## 4 Performance Evaluation
### Fault Tolerance  
In this subsection, the fault tolerance of F-HHT is illustrated. For this purpose, a short-range scenario (#1) and a long-range scenario (#4) were utilized. In the experiments, various numbers of faulty sensors were used; the faulty sensors provided outlier measurements with a standard deviation 100 times larger than that of the correct sensors. The faulty sensors were selected randomly, but the reference sensor \(S_{1}\) was never faulty.  
The measured RMSE values for the LS and F-HHT algorithms, as a function of the number \(N_{\mathit{oulier}}\) of outliers, measured using 1000 independent experiments, are shown in Table 6. In the tests, the LS algorithm was found to be highly sensitive to outliers: the presence of even a single outlier significantly increased the error level. The F-HHT, however, proved to be much more robust: in the near-range scenario, the error did not significantly increase when up to two faulty sensors were present. In the far-range scenario, one faulty sensor was tolerated well (the error level increased from 2.1 m to 2.8 m), while two faulty sensors increased the error to 7.5 m. Notice that in the near-range case the LS solver behaved reasonably well in the presence of outliers, while in the far-range case its results were practically useless. The higher sensitivity to outliers in the far-range case was caused by the high GDOP.  
In another experiment, a large number of sensors was utilized, as shown in Figure 9. The setup, containing \(N=57\) sensors, was used in a real shooter localization system [33]. In the current simulation, the exact measurements were generated for source position (25, 20), shown by a green x in Figure 9, then white noise with \(\sigma_{i,t}=2\) ms (approximately equivalent to \(\sigma_{i,1}=1\) m) was added to each measurement. Then, a random set of \(N_{\mathit{oulier}}\) measurements was selected and a large noise with \(\sigma_{i,t}=200\) ms was added to them to create the outlier measurements. The measurement sets, including both correct measurements and outliers,were used as inputs for the LS and F-HHT algorithms. The number of outliers, \(N_{\mathit{oulier}}\), was varied between 0 and 55, and for each value of \(N_{\mathit{oulier}}\) 1000 independent experiments were conducted. Figure 10 shows the theoretical error level \(\sqrt{CRLB}\) and the measured RMSE values for the LS and F-HHT algorithms on a logarithmic scale as a function of \(N_{\mathit{oulier}}\). Notice that the LS solver diverged in many cases, thus a constrained version (fmincon in MATLAB) was utilized during testing, which ensured that the solution remained within the search area depicted in Figure 9.  
According to Figure 10, the LS solver did not tolerate any outliers, whereas the F-HHT method performed very well up to 30-40 outliers out of 57 measurements, which is remarkably robust behavior.  
Note that in the experiments, the reference sensor was never faulty. If this cannot be guaranteed in practice, the role of the reference sensor must be rotated, and the solution with the highest HHT value must be selected. This process will be illustrated in the next subsection.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{\(N_{\mathit{oulier}}\)} & \multicolumn{2}{c}{**RMSE (m)**} \\ \cline{3-4}  & & **LS** & **F-HHT** \\ \hline  & 0 & 0.07 & 0.07 \\  & 1 & 2.0 & 0.08 \\ \#1 & 2 & 2.9 & 0.10 \\  & 3 & 3.5 & 1.4 \\  & 4 & 4.1 & 11.8 \\ \hline  & 0 & 2.1 & 2.1 \\  & 1 & 80 & 2.8 \\ \#4 & 2 & 120 & 7.5 \\  & 3 & 143 & 41 \\  & 4 & 181 & 193 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Localization error of LS and F-HHT in the presence of outlier measurements.  
Figure 9: Sensor setup with 57 sensors.