## 4 GDOP of the Proposed Hybrid Positioning
Therefore, in order to make an unbiased and efficient estimator, the vector \(\Delta u\) cannot be calculated based on the ordinary least square (OLS) but weighted least square (WLS) as the following expression:  
\[\Delta u=\left(A^{T}WA\right)^{-1}A^{T}We. \tag{39}\]  
where the weighting matrix W could be given by covariance matrix of error vector \(e\). Therefore, we could get a covariance matrix of error vector \(E\) as:  
\[Cov(E)=E\left(\left[ \begin{array}{ccccc}e_{1}\\ e_{2}\\ e_{3}\\ \vdots\\ e_{M}\end{array} \right]\left[ \begin{array}{ccccc}e_{1}&e_{2}&e_{3}&\ldots&e_{M }\\ \vdots&e_{M}\end{array} \right]\right)=\sigma_{tdda}^{2}\left[ \begin{array}{ cccc}k^{2}&0&0&\ldots&0\\ 0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\end{array} \right]=\sigma_{tdda}^{2}\Sigma \tag{40}\]Therefore, the weighting matrix \(W\) in Equation (39) could be given as  
\[W=\Sigma^{-1}=\left[ \begin{array}{ccccc}\frac{1}{k^{2}}&0&0&\ldots&0\\ 0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\end{array} \right] \tag{41}\]  
Expressing matrix \(A^{T}W\) as  
\[A^{T}W=\left[ \begin{array}{ccccc}\alpha_{1}&\alpha_{2}&\alpha_{3}&\ldots& \alpha_{M}\\ \beta_{1}&\beta_{2}&\beta_{3}&\ldots&\beta_{M}\end{array} \right]\left[ \begin{array}{ccccc}\frac{1}{k^{2}}&0&0&\ldots&0\\ 0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\end{array} \right]=\left[ \begin{array}{ccccc}\frac{\alpha_{ 1}}{k^{2}}&\alpha_{3}&\alpha_{3}&\ldots&\alpha_{M}\\ \frac{\beta_{1}}{k^{2}}&\beta_{2}&\beta_{3}&\ldots&\beta_{M}\end{array} \right] \tag{42}\] \[=\left[ \begin{array}{ccccc}a_{1}&a_{2}&a_{3}&\ldots&a_{M}\\ b_{1}&b_{2}&b_{3}&\ldots&b_{M}\end{array} \right]\]  
From Equation (39), \(e_{x}\) can be calculated by:  
\[e_{x}=\left[\left(A^{T}WA\right)^{-1}\right]_{1,1}\sum\limits_{i=1}^{M}a_{i}e _{i}+\left[\left(A^{T}WA\right)^{-1}\right]_{1,2}\sum\limits_{i=1}^{M}b_{i}e_{ i}=\frac{s_{1}e_{1}}{k^{2}}+\sum\limits_{i=2}^{M}s_{i}e_{i} \tag{43}\]  
where  
\[s_{i}=\left[\left(A^{T}WA\right)^{-1}\right]_{1,1}a_{i}+\left[\left(A^{T}WA \right)^{-1}\right]_{1,2}b_{i}. \tag{44}\]  
The mean of \(e_{x}\) could be calculated by:  
\[E(e_{x})=E\left(s_{1}e_{1}+\sum\limits_{i=2}^{M}s_{i}e_{i}\right)=E(s_{1}e_{1 })+\sum\limits_{i=2}^{M}E(s_{i}e_{i})=s_{1}E(e_{1})+\sum\limits_{i=2}^{M}s_{i }E(e_{i})=0 \tag{45}\]  
The variance of \(e_{x}\) could be calculated by:  
\[ \begin{array}{rl}\sigma_{x}^{2}=Var(e_{x})=&E\left(e_{x}^{2}\right)-E(e_{x} )^{2}=E\left(e_{x}^{2}\right)=E\left(\left(s_{1}e_{1}+\sum\limits_{i=2}^{M}s_ {i}e_{i}\right)^{2}\right)\\ &=E\left(\left(s_{1}e_{1}\right)^{2}+\sum\limits_{i=2}^{M}\left(s_{i}e_{i} \right)^{2}\right)=E\left(\left(s_{1}e_{1}\right)^{2}\right)+\sum\limits_{i=2} ^{M}E\left(\left(s_{i}e_{i}\right)^{2}\right)\\ &=\sigma_{tot}^{2}s_{1}{}^{2}+\sigma_{tota}^{2}\sum\limits_{i=2}^{M}s_{i}{}^{ 2}=k^{2}\sigma_{tota}^{2}s_{1}{}^{2}+\sigma_{tota}^{2}\sum\limits_{i=2}^{M}s_{ i}{}^{2}.\end{array}  \tag{46}\]  
It is easy to obtain the mean and variance of \(e_{y}\) using the above equations:  
\[E\left(e_{y}\right)=0,\ \sigma_{y}^{2}=Var\left(e_{y}\right)=k^{2}\sigma_{tota}^{2 }s_{1}{}^{2}+\sigma_{tota}^{2}\sum\limits_{i=2}^{M}s_{i}{}^{2} \tag{47}\]  
where:  
\[g_{i}=\left[\left(A^{T}WA\right)^{-1}\right]_{2,1}a_{i}+\left[\left(A^{T}WA \right)^{-1}\right]_{2,2}b_{i} \tag{48}\]  
GDOP was been definite as the ratio of the accuracy of a position fix to the accuracy of measurements [34]. In order to directly compare the GDOP of the proposed hybrid positioning and the GDOP of TDOA-only positioning, we redefine the GDOP of the hybrid positioning usingmultiple TDOAs and single TOA as the ratio of the accuracy of a position fix to the accuracy of TDOA measurements. The GDOP could be calculated as:  
\[GDOP=\sqrt{\frac{\sigma_{x}^{2}+\sigma_{y}^{2}}{\sigma_{tola}^{2}}}=\sqrt{k^{2}({ s_{1}}^{2}+{g_{1}}^{2})+\sum_{i=2}^{M}({s_{i}}^{2}+{g_{i}}^{2})}. \tag{49}\]