# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## 3 New Compact Dual Source CRB for Delay and Doppler Estimation with Band-Limited Signals
### Interference Fisher Information Matrix Terms  
From the Slepian-Bangs formula the non-diagonal blocks of the FIM, referred to as interference Fisher information matrices, are expressed as follows:  
\[\mathbf{F}_{\theta_{1},\theta_{0}|e}\left(\mathbf{e}\right)=\frac{2F_{s}}{\sigma_ {n}^{2}}\text{Re}\left\{\mathbf{Q}_{1}\mathbf{W}^{\Delta}\mathbf{Q}_{0}^{H} \right\}\, \tag{13}\]  
where \(\mathbf{Q}_{0}\), \(\mathbf{Q}_{1}\) are defined in (A9) and \(\mathbf{W}^{\Delta}\) is defined as  
\[\mathbf{W}^{\Delta}=\left[\begin{array}{ccc}W_{1,1}^{\Delta}&W_{1,2}^{ \Delta}&W_{1,3}^{\Delta}\\ W_{2,1}^{\Delta}&W_{2,2}^{\Delta}&W_{2,3}^{\Delta}\\ W_{3,1}^{\Delta}&W_{3,2}^{\Delta}&W_{3,3}^{\Delta}\end{array}\right]e^{i \Delta\psi}e^{-i\omega_{c}\Delta\eta_{0}}\, \tag{14}\]  
where \(\Delta b\) and \(\Delta\psi\) are defined in (A15) and with the different components of the matrix \(\mathbf{W}^{\Delta}\) expressed w.r.t the baseband signal samples given by  
\[W_{1,1}^{\Delta} =\frac{1}{F_{s}}\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{ \Delta,0}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s}\, \tag{15}\] \[W_{1,2}^{\Delta} =\frac{1}{F_{s}^{2}}\mathbf{s}^{H}\mathbf{D}\mathbf{U}(\Delta b) \mathbf{V}^{\Delta,0}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s}\,\] (16) \[W_{1,3}^{\Delta} =-\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{\Delta,1}\left( \frac{\Delta\tau}{T_{s}}\right)\mathbf{s}+\frac{i\omega_{c}\Delta b}{F_{s}} \mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{\Delta,0}\mathbf{s}\,\] (17) \[W_{2,1}^{\Delta} =\frac{1}{F_{s}^{2}}\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^ {\Delta,0}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{D}\mathbf{s}\,\] (18) \[W_{2,2}^{\Delta} =\frac{1}{F_{s}^{3}}\mathbf{s}^{H}\mathbf{D}\mathbf{U}(\Delta b) \mathbf{V}^{\Delta,0}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{D}\mathbf{s}\,\] (19) \[W_{2,3}^{\Delta} =-\frac{1}{F_{s}}\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{ \Delta,1}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{D}\mathbf{s}+\frac{i \omega_{c}\Delta b}{F_{s}^{2}}\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{ \Delta,0}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{D}\mathbf{s}\,\] (20) \[W_{3,1}^{\Delta} =\mathbf{s}^{H}\mathbf{U}(\Delta b)\mathbf{V}^{\Delta,1}\left( \frac{\Delta\tau}{T_{s}}\right)\mathbf{s}\,\] (21) \[W_{3,2}^{\Delta} =\frac{1}{F_{s}}\mathbf{s}^{H}\mathbf{D}\mathbf{U}(\Delta b) \mathbf{V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s}\,\] (22) \[W_{3,3}^{\Delta} =F_{s}\mathbf{s}^{H}\mathbf{U}\left(\Delta b\right)\mathbf{V}^{ \Delta,2}\left(\frac{\Delta\tau}{T_{s}}\right)\mathbf{s}+j\omega_{c}\Delta bs^{ H}\mathbf{U}(\Delta b)\mathbf{V}^{\Delta,1}\left(\frac{\Delta\tau}{T_{s}} \right)\mathbf{s}\, \tag{23}\]  
where \(\mathbf{s}\), the baseband sample vector, is defined in (A36), \(\mathbf{D}\) in (A39), \(\mathbf{U}\) in (A38), \(\mathbf{V}^{\Delta,0}\) in (A40), \(\mathbf{V}^{\Delta,1}\) in (A41) and \(\mathbf{V}^{\Delta,2}\) in (A42).  
Proof.: refer to Appendix A.  
An interesting and reassuring fact is that when the difference between LOS and NLOS signals (i.e., signal 0 and signal 1) is set to zero: \(\Delta\tau=\tau_{1}-\tau_{0}=0\), \(\Delta b=b_{1}-b_{0}=0\), \(\Delta\phi=\phi_{1}-\phi_{0}=0\), \(\rho_{1}=\rho_{0}\), we have \(\mathbf{Q}_{0}=\mathbf{Q}_{1}\) and \(\mathbf{W}^{\Delta}=\mathbf{W}\). This result could be expected but the FIM terms equationsobtained confirm it. Indeed, if both signals are indistinguishable, they must have the same FIM and their interference FIM must coincide (total interference between both signals).