## 2 Principle
### STBC-Assisted RLS Algorithms  
One of the primary issues regarding the conventional RLS implementation might be its comparatively sophisticated computational complexity, attributable to the recursive updating with a growing memory. Henceforth, in order to accomplish a quasi-RLS performance at a quasi-LMS cost, the STBC technique can be applied to aid the RLS approach, which is defined as follows: assuming multiple copies of data stream are transmitted across a number of spatial modes in an FMF transmission, some of the received copies are better than the others under the same scattering, reflection, or thermal noise; thus, the various received versions of the data can be exploited to increase the overall system reliability [30]. The space and time allocation for the STBC-aided RLS algorithm can be briefly represented as shown in Figure 7, as a matrix of spatial channels and time slots, whereby each element is the modulated symbol to be transmitted in mode \(i\) at time slot \(j\). We may choose different combinations, and we can apply one or more of received copies to properly decode the received signal and, therefore, achieve better performance. The architecture of the adaptive STBC-RLS equalizer in an m-mode FMF transmission system is illuminated in Figure 8. At coherent detection, the serial-to-parallel (S/P) converters split each data sequence \(\mathrm{y_{1}},\ldots\mathrm{y_{m}}\) into even/odd sequences, while two consecutive blocks are concatenated. After the FFT, the samples of the k-th block are overlapped with the (k \(-\) 1)th block at frequency k, which are further alienated into sequential blocks \(\mathrm{y_{q}^{P}(k)}\), where superscript P specifies an even or odd sequence. Each block is then converted to the frequency domain with FFT as \(\mathrm{Y_{q}^{P}(k)}\), which goes through an inversed channel filter for the data matrix production with the even or odd tributaries, while \(\mathrm{H_{qj}^{P}(k)}\) stands for the inversed channel filter in Jones-vector notation, while both q and j denote mode indices between 1 and m. After an adaptive MIMO equalization, the carrier recovery is applied to mitigate  
Figure 6: Schematic of an MDM system utilizing the MIMO FDE [31]. the laser phase noise, the outputs of which are transformed back to the time domain using inverse FFT (IFFT), compared to the desired response to generate an error vector, and to update the receiver coefficients in a training mode until they converge. Then, it switches to a decision-directed mode, whereby reconstructed data are transformed back using FFT through the zero padding to update the coefficients [31]. The weight update of the STBC-aided RLS algorithm follows the rules below, whereas the corresponding equalizer coefficients are updated every two blocks (\(\mathrm{k}+2\)) instead of each block (\(\mathrm{k}+1\)) in the recursion. \[\mathrm{W}(k+2)=\mathrm{W}(k)+\left( \begin{array}{cc}\mathrm{P}_{k+2}&0\\ 0&\mathrm{P}_{k+2}\end{array} \right)(\mathrm{U}_{k+2})^{*}\cdot[\mathrm{D}_{k+ 2}-\mathrm{U}_{k+2}\cdot\mathrm{W}(k)], \tag{9}\]  
where \(\mathrm{U}_{k+2}\) denotes the Alamouti matrix containing the received symbols from block \(\mathrm{k}+2\) and \(\mathrm{k}+3\), and \(\mathrm{D}_{k+2}\) represents the desired response vector for training and decision-directed tracking [32], while the superscript \({}^{*}\) indicates the complex conjugation. Meanwhile, the diagonal term \(\mathrm{P}_{k+2}\) is given by  
\[\mathrm{P}_{k+2}=\beta^{-1}\cdot\left[\mathrm{P}_{k}-\beta^{-1}\cdot\mathrm{P} _{k}\cdot\Omega_{k+2}\cdot\mathrm{P}_{k}\right], \tag{10}\]  
where \(\beta\) is the forgetting factor, and the term \(\Omega_{k+2}\) is given by  
\[\Omega_{k+2}=\left\{\mathbf{Diagonal}[|\mathrm{V}(k)|^{2}+|\mathrm{V}(k+1)|^{ 2}]^{-1}+\beta^{-1}\cdot\mathrm{P}_{k}\right\}^{-1}, \tag{11}\]  
where \(\mathrm{V}(\mathrm{k})\) stands for the FFT of sampled received signals.