## 4 GDOP of the Proposed Hybrid Positioning  
GDOP is the ratio of the accuracy limitation of a position fix to the accuracy of measurements [21]. A positioning system has a small GDOP means that the positioning is accurate. In order to compare with GDOP of TDOA-only positioning, the GDOP of the proposed hybrid positioning is defined as the ratio of the positioning accuracy to the TDOA ranging accuracy. In this section, the GDOP of the hybrid positioning based on multiple TDOAs and single TOA is derived. To achieve the GDOP of the hybrid TDOA and TOA positioning of the system described in Section 2, the difference between TDOA measurements and real distances difference when UE is at the location \((x,y)\) can be expressed as:  
\[f_{i}(x,y)=d_{i,1}-r_{1}+r_{i},\ i=2,3,\ldots,M. \tag{29}\]  
Difference between TOA measurements and real distances when UE is at the location \((x,y)\) is expressed as  
\[f_{1}(x,y)=d_{1}-r_{1}. \tag{30}\]  
Using a first order Taylor series to approximate Equations (6) and (7), the approximation of \(f_{i}(x,y)\) at location \(\Big{(}x_{p},y_{p}\Big{)}\) can be obtained as  
\[f_{i}(x,y)\approx f_{i}\Big{(}x_{p},y_{p}\Big{)}+\Big{(}x-x_{p}\Big{)}\frac{ \partial f_{i}\Big{(}x_{p},y_{p}\Big{)}}{\partial x}+\Big{(}y-y_{p}\Big{)} \frac{\partial f_{i}\Big{(}x_{p},y_{p}\Big{)}}{\partial y},\ i=1,2,3,\ldots,M. \tag{31}\]  
When there are no ranging errors, (9) can be obtained from (8). \[0=f_{i}\Big{(}x_{p},y_{p}\Big{)}+\Big{(}x^{0}-x_{p}\Big{)}\frac{\partial f_{i }\Big{(}x_{p},y_{p}\Big{)}}{\partial x}+\Big{(}y^{0}-y_{p}\Big{)}\frac{ \partial f_{i}\Big{(}x_{p},y_{p}\Big{)}}{\partial y} \tag{32}\]  
where \(\Big{(}x^{0},y^{0}\Big{)}\) is UE's real location. To simplify the mathematical derivations, we use \(e_{1}\) to represent the TOA ranging error. Moreover, \(e_{i}\), \(i=2,\ldots,M\), is used to represent TDOA ranging errors between the signal from ANT \(i\) and signal from ANT 0, which is represented as \(e_{i,1}\) in Section 2. As the measurements have ranging errors \(e_{i}\), Equation (33) can be obtained from (31):  
\[e_{i}=f_{i}\Big{(}x_{p},y_{p}\Big{)}+\Big{(}x-x_{p}\Big{)}\frac{\partial f_{i }\Big{(}x_{p},y_{p}\Big{)}}{\partial x}+\Big{(}y-y_{p}\Big{)}\frac{\partial f_ {i}\Big{(}x_{p},y_{p}\Big{)}}{\partial y} \tag{33}\]where \((x\prime,y\prime)\) is the position estimation result. Let position estimation error vector \(\Delta u=\left(e_{x},e_{y}\right)^{T}\),\(e_{x}=x\prime-x^{0},e_{y}=y\prime-y^{0}\). Subtracting Equation (33) from Equation (32), the equation of \(e_{i}\) and \(\left(e_{x},e_{y}\right)^{T}\) can be obtained as:  
\[e_{i}=\ \frac{\partial f_{i}\Big{(}x_{p},y_{p}\Big{)}}{\partial x}e_{x}+\frac{ \partial f_{i}\Big{(}x_{p},y_{p}\Big{)}}{\partial y}e_{y}. \tag{34}\]  
For TOA ranging error \(e_{0}\), it is easy to obtain partial differential equations:  
\[\alpha_{1}=\frac{\partial f_{1}\Big{(}x_{p},y_{p}\Big{)}}{\partial x}=\frac{x _{1}-x_{p}}{r_{1}}\beta_{1}=\frac{\partial f_{1}\Big{(}x_{p},y_{p}\Big{)}}{ \partial y}=\frac{y_{1}-y_{p}}{r_{1}}. \tag{35}\]  
For TDOA ranging error \(e_{i}\), partial differential equations is:  
\[ \begin{array}{c}\alpha_{i}=\frac{\partial f_{i}\big{(}x_{p},y_{p}\big{)}}{ \partial x}=\frac{x_{i}-x_{p}}{r_{i}}-\frac{x_{1}-x_{p}}{r_{1}}\\ \beta_{i}=\frac{\partial f_{i}\big{(}x_{p},y_{p}\big{)}}{\partial y}=\frac{y_ {i}-y_{p}}{r_{i}}-\frac{y_{1}-y_{p}}{r_{1}}\\ i=2,3,\ldots,M.\end{array}  \tag{36}\]  
Therefore, Equation (34) can be formulated as the following matrix form:  
\[A\Delta u=e \tag{37}\]  
where:  
\[A=\left[ \begin{array}{ccc}\alpha_{1}&\beta_{1}\\ \alpha_{2}&\beta_{2}\\ \alpha_{3}&\beta_{3}\\ \vdots&\vdots\\ \alpha_{M}&\beta_{M}\end{array} \right]\Delta u=\left[ \begin{array}{c}e_{x} \\ e_{y}\end{array} \right]e=\left[ \begin{array}{c}e_{1}\\ e_{2}\\ e_{3}\\ \vdots\\ e_{M}\end{array} \right] \tag{38}\]  
Different from the GDOP calculation process in [21], TOA and TDOA measurements ranging errors have different variances.