## 9 Stokes Space Polarization and Field Recovery Principle
### Polarization Recovery Techniques in the Direct Detection Stokes Vector Receiver
#### 9.1.1 Adaptive MIMO Equalization  
An extensively employed polarization recovery technique for the PDM-COH systems is an adaptive MIMO equalization. Furthermore, to facilitate a linear-equalization-based polarization recovery in the Stokes space modulation systems with DD-SVR, the associated signals have to be linear to the Stokes parameters. For instance, a \(3\times 2\) MIMO equalizer is usually applied after Stokes vector detection in the PDM-PAM system. This helps the recovery of PAM signals that are modulated on the \(|X|^{2}\) and \(|Y|^{2}\), which are analogous to \(S_{0}\) and \(S_{1}\). Comparable with some 3D Stokes space modulation formats in which MIMO equalization is modified for the polarization recovery, the Stokes space signals can present a relaxed condition by being finite discrete values that can support digital equalization. Moreover, regarding the Stokes space field recovery, the SCOH-SSM is capable of employing a \(3\times 2\) equalizer for the polarization recovery. This is as a result of the complex signal that is linearly modulated on \(S_{2}\) and \(S_{3}\)[13].  
An instance of adaptive equalization is the blind adaptive equalization (BA-E), which is another viable approach for tracking the polarization rotation in the SV-DD scheme [24; 39; 146]. As illustrated in Figure 20a, this can be realized through the simultaneous implementation of residual chromatic dispersion and other linear distortion compensations along with the complex signal development. The adaptive equalizer is based on the least mean squares algorithm. The three inputs (see Figure 20a) are to obtain the three components of the received Stokes vector [11]. The signal \(s(k)\) is the obtained output after polarization rotation, where \(k\) is the time index. The equalizer coefficients \((c_{1},c_{2}\) and \(c_{3})\) are related to the rotation matrix inverse \(\left(\text{RM}\right)^{-1}\) as \(c_{n}=h_{2n}+jh_{3n}\) (\(n=1,2,3\)), where \(h_{mn}\) denotes the matrix element of \(\left(\text{RM}\right)^{-1}\) of the \(m\)th row and \(n\)th column [24; 39].  
The procedures involved in the adaptive algorithm can be broadly classified into the steps depicted in Figure 21[11; 24; 39]. The instantaneous squared error is minimized by the algorithm at new time \(k+1\). Then, the filter coefficients' deviation from the previous evaluation at time \(k\) is minimized as [11; 24]:  
\[\min_{c(k+1)}\left\{\|e(k+1)\|^{2}+\left(1/(\mu-1)\right)\|\mathbf{\mathcal{R}}(k+ 1)\|^{2}\|c(k+1)-c(k)\|^{2}\right\} \tag{47}\]  
Moreover, an experimental implementation of the algorithm for a single-carrier SV-DD was demonstrated in [39].  
Figure 20: SV-RM (**a**) blind adaptive equalization (BA-E) and (**b**) SV-RM training symbol structures (adapted from [11; 24]).  
The filter coefficients are first initialized at time \(k\). This is represented by \(c_{n}(k)\), \((n=1,2\) or \(3)\), with only the central tap weight set to unitary, while all other tap weights are set to zero. To speed up the system convergence, the training-assisted equalization scheme is usually adopted for the initialized coefficients decision;
* The received Stokes vector can then be equalized by \(s(k)=[c_{1},c_{2},c_{3}]\cdot[S_{1},S_{2},S_{3}]^{T}\);
* The instantaneous error, \(e(k)=d(k)-s(k)\), is estimated, where \(d(k)\) denotes the constellation point that is closest to \(s(k)\);
* The filter coefficients are then updated as \(c_{n}(k+1)=c_{n}(k)+\mu\cdot e(k)\cdot S_{n}(k)/\left\|\mathbf{\mathcal{R}}(k) \right\|^{2}\)\((n=1,2,3)\), where \(\mu\) denotes the adaptive filter convergence parameter (step size), \(\left\|\cdot\right\|^{2}\) represents the \(L_{2}\)-norm of the braced vector, and \(\mathbf{\mathcal{R}}(k)=[S_{1}(k),S_{2}(k),S_{3}(k)]\) is the received SV at the \(k\)th sampling point.