# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## Appendix A Details of the Derivation of the Fisher Information Matrix
### Bounds Derivation
\right]e^{j\Delta \varphi}e^{-j\omega_{c}\Delta h\tau_{0}}, \tag{104}\]  
with  
\[W_{1,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t, \tag{105}\] \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (106) \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (107) \[W_{2,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})s(t- \tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (108) \[W_{2,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})(t- \tau_{0})s(t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (109) \[W_{2,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})s(t- \tau_{1})s^{(1)}(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (110) \[W_{3,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s^{(1)}(t-\tau_{1} )s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (111) \[W_{3,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s^{(1 )}(t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\] (112) \[W_{3,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s^{(1)}(t-\tau_{1 })s^{(1)}(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t. \tag{113}\]  
Exploiting the Fourier transform properties over the hermitian product and the relations recalled in Appendix A.1, one can work these integral expressions as follows:  
\[W_{1,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\Delta\pi f_{s}\Delta ht}\mathrm{d}t=s(u-\Delta\tau)s(u)^{*}e^{- j2\pi f_{s}\Delta hu}\mathrm{d}u,\] \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]0  
and using the sum definition of the Fourier transform (100),  
\[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]1Following the same procedure we have that,  
\[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]2 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]3 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]4 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]5  
\[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]6 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]7 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]8 \[W_{1,2}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{0})s( t-\tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]9 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]0 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]1 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]2 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]3 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]4 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]5 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]6 (102) \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]7 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]8 \[W_{1,3}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}s(t-\tau_{1})s( t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]9 \[W_{2,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})s(t- \tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]0 \[W_{2,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})s(t- \tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]1 \[W_{2,1}^{\Delta} =e^{j\omega_{c}\Delta h\tau_{0}}\int_{\mathbb{R}}(t-\tau_{1})s(t- \tau_{1})s(t-\tau_{0})^{*}e^{-j\omega_{c}\Delta ht}\mathrm{d}t,\]2  
where  
\[\mathbf{s} =\left(