## 6 Probability of Bit Error  
We estimate how the non-ideal equivalent baseband transfer function of the direct channel, and the presence of interference coming from the orthogonal channel, affect the probability of bit error in the direct channel. We compare the ideal case, i.e., no rain attenuation, to the case with rain attenuation. We assume a Nyquist impulse with roll-off factor \(r=1\) and consider only the amplitude and phase distortions due to two simultaneous impulses present in the direct and orthogonal channels. In other words, we do not consider the interference due to several contiguous impulses in the direct and cross-channels. As we assume \(r=1\), the tails of these contiguous impulses should be at a minimum; however, only a full Monte Carlo simulation with actual random sequences of bits would give a clearer insight into the effects of these linear distortions. This issue will be the topic of future work. For assessing how the non-ideal equivalent baseband transfer function of the direct channel and the presence of interference affect the probability of bit error in the direct channel, we have calculated the ratio between the probability of bit error with rain attenuation, \(P_{A}(\varepsilon)\), and probability of bit error \(P(\varepsilon)=10^{-2},\;10^{-3},\;10^{-4},\;10^{-5},\;10^{-6}\) with no rain attenuation:  
\[\rho=\frac{P_{A}(\varepsilon)}{P(\varepsilon)} \tag{15}\]  
In other words, with \(\rho\), we compare the probability of bit error due to the SNR in ideal conditions to that due to the SNR in rain conditions, by assuming the same system noise. Usually \(\rho>1\), and rarely \(\rho<1\). For these calculations, we have first considered the average transfer functions. Figure 13 shows the results obtained in two cases: (a) without interference and (b) with interference due to an impulse with the opposite sign of that of the direct channel (assumed positive); and with interference due to an impulse with the same sign of that of the direct channel, at Spino d'Adda. In other words, these two cases, at least for the average transfer functions, give the lower and upper bounds to \(\rho\). Appendix B shows similar results for the other sites. We can observe the following general features. With no interference, \(\rho\) increases as rain attenuation increases. With interference, there are significant differences as \(\rho\) tends to decrease when the impulses are of different signs and tends to increase when the impulses are of the same sign, which was not expected, therefore implying a significant amplitude distortion of the interfering impulse coming from the cross channel, as seen in Equation (7). In other words, at the sampling time \(t=0\), the interfering impulse may not be at its maximum value. When the impulses are of opposite sign, \(\rho\) approaches a minimum value at \(A\approx 20\) dB. With impulses of equal sign, \(\rho\approx 5\sim 10\) at \(A\approx 10\), \(\rho\approx 10\) at \(A\approx 20\), and \(\rho\approx 100\) at \(A\approx 30\). In all cases, \(\rho\) increases as \(P(\varepsilon)\) decreases. Now, as a decade increase in the probability of bit error means an equivalent reduction of roughly 1 dB in the SNR [18, 19], we can conclude that, considering only average transfer functions, the linear distortions due to rain can give rise to an average loss of \(1\sim 2\) dB in the radio link budget in a 10-GHz bandwidth. If we consider the SNR variations of \(-1\) standard deviation about the average SNR (i.e., the SNR gets smaller) more loss is found in the direct channel with no interference. Figure 14 shows the findings. Now, we can notice that even without interference, \(\rho>10\) at \(P(\varepsilon)=10^{-6}\). At Tampa, it can even reach, \(|\rho=100\) for \(A=20\) dB at \(P(\varepsilon)=10^{-6}\), therefore giving \(P_{A}(\varepsilon)\approx 10^{-4}\) and an SNR loss of about 2 dB just for these variations. If we also consider interference, the loss can reach \(3\sim 4\) dB. Figure 13: Spino d’Adda.