## 2 Spatial Polarimetric Time-Frequency Distribution  
### Polarization Modeling  
In Figure 1, the electric field of transverse electromagnetic (TEM) waves incoming to the array can be expressed as  
\[ \begin{split}\mathbf{E}(t)&=E_{\theta}(t)\mathbf{\theta} +E_{\phi}(t)\mathbf{\phi}=\big{[}E_{\theta}(t)\cos(\theta)\cos(\phi)-E_{\phi}(t) \sin(\phi)\big{]}\mathbf{x}\\ &+\big{[}E_{\theta}(t)\cos(\theta)\sin(\phi)+E_{\phi}(t)\cos( \phi)\big{]}\mathbf{y}+E_{\theta}(t)\sin(\theta)\mathbf{z},\end{split}  \tag{1}\]  
where \(\mathbf{\phi}\) and \(\mathbf{\theta}\) are the azimuth and elevation spherical unit vectors of the signals. The \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\) are, respectively, rectangular coordinate unit vectors along the \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\) directions. For generality and succinctness, the signal and arrays are assumed in the \(\mathbf{x}\)-\(\mathbf{y}\) plane and the \(\mathbf{y}\)-\(\mathbf{z}\) plane, respectively. Then, \(\theta=90^{\circ}(\mathbf{\theta}=-\mathbf{z})\), and  
\[\mathbf{E}(t)=-E_{\phi}(t)\sin(\phi)\mathbf{x}+E_{\phi}(t)\cos(\phi)\mathbf{y }+E_{\theta}(t)\mathbf{z}. \tag{2}\]\(s(t)\) is expressed as the source amplitude measured by the receiving reference antenna sensor. The polarization auxiliary angle is \(\gamma\in[0,\pi/2]\) and the polarization phase difference is \(\eta\in(-\pi,\pi]\). \(s^{[p]}(t)\) and \(s^{[h]}(t)\) are the vertical and horizontal polarization components of the signal source. \(E_{\theta}(t)\) and \(E_{\phi}(t)\) can be expressed by spherical fields as  
\[E_{\theta}(t)=s^{[p]}(t)=s(t)\cos(\gamma),E_{\phi}(t)=s^{[h]}(t)=s(t)\sin( \gamma)e^{i\eta}. \tag{3}\]  
A signal is linearly polarized if \(\eta=0^{\circ}\) or \(\eta=180^{\circ}\). Bringing (3) into (2) yields  
\[\mathbf{E}(t)=s(t)[-\cos(\gamma)\sin(\phi)\mathbf{x}+\cos(\phi)\sin(\gamma)e^ {i\eta}\mathbf{y}+\cos(\gamma)\mathbf{z}]. \tag{4}\]  
The \(N\) signals are assumed entering on the \(M\) dual-polarization antenna array. The \(n\)th source's dual-polarized components are  
\[s_{n}^{[p]}(t)=s_{n}(t)\cos(\gamma_{n})=c_{nv}s_{n}(t),s_{n}^{[h]}(t)=s_{n}(t) \sin(\gamma_{n})e^{i\eta_{n}}=c_{nh}s_{n}(t), \tag{5}\]  
where the parameters \(c_{nv}=\cos(\gamma_{n})\) and \(c_{nh}=\sin(\gamma_{n})e^{i\eta_{n}}\) describe the vertical and horizontal polarization coefficients. The signal source incident on the \(m\)th dual-polarization antenna sensor expresses  
\[\underline{y}_{m}(t) =\left[y_{m}^{[p]}(t),y_{m}^{[h]}(t)\right]^{\mathrm{T}}=\sum_{n= 1}^{N}\left[a_{\mathit{min}}^{[p]}\mathbf{E}_{n}\cdot\mathbf{z},a_{\mathit{ min}}^{[h]}\mathbf{E}_{n}\cdot\mathbf{y}\right]^{\mathrm{T}} \tag{6}\] \[=\sum_{n=1}^{N}\left[a_{\mathit{min}}^{[p]}s_{n}^{[p]}(t),a_{ \mathit{min}}^{[h]}s_{n}^{[h]}(t)\cos(\phi_{n})\right]^{\mathrm{T}},\]  
where \({}^{n}\cdot\cdot\)\({}^{n}\) is the dot product and \(\mathbf{E}_{n}\) is the \(n\)th signal source electric field vector. \(a_{\mathit{min}}^{[v]}\) and \(a_{\mathit{min}}^{[h]}\) are the \(m\)th elements of the vertical and horizontal polarization steering vectors, \(\mathbf{a}^{[v]}(\phi_{n})\) and \(\mathbf{a}^{[h]}(\phi_{n})\), respectively. It is assumed that \(\mathbf{a}^{[v]}(\phi)\) and \(\mathbf{a}^{[h]}(\phi)\) are normalized \(\left(\left\|\mathbf{a}^{[v]}(\phi)\right\|^{2}=\left\|\mathbf{a}^{[h]}(\phi )\right\|^{2}=M)\) and the array has been calibrated. In this paper, the \(\cos(\phi_{n})\) terms of the horizontal polarization array manifold can be introduced in the array calibration. Therefore, they will not be considered. Equation (6) is simplified as  
\[\underline{y}_{m}(t) =\left[a_{\mathit{min}}^{[p]}s_{n}^{[p]}(t),a_{\mathit{min}}^{[h] }s_{n}^{[h]}(t)\right]^{\mathrm{T}}=s_{n}(t)\left(\left[ \begin{array}{cc}a_ {\mathit{min}}^{[p]}&a_{\mathit{min}}^{[h]}\\ \end{array} \right]^{\mathrm{T}}\odot\left[ \begin{array}{cc}c_{nv}&c_{nh}\\ \end{array}