## 3 Proposed Extreme Learning Machine Algorithm for Laser Phase-Noise Reduction Purposes  
An equalizer based on the standard ELM can be robust against the multipath effects introduced by the wireless channel [13; 21; 22]. In our work, the improved ELM algorithm is used to increase the RF-linewidth tolerance in CO-OFDM signals, by considering pilot and data subcarriers as training and testing samples, respectively. This noise-mitigation mechanism takes the next advantages: (i) the ELM learning results in a determined linear system [20] and (ii) the position and information for all OFDM pilots are known in the receiver during one symbol period [37]. Unlike the PAE, in the ELM equalizer to obtain a superior accuracy in the testing stage and, hence, a minimum BER, the reference tones must share the in-phase (the real part of the symbol in the constellation diagram) and quadrature componentsof the M-ary subcarrier modulation format, which in turn must be equal or greater than these. It is desired that each constellation point has the same number of pilot subcarriers so that the BER does not increase for certain data streams. For instance, a QPSK mode requires at least 4 pilots, where both data and pilots are modulated in [1+1j, 1-1j, -1-1j, and -1+1j] (its constellation alphabet). By considering practical relationships between data and pilots for spectral efficiency purposes, the proposed equalizer is not useful in the situation of a few tones together with high constellations; here, PAE should be considered [4; 6; 7]. Notice that in the ELM algorithm as in the PAE method, no additional blocks in the modem as well as no extra overhead in the signal are necessary for getting the phase-error mitigation and channel correction. Namely, the effective transmission rate does not decrease. On the contrary, the spectral efficiency is affected by the use of specific phase-noise mechanisms [8; 9]. The reason behind these benefits comes from the exploitation of the common OFDM in the equalizer stage. In this work, the ELM architecture consists of 2 real input (one for the real part and one for the imaginary part of the points in the constellation), \(L\) neurons in the hidden layer defined in the real domain, and 1 complex output (the constellation symbol), see Figure 2 where constants and variables may be identified. After the FFT module, pilot and data subcarriers are separated as well as serialized to accelerate and reinforce the learning procedure. The real-complex (RC) ELM by assuming the pilot subcarriers as training data is introduced for the first time since it could be directly extended to a semisupervised machine learning algorithm [41; 42] for enhancing the system performance, but its realization is left for future research. Remember that the origin of phase error in the diverse subcarriers comes from the same oscillators and, hence, the inclusion of some data in the training step should be an overall positive impact. In addition, the conditions of the input weight vectors and biases based on a certain activation function in the complex plane (a fully-complex ELM) [18; 19] do not occur in the RC-ELM where a real activation function is considered, by facilitating its realization. Consequently, the output weights thanks to pilot subcarriers acquire the form of:  
\[\beta_{L\times 1}=H[\overline{\Re\{a_{p}(i)\}},\overline{\Im\{a_{p}(i)\}}]_{L \times N_{p}}^{\dagger}P[a_{p}(i)]_{N_{p}\times 1}, \tag{11}\]  
where \(\Im\) corresponds to the imaginary part of a complex function and \(i\) varies from 0 to \(N_{p}-1\). Expression (11) synthesizes the learning step, where the number of hidden nodes \(L\), the input weights \(w_{j}\) and biases \(b_{j}\) according to any continuous probability distribution, the activation function \(g(\cdot)\), and the regularization parameter \(C\) must be firstly specified. As will be observed in Section 4.1, the consideration of \(C\) becomes very important because it allows to reduce the impact of the laser phase noise on the BER.