# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 5 Simulations  
In performing the simulations, it was assumed that the time delay between FeNB\({}_{i}\) and FeNB\({}_{0}\) was successfully measured with a detection probability of P\({}_{\rm d}\) = 0.9, while the false alarm probability was equal to P\({}_{\rm FA}\) = 10\({}^{-4}\). Moreover, the signal-to-noise ratio (SNR) of the TDE receivers with a linear optimum filter was assumed to be \(-\)13 dB over the Rayleigh channels, with P\({}_{\rm d}\) = 0.9 and P\({}_{\rm FA}\) = 10\({}^{-4}\). The simulations considered the outdoor geo-location scenario shown in Figure 1, consisting of 16 FeNBs located in adjacent multi-floor buildings. The total number of four FeNBs in the coverage range of the target MUE is \(P\) = 1820. Moreover, the moving MUE was assumed to be located between two buildings and positioned at the boundary of a macrocell network. The coordinates of the MUE trajectory are (60, 0, 0), (60, 10, 0), (60, 20, 0), (60, 30, 0), (60, 40, 0), (60, 50, 0), (60, 60, 0), (60, 70, 0), (60, 80, 0), and (60, 90, 0). In simulations, the SC-FDMA signal was considered as it is used by LTE MUE as an uplink access scheme [7; 15]. The SC-FDMA waveforms are specified with bandwidth 10 MHz, subcarrier spacing 15 KHz, FFT/IFFT size 1024, and preamble length 320. The sixteen FeNBs in accordance with their coordinate information are shown in Table 2. (Note that the coordinates are expressed in units of meters.) Any four FeNBs constituted a quadrilateral group. Moreover, the MUE is assumed to be moving between two adjacent buildings. The simulation tool is Matlab. After the simulations, the performance of the proposed NMLP geometry classifier using the multi-station TDOA MUE localization scheme was numerically evaluated. \begin{table}
\begin{tabular}{l l} \hline \hline FeNB\({}_{1}\): (0, 0, 30) & FeNB\({}_{5}\): (0, 0, 40) \\ FeNB\({}_{2}\): (0, 120, 30) & FeNB\({}_{6}\): (0, 120, 60) \\ FeNB\({}_{3}\): (120, 0, 30) & FeNB\({}_{7}\): (120, 0, 20) \\ FeNB\({}_{4}\): (120, 120, 30) & FeNB\({}_{8}\): (120, 120, 40) \\ FeNB\({}_{9}\): (0, 60, 5) & FeNB\({}_{13}\): (0, 0, 20) \\ FeNB\({}_{10}\): (0, 60, 60) & FeNB\({}_{14}\): (0, 15, 30) \\ FeNB\({}_{11}\): (120, 0, 0) & FeNB\({}_{15}\): (0, 30, 40) \\ FeNB\({}_{12}\): (120, 100, 60) & FeNB\({}_{16}\): (0, 45, 50) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coordinates of sixteen FeNBs. Figure 6: GDOP for different distances. The simulations are performed as follows. First, the IGT algorithm is used to calculate the training data of all quadrilateral FeNBs. Then, the NMLP geometry classifier is trained to determine the optimal shape by searching all of the quadrilateral sets of FeNBs within the detection range of the MUE signal. The weight matrices of MLP networks for the first, second, and third layers are \(\mathbf{W}_{mn}^{(2)}\), \(\mathbf{W}_{mn}^{(3)}\), and \(\mathbf{W}_{nm}^{(4)}\), respectively. The matrix dimensions are 12 \(\times\) 6, 6 \(\times\) 6 and 6 \(\times\) 5, respectively. The desired outputs obtained from the training results are classified as five different shapes, which include [0.99, 0.01, 0.01, 0.01, 0.01], [0.01, 0.99, 0.01, 0.01, 0.01], [0.01, 0.01, 0.99, 0.01, 0.01], [0.01, 0.01, 0.01, 0.99, 0.01], and [0.01, 0.01, 0.01, 0.01, 0. 99] for the square, lozenge, IQ, straight-line, and non-planar shapes, respectively. Finally, the optimum geometry disposition decision criteria are used to determine the best accuracy of the TDOA positioning measures. Figure 7 shows that the MLP network using six by six neurons for the hidden layers has the shortest convergence time. It achieved a classification accuracy of 98.0659%. Figure 8 shows that the NMLP geometry classifier can improve the convergent time, and the classification accuracy achieved was 98.0330%. Table 3 shows the estimated NMLP network outputs of five neurons for five different shapes.