## 2 Background
### Coherent Optical OFDM Network  
CO-OFDM is realized via the OFDM modulation in the optical domain accompanied by the coherent detection. As mentioned in the Section 1, it exposes excellent results in terms of the receiver sensitivity and spectral efficiency. Figure 1a shows a simplified model of a CO-OFDM system [28], which consists of five parts: an OFDM modulator, a RF to optical upconverter, a fiber link, an optical to RF down-converter, and an OFDM demodulator. At the transmitter end, the input serial data bits are transformed into \(N_{D}\) data pipes and mapped onto corresponding constellation symbols. \(N_{P}\) pilot subcarriers, termed as reference tones, are periodically inserted along the data streams. The digital time-domain signal is obtained by utilizing the inverse FFT, which is subsequently added with a cyclic prefix (CP) to combat with inter-symbol interference resulted from multipath channels, and converted into real-time waveform by using an ideal digital-to-analog converter (DAC) under a fixed sampling rate [29; 30], refer to the OFDM transmitter (Figure 1b). The baseband signal of an OFDM symbol can be written as: \begin{table}
\begin{tabular}{c c} \hline Function & \(g(x)\) \\ \hline \hline \begin{tabular}{c} Sigmoid (SIG) \\ \end{tabular} & \(\frac{1}{1+\exp(-x)}\) \\ \hline \begin{tabular}{c} Cosine (COS) \\ \end{tabular} & \(cos(x)\) \\ \hline \begin{tabular}{c} Hard limit (HL) \\ \end{tabular} & \(\left\{\begin{array}{cl} 1\text{ if }x\geq 0\\ 0\text{ otherwise }\end{array}\right.\) \\ \hline \begin{tabular}{c} Triangular basis (TB) \\ \end{tabular} & \(\max(1-|x|,0)\) \\ \hline \
\begin{tabular}{c} Radial basis (RB) \\ \end{tabular} & \(\exp(-x^{2})\) \\ \hline
\begin{tabular}{c} Hyperbolic tangent (HT) \\ \end{tabular} & \(\frac{1-\exp(-x)}{1+\exp(-x)}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Popular activation functions in extreme learning machines (ELMs). \[x_{1}(t)=\sum_{k=0}^{N_{\rm SC}-1}a(k)\exp{(j2\pi f_{k}t)}, \tag{3}\]  
where \(a(k)\) and \(f_{k}\) respectively represent the transmitted data and subcarrier frequency at the \(k\)th subcarrier and \(N_{\rm SC}=N_{D}+N_{P}\) implies the total number of subcarriers. In terms of the subcarrier index, \(a(k)\) should be either modulated \(a_{d}(k)\) or unmodulated \(a_{p}(k)\). As mentioned, reference tones \(a_{p}(k)\) are periodically added among data subcarriers \(a_{d}(k)\) throughout the OFDM symbols, see the OFDM frame of Figure 1b for clarification purposes. \(1/T=f_{k}-f_{k-1}\) establishes the separation between adjacent subcarriers in order to that subcarriers become orthogonal to each other, where \(T\) defines the duration of a single OFDM symbol. Figure 1: (**a**) Block diagram of a coherent optical orthogonal frequency division multiplexing (CO-OFDM) system, where its five parts are noticed. BPF: band pass filter. (**b**) Structure of the generic OFDM modem whose differences occur in the pilots’ information as well as the equalization stage. ADC: analog-to-digital converter. The optical-upconversion stage is then realized, where the phase noise variation due to laser oscillator occurs. The modulated signal by a single-frequency laser \(E_{L}(t)\) results in References [4; 31]:  
\[x_{2}(t) = E_{L}(t)x_{1}(t) \tag{4}\] \[= A_{L}\exp\left\{j[2\pi f_{L}t+\phi_{1}(t)]\right\}\sum_{k=0}^{N_{ \mathrm{SC}}-1}a(k)\exp\left(j2\pi f_{k}t\right),\]  
where \(A_{L}\) denotes the amplitude of the optical tone, \(f_{L}\) refers to the optical carrier frequency, and \(\phi_{1}(t)=\int_{-\infty}^{t}\delta w_{1}(\tau)d\tau\) is the laser phase noise. \(\delta w_{1}(t)\) is frequency noise with zero mean and autocorrelation function equals to \(2\pi\Delta v_{1}\delta(\tau)\), where \(\Delta v_{1}\) and \(\delta(\tau)\) are respectively the laser linewidth parameter and Dirac delta function.