## 5 Simulation Results  
We consider a center hexagonal cell with six adjacent hexagonal cells of the same size. Each cell has a radius of 5000 m and the MS locations are uniformly distributed in the center cell [32]. There are seven BSs in cellular communication systems. The serving BS, that is, BS\({}_{1}\), is located at (0, 0, 0.2 km). The heights of the other six BSs, BS\({}_{2}\), BS\({}_{3}\), BS\({}_{4}\), BS\({}_{5}\), BS\({}_{6}\), and BS\({}_{7}\), are 0.15 km, 0.14 km, 0.16 km, 0.12 km, 0.11 km, and 0.13 km, respectively. The height of MS is assumed to be at uniformly distributed over (0, 0.06 km). The NLOS propagation model is based on the uniformly distributed noise model [33], in which the TOA measurement error is assumed to be uniformly distributed over (0,\(U_{i}\)), for \(i\!=\!1,2,...7\) where \(U_{i}\) is the upper bound. The variables are chosen as follows: \(U_{1}=0.2\) km, \begin{table}
\begin{tabular}{c c c c c} \hline  & **Additions** & **Multiplications** & **Division** & **Square Root** \\ \hline \(M\) & 30 & 55 & 0 & 0 \\ \(M^{2}\) & 60 & 98 & 0 & 0 \\ \(M^{3}\) & 60 & 90 & 0 & 0 \\ \(\det(M)\) & 60 & 77 & 0 & 0 \\ WGDOP & 11 & 6 & 1 & 1 \\ \hline Total & 221 & 326 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: The complexity of WGDOP calculation when the five measurements with different error variances. \(U_{2}=0.4\) km, \(U_{3}=0.35\) km, \(U_{4}=0.7\) km, \(U_{5}=0.3\) km, \(U_{6}=0.5\) km and \(U_{7}=0.35\) km. The reciprocal of the square root of an upper bound of the NLOS errors is set to be diagonal elements of the weight matrix \(W\). Therefore, the previously proposed BSs selection criterion chooses the serving BS first and combines it with three optimal measurements to form a subset [27]. The number of the subsets is reduced from 35 (\(C_{4}^{7}\)) to 20 (\(C_{3}^{6}\)) and the calculation load can be relaxed. The subset of the serving BS and three optimal BSs with minimum WGDOP is used to estimate the mobile station (MS) location. The WGDOP residual is defined as the difference between the WGDOP value by matrix inversion and the estimation methods. In order to verify the superior properties of the proposed formula, we compare the results of WGDOP calculation accuracy for all methods and matrix inversion method. Figure 2 shows cumulative distribution functions (CDFs) of the average WGDOP residual for the proposed formula, the previously proposed method [30] and Rprop-based algorithm for Type 6 [27] when four measurements are available. From simulation results, the proposed method and matrix inversion method provide nearly identical WGDOP estimation. The average WGDOP residual of the proposed formula is 2.024245 \(\times\) 10\({}^{-12}\). The proposed efficient formula provides very precise solution of WGDOP calculation. Table 3 shows average WGDOP residual for all methods. Compared to other previously proposed methods, the proposed method always yields much better WGDOP residual than the previously proposed methods. \begin{table}
\begin{tabular}{l c} \hline \hline \multicolumn{1}{c}{**Method**} & \multicolumn{1}{c}{**Average WGDOP Residual**} \\ \hline Proposed formula & 2.024245 \(\times\) 10\({}^{-12}\) \\ Previously proposed method [30] & 1.694013 \(\times\) 10\({}^{-11}\) \\ Previously proposed Rprop-based algorithm for Type 6 [27] & 0.238554 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of average WGDOP residual for all methods when four measurements are available. Figure 2: The CDFs curves of WGDOP residual for all methods when four measurements are available. The improvement in WGDOP accuracy using the proposed formula can also be seen, the CDF curves of the average WGDOP residual for all methods when five BSs are available for location purposes, as shown in Figure 3. The proposed formula can give much better WGDOP estimation as compared with the other previously proposed methods. Table 4 shows the average WGDOP residual for all methods under the constraint that the MS can be heard by five BSs.