## 4 Performance Evaluation for Different Adaptive Filtering Schemes  
In this section, the convergence speed comparison amongst the conventional LMS algorithm, the noise PSD directed (NPD) LMS algorithm, the RLS algorithm, and the proposed STBC-RLS algorithm for PDM-QPSK signal is exemplified in Figure 13, whereas the normalized mean square error (NMSE) represents the average square errors of each block after the MIMO equalization. The adaptation rate and step size of four algorithms were set to their optimal values; the block length was 8192 samples using 50% overlap with OSNR set at 18 dB. In the meantime, the inset on the right displays the constellation diagram before the STBC compensation, as the received signal was severely distorted, attributable to mode coupling and delay, while the inset on the left indicates the constellation diagram after applying the STBC-RLS compensation, as the modal distortions were remarkably compensated for by the MIMO equalizers over a 30-km FMF. As for the computational complexity per data symbol for the MIMO FDE, which is mainly determined by the number of complex multiplications, the RLS and LMS algorithms would consume 148 and 302 complex multiplications, respectively, while the proposed STBC-RLS would require 172 complex multiplications, which is roughly a 16.2% increase in hardware complexity compare to LMS, but 75.6% less than that of the RLS approach. To further upsurge the spectral efficiency of the next-generation SDM transmission systems, the convergence speed comparisons between different algorithms for PDM-16 QAM and PDM-64 QAM signals are presented in Figures 14 and 15, respectively, whereas the comparison between different modulation formats is summarized in Table 1.  
From the plot above, we can see that RLS converges faster to a lower asymptotic NMSE than LMS, because it has a growing memory due to the forgetting factor. The NPD-LMS algorithm can achieve a faster convergence than the traditional LMS, because it adopts variable bin-wise step size to render posterior error of every frequency bin convergent to the background noise of the AWGN channels. Still, the NMSE convergence of the proposed STBC-RLS algorithm seems a bit inferior to that of the conventional RLS in the beginning, for a smaller block size is used for training than the data block size to reduce the system overhead at the expense of a minor loss of performance. Subsequently, to  
Figure 13: The normalized mean square error (NMSE) vs. the number of fast Fourier transform (FFT) blocks for various algorithms for a polarization-division multiplexing (PDM) quadrature phase shift keying (QPSK) signal; inset: constellation diagrams before (**right**) and after (**left**) the STBC-RLS compensation [27]. achieve \(-10\) dB steady NMSE, which is equal to a 9.8-dB Q-value, the LMS and NPD-LMS schemes need 55 and 47 FFT blocks, while the RLS and STBC-RLS approaches require roughly 27 and 31 blocks, respectively; henceforward, the convergence rate would be enhanced by 50.9% and 43.6%, respectively. From the plots, we can see that with higher-order modulation formats, the advantage of RLS convergence rate over that of the LMS becomes even larger, owing to a growing memory, while the difference in NMSE between the proposed STBC-RLS algorithm and the conventional RLS shrinks, which indicates that such proposed adaptive receivers could lower system overhead requirements for higher-order modulation formats. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Modulation** & **\# of Symbols** & **Levels per** & **\# of Bits per Symbol** & **Corresponding** \\
**Format** & **Transmitted** & **Carrier** & **Contained** & **Data Rate** \\ \hline
**QPSK** & \(4=2^{2}\) & 2 & 2 & 112 Gb/s \\
**16 QAM** & \(16=2^{4}\) & 4 & 4 & 224 Gb/s \\
**64 QAM** & \(64=2^{6}\) & 8 & 6 & 336 Gb/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of multiple modulation formats [21]. QPSK—quadrature phase shift keying; QAM—quadrature amplitude modulation.