# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 4 Optimum Geometry Disposition Decision Criteria  
It was noted that the optimum quadrilateral shapes, consisting of four FeNBs, can be selected to measure the location of the moving MUE. The square is the best, the lozenge is next, and the IQ is the worst. When two or more of the same quadrilateral shapes are selected, the following three optimum geometry disposition decision criteria are considered, to determine the best accuracy of the TDOA positioning measures. The GDOP formulas of three optimum geometry disposition decision criteria are derived. Since the location of four FeNBs is fixed, the quadrilateral shape of the FeNBs does not change due to the movement of the MUE. Therefore, in order to simplify the proof of selecting the optimum shapes of four FeNBs, three different criteria are proved in the two dimensional space, and the results are still valid in the three dimensional space. Criterion 1: Square optimum geometry disposition decision criterion depends on the distance between the MUE and the square center. When the distance is shorter, the positioning accuracy is better. Let the distance from the MUE to the \(i\)th FeNB be given by [11]:  
\[r_{i,TDOA}=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}},\;i=0,1,2,3 \tag{22}\]  
The differential distance is obtained from subtracting two TDOA measurements between FeNB\({}_{i}\) and FeNB\({}_{0}\). \[\Delta r_{i,TDOA}=r_{i,TDOA}-r_{0,TDOA},\;i=1,2,3 \tag{23}\]  
The TDOA measurements for three pairs of FeNBs are given as:  
\[t_{i}-t_{0}=\frac{\Delta r_{i,TDOA}}{c}+\frac{\left(\varepsilon_{i}- \varepsilon_{0}\right)}{c}i=1,2,3 \tag{24}\]  
where \(c\) is the light velocity and the measurement error is expressed as:  
\[\varepsilon=\left[ \begin{array}{c}\varepsilon_{0}\\ \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\end{array} \right] \tag{25}\]It is assumed that the errors in all actual measurements are random, independent, and have an identical root mean square (RMS) value \(\sigma_{r}^{2}\). \[ \begin{array}{l}E[\varepsilon_{i}]=\bar{\epsilon}\\ E[\varepsilon_{i}\varepsilon_{j}]=0\\ E[\varepsilon_{i}^{2}]=\sigma_{r}^{2}+\bar{\epsilon}^{2},i=0,1,2,3\\ \sigma_{r}^{2}=E\Big{[}(\varepsilon_{i}-E[\varepsilon_{i}])^{2}\Big{]}=E \big{[}\varepsilon_{i}^{2}\big{]}-\bar{\epsilon}^{2}\end{array}  \tag{26}\]  
The TDOA measurement errors for three pairs of FeNBs can be expressed as the matrix form. \[\left[ \begin{array}{c}(\varepsilon_{1}-\varepsilon_{0})\\ (\varepsilon_{2}-\varepsilon_{0})\\ (\varepsilon_{3}-\varepsilon_{0})\end{array} \right]=\left[ \begin{array}{ cccc}-1&1&0&0\\ -1&0&1&0\\ -1&0&0&1\end{array} \right]\left[ \begin{array}{c}\varepsilon_{0}\\ \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\end{array} \right]=A\varepsilon \tag{27}\]  
The error covariance matrix is:  
\[ \begin{array}{ll}Q&=E\Big{[}(A\varepsilon)(A\varepsilon)^{T}\Big{]}=AE \big{[}\varepsilon\bar{\epsilon}^{T}\big{]}A^{T}=(\sigma_{r}^{2}+\bar{ \epsilon}^{2})AA^{T}\\ &=(\sigma_{r}^{2}+\bar{\epsilon}^{2})\left[\begin{array}{ccc}2&1&1\\ 1&2&1\\ 1&1&2\end{array}\right]\end{array}  \tag{28}\]  
The partial derivatives of the noise free measurement Equations (23) with respect to the unknown MUE coordinates \((x,y)\) are given as:  
\[d\Delta r_{i}=(\frac{x-x_{i}}{r_{i,TDOA}}-\frac{x-x_{0}}{r_{0,TDOA}})dx+(\frac {y-y_{i}}{r_{i,TDOA}}-\frac{y-y_{0}}{r_{0,TDOA}})dy;\;i=1,2,3 \tag{29}\]  
The matrix is expressed as:  
\[\left[ \begin{array}{c}d\Delta r_{1,TDOA}\\ d\Delta r_{2,TDOA}\\ d\Delta r_{3,TDOA}\end{array} \right]=\left[ \begin{array}{c}\frac{x-x_{1}}{r _{1,TDOA}}-\frac{x-x_{0}}{r_{o,TDOA}}+\frac{y-y_{1}}{r_{1,TDOA}}-\frac{y-y_{ 0}}{r_{o,TDOA}}\\ \frac{x-x_{2}}{r_{2,TDOA}}-\frac{x-x_{0}}{r_{o,TDOA}}+\frac{y-y_{2}}{r_{2,TDOA }}-\frac{y-y_{0}}{r_{o,TDOA}}\\ \frac{x-x_{3}}{r_{3,TDOA}}-\frac{x-x_{0}}{r_{o,TDOA}}+\frac{y-y_{3}}{r_{3,TDOA }}-\frac{y-y_{0}}{r_{o,TDOA}}\end{array} \right]\left[ \begin{array}{c}dx\\ dy\end{array} \right] \tag{30}\]  
where:  
\[H=\left[