## 2 Background
### Coherent Optical OFDM Network
The optical OFDM signal convoys via the fiber optical previously being combined with a local oscillator at the coherent receiver, the received signal acquires the form of [8]:  
\[x_{3}(t)=h(t)*x_{2}(t), \tag{5}\]  
with \(h(t)\) and \({}^{\ast}\) being the impulse response of the optical channel and convolution operation, respectively. Each fiber span consists of chromatic dispersion, numerous stages of high birefringence devices, and polarization dependent on loss elements [28; 32]. Taking into account that the phase error resulted by the effect of the fiber chromatic dispersion (the fiber-link penalty in terms of phase noise) results in a time-invariant phase rotation [4; 33] that may be eliminated with a single complex multiplication at the OFDM receiver [34; 35], a flat channel response is possible to assume that is, \(x_{3}(t)=\delta(t)*x_{2}(t)=A_{L}\exp\left\{j[2\pi f_{L}t+\phi_{1}(t)]\right\} \sum_{k=0}^{N_{\mathrm{SC}}-1}a(k)\exp\left(j2\pi f_{k}t\right)\). This situation makes strict sense if the RF is much greater than the bandwidth of the OFDM signal [24; 30], which is very common for OFDM signals in the optical domain. Besides, our study is focused on the introduction of the improved ELM to enhance the CO-OFDM performance affected by laser phase noises; consequently, linear distortions (chromatic dispersion) and non-linear impairments (fiber non-linearity, intra-channel nonlinearity, among others) [36] are discarded. As future research, results in the case of optical channel model presence by including both linearities and non-linearities must be considered, where the impact of the chromatic dispersion and fiber nonlinearities in the BER depends on the subcarrier position. By considering that the coherent detection implies of heterodyning the optical signal with a continuous-wave optical field \(E_{LO}(t)\) previously it falls on the photodetector, the electrical current may be written as [7; 32]:  
\[x_{4}(t) = R\mid E_{LO}(t)+x_{3}(t)\mid^{2}+n(t) \tag{6}\] \[= R\mid A_{LO}\exp\left\{j[2\pi f_{LO}t+\phi_{2}(t)]\right\}+A_{L} \exp\left\{j[2\pi f_{L}t+\phi_{1}(t)]\right\}\] \[\times \sum_{k=0}^{N_{\mathrm{SC}}-1}a(k)\exp\left(j2\pi f_{k}t\right) \mid^{2}+n(t)\] \[= R(A_{LO}^{2}+A_{L}^{2}\mid\sum_{k=0}^{N_{\mathrm{SC}}-1}a(k)\exp \left(j2\pi f_{k}t\right)\mid^{2}\] \[x_{1}(t)=\sum_{k=0}^{N_{\rm SC}-1}a(k)\exp{(j2\pi f_{k}t)}, \tag{3}\]0  
where \(R\) corresponds to the responsivity of the photodiode, \(A_{LO}\), \(f_{LO}\) and \(\phi_{2}(t)\) respectively denote the amplitude, frequency, and phase noise of the local oscillator, \(\Re\) is the real part function, \(f_{RF}=f_{L}-f_{LO}\) implies the RF, and \(\phi_{RF}(t)=\phi_{1}(t)-\phi_{2}(t)\) is the RF phase noise, whose linewidth \(\Delta v_{RF}\) comes from the addition of the laser linewidth and local oscillator linewidth, which in turn is evident via subcarriers characterized by Lorentzian spectra (see Figure 10) [4; 31]. Additionally, \(n(t)\) refers to the AWGN signal for accounting thermal and shot noises, still present in a perfect receiver [37]. After secreting the low frequencies, transforming to an analytic representation with the assistance of the Hilbert transform,and converting to the baseband frequency, the normalized input to the OFDM demodulator is given by [7]:  
\[x_{1}(t)=\sum_{k=0}^{N_{\rm SC}-1}a(k)\exp{(j2\pi f_{k}t)}, \tag{3}\]1  
where \(n^{*}(t)\) represents the baseband AWGN signal, which may be clearly observed through the SNR at the frequency domain (see Figure 9). By assuming a perfect time and frequency synchronization at the demodulation process, the OFDM symbol is under-sampled with the fixed sampling rate to simulate the ADC process, CP is removed, and FFT is performed, see Figure 1b.