## 2 Spatial Polarimetric Time-Frequency Distribution
### Spatial Polarimetric Time-Frequency Distributions  
The fundaments of STFDs are discussed. The Cohen classes, especially the Wigner-Ville distribution (WVD) and its peculiarities, are examined. The WVD formulation of a signal \(x(t)\) is given by  
\[D_{xx}(t,f)=\iint\varphi(t-u,\tau)x\Big{(}u+\frac{\tau}{2}\Big{)}\times x^{*} \Big{(}u-\frac{\tau}{2}\Big{)}\mathrm{e}^{-\mathrm{i}2\pi f\tau}\mathrm{d}u \mathrm{d}\tau, \tag{8}\]  
where \(t\) and \(f\) are, respectively, the time and frequency indexes, \(\mathrm{j}\) is the imaginary unit and \({}^{*}\) implies complex conjugate. The kernel \(\varphi(t,\tau)\) is a function of the time and lag variables and uniquely defines the TFDs. In this paper, all the integrals are from \(-\infty\) to \(\infty\). With the \(M\) dual-polarized antennas array, the data vector of each polarization is expressed as  
\[ \begin{split}\mathbf{x}^{[i]}(t)&=\Big{[}x_{1}^{[i] }(t),x_{2}^{[i]}(t),\dots,x_{M}^{[i]}(t)\Big{]}^{\mathrm{T}}=\mathbf{y}^{[i]}( t)+\mathbf{n}^{[i]}(t)\\ &=\mathbf{A}^{[i]}(\boldsymbol{\Phi})\mathbf{s}^{[i]}(t)+\mathbf{ n}^{[i]}(t)\,(i=v\text{ or }h),\end{split}  \tag{9}\]  
where \(\mathbf{A}^{[i]}(\boldsymbol{\Phi})=[\mathbf{a}_{1},\mathbf{a}_{2},..., \mathbf{a}_{M}]\)=\([\mathbf{a}(\phi_{1}),\mathbf{a}(\phi_{2}),...,(\phi_{M})]\) and \(\mathbf{a}(\phi_{n})=-2\pi(n-1)d\sin\phi/\lambda\). The array is a uniform linear array, and the array manifold vector is \(\mathbf{A}\), where each element represents the wavepath difference reached by the electromagnetic wave. Therefore, the DOA of the signal can be derived from the array manifold. The autopolarization STFDs matrix of the data vector is defined as  
\[\mathbf{D}_{\mathbf{x}^{[i]}\mathbf{x}^{[i]}}(t,f)=\iint\varphi(t-u,\tau) \mathbf{x}^{[i]}\Big{(}u+\frac{\tau}{2}\Big{)}\times\Big{(}\mathbf{x}^{[i]} \Big{(}u-\frac{\tau}{2}\Big{)}\Big{)}^{\mathrm{H}}\mathrm{e}^{-\mathrm{i}2 \pi f\tau}\mathrm{d}u\mathrm{d}\tau, \tag{10}\]  
where H is conjugate transpose. In the noise-free circumstance, (10) can be described as  
\[\mathbf{D}_{\mathbf{x}^{[i]}\mathbf{x}^{[i]}}(t,f)=\mathbf{A}^{[i]}( \boldsymbol{\Phi})\mathbf{D}_{\mathbf{s}^{[i]}\mathbf{s}^{[i]}}(t,f)\Big{(} \mathbf{A}^{[i]}(\boldsymbol{\Phi})\Big{)}^{\mathrm{H}}. \tag{11}\]  
Similarly, the cross-polarization STFDs matrix between two different polarization data vectors can be described as  
\[\mathbf{D}_{\mathbf{x}^{[i]}\mathbf{x}^{[i]}}(t,f)=\iint\varphi(t-u,\tau) \mathbf{x}^{[i]}\Big{(}u+\frac{\tau}{2}\Big{)}\times\Big{(}\mathbf{x}^{[i]} \Big{(}u-\frac{\tau}{2}\Big{)}\Big{)}^{\mathrm{H}}\mathrm{e}^{-\mathrm{i}2\pi f \tau}\mathrm{d}u\mathrm{d}\tau, \tag{12}\]  
which becomes  
\[\mathbf{D}_{\mathbf{x}^{[i]}\mathbf{x}^{[i]}}(t,f)=\mathbf{A}^{[i]}( \boldsymbol{\Phi})\mathbf{D}_{\mathbf{s}^{[i]}\mathbf{s}^{[i]}}(t,f)\Big{(} \mathbf{A}^{[i]}(\boldsymbol{\Phi})\Big{)}^{\mathrm{H}}. \tag{13}\]  
On the basis of (9), the extended data vector of both polarizations can be established thus:  
\[ \begin{split}\mathbf{x}(t)&=\begin{bmatrix}\mathbf{x} ^{[i]}(t)\\ \mathbf{x}^{[i]}(t)\end{bmatrix}=\begin{bmatrix}\mathbf{A}^{[i]}(\boldsymbol{ \Phi})&\mathbf{0}\\ \mathbf{0}&\mathbf{A}^{[i]}(\boldsymbol{\Phi})\end{bmatrix}\begin{bmatrix} \mathbf{s}^{[i]}(t)\\ \mathbf{s}^{[i]}(t)\end{bmatrix}+\begin{bmatrix}\mathbf{n}^{[i]}(t)\\ \mathbf{n}^{[i]}(t)\end{bmatrix}\\ =\begin{bmatrix}\mathbf{A}^{[i]}(\boldsymbol{\Phi})&\mathbf{0}\\ \mathbf{0}&\mathbf{A}^{[i]}(\boldsymbol{\Phi})\end{bmatrix}\begin{bmatrix} \mathbf{Q}^{[i]}\\ \mathbf{Q}^{[i]}\end{bmatrix}\mathbf{s}(t)+\begin{bmatrix}\mathbf{n}^{[i]}(t)\\ \mathbf{n}^{[i]}(t)\end{bmatrix}=\mathbf{B}(\boldsymbol{\Phi})\mathbf{Q} \mathbf{s}(t)+\mathbf{n}(t),\end{split}  \tag{14}\]  
where  
\[\mathbf{B}(\boldsymbol{\Phi})= \begin{bmatrix}\mathbf{A}^{[i]}(\boldsymbol{ \Phi})&\mathbf{0}\\ \mathbf{0}&\mathbf{A}^{[i]}(\boldsymbol{\Phi})\end{bmatrix} , \tag{15}\]  
is a block diagonal matrix and  
\[\mathbf{Q}=\left[ \begin{array}{c}\mathbf{Q}^{[i]}\\ \mathbf{Q}^{[i]}\end{array}