## 3 The Proposed Algorithm
### NC-Euler-ESPRIT-DPD  
On the basis of the method of converting complex number calculation into real number operation [29], Equation (15) can be rewritten:  
\[\mathbf{z}_{g}(k)=\left[ \begin{array}{c}-\frac{\mathbf{r}_{g}(k)-\mathbf{r}_{g}^{2}(k)} {2f}\\ \frac{\mathbf{r}_{g}(k)+\mathbf{r}_{g}^{2}(k)}{2}\end{array} \right]=\left[ \begin{array} []{c}-\text{Im}\big{(}\mathbf{A}_{g}\mathbf{\Phi}\big{)}\\ \text{Re}\big{(}\mathbf{A}_{g}\mathbf{\Phi}\big{)}\end{array} \right]\mathbf{\Lambda}_{g} \mathbf{s}_{g}^{(R)}(k)+\left[ \begin{array}{c}-\text{Im}\big{(}\mathbf{n}_{g}(k) \big{)}\\ \text{Re}\big{(}\mathbf{n}_{g}(k)\big{)}\end{array} \right]=\mathbf{H}_{g}\mathbf{s}_{0,g}( k)+\mathbf{n}_{1,g}(k), \tag{34}\]  
where \(\mathbf{H}_{g}\) can be considered as a direction matrix  
\[\mathbf{H}_{g}=\left[ \begin{array}{c}-\text{Im}\big{(}\mathbf{A}_{g}\mathbf{\Phi}\big{)} \\ \text{Re}\big{(}\mathbf{A}_{g}\mathbf{\Phi}\big{)}\end{array} \right]\in\mathbb{R}^{2M \times Q}, \tag{35}\]\[\mathbf{n}_{1,\mathcal{G}}(k)=\left[ \begin{array}{c}-\text{Im}\big{(}\mathbf{n}_{g}(k) \big{)}\\ \text{Re}\big{(}\mathbf{n}_{g}(k)\big{)}\end{array} \right]\in\mathbb{R}^{2M\times 1}, \tag{36}\]  
\((\cdot)^{*}\), \(\text{Re}(\cdot)\) and \(\text{Im}(\cdot)\) indicate conjugate, real part and imaginary part, respectively. As \(\gamma_{g,\mathcal{A}}=\frac{2\pi d}{\lambda}\frac{\mathbf{n}_{g}(1)-\mathbf{n}_{g}(1) }{||\mathbf{n}_{g}-\mathbf{n}_{g}||}\) is defined above, direction matrix \(\mathbf{H}_{g}\) can be expressed as  
\[\mathbf{H}_{g} = \begin{bmatrix}\mathbf{h}_{g1}\big{(}\gamma_{g,1},\mathbf{\varphi}_{1} \big{)}&\mathbf{h}_{g1}\big{(}\gamma_{g,2},\mathbf{\varphi}_{2}\big{)}&\cdots&\mathbf{h}_ {g1}\big{(}\gamma_{g,\mathcal{O}},\mathbf{\varphi}_{Q}\big{)}\\ \mathbf{h}_{g2}\big{(}\gamma_{g,1},\mathbf{\varphi}_{1}\big{)}&\mathbf{h}_{g2}\big{(} \gamma_{g,2},\mathbf{\varphi}_{2}\big{)}&\cdots&\mathbf{h}_{g2}\big{(}\gamma_{g, \mathcal{O}},\mathbf{\varphi}_{Q}\big{)}\end{bmatrix}  \tag{37}\] \[= \begin{bmatrix}\mathbf{h}_{g}\big{(}\gamma_{g,1},\mathbf{\varphi}_{1} \big{)}&\mathbf{h}_{g}\big{(}\gamma_{g,2},\mathbf{\varphi}_{2}\big{)}&\cdots&\mathbf{h}_ {g}\big{(}\gamma_{g,\mathcal{O}},\mathbf{\varphi}_{Q}\big{)}\end{bmatrix} ,\]  
where  
\[\mathbf{h}_{g1}(\gamma_{g,\mathcal{A}},\mathbf{\varphi}_{q})=\left[ \begin{array}{c} \sin\varphi_{q}\\ \sin\big{(}\gamma_{g,\mathcal{A}}+\mathbf{\varphi}_{q}\big{)}\\ \vdots\\ \sin\big{(}(M-1)\gamma_{g,\mathcal{A}}+\mathbf{\varphi}_{q}\big{)}\end{array}  \right], \tag{38}\]  
\[\mathbf{h}_{g2}(\gamma_{g,\mathcal{A}},\mathbf{\varphi}_{q})=\left[ \begin{array}{c} \cos\varphi_{q}\\ \cos\big{(}\gamma_{g,\mathcal{A}}+\mathbf{\varphi}_{q}\big{)}\\ \vdots\\ \cos\big{(}(M-1)\gamma_{g,\mathcal{A}}+\mathbf{\varphi}_{q}\big{)}\end{array}  \right] \tag{39}\]  
Then, we can obtain the signal subspace \(\mathbf{U}_{g}^{(s)}\) according to Equations (23), (24) and (34). As signal subspace and noise subspace are orthogonal, we can get \(span\big{\{}\mathbf{H}_{g}\big{\}}=span\Big{\{}\mathbf{U}_{g}^{(s)}\Big{\}}\), i.e., there is a reversible matrix \(\mathbf{T}\) that makes  
\[\mathbf{U}_{g}^{(s)}=\mathbf{H}_{g}\mathbf{T} \tag{40}\]  
Define reversible matrix  
\[\mathbf{T}_{3}= \begin{bmatrix}\mathbf{T}_{b3}&\mathbf{O}\\ \mathbf{O}&\mathbf{T}_{b3}\end{bmatrix} \in\mathbb{R}^{2(M-1)\times 2M}, \tag{41}\]  
\[\mathbf{T}_{4}= \begin{bmatrix}\mathbf{O}&\mathbf{T}_{b4}\\ -\mathbf{T}_{b4}&\mathbf{O}\end{bmatrix} \in\mathbb{R}^{2(M-1)\times 2M}, \tag{42}\]  
where  
\[\mathbf{T}_{b3}= \begin{bmatrix}1&1\\ &1&1\\ &&\ddots&\ddots\\ &&&1&1\end{bmatrix} \in\mathbb{R}^{(M-1)\times M}, \tag{43}\]  
\[\mathbf{T}_{b4}= \begin{bmatrix}1&-1\\ &1&-1\\ &&\ddots&\ddots\\ &&&1&-1\end{bmatrix} \in\mathbb{R}^{(M-1)\times M}, \tag{44}\]  
\(\mathbf{O}\in\mathbb{R}^{(M-1)\times M}\) is a zero matrix. Then, we can obtain  
\[\mathbf{T}_{3}\mathbf{H}_{g}=\left[ \begin{array}{c}-\mathbf{T}_{b3}\text{Im}\big{(}\mathbf{ A}_{g}\Phi\big{)}\\ \mathbf{T}_{b3}\text{Re}\big{(}\mathbf{A}_{g}\Phi\big{)}\end{array}