## 4 Results and Discussion
### Parameters' Optimization of the Extreme Learning Machine  
Initially, we discover the parameters (the number of hidden nodes, regularization parameter, and activation function) of the proposed ELM to avoid the system degradation owing to laser phase noise. AWGN is discarded for the moment. Taking into account the SIG activation function, Figure 3 depicts the contour plots of the BER metric as a function of the regularization parameter and number of hidden neurons according to \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Subcarrier modulation format & BPSK, QPSK, and 16 QAM \\ Bit rate & 10 Gbps \\ Number of data & 112 \\ Number of pilots & 16 \\ FFT size & 128 \\ CP length & 1/10 \\ Sample rate & 100 Gsps \\ Number of bits & \(10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used for the CO-OFDM signal. Figure 2: Structure of the ELM equalizer. (a-c) BPSK, (d-f) QPSK, and (g-i) 16 QAM constellations. This function guarantees the universal approximation capability of SLFNs under any learning algorithm, such as the ELM [10; 20]. Throughout the manuscript, the input weights and biases are generated from the range \([-1,1]\) based on uniform distribution [24; 41]. For each subcarrier modulation format, three outcomes are observed by varying the RF linewidth. As mentioned, the linewidth parameter depends on the number of points in the constellation to determine the system performance around the FEC limit. The BER deteriorates as the bandwidth narrows and the laser phase noise increases, just meaning that a format with higher bit-rate efficiency needs more pure subcarriers at the frequency domain. In detail, the penalty of the adoption of a certain subcarrier modulation format in the BER metric is explained later through normalized linewidths for realizing overall conclusions. For each symbol constellation and phase-noise level, the superior success rate in the testing stage happens in an irregular region dependent on the ELM parameters. In general terms, for coefficient penalties between \(2^{0}\) and \(2^{5}\) and number hidden nodes greater than the number learning samples (pilot subcarriers), the system performance exhibits important enhancement. The highlighted zone in terms of the BER extends for broad lasers. In the following, \(C=2^{2.5}\) as well as \(L=24\) are adopted as sub-optimal parameters in order to convert these invariants to the symbol constellation and/or mixed linewidth, and get away from the boundaries of the BER degradation. As mentioned in Section 2.1, for further enhancing the ELM behavior, the regularization parameter is added, which is utilized to balance structural and empirical risks. In the absence of this (\(C=\infty\), refer to the Equation (2)), for (a) BPSK, (b) QPSK, and (c) 16 QAM subcarrier modulation formats, Figure 4 depicts the BER against the number of hidden nodes with the combined linewidth as parameter. In other words, here, the inverse of the hidden layer output matrix is determined by employing the singular value method, which averts matrix requirements as well as time consumption [20]. For 16 hidden neurons, the signal deterioration is maximum. Instead, the better performances respectively occur with 4 and 10 hidden nodes for BPSK/QPSK and 16 QAM constellations. By contrasting Figures 3 and 4, it is clear that the penalty-coefficient insertion in a standard ELM improves its learning capability, which is more notorious as the constellation points augment, but with the obvious cost of increasing its complexity. Evidently, results of Figure 3 with \(C=2^{10}\) and Figure 4 where \(C=\infty\) expose similar behaviors. Taking into account the 16 QAM signaling and 50-kHz mixed linewidth, for instance, the \(log_{10}\)(BER) goes from \(-2.6\) with 10 hidden neurons and without regularization parameter to \(-3.2\) subject to 18 hidden nodes and a positive constant equals to \(2^{5}\).