## 3 Time-Frequency Point Selection
### Time-Frequency Points Selection Procedures  
By selecting cross term T-F points, the data cross-source TFDs will have the following structure:  
\[\underline{\mathbf{D}}_{\mathbf{xxc}}(t,f)=\mathbf{U}\mathbf{D}_{\mathbf{ssc}} (t,f)\mathbf{U}^{\mathrm{H}}, \tag{30}\]  
where \(\mathbf{D}_{\mathbf{ssc}}(t,f)\) is antidiagonal. The missing unitary matrix \(\mathbf{U}\) is uniquely retrieved by the joint antidiagonalization (JAD) of a combined set \(\{\underline{\mathbf{D}}_{\mathbf{xxc}}(t_{i},f_{i})|i=1,\ldots,q\}\) of \(q\) cross-source TFDs matrices. The JAD is explained by first noting that the problem of the antidiagonalization of a single \(N\times N\) matrix \(\mathbf{N}\) is equivalent (this is due to the fact that the Frobenius norm of a matrix is constant under unitary transform) to the maximization of the criterion  
\[C(\mathbf{N},\mathbf{V})\stackrel{{\mathrm{def}}}{{=}}-\sum_{i=1 }^{N}\left|\mathbf{v}_{i}^{H}\mathbf{N}\mathbf{v}_{i}\right|^{2}, \tag{31}\]  
over the set of unitary matrices \(\mathbf{V}=[\mathbf{v}_{1},\cdots,\mathbf{v}_{n}]\). Hence, the JAD of a set \(\{\mathbf{N}_{k}\mid k=1,\ldots,q\}\) of \(q\) matrices is defined as the maximization of the JAD criterion,  
\[C(\mathbf{V})\stackrel{{\mathrm{def}}}{{=}}\sum_{k=1}^{q}C( \mathbf{N}_{k},\mathbf{V})=-\sum_{k=1}^{q}\sum_{i=1}^{N}\left|\mathbf{v}_{i}^ {H}\mathbf{N}_{k}\mathbf{v}_{i}\right|^{2}, \tag{32}\]  
under the same unitary constraint. A Jacobi-like algorithm has been derived for the maximization of the JAD criterion (32). The specific process is described below. (1) Determine the whitening matrix \(\hat{\mathbf{W}}\) from the eigendecomposition of an estimate of the data covariance matrix. (2) Compute the TFDs of the array output according to (22). (3) Select a set of T-F points (usually corresponding to the high amplitude points of the signal T-F transform), and then distinguish between auto term and cross term points using the above selection procedure. (4) Determine the unitary matrix \(\hat{\mathbf{U}}\) by maximizing the JD/JAD criterion applied to the whitened TFD matrices computed at the selected T-F points. (5) Obtain an estimate of the mixture matrix \(\hat{\mathbf{A}}\) as \(\hat{\mathbf{A}}=\hat{\mathbf{W}}^{\mathrm{H}}\hat{\mathbf{U}}\), where the superscript \(\#\) denotes the pseudoinverse, and an estimate of the source signals \(\hat{\mathbf{s}}(t)\) as \(\hat{\mathbf{s}}(t)=\hat{\mathbf{U}}^{H}\mathbf{W}\mathbf{x}(t)\). The success of the JD or JAD of TFDs matrices in determining the unitary matrix \(\mathbf{U}\) depends strongly on the correct selection of the auto and cross terms points. It is crucial to have a selection procedure that is able to distinguish between auto and cross terms points based only on the TFDs matrices of the observations. Here, we propose a selection approach that exploits the antidiagonal structure of the cross-source TFDs matrices. More precisely, we have  
\[\mathrm{Trace}(\underline{\mathbf{D}}_{\mathbf{xxc}}(t,f))=\mathrm{Trace} \Big{(}\mathbf{U}\mathbf{D}_{\mathbf{ssc}}(t,f)\mathbf{U}^{H}\Big{)}=\mathrm{ Trace}(\mathbf{D}_{\mathbf{ssc}}(t,f))\approx 0. \tag{33}\]  
The T-F points' automatic selection procedure is generally difficult, as described in the following. In a whitening case, a matrix's trace invariance for a unitary transform is utilized to judge the cross terms' existence. The matrices for the second type of T-F points require the following condition:  
\[\frac{\mathrm{trace}\Big{\{}\underline{\mathbf{D}}_{\mathbf{xx}}\Big{(}k_{t}, k_{f}\Big{)}\Big{\}}}{\Big{\|}\underline{\mathbf{D}}_{\mathbf{xx}}\Big{(}k_{t}, k_{f}\Big{)}\Big{\|}}<\epsilon, \tag{34}\]  
where \(\mathrm{trace}|\) is matrix trace, \(\epsilon\) is a positive small user-defined value and \(\|\|\) is the Frobenius norm. The noise influence is further reduced by averaging the selected SPTFDs of cross terms. Most procedures take advantage of a diagonal rank-one matrix for the auto terms. The emphasis of this paper is to make use of the off-diagonalization characteristic of the cross terms matrix.