## 5 Statistical Assumption and Effects of Array Model Errors
\begin{array}{l}\mathrm{blkdiag}[\mathbf{\hat{x}}_{1}\ \ \mathbf{\hat{x}}_{2}\ \ \cdots\ \mathbf{\hat{x}}_{N}]=\mathrm{blkdiag}[\overline{x}_{1,0}\ \ \mathbf{\hat{x}}_{2,0}\ \ \cdots\ \mathbf{\hat{x}}_{N,0}]+\mathrm{blkdiag}[\overline{\varepsilon}_{1}\ \ \mathbf{\hat{x}}_{2}\ \ \cdots\ \mathbf{\hat{x}}_{N}]\\ +\mathrm{blkdiag}[(\mathbf{r}_{1}\otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{1}\ \ (\mathbf{r}_{2}\otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{2}\ \ \cdots\ (\mathbf{r}_{N}\otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{N}]\Leftrightarrow\mathbf{ \hat{X}}=\overline{\mathbf{X}}_{0}+\overline{\mathbf{E}}+\widetilde{\mathbf{\Psi}}\,,\end{array}  \tag{20}\]  
where  
\[\widetilde{\mathbf{\Psi}}=\mathrm{blkdiag}[(\mathbf{r}_{1}\otimes\mathbf{I}_{M})\widetilde {\mathbf{\varphi}}_{1}\ \ (\mathbf{r}_{2}\otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{2}\ \ \cdots\ \ (\mathbf{r}_{N}\otimes\mathbf{I}_{M})\widetilde{\mathbf{\varphi}}_{N}]\,. \tag{21}\]  
In the presence of array model errors, the emitter position is actually determined by  
\[\max_{\mathbf{p}}\hat{J}(\mathbf{p})=\max_{\mathbf{p}}\lambda_{\max}\{\mathbf{\hat{B}}^{\mathrm{ H}}(\mathbf{p})\mathbf{\hat{B}}(\mathbf{p})\}\,, \tag{22}\]  
where \(\mathbf{\hat{B}}(\mathbf{p})=\mathbf{A}^{\mathrm{H}}(\mathbf{p})\mathbf{\hat{X}}\). We assume the optimal solution to (22) is \(\mathbf{\hat{p}}\) and its estimate error is \(\widetilde{\mathbf{p}}=\mathbf{\hat{p}}-\mathbf{p}\). It is evident that the estimate error \(\widetilde{\mathbf{p}}\) depends on both sensor noise and array model errors. Insubsequent sections, the statistical performance of \(\widetilde{\mathbf{p}}\) is derived under the combined effects of the two sources of error. For convenience in later formulae, we proceed by defining two error vectors  
\[\overline{\mathbf{\varepsilon}}_{\text{c}}=\left[ \begin{array}{c}\overline{\mathbf{ \varepsilon}}\\ \overline{\mathbf{\varepsilon}}^{*}\end{array} \right],\widetilde{\mathbf{\varphi}}_{ \text{c}}=\left[ \begin{array}{c}\widetilde{\mathbf{\varphi}}\\ \widetilde{\mathbf{\varphi}}^{*}\end{array} \right], \tag{23}\]  
where  
\[\overline{\mathbf{\varepsilon}}=\left[\overline{\mathbf{\varepsilon}}_{1}^{\text{H}} \ \overline{\mathbf{\varepsilon}}_{2}^{\text{H}}\ \cdots\ \overline{\mathbf{\varepsilon}}_{N}^{\text{H}}\right]^{\text{H}}=\overline{\mathbf{ \varepsilon}}\mathbf{1}_{N\times 1}\,,\,\widetilde{\mathbf{\varphi}}=\left[ \widetilde{\mathbf{\varphi}}_{1}^{\text{H}}\ \widetilde{\mathbf{\varphi}}_{2}^{\text{H}}\ \cdots\ \widetilde{\mathbf{\varphi}}_{N}^{\text{H}}\right]^{\text{H}}. \tag{24}\]  
Obviously, \(\overline{\mathbf{\varepsilon}}_{\text{c}}\) and \(\widetilde{\mathbf{\varphi}}_{\text{c}}\) are related to sensor noise and array model errors, respectively. Further, we define two permutation matrices  
\[\mathbf{I}_{\overline{\mathbf{\varepsilon}}}=\left[ \begin{array}{cc}\mathbf{O}_{MNK \times MNK}&\mathbf{I}_{MNK}\\ \mathbf{I}_{MNK}&\mathbf{O}_{MNK\times MNK}\end{array} \right],\,\mathbf{I}_{\widetilde{ \mathbf{\varphi}}}=\left[ \begin{array}{cc}\mathbf{O}_{MN\times MN}&\mathbf{I}_{MN}\\ \mathbf{I}_{MN}&\mathbf{O}_{MN\times MN}\end{array} \right], \tag{25}\]  
It can then be easily checked from (23) and (25) that \(\overline{\mathbf{\varepsilon}}_{\text{c}}=\mathbf{\Pi}_{\overline{\mathbf{\varepsilon}} }\overline{\mathbf{\varepsilon}}_{\text{c}}^{*}\) and \(\widetilde{\mathbf{\varphi}}_{\text{c}}=\mathbf{\Pi}_{\widetilde{\mathbf{\varphi}}} \widetilde{\mathbf{\varphi}}_{\text{c}}^{*}\). In addition, it is straightforward to deduce from (17), (21), and (24) that \(\overline{\mathbf{E}}=O(||\widetilde{\mathbf{\varepsilon}}||_{2})\) and \(\widetilde{\mathbf{\Psi}}=O(||\widetilde{\mathbf{\varphi}}||_{2})\).