# Novel Equalization Techniques for Space Division Multiplexing Based on Stokes Space Update Rule
## 3 Equalizers for SDM
### Single-Mode 2 \(\times\) 2 Dynamic Equalization
#### 3.1.2 Stokes Space Algorithm  
The Stokes space transformation has been actively used for characterizing signal and its impairments [22]. The Stokes transformation converts the 2-D complex Jones space, \(\{x,y\}\in\Im\), into a 3-D real space: \(\{S_{1},S_{2},S_{3}\}\in\mathbb{R}\):  
\[\begin{array}{l}S_{1}(n)=|x(n)|^{2}-|y(n)|^{2},\\ \\ S_{2}(n)=2\Re\{x(n)y^{*}(n)\},\\ \\ S_{3}(n)=2\Im\{x(n)y^{*}(n)\}.\end{array} \tag{6}\]  
Although one dimension is lost in the Stokes transformation, its transformation can be effectively used for tracking signal impairments. Introduced by [6], the Stokes Space Algorithm (SSA) equalizer takes advantage of the Stokes transformation to calculate the error function in that domain. As a consequence of this non-injective transformation, the number of points of the constellation of the transformed stokes space are reduced by a factor of approximately four [23].  
The advantages of Stokes space is that it is insensitive to frequency offsets and phase rotations simultaneously introduced in both polarizations [6]. The SSA computes the error function as described by:  
\[f_{SSA}(h(n))=\left|\left|\overline{S}(n)-\hat{S}(n)\right|\right|^{2}, \tag{7}\]  
where \(\overline{S}(n)\) and \(\hat{S}(n)\) are the vectors of the equalized and expected signal respectively expressed in the Stokes space.  
For illustrative purposes, we include the comparison between the constellations in the Euclidean space, Figure 1a-c, and in the Stokes space, Figure 1d-f using the representation proposed by [23].  
Moreover, the update coefficients of each polarization can be written as:  
\[\begin{split} C_{1}(n)&=4\big{[}\overline{S}_{1}( n)-\hat{S}_{1}(n)\big{]}\overline{x}(n)+4\big{[}\big{(}\overline{S}_{2}(n)- \hat{S}_{2}(n)\big{)}+\sqrt{-1}\big{(}\overline{S}_{3}(n)-\hat{S}_{3}(n) \big{)}\big{]}\overline{y}(n),\\ C_{2}(n)&=4\Big{[}\big{(}\overline{S}_{2}(n)-\hat{ S}_{2}(n)\big{)}-\sqrt{-1}\big{(}\overline{S}_{3}(n)-\hat{S}_{3}(n)\big{)} \Big{]}\overline{x}(n)-4\big{[}\overline{S}_{1}(n)-\hat{S}_{1}(n)\big{]} \overline{y}(n).\end{split} \tag{8}\]  
Figure 1: Examples of different single- and dual-polarization (SP and DP) constellations in the Euclidean and Stokes space. Since the Stokes transformation is a 3-D real space, colors are used to identify \(S_{1}\) dimension. We plot only one dimension of the Euclidean space since it is a 2-D complex space where both dimensions have the same constellation. (**a**) SP-QPSK in Euclidean space; (**b**) SP-16 QAM in Euclidean space; (**c**) SP-64 QAM in Euclidean space; (**d**) DP-QPSK in Stokes space; (**e**) DP-16 QAM in Stokes space; (**f**) DP-64 QAM in Stokes space (legend is omitted and colors are repeated for different cases of \(S_{1}\)).  
Since the Stokes transformation is a nonlinear transformation, the DD algorithm based on minimum distance results in a suboptimal implementation. Alternatively, a maximum likelihood (ML) rule can be applied at the expense of increasing computational complexity [24]:  
\[\hat{S}(n)=\max_{k}^{-1}f_{S|\mathbf{S}(k)}\Big{(}\overline{S}(n)\Big{|}\mathbf{S}_{(k) }\Big{)}=\max_{k}^{-1}[-\frac{\left|\left|\overline{S}_{(k)}\right|\right|}{2 \sigma^{2}}+logl_{0}\left(\frac{\sqrt{\left|\left|\mathbf{S}_{(k)}\right|\right| \left|\overline{S}(n)\right|\right|}}{\sigma^{2}}\cos\left(\frac{\theta_{(k)}} {2}\right)\right)]\,, \tag{9}\]  
where \(I_{0}\) is the modified Bessel function of order zero, \(\theta_{(k)}\) is the angle between the Stokes vectors \(\mathbf{S}_{(k)}\) and \(\overline{S}(n)\), and \(\sigma^{2}\) is the noise variance. Simplified approximations of the previous formula are also available for decreasing the computational complexity [6].