## 8 Cramer-Rao Bound on Covariance Matrix of Localization Errors
\begin{array}{l}\mathbf{\Omega}_{\mathbf{p}}=\frac{\partial\overline{\mathbf{x}} _{0}}{\partial\mathbf{p}^{\rm T}}=\mathrm{diag}[\mathbf{\beta}\otimes\mathbf{1}_{MK\times 1 }]\cdot\mathrm{diag}[\mathbf{1}_{N\times 1}\otimes\overline{\mathbf{s}}\otimes\mathbf{1}_{M \times 1}]\cdot\frac{\partial\mathbf{a}^{\prime}(\mathbf{p})}{\partial\mathbf{p}^{\rm T}},\\ \mathbf{\Omega}_{\rm Re\{\mathbf{\beta}\}}=\frac{\partial\overline{\mathbf{x}}_{0}}{\partial \left(\mathrm{Re}\{\mathbf{\beta}\}\right)^{\rm T}}=\mathrm{diag}[\mathbf{1}_{N\times 1 }\otimes\overline{\mathbf{s}}\otimes\mathbf{1}_{M\times 1}]\cdot\mathrm{blkdiag}[\mathbf{a}_{1} ^{\prime}(\mathbf{p})\ a_{2}^{\prime}(\mathbf{p})\ \cdots\ a_{N}^{\prime}(\mathbf{p})]\,,\\ \mathbf{\Omega}_{\rm Im\{\mathbf{\beta}\}}=\frac{\partial\overline{\mathbf{x}}_{0}}{ \partial\left(\mathrm{Im}\{\mathbf{\beta}\}\right)^{\rm T}}=\mathrm{j}\cdot\frac{ \partial\overline{\mathbf{x}}_{0}}{\partial\left(\mathrm{Re}\{\mathbf{\beta}\}\right) ^{\rm T}}=\mathrm{j}\cdot\mathrm{diag}[\mathbf{1}_{N\times 1}\otimes\overline{\mathbf{s}}\otimes\mathbf{1}_{M \times 1}]\cdot\mathrm{blkdiag}[\mathbf{a}_{1}^{\prime}(\mathbf{p})\ a_{2}^{\prime}(\mathbf{p})\ \cdots\ a_{N}^{\prime}(\mathbf{p})]\,,\\ \mathbf{\Omega}_{\rm Re\{\mathbf{\overline{\mathbf{s}}}\}}=\frac{\partial\overline{\mathbf{x}} _{0}}{\partial\left(\mathrm{Re}\{\mathbf{\overline{\mathbf{s}}}\}\right)^{\rm T}}= \mathrm{diag}[\mathbf{\beta}\otimes\mathbf{1}_{MK\times 1}]\cdot\mathbf{a}(\mathbf{p})\,,\\ \mathbf{\Omega}_{\rm Im\{\mathbf{\overline{\mathbf{s}}}\}}=\frac{\partial\overline{\mathbf{x}} _{0}}{\partial\left(\mathrm{Im}\{\mathbf{\overline{\mathbf{s}}}\}\right)^{\rm T}}= \mathrm{j}\cdot\frac{\partial\overline{\mathbf{x}}_{0}}{\partial\left(\mathrm{Re} \{\mathbf{\overline{\mathbf{s}}}\}\right)^{\rm T}}=\mathrm{j}\cdot\mathrm{diag}[\mathbf{ \beta}\otimes\mathbf{1}_{MK\times 1}]\cdot\mathbf{a}(\mathbf{p})\,,\end{array} \right.\right. \tag{80}\]  
where  
\[\left\{ \begin{array}{l}\mathbf{a}^{\prime}(\mathbf{p})=\left[\mathbf{a}_{n}^{\prime\rm H }(\mathbf{p})\ \mathbf{a}_{2}^{\prime\rm H}(\mathbf{p})\ \cdots\ \mathbf{a}_{N}^{\prime\rm H}(\mathbf{p})\right]^{\rm H},\\ \frac{\partial\mathbf{a}^{\prime}(\mathbf{p})}{\partial\mathbf{p}^{\rm T}}=\left[\left( \frac{\partial\mathbf{a}_{n}^{\prime}(\mathbf{p})}{\partial\mathbf{p}^{\rm T}}\right)^{\rm H }\left(\frac{\partial\mathbf{a}_{n}^{\prime}(\mathbf{p})}{\partial\mathbf{p}^{\rm T}} \right)^{\rm H}\ \cdots\ \left(\frac{\partial\mathbf{a}_{n}^{\prime}(\mathbf{p})}{\partial\mathbf{p}^{\rm T}} \right)^{\rm H}\right]^{\rm H},\\ \mathbf{a}_{n}^{\prime}(\mathbf{p})=\left[\mathbf{a}_{n,1}^{\prime\rm H}(\mathbf{p})\ \mathbf{a}_{n,2}^{ \prime\rm H}(\mathbf{p})\ \cdots\ \mathbf{a}_{n,K}^{\prime\rm H}(\mathbf{p})\right]^{\rm H}=\mathbf{a}_{n}(\mathbf{p})\mathbf{1}_{K \times 1}\quad(1\leq n\leq N)\,.\end{array} \right. \tag{81}\]  
Note that only the \(\mathbf{p}\) corner of the CRB matrix is of interest here. However, it is easily observed from (78) that matrix \(\mathbf{CRB}(\mathbf{\mu}_{\rm a})\) does not exhibit a block-diagonal structure, because there might be correlation between the parameters. Hence, it is somewhat difficult to obtain the CRB for position vector \(\mathbf{p}\). To overcome this difficulty, we adopt the idea of [59; 67] to redefine a parameter vector whose CRB matrix becomes block-diagonal. The new parameter vector is defined as  
\[\overline{\mathbf{\mu}}_{\rm a}=\left[\mathbf{p}^{\rm T}\ \mathbf{\beta}_{\rm r}^{\rm T}\ \left(\mathrm{Re}\{\overline{\mathbf{s}}\}+\mathrm{Re}\{\mathbf{W}_{1}\}\cdot\mathbf{p}+ \mathrm{Re}\{\mathbf{W}_{2}\}\cdot\mathbf{\beta}_{\rm r}\right)^{\rm T}\ \left(\mathrm{Im}\{\overline{\mathbf{s}}\}+\mathrm{Im}\{\mathbf{W}_{1}\}\cdot\mathbf{p}+ \mathrm{Im}\{\mathbf{W}_{2}\}\cdot\mathbf{\beta}_{\rm r}\right)^{\rm T}\right]^{ \rm T}, \tag{82}\]  
where  
\[\left\{