## 4 Experimental Interference  
Numerical results have been obtained by means of time-domain simulations of the system described in Figure 7 in MATLAB/Simulink(r). We have simulated the transmission of a sequence of \(10^{4}\) symbols drawn from a quadrature phase-shift keying (QPSK) modulation of long sequences of independent bits at the bit rate \(\rho\) (bits per second, bps) according to the value of the roll-off factor \(r\). Since the baseband width is fixed to \(B=5\) GHz, the bit rate is a function of \(r\), given by  
\[\rho=\frac{2B}{1+r} \tag{8}\]  
We have simulated bit rates corresponding to \(r=0.25\), i.e., \(\rho=8\) Gbps (gigabits per second); \(r=0.50\), \(\rho=6.67\) Gbps; \(r=1\), \(\rho=5\) Gbps. A numerical approximation of the continuous-time square-root Nyquist filter \(\sqrt{S(f)}\) has been implemented by considering a windowing of 16 symbol intervals and an oversampling factor of 4 compared to the symbol rate. Because we refer to normalized values, the algebraic sum of total interference \(s_{sd}+s_{sq}\) gives the factor by which the SNR \(\Gamma\) obtainable in the ideal channel (Figure 6) must be multiplied to obtain the SNR in the presence of ISI and QI. Therefore, we defined the channel factor \(\mu\):  
\[\mu=s_{sd}+s_{sq} \tag{9}\]  
The SNR obtainable with interference is given by  
\[\gamma=\mu^{2}\Gamma \tag{10}\]  
\[\gamma(\text{dB})=10\times\log_{10}\mu^{2}+10\times\log_{10}\Gamma\]  
In applying these equations, we assume that the channel Gaussian noise power is constant, although it should slightly increase with rain attenuation. Our main results are the histograms (relative frequency) of the sampler output amplitude (dB) first but only with ISI and then only with ISI and QI, i.e., with the total interference. To show the dependence on the rain attenuation calculated at the center frequency (i.e.,  
Figure 7: Flowchart of the quadrature baseband receiver in rain attenuation. \(S(f)\) is the two-sided spectrum of the Nyquist reference pulse assumed to be positive, \(\sqrt{S(f)}\) is the matched filter and \(n(t)\) is the receiver total additive Gaussian white noise for each channel of equal power. 80 GHz), we have considered the cases in which rain attenuation (and corresponding time delay) was in the following ranges: \(9\leq A_{80\text{ GHz}}\leq 11\) dB (referred to, for short, as \"10\" dB), \(18\leq A_{80\text{ GHz}}\leq 22\) dB (\"20\" dB) and \(27\leq A_{80\text{ GHz}}\leq 33\) dB (\"30\" dB). Figure 8 shows the relative frequency histogram of the channel factor \(\mu\) (dB) for \(r=0.25\), Figure 9 for \(r=0.50\) and Figure 10 with \(r=1\). From these results, we can draw the following general conclusions:  
1. The three sites, although in different climatic zones, are practically indistinguishable in all cases. 2. All histograms show even symmetry; therefore, indicating that for about 50% of the time, we can consider the channel factor to be either \(\mu<0\) (dB), therefore, \(\gamma<\Gamma\); or \(\mu>0\) (dB), therefore, \(\gamma>\Gamma\). 3. Histograms with only ISI and with ISI + QI are distinguishable. 4. With ISI only, the lowest range of attenuation (10 dB) shows more marked peaks at \(\mu=0\) dB and \(\mu\approx\pm 0.3\) dB. These peaks are largely smoothed when QI is also added. Figure 8: Relative frequency histogram of the channel factor \(\mu\) (dB) with ISI (**upper panel**) and with ISI and QI (**lower panel**). **Color key**: blue = Spino d’Adda; green = Madrid; red = Tampa. **Curve key**: continuous \(9\leq A_{80\text{ GHz}}\leq 11\) dB; dashed \(18\leq A_{80\text{ GHz}}\leq 22\) dB; \({}^{\prime}+^{\prime}27\leq A_{80\text{ GHz}}\leq 33\) dB. Roll-off factor \(r=0.25\). The roll-off factor \(r\) plays a small role because the values of the peaks change a little, especially for \(r=1\). Because of these general findings, in Figure 11, we have drawn the average relative frequency histograms by distinguishing only the three attenuation ranges.