## 4 Results and Discussion
### Parameters' Optimization of the Extreme Learning Machine
Namely, there is a \(log_{10}\)(BER) improvement of \(-0.6\), but the dimension of the hidden layer output matrix \((H)\) and its Moore-Penrose generalized inverse \((H^{\dagger})\) possesses an increase of 8 rows and 8 columns and additionally the summation of the \(C\) rectangular matrix \((C/I_{18\times 18})\) must be computed in the learning procedure, refer to expression (2). Note that in our previous study (ELMs to reduce phase error in radio-over-fiber OFDM schemes with direct detection [24]), \(H^{\dagger}\) does not consider the regularization parameter and, consequently, its inclusion could be explored in a next paper. Indeed, this constant is either discarded or not optimized in the rest of the manuscripts about OFDM schemes with the ELM algorithm as equalizer [13; 21; 22; 23]. In the next Subsection, an analysis of the complexity of the proposed ELM and other phase-error mitigation mechanisms is presented. Figure 3: According to diverse radio-frequency (RF) linewidths and (**a–c**) binary phase-shift keying (BPSK), (**d–f**) quadrature phase-shift keying (QPSK), and (**g–i**) 16 QAM subcarrier modulation formats, the BER in terms of the regularization parameter and number of hidden neurons by considering the SIG activation function. Figure 5 afterward illustrates the BER as a function of the RF linewidth with the most used activation function as parameter (see Table 1) for the studied subcarrier modulation formats. The impact of the activation function on the system performance augments as the constellation symbols increase and/or phase noise becomes more predominant, where the SIG function outperforms the rest of the functions. The HT function, which is also SIG (s-shaped), has practically the same behavior that the SIG function. In the same context, the worst activation function depends on the number of points in the constellation. For example, the HL and TB functions provide poor results for the BPSK and QPSK modes, respectively. If the equalizer outcome will be a real number, these observations could be attributed to the testing root mean square error, which is commonly used to quantify the effectiveness of any classifier or regressors [10; 11; 12; 14; 15; 16; 17; 20; 41; 42]. Unfortunately, the proposed phase-noise suppression technique provides an output in the complex plane (the points in the constellation). The root mean square EVM is hence employed. It manages to measure how far a symbol constellation is from its perfect location in telecommunication networks [45; 46]. In Figure 6, histogram constellations together with EVM values at training as well as testing stages are displayed. For the sake of illustration, the subcarrier modulation format and combined linewidth correspond to 16 QAM and 100 kHz, respectively. It can be seen that EVMs in the testing phase explain the superiority in terms of the BER of SIG and HT functions, whose constellations manifest low dispersion. At the same time, the poor results by the use of TB and RB functions make sense, whose constellation shapes do not remain. To further gain insight on the explanation behind these performances, the histogram constellation in the learning stage for each activation function is exposed. While SIG and HT functions present the most regular constellations, the other functions are characterized by scattered symbols; the EVM metric obviously affirms this note. In the rest of the paper, the SIG as activation function is adopted. Figure 4: According to (**a**) BPSK, (**b**) QPSK, and (**c**) 16 QAM constellations, the bit error rate (BER) in terms of the number of hidden nodes with the combined linewidth as parameter. The penalty coefficient is not considered. Finally, the BER in terms of the product between the RF linewidth and the OFDM symbol period for the studied subcarrier modulation formats is displayed in Figure 7. This normalization is done in order to clearly establish the cost of increasing the spectral efficiency on the system performance as long as the bit rate remains constant.