}{3}}\) where c is a constant. In log-log scale the latter function is a straight line with \(-8/3\) slope and verifies that the spectra of downlink scintillation decreases with \(-80/3\) dB/decade.  
Stochastic differential equations (SDEs) driven by fractional Brownian motion are used to reproduce such processes as reported in [20]. The requested Gaussian process is the solution of the following SDE [20]:  
\[dX_{t,1}=-\lambda_{s}X_{t,1}dt+\sigma dB_{H} \tag{16}\]  
Figure 3: The power spectral density (PSD) of scintillation decreases with \(-80/3\) dB/decade.  
where \(X_{t,1}\) is given by:  
\[X_{t,1}(t)=e^{-\lambda_{s}t}\sigma\int\limits_{0}^{t}e^{\lambda_{s}u}dB_{u}^{H} \tag{17}\]  
The \(\lambda_{s}\) and \(\sigma\) depend on the dynamic parameters of the stochastic process and its long-term statistics, respectively [20]. (\(0<H<1\)) is the Hurst exponent [20].  
The log-amplitude time series \(X_{s}(t)\) can now be expressed as:  
\[X_{s}(t)=-\sigma_{x}^{2}+\sigma_{x}X_{t,1}(t) \tag{18}\]  
where  
\[\sigma_{x}^{2}=\frac{1}{4}\ln(\sigma_{I}^{2}+1) \tag{19}\]