## 5 Simulation Results and Analysis
### Analysis and Comparison of the Proposed Localization Algorithm
#### 5.2.2 Simulation Results in Different Regions  
To compare the positioning accuracy of the three localization algorithms in different regions, the two groups of selected points in Figure 4 were used for simulation. The positioning simulation is performed 10,000 times at each test point. For each test point group, the MSE of the positioning errors for each group test points was calculated and denoted as  
\[\text{MSE}_{all}=\frac{\sum_{p=1}^{N}\sum_{i=1}^{10,000}\left(\left(x_{p,i}-x_ {p}\right)^{2}+\left(y_{p,i}-y_{p}\right)^{2}\right)}{N\times 10,000} \tag{53}\]  
where \(x_{p}\) and \(y_{p}\) are coordination of the \(p\)th test point in the group, \(x_{p,i}\) and \(y_{p,i}\) are the positioning result of the \(i\)th calculation at the \(p\)th test point. The \(\text{MSE}_{all}\) of the two groups' positioning results is counted separately. The MSEs and CRLBs of positioning results with different TDOA ranging error of the two groups are shown in Figures 13 and 14. Figure 14: Accuracy of different positioning method at points outside green quadrilateral; (**a**) positioning error MSE when the standard deviation of TOA ranging error is 0.3 m; (**b**) positioning error MSE when the standard deviation of TOA ranging error is 1.5 m.  
Figure 13: Accuracy of different positioning method at points inside green quadrilateral; (**a**) positioning error MSE when the standard deviation of TOA ranging error is 0.3 m; (**b**) positioning error MSE when the standard deviation of TOA ranging error is 1.5 m.  
Figure 13 shows the MSE\({}_{all}\) of the positioning results and CRLBs at test points inside the green square. The result shows that the CRLB of the proposed hybrid positioning is close to the CRLB of TDOA-only positioning. When \(\sigma_{toa}=0.3\) m, the MSE\({}_{all}\) of proposed algorithm approach CRLB. When \(\sigma_{toa}=1.5\) m, the accuracy of the proposed algorithm is slightly higher than the CRLB while the standard deviation of TDOA ranging error is relatively small but has not yet approached zero. For the proposed algorithm with different TOA ranging error, the positioning results with \(\sigma_{toa}=0.3\) m are very slightly better than the positioning results with \(\sigma_{toa}=1.5\) m. However, regardless of the different performance caused by the TOA ranging error, the positioning error of the proposed hybrid positioning method is significantly lower than the other two TDOA-only positioning methods. Figure 14 shows the MSE\({}_{all}\) of positioning results and CRLBs at test points outside the green square. The result shows that the hybrid positioning has a lower CRLB than TDOA-only positioning. The MSE\({}_{all}\) of the proposed algorithm approaches CRLB when TDOA ranging error is small. Even when the TDOA ranging error is higher, the positioning error of the proposed algorithm is also slightly higher than the CRLB and is significantly lower than the other two TDOA-only positioning methods. The positioning results of the proposed algorithm with \(\sigma_{toa}=0.3\) m are slightly better than it with \(\sigma_{toa}=1.5\) m when the standard deviation of TDOA ranging error is around 0.5 m.  
The algorithms performance at each selected points in Figure 4 is also simulated as in Section 5.2.1. Different from Section 5.2.1, the square roots of MSE, called the root mean square error (RMSE), for different algorithms at every selected point are calculated. The CDF of the RMSE for the Chan-Ho, BiasRed, proposed algorithm with \(\sigma_{toa}=1.5\) m, proposed algorithm with \(\sigma_{toa}=0.3\) m and CRLBs are shown in Figure 15, when \(\sigma_{toa}=1\) m.  
As shown in Figure 15b, it is obviously that, regardless of \(\sigma_{toa}=0.3\) or 1.5 m, performance of proposed algorithm is better than Chan-Ho and BiasRed algorithms.