## 4 Numerical Results
### Probability of Blocking an Authorized Node  
First of all, we are interested in evaluating the false spoofing detection capability of the system. It is related to the accuracy (i.e., the capability of the classification method of correctly classifying the authorized nodes) of the classification method as \(P_{ban}=1-Accuracy\). Indeed, if the classification is not correct, an authorized node is erroneously associated to a different class, and the ID check fails. In the basic scenario, we refer to the model channel parameters described before: the scatters' speed is in the range [0-5] km/h, \(\sigma_{\tau}=1/\lambda\) with \(\lambda=1.664\cdot 10^{7}\) and \(\sigma_{AoA}=1.5849\)[36]. However, in order to test the effectiveness of the classification also under more challenging conditions, we have also considered the following different scenarios:  
* _Scenario A1_--nodes are randomly placed in \(\mathcal{A}\) according to a bidimensional probability distribution. The Doppler spread is related to a scatterers' movement in the range [0-5] km/h;
* _Scenario A2_--nodes are randomly placed as in A1, but scatterers' speeds are increased in the range [0-15] km/h;
* _Scenario A3_--nodes and speeds are set as in A2, but also the angle and delay spread are increased, considering a variance that is three times the original one;
* _Scenario B1_--nodes are placed in clusters as shown in Figure 4, and the signals experience the Doppler effect under the same conditions as case A1;
* _Scenario B2_--clustered nodes are paired with the same environmental conditions of case A3. Different datasets were created for each scenario to train the machine. In particular, for each node, 5000 impulse responses were sampled and of those, the first 100 were used as the training dataset, while the rest of them were used to evaluate the performance of the classifier. As shown in Figure 5 for the CART algorithm (similar results were derived also for the random forest algorithm), the value of 100 for the training sequence length was selected because an increase does not provide a noticeable performance improvement. Moreover, until the length of 80 (it is more evident with very short lengths, 5/10) we can note an overfitting effect due to the fact that with a few data, the training is too fitted on these; hence, there is a consequent significant loss of performance after training. First of all, we evaluated the performance of the CART algorithm, varying the number of splits in the range [20-60]. We saw that in basic scenarios, there is not a high variance of the achieved values with the number of splits. Differently, when the Doppler, variance of angle and delay spread increase, a higher number of splits is beneficial. In general, 20 splits is a good trade off. Table 1 reports the maximum and minimum values of the _classification accuracy_ for different scenarios and the number of splits for which these values are reached. Figure 4: Example of the nodes position in a clustered scenario. Figure 5: CART classification accuracy for different training sequence lengths in A1 scenario. The following results were derived, assuming a CART classifier with a maximum of 20 splits and a random forest classifier composed of five trees of the same size as the CART one. Figure 6 shows the accuracy of the two classifiers on the A1 dataset for the whole sample sequence of 5000 samples (graphs starts at the 100-th sample because the first 100 samples are used for training). Both classifiers show a good stability with little loss of accuracy over time, with an average percentage of correct classification around 95% (i.e., in average \(P_{\mathit{fstd}}=5\%\)). There are not significant differences in the performance of CART and random forest algorithms; the gain of random forest is substantially negligible. The accuracy averaged on the whole dataset for all scenarios is reported in Table 2.  
Results in Table 2 show that only in scenario B2 is there a noticeable reduction in the accuracy, which goes down around 88-89%.