## 4 Performance Evaluation
### Error Analysis  
The performance of the F-HHT will be compared to the theoretical limit of the Cramer-Rao lower bound (CRLB). The CRLB of our 2.5D TDOA problem, where the sensors are in 3D but the source position is searched for in a plane, can be derived using the method described in [27; 47], as follows:  
\[CRLB(x,y)=trace\Big{(}F^{-1}\Big{)}, \tag{27}\]  
where \(F\) is the Fisher information matrix  
\[F=\frac{1}{c^{2}}\mathrm{G}^{T}Q^{-1}G, \tag{28}\]  
with  
\[Q=\sigma_{t,1}^{2}1_{N-1}1_{N-1}^{T}+E_{N-1}\Sigma_{N-1}^{2}, \tag{29}\]  
\[\Sigma_{N-1}^{2}=\Big{[}\sigma_{t,2}^{2},\ \sigma_{t,2}^{2},\ \ldots,\sigma_{t,N}^{2} \Big{]}^{T}, \tag{30}\]  
\[G=\begin{bmatrix}\bar{s}_{2}^{T}-\bar{s}_{1}^{T}\\ \vdots\\ \bar{s}_{N}^{T}-\bar{s}_{1}^{T}\end{bmatrix}, \tag{31}\]  
\[\bar{s}_{i}^{T}=\frac{[x,\ y]-[x_{i},\ y_{i}]}{\left\|[x,\ y,0]-[x_{i},\ y_{i} ]\right\|_{2}},\ 1\leq i\leq N \tag{32}\]  
and \(1_{k}\) is the vector of length \(k\) containing 1 s and \(E_{k}\) is the identity matrix of size \(k\times k\).  
The results for all scenarios are shown in Table 3. The table shows the square root of the CRLB and the measured root mean square error (RMSE) for all algorithms, using 1000 independent experiments in each scenario.  
In all cases, the measured errors of LS, H-HHT, and F-HHT were close to the theoretical limit. The LS algorithm was the most accurate, which is probably due to the fact that the search of the HHT was performed on a finite grid. The F-HHT provided results almost as accurate as the LS method, when the final grid size was set to a sufficiently small value.  
The impact of grid size can be observed in experiments #2 and #3. The theoretical error limit was approximately 0.6m, so the grid size of 0.1m in experiment #2 was safely below it, while in experiment #3, the grid size of 1m exceeded the error limit. As expected, the estimation error in experiment #2 was closer to the theoretical limit, while in experiment #3 the error was significantly higher. The distinction between experiments #5 and #6 is less apparent, as both grid sizes (1 m and 5 m) were significantly below the theoretical error level (20 m).  
Note that for the R-HHT and H-HHT algorithms, there is an additional parameter for the number of trials. This parameter defines the number of random test positions where (11) is evaluated. Increasing the number of trials decreases the error level, but the increase in accuracy is small near the theoretical level. Therefore, a small increase in accuracy  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{\(\surd\)**CRLB**} & \multicolumn{4}{c}{**RMSE**} \\ \cline{3-6}  & & **LS** & **R-HHT** & **H-HHT** & **F-HHT** \\ \hline \#1 & 0.064 m & 0.071 m & 0.080 m & 0.079 m & 0.075 m \\ \#2 & 0.64 m & 0.69 m & 0.72 m & 0.71 m & 0.70 m \\ \#3 & 0.64 m & 0.68 m & 0.80 m & 0.82 m & 0.81 m \\ \#4 & 1.98 m & 2.03 m & 2.06 m & 2.32 m & 2.04 m \\ \#5 & 19.8 m & 21.0 m & 21.3 m & 21.6 m & 21.1 m \\ \#6 & 19.8 m & 21.0 m & 21.3 m & 21.8 m & 21.2 m \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of localization errors.  
requires a large increase in trial numbers. To ensure a fair performance comparison, the trial numbers of R-HHT and H-HHT were adjusted so that the error levels of R-HHT, H-HHT, and F-HHT were approximately equal (see Table 3). This allows for a comparison of the required computational power in the next section.