# Factorial Design Analysis for Localization Algorithms
## 4 Results
### Impact of Noise Level  
Figure 5 shows the MSE variation for different noise levels (\(\sigma_{dB}\)); this factor has greater impact on the Hyperbolic and WLS Multilateration range-based localization techniques than on the range-free localization techniques, since these algorithms show greater MSE variation for different node densities (\(\rho\)). The noise level (\(\sigma_{dB}\)) is a factor that has greater impact on the distance separating the NOI from the RNs than on the RSS respectively; thus, the greater the noise level (\(\sigma_{dB}\)), the greater the error  
Figure 4: CDF vs. Localization Error for different node densities (\(\rho\)).  
Figure 3: MSE vs. Node Density (\(\rho\)) for different noise levels (\(\sigma_{dB}\)).  
of the distance separating the NOI from the RNs, than the error of the RSS respectively, which is observed using Equation (21); therefore, the localization error is greater in the range-based localization algorithms than in the range-free algorithms, as shown for both cases in Figure 5.  
\[\widetilde{d}=10^{\left(\frac{A-\text{RSS}_{dB}}{10\eta}\right)}=d10^{\left( \frac{\widetilde{d}}{10\eta}\right)} \tag{21}\]  
where \(\widetilde{d}\) is the separation range between the NOI and the RNs, \(\text{RSS}_{dB}\) is the power received in dB for the shadowing effects obtained in Equation (19) and \(d\) is the true Euclidean distance between the NOI and the RNs.  
Figure 6 shows the CDF behavior of the localization techniques. As shown in Figure 6a, the noise level has greater impact on the Hyperbolic algorithm, as this algorithm shows greater CDF variation for different node densities (\(\rho\)). Figure 6b shows that the Hyperbolic algorithm shows a high level of CDF variation for high noise levels (\(\sigma_{dB}=10\) dB), with a higher node density (\(\rho=0.0009\) nodes/m\({}^{2}\)). In this figure, with low noise levels (\(\sigma_{dB}=2\) dB), the CL and REWL range-free localization algorithms show greater CDF variation than the Hyperbolic algorithm, meaning that node density (\(\rho\)) is a factor that shows greater impact on the range-free localization algorithms.  
The results obtained show that the path-loss exponent (\(\eta\)) and noise level (\(\sigma_{dB}\)) factors show greater impact on the MSE of the range-based algorithms, because as both factors increase, they have more and more impact on the error of the distance separating the NOI from the RNs while the node  
Figure 5: MSE vs. Noise level (\(\sigma_{dB}\)) for different node densities (\(\rho\)).  
Figure 6: CDF vs. Localization Error for different noise levels (\(\sigma_{dB}\)).  
density shows greater impact on the MSE of the range-free algorithms. As the path-loss exponent (\(\eta\)) increases, the range-based localization algorithms show less MSE than the range-free algorithms, since the range-based algorithms have greater precision in localizing the NOI. Considering high noise levels (\(\sigma_{dB}\)), the range-based localization algorithms show greater MSE than the range-free algorithms, since this factor shows greater impact on the range-based algorithms. The number of RNs is a factor that shows a very small impact on the MSE and the CDF of the localization algorithms; the results obtained show that 5 RNs are enough to localize the NOI in the proposed evaluation scenario.