# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 1 Introduction
For quadrature phase shift keying (QPSK) signaling, the superiority of the complex ELM over the complex minimal resource allocation network, complex radial basis function network, and complex back-propagation equalizers in terms of the symbol error rate and the learning speed is revealed, where a complex non-minimum-phase channel model channel of order three is taken into account [12]. In an additive white Gaussian noise (AWGN) channel, the standard ELM algorithm is utilized as a novel classification framework to perform correct symbol detection for amplitude phase shift keying constellations [16] by showing the same symbol and bit error rates as the well-known max log maximum a posteriori method (the optimal solution for symbol detection in single carrier). For OFDM signals over a frequency selective fading channel subject to diverse values of the signal-to-noise ratio (SNR), an algorithm that employs real-valued ELM that jointly solves the issue of equalization and symbol detection for QPSK and 16 quadrature amplitude modulation (16-QAM) formats is presented [17]. Simulation results show that the proposed technique outperforms other learning-based equalizers in terms of the symbol error rate and the training speed. These results are obtained in the following situation: data subcarriers are split into training and testing data by decreasing the overall throughput; pilot symbols that always exist in the OFDM signals are not exploited. For coherent optical OFDM, the performance analysis of Wilcoxon-based machine learning nonlinear equalizers is exposed [18], by reporting that the nonlinear equalizers trained with Wilcoxon multilayer perceptron, Wilcoxon generalized radial basis function, and Wilcoxon robust ELM give enhanced performance in terms of the Q-factor as compared to non-robust algorithms (based on minimization of the least squares error principle). Here, the algorithm based on the ELM converges faster than the other algorithms. In our work, phase noise estimation methods based on ELMs in the real and complex domains are presented in the context of direct-detection RoF-OFDM signals with QPSK as the subcarrier modulation format. The SLFN for the fully-complex ELM has a single input and output to reinforce the learning stage; the method only trains the pilot subcarriers to make the output of the network converge extremely fast, and phase and amplitude noises in the data symbols are simply compensated by multiplexing the output weight vector obtained from pilots and the hidden layer output matrix of the data subcarriers. For the real ELM algorithm, the complex OFDM signal is divided into its real and imaginary components and then fed into two different real-valued ELMs. The main contributions of the manuscript can be summarized as follows: (1) The ELM algorithms in the real and complex domains for phase noise estimation are introduced for the first time, by exploiting the pilots to achieve a real-time learning stage, as well as to maintain the spectral efficiency. (2) In terms of the bit error rate (BER), as well as the time simplicity, the superiority of the fully-complex ELM among the feasible pilot-assisted equalization and the real ELM is revealed for numerous laser linewidths, SNRs, transmission distances, radio frequencies, and OFDM symbol periods. Furthermore, the complex ELM almost achieves the same BER as the CPE compensation, which decreases the transmission rate. (3) For a certain relationship between the laser phase noise and the duration of a single OFDM symbol, any phase noise compensation may reduce the BER more for a narrow laser linewidth or a long OFDM symbol period, as long as there is partial phase-decorrelation between the combined fields. Therefore, by using the complex ELM with the assistance of pilot subcarriers at the training phase, better and faster OFDM modems could be designed and implemented for the topic and future communication systems. The rest of the paper is organized as follows.