## 8 Cramer-Rao Bound on Covariance Matrix of Localization Errors
\begin{array}{l}\mathbf{W}_{1}=\mathbf{\Omega}_{\rm Re}\{\overline{\mathbf{s}}\} \mathbf{\Omega}_{\mathbf{p}}\,,\\ \mathbf{W}_{2}=\mathbf{\Omega}_{\rm Re}\{\overline{\mathbf{s}}\}\cdot\left[\mathbf{\Omega}_{ \rm Re}\{\overline{\mathbf{s}}\}\ \mathbf{\Omega}_{\rm Im}\{\mathbf{\beta}\}\right].\end{array} \right. \tag{83}\]  
It is worth highlighting that because the vector \(\overline{\mathbf{\mu}}_{\rm a}\) includes the source location parameters, it is meaningful to derive the CRB matrix for \(\overline{\mathbf{\mu}}_{\rm a}\). In addition, there is a one-to-one mapping between the new and old vectors \(\overline{\mathbf{\mu}}\) and \(\overline{\mathbf{\mu}}_{\rm a}\). The relationship between them can be written in matrix form as \[\overline{\mu}_{\rm a}=J\mu_{\rm a}=\left[ \begin{array}{cccc}\mathbf{I}&O&O&O\\ \mathbf{O}&\mathbf{I}&O&O\\ \mathrm{Re}\{W_{1}\}&\mathrm{Re}\{W_{2}\}&\mathbf{I}&O\\ \mathrm{Im}\{W_{1}\}&\mathrm{Im}\{W_{2}\}&O&\mathbf{I}\end{array} \right].\mu_ {\rm a}\,, \tag{84}\]  
where  
\[\mathbf{J}=\left[ \begin{array}{cccc}\mathbf{I}&O&O&O\\ \mathbf{O}&\mathbf{I}&O&O\\ \mathrm{Re}\{W_{1}\}&\mathrm{Re}\{W_{2}\}&\mathbf{I}&O\\ \mathrm{Im}\{W_{1}\}&\mathrm{Im}\{W_{2}\}&O&\mathbf{I}\end{array} \right]. \tag{85}\]  
Then, combining the results in Proposition 3 and (84), the CRB matrix for \(\overline{\mu}_{\rm a}\) is given by  
\[\mathbf{CRB}(\overline{\mu}_{\rm a})=\mathbf{J}\cdot\mathbf{CRB}(\mathbf{\mu}_{\rm a} )\cdot\mathbf{J}^{\rm T}=\frac{\sigma_{\rm e}^{2}}{2}\cdot(\mathrm{Re}\{{(\mathbf{ \Omega}_{\mu_{\rm a}}\mathbf{J}^{-1})}^{\rm H}(\mathbf{\Omega}_{\mu_{\rm a}}\mathbf{J}^{- 1})\})^{-1}\,, \tag{86}\]  
where  
\[\mathbf{J}^{-1}=\left[ \begin{array}{cccc}\mathbf{I}&\mathbf{O}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{I}&\mathbf{O}&\mathbf{O}\\ -\mathrm{Re}\{W_{1}\}&-\mathrm{Re}\{W_{2}\}&\mathbf{I}&\mathbf{O}\\ -\mathrm{Im}\{W_{1}\}&-\mathrm{Im}\{W_{2}\}&\mathbf{O}&\mathbf{I}\end{array} \right]. \tag{87}\]  
Combining (79), (83), and (87) leads to the orthogonal projection matrix  
\[\mathbf{\Omega}_{\mu_{\rm a}}\mathbf{J}^{-1}=\left[\begin{array}{cccc}\mathbf{T}^{ \perp}\Big{[}\mathbf{\Omega}_{\mathrm{Re}\{\overline{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bm  
**Remark 12**. : _The diagonal elements of \(\mathbf{CRB}(\mathbf{p})\) give the bounds for the estimation variance of the components in \(\mathbf{p}\) when the array manifold is perfectly calibrated._  
**Remark 13**. : _The trace of \(\mathbf{CRB}(\mathbf{p})\) is the bound for the localization MSE in the absence of array model errors._  
**Remark 14**. : _Although there is no rigorous proof, it is expected that the trace of \(\mathbf{P}_{1}\) is asymptotically close to that of \(\mathbf{CRB}(\mathbf{p})\). The reason for this is that the least square estimator in (5) is equivalent to the MLE, which is statistically efficient under the Gaussian noise model._  
**Remark 15**. : _By comparing the trace of \(\mathbf{CRB}(\mathbf{p})\) with that of \(\mathbf{P}\), we can assess the expected degradation of the emitter location accuracy with respect to the amount of array model error. If the difference is significant, it can be concluded that the DPD method in [29] is sensitive to array model errors._