# 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
### Problem Formulation  
The parameters to be estimated are gathered in the following vector: \(\mathbf{e}^{T}=[\sigma_{n}^{2},\mathbf{\eta}_{0}^{T},\rho_{0},\phi_{0},\mathbf{\eta}_{1}^{T },\rho_{1},\phi_{1}]\). From (5), we can write \(\mathbf{x}\sim\mathcal{CN}\left(\mathbf{A}(\mathbf{\eta}_{0},\mathbf{\eta}_{1})\mathbf{ \alpha},\sigma_{n}^{2}\mathbf{I}_{N}\right)\) and express the probability density function (pdf) as,  
\[p(\mathbf{x},\mathbf{\epsilon})=\frac{1}{(\pi\sigma_{n}^{2})^{N}}e^{-\frac{1}{\sigma _{n}^{2}}\left\|\mathbf{x}-\mathbf{A}(\mathbf{\eta}_{0},\mathbf{\eta}_{1})\mathbf{\alpha} \right\|^{2}}. \tag{6}\]  
The corresponding CRB for the estimation of \(\mathbf{\epsilon}\) is defined as the inverse of the Fisher Information matrix (FIM) [43],  
\[\mathbf{CRB}_{\mathbf{\epsilon}|\mathbf{\epsilon}}=\mathbf{F}_{\mathbf{\epsilon}|\mathbf{ \epsilon}}^{-1}(\mathbf{\epsilon}),\ \mathbf{F}_{\mathbf{\epsilon}|\mathbf{\epsilon}}(\mathbf{\epsilon})=-E\left[\frac{ \partial^{2}\ln p(\mathbf{x},\mathbf{\epsilon})}{\partial\mathbf{\epsilon}\partial \mathbf{\epsilon}^{T}}\right]. \tag{7}\]  
Note that for the Gaussian CSM of interest the Slepian-Bangs formula [21] provides the direct evaluation of the FIM terms,  
\[\left(\mathbf{F}_{\mathbf{\epsilon}|\mathbf{\epsilon}}\left(\mathbf{\epsilon}\right) \right)_{k,l}\triangleq F_{\epsilon_{k}\mathbf{\epsilon}|\mathbf{\epsilon}}\left(\bm {\epsilon}\right)=\frac{2}{\sigma_{n}^{2}}\text{Re}\left\{\left(\frac{\partial \mathbf{A}\mathbf{\alpha}}{\partial\mathbf{\epsilon}_{k}}\right)^{H}\left(\frac{ \partial\mathbf{A}\mathbf{\alpha}}{\partial\mathbf{\epsilon}_{l}}\right)\right\}+ \frac{N}{\sigma_{n}^{4}}\frac{\partial\sigma_{n}^{2}}{\partial\mathbf{\epsilon}_{ k}}\frac{\partial\sigma_{n}^{2}}{\partial\mathbf{\epsilon}_{l}}\, \tag{8}\]  
where the noise power appears to be independent from the other parameters, and the FIM reduces to  
\[\mathbf{F}_{\mathbf{\epsilon}|\mathbf{\epsilon}}(\mathbf{\epsilon})=\left[\begin{array}[] {cc}F_{\sigma_{n}^{2}|\mathbf{\epsilon}}(\mathbf{\epsilon})&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{F}_{\mathbf{\theta}_{0}|\mathbf{\epsilon}}(\mathbf{\epsilon})&\mathbf{F}_{ \mathbf{\theta}_{0},\mathbf{\theta}_{1}|\mathbf{\epsilon}}(\mathbf{\epsilon})\\ \mathbf{0}&\mathbf{F}_{\mathbf{\theta}_{1},\mathbf{\theta}_{0}|\mathbf{\epsilon}}(\mathbf{\epsilon })&\mathbf{F}_{\mathbf{\theta}_{1}|\mathbf{\epsilon}}(\mathbf{\epsilon})\end{array}\right]\, \tag{9}\]  
where:  
\(\bullet\)\(F_{\sigma_{n}^{2}|\mathbf{\epsilon}}(\mathbf{\epsilon})=N/\sigma_{n}^{4}\).  
\(\bullet\)\(\mathbf{F}_{\mathbf{\theta}_{0}|\mathbf{\epsilon}}(\mathbf{\epsilon})\) and \(\mathbf{F}_{\mathbf{\theta}_{1}|\mathbf{\epsilon}}(\mathbf{\epsilon})\) correspond to the FIMs of the signals when they are totally decoupled. These matrices have been derived and studied in [16] without the Doppler frequency estimation, and in [17,18] for the general Gaussian CSM. The main results in [17,18] concerning the single source CSM FIM terms are summarized in Section 3.2.  
\(\bullet\)\(\mathbf{F}_{\mathbf{\theta}_{0},\mathbf{\theta}_{1}|\mathbf{\epsilon}}(\mathbf{\epsilon})= \mathbf{F}_{\mathbf{\theta}_{1}\mathbf{\theta}_{0}|\mathbf{\epsilon}}(\mathbf{\epsilon})^{T}\) characterizes the interference between both signals. The derivation of such FIM terms is given in Section 3.3.