# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 5 Results
From 10 hidden neurons to forward, the ELM effectively diminished phase error. With a single hidden neuron, the novel method was useless. Instead, for the complex ELM (see Figure 3b), the BER depended on the decorrelation degree between the mixed fields in the following form: the BER rapidly and slowly augmented for time delays less and greater than the OFDM symbol period, respectively, and the system performance revealed its worst result when the combined tones completely lost the phase coherence. The proposed technique in the presence of two hidden neurons was highlighted in terms of phase noise correction. The reasons behind the different system performances of the real and complex ELMs may be explained via the effectiveness of the phase noise compensator (see the constellation histograms in Figure 4) and the evolution of the ICI term (see the zoom of the power spectral densities of a demodulated OFDM signal in Figure 5) in terms of the time delay parameter; but its detailed study is left for future analysis. On the other hand, Figure 6 depicts the BER as a function of the time delay with the activation function as a parameter for (a) fully-real and (b) fully-complex ELMs in the presence of their best number of hidden neurons, previously found. The input weights and biases of the real ELM again obeyed a uniform distribution from \(-1\)-\(1\), and those of the complex ELM were inside the radius of the converge of the complex mapping function used [13]. For the real ELM (see Figure 6a), sigmoid (SIG) and hyperbolic tangent (HT) functions outperformed the rest of the mapping functions: cosine (COS), hard limit (HL), triangular basis (TB), and radial basis (RB), by reducing the chromatic dispersion effect in the BER; the performance of the TB function was disastrous. For the complex ELM (see Figure 6b), in the same context, the arcsin (AS), sinh (SH), tanh (TH), sin (S), and tan (T) functions were superior to the arcsinh (ASH), arctanh (ATH), and arctan (AT) functions. Note that according to the simulations, this optimization process led to the same number of hidden nodes and mapping functions for other laser phase noises and AWGN levels. Afterward, we contrast in terms of improving the symbol constellation detection the introduced schemes with the feasible pilot-assisted equalization [9] and without phase error mitigation as the BER limiting references. It is worth noting that this comparison was not carried out with pilot tone phase-noise suppression [22] and post phase-noise suppression [25] algorithms (the most famous compensators), since the former decreases the bit-rate efficiency, and the latter increases the digital-signal-processing complexity and cost. Instead, our proposals were highlighted by no additional overhead, as well as no modification of the standard OFDM signal; refer to Section 4. Additionally, the complex ELM was better than the pilot-based channel correction in terms of the complexity evaluation, as seen below. In the scenario of a broad laser linewidth and/or a long  
Figure 3: The BER in terms of the time delay with the number of hidden neurons as the parameter for (**a**) the real ELM with the sigmoid mapping function and (**b**) the complex ELM with the tanh mapping function. fiber length, time delay pre-compensations between the central station and each base station [30], self-homodyne RF receivers [31], or phase-insensitive RF envelope detectors [32] could be implemented to eliminate the impact of phase error in the system performance. These solutions massively increased the cost of the RoF network design. On the other hand, the CPE compensation [33] may be useful for low phase errors prone to short OFDM symbol periods, but with the cost of needing an extra preamble in every OFDM frame for equalization purposes. Both BER upper bonds were also exposed for comparison purposes. Phase-insensitive results were analytically found [34], while the BERs resulted from the CPE compensation were simulated.