## 6 MSE of Direct Position Determination Method in Presence of Array Model Errors
### Second-Order Perturbation Analysis on the Cost Function  
Generally, first-order analysis is applied to predict the statistical performance of an estimator. The reason is that this analysis method gives the linear relationship between the estimation errors and measurement noise as well as model errors. As a result, the theoretical MSE of the estimator can be obtained according to statistical assumptions of the two sources of error. Moreover, first-order analysis is valid in most cases, provided that the error levels are not too high. In this paper, we employ this approach to derive the performance of the DPD estimator described above. For this purpose,first-order perturbation analysis is performed on the first derivative of the objective function in (22), or alternatively, second-order perturbation analysis is performed on the cost function in (22). Herein, because the analytical expression for the derivative of the cost function is rather complex, we prefer the second approach. First, performing second-order perturbation analysis on matrix \(\tilde{\mathbf{B}}(\hat{\mathbf{p}})=\mathbf{A}^{\rm H}(\hat{\mathbf{p}})\widehat{\mathbf{X}}\) leads to  
\[\tilde{\mathbf{B}}(\hat{\mathbf{p}})=\mathbf{A}^{\rm H}(\hat{\mathbf{p}})\widehat{\mathbf{X}}=\mathbf{B }_{0}+\widetilde{\mathbf{B}}^{(1)}+\widetilde{\mathbf{B}}^{(2)}+o(||\widetilde{\mathbf{ \xi}}||_{2}^{2})\,, \tag{28}\]  
where \(\widetilde{\mathbf{\xi}}=[\widetilde{\mathbf{p}}^{\rm T}\ \overline{\mathbf{\epsilon}}^{ \rm T}\ \widetilde{\mathbf{\varphi}}^{\rm T}]^{\rm T}\) consists of all error vectors, and  
\[\left\{ \begin{array}{l}\mathbf{B}_{0}=\mathbf{A}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{ 0}\\ \widetilde{\mathbf{B}}^{(1)}=\mathbf{A}^{\rm H}(\mathbf{p})\overline{\mathbf{E}}+\mathbf{A}^{\rm H }(\mathbf{p})\overline{\mathbf{\Psi}}+\sum\limits_{l=1}^{L}<\overline{\mathbf{p}}>_{l} \cdot\dot{\mathbf{A}}_{l}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{0}=O(||\mathbf{\xi}||_{2} )\\ \widetilde{\mathbf{B}}^{(2)}=\sum\limits_{l=1}^{L}<\overline{\mathbf{p}}>_{l}\cdot \dot{\mathbf{A}}_{l}^{\rm H}(\mathbf{p})\overline{\mathbf{E}}+\sum\limits_{l=1}^{L}< \overline{\mathbf{p}}>_{l}\cdot\dot{\mathbf{A}}_{l}^{\rm H}(\mathbf{p})\overline{\mathbf{ \Psi}}+\frac{1}{2}\cdot\sum\limits_{l_{1}=1}^{L}\sum\limits_{l_{2}=1}^{L}< \overline{\mathbf{p}}>_{l_{1}}\cdot\dot{\mathbf{\xi}}^{\rm H}_{l_{1}l_{2}}(\mathbf{p}) \overline{\mathbf{x}}_{0}=O(||\mathbf{\xi}||_{2}^{2})\end{array} \right. \tag{29}\]  
with  
\[\dot{\mathbf{A}}_{l}(\mathbf{p})=\frac{\partial\mathbf{A}(\mathbf{p})}{\partial<\mathbf{p}>_{l}}\,\ \ddot{\mathbf{A}}_{l_{1}l_{2}}(\mathbf{p})=\frac{ \partial^{2}\mathbf{A}(\mathbf{p})}{\partial<\mathbf{p}>_{l_{1}}\partial<\mathbf{p}>_{l_{2}}}\,. \tag{30}\]  
The explicit expressions for \(\dot{\mathbf{A}}_{l}(\mathbf{p})\) and \(\ddot{\mathbf{A}}_{l_{1}l_{2}}(\mathbf{p})\) are given in Appendix A. It is seen from (28) and (29) that \(\widetilde{\mathbf{B}}^{(1)}\) and \(\widetilde{\mathbf{B}}^{(2)}\) collect all first- and second-order perturbation terms, respectively. It is deduced from (28) that  
\[\hat{\mathbf{B}}^{\rm H}(\hat{\mathbf{p}})\hat{\mathbf{B}}(\hat{\mathbf{p}})=\mathbf{C}_{0}+\mathbf{C} ^{(1)}+\mathbf{C}^{(2)}+o(||\widetilde{\mathbf{\xi}}||_{2}^{2})\,, \tag{31}\]  
where  
\[\left\{ \begin{array}{l}\mathbf{C}_{0}=\mathbf{B}_{0}^{\rm H}\mathbf{B}_{0}\,,\ \mathbf{C}^{(1)}=\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}+\widetilde{\mathbf{B}}^{( 1)\rm H}\mathbf{B}_{0}=O(||\mathbf{\xi}||_{2})\,,\\ \mathbf{C}^{(2)}=\widetilde{\mathbf{B}}^{(1)\rm H}\widetilde{\mathbf{B}}^{(1)}+\mathbf{B}_{0}^ {\rm H}\widetilde{\mathbf{B}}^{(2)}+\widetilde{\mathbf{B}}^{(2)\rm H}\mathbf{B}_{0}=O(|| \mathbf{\xi}||_{2}^{2})\,.\end{array} \right. \tag{32}\]  
From (31) and (32) we observe that \(\mathbf{C}^{(1)}\) and \(\mathbf{C}^{(2)}\) consist of all first- and second-order perturbation terms, respectively.