# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 4 Optimum Geometry Disposition Decision Criteria
\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] \tag{31}\]  
When the error covariance matrix is not diagonal, as in Equation (28), the GDOP will be given by:  
\[\text{GDOP}=\sqrt{trace(G)} \tag{32}\]  
where the matrix \(G\) is the variance of the linear optimum unbiased estimator divided by the variance of measurement noise [17]. \[G=\frac{1}{\sigma_{r}^{2}}(H^{T}Q^{-1}H)^{-1} \tag{33}\]For the simplification of proving the square optimum geometry disposition decision criterion, the coordinates of FeNB\({}_{0}\) (1, 0), FeNB\({}_{1}\) (0, 1), FeNB\({}_{2}\) (\(-1,0\)), FeNB\({}_{3}\) (0, \(-1\)), and MUE (\(x,y\)) are substituted into Equation (29), to yield:  
\[H=\left[ \begin{array}{cc}\dfrac{x}{\sqrt{x^{2}+\left(y-1\right)^{2}}}-\dfrac {x-1}{\sqrt{\left(x-1\right)^{2}+y^{2}}}&\dfrac{y-1}{\sqrt{x^{2}+\left(y-1 \right)^{2}}}-\dfrac{y}{\sqrt{\left(x-1\right)^{2}+y^{2}}}\\ \dfrac{x+1}{\sqrt{\left(x+1\right)^{2}+y^{2}}}-\dfrac{x-1}{\sqrt{\left(x-1 \right)^{2}+y^{2}}}&\dfrac{y}{\sqrt{\left(x+1\right)^{2}+y^{2}}}-\dfrac{y}{ \sqrt{\left(x-1\right)^{2}+y^{2}}}\\ \dfrac{x}{\sqrt{x^{2}+\left(y+1\right)^{2}}}-\dfrac{x-1}{\sqrt{\left(x-1 \right)^{2}+y^{2}}}&\dfrac{y+1}{\sqrt{x^{2}+\left(y+1\right)^{2}}}-\dfrac{y}{ \sqrt{\left(x-1\right)^{2}+y^{2}}}\end{array} \right] \tag{34}\]  
When the MUE moves from (1, 0) to (0.1, 0), the GDOPs for different average TDOA position measurement errors (0, 1, 2, 3, 4 m) are computed and these are shown in Figure 4. It shows that the GDOP decreases with the distance between the MUE and the square center. The GDOP also decreases with the average TDOA position measurement error. Criterion 2: Lozenge optimum geometry disposition decision criterion depends on the complementary angle difference (CAD), which is the difference between two complementary angles of a lozenge shape. When the CAD of the lozenge optimum geometry disposition is smaller, the positioning accuracy is better. For simplification, the GDOP formula is derived by substituting the coordinates of FeNB\({}_{0}\) (1, 0), FeNB\({}_{2}\) (\(-1,0\)), FeNB\({}_{1}\) (0, \(N\)), FeNB\({}_{3}\)(0, \(-N\)), and MUE (0.1, 0) into Equation (31), to yield:  
\[H=\left[ \begin{array}{cc}\dfrac{1}{10\sqrt{N^{2}+0.01}}+1&\dfrac{-N}{\sqrt{N ^{2}+0.01}}\\ 2&0\\ \dfrac{1}{10\sqrt{N^{2}+0.01}}+1&\dfrac{N}{\sqrt{N^{2}+0.01}}\end{array} \right] \tag{35}\]  
Figure 4: GDOP for different distances. Substituting Equations (28) and (35) into Equation (33) yields:  
\[G=\left[ \begin{array}{cc}\frac{(5+\bar{\epsilon}^{2})(N^{2}+0.01)}{200N^{2}+3}&0 \\ 0&\frac{(5+\bar{\epsilon}^{2})(N^{2}+0.01)}{10N^{2}}\end{array} \right] \tag{36}\]  
Then the GDOP of lozenge shape is expressed as:  
\[ \begin{array}{cc}GDOP&=G_{1,1}+G_{2,2}\\ &=\frac{(\bar{\epsilon}^{2}+5)(N^{2}+0.01)}{200N^{2}+3}+\frac{(\bar{\epsilon} ^{2}+5)(N^{2}+0.01)}{10N^{2}}\end{array}  \tag{37}\]  
The CAD formula is derived as:  
\[\text{CAD}=2(\arccos(\frac{1}{\sqrt{1+N^{2}}})-\arccos(\frac{N}{\sqrt{1+N^{2}}})) \tag{38}\]  
Figure 5 is generated from Equations (37) and (38), where \(N\) varies from 0 to 1 and \(\sigma_{r}^{2}=5\). It shows that the GDOP decreases with the CAD. The GDOP also decreases with the average TDOA position measurement error. When \(N=1\), CAD = 0\({}^{\circ}\). This is the square shape, which has a smaller GDOP value than the lozenge shape. When N = 0, CAD = 180\({}^{\circ}\). This is the straight line, which has the largest GDOP value. Criterion 3: Irregular-quadrilateral optimum geometry disposition decision criterion depends on the distance between the diagonal intersection and the center of the IQ. When the distance is shorter, the positioning accuracy of the IQ optimum geometry disposition is better.