## 4 Passband and Baseband Transfer Functions  
Figures 4 and 5 show the baseband receiver in ideal and in rain conditions. \(S(f)\) is the two-sided spectrum of the Nyquist impulse, \(\sqrt{S(f)}\) is the matched filter, and \(n(t)\) is the receiver total additive Gaussian white noise.  
Figure 3: Annual probability distributions (%) \(P(R)\) at Spino d’Adda, Madrid, Tampa, and Prague.  
the real and imaginary parts of \(H(\psi)\) are given by  
\[H_{r}(\psi)=|H(\psi)|\cos(\varphi(\psi)) \tag{14}\]  
\[H_{i}(\psi)=|H(\psi)|\sin(\varphi(\psi))\]  
From Equation (14), Equation (5) (direct channel), and Equation (7) (orthogonal channel interference), we can calculate the baseband equivalent transfer functions.  
An example illustrates calculations and findings. Figure 6 shows an SST-simulated rain attenuation event at Spino d'Adda at 80 GHz and the relative rain attenuation at the extremes of the 10-GHz bandwidth centered at 80 GHz. In this particular event, the maximum rain attenuation is 32.6 dB and at the extremes of the bandwidth, it can change by about \(0.7\sim 0.9\) dB. This behavior is, of course, due to the increase in rain attenuation with frequency, and this fact can produce amplitude distortions in baseband signals.  
For this particular rain event, we have calculated the instantaneous (1-min) passband transfer function at the time of maximum rain attenuation. Figure 8 shows the real and imaginary parts of the passband transfer function, as shown in Equation (12).  
Figure 7 shows the time delay at 80 GHz and the relative time delay at the extremes of the bandwidth. The time delay is very small, as was also shown in [16], but more importantly, it is not a constant across the bandwidth, and this fact can produce phase distortions in the baseband signal.  
Figure 6: SST-simulated rain attenuation event at Spino d’Adda at 80 GHz (8 May 2000, starts at 0:48 AM local time). **Upper panel**: Rain attenuation at 80 GHz (circular polarization). **Lower panel**: Relative attenuation at the extreme of a 10-GHz bandwidth.  
In Figure 8, we can observe that \(H_{r}(\psi)\) is neither a constant nor an even function about \(\psi=f_{c}\); therefore, the output of the direct channel is distorted in amplitude. \(H_{i}(\psi)\) is, to a first approximation, a linear odd function; therefore, the output of the direct channel should be lightly distorted in phase at this particular time. As for the interference produced by the orthogonal channel on the direct channel (cosine), it will be present because also in this channel, \(H_{r}(\psi)\), as observed, is not an even function. \(H_{i}(\psi)\) might produce a small phase distortion.  
Figure 8: Real (**upper panel**) and imaginary (**lower panel**) parts of the passband transfer function at the time of maximum rain attenuation (32.6 dB) in the event shown in Figures 6 and 7.  
Figure 7: SST-simulated rain attenuation event at Spino d’Adda at 80 GHz GHz (8 May 2000, starts at 0:48 AM local time). **Upper panel**: Time delay at 80 GHz (circular polarization). **Lower panel**: Relative time delay at the extreme of a 10-GHz bandwidth.  
In the next section, we report averages and standard deviations of passband and equivalent baseband transfer functions for several rain attenuation ranges. These data will be needed to compensate, on average, for these linear distortions.