## 4 Evaluation
### Simulation Setup of the Optical Channel  
According to the analysis presented in [28], the mean SNR of the electrical signal that is direct-detected by the PD that is placed on-board the satellite is given by  
\[\mathrm{SNR_{e,pd}}=\frac{\mathbb{E}\{|i_{\mathrm{d}}(t)|^{2}\}}{\mathbb{E}\{ |n_{\mathrm{o}}(t)|^{2}\}}\approx\frac{I_{D}^{2}\,\beta^{2}}{\mathbb{E}\{|n_{ \mathrm{o}}(t)|^{2}\}}\quad\beta\ll 1, \tag{28}\]  
where  
\[I_{\mathrm{D}}=\mathbb{E}\{i_{\mathrm{D}}(t)\}=\mu\,\frac{G_{\mathrm{o,LK}}\, G_{\mathrm{o,rx}}\,G_{\mathrm{o,edfa}}}{L_{\mathrm{o,fd}}\,L_{\mathrm{o,atm}}\,L_{ \mathrm{o,bsl}}\,L_{\mathrm{o,sys}}}\,\frac{P_{\mathrm{o,ld}}}{2} \tag{29}\]  
is the DC component of the time-varying electrical current \(i_{\mathrm{D}}(t)\) that the PD generates when being excited by the intensity modulated optical signal, \(\beta\) is the intensity modulation index, and  
\[\mathbb{E}\{|n_{\mathrm{o}}(t)|^{2}\}=\mathbb{E}\{|i_{\mathrm{shot}}(t)|^{2} \}+\mathbb{E}\{|i_{\mathrm{thermal}}(t)|^{2}\}+\mathbb{E}\{|i_{\mathrm{rin}}(t )|^{2}\}+\mathbb{E}\{|i_{\mathrm{beat}}(t)|^{2}\} \tag{30}\]  
includes the contribution of all noise sources in the optical feeder link, namely the _shot noise_ sources, _thermal noise_, _Relative Intensity Noise_ (RIN) of LD, and _beat noise_[25]. Note that shot noise term includes the contribution of the received optical signal, the Amplified Spontaneous Emission (ASE) noise, the background optical noise and the dark current noise, whereas the beat noise term accounts the effect of combining the received optical signal with the ASE noise. When the received optical power is between \(-90\) and \(-20\,\mathrm{dBW}\), it can be shown that the beat noise between received optical signal and ASE noise dominates the SNR of the optical feeder link [47]. In this situation,  
\[\mathbb{E}\{|n_{\mathrm{o}}(t)|^{2}\}\approx\mathbb{E}\{|i_{\mathrm{beat}}(t)| ^{2}\}=i_{\mathrm{sig-sp}}^{2}+i_{\mathrm{sp-sp}}^{2}\approx i_{\mathrm{sig- sp}}^{2}=4\,I_{\mathrm{D}}\,I_{\mathrm{ase}}\left(B_{\mathrm{e}}/B_{\mathrm{o}} \right), \tag{31}\]  
where \(B_{\mathrm{o}}\) is the bandwidth of the optical signal at the PD input, \(B_{\mathrm{e}}\) is the bandwidth of the electrical signal at the PD output, and \(I_{\mathrm{ase}}=\mu\,G_{\mathrm{o,edfa}}\,P_{\mathrm{ase}}\) is the DC component generated by the ASE noise, whose equivalent noise power at the input of the EDFA is given by \(P_{\mathrm{ase}}=\rho_{\mathrm{ase}}\,B_{\mathrm{o}}\). Table 2 summarizes the parameters of the optical feeder link, taking into account both the optical gains and losses, as well as the different sources of optical noise [25; 48]. Note that the values of the parameters that appear in this table have been determined taking into account the different phenomena described in Section 3, when modeling the contribution of each of them in the space-to-ground optical link budget. The effect of any other parameter not listed in this table is considered as negligible. When \(L_{\mathrm{o,atm}}=0\,\mathrm{dB}\) (i.e., clear-sky conditions), the DC current at the PD output is \(I_{\mathrm{D}}=75.68\,\mathrm{mA}\), whereas the DC current generated by the ASE noise is \(I_{\mathrm{ase}}=0.125\,\mathrm{mA}\) regardless of the weather. Thus, when we set the intensity modulation index \(\beta=0.5\), the SNR of the electrical signal at the PD output becomes  
\[\mathrm{SNR_{e,pd}}[\mathrm{dB}]=25.01[\mathrm{dB}]-L_{\mathrm{o,atm}}[ \mathrm{dB}]. \tag{32}\]  
Note that for larger intensity modulation indexes \(\beta\), the non-linear distortion introduced by the MZM of the optical transmitter becomes more notable. For those situation, the use of Digital Pre-Distortion (DPD) is necessary to keep non-linear distortion under control [25]. Figure 8 shows the Cumulative Distribution Function (CDF) of the power loss that turbulence-induced beam wander and beam scintillation introduce on the received optical power of the optical uplink transmission in the satellite forward link (see Section 3).