# Geolocation and Tracking by TDOA Measurements Based on Space-Air-Ground Integrated Network
## 2 Theoretical Performance
### Crlb
#### 2.2.1 CRLB in ECEF Coordinate System  
Define the unknown parameter as \(\theta_{1}=\left[\theta_{u}^{T},\theta_{nav}^{T}\right]^{T}\), where the target's location vector is \(\theta_{u}=\left[\mathbf{u}_{\theta}^{T},\dot{\mathbf{u}}_{\theta}^{T}\right] ^{T}\), and \(\theta_{nav}=\mathbf{z}_{nav}^{o}\) is the location vector of the observers. The FIM of the unknown parameter without using the constrained conditions in (1) can be derived as  
\[\mathbf{I}_{1}(\theta_{1})=\mathbf{J}_{1}^{T}\mathbf{C}^{-1}\mathbf{J}_{1} \tag{7}\]  
where  
\[\mathbf{J}_{1}=\frac{\partial\mathbf{z}}{\partial\theta_{1}}=\begin{bmatrix} \mathbf{J}_{u}&\mathbf{J}_{t,nav}\\ 0&\mathbf{E}_{nav}\end{bmatrix} \tag{8}\]  
\[\mathbf{J}_{u}=\frac{\partial\mathbf{z}_{t}}{\partial\theta_{u}}\mathbf{J}_{t,nav}=\frac{\partial\mathbf{z}_{t}}{\partial\theta_{nav}}\mathbf{E}_{nav}= \frac{\partial\mathbf{z}_{nav}}{\partial\theta_{nav}} \tag{9}\]  
Then, the CRLB of \(\theta_{1}\) without using the constraint conditions is  
\[\text{CRLB}(\theta_{1})=\begin{cases}\begin{bmatrix}\mathbf{J}_{u}&\mathbf{J }_{t,nav}\\ 0&\mathbf{E}_{nav}\end{bmatrix}^{T}\begin{bmatrix}\mathbf{C}_{t}^{-1}&0\\ 0&\mathbf{C}_{nav}^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{J}_{u}&\mathbf{J}_{ t,nav}\\ 0&\mathbf{E}_{nav}\end{bmatrix}^{-1}=\begin{bmatrix}\mathbf{A}_{1}&\mathbf{U}_{1} \\ \mathbf{V}_{1}&\mathbf{D}\end{bmatrix}^{-1} \tag{10}\]  
Hence, the CRLB of \(\theta_{u}\) without constrained conditions is  
\[\text{CRLB}(\theta_{u})=\mathbf{A}_{1}^{-1}  
\[\frac{\partial\textbf{u}_{k}}{\partial\textbf{u}_{o}}=\textbf{I}_{3 \times 3}\frac{\partial\textbf{u}_{k}}{\partial\ddot{\textbf{u}}_{o}}=(k-1)T \times\textbf{I}_{3\times 3} \tag{19}\] \[\textbf{J}_{t,nav,k}=\frac{\partial\textbf{z}_{t,k}}{\partial \textbf{z}_{nav,k}}=\left[\frac{\partial t_{1,0,k}}{\partial\textbf{z}_{nav,k} };\frac{\partial t_{2,0,k}}{\partial\textbf{z}_{nav,k}};\dots;\frac{\partial t _{M-1,0,k}}{\partial\textbf{z}_{nav,k}}\right]\] (20) \[\frac{\partial t_{j,0,k}}{\partial\textbf{z}_{nav,k}}=\left[\frac{ \textbf{u}_{k}^{T}-\textbf{s}_{0,k}^{T}}{c\left\|\textbf{u}_{k}-\textbf{s}_{ 0,k}\right\|},0_{1\times 3(j-1)},-\frac{\textbf{u}_{k}^{T}-\textbf{s}_{j,k}^{T}}{c \left\|\textbf{u}_{k}-\textbf{s}_{j,k}\right\|},0_{1\times 3(M-1-j)}\right] \tag{21}\]  
The constrained conditions in (1) can be rewritten as  
\[G(\theta_{1})=\begin{bmatrix}\textbf{u}_{0}^{T}\textbf{u}_{0}-(N+h)^{2}\\ \textbf{u}_{0}^{T}\ddot{\textbf{u}}_{0}\end{bmatrix} \tag{22}\]  
The gradient matrix of (22) with respect to the unknown parameter \(\theta_{1}\) is  
\[\textbf{F}=\begin{bmatrix}\frac{\partial G(\theta_{1})}{\partial\theta_{1}^{T }}\end{bmatrix}^{\rm T}=\begin{bmatrix}\textbf{F}_{u}\\ \textbf{O}\end{bmatrix} \tag{23}\]  
where  
\[\textbf{F}_{u}=\begin{bmatrix}\frac{\partial G(\theta_{1})}{\partial\theta_{u} ^{T}}\end{bmatrix}^{\rm T}=\begin{bmatrix}2\textbf{u}_{0}^{T}&0\\ \vdots\ddot{\textbf{u}}_{0}^{T}&\textbf{u}_{0}^{T}\end{bmatrix}^{\rm T} \tag{24}\]  
The constrained CRLB of \(\theta_{1}\) is derived as [31]  
\[\text{CRLB}(\theta_{1})=\textbf{I}_{1}^{-1}-\textbf{I}_{1}^{-1}\textbf{F} \Big{(}\textbf{F}^{\rm T}\textbf{I}_{1}^{-1}\textbf{F}\Big{)}^{-1}\textbf{F} ^{\rm T}\textbf{I}_{1}^{-1} \tag{25}\]