## 9 Stokes Space Polarization and Field Recovery Principle
### Polarization Recovery Techniques in the Direct Detection Stokes Vector Receiver
#### 9.1.2 Training-Assisted Polarization Characterization  
In a situation where the Stokes space signal distributions do not present discrete constellation information for the Stokes space modulation formats, certain special polarization training sequences have to be transmitted to facilitate the Stokes vector receiver in characterizing polarization variation. The three orthogonal bases in the Stokes space are the widely employed training sequences [8,13,24]. As depicted in Figure 20b, the training-assisted equalization (TA-E) scheme implementation entails the division of one training period into three time slots while the carrier power is kept constant. In this context, in the time domain, a specific training pattern can be appended before each frame of the signal. For instance, in the first time slot, while the carrier power is kept constant, no signal will be transmitted. Consequently, the Jones vector is for the slot \((0,1)\). Then, the corresponding Stokes vector for the Jones space representation of \((0,1)\) is given as, \((-1,0,0)\). Therefore, the received Stokes vector can be used to acquire the first column of the SV-RM in the first time slot as [8,24]:  
\[\left[ \begin{array}{c}S_{1}\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{ccc}r_{11}&\times&\times\\ r_{21}&\times&\times\\ r_{31}&\times&\times\end{array} \right]\left[ \begin{array}{c}-1\\ 0\\ 0\end{array} \right] \tag{48}\]  
where \(r_{ij}\) denotes the rotation matrix element of the \(i\)th row and \(j\)th column. In the second time slot, a real signal with constant power as that of the carrier can be transmitted. This gives rise to a Jones vector of \((1,1)\), and using the same approach discussed earlier, the Stokes vector of \((0,1,0)\) will be obtained. Based on this, the second column of the SV-RM can be acquired. Furthermore, in the third time slot, the Jones vector of \((i,1)\) resulting in the Stokes vector of \((0,0,1)\) can be employed to acquire the third column of the SV-RM. Therefore, at the transmitter, three orthogonal training symbols with the Jones space representation \((0,1)\), \((1,1)\), and \((i,1)\) with the three corresponding Stokes vector bases \((-1,0,0)\), \((0,1,0)\), and \((0,0,1)\), respectively, are launched. It should be noted that the training symbols' transmitted Stokes vector (\(\mathbb{T}\)) and the training symbols' received Stokes vector (\(\mathbb{R}\)) are related by:  
\[\mathbb{R}=\mathbf{\mathcal{H}}\cdot\mathbb{T}, \tag{49}\]  
where \(\mathbf{\mathcal{H}}\) is the \(3\times 3\) rotation matrix in the Stokes space. Figure 21: Adaptive algorithm procedure. Therefore, the transmitted Stokes vector can be realized by multiplying the received Stokes vector with the inverse of the acquired rotation matrix as (\(\mathbb{T}=\mathbf{\mathcal{H}}^{-1}\cdot\mathbb{R}\)) [8; 24]:  
\[\left[ \begin{array}{c}|X_{T}|^{2}-|Y_{T}|^{2}\\ 2\Re(X_{T}\cdot Y_{T}\pi^{*})\\ 2\Im(X_{T}\cdot Y_{T}\pi^{*})\end{array} \right]=\left[ \begin{array}{ccc}r_{ 11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{array} \right]^{-1}\cdot\left[ \begin{array}{c}|X_{R}| ^{2}-|Y_{R}|^{2}\\ 2\Re(X_{R}\cdot Y_{R}\pi^{*})\\ 2\Im(X_{R}\cdot Y_{R}\pi^{*})\end{array} \right] \tag{50}\]  
where \(T\) and \(R\) denote the transmitter and receiver, respectively, and \((\cdot)^{-1}\) is the inverse operation. Furthermore, based on the estimation of the Stokes vector rotation matrix (\(\mathbf{\mathcal{H}}=\mathbb{R}\cdot\mathbb{T}^{-1}\)), the received signal Stokes vector can be recovered back to the transmitted polarization state as \(\mathbf{\mathcal{T}}=\mathbf{\mathcal{H}}^{-1}\cdot\mathbf{\mathcal{R}}\). Moreover, when a polarization division multiplexed signal is sent at the transmitter, the entire Stokes vector components will be the second order term of the signal, which results in a nonlinear channel response. Therefore, the signal will be considerably impaired through the spectrum power fading in the presence of the chromatic dispersion.