# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## Appendix A Joint Time-Delay and Phase Estimation Cramer-Rao Bound  
Recall that the observation model of interest in Equation (1) is  
\[\mathbf{x}=\alpha\mathbf{c}\left(\tau\right)+\mathbf{n},\quad\mathbf{x}, \mathbf{n}\in\mathbb{C}^{N},\;\alpha\in\mathbb{C},\mathbf{n}\sim\mathcal{CN} \left(\mathbf{0},\sigma_{n}^{2}\mathbf{I}_{N}\right), \tag{1}\]  
which can be reparameterized as  
\[\mathbf{x}=\rho\mathbf{c}^{\prime}\left(\theta\right)+\mathbf{n},\quad\mathbf{ c}^{\prime}\left(\theta\right)=\mathbf{c}\left(\tau\right)e^{i\varphi},\;\rho \in\mathbb{R}^{+},\boldsymbol{\theta}^{\top}=\left(\varphi,\tau\right), \tag{2}\]or using a real and imaginary part representation, \(\forall\mathbf{A}\in\mathbb{C}^{Q\times R},\mathbf{\underline{A}}= \begin{bmatrix} \operatorname{Re}\left\{\mathbf{A}\right\}\\ \operatorname{Im}\left\{\mathbf{A}\right\}\end{bmatrix} \),  
\[\mathbf{\underline{x}}=\rho\underline{\mathbf{c}}^{\prime}\left(\boldsymbol{ \theta}\right)+\underline{\mathbf{n}},\quad\underline{\mathbf{n}}\sim\mathcal{ N}\left(\mathbf{0},\frac{\sigma_{n}^{2}}{2}\mathbf{I}_{2N}\right)\,, \tag{10}\]  
where the unknown deterministic parameters to be estimated are \(\boldsymbol{\epsilon}^{\top}=\left(\sigma_{n}^{2},\rho,\boldsymbol{\theta}^ {\top}\right)\). The corresponding CRB is given by the Slepian-Bangs Formula [39],  
\[\mathbf{\underline{x}}\sim\mathcal{N}\left(\mathbf{m}_{\underline {x}}\left(\boldsymbol{\epsilon}\right),\mathbf{C}_{\underline{x}}\left( \boldsymbol{\epsilon}\right)\right)\Rightarrow\mathbf{CRB}_{\boldsymbol{ \epsilon}}=\mathbf{F}_{\boldsymbol{\epsilon}}^{-1},\\ \left(\mathbf{F}_{\boldsymbol{\epsilon}}\right)_{k,l}=\frac{ \partial\mathbf{m}_{\underline{x}}\left(\boldsymbol{\epsilon}\right)^{T}}{ \partial\epsilon_{k}}\mathbf{C}_{\underline{x}}^{-1}\left(\boldsymbol{ \epsilon}\right)\frac{\partial\mathbf{m}_{\underline{x}}\left(\boldsymbol{ \epsilon}\right)}{\partial\epsilon_{l}}+\frac{1}{2}tr\left(\mathbf{C}_{ \underline{x}}^{-1}\left(\boldsymbol{\epsilon}\right)\frac{\partial\mathbf{C} _{\underline{x}}\left(\boldsymbol{\epsilon}\right)}{\partial\epsilon_{k}} \mathbf{C}_{\underline{x}}^{-1}\left(\boldsymbol{\epsilon}\right)\frac{ \partial\mathbf{C}_{\underline{x}}\left(\boldsymbol{\epsilon}\right)}{ \partial\epsilon_{l}}\right). \tag{11}\]  
For the sake of simplicity in the following equations, we note \(\boldsymbol{\epsilon}^{\prime}\triangleq\underline{\boldsymbol{\epsilon}}^{ \prime}\left(\boldsymbol{\theta}\right)\), \(\boldsymbol{\epsilon}\triangleq\mathbf{c}\left(\tau\right)\). Considering the model in Equation (10), \(\mathbf{m}_{\underline{x}}\left(\boldsymbol{\epsilon}\right)=\rho\underline{ \boldsymbol{\epsilon}}^{\prime}\left(\boldsymbol{\theta}\right)\) and \(\mathbf{C}_{\underline{x}}\left(\boldsymbol{\epsilon}\right)=\frac{\sigma_{n }^{2}}{2}\mathbf{I}_{2N}\), then  
\[\mathbf{CRB}_{\boldsymbol{\epsilon}}=\left[ \begin{array}{cc}\frac{\sigma_{ n}^{4}}{N}&\mathbf{0}^{T}\\ \mathbf{0}&\mathbf{CRB}_{\left(\rho,\boldsymbol{\theta}\right)}\end{array}  \right],\quad\mathbf{CRB}_{\left(\rho,\boldsymbol{\theta}\right)}=\frac{ \sigma_{n}^{2}}{2}\left[ \begin{array}{cc}\left\|\boldsymbol{\epsilon}^{ \prime}\right\|^{2}&\rho\underline{\boldsymbol{\epsilon}}^{\prime T}\frac{ \partial\underline{\boldsymbol{\epsilon}}^{\prime}}{\partial\overline{ \boldsymbol{\epsilon}}^{\prime}}\\ \rho\frac{\partial\underline{\boldsymbol{\epsilon}}^{\prime}}{\partial \overline{\boldsymbol{\epsilon}}^{\prime}}\underline{\boldsymbol{\epsilon}}^{ \prime}&\rho^{2}\frac{\partial\underline{\boldsymbol{\epsilon}}^{\prime}}{ \partial\overline{\boldsymbol{\epsilon}}^{\prime}}\frac{\partial \underline{\boldsymbol{\epsilon}}^{\prime}}{\partial\overline{\boldsymbol{ \epsilon}}^{\prime}}\end{array} \right]^{-1}. \tag{12}\]  
Using the block matrix inversion lemma ([40]),  
\[\left[ \begin{array}{cc}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{array} \right]^{-1}=\left[