## 3 Time-Frequency Point Selection
### Time-Frequency Point Properties and Categorization  
The SPTFDs framework requires the signal T-F characteristics to satisfy one or two conditions with adequate difference:  
1. The T-F points correspond to auto terms. That is, if \(D_{s_{i}s_{j}}(t,f)=\left(\mathbf{D_{ss}}(t,f)\right)_{ij}\), then \[D_{s_{i}s_{j}}(t_{nt},f_{n})=\delta_{i,j}D_{i,j,n}.\] (25) There is at least an \(n\)th T-F point for each \(N\) signal, such that \(D_{i,j,n}\
eq 0\). \(\delta_{i,j}\) represents the Kronecker delta. \(D_{i,j,n}\) is the value of the SPTFDs between the signals \(s_{i}\) and \(s_{j}\) (or \(s_{i}\)) at the T-F point. 2. The T-F points correspond to cross terms. That is, \[D_{s_{i}s_{j}}(t_{nt},f_{n})=\big{(}1-\delta_{i,j}\big{)}D_{i,j,n}.\] (26)  
In the SPTFDs framework, the two assumptions imply that the signal T-F signatures could not overlap strongly. For DOA estimation algorithms, the adequately different signatures represent the signal's known discriminating property. The virtues of the T-F-based polarization DOA estimation algorithm may be merely reflected when reasonable T-F points are selected for SPTFDs matrix establishment. The algorithm's key point is how to select reasonable T-F points. This paper's objective is the polarization DOA estimation of the weak nonstationary signal from strong nonstationary signal interference. The signal source SPTFDs matrix is  
\[\mathbf{D_{ss}}(t,f)= \begin{pmatrix}\mathbf{D_{s_{1}s_{1}}}(t,f)&\mathbf{D_{s_{ 1}s_{2}}}(t,f)\\ \mathbf{D_{s_{2}s_{1}}}(t,f)&\mathbf{D_{s_{2}s_{2}}}(t,f)\end{pmatrix} . \tag{27}\]  
The \(\mathbf{s_{1}}(t)\) and \(\mathbf{s_{2}}(t)\) are two polarization linear frequency modulation signals. The four types of T-F points are described. The first type of T-F point is merely in connection with auto terms. The TFDs matrix possesses a rank-one diagonal structure for those points. The second type of T-F point is merely in connection with cross terms. The TFDs matrix possesses an off-diagonal (a matrix is said to be off-diagonal if its diagonal entries are zeros) structure for those points. Because their diagonal elements are zeros, the matrix is considered to be off-diagonal. The third type of T-F point is in connection with both auto terms and cross terms, however. The auto and cross terms do not exist in the fourth type of T-F point. The TFDs matrices of the third and fourth types of T-F points have no clear algebraic structure for those points. Therefore, they can not be straightforwardly utilized. The diagonal and off-diagonal structures are frequently destroyed when the signals are mixed. For DOA estimation, the first and second types of T-F points are effective and important. Because there is effect in this situation, the third and fourth T-F points should be abnegated. When there are both strong and weak nonstationary signals, in this paper, the cross terms of SPTFDs are exploited to estimate weak nonstationary signal DOA. Because of the high outstanding off-diagonal algebraic structure of the second type of T-F points, a novelty polarization DOA estimation algorithm will exploit the SPTFDs matrix decomposition technology of T-F cross terms. The relationship between cross terms' SPTFDs and weak signal DOA is derived as follows. After cross terms extraction, the source time-frequency distribution matrix is  
\[\mathbf{D_{ssc}}(t,f)= \begin{pmatrix}\mathbf{0}&\mathbf{D_{s_{1}s_{2}}}(t,f)\\ \mathbf{D_{s_{2}s_{1}}}(t,f)&\mathbf{0}\end{pmatrix} . \tag{28}\]  
Further, the SPTFDs matrix of cross terms is obtained as follows. \[\mathbf{D_{xxc}}(t,f) = \begin{bmatrix}\widetilde{\mathbf{a}}(\phi_{1})&\widetilde{ \mathbf{a}}(\phi_{2})\end{bmatrix} \times\begin{pmatrix}\mathbf{0}&\mathbf{D_{s _{1}s_{2}}}(t,f)\\ \mathbf{D_{s_{2}s_{1}}}(t,f)&\mathbf{0}\end{bmatrix} \begin{bmatrix} \widetilde{\mathbf{a}}^{\mathrm{H}}(\phi_{1})\\ \widetilde{\mathbf{a}}^{\mathrm{H}}(\phi_{2})\end{bmatrix}