# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## Appendix B Details on the Constrained Fisher Information Matrix  
We consider the vector of unknown parameters transform (29). The corresponding jacobian, initially defined in (30), can be written as follows,  
\[\frac{\partial\epsilon(\xi)}{\partial\xi^{T}}=\left[\begin{array}{ccc}1& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}_{4}&\mathbf{0}_{4\times 3}\\ \mathbf{0}&\mathbf{E}_{1}&\mathbf{E}_{3}\end{array}\right],\ \mathbf{E}_{1}=\left[\begin{array}{ccc}0&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right],\ \mathbf{E}_{3}=\left[\begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]. \tag{107}\]  
Using this representation, it is easy to develop (32) as,  
\[\mathbf{F}_{\xi|\xi}=\left(\frac{\partial\epsilon(\xi)}{\partial\xi^{T}}\right) ^{T}\mathbf{F}_{e|e}\frac{\partial\epsilon(\xi)}{\partial\xi^{T}}=\left[ \begin{array}{ccc}F_{\sigma_{0}^{2}|\xi}(\xi)&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{F}_{\theta_{0}|\xi}(\xi)&\mathbf{F}_{\theta_{0},\tau_{1}, \rho_{1},\rho_{1}|\xi}(\xi)\\ \mathbf{0}&\mathbf{F}_{\tau_{1},\rho_{1},\rho_{1},\rho_{1}|\xi}(\xi)&\mathbf{F} _{\tau_{1},\rho_{1},\rho_{1}|\xi}(\xi)\end{array}\right]. \tag{108}\]  
with  
\[F_{\sigma_{0}^{2}|\xi}(\xi) =F_{\sigma_{0}^{2}|\epsilon}(\epsilon), \tag{109}\] \[\mathbf{F}_{\theta_{0}|\xi}(\xi) =\mathbf{F}_{\theta_{0}|\epsilon}(\epsilon)+\mathbf{E}_{1}\mathbf{ F}_{\theta_{1}|\epsilon}(\epsilon)\mathbf{E}_{1}+\mathbf{E}_{1}\mathbf{F}_{\theta_{1}, \theta_{0}|\epsilon}(\epsilon)+\mathbf{F}_{\theta_{0},\theta_{1}|\epsilon}( \epsilon)\mathbf{E}_{1},\] (110) \[\mathbf{F}_{\tau_{1},\rho_{1},\rho_{1}|\xi}(\xi) =\mathbf{E}_{3}^{T}\mathbf{F}_{\theta_{1}|\epsilon}(\epsilon) \mathbf{E}_{3},\] (111) \[\mathbf{F}_{\tau_{1},\rho_{1},\rho_{1},\rho_{1},\rho_{0}|\xi}( \xi) =\left(\mathbf{F}_{\theta_{0},\tau_{1},\rho_{1},\rho_{1}|\xi}(\xi) \right)^{T}=\mathbf{E}_{3}^{T}\mathbf{F}_{\theta_{1},\theta_{0}|\epsilon}( \epsilon)+\mathbf{E}_{3}^{T}\mathbf{F}_{\theta_{1}|\epsilon}(\epsilon)\mathbf{E }_{1}. \tag{112}\]The constrained FIM can directly be computed using the (10) and (13). Besides, the expression of \(\mathbf{W}^{\Delta}\) components is simplified by the assumption \(b_{0}=b_{1}\) as follows,  
\[W^{\Delta}_{1,1} =\frac{1}{F_{\mathrm{s}}}\mathbf{s}^{H}\mathbf{V}^{\Delta,0}\left( \frac{\Delta\tau}{T_{s}}\right)\mathbf{s},\;W^{\Delta}_{1,2}=\frac{1}{F_{ \mathrm{s}}^{2}}\mathbf{s}^{H}\mathbf{D}\mathbf{V}^{\Delta,0}\left(\frac{ \Delta\tau}{T_{s}}\right)\mathbf{s},\;W^{\Delta}_{1,3}=-\mathbf{s}^{H}\mathbf{ V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s},\] \[W^{\Delta}_{2,1} =\frac{1}{F_{\mathrm{s}}^{2}}\mathbf{s}^{H}\mathbf{V}^{\Delta,0} \left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{Ds},\;W^{\Delta}_{2,2}=\frac{1}{F_ {\mathrm{s}}^{3}}\mathbf{s}^{H}\mathbf{D}\mathbf{V}^{\Delta,0}\left(\frac{ \Delta\tau}{T_{s}}\right)\mathbf{Ds},\;W^{\Delta}_{2,3}=-\frac{1}{F_{\mathrm{s }}}\mathbf{s}^{H}\mathbf{V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s}}\right) \mathbf{Ds},\] \[W^{\Delta}_{3,1} =\mathbf{s}^{H}\mathbf{V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s} }\right)\mathbf{s},\;W^{\Delta}_{3,2}=\frac{1}{F_{\mathrm{s}}}\mathbf{s}^{H} \mathbf{D}\mathbf{V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s},\;W^{\Delta}_{3,3}=F_{\mathrm{s}}\mathbf{s}^{H}\mathbf{V}^{\Delta,2}\left( \frac{\Delta\tau}{T_{s}}\right)\mathbf{s}. \tag{100}\]