# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 2 Orthogonal Frequency Division Multiplexing Radio over Fiber Systems  
To begin, the block diagram of the direct-detection RoF system focused on the OFDM modem and the simulation parameters is displayed in Figure 1. A single baseband OFDM symbol with a symbol duration of \(T_{s}\) is given by [19]:  
\[s(t)=\sum_{k=1}^{N}c_{k}\exp(j2\pi kt/T_{s}), \tag{1}\]  
with \(N\) and \(c_{k}\) being the number of subcarriers and the symbol constellations, respectively. The number of subcarriers comes from the summation of the number of pilot and data signals (\(N=Np+Nd\)). Depending on the subcarrier position, a symbol constellation takes the value of either the data symbol \(d_{k}\) or the pilot symbol \(p_{k}\)[20]. If the OFDM-based RoF system is characterized by correlated-phase fields at the central station (the direct optical detection approach), the input to the OFDM demodulator is corrupted by (i) phase noise due to laser linewidth, as well as chromatic dispersion [9] and (ii) AWGN due to thermal and shot noises [21]. After the photo detection and down-conversion processes, the RoF-OFDM signal may be written as [22]:  
\[r(t)=s(t)\exp\{j[\phi_{0}+\Delta\phi(t)]\}+n^{*}(t), \tag{2}\]  
where \(\phi_{0}=2\pi(f_{RF}-f_{0})\tau\) denotes the phase rotation as a consequence of the time delay parameter \(\tau\) and \(f_{RF}\) and \(f_{0}\) are the radio frequency and the optical carrier, respectively. It is worth mentioningthat \(\tau\) models the chromatic fiber dispersion effect as long as the OFDM bandwidth is much smaller than the radio frequency [7] and can be interpreted as the phase-decorrelation degree between the mixed tones in the photo detection process [9]. \(\Delta\phi(t)=\phi(t)-\phi(t+\tau)\) refers to the differential phase noise, where \(\phi(t)\) is phase noise modeled from a Wiener process quantified by the laser linewidth \(\Delta\
u\). Note that all stages experimenting with the RoF-OFDM signal from the central station to a certain base station were explained in our previous work [9]. The authors in [23] revealed that in terms of the time delay parameter, the heterodyning field correlates perfectly (\(\tau=0\)), decorrelates partially (\(0<\tau<1/\Delta\
u\)), or decorrelates completely (\(\tau>1/\Delta\
u\)). AWGN is represented by \(n^{*}(t)\), which is the complex base-band signal. In order to focus on the impact of phase noise in the system performance, Expression (2) makes sense for the case of linear field modulation without the presence of the optical carrier at the intermediate frequency. In the demodulation step, each OFDM symbol is sampled at the sampling rate in order to mimic an analog-to-digital converter (ADC), convert into \(N\) parallel data pipes (the serial-to-parallel process (S/P)), and demodulated via the fast Fourier transform (FFT) module, whose size is also given by \(N\) (i.e., DC subcarrier and null data are not considered for the sake of simplicity). Therefore, the \(k^{\text{th}}\) symbol constellation (a single QPSK symbol during the OFDM symbol period) acquires the form of [19]:  
\[\overline{c_{k}} = \frac{\exp(j\phi_{0})}{N}\sum_{m=1}^{N}c_{m}\sum_{n=0}^{N-1}\exp \left\{j\Big{[}\frac{2\pi(k-m)n}{N}+\Delta\phi(n)\Big{]}\right\}+n_{k}^{*} \tag{3}\] \[= \exp(j\phi_{0})\sum_{m=1}^{N}c_{m}\Delta\phi(k-m)+n_{k}^{*}\] \[= [c_{k}\Phi(0)+\sum_{m=1,m\
eq k}^{N}c_{m}\Delta\Phi(k-m)]\exp(j \phi_{0})+n_{k}^{*},\]  
where \(\Delta\phi(n)\) refers to the discrete-time converted phase fluctuation, \(n_{k}\) represents the AWGN at the \(k^{\text{th}}\) subcarrier, and \(\Delta\Phi(p)=\frac{1}{N}\sum_{n=0}^{N-1}\exp\left\{j\Big{[}\frac{2\pi pm}{N} +\Delta\phi(n)\Big{]}\right\}\), i.e., is the FFT of \(\Delta\phi(n)\). It is noted that (i) the received \(k^{\text{th}}\) symbol constellation shows a rotation by the first term (CPE), as well as a dispersion by the second term (ICI), and the last term (AWGN) and (ii) all subcarriers are closely related. Taking into  
Figure 1: Block diagram of the simulated RoF system.