# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## Appendix A Joint Time-Delay and Phase Estimation Cramer-Rao Bound
\right),\]which lead to  
\[\mathbf{\Phi}_{\mathbf{\theta}}=\frac{\frac{\partial\mathbf{\epsilon}^{H}}{\partial\mathbf{\theta} ^{T}}}{\frac{\partial\mathbf{\epsilon}^{H}}{\partial\mathbf{\theta}^{T}}}-\frac{1}{ \xi^{T}\xi}\left(\mathbf{\epsilon}^{T}\frac{\partial\mathbf{\xi}^{C}}{\partial\mathbf{ \theta}^{T}}\right)^{T}\left(\mathbf{\epsilon}^{T}\frac{\partial\mathbf{\epsilon}^{T} }{\partial\mathbf{\theta}^{T}}\right)=\left[ \begin{array}{cc}\left\|\mathbf{ \epsilon}\right\|^{2}&\text{Im}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{ \epsilon}}{\partial\mathbf{\epsilon}}\right\}\\ \text{Im}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{ \epsilon}}\right\}&\text{Re}\left\{\frac{\partial\mathbf{\epsilon}^{H}}{\partial \mathbf{\epsilon}}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\epsilon}}\right\}- \frac{\text{Re}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial \mathbf{\epsilon}}\right\}^{2}}{\left\|\mathbf{\epsilon}\right\|^{2}}\end{array}  \right]. \tag{10}\]  
Finally, using again Equation (10),  
\[\mathbf{\Phi}_{\mathbf{\theta}}^{-1}=\left[ \begin{array}{cc}C_{1}^{-1}&-A_{11}^{-1} A_{12}C_{2}^{-1}\\ -C_{2}^{-1}A_{21}A_{11}^{-1}&C_{2}^{-1}\end{array} \right]\]  
\[C_{2}=\text{Re}\left\{\frac{\partial\mathbf{\epsilon}^{H}}{\partial\mathbf{\epsilon}} \frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right\}-\frac{\text{Re}\left\{ \mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right\}^{2}}{ \left\|\mathbf{\epsilon}\right\|^{2}}-\frac{\text{Im}\left\{\mathbf{\epsilon}^{H} \frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right\}^{2}}{\left\|\mathbf{ \epsilon}\right\|^{2}}=\frac{\partial\mathbf{\epsilon}^{H}}{\partial\mathbf{\epsilon} }\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}-\frac{\left(\mathbf{\epsilon}^{ H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right)^{H}\left(\mathbf{\epsilon}^{H} \frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right)}{\left\|\mathbf{\epsilon} \right\|^{2}}=\frac{\partial\mathbf{\epsilon}^{H}}{\partial\mathbf{\tau}}\mathbf{\Pi}_{ \mathbf{\mathbf{\epsilon}}}^{\perp}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\]  
which allow us to express the CRB as  
\[\text{CRB}_{\mathbf{\theta}} = \left[ \begin{array}{cc}\text{CRB}_{\mathbf{\varphi}}&\text{CRB}_{ \text{r},\mathbf{\varphi}}\\ \text{CRB}_{\text{r},\mathbf{\varphi}}&\text{CRB}_{\text{r}}\end{array} \right], \tag{11}\] \[\text{CRB}_{\text{r}} = \frac{\sigma_{n}^{2}}{2\rho^{2}}\frac{1}{\frac{\mathbf{\epsilon}(\tau )^{H}\Pi_{\mathbf{\xi}(\tau)}^{\perp}\frac{\partial\mathbf{\epsilon}(\tau)}{\partial \mathbf{\tau}^{T}}}},\] (12) \[\text{CRB}_{\mathbf{\varphi}} = \frac{\sigma_{n}^{2}}{2\rho^{2}}\frac{1}{\left\|\mathbf{\epsilon} \left(\tau\right)\right\|^{2}}+\text{CRB}_{\text{r}}\frac{\text{Im}\left\{ \mathbf{\epsilon}\left(\tau\right)^{H}\frac{\partial\mathbf{\epsilon}(\tau)}{\partial \mathbf{\epsilon}}\right\}^{2}}{\left\|\mathbf{\epsilon}\left(\tau\right)\right\|^{4}},\] (13) \[\text{CRB}_{\text{r},\mathbf{\varphi}} = -\text{CRB}_{\text{r}}\frac{\text{Im}\left\{\mathbf{\epsilon}\left( \tau\right)^{H}\frac{\partial\mathbf{\epsilon}(\tau)}{\partial\mathbf{\tau}}\right\}}{ \left\|\mathbf{\epsilon}\left(\tau\right)\right\|^{2}}. \tag{14}\]  
Using the relation in Equation (11),  
\[\text{CRB}_{\mathbf{\rho}} = \frac{\sigma_{n}^{2}}{2}\left(\mathbf{\epsilon}^{H}\mathbf{\epsilon}- \left(\begin{array}{cc}0\\ \text{Re}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau} }\right\}\right)^{T}\left(\text{Re}\left\{\frac{\partial\mathbf{\epsilon}^{H}}{ \partial\mathbf{\epsilon}}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}}\right\}- \frac{\text{Im}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{ \epsilon}}\right\}^{2}}{\mathbf{\epsilon}^{H}\mathbf{\epsilon}}\right)^{-1}\left( \begin{array}{cc}0\\ \text{Re}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\tau}} \right\}\end{array}