## 4 Design with Extra Fixed Power Margin  
In Section 3 the method is implemented at the onset of rainfall. In this section, we suppose that a power margin \(S\) (dB) is already implemented in the link so that the method is applied only in the intervals in which the attenuation threshold \(S\) is exceeded, i.e., when \(A(t)\geq S\geq 0\). The theory recalled in Section 2 can be fully applied to this case by considering the variable \(A^{*}(t)=A(t)-S\) and applying Equations (8)-(12) to the conditional probability distribution \(P_{c}(A^{*})=P(A^{*})/P_{a}^{*}\) in which \({P_{a}}^{*}=P(S)\), all calculated from the data drawn in Figure 2. Of course, if \(S=0\), we obtain the results of Section 3. Figure 6 shows the mean and bounds of the efficiency versus \(S\), at Spino d'Adda. Figure 7 shows the corresponding values of the extra power margin and Figure 8 shows the corresponding values of the bandwidth expansion factor. Similar curves are obtained for the other sites (these are not shown, for the sake of brevity). Figures 9 and 10 show the curves obtained for all sites by considering only the mean efficiency. We can observe trends very similar to those of Spino d'Adda, therefore confirming a general behavior of efficiency, and its related parameters, versus \(S\). However, differences from site to site are clearly evident, with Tampa being the \"worst\" site. From the figures concerning Spino d'Adda, we can observe the following interesting findings, which are also applicable to all of the other sites:  
1. The efficiency is at its maximum at lower thresholds, but not at \(S=0\) dB. 2. The extra power margin is at its minimum in accordance with efficiency. 3. The bandwidth expansion factor is at its minimum in accordance with efficiency. 4. The efficiency is at its minimum (worst case) at about \(S\approx 20\) dB. 5. For \(S\gg 20\) dB, another maximum efficiency occurs, but this range is not attractive because of the very large power margin. Figure 6: Mean (blue line) and upper (green line), lower (red line) bounds of the efficiency \(\eta\) versus the threshold \(S\), at Spino d’Adda. Figure 11 shows details at the low threshold range, which suggests an optimum use of the method. Here, it is interesting to notice that there is no sensible difference in the extra power factor (left panel) in the range \(0\leq S\lesssim 1.5\) dB (right panel), while minima are clearly evident in the bandwidth expansion factor in the range \(S\approx 0.5\,\sim 1.5\) dB. In other words, at any site, the method can be applied when \(A(t)\gtrsim 0.5\,\sim 1.5\) dB, and therefore a power margin \(0.5\,\sim 1.5\) dB should be provided before applying the method. For example, for Tampa geographical area (worst site) the final total power margin necessary would be \(1.5+5.9=7.4\) dB, with a bandwidth expansion factor \(\approx 2.8\), with \(S=1.5\) dB being the threshold of minimum bandwidth expansion factor (Figure 11, right panel) and \(\sim\)5.9 dB the extra power factor (Figure 11, left panel). Figure 8: Maximum bandwidth expansion factor given by the mean (blue line), upper (green line) and lower (red line) bounds of the efficiency, versus the threshold \(S\), at Spino d’Adda. Figure 7: Extra power factor given by the mean (blue line), upper (green line) and lower (red line) bounds of the efficiency, versus the threshold \(S\), at Spino d’Adda. Table 5 reports these optimum values for all sites. The total power margin ranges from 2.0 dB (Norman) to 7.4 dB (Tampa) and the bandwidth expansion factor ranges from 1.70 (Prague) to 2.80 (Tampa). As already observed, the power margin required is very small for all sites, making it well within the capabilities of current technology. The bandwidth expansion factor per se is not very large, but it can be relatively large if the clear--sky bandwidth is already large for the current technology. Figure 10: Maximum bandwidth expansion factor given by the mean (blue line), upper (green line) and lower (red line) bounds of the efficiency, versus the threshold \(S\), at the indicated sites.