## 4 Results and Discussion
### Performance Evaluation  
In order to analyze the behavior of the RC-ELM, in Figure 8, we display the BER at SNRs ranging from 0 to 25 dB with the RF linewidth as parameter for five techniques that reduce the laser phase noise: feasible PAE [4; 6; 7], fully-real ELM (R-ELM) [24], ELM in the complex plane (C-ELM) [24], new ELM algorithm, and conventional CPE compensation [8; 40]. One again, (a) BPSK, (b) QPSK, and (c) 16 QAM subcarrier modulation formats are taking into account for a complete study of CO-OFDM signals. As mentioned at the end of Section 2.2, the PAE is included in most OFDM modems since it can control frequency-selective impairments by the linear-interpolation of pilot subcarriers in the frequency domain [5] and, furthermore, PAE is able to diminish the low frequencies of phase-noise spectrum. Taking into account that ELMs are suitable for channel estimation and correction purposes [9; 13; 21; 22], ELMs in real and complex planes are recently introduced to eliminate the chromatic-dispersion effect in direct-detection OFDM-based radio-over-fiber systems with the QPSK signaling [24], where the C-ELM is distinguished by low complexity as well as high precision. \(L=10\) and SIG activation function, and \(L=2\) and HT activation function are adopted for the R-ELM and C-ELM, respectively. These parameters are the ones that obtain the BER enhancement. Notice that here, the penalty coefficient (\(C\)) does not exist in the inverse matrix calculation (\(H^{\dagger}\)). With the cost of some preambles to assistant frame synchronization as well as channel estimation (an extra bandwidth consumption), the CPE term (see Equation (8)) may be effectively mitigated by approximating the mean phase rotation of each OFDM symbol from reference tones and rotating the received constellation points back [9], namely the pilots are only destined for reducing phase noise. As upper and lower bounds, the system performance without phase-error reduction (the worst case) and no susceptible to phase noise (the ideal situation) are depicted, respectively. These limits correspond to theoretical results [3] and are only represented with the black color. As expected, the BER enhances as AWGN disappears and/or the combined parameter narrows. The reasons behind these facts result evident via the power spectral densities of the received OFDM-based CO signal, exposed in Figures 9 and 10. The former comes from a phase-noise free simulation, while the latter is a theoretical result without considering CP and AWGN [4; 43]. For demonstration purposes, a zoom-in some QPSK subcarriers is depicted, whose OFDM symbol period corresponds to \(22\times 10^{-9}\) s. On the one hand, the OFDM and AWGN signals overlap inversely proportional to the SNR value, see Figure 9. On the other hand, as the RF-linewidth widens, the phase-noise strength in any subcarrier bandwidth increases, refer to Figure 10. Consequently, a spectrally purer and more powerful CO-OFDM signal leads to the best system performance. Figure 7: The BER in terms of the normalized RF linewidth according to the the modulation modes. Figure 8: By considering diverse phase-noise compensation techniques for the studied subcarrier modulation formats, the BER against the SNR with the RF linewidth as parameter. For an SNR of 20 dB and different combined linewidths, Subfigures show constellations. In terms of the phase-error reduction, the classic CPE compensation, C-ELM, and proposed method outperform the benchmark PAE, showing more advantages for inefficient spectrally constellations, high SNR values, and regular RF linewidths. In this context, the former is a bit more effective for all CO-OFDM signals, and the C-ELM and RC-ELM expose negligible differences in the performance of the proposed method. Figure 10: By considering an unmodulated QPSK-OFDM signal without CP and AWGN, its frequency profile in the last subcarriers as the combined linewidth increases. Figure 9: Taking into account a phase-noise free QPSK-OFDM signal, its positive spectrum as a function of the SNR values.