## 4 Performance Evaluation
### Evaluation Time  
In this subsection, the evaluation time of F-HHT will be compared to that of other methods. The run times of the algorithms are listed in Table 4. As mentioned before, the trial numbers of the R-HHT and H-HHT were selected to achieve accuracy similar to that of F-HHT. The run times of the R-HHT and H-HHT were measured for settings corresponding to the accuracy levels, shown in Table 3. The run times were measured on a computer featuring an Intel i5 processor with a clock frequency of 1.6 GHz and 24 GB of RAM, using Matlab (without parallel processing). The trial numbers are also shown in Table 5, along with the theoretical number of grid points \(N_{grid}\) for a full grid with a grid size of \(D_{final}\). In the tests, the LS solution converged in approximately the same amount of time in all scenarios. However, the evaluation times of HHT-based algorithms were spread over a wider range due to two effects: GDOP and measurement noise. The primary effect is GDOP: small GDOP results in sharp peaks in the HHT, while large GDOP widens the peaks (see Figure 6 for an illustration of this effect). The measurement noise also blurs the peaks, resulting in smaller peaks with gentler slopes. The F-HHT algorithm performs very well when the peaks are sharp: in the near-range scenarios (#1, #2, #3) F-HHT was the fastest of all algorithms. In these cases, the elimination steps quickly excluded most of the search area, resulting in the processing of a small number of tiles and fast evaluation times. In the long-range cases (with wider peaks), the elimination process of F-HHT was less efficient: a larger area was still active while the grid size decreased during the iteration, resulting in the processing of a large number of small tiles and thus longer evaluation times. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multicolumn{4}{c}{**Run Time (ms)**} \\ \cline{2-5}  & **LS** & **R-HHT** & **H-HHT** & **F-HHT** \\ \hline \#1 & 5.4 & \(59\times 10^{3}\) & \(4.3\times 10^{3}\) & 2.6 \\ \#2 & 4.5 & \(2.6\times 10^{3}\) & 66 & 3.4 \\ \#3 & 4.4 & \(539\) & 48 & 2.3 \\ \#4 & 4.4 & \(59\times 10^{3}\) & 839 & 41 \\ \#5 & 3.7 & 513 & 30 & 16 \\ \#6 & 3.8 & 513 & 6.3 & 7.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of evaluation times. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{\(N_{grid}\)} & \multicolumn{3}{c}{**Trials**} \\ \cline{2-5}  & & **R-HHT** & **H-HHT** & **F-HHT** \\ \hline \#1 & \(2.5\times 10^{7}\) & \(7\times 10^{7}\) & \(5\times 10^{6}\) & \(2.5\times 10^{3}\) \\ \#2 & \(2.5\times 10^{7}\) & \(3\times 10^{6}\) & \(4\times 10^{4}\) & \(1.9\times 10^{3}\) \\ \#3  
The evaluation time of the R-HHT is dependent on several factors, including the CRLB, the size of the search area, and the required accuracy, i.e., how close the required error is to the CRLB. Note that R-HHT takes random picks from the search area: the larger the search area, the more picks are needed to obtain a hit close to the optimum (the target area). The smaller the CRLB and the closer we try to get to the theoretical optimum, the smaller the target area, necessitating a higher number of trials. The results illustrate these effects. In scenarios #5 and #6, the CRLB and the required accuracy were the same, resulting in the same number of trials. In scenario #4, the CRLB was smaller, thus the trial number increased. In scenarios #1, #2, and #3, the effect of accuracy is also visible. The CRLB is smaller in #1 and larger in #2 and #3. As a result, the trial numbers in #2 and #3 are smaller than in the case of #1. In scenario #2, higher accuracy was required than in scenario #3 (while having the same CRLB), thus the trial number in #2 was higher. The impact of the search space can also be observed: e.g., in #4, the CRLB is smaller than in #1, but the increase in the search space compensates for this effect, resulting in both scenarios having approximately the same number of trials.