# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 2 Femtocell Positioning Network Model  
The scenarios under study include the combination of a macro cell with femtocells in outdoor environments, which are operated at the same frequency band. It is assumed that the interference coordination and smart digital beam forming techniques may be needed to solve the problems of co-channel interference and signal attenuations. Figure 1 shows an illustrative femtocellnetwork consisting of multiple FeNBs deployed throughout two adjacent multi-floor buildings. An assumption is made that the FeNBs are connected to a cloud-computing platform through the Internet. Moreover, the coordinates of each FeNB and the training data are pre-stored in the database of the cloud-computing platform. The training data of the NMLP geometry classifier are determined using an IGT algorithm based on the TDOA measurements obtained by each FeNB and the GDOP metric. The architecture of the NMLP geometry classifier is realized in the server of the cloud-computing platform to perform the searching of the optimal set of FeNBs for positioning purposes. The uplink transmissions from the MUE are conventionally handled using the Single Carrier-frequency division multiple access (SC-FDMA) scheme [15]. Moreover, the signals transmitted from the MUE are detected by digital time delay estimation (TDE) receivers [16], utilizing a digital cross-correlation technique in the frequency domain. In the multi-FeNB TDOA positioning system proposed in this study, the received signal of each FeNB is sent through the Internet to a reference FeNB, which calculates the TDOA between itself and each FeNB such that the position of the MUE can be further derived. Assume that the moving MUE emits an SC-FDMA signal \(\mathrm{s}[n]\) in 3D space. The received signal at the reference FeNB (FeNB\({}_{0}\)) thus has the form:  
\[x_{0}[n]=s_{0}[n]+w_{0}[n],n\in[0,M-1] \tag{1}\]  
where \(w_{0}[n]\) is the discrete white Gaussian noise at FeNB\({}_{0}\); and the variable \(n\) is defined as the sampling instant. The received signals at three other arbitrarily-chosen FeNBs are given by:  
\[ \begin{split} x_{i}[n]&=s_{i}[n-D_{i0}]+w_{i}[n]\\ &\quad\quad\quad\quad n\in[0,M-1],\ i=1,2,3\end{split}  \tag{2}\]  
where \(w_{i}[n]\) is the discrete white Gaussian noise at FeNB\({}_{i}\). The discrete time delay between FeNB\({}_{i}\) and FeNB\({}_{0}\) is estimated as:  
\[\hat{l}_{i}=D_{i0},\ i=1,2,3 \tag{3}\]  
Four-FeNB TDOA localization schemes calculate the possible MUE location based on the TDOA measurements of three FeNB receiver pairs lying on separate hyperbola [10]. In other words, the MUE location is obtained by solving three hyperbolic equations, i.e. :  
Figure 1: Outdoor geo-location scenario between two multi-floor buildings. \[ \begin{array}{ll}r_{i,0}=&\sqrt{\left(\hat{x}-x_{0}\right)^{2}+\left(\hat{y}-y_{0 }\right)^{2}+\left(\hat{z}-z_{0}\right)^{2}}\\ &-\sqrt{\left(\hat{x}-x_{i}\right)^{2}+\left(\hat{y}-y_{i}\right)^{2}+\left( \hat{z}-z_{i}\right)^{2}};\;i=1,2,3\end{array}  \tag{4}\]  
where \((x_{i},y_{i},z_{i})\) and \((x_{0},y_{0},z_{0})\) are the coordinates of the \(i\)th and reference FeNBs, respectively; and \((\hat{x},\hat{y},\hat{z})\) is the estimated MUE location. In addition, \(r_{i,0}\) is the differential distance between the MUE and the master FeNB, and the MUE and FeNB\({}_{i}\), respectively, and is obtained from the measured time delay between FeNB\({}_{i}\) and FeNB\({}_{0}\) as:  
\[ \begin{array}{ll}r_{i,0}=\frac{cD_{i0}}{Mf_{s}}+e_{r_{i,0}}=c\hat{t}_{i,0}+ e_{r_{i,0}}=c(\hat{\tau}_{i}-\hat{\tau}_{0})+e_{r_{i,0}}\\ &i=1,2,3\end{array}  \tag{5}\]  
where \(M\) is the size of the FFT, \(\hat{\tau}_{i,0}\) is the analog time delay between FeNB\({}_{i}\) and FeNB\({}_{0}\), and \(f_{s}\) is the sampling frequency of the analog-to-digital converter. Finally, \(e_{r_{i,0}}\) is the measurement error with standard deviation \(\sigma_{e_{i}}\).