## 2 Hybrid Source Localization Based on TOA and TDOA
### Analysis of Positioning Accuracy Analysis  
#### 2.2.1 Least Squares Localization Error Analysis  
The precision of positioning is determined by two factors: the measurement error and the geometric factor of the positioning error, which are usually coupled together. The geometric factor of the measurement error and the positioning error are usually mixed together, but can be considered separately and independently only when the measurement noise of each signal source is independent and has the same variance. Independent consideration of the geometric factors of measurement error and positioning error facilitates the separate evaluation of the station placement scheme, and the geometric factor of positioning error is usually expressed as GDOP, which can also be expressed as HDOP (horizontal) or DOP in the direction of the other axis. The measurement error generally includes the station location positioning error of each station and the measurement error of TOA and TDOA, which are considered to be independent of each other, and the measurement error of TOA and TDOA includes the signal propagation error, synchronization error between measurement stations, a random error caused by receiver noise, Doppler effect error, etc.  
Considering the error terms of TDOA, it is obtained from (2) that  
\[\begin{array}{l}d(r_{TDOA,\;i})=dr_{i}-dr_{0}=d(\sqrt{\left(x-x_{i}\right)^{ 2}+\left(y-y_{i}\right)^{2}+\left(z-z_{i}\right)^{2}}\\ -\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^ {2}})\\ =\left(g_{ix}-g_{0x}\right)dx+\left(g_{iy}-g_{0y}\right)dy+\left(g_{iy}-g_{0y} \right)dz-h_{i}+h_{0},\;i=1,2,3\end{array} \tag{10}\]  
where  
\[g_{ix}=\frac{x-x_{i}}{\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2} +\left(z-z_{i}\right)^{2}}} \tag{11}\]  
\[g_{iy}=\frac{y-y_{i}}{\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2} +\left(z-z_{i}\right)^{2}}} \tag{12}\]  
\[g_{iz}=\frac{z-z_{i}}{\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2} +\left(z-z_{i}\right)^{2}}} \tag{13}\]  
\[h_{i}=g_{ix}dx_{i}+g_{iy}dy_{i}+g_{iz}dz_{i} \tag{14}\]  
\(d(r_{TDOA\;i})\) represents the measurement error of distance difference, \(dx_{i}dy_{i}\), \(dz_{i}\) are the measurement errors of the station positions in the three-coordinate axis. \(dx_{i}dy_{i}\) are signal source \(T\) positioning errors. Combine the three equations in (10), and let  
\[\begin{array}{l}\mathbf{G}_{TDOA}=\begin{pmatrix}g_{1x}-g_{0x}&g_{1y}-g_{0y}&g_{ 1z}-g_{0z}\\ g_{2x}-g_{0x}&g_{2y}-g_{0y}&g_{2z}-g_{0z}\\ g_{3x}-g_{0x}&g_{3y}-g_{0y}&g_{3z}-g_{0z}\end{pmatrix}\\ \\ \mathbf{dx_{s}}_{\text{TDOA}}=\begin{pmatrix}h_{1}-h_{0}\\ h_{2}-h_{0}\\ h_{3}-h_{0}\end{pmatrix},\;\mathbf{dx}=\begin{pmatrix}dx\\ dy\\ dz\end{pmatrix}\\ \\ \mathbf{dr_{TDOA}}=\begin{pmatrix}d(r_{TDOA,\;1})\\ d(r_{TDOA,2})\\ d(r_{TDOA,3})\end{pmatrix}=\begin{pmatrix}dr_{1}-dr_{0}\\ dr_{2}-dr_{0}\\ dr_{3}-dr_{0}\end{pmatrix}\\ \\ \mathbf{G}_{TDOA}dx=d\mathbf{r_{TDOA}}+d\mathbf{x_{s}}_{\text{TDOA}} \tag{15}\]  
Considering the error terms of TOA, from (3) we can obtain  
\[dr_{i}=d\left(\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}+\left(z -z_{i}\right)^{2}}\right)=g_{ix}dx+g_{iy}dy+g_{iz}dz-h_{i},\;i=4,5,6 \tag{16}\]  
\[\mathbf{C}=E[(d\mathbf{r}+d\mathbf{x}_{s})\Big{(}d\mathbf{r}+d\mathbf{x}_{s})^{T}]=\begin{pmatrix} \sigma^{2}&0&0&0&0&0\\ 0&\sigma^{2}&0&0&0&0\\ 0&0&\sigma^{2}&0&0&0\\ 0&0&0&2\sigma^{2}&\sigma^{2}&\sigma^{2}\\ 0&0&0&\sigma^{2}&2\sigma^{2}&\sigma^{2}\\ 0&0&0&\sigma^{2}&\sigma^{2}&2\sigma^{2}\end{pmatrix} \tag{25}\]  
The covariance matrix of positioning error \(\sigma^{2}\) based on least square estimation is denoted as  
\[\sigma^{2}=E\Big{[}dxd\mathbf{x}^{T}\Big{]}=\left(\mathbf{G}^{T}\mathbf{G}\right)^{-1}\bm {G}^{T}\mathbf{CG}\left(\mathbf{G}^{T}\mathbf{G}\right)^{-1} \tag{26}\]  
The positioning error can be denoted as \(\mathit{Tr}\big{(}\sigma^{2}\big{)}\).