## 5 Simulation Results and Analysis
### Analysis and Comparison of the Proposed Localization Algorithm
#### 5.2.1 Simulations at Selected Point  
According to the GDOP analysis in Section 5.1, we selected three test points at the right edge, the center, and the left edge of the scenario. Scenario shown in Figure 3 is used, and three test points are selected for the simulation, which is shown in Figure 9.  
Figure 9: Selected Test Points: the blue dots are selected three test points; the red dots are the location of ANTs. The simulation was performed 10,000 times at each test point. TDOA and TOA ranging errors were the zero-mean Gaussian distribution but with different standard deviations. 5G system could provide a TOA ranging accuracy with standard deviation \(\sigma_{tau}=0.3\ m\) or \(\sigma_{t\alpha}=1.5\ m\)[25] and provide a TDOA ranging accuracy with a best \(\sigma_{t\alpha}=0.03\ m\)[13]. In each simulation, we calculated the Mean Square Error (MSE) of the positioning errors for each test points as  
\[\text{MSE}=\frac{\sum_{i=1}^{10,000}\Bigl{(}(x_{i}-x_{t})^{2}+(y_{i}-y_{t})^{ 2}\Bigr{)}}{10,00} \tag{52}\]  
where \(x_{t}\) and \(y_{t}\) are the coordinate of the test point, \(x_{i}\) and \(y_{i}\) are the positioning result of the ith calculation. Figure 10 shows the MSE of the positioning results and the calculated CRLBs at test point A. The result shows that the positioning using multiple TDOAs and single TOA has a lower CRLB than TDOA-only positioning. Regardless of whether \(\sigma_{t\alpha}=1.5\) or \(\sigma_{t\alpha}=0.3\ m\), the MSE of proposed algorithm approaches CRLB when TDOA ranging error is small. When the TDOA ranging error is high, the positioning error of the proposed localization algorithm is higher than the CRLB but is still significantly lower than the other two TDOA-only closed-form algorithms. For the proposed algorithm, the positioning results with \(\sigma_{t\alpha}=0.3\ m\) are slightly better than the positioning results with \(\sigma_{t\alpha}=1.5\ m\) when the standard deviation of TDOA ranging error is around \(0.5\ m\). Figure 11 shows the MSE of positioning results and CRLBs at test point B. The result shows that the CRLB of the proposed hybrid positioning is close to the CRLB of TDOA-only positioning at test point B. Regardless of whether \(\sigma_{t\alpha}=1.5\) or \(\sigma_{t\alpha}=0.3\ m\), the MSE of 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 only slightly higher than the CRLB. For the proposed algorithm, the positioning results with \(\sigma_{t\alpha}=0.3\ m\) are slightly better than the positioning results with \(\sigma_{t\alpha}=1.5\ m\). However, regardless of the different performance caused by TOA ranging error, the positioning error of the proposed algorithm is lower than the other two TDOA-only positioning methods. Figure 10: Accuracy of different positioning method at point A; (**a**) positioning error mean square 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 12 shows the MSE of positioning results and CRLBs at test point C. The result shows that the proposed hybrid positioning has a lower CRLB than TDOA-only positioning at test point C. Regardless of whether \(\sigma_{taa}=1.5\) or \(\sigma_{taa}=0.3\) m, the MSE of 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 very slightly higher than the CRLB. For the different results in (a) and (b), the proposed algorithm is more accurate while \(\sigma_{taa}=0.3\) m than \(\sigma_{taa}=1.5\) m, when the standard deviation of TDOA ranging error is around 0.5 m. However, regardless of the different performance caused by TOA ranging error, the positioning error of the proposed algorithm is significantly lower than the other two TDOA-only positioning methods.