# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 3 Robust Whitening  
The conventional whitening process utilizes the zero time delay correlation matrix of the observed signal, which cannot eliminate the effect of additive noise. In [31; 32], a robust whitening algorithm is proposed using the property that the time delay correlation matrix is insensitive to white noise. The time delay correlation matrix of the observed signal is  
\[\mathbf{R}_{x}(\tau)=E\biggl{[}\mathbf{x}(t)\mathbf{x}^{H}(t-\tau)\biggr{]}=\mathbf{A}\mathbf{R}_{ s}(\tau)\mathbf{A}^{H},\tau\
eq 0 \tag{14}\]  
where \(\tau\) is the time delay and \(\mathbf{R}_{s}(\tau)\) is the time delay correlation matrix of the source signal. Since the signals are all complex, \(\tilde{\mathbf{R}}_{x}(\tau)\) should be replaced by the ergodic matrix, i.e.,  
\[\tilde{\mathbf{R}}_{x}(\tau)=\frac{1}{2}\Bigl{(}\mathbf{R}_{x}(\tau)+\mathbf{R}_{x}^{H}( \tau)\Bigr{)} \tag{15}\]  
However, \(\tilde{\mathbf{R}}_{x}(\tau)\) is not a positive definite matrix. The literature [32] demonstrated that when the number of delay-dependent matrices and observed signals is equal, a linear combination of multiple delay-dependent matrices can form a positive definite matrix, i.e.,  
\[\mathbf{C}=\sum\limits_{m=1}^{M}\beta_{m}\mathbf{R}_{x}(\tau_{m}) \tag{16}\]  
where \(\mathbf{\beta}=[\beta_{1},\beta_{2},\cdots,\beta_{M}]\in\mathbb{R}^{K}\) is the real vector, i.e., the coefficients of the time delay correlation matrices. \(\tau_{m}(1\leq m\leq M)\) is the time delay sequence. Thus, the key to the algorithm lies in how to determine \(\mathbf{\beta}\). \(\tilde{\mathbf{R}}_{x}(\tau_{m})\) is an ergodic matrix, so \(\mathbf{C}=\sum\limits_{m=1}^{M}\beta_{m}\tilde{\mathbf{R}}_{x}(\tau_{m})\) is also an ergodic matrix. The minimum eigenvalue of the positive definite matrix should be greater than 0. A new method of finding \(\mathbf{\beta}\) is as follows. Let the set \(\mathcal{C}\) be all linear combinations of \(\tilde{\mathbf{R}}_{x}(\tau_{m})\), i.e.,  
\[\mathcal{C}=\left\{\sum\limits_{k=1}^{K}\beta_{k}\tilde{\mathbf{R}}_{x}(\tau_{k}) \mid\beta\in\mathbb{R}^{K}\right\} \tag{17}\]  
When \(\mathbf{C}=\sum\limits_{k=1}^{K}\beta_{k}\mathbf{R}_{x}(\tau_{k})\) is a positive definite matrix, the minimum eigenvalue of \(\mathcal{C}\) should be no less than a given threshold \(\sigma\in\mathbb{R}^{+}\), i.e., \(\mathcal{O}_{\sigma}^{+}\cap\mathcal{C}\
eq\varnothing\), where \(\mathcal{O}_{\sigma}^{+}\) represents a set of minimum eigenvalue matrices. It is equivalent to minimizing the following optimization problem. \[\min F(\mathbf{O},\mathbf{C})=\parallel\mathbf{O}-\mathbf{C}\parallel,\ \mathbf{O}\in\mathcal{O}_{ \sigma}^{+},\mathbf{C}\in\mathcal{C} \tag{18}\]  
\(\mathcal{H}=span\bigl{\{}\tilde{\mathbf{R}}_{x}(\tau_{1}),\cdots,\tilde{\mathbf{R}}_{ x}(\tau_{K})\bigr{\}}\) is the subspace of \(\tilde{\mathbf{R}}_{x}(\tau_{k})\) and \(\mathcal{O}_{\mathcal{H}}(\mathbf{O})\) is the orthogonal projection operator from \(\mathbf{O}\) to \(\mathcal{H}\). \(\mathbb{A}\) is the \(M^{2}\times K\) tensor, i.e.,  
\[\mathbb{A}=\bigl{[}vec\bigl{(}\mathbf{R}_{x}(\tau_{1})\bigr{)},\cdots,vec\bigl{(} \mathbf{R}_{x}(\tau_{K})\bigr{)}\bigr{]} \tag{19}\]  
\(\mathcal{O}_{\mathcal{H}}\) is  
\[\mathcal{O}_{\mathcal{H}}=\mathbb{A}\Bigl{(}\mathbb{A}^{H}\mathbb{A}\Bigr{)}^{ -1}\mathbb{A}^{H} \tag{20}\]Initialize \(\mathbf{C}\in\mathbf{C}\), \(\mathbf{\beta}=[1,1,\cdots,1]\), and calculate the eigenvalue decomposition \([\mathbf{U}_{C},\mathbf{\Lambda}_{C}]\) of \(\mathbf{C}\). If the minimum eigenvalue is less than the given threshold \(\sigma\), then update \(\mathbf{O}\) and \(\mathbf{C}\), respectively, according to the following equation. \[\mathbf{\Lambda}_{\text{P}}=\text{max}(\mathbf{\Lambda}_{\text{C}},\sigma\mathbf{I}) \tag{21}\]  
\[\mathbf{O}=\mathbf{U}_{\text{C}}\mathbf{\Lambda}_{\text{o}}\mathbf{U}_{\text{C}}^{H} \tag{22}\]  
\[\mathbf{C}=vec^{-1}(O_{\mathcal{H}}(\mathbf{O})) \tag{23}\]  
where \(\mathbf{I}\) is the unit matrix, until the minimum eigenvalue of \(\mathbf{C}\) is greater than or equal to the given threshold \(\sigma\).