## 4 Polarimetric Time-Frequency DOA Estimation Using Dual Algorithms  
The TF-MUSIC algorithm has recently been proposed for improving the spatial resolution of a good T-F peculiarity signal [35]. The algorithm in this paper provides the extension for applying TF-MUSIC based on cross terms to polarization arrays. Consider the spatial steering matrix  
\[\mathbf{F}(\phi)=\frac{1}{\sqrt{M}} \begin{bmatrix}\mathbf{a}^{[x]}(\phi)& \mathbf{0}\\ \mathbf{0}&\mathbf{a}^{[h]}(\phi)\end{bmatrix} , \tag{35}\]  
corresponding to the signal DOA. Because \(\left\|\mathbf{a}^{[i]}(\phi)\right\|^{2}=M\), \(\mathbf{F}^{\mathrm{H}}(\phi)\mathbf{F}(\phi)\) represents the identity matrix. To search in the spatial and polarimetric joint domains, the spatial polarization vector is defined as:  
\[\mathbf{f}(\phi,\mathbf{c})=\frac{\mathbf{F}(\phi)\mathbf{c}}{\|\mathbf{F}( \phi)\mathbf{c}\|}=\mathbf{F}(\phi)\mathbf{c}, \tag{36}\]  
where \(\mathbf{c}= \begin{bmatrix}c_{\mathrm{p}}&c_{h}\end{bmatrix} ^{\mathrm{T}}\) represents a unit norm vector of the unknown polarization coefficient. In (36), \(\left\|\mathbf{F}(\phi)\mathbf{c}\right\|= \begin{bmatrix}\mathbf{c}^{H} \mathbf{F}^{\mathrm{H}}(\phi)\mathbf{F}(\phi)\mathbf{c}\end{bmatrix} ^{1/2}= \left(\mathbf{c}^{H}\mathbf{c}\right)^{1/2}=1\). The PTF-MUSIC spectrum is provided via the following function:  
\[P(\phi)= \begin{bmatrix}\min_{\mathbf{c}}\mathbf{f}^{\mathrm{H}}(\phi,\mathbf{ c})\mathbf{U}_{n}\mathbf{U}_{n}^{\mathrm{H}}\mathbf{f}(\phi,\mathbf{c}) \end{bmatrix} ^{-1}= \begin{bmatrix}\min_{\mathbf{c}}\mathbf{c}^{\mathrm{H}} \mathbf{F}^{\mathrm{H}}(\phi)\mathbf{U}_{n}\mathbf{U}_{n}^{\mathrm{H}} \mathbf{F}(\phi)\mathbf{c}\end{bmatrix} ^{-1}, \tag{37}\]  
where \(\mathbf{U}_{n}\) is the noise subspace, which is from the SPTFDs matrix of the selected T-F points. Selecting these points from high energy concentration regions can enhance the signal-to-noise ratio (SNR), which makes the proposed algorithm more robust than the conventional MUSIC counterparts. In (37), by obtaining the smallest eigenvalue of \(\mathbf{F}^{\mathrm{H}}(\phi)\mathbf{U}_{n}\mathbf{U}_{n}^{\mathrm{H}} \mathbf{F}(\phi)\), the items in parentheses are the fewest. This algorithm can execute the simple \(2\times 2\) matrix eigendecomposition, thus avoiding abundant polarization dimensionality operations. Namely, the PTF-MUSIC spectrum can be simplified as  
\[P(\phi)=\lambda_{\min}^{-1} \begin{bmatrix}\mathbf{F}^{\mathrm{H}}(\phi) \mathbf{U}_{n}\mathbf{U}_{n}^{\mathrm{H}}\mathbf{F}(\phi)\end{bmatrix} , \tag{38}\]  
where \(\lambda_{\min}[\cdot]\) represents solving the smallest eigenvalue. The signal DOA estimation value is the angle coordinate corresponding to the spectrum's highest value. For each DOA \(\phi_{n}\) of \(N\) signals, \(n=1,2,\ldots,N\), each signal polarization parameter can be estimated via  
\[\mathbf{\hat{c}}(\phi_{n})=\mathbf{v}_{\min} \begin{bmatrix}\mathbf{F}^{ \mathrm{H}}(\phi_{n})\mathbf{U}_{n}\mathbf{U}_{n}^{\mathrm{H}}\mathbf{F}(\phi_ {n})\end{bmatrix} , \tag{39}\]  
where \(\mathbf{v}_{\min}[\cdot]\) is the eigenvector for the minimal eigenvalue \(\lambda_{\min}[\cdot]\). For achieving rotational invariance in the array, the \(M\)-element array can be divided into two \(M-1\)-element overlapping subarrays. The first and second subarray include, respectively, the left-most and right-most \(M-1\) antennas, as shown in Figure 2. In addition, the two subarrays' steering matrices of the identically polarized sensors are, respectively, \(\mathbf{A}_{1}\) and \(\mathbf{A}_{2}\). Namely,  
\[ \begin{bmatrix}\mathbf{A}_{2}\mathbf{Q}^{[y]}\\ \mathbf{A}_{2}\mathbf{Q}^{[h]}\end{bmatrix} = \begin{bmatrix}\mathbf{A}_{1} \mathbf{Q}^{[y]}\\ \mathbf{A}_{1}\mathbf{Q}^{[h]}\end{bmatrix}