# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 5 Results
According to the previous performances, the real ELM composed of 10 hidden neurons prone to the sigmoid mapping function and the complex ELM together with two hidden neurons affected by the tanh activation function were adopted. For the case of not fighting phase error (the worst scenario), the results were theoretically obtained [9]. Note that in terms of sensitivity, complexity, and capacity, a survey regarding phase noise reduction mechanism for optical OFDM communications is a research opportunity. Firstly, Figure 7 shows the BER as the heterodyning tones became decorrelated with the laser linewidth as the parameter and without AWGN. The superiority of the complex ELM among the pilot-assisted equalizer and the real ELM in terms of phase noise correction is clear, which is more notorious for narrow lasers. The advantage of the complex ELM over the real ELM can be attributed to only the former combining the quadrature and in-phase components at the learning step; contrast Algorithms 1 and 2. Notice that below the FEC threshold, the real ELM outperformed the pilot-based channel correction; otherwise, the opposite happened. With the benefit of an effective bit rate, the algorithm ELM in the complex domain almost obtained the same BER as the CPE  
Figure 4: According to the time delay parameter, constellation histograms for **(a)** the real ELM with 10 hidden neurons and **(b)** the complex ELM with two hidden neurons. Figure 5: Zoom of the power spectral density of a demodulated OFDM signal as a function of the chromatic dispersion effect. compensation. As the laser linewidth increased, these system performances tended to match. The BER led to augmentation when phase noise was not combated, while in the presence of phase-insensitive methods, the BER acquired its best performance. Lastly, Figure 8 displays the system performance as a function of the decorrelation degree between the mixed tones with the SNR as the parameter and a 0.5-MHz laser linewidth according to the different phase error compensation methods. In this scenario, all previous results occurred again with the following observations: (1) The complex ELM improvement increased as AWGN disappeared. By replacing the pilot-assisted equalization by the fully-complex ELM when the heterodyning fields were phase-uncorrelated (\(\tau>2\times 10^{-6}\)), for demonstration purposes, the BER decreased from \(3.5\times 10^{-2}\) to \(1.2\times 10^{-2}\) with an SNR equal to 12 dB for the AWGN channel, while from \(2\times 10^{-2}\) to \(4\times 10^{-3}\) in the absence of AWGN (SNR = \(\infty\)dB). This BER improvement also happened for the different simulated SNR values, being more relevant for high SNRs as mentioned, and along the time delay axis. (2) As the SNR value increased and/or the time delay parameter decreased, the CPE compensation performed better than the ELM complex. (3) For high partially phase-correlated tones, the fact of reducing the phase noise helped to minimize the BER result of phase-insensitive methods owing to the phase-noise mitigation combating AWGN as well. Figure 6: With the best number of hidden neurons, the system performance as the combined fields become phase-decorrelated with the activation function as the parameter for (**a**) real and (**b**) complex ELMs. Figure 7: For different phase noise mitigation methods, the BER in terms of the amount of phase correlation between the beating tones with the laser linewidth as the parameter and without AWGN. In Table 2, we finally report the complexity of the new mechanisms with the pilot-assisted equalization technique by measuring the elapsed time (expressed in ms) at the equalization stage of the OFDM demodulator, where the design and implementation could change. As mentioned, this methodology was based on previous works due to simplicity [26; 29]. The 95% confidence interval [28] of this metric was obtained via \(10^{3}\) runs in the next computer system: Intel (R) Core (TM) i5 at 2.6 GHz with 4 GB RAM.