## 5 Simulation Results and Analysis
### Analysis and Comparison of GDOP  
To compare the GDOP of the TDOA-only positioning and the GDOP of the hybrid positioning using multiple TDOAs and single TOA in the same indoor scenario. 2889 points is selected every 0.5 m, as shown in Figure 4. We divide the selected points into two groups. One group contains the points within the quadrilateral surrounded by connections between ANTs, which is the green quadrilateral in Figure 4. The other group contains the points outside the quadrilateral. 2292 points are outside the green quadrilateral and 597 points are inside the green quadrilateral. The GDOPs of the hybrid positioning method based on three TDOAs and one TOA with different ratio \(k\) values are calculated at each points. The \(k\) is the ratio of TOA ranging error standard deviation to TDOA ranging error standard deviation definite in Section 4. We use four different \(k\) values and the result is shown in Figure 5.  
Figure 4: Selected points (blue points) in the developed indoor two-dimensional (2D) simulation scenario. The red dots are ANTs, the green lines are connections of ANTs. Figure 5: _Cont._  
Figure 6 shows the GDOP of TDOA-only positioning in the same scenario. The TDOA-only positioning is based on three TDOA measurements which are the same as the proposed hybrid positioning method used above. The GDOP calculation method for TDOA-only positioning is similar to that GDOP calculation method for hybrid positioning described in Section 5 but does not contain the TOA measurement terms. The calculation method is the same as the method described in the definition of GDOP [21]. It can be easily seen from the comparison between Figures 5 and 6 that the GDOP of the TDOA-only positioning is not much different from the hybrid positioning in the center region (the quadrilateral area surrounded by the four ANTs). In other regions, the GDOP of hybrid positioning is significantly smaller than the TDOA-only positioning. However, the area of the center region only accounts for one-fifth of the total area. Assuming that the distribution of UEs is uniform, in most cases, using the hybrid positioning method will increase the positioning accuracy limitation. Figure 5: Geometric dilution of precision (GDOP) of the proposed hybrid time-difference-of-arrival (TDOA) and time-of-arrival (TOA) positioning in the developed indoor simulation scenario. The red dots are ANTs. Figure 6: GDOP of TDOA positioning in the developed indoor 2D simulation scenario. The red dots are ANTs. The cumulative distribution functions (CDFs) of GDOPs is obtained and shown in Figure 7.  
From Figure 7, it is obvious that the GDOP values of the proposed hybrid positioning are significantly lower than the TDOA-only positioning. It can be seen in (16) that the greater the ratio \(k\) is, the smaller the weight of the TOA ranging error item in the weighting matrix (18) is, and the closer the GDOP calculated is to the GDOP of TDOA-only positioning. The result shows that it is possible to effectively improve the positioning performance by adding a TOA measurement. To compare the GDOPs of the two kinds of positioning in the two different regions as shown in Figure 4, we separately counted the GDOPs of two kinds of positioning at points in the two groups. The CDFs of the GDOPs are shown in Figure 8.  
As shown in Figure 8a, at the points outside the green quadrilateral, GDOPs of the proposed hybrid positioning are significantly lower than the TDOA-only positioning. Even when the standard deviation of TOA ranging error is 10 times the TDOA ranging error, the 24.85% points still have a higher GDOP of the TDOA-only positioning than the highest GDOP of the proposed hybrid positioning. Figure 8: The GDOP CDF of different positioning methods; (**a**) the CDF of GDOPs at points outside the green quadrilateral; (**b**) the CDF of GDOPs at points inside the green quadrilateral, the curves for TDOA and Hybrid k = 10 are identical. Figure 4.0 The cumulative distribution function (CDF) of different positioning methods GDOPs at all selected points.