# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## 6 Further Insights and Outlooks  
### Comparison with Existing Literature  
As previously mentioned, a formulation of the CRBs in the dual source context was proposed in [28]. In this reference, authors tackled the study of the best achievable performance in the estimation of a receiver altitude using GNSS-R. Starting from the Slepian-Bangs formula, they derived a general FIM that is similar to the one presented in Section 3. The main difference between the two formulations is that in [28], the study is done under the assumption of a static, ground-based receiver. This assumption allows the authors to set the two Doppler shifts at the same value. One can find a way to connect the results presented in Section 3 with the one proposed in [28]. Indeed, setting an equality such as \(b_{0}=b_{1}\), which constrains the current model, can be seen as a reparameterization of the vector of unknown parameters \(\mathbf{\epsilon}\)[31]. We therefore consider the following vector transform,  
\[b_{0}=b_{1}\Leftrightarrow\mathbf{\epsilon}=\mathbf{\epsilon}(\mathbf{\xi}), \tag{29}\]  
where  
\[\mathbf{\epsilon} =\left[ \begin{array}{cccccccc}\sigma_{n}^{2}&\tau_{0}&b_{0}& \rho_{0}&\phi_{0}&\tau_{1}&b_{1}&\rho_{1}&\phi_{1}\\ \mathbf{\xi}&=\left[\begin{array}{cccccccc}\sigma_{n}^{2}&\tau_{0}&b_{0}&\rho_ {0}&\phi_{0}&\tau_{1}&\rho_{1}&\phi_{1}\\ \end{array}\right]^{T}.\end{array} \right]^{T},\]  
From this, if we note the 9-by-9 unity matrix \(\mathbf{I}_{9}=\left[\mathbf{e}_{1}\ldots\mathbf{e}_{9}\right]\) with \(\mathbf{e}_{i}\) the column unit vector with 1 at the \(i\)th component and 0 elsewhere, the relation (29) can be explicitly written as follows,  
\[\mathbf{\epsilon}(\mathbf{\xi})=\left[ \begin{array}{cccccccc}\mathbf{e}_{1}&\mathbf{ e}_{2}&\mathbf{e}_{3}+\mathbf{e}_{7}&\mathbf{e}_{4}&\mathbf{e}_{5}&\mathbf{e}_{6}& \mathbf{e}_{8}&\mathbf{e}_{9}\\ \end{array} \right]\mathbf{\xi}=\frac{\partial\mathbf{\epsilon}(\mathbf{\xi})}{\partial \mathbf{\xi}^{T}}\mathbf{\xi}. \tag{30}\]  
Now if we consider a reparameterization of the pdf \(p(\mathbf{\kappa};\mathbf{\epsilon})\) for the unknown parameters \(\mathbf{\epsilon}\) where  
\[\mathbf{\epsilon}=\mathbf{\epsilon}(\mathbf{\xi}),\quad\dim\left\{\mathbf{\xi}\right\}=8, \quad\dim\left\{\mathbf{\epsilon}\right\}=9,\]  
then \(p(\mathbf{\kappa};\mathbf{\xi})=p(\mathbf{\kappa};\mathbf{\epsilon}(\mathbf{\xi}))\) and by use of the derivation chain rule identity,  
\[\frac{\partial\ln p(\mathbf{\kappa};\mathbf{\xi})}{\partial\mathbf{\xi}^{T}}=\frac{ \partial\ln p(\mathbf{\kappa};\mathbf{\epsilon}(\mathbf{\xi}))}{\partial\mathbf{\xi}^{T}} \frac{\partial\mathbf{\epsilon}(\mathbf{\xi})}{\partial\mathbf{\xi}^{T}}\Leftrightarrow \frac{\partial\ln p(\mathbf{\kappa};\mathbf{\xi})}{\partial\mathbf{\xi}}=\left(\frac{ \partial\mathbf{\epsilon}(\mathbf{\xi})}{\partial\mathbf{\xi}^{T}}\right)^{T}\frac{Developing the right-hand side of (31), one can obtain the following FIM (refer to Appendix B for more details on the different matrices),  
\[\mathbf{F}_{\xi|\xi}(\xi)=\left[ \begin{array}{ccc}F_{\sigma_{\xi}^{2}|\xi}\left( \xi\right)&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{F}_{\theta_{0}|\xi}\left(\xi\right)&\mathbf{F}_{\theta_{0},\tau_{1},\rho_{1},\rho_{1}|\xi}\left(\xi\right)\\ \mathbf{0}&\mathbf{F}_{\tau_{1},\rho_{1},\rho_{1},\rho_{1}|\xi}\left(\xi\right) &\mathbf{F}_{\tau_{1},\rho_{1},\rho_{1}|\xi}\left(\xi\right)\end{array} \right]. \tag{32}\]  
It appears that from the CRB presented in this contribution, it is possible to deduce the specific scenario studied in [28]. However, in other scenarios, the Doppler shifts of the LOS and the NLOS signals may be different (e.g., if the signal is reflected from a moving object [47]) and a more general approach is needed, for which the result in [28] is not valid and the proposed CRB provides an answer. Another key difference between the two contributions is that the results in the present contribution are expressed directly using the baseband signal samples, whereas in [28] they are expressed with the signal Power Spectral Density (PSD).