## 2 Rain Attenuation and Phase Delay Due to Rain in Zenith Propagation Paths  
To calculate the complex passband (radio frequency) and complex baseband equivalent transfer functions of direct and orthogonal channels in DSB-SC modulation, defined and discussed in [4] and based on classical linear modulation theory (e.g., [19; 20]), we must know the time series \(A(t)\) (dB) and \(\Phi(\mathrm{t})\) (degrees). Both are calculated with the SST [18], as shown in [4], with the modelling of [21; 22; 23]. Figure 1 shows the average annual probability distribution \(P(R)\) of exceeding \(R\) (mm/h, averaged in 1 min) measured at Spino d'Adda, Madrid and Tampa. The different climatic rain conditions at these sites are clearly evident if the rain rates exceeded at equal probability are compared. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Site** & **Latitude N (\({}^{\circ}\))** & **Longitude E (\({}^{\circ}\))** & **Altitude \(H_{S}\) (km)** & **Precipitation Height \(H_{R}\) (km)** & **Rain Rate Data Bank (Years)** \\ \hline Spino d’Adda (Italy) & 45.4 & 9.5 & 0.084 & 3.341 & 8 \\ Madrid (Spain) & 40.4 & 356.3 & 0.630 & 3.001 & 8 \\ Tampa (Florida) & 28.1 & 277.6 & 0.050 & 4.528 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geographical coordinates, altitude (km), rain height \(H_{R}\) (km), number of years of continuous rain rate time series measurements at the indicated sites. Figure 2a shows the annual probability distribution \(P(A)\) of exceeding \(A\) (dB) calculated with Equation (8) of Reference [4], and Figure 2b shows the corresponding annual probability distribution \(P(\Phi)\) of exceeding \(\Phi\) (degrees) calculated with Equation (9) of Reference [4]. Figure 1: Annual probability distribution (%) \(P(\mathbb{R})\) of exceeding the value indicated in abscissa at Spino d’Adda, Madrid and Tampa. Figure 2: (**a**) Annual probability distribution (%) \(P(\mathbb{A})\) of exceeding the value indicated in abscissa at Spino d’Adda, Madrid and Tampa; (**b**) annual probability distribution (%) \(P(\Phi)\) of exceeding the value indicated in abscissa at Spino d’Adda, Madrid and Tampa, at 80 GHz and with circular polarization. In the following, we consider the relative phase delay (degrees) at radio frequency \(\psi\):  
\[\varphi(\psi)=\Phi(\psi)-\Phi(\psi=f_{c}) \tag{1}\]  
We defined the time delay (picoseconds):  
\[T(t)=\frac{1000}{360\psi}\Phi(t) \tag{2}\]  
where \(\psi\) is measured in GHz and the relative time delay is given by  
\[\tau(f)=T(\psi)-T(\psi=f_{c}) \tag{3}\]  
We defined the normalized magnitude \(|H(\psi)|\) of the passband transfer function \(H(\psi)\), at time \(t\), as  
\[|H(\psi)|=10^{\frac{A(f_{c})-A(\varphi)}{20}} \tag{4}\]  
The real and imaginary parts of \(H(\psi)\) are given by  
\[H_{t}(\psi)=|H(\psi)|\cos(\varphi(\psi)) \tag{5}\]  
\[H_{i}(\psi)=|H(\psi)|\sin(\varphi(\psi))\]  
From Equation (5), according to Reference [4] (see Equations (5)-(7) of [4]), we obtain the equivalent baseband complex transfer function of the direct channel as  
\[H_{ed}(f)=H_{edr}(f)+jH_{edi}(f) \tag{6}\]  
and the equivalent baseband complex transfer function that couples it to the orthogonal channel:  
Figure 1:0  
Now, from \(A(t)\) and \(\Phi(t)\), the transfer function can be calculated at any time \(t\), as we will recall in Section 3. Notice that, since \(R(t)\) is averaged in 1 min, \(A(t)\) and \(\Phi(t)\) are also \"averaged\" in 1 min, so that the time-varying transfer function changes every minute and, therefore, very slowly compared to any digital signal so that, in every minute, the Gaussian channel is affected by the same in-band rain attenuation and phase delay (see [4]). Moreover, we assume that at any time \(t\), the channel SNR is larger than the minimum required by the bit error probability constraint; therefore, the rain fade, in order to be compensated with power control or with other techniques, is that which is measured at the highest frequency of the upper side-band, i.e., at 85 GHz in our exercise.