## 1 Introduction  
For optical fiber transmissions, intersymbol interference originated by polarization-mode dispersion and chromatic dispersion becomes relevant as the bit rate achieves the order of various tens of Gbps [1]. As an efficient alternative to mitigate this issue, coherent optical (CO) orthogonal frequency division multiplexing (OFDM) systems have attracted the attention of the scientific community because intersymbol interference may be eliminated as long as the guard interval occupies enough space of the OFDM symbol period [2]. Additionally, next-generation Tbps, bandwidth variable elastic optical systems can be a direct consequence of the CO-OFDM technology by its high receiver sensitivity as well as its effective transmission rate. Nevertheless, both coherent detection and OFDM are prone to phase noise due to the phase mismatch between the laser oscillators at the transmitter and receiver sides and the relatively long OFDM symbol duration compared to that of the single carrier communications. In the CO-OFDM signal, laser phase noise produces two clearly effects, termed as common phase error (CPE) and inter-carrier interference (ICI) [3]. CPE is relatively easy to diminish contrasted with ICI since the former induces the same rotation in all constellation symbols during the OFDM symbol, instead the latter behaves like Gaussian noise and depends on the subcarrier frequency. Fortunately, the CPE may be controlled through the pilot-assisted equalization (PAE) method [4; 5; 6; 7], which is usually inserted in OFDM modems in order to correct the multipath channel effects. Here, the channel frequency response for data subcarriers is estimated by interpolation from the values provided by pilot subcarriers. Other phase-noise reduction techniques, such as the pilot-added scheme [8] that realizes ideal CPE compensation and the RF-pilot-based mechanism [9] that completely removes phase noise fluctuations (pioneering studies for optical OFDM transmissions), employ not only various pilots dedicated to phase-noise issue but also some preambles for channel equalization purposes. By comparing with the PAE technique, the spectral efficiency of these methods hence suffer a diminution. In a general sense, optimal approaches for suppressing the laser phase noise for CO-OFDM systems are still required in order to enhance the bit error rate (BER) especially in the situation of the low-cost broad-linewidth lasers are used. [PERSON] et al. [10] introduced the extreme leaning machine (ELM) algorithm in 2006, which is a novel technique of machine learning focused on single-hidden layer feedforward networks (SLFNs). The ELM is well-known for their fast training capability and remarkable performance. The main feature of this algorithm is that the parameters of the input-hidden-layer are randomly originated and do not necessitate to be adjusted. Therefore, the output weights in the training process simplify into a regularized least problem, and of which result comes from a closed equation. Enhancements and extensions of the standard ELM such as improving the stability [11] and compactness [12] of the ELM, ELMs for online sequential [13] and imbalanced [14] data, Bayesian [15], fuzzy [16], Wavelet [17], or complex [18; 19] ELMs have been demonstrated excellent generation performance, superior efficiency, and less computational complexity, by giving solutions for regression and classification problems in diverse practical areas (medical, remote sensing, control and robotics, image/video processing, time series analysis, text classification and understanding, telecommunication, chemical process, and computer vision). Hence, the issues of convergence ability, generalization, over-fitting, local minima, and/or parameter adjustment (learning rate, learning epochs, among others) presented in support vector machines as well as back-propagation artificial neural networks do not happen in ELMs [10; 20].