# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## Appendix A Joint Time-Delay and Phase Estimation Cramer-Rao Bound
\right)^{T}\right)^{-1}\] \[= \frac{\sigma_{n}^{2}}{2}\left(\mathbf{\epsilon}^{H}\mathbf{\epsilon}-\frac {\text{Re}\left\{\mathbf{\epsilon}\left(\tau\right)^{H}\frac{\partial\mathbf{\epsilon}}{ \partial\mathbf{\epsilon}}\right\}^{2}}{\text{Re}\left\{\frac{\mathbf{\epsilon}^{H}}{ \partial\mathbf{\epsilon}}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\epsilon}}\right\}^ {2}}\frac{\left\|\mathbf{\epsilon}^{H}\right\|^{2}}{\mathbf{\epsilon}^{H}\mathbf{\epsilon}} \left(1+\frac{\text{Re}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{ \partial\mathbf{\epsilon}}\right\}^{2}}{\mathbf{\epsilon}^{H}\mathbf{\epsilon}\mathbf{\epsilon}} \right)^{2}\right)^{-1}=\frac{\sigma_{n}^{2}}{2}\frac{1}{\sigma^{2}}\left(1+ \frac{\text{Re}\left\{\mathbf{\epsilon}^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{ \epsilon}}\right\}^{2}}{\mathbf{\epsilon}^{H}\mathbf{\epsilon}\mathbf{\epsilon}}\right)\] \[= \frac{\sigma_{n}^{2}}{2\left\|\mathbf{\epsilon}\left(\tau\right)\right\| ^{2}}+\rho^{2}CRB_{\text{r}}\frac{\text{Re}\left\{\mathbf{\epsilon}\left(\mathbf{ \epsilon}^{H}\right)^{H}\frac{\partial\mathbf{\epsilon}}{\partial\mathbf{\epsilon}}\right\}^ {2}}{\left\|\mathbf{\epsilon}\left(\tau\right)\right\|^{4}}.\]  
Moreover, from [16],  
\[\lim_{\text{min}\left(N_{1},N_{2}\right)\rightarrow\infty}\left\| \mathbf{\epsilon}\left(\tau\right)\right\|^{2} = F_{\text{s}}\mathbf{w}_{1}=F_{\text{s}}\left(\frac{1}{F_{\text{s}}} \mathbf{\epsilon}^{H}\mathbf{\epsilon}\right)=\mathbf{\epsilon}^{H}\mathbf{\epsilon},\] \[\lim_{\text{min}\left(N_{1},N_{2}\right)\rightarrow\infty}\left\| \frac{\partial\mathbf{\epsilon}\left(\tau\right)}{\partial\mathbf{\tau}}\right\|^{2} = F_{\text{s}}\mathbf{w}_{2}=F_{\text{s}}\left(F_{\text{s}}\mathbf{\epsilon}^{H}\mathbf{ \
abla}\mathbf{\epsilon}\right)=F_{\text{s}}^{2}\mathbf{\epsilon}^{H}\mathbf{\
abla}\mathbf{ \epsilon},\] \[\lim_{\text{min}\left(N_{1},N_{2}\right)\rightarrow\infty}\mathbf{ \epsilon}\left(\tau\right)^{H}\frac{\partial\mathbf{\epsilon}\left(\tau\right)}{ \partial\mathbf{\tau}} = F_{\text{s}}\mathbf{w}_{3}=F_{\text{s}}\left(\mathbf{\epsilon}^{H}\mathbf{\Lambda} \mathbf{\epsilon}\right)=F_{\text{s}}\mathbf{\epsilon}^{H}\mathbf{\Lambda}\mathbf{\epsilon},\]and we finally obtain the CRB from the signal samples as  
\[\mathbf{CRB}_{\theta} = \left[ \begin{array}{cc}\mathrm{CRB}_{\varphi}&\mathrm{CRB}_{\tau, \varphi}\\ \mathrm{CRB}_{\tau,\varphi}&\mathrm{CRB}_{\tau}\end{array} \right], \tag{100}\] \[\mathrm{CRB}_{\tau} = \frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\frac{1}{F_{s}^{2}}\left( \frac{\epsilon^{H}\mathbf{v}}{\epsilon^{H}\mathbf{c}}-\mathrm{Im}\left\{ \frac{\epsilon^{H}\mathbf{A}\mathbf{c}}{\epsilon^{H}\mathbf{c}}\right\}^{2} \right)^{2}\right),\] (101) \[\mathrm{CRB}_{\varphi} = \frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\left(1+\frac{\mathrm{Im} \left\{\frac{\epsilon^{H}\mathbf{A}\mathbf{c}}{\epsilon^{H}\mathbf{c}}\right\} ^{2}}{\frac{\epsilon^{H}\mathbf{v}}{\epsilon^{H}\mathbf{c}}-\mathrm{Im}\left\{ \frac{\epsilon^{H}\mathbf{A}\mathbf{c}}{\epsilon^{H}\mathbf{c}}\right\}^{2}} \right),\] (102) \[\mathrm{CRB}_{\tau,\varphi} = \frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\frac{\mathrm{Im}\left\{ \frac{\epsilon^{H}\mathbf{A}\mathbf{c}}{\epsilon^{H}\mathbf{c}}\right\}}{F_{s }\left(\frac{\epsilon^{H}\mathbf{v}}{\epsilon^{H}\mathbf{c}}-\mathrm{Im}\left\{ \frac{\epsilon^{H}\mathbf{A}\mathbf{c}}{\epsilon^{H}\mathbf{c}}\right\}^{2} \right)^{2}},\] (103) \[\mathrm{CRB}_{\rho} = \frac{\sigma_{n}^{2}}{2\epsilon^{H}\mathbf{c}}. \tag{104}\]