# Wideband Mixed Signal Separation Based on Photonic Signal Processing
## 2 Interference Management
### Related Work
#### 2.1.2 Digital System  
The digital system measures the statistical pattern with the under-sampled signal and controls the photonic circuits with tunable weights and delays (TD and TA in Figure 2) to perform the separation. Figure 3 shows the mixed signal \(x_{1}\) and \(x_{2}\). Both \(s_{oi}\) and \(s_{int}\) have Gaussian distribution. The BSS is to solve the inverse of the mixing matrix \(A\) and include two steps: PCA and ICA [20; 21; 22], symbols used in this section are listed in Table 1 in Appendix A.  
\[A^{-1}=VU\Sigma U^{-1} \tag{2}\]  
where \(U\Sigma U^{-1}\) represents the PCA, and \(V\) represents the ICA. To perform PCA, the 2 nd order moments of the mixed signal with different weights are calculated. By using the tunable optical attenuators in the photonic circuit (TA in Figure 2), the following weight is added to the mixed signal:  
\[x_{PCA}=\cos(\theta)x_{1}+\sin(\theta)x_{2} \tag{3}\]  
where \(\theta\) is defined in Figure 3. The second order moment of \(x_{PCA}\) is a function of \(\theta\) (red curve in Figure 3), and is in the form of:  
\[E\Big{(}x_{PCA}^{2}\Big{)}=q_{1}+q_{2}\cos[2(\theta-\theta_{0})] \tag{4}\]  
where \(E\big{(}x_{PCA}^{2}\big{)}\) is the expectation or time average of \(x_{PCA}^{2}\),which is referred to as a two-petal epitrochoid, has a clear relation to the principal components [22]. The first principal components vector angle (direction) is \(\theta_{0}\), and the second is orthogonal. The \(q\) and \(\theta\) parameters describe all the covariance properties of the joint distribution. \(q_{1}+q_{2}\) is the magnitude of the first component, and \(q_{1}-q_{2}\) is the magnitude of the second principal component. The second order moment \(E\big{(}x_{PCA}^{2}\big{)}\) can be measured with different weights \([\cos(\theta),~{}\sin(\theta)]\) added to the mixed signal \(x_{1}\) and \(x_{2}\), and with three random \(\theta\) values. The diagonal elements for \(\Sigma\) should be the principal components of the PCA, \(q_{1}+q_{2}\) and \(q_{1}-q_{2}\). After being normalized, the first element is 1 and last element is \(\sqrt{(q_{1}+q_{2})/(q_{1}-q_{2})}\) \begin{table}
\begin{tabular}{c c} \hline
**Interference Bandwidth** & **Cancellation Ratio** \\ \hline
200 MHz & 60 dB \\
1 GHz & 36 dB \\
3 GHz & 30 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Photonic interference cancellation ratio at different bandwidths. With three random \(\theta\) values, the three unknown parameters \(q_{1}\), \(q_{2}\), and \(\theta_{0}\) can be determined. PCA and whitening are performed with \(q_{1}\), \(q_{2}\), and \(\theta_{0}\):  
\[U=\left[ \begin{array}{cc}\cos(\theta_{0})&-\sin(\theta_{0})\\ \sin(\theta_{0})&\cos(\theta_{0})\end{array} \right] \tag{5}\]  
\[\Sigma=\left[ \begin{array}{cc}1&0\\ 0&\sqrt{\frac{q_{1}+q_{2}}{q_{1}-q_{2}}}\end{array} \right] \tag{6}\]  
After whitening, the signal is changed to \(X^{\prime}=[x^{\prime}1,\;\;x^{\prime}2]=U\Sigma U^{-1}X\), where \(U\) is the rotation matrix by angle \(\theta_{0}\), \(\Sigma\) is rectangular diagonal matrix, and \(X\) the original received signal, \(x^{\prime}1\) and \(x^{\prime}2\) mean the whitened received signal from Receiver 1 and 2, \(\cos(\phi)\) and \(\sin(\phi)\) are the corresponding weights from each receiver. To perform ICA, fourth-order moments (kurtosis) of the whitened signal, which take the form of a four-patal epitrochoid, are needed, with added weight:  
\[x_{ICA}=\cos(\phi)x^{\prime}1+\sin(\phi)x^{\prime}2 \tag{7}\]  
\[\left(x^{4}_{ICA}\right)=p_{1}+p_{2}\cos[2(\phi-\phi_{0})]+p_{3}\cos[4(\phi- \phi_{0})] \tag{8}\]  
The fourth-order moment can be parameterized by weight vector angle using unit normal weight vectors, \(\phi_{0}\) is set as the new principal component vector angle. With four random \(\phi\) values, the unknown parameters \(p_{1}\), \(p_{2}\), \(p_{3}\) and \(\phi_{0}\) are determined, and ICA can be performed:  
\[V=\left[ \begin{array}{cc}\cos(\phi_{0})&-\sin(\phi_{0})\\ \sin(\phi_{0})&\cos(\phi_{0})\end{array}