# True Equalization of Polarization-Dependent Loss in Presence of Fast Rotation of SOP +
## 3 The True Equalization of PDL in the Presence of Fast RSOP
### The Power Equalization and RSOP Tracking
For the RSOP compensation matrix, we choose the three-parameter model, which represents the real RSOP in the optical fiber [23], instead of the two-parameter one that has been misused for a period of time [3; 4; 5; 6]. \[\mathbf{R}_{RSOP}=\left( \begin{array}{cc}cos\kappa e^{-j\xi}& sin\kappa e^{j\eta}\\ -sin\kappa e^{-j\eta}&cos\kappa e^{j\xi}\end{array} \right)\] (10) where \(\kappa\), \(\xi\), and \(\eta\) are three time-varying angles. 2. Based on the RSOP compensation matrix in Equation (10), we decide the state vector as \[\mathbf{x}=(\kappa,\eta,\xi)^{T}\] (11) It reflects the three independent parameters of RSOP. 3. The choice of measurement vector \(h(\mathbf{x})\) is based on the location of the ideal constellation points after compensation. As for the Golden-coded QPSK signal, the ideal constellation points are located at three rings with the radii \(r_{1}\), \(r_{2}\), and \(r_{3}\), as shown in Figure 5b. Thus, we design \(h(\mathbf{x})\) as \[h(\mathbf{x})=\left[ \begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array} []{c}u_{x}u_{x}^{*}-r_{1}^{2}\end{array}\end{array}\end{array}\end{array}  \end{table}
Table 1: The choice of measurement vector \(h(\mathbf{x})\) is based on the location of the ideal constellation points after compensation. As for the Golden-coded QPSK signal, the ideal constellation points are located at three rings with the radii \(r_{1}\), \(r_{2}\), and \(r_{3}\), as shown in Figure 5b. Thus, we design \(h(\mathbf{x})\) as \[h(\mathbf{x})=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c} \begin{array} []{c}u_{x}u_{x}^{*}-r_{2}^{2}\end{array}   
combines the modulated symbols on two orthogonal polarizations. Here, we focus on the Golden code, a kind of polarization-time code whose coding matrix is:  
\[\mathbf{M}=\frac{1}{\sqrt{5}}\left[ \begin{array}{cc}\alpha(a+\theta b)&\alpha(c+ \theta d)\\ i\overline{a}\Big{(}c+\overline{\theta}d\Big{)}&\overline{\alpha}\Big{(}a+ \overline{\theta}b\Big{)}\end{array} \right] \tag{14}\]  
where \(\theta=\Big{(}1+\sqrt{5}\Big{)}/2\), \(\overline{\theta}=\Big{(}1-\sqrt{5}\Big{)}/2\), \(\alpha=1+i-i\theta\), \(\overline{\alpha}=1+i-i\overline{\theta}\), and \(a,b,c,d\) are data symbols. The first row represents the code modulated on the x-polarization and the second row represents the code on the orthogonal y-polarization. The Golden code does not introduce any redundancy compared to the uncoded case, because four symbols are transmitted on two polarization tributaries during two symbol periods. At the receiver, the optimal method to recover the information data is the maximum likelihood (ML) decoder, which is to estimate the transmitted symbols through minimizing the following distance with the received symbols \(\mathbf{Y}\). \[\widetilde{\mathbf{M}}=\operatorname*{argmin}_{\mathbf{M}\in A}\lVert\mathbf{Y}-\mathbf{H} \mathbf{M}\rVert^{2} \tag{15}\]  
where A is the alphabet and \(\mathbf{H}\) is the channel state information matrix, which need to be obtained at the receiver. Now, the three steps have been completed: Obtaining the decoupling model when RSOP and PDL are co-existing, equalizing the power imbalance in Stokes space combined with tracking RSOP by the Kalman filter, and recovering BER asymmetry using the polarization-time code. Therefore, the PDL in the presence of RSOP is truly equalized.