## 5 Rain Rate Statistics  
We performed a numerical calculation of the probability for signal error weighted by the pdf for rain (Equation (20)), and we compared this to the probability without considering the pdf for rain. The probability as developed in Equation (20) was numerically calculated for representative values of 64 QAM (\(n=6\) in the equations.) For the constants, we chose values of \(k\) and \(\alpha\) from the ITU recommendation (ITU 838-3). We present here the results for 80 GHz, since our interest is in high frequencies. However, the same calculation was performed for additional frequencies giving similar results. Distance \(z_{0}\) was taken as 1 km, as representing a variety of applications, and \(\gamma_{0}\) taken as 500. We performed the simulation using alternate values and frequencies and found that it did not bring forth a significant difference in the results. In order to complete the simulation using concrete values for rain rate (\(m_{r}\), from Equation (20)), it was necessary for numerical purposes to create an array of values. Therefore, a vector was created for \(m_{r}\) ranging from 1 to 300 mm/h, giving a great range of possible values for rain rate. Accordingly, a vector of equal length for standard deviation was obtained as follows: actual measured values for m and \(\sigma\) were taken from [26], and the average ratio of \(m/\sigma\) was calculated for the wide variety of values found there, with synthetic extrapolations to further explore the validity of the results. (See also [42; 43; 44; 45]). It was found that, on average, \(\sigma=m/0.5\). This was used to create a vector for \(\sigma\) matching the rate vector. These were then given logarithmic transformation as in (2) and (3), above. The result of the simulation is shown in Figure 2, which presents a logarithmic plot of the probability for symbol error with and without considering the rain rate. In the figure, the log of the average probability (red solid line) is shown and compared to the probability as calculated in our model, where the pdf for rain is taken into account. The interesting part of the graph is on the left, for low rain rate, as the operating region for effective signal transmission is for a rain rate of less than 15 mm/h. In this regime, we see that without taking the probability of rain into account, the probability of symbol error is negligible, as shown in above. However, if we weight this with the pdf for rain, as is done in Equation (20), the probability for symbol error is considerably greater. Therefore, it appears vital to take into account the probability for rain rate in considering symbol error. This trend is reversed for rain rates of 30 mm/h and greater, but for such high rain rates, in any case, the error probability is too high for useful transmission. Simulations for a range of other frequencies appear in Figure 3 and show similar disparities in the relevant rain rates of under 40 mm/h. Table 3 shows the SNR required for various error probabilities in both cases, for a rain rate of 10 mm/h for the given set of parameters. The main conclusion from the table is that  
Figure 3: Logarithmic plot comparing probability for symbol error for a range of frequencies. The red solid line is the log of average probability as in (6), and the blue dotted line is the probability considering the pdf for rain, (20). For low rain rate, which would be the practical range for signal transmission, the probability for error is higher if the rain rate is considered. The calculation was performed for \(M=64\) and for **(a)** 50 GHz, **(b)** 100 GHz, **(c)** 120 GHz, and **(d)** 150 GHz. For higher frequencies, atmospheric disturbance will be more significant, as discussed in the text. Figure 2: Logarithmic plot comparing probability for symbol error. The red solid line is the log of average probability as in (6), and the blue dotted line is the probability considering the pdf for rain, (20).