## 3 The Proposed Algorithm  
### Nc-ESPRIT-DPD  
From [25], the non-circular signal can be written in the following form  
\[\mathbf{s}_{g}(k)=\mathbf{\Phi}\mathbf{s}_{g}^{(R)}(k), \tag{11}\]  
where \(\mathbf{\Phi}\) represents the NC phase matrix  
\[\mathbf{\Phi}=\begin{bmatrix}e^{-j\varphi_{1}}&0&\cdots&0\\ 0&e^{-j\varphi_{2}}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\cdots&0&e^{-j\varphi_{Q}}\end{bmatrix}\in\mathbb{C}^{Q\times Q}, \tag{12}\]  
\[\mathbf{s}_{g}^{(R)}(k)=\begin{bmatrix}s_{g,1}^{(R)}(k),s_{g,2}^{(R)}(k),\cdots,s _{g,Q}^{(R)}(k)\end{bmatrix}^{T}\in\mathbb{R}^{Q\times 1}, \tag{13}\]  
where \(\varphi_{q}\) represents NC phase of the \(q\)th source and \(s_{g,q}^{(R)}(k)\) stands for the amplitude of the signal. Thus, Equation (6) can be rewritten according to Equation (11):  
\[\mathbf{r}_{g}(k)=\mathbf{A}_{g}(\mathbf{p})\mathbf{\Lambda}_{g}\mathbf{\Phi}\mathbf{s}_{g}^{(R)}(k)+ \mathbf{n}_{g}(k)\in\mathbb{C}^{M\times 1} \tag{14}\]  
The received signal can be extended by making full use of the elliptic covariance information of NC sources [34]:  
\[\mathbf{z}_{g}(k)=\begin{bmatrix}\begin{array}{c}\mathbf{r}_{g}(k)\\ \mathbf{I}\mathbf{r}_{g}^{*}(k)\end{array}\end{bmatrix}=\begin{bmatrix}\begin{array}{c }\mathbf{A}_{g}\mathbf{\Phi}\\ \mathbf{I}\mathbf{A}_{g}^{*}\mathbf{\Phi}^{*}\end{array}\end{bmatrix}\mathbf{\Lambda}_{g}\mathbf{ s}_{g}^{(R)}(k)+\begin{bmatrix}\begin{array}{c}\mathbf{n}_{g}(k)\\ \mathbf{I}\mathbf{n}_{g}^{*}(k)\end{array}\end{bmatrix}=\mathbf{B}_{g}\mathbf{s}_{0,g}(k)+\mathbf{ n}_{0,g}(k)\in\mathbb{C}^{2M\times 1}, \tag{15}\]  
Figure 1: Scene of multiple arrays positioning.  
Define row exchange matrix  
\[\mathbf{J}_{1}=\begin{bmatrix}T_{b1}&\mathbf{O}\\ \mathbf{O}&\mathbf{T}_{b1}\end{bmatrix}\in\mathbb{R}^{2(M-1)\times 2M}, \tag{26}\]  
\[\mathbf{J}_{2}=\begin{bmatrix}T_{b2}&\mathbf{O}\\ \mathbf{O}&\mathbf{T}_{b2}\end{bmatrix}\in\mathbb{R}^{2(M-1)\times 2M}, \tag{27}\]  
where  
\[\mathbf{T}_{b1}=\begin{bmatrix}0&1&&\\ 0&1&&\\ \vdots&&\ddots&\\ 0&&&1\end{bmatrix}\in\mathbb{R}^{(M-1)\times M}, \tag{28}\]  
\[\mathbf{T}_{b2}=\begin{bmatrix}1&&&0\\ &1&&0\\ &\ddots&\vdots\\ &&&1&0\end{bmatrix}\in\mathbb{R}^{(M-1)\times M}, \tag{29}\]  
\(\mathbf{O}\in\mathbb{R}^{(M-1)\times M}\) is a zero matrix. Then, we can obtain  
\[\mathbf{J}_{1}\mathbf{U}_{g}^{(s)}\mathbf{T}=\mathbf{J}_{2}\mathbf{U}_{g}^{(s)}\mathbf{T}\mathbf{T}_{g}, \tag{30}\]  
\[\big{(}\mathbf{J}_{2}\mathbf{U}_{g}^{(s)}\big{)}^{+}\mathbf{J}_{1}\mathbf{U}_{g}^{(s)}=\mathbf{T} \mathbf{T}_{g}\mathbf{T}^{-1}, \tag{31}\]  
where \((\cdot)^{+}\) denotes general Moore-Penrose inverse and  
\[\mathbf{\Gamma}_{g}=\begin{bmatrix}e^{i\big{(}\gamma_{g,1}\big{)}}&&&\\ &e^{i\big{(}\gamma_{g,2}\big{)}}&&\\ &&\ddots&\\ &&&e^{i\big{(}\gamma_{g,Q}\big{)}}\end{bmatrix} \tag{32}\]  
Then, \(\gamma_{g,q}(q=1,2,\cdots Q)\) could be easily acquired according to Equation (32). In view of the above definition, we can know that the phase of \(\gamma_{g,q}\) contains the azimuth information of targets. Then, by combining the azimuth information of \(G\) observation stations, we can obtain the position information of targets. The joint spectral function of NC-ESPRIT-DPD algorithm can be constructed as  
\[\mathbf{f}_{NC-ESPRIT}(\mathbf{p})=1/\sum_{g=1}^{G}\min_{q}\ |\mu_{g,q}-e^{i\big{(} \frac{2\pi d}{\lambda}\frac{\mathbf{\Psi}_{g}(1)-\mathbf{\Psi}_{g}(1)}{\mathbf{\Psi}_{g}- \mathbf{\Psi}_{g}(1)}\big{)}}|,\ q\in[1,Q], \tag{33}\]  
where the \(q\)th diagonal element of diagonal matrix \(\mathbf{\Gamma}_{g}\) is \(\mu_{g,q}\) and \(|\cdot|\) indicates absolute value.