# Methodology for Simulating 5G and GNSS High-Accuracy Positioning
## Appendix A CRLB on mmWave Channel Parameters
Then each component of the FIM is given by (in following we use \(\mathbf{a}_{r}\) and \(\mathbf{a}_{t}\) to shorten the notation of \(\mathbf{a}_{r}(\theta),\mathbf{a}_{t}(\phi)\))  
\[J_{\tau,\tau}^{n}=\frac{P_{c}}{\sigma^{2}}\Re\left\{\mathbb{E}_{\mathbf{a}}\left\{ \frac{\partial y_{n,m}^{\mathrm{H}}}{\partial\tau_{0}}\frac{\partial y_{n,m}} {\partial\tau_{0}}\right\}\right\}=\frac{P_{c}}{\sigma^{2}}|h_{0}|^{2}(2\pi \Delta_{f})^{2}n^{2}|q_{w}\|^{2}\mathrm{tr}\left(\mathbf{\Gamma}_{f}\mathbf{\Sigma}_{n }\right), \tag{16}\]  
with \(\mathbf{\Gamma}_{f}=\mathbf{F}^{\mathrm{H}}\mathbf{a}_{T}\mathbf{a}_{T}^{H} \mathbf{F}\) and \(\mathbf{\Sigma}_{n}\) is the expected covariance of the transmitted symbols over the \(n\)-th sub-carrier, for instance, \(\mathbf{\Sigma}_{n}=\mathbf{I}\) for QAM independent symbols, and note that the expectation in \(\omega\) can be removed as all partial derivatives are independent of the frequency;  
\[J_{\theta,\theta}^{n}=\frac{P_{c}}{\sigma^{2}}\Re\left\{\mathbb{E}_{\mathbf{a}} \left\{\frac{\partial y_{n,m}^{\mathrm{H}}}{\partial\theta_{0}}\frac{\partial y _{n,m}}{\partial\theta_{0}}\right\}\right\}=\frac{P_{c}}{\sigma^{2}}|h_{0}|^{ 2}\|\dot{q}_{w}\|^{2}\mathrm{tr}\left(\mathbf{\Gamma}_{f}\mathbf{\Sigma}_{n}\right), \tag{17}\]  
with \(\dot{q}_{w}=\mathbf{w}^{\mathrm{H}}\mathbf{D}_{r}\mathbf{a}_{r}\) and \(\mathbf{\Gamma}_{f}=\mathbf{F}^{\mathrm{H}}\mathbf{a}_{T}\mathbf{a}_{T}^{\mathrm{ H}}\mathbf{F}\);\[J_{\phi,\phi}^{n}=\frac{P_{c}}{\sigma^{2}}\Re\left\{\mathbb{E}_{a}\left\{\frac{ \partial y_{n,m}^{H}}{\partial\theta_{0}}\frac{\partial y_{n,m}}{\partial h_{I} }\right\}\right\}=\frac{P_{c}}{\sigma^{2}}|h_{0}|^{2}\|q_{w}\|^{2}\text{tr} \left(\Gamma_{df}\Sigma_{n}\right), \tag{100}\]  
with \(\mathbf{\Gamma}_{df}=\mathbf{F}^{H}\mathbf{D}_{t}\mathbf{a}_{T}\mathbf{a}_{T}^{H} \mathbf{D}_{t}^{H}\mathbf{F}\);  
\[J_{h_{I0}h_{I0}}^{n}=\frac{P_{c}}{\sigma^{2}}\Re\left\{\mathbb{E}_{a}\left\{ \frac{\partial y_{n,m}^{H}}{\partial h_{I0}}\frac{\partial y_{n,m}}{\partial h _{I0}}\right\}\right\}=\frac{P_{c}}{\sigma^{2}}|q_{w}\|^{2}\text{tr}\left( \Gamma_{f}\Sigma_{n}\right), \tag{101}\]  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]0  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]1  
with  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]2  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]3 \[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]4 \[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]5  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]6 \[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]7 \[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]8  
\[\frac{\partial}{\partial\tau_{0}}y_{n,m} =-j\frac{\sqrt{P_{c}}}{\sigma}2\pi\Delta_{f}h_{0}ne^{-j2\pi n \Delta_{f}\tau_{0}}\mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{ a}_{n}, \tag{11}\]9 \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]0 \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]1Summarizing all the above derivations, the non-zero FIM components are given by  
\[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]2 \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]3 (112) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]4 (113) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]5 (114) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]6 (115) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]7 (116) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]8 (117) \[\frac{\partial}{\partial\theta_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{D}_{r}(\theta_{0})\mathbf{a}_{R}(\theta_{ 0})\mathbf{a}_{T}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n}\]9 (118) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]0  
with  
\[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]1 \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]2 (121) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]3 (122) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]4 (123) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]5 (124) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]6 (125) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]7 (126) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]8 (127) \[\frac{\partial}{\partial\phi_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}h_{0}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{a}_{R}(\theta_{0})\mathbf{a}_{T}^{\mathrm{ H}}(\phi_{0})\mathbf{D}_{I}^{\mathrm{H}}(\phi_{0})\mathbf{F}\mathbf{a}_{n},\]9 (128) \[\frac{\partial}{\partial h_{0}}y_{n,m} =\frac{\sqrt{P_{c}}}{\sigma}e^{-j2\pi n\Delta_{f}\tau_{0}} \mathbf{w}_{m}^{\mathrm{H}}\mathbf{H}_{0}\mathbf{F}\mathbf{a}_{n},\]0