## 4 Direct Position Determination Method  
This section introduces the DPD method presented in [29]. The optimization model for direct localization is established according to the least square criterion, which can be formulated as  
\[\min_{\mathbf{p},\{\bar{\beta}_{n}\},\{\bar{\beta}_{n}\},\{\bar{\gamma}_{k}\}\sum \limits_{n=1}^{N}\sum\limits_{k=1}^{K}||\overline{\mathbf{x}}_{n,k}-\beta_{n}\mathbf{a }_{n}(\mathbf{p})\overline{s}_{k}\cdot\exp\{-\mathrm{j}\omega_{k}(\tau_{n}(\mathbf{p}) +t_{0})\}||_{2}^{2}=\min_{\mathbf{p},\{\bar{\beta}_{n}\},\{\bar{\gamma}_{k}\}_{n }=1}\|\overline{\mathbf{x}}_{n}-(\overline{\mathbf{s}}_{n}^{\prime}\otimes\mathbf{a}_{n}( \mathbf{p}))\beta_{n}||_{2}^{2}\,, \tag{5}\](5)  
where  
\[\left\{ \begin{array}{l}\overline{\mathbf{x}}_{n}=[\overline{\kappa}_{n,1}^{\rm H}\ \ \overline{\kappa}_{n,2}^{\rm H}\ \ \cdots\ \overline{\kappa}_{n,K}^{\rm H}]^{\rm H},\\ \overline{\mathbf{s}}_{n}^{\prime}=[\overline{s}_{1}\cdot\exp\{-{\rm j}\omega_{1} (\tau_{n}(\mathbf{p})+t_{0})\}\ \ \overline{s}_{2}\cdot\exp\{-{\rm j}\omega_{2}(\tau_{n}(\mathbf{p})+t_{0})\}\ \ \cdots\ \ \overline{s}_{K}\cdot\exp\{-{\rm j}\omega_{K}(\tau_{n}(\mathbf{p})+t_{0})\}]^{\rm T}.\end{array} \right. \tag{6}\]  
Obviously, (5) is a multidimensional nonlinear minimization problem. A direct minimization involves a search over the parameter space and is computationally prohibitive. The technique of the separation of variables can be applied to simplify the optimization problem. First, the channel attenuation scalar \(\beta_{n}\) that minimizes (5) is given by  
\[\beta_{n}=\frac{1}{||\mathbf{a}_{n}(\mathbf{p})||_{2}^{2}\cdot||\overline{\mathbf{s}}_{n}^ {\prime}||_{2}^{2}}\cdot(\overline{\mathbf{s}}_{n}^{\prime}\otimes\mathbf{a}_{n}(\mathbf{ p}))^{\rm H}\overline{\mathbf{x}}_{n}\,. \tag{7}\]  
It can be assumed, without loss of generality, that \(||\mathbf{a}_{n}(\mathbf{p})||_{2}=||\overline{\mathbf{s}}_{n}^{\prime}||_{2}=1\). Then, substituting (7) into (5) and applying algebraic manipulations leads to the concentrated problem [29]  
\[\max_{\mathbf{p},\overline{\mathbf{s}}}\ \overline{\mathbf{s}}^{\rm H}\Bigg{(}\sum_{n=1}^{N} \mathbf{A}_{n}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{n}\overline{\mathbf{x}}_{n}^{\rm H} \mathbf{A}_{n}(\mathbf{p})\Bigg{)}\overline{\mathbf{s}}\,, \tag{8}\]  
where  
\[\left\{ \begin{array}{l}\mathbf{A}_{n}(\mathbf{p})={\rm blkdiag}\Big{[}\mathbf{a}_{n,1}^{ \prime}(\mathbf{p})\ \ \mathbf{a}_{n,2}^{\prime}(\mathbf{p})\ \ \cdots\ \ \mathbf{a}_{n,K}^{\prime}(\mathbf{p})\Big{]}\\ \overline{\mathbf{s}}=[\overline{s}_{1}\cdot\exp\{-{\rm j}\omega_{1}t_{0}\}\ \ \overline{s}_{2}\cdot\exp\{-{\rm j}\omega_{2}t_{0}\}\ \ \cdots\ \ \overline{s}_{K}\cdot\exp\{-{\rm j}\omega_{K}t_{0}\}]^{\rm T}\end{array} \right. \tag{9}\]  
with \(\mathbf{a}_{n,k}^{\prime}(\mathbf{p})=\mathbf{a}_{n}(\mathbf{p})\cdot\exp\{-{\rm j}\omega_{k }\tau_{n}(\mathbf{p})\}\). According to quadratic form theory, the cost function in (8) is maximized by selecting the vector \(\overline{\mathbf{s}}\) as the eigenvector corresponding to the largest eigenvalue of matrix \(\sum\limits_{n=1}^{N}\mathbf{A}_{n}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{n}\overline {\mathbf{x}}_{n}^{\rm H}\mathbf{A}_{n}(\mathbf{p})\). Therefore, (8) reduces to  
\[\max_{\mathbf{p}}J(\mathbf{p})=\max_{\mathbf{p}}\lambda_{\max}\{\mathbf{B}(\mathbf{p})\mathbf{B}^{\rm H }(\mathbf{p})\}=\max_{\mathbf{p}}\lambda_{\max}\{\mathbf{B}^{\rm H}(\mathbf{p})\mathbf{B}(\mathbf{p}) \}\,, \tag{10}\]  
where  
\[\mathbf{B}(\mathbf{p})=[\mathbf{A}_{1}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{1}\ \ A_{2}^{\rm H}(\mathbf{p}) \overline{\mathbf{x}}_{2}\ \ \cdots\ \ \mathbf{A}_{N}^{\rm H}(\mathbf{p})\overline{\mathbf{x}}_{N}]=\mathbf{A}^{\rm H}(\mathbf{p}) \overline{\mathbf{x}} \tag{11}\]  
with  
\[\left\{