# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 5 Results  
As mentioned, the RoF-OFDM schemes with direct detection allow the use of commercial cost-effective DFB lasers and offer the advantage of compact and simple receivers. Unfortunately, as the fiber length between the central station and the base station increases, the transmitted tones (the optical carrier and the sideband OFDM signal) lose their phase coherency; inducing important phase noise in the photo-generated current. This phenomenon comes from the chromatic fiber dispersion, which may be modeled via the time delay parameter as follows [22]:  
\[\tau=DL\frac{c}{f_{0}^{2}}f_{RF}, \tag{10}\]  
where \(D\) is the dispersion parameter, \(L\) denotes the transmission distance, \(c\) represents the light speed in vacuum, and \(f_{0}\) refers to the operating frequency. The expression (10) makes sense in the situation of narrow OFDM signals and high radio frequencies [7]. In order to not restrict the observations to some value of the millimeter frequency given a certain fiber length or vice versa (see Expression (10)), we initially studied the OFDM tolerance to laser phase noise and AWGN as the time delay increases in RoF systems with the assistance of real and complex ELMs in the demodulation stage. To this end, Table 1 illustrates the simulation parameters used, which were based on a recent work [6] that took into account typical RoF-OFDM signals [6; 9; 22; 25]. The considered laser linewidths were in the order of MHz to model low-priced DFB lasers [5], by allowing reaching a BER threshold of \(10^{-3}\)[26] at certain time delays. The post forward error correction (FEC) BER could reach values in the order of \(10^{-15}\) by utilizing modern FEC architectures [27]; however, no channel coding was taken into account in the manuscript. The BER was estimated via Monte Carlo simulations with the direct error counting method, i.e., 10 runs of \(10^{6}\) bits were performed for having a confidence interval of 95% with an uncertainty factor of two on the error rate scale [28]. On the other hand, following an easy methodology employed in [26; 29], we realized the complexity evaluation of these new approaches by estimating the average elapsed time at the OFDM equalizer. In the next section, results are discussed and, consequently, exposed in terms of the relationship between the radio frequency and the distance from the central station to the base station. We first determined the number of the hidden neurons and the mapping function of the ELMs that enhanced the BER, by setting the laser linewidth to 0.5 MHz and by discarding AWGN for the moment. As mentioned, this laser linewidth parameter can represent a typical DFB laser and was chosen in order to observe all the system performance evolution (from BER = \(10^{-6}\) to BER = \(10^{9}\)) as the chromatic dispersion effect increases. On the one hand, Figure 3 displays the BER in terms of the time delay parameter with the number of the 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. These activation functions were employed for demonstration purposes. The input weight vectors and biases of the fully-real and fully-complex ELMs were randomly chosen according to the uniform distribution on (\(-\)1,1) [15] and from a complex area centered at the origin with the radius set as \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Subcarrier modulation format & QPSK \\ Number of data subcarriers & 120 \\ Number of pilot subcarriers & 8 \\ FFT size & 128 \\ Symbol period & 64 ns \\ Power signal & \(-\)7 dBm \\ Sampling rate & 100 GHz \\ Dispersion coefficient & 16.75 ps/nm-km \\ Optical carrier & 193.1 THz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. 0.005 [13], respectively. For the real ELM (see Figure 3a), the system performance deteriorated as the amount of phase correlation between the beating tones increased until these fields were completely decorrelated, where the BER reached its maximum value.