# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## Appendix A Joint Time-Delay and Phase Estimation Cramer-Rao Bound
\begin{array}{ cc}\mathbf{C}_{1}^{-1}&-\mathbf{A}_{11}^{-1}\mathbf{A}_{12}\mathbf{C}_{2}^{-1}\\ -\mathbf{C}_{2}^{-1}\mathbf{A}_{21}\mathbf{A}_{11}^{-1}&\mathbf{C}_{2}^{-1} \end{array} \right], \tag{13}\]  
where \(\mathbf{C}_{1}=\mathbf{A}_{11}-\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{A}_ {21}\), \(\mathbf{C}_{2}=\mathbf{A}_{22}-\mathbf{A}_{21}\mathbf{A}_{11}^{-1}\mathbf{A}_ {12}\) and \(\mathbf{C}_{1}^{-1}=\mathbf{A}_{11}^{-1}+\mathbf{A}_{11}^{-1}\mathbf{A}_{12} \mathbf{C}_{2}^{-1}\mathbf{A}_{21}\mathbf{A}_{11}^{-1}\), then  
\[\mathbf{CRB}_{\left(\rho,\boldsymbol{\theta}\right)}=\left[ \begin{array}{cc} \operatorname{CRB}_{\rho}&\mathbf{CRB}_{\rho,\boldsymbol{\theta}}\\ \mathbf{CRB}_{\rho,\boldsymbol{\theta}}^{T}&\mathbf{CRB}_{\boldsymbol{\theta}} \end{array} \right],\]  
with  
\[\operatorname{CRB}_{\rho}=\frac{\sigma_{n}^{2}}{2}\left(\boldsymbol{\epsilon}^{ \prime T}\mathbf{\Pi}_{\frac{\partial\underline{\boldsymbol{\epsilon}}^{ \prime}}{\partial\overline{\boldsymbol{\epsilon}}^{\prime}}}^{\perp}\boldsymbol{ \epsilon}^{\prime}\right)^{-1},\quad\mathbf{CRB}_{\boldsymbol{\theta}}=\frac{ \sigma_{n}^{2}}{2\rho^{2}}\boldsymbol{\Phi}_{\boldsymbol{\theta}}^{-1},\quad \boldsymbol{\Phi}_{\boldsymbol{\theta}}^{-1}=\left(\frac{\partial\underline{ \boldsymbol{\epsilon}}^{\prime}}{\partial\overline{\boldsymbol{\theta}}^{T}} \right)^{T}\mathbf{\Pi}_{\underline{\boldsymbol{\epsilon}}^{\perp}}^{\perp} \frac{\partial\underline{\boldsymbol{\epsilon}}^{\prime}}{\partial\overline{ \boldsymbol{\theta}}^{T}}. \tag{14}\]  
Using the following equality  
\[\mathbf{\underline{A}}^{T}\mathbf{\underline{B}}=\operatorname{Re}\left\{ \mathbf{A}\right\}^{T}\operatorname{Re}\left\{\mathbf{B}\right\}+ \operatorname{Im}\left\{\mathbf{A}\right\}^{T}\operatorname{Im}\left\{\mathbf{B} \right\}=\operatorname{Re}\left\{\mathbf{A}^{H}\mathbf{B}\right\},\]  
we have that  
\[\boldsymbol{\underline{\boldsymbol{\epsilon}}^{\prime T}}\boldsymbol{ \underline{\boldsymbol{\epsilon}}^{\prime}} = \mathbf{c}^{H}\mathbf{c}\] \[\left(\frac{\partial\underline{\boldsymbol{\epsilon}}^{\prime}}{ \partial\overline{\boldsymbol{\theta}}^{T}}\right)^{T}\frac{\partial\underline{ \boldsymbol{\epsilon}}^{\prime}}{\partial\overline{\boldsymbol{\theta}}^{T}} = \operatorname{Re}\left\{\left[ \begin{array}{cc}-je^{-j\varphi} \mathbf{c}^{H}\\ e^{-j\varphi}\frac{\partial\underline{\boldsymbol{\epsilon}}^{H}}{\partial \overline{\boldsymbol{\epsilon}}^{H}}\end{array} \right]\left[ \begin{array}{ cc}je^{j\varphi}\mathbf{c}&e^{j\varphi}\frac{\partial\underline{\boldsymbol{\epsilon}}}{ \partial\overline{\boldsymbol{\epsilon}}^{\prime}}\\ e^{-j\varphi}\frac{\partial\underline{\boldsymbol{\epsilon}}^{H}}{\partial \overline{\boldsymbol{\epsilon}}^{\prime}}\end{array} \right]\right\}= \operatorname{Re}\left\{\left[ \begin{array}{cc}\left\|\mathbf{c}\right\|^{2}&- j\mathbf{c}^{H}\frac{\partial\underline{\boldsymbol{\epsilon}}}{\partial\overline{ \boldsymbol{\epsilon}}^{\prime}}\\ j\frac{\partial\underline{\boldsymbol{\epsilon}}^{H}}{\partial\overline{ \boldsymbol{\epsilon}}^{H}}\mathbf{c}&\frac{\partial\underline{\boldsymbol{ \epsilon}}}{\partial\overline{\boldsymbol{\epsilon}}^{H}}\frac{\partial \underline{\boldsymbol{\epsilon}}}{\partial\overline{\boldsymbol{\epsilon}}}\end{array}  \right]\right\}\] \[\boldsymbol{\underline{\boldsymbol{\epsilon}}^{\prime T}}\frac{\partial \underline{\boldsymbol{\epsilon}}^{\prime}}{\partial\overline{\boldsymbol{\theta}}^{T}} = \operatorname{Re}\left\{e^{-j\varphi}\mathbf{c}^{H}\left[ \begin{array}{ cc}je^{j\varphi}\mathbf{c}&e^{j\varphi}\frac{\partial\underline{\boldsymbol{\epsilon}}}{ \partial\overline{\boldsymbol{\epsilon}}}\end{array} \right]\right\}=\left( \begin{array}{cc}0&\operatorname{Re}\left\{\mathbf{c}^{H}\frac{\partial \underline{\boldsymbol{\epsilon}}}{\partial\overline{\boldsymbol{\epsilon}}}\right\} \end{array}