## 6 MSE of Direct Position Determination Method in Presence of Array Model Errors
### Second-Order Perturbation Analysis on the Cost Function
\begin{array}{rl}\widetilde{\mathbf{p}}&=\arg\max_{\mathbf{z}\in\mathbb{R}^{\perp \times 1}}\left\{\begin{array}{c}\mathbf{h}_{1}^{\rm H}(\mathbf{p})\widetilde{ \mathbf{\varepsilon}}_{\rm c}+\mathbf{h}_{2}^{\rm H}(\mathbf{p})\widetilde{\mathbf{\varphi}}_{ \rm c}+\mathbf{h}_{3}^{\rm T}(\mathbf{p})\mathbf{z}+\widetilde{\mathbf{\varepsilon}}_{\rm c}^{ \rm H}\mathbf{H}_{1}(\mathbf{p})\overline{\mathbf{\varepsilon}}_{\rm c}+\widetilde{\mathbf{ \varphi}}_{\rm c}^{\rm H}\mathbf{H}_{2}(\mathbf{p})\widetilde{\mathbf{\varphi}}_{\rm c}\\ +\mathbf{z}^{\rm T}\mathbf{H}_{3}(\mathbf{p})\mathbf{z}+\widetilde{\mathbf{\varepsilon}}_{\rm c}^{ \rm H}\mathbf{H}_{4}(\mathbf{p})\widetilde{\mathbf{\varphi}}_{\rm c}+\widetilde{\mathbf{ \varepsilon}}_{\rm c}^{\rm H}\mathbf{H}_{5}(\mathbf{p})\mathbf{z}+\widetilde{\mathbf{\varphi}}_ {\rm c}^{\rm H}\mathbf{H}_{6}(\mathbf{p})\mathbf{z}\end{array}\right\}\\ &=\arg\max_{\mathbf{z}\in\mathbb{R}^{\perp\times 1}}\{\mathbf{h}_{3}^{\rm T}(\mathbf{p}) \mathbf{z}+\mathbf{z}^{\rm T}\mathbf{H}_{3}(\mathbf{p})\mathbf{z}+\widetilde{\mathbf{\varepsilon}}_{ \rm c}^{\rm H}\mathbf{H}_{5}(\mathbf{p})\mathbf{z}+\widetilde{\mathbf{\varphi}}_{\rm c}^{\rm H }\mathbf{H}_{6}(\mathbf{p})\mathbf{z}\}\,,\end{array}  \tag{48}\]  
which further implies  
\[ \begin{array}{rl}\widetilde{\mathbf{p}}&=-\frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{ H}_{5}^{\rm T}(\mathbf{p})\overline{\mathbf{\varepsilon}}_{\rm c}^{*}-\frac{1}{2}\mathbf{H}_{3}^ {-1}(\mathbf{p})\mathbf{H}_{6}^{\rm T}(\mathbf{p})\widetilde{\mathbf{\varphi}}_{\rm c}^{*}- \frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{h}_{3}(\mathbf{p})\\ &=-\frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{H}_{5}^{\rm T}(\mathbf{p})\overline{\mathbf{ \varepsilon}}_{\rm c}^{*}-\frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{H}_{6}^{\rm T} (\mathbf{p})\widetilde{\mathbf{\varphi}}_{\rm c}^{*}=O\!\left(\left\|\left[\begin{array} []{c}\overline{\mathbf{\varepsilon}}\\ \widetilde{\mathbf{\varphi}}\end{array}\right]\right\|_{\!2}\right).\end{array}  \tag{49}\]  
The second equality in (49) follows from (47). In (49), the linear relationship between the localization error \(\widetilde{\mathbf{p}}\) and the sensor noise \(\overline{\mathbf{\varepsilon}}_{\rm c}\) as well as the array model error \(\widetilde{\mathbf{\varphi}}_{\rm c}\) is formulated. It is easily observed from (49) that the positioning error vector \(\widetilde{\mathbf{p}}\) consists of two terms. The first term is associated with the sensor noise, which can be described as  
\[\widetilde{\mathbf{p}}_{1}=-\frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{H}_{5}^{\rm T}( \mathbf{p})\overline{\mathbf{\varepsilon}}_{\rm c}^{*}=O(||\widetilde{\mathbf{\varepsilon }}||_{2})\,. \tag{50}\]  
The second term is due to the array model errors, which can be written as  
\[\widetilde{\mathbf{p}}_{2}=-\frac{1}{2}\mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{H}_{6}^{\rm T}( \mathbf{p})\widetilde{\mathbf{\varphi}}_{\rm c}^{*}=O(||\widetilde{\mathbf{\varphi}}||_{2} )\,. \tag{51}\]  
According to the statistical assumptions in Sections 3 and 5, it is concluded that the localization error \(\widetilde{\mathbf{p}}\) is asymptotically Gaussian distributed with a zero mean and a covariance matrix given by  
\[\mathbf{p}=\mathbb{E}[\widetilde{\mathbf{p}}\widetilde{\mathbf{p}}^{\rm T}]=\tfrac{1}{4} \mathbf{H}_{3}^{-1}(\mathbf{p})\mathbf{H}_{5}^{\rm T}(\mathbf{p})\cdot\mathbb{E}[\overline{ \mathbf{\varepsilon}}_{\rm c}^{*}\overline{\mathbf{\varepsilon}}_{\rm c}^{\rm H}] \cdot\mathbf{H}_{5}(\mathbf{p})\mathbf{H}_{3}^{-\rm T}(\mathbf{p})+\tfrac{1}{4}\mathbf{H}_{3}^{-1}  
The second part, due to the array model errors, is given by  
\[\mathbf{P}_{2}=\tfrac{1}{4}\mathbf{H}_{3}^{-1}(\boldsymbol{p})\mathbf{H}_{6}^{T}( \boldsymbol{p})\cdot\left[\begin{array}{cc}\text{blkdiag}[\boldsymbol{\Phi}_{ 1}^{(1)*}\ \boldsymbol{\Phi}_{2}^{(1)*}\ \cdots\ \boldsymbol{\Phi}_{N}^{(1)*}]&\text{blkdiag}[\boldsymbol{\Phi}_{1}^{(2)*}\ \boldsymbol{\Phi}_{2}^{(2)*}  
Consequently, the first localization SP can be determined by  
\[