## 1 Introduction
The merits owned by the ELM algorithm make it a suitable alternative to perform channel equalization in OFDM schemes in wireless [13; 21; 22] and optical fiber [23; 24] communications. On the one hand, a fully real-valued ELM to perform equalization and symbol detection processesin OFDM-based systems is introduced in Reference [21]. In the following, the methodology and results of this study are briefly exposed. After the fast Fourier transform (FFT) module at the demodulator, data are separated into training and testing parts, where the first set is used to estimate the channel response thanks to the Moore-Penrose generalized inverse of the hidden layer output matrix. Consequently, pilot subcarriers are not exploited. Over a time-varying Rayleigh fading channel for the quadrature phase-shift keying (QPSK), 16-ary quadrature amplitude modulation (16 QAM), and 64-ary QAM modes, the ELM exhibits significant improvement in terms of the symbol error rates and has lower computations than the following learning-based equalization schemes--the fully complex-valued ELM [25], complex-valued radial basis function network [25], complex-valued minimal resource allocation network [25], k-nearest neighbor approach [25], and back propagation [26] and stochastic gradient [27] neural networks. For QPSK-OFDM networks, an online fully-complex ELM based channel estimation and equalization scheme robust to fading channels based on the 802.11g standard and the nonlinear distortion originated from a high-power amplifier is proposed by [PERSON] et al. [13]. By combining the least square-based decision-directed channel estimation and the ELM algorithm, the received/equalized pilots may be employed to train the SSFN without pre-training and feedback connection between the transmitter and receiver. The ELM defined in the complex domain demonstrates the best BER metric contrasting with the minimum mean-square error and least square channel estimators as well as a deep neural network-based scheme. Methods that amalgamate real and complex ELM regressors for equalization and minimum-distance based symbol slicers for symbol detection are presented in Reference [22] for QPSK-OFDM communications over strong frequency selective channels. To find the output weights matrix, pilot signals are utilized. With the assistance of 1000 hidden neurons, simulations establish that compared to the previous ELM based methods [13; 21], the multiple split-complex ELM owns the benefit of higher detection accuracy with a slight complexity increase. On the other hand, the multi-layer perceptron, generalized radial basis function, and robust ELM are proposed in Reference [23] for mitigating nonlinearities in CO-OFDM systems under the 16 QAM signaling. All of these equalizers are trained with the Wilcoxon learning algorithm in order to the outliers do not depend on the training data. While the Wilcoxon robust ELM possesses a good generalization capability and fast training speed, the Wilcoxon generalized radial basis function gives the highest value of Q-factor among the rest of the algorithms as well as non-robust approaches based on minimization of least square error principle. ELMs in the real and complex planes for decreasing the chromatic dispersion effect in direct-detection OFDM-based radio-over-fiber systems are revealed by [PERSON] et al. [24]. Immediately, its ELM algorithm is summarized. Based on the comb-type pilot arrangement in an OFDM packet, pilot subcarriers are exploited during each OFDM symbol period in order to obtain a real-time equalization without reducing the effective bit rate. To prevent time limitations and algebraic requirements related to the hidden layer output matrix in the training process, the singular value decomposition approach is adopted [20]. Simulated results report that for the QPSK constellation subject to an additive white Gaussian noise (AWGN) channel, the fully-complex ELM outperforms the fully-real ELM and feasible PAE [4; 7] in terms of the BER as well as the complexity modem.