## 3 Applications of the CDS for the Fiber Optic Link
### Perceptor
#### 3.1.1 Three-Layered Bayesian Generative Model  
Figure 8 shows the three-layered Bayesian modeling for a long-haul fiber optic system inspired by the human brain. Suppose that, initially, there exists no statistical model of the system in a model library. Then, the three-layered Bayesian modeling block extracts the statistical model of the system. We illustrate the modeling process by performing the numerical simulation of the OFDM system. The simulation parameters are presented in Table 1.  
The signal propagation in an optical fiber is described by the nonlinear Schrodinger equation (NLSE), which can be solved using the standard split-step Fourier scheme [32; 33]. The output of the fiber optic link passes through a coherent receiver and then the Fast-Fourier transform (FFT) is  
Figure 8: Three layered Bayesian modeling inspired by the brain for modeling fiber optic link. applied to demultiplex the subcarriers of the OFDM data (See Section 5 for more details). The channel estimation and compensation of linear impairments is done using the one tap linear equalizer that uses 16 training frames to mitigate linear distortion [33]. The output of the linear equalizer is passed to the preceptor. \(X_{n}^{k}\) represents the transmitted QAM-16 symbols which takes any one of the values \(X_{t}\) from the following set \(\{\pm 3\pm 3j,\ \pm 1\pm 1j,\ \pm 1\pm 3j,\ \pm 3\pm 1j\}\), with equal probability at the \(k\)th cycle and at discrete time \(n\), and \(t\) is the index denoting one of the 16 possible symbols. \(\gamma_{n}^{k}\) represents the corresponding received symbols after linear equalization. The three-layered Bayesian modeling consists of three layers, as explained below. Layer I  
Figure 9 shows the in-phase (I) and quadrature (Q) of the received data \(Y_{n}^{k}\). We define 96% probability box (or 99% probability box depending on the application), which means that 96% of the received data is inside this box when QAM-16 data is sent. In addition, the system maps and normalizes all receiving symbols outside of this box to its borders. For example, if the received symbols \(\gamma_{n}^{k}\) is \(8+9j\), then it is mapped to \(6+6j\) when the 96% box is used. The normalized data \(\overline{Y}_{n}^{k}\) (i.e., \(6+6j\) in this example) is sent to Layer II (Figure 8) for further processing. The BER is used to adaptively select the borders of the probabilistic box. For lower computational cost and faster modeling, a probability box with smaller area is better. For example, the computational cost associated with the 90% probabilistic box is a lower than the 99% probabilistic box. However, this leads to inaccurate results and higher BER. Therefore, the BER is used as a feedback to choose the optimum probability box for which the computational cost is lower with acceptable high accuracy (in this example, it is 96% probabilistic box, see Figure 9). \begin{table}
\begin{tabular}{c c} \hline
**Simulation Parameters** & **Value** \\ \hline Fiber dispersion coefficient (\(\beta_{2}\)) & \(-22.1\) ps\({}^{2}\)/km \\ Fiber nonlinear coefficient (\(\gamma\)) & \(1.1\) W\({}^{-1}\)km\({}^{-1}\) \\ Fiber loss coefficient (\(\alpha\)) & \(0.2\) dB/km \\ Number of fiber spans (\(N\)) & \(20\) \\ Span length (\(L_{span}\)) & \(80\) km \\ Noise figure (\(NF\)) & \(4.77\) dB \\ Number of OFDM subcarriers per frame & \(256\) \\ Subcarrier modulation & QAM-16 \\ Guard intervals (cyclic prefix) & \(1.86\) ns \\ Number of data frames & \(1024\) \\ Line width of transmitter laser/LO & \(22\) kHz \\ Data carrying subcarriers per frame & \(126\) \\ Over sampling factor & \(2\) \\ Number of pilot subcarrier & \(2\) \\ Wavelength (\(\lambda\)) & \(1550\) nm \\ Length of PRBS & \(2^{16}-1\) \\ Fiber type & Standard single mode fiber (SSMF) \\ \hline \end{tabular}
\end{table} 
Table 1: The numerical simulation parameters of the orthogonal frequency-division multiplexing (OFDM) system. In Layer II, the received symbols are discretized with discretization steps \(\Delta x\) and \(\Delta y\).