## 2 Mathematical Model  
### Polarization Mode  
As depicted in Figure 1, a transverse electric (TE) wave is incident from the direction \((\theta,\varphi)\) with respective to the reference point \(O\), where \(\theta\in[0,\pi/2]\) represent the pitching angle and \(\varphi\in[0,2\pi)\) denote the azimuth angle. Moreover, TE wave refers to an electromagnetic wave in which the electric field is perpendicular to the propagation direction. The pitching angle refers to the acute angle between the direction of the incoming signal and the normal of the antenna. The azimuth angle refers to the angle between the projection of the incoming signal on the antenna and the reference direction, which is artificially specified. Define the transient electric field vector in plane \(\Theta\) as \(\overset{\rightarrow}{E}(t)\) and it can be written as  
\[\overset{\rightarrow}{E}(t)=\xi_{\mathrm{h}}(t)\overset{\rightarrow}{E}_{ \mathrm{h}}+\xi_{\mathrm{v}}(t)\overset{\rightarrow}{E}_{\mathrm{v}}, \tag{1}\]  
where \((\overset{\rightarrow}{E}_{\mathrm{h}},\overset{\rightarrow}{E}_{\mathrm{v}})\) represents a pair of the orthonormal vector in the plane \(\Theta\), and \(\xi_{\mathrm{h}}(t)\), \(\xi_{\mathrm{v}}(t)\) respectively denote the transient projection values along the \(\overset{\rightarrow}{E}_{\mathrm{h}}\) and \(\overset{\rightarrow}{E}_{\mathrm{v}}\).  
According to the orientation of the end point of the transient electric field vector, the TE wave can be classified into linear polarization, circular polarization, and elliptical polarization (EP). Moreover, linear polarization can be classified into horizontal polarization (HP) and vertical polarization (VP). Circular polarization can be classified into right-handed circular polarization (RHCP) and left-handed  
Figure 1: Transverse electric wave.  
circular polarization (LHCP). To describe the polarization mode of the TE wave, the polarization parameter \((\gamma,\eta)\) is adopted where \(\gamma\in[0,\pi/2]\) represents the amplitude ratio between the horizontal component and the vertical component of the electric field and meanwhile \(\eta\in[0,2\pi)\) denotes the phase difference between the horizontal component and the vertical component of the electric field. The polarization modes mentioned above can be defined by the polarized parameters in Table 1.  
The polarization information of the TE wave can be distinguished by the electric vector sensor (EVS). Complete EVS contains three concentric and mutually perpendicular dipoles, while the dual-polarized EVS that consists of a pair of crossed dipoles is utilized most widely in practice. Note that multiple dual-polarized EVSs arrayed in space can form the DPSA. The TE wave received by the dual-polarized EVS can be given by  
\[\mathbf{x}(t)=\left[\begin{array}{c}s_{x}(t)\\ s_{z}(t)\end{array}\right]=\left[\begin{array}{cc}-\sin\varphi&\cos\theta \cos\varphi\\ \cos\varphi&\cos\theta\sin\varphi\end{array}\right]\left[\begin{array}{c} \cos\gamma\\ \sin\gamma e^{\eta\eta}\end{array}\right]s(t), \tag{2}\]  
where \(s_{x}(t)\) is the signal received by the dipole along the \(x\) axis, \(s_{z}(t)\) is the signal received by the dipole along the \(z\) axis, and \(s(t)\) denotes the envelope of the incident TE wave. One can find that the received information is not only determined by the polarization of the TE wave, but also related to the DOA of the incident signal.