# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 5 Simulations
The estimated NMLP geometry classifier outputs are [0.9418, 0.0186, 0.0587, 0.0145, 0.0173], [0.0504, 0.8951, 0.0751, 0.0291, 0.0912], [0.0124, 0.0186, 0.9427, 0.0624, 0.0174], [0.0107, 0.0181, 0.0114, 0.9755, 0.0130], and [0.0178, 0.0136, 0.0584, 0.0175, 0.9581] which are classified as square, lozenge, IQ, straight-line, and non-planar shapes, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Neuron 1** & **Neuron 2** & **Neuron 3** & **Neuron 4** & **Neuron 5** \\  & **Output** & **Output** & **Output** & **Output** & **Output** \\ \hline square & 0.9418 & 0.0186 & 0.0587 & 0.0145 & 0.0173 \\ lozenge & 0.0504 & 0.8951 & 0.0751 & 0.0291 & 0.0912 \\ IQ & 0.0124 & 0.0186 & 0.9427 & 0.0624 & 0.0174 \\ straight line & 0.0107 & 0.0181 & 0.0114 & 0.9755 & 0.0130 \\ not co-plane & 0.0178 & 0.0136 & 0.0584 & 0.0175 & 0.9581 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimated NMLP output for different geometries. Figure 7: MSE of the MLP network using a different number of neurons for the first and second layers. The classification results after training are shown in Table 4, which shows the number of geometry classifications for three cases. For all FeNBs, a unique square geometry is selected by the proposed NMLP geometry classifier to estimate the current MUE position obtained using the TDOA scheme. Here, the TDOA measurement error of each FeNB is assumed to be Gaussian distributed with a mean of zero and a standard deviation of \(\sigma_{e_{i}}=5\) m. Moreover, the true coordinates of the MUE are assumed to be (60, 50, 0). The GDOP value is calculated as 1.533 m. For the case not including FeNB\({}_{4}\) in the adjacent multi-floor buildings, \(P\) reduces to 1465, and two lozenge geometry groups are selected by the proposed NMLP geometry classifier. The CAD of the first lozenge geometry group is 0.8\({}^{\circ}\) and the CAD of the second lozenge group is 3.1\({}^{\circ}\). The GDOP value of the former group is 2.012 m and the GDOP value of the latter group is 2.432 m. Then, four FeNBs of a lozenge geometry with a GDOP value of 2.012 m are determined by the proposed NMLP geometry classifier. The decision criterion 2 is verified. The positioning accuracy is better when the CAD of the lozenge optimum geometry disposition is smaller. The TDOA measurement error of the lozenge geometry group is larger than the square geometry group if the FeNB\({}_{4}\) is not deployed in the adjacent multi-floor buildings. For the case not including FeNB\({}_{4}\) and FeNB\({}_{8}\) in the adjacent multi-floor buildings, \(P\) reduces to 1001, and all of the square and lozenge geometry groups will be removed. In accordance with the IQ optimum geometry disposition decision criterion, the IQ group with the minimum GDOP value of 4.354 will be selected from 263 IQ groups. When the MUE moves from the position (60, 0, 0) to (60, 90, 0), the trajectory of the moving path is shown in Figure 1. The proposed NMLP geometry classifier determines the first three best shapes for the three cases. The GDOP values of the first three best quadrilateral shapes for all FeNBs are listed in Table 5, which shows that the GDOP of the square geometry group is smaller when the distance between the MUE and the square center is shorter. The GDOP of the square geometry group is smaller than the lozenge geometry. When two lozenge geometry groups are selected by the proposed NMLP geometry classifier, the GDOP for the CAD of 0.8\({}^{\circ}\) is smaller than the CAD of 3.1\({}^{\circ}\). In this \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Square** & **Lozenge** & **IQ** & **Straight Line** & **Not Co-Plane** \\ \hline all FeNBs & 1 & 6 & 217 & 4 & 1593 \\ FeNB 4 not included & 0 & 2 & 139 & 4 & 1220 \\ FeNB4,8 not included & 0 & 0 & 263 & 5 & 733 \\ \hline \end{tabular}
\end{table} 
Table 4: Number of geometry classifications. Figure 8: MSE comparison between NMLP and MLP networks for six by six neurons in the first and second layers. case, the former is determined.