## 3 Applications of the CDS for the Fiber Optic Link
### Perceptor
#### 3.1.1 Three-Layered Bayesian Generative Model
For simplicity, we have assumed \(\Delta x=\Delta y\) and we define precision factor (PF) = \(10\Delta x=10\Delta y\). The normalized received symbols, \(\overline{Y}_{n}^{k}\) that are inside the shaded box (see Figure 10) are mapped to the center of the square as \((x_{l},~{}y_{m})\), where \(l\) is the index for in-phase and \(m\) is the index for quadrature. For example, if \(x_{l}=1\), \(y_{m}=2\) and \(\overline{Y}_{n}^{k}=1.1+2.05j\), it is mapped to \(\hat{Y}_{n}^{k}=1+2j\). The discretized symbols, \(\hat{Y}_{n}^{k}\) are sent to Layer III for probability estimation. This discretization is inspired by the brain. Our brain monitors the temperature, light, and other environmental conditions and maps this low-level representation to our initial perception, which is a high-level representation such as day or night, cold or warm [34]. The low-level representation requires a huge amount of data, which is mapped to an important set of data (initial perception) that takes significantly less memory. This discretization can be more complicated if we use non-uniform discretization. Figure 11 shows the comparison between the CDS system with PF = 0.5, 1, 2, 5 and the conventional system for 52 Gb/s data rate (details on conventional system and BER calculation are provided in Section 5.1). Thus, a lower PF means a larger BER improvement and higher model accuracy. However, it leads to higher computational cost. The Layer II also receives the BER as top-bottom attention and it can choose the PF based on the BER. Layer II will send the discretized received symbols as \(\hat{Y}_{n}^{k}\) to Layer III. Figure 10: Discretization of received symbols. Figure 9: Received symbols (\(\hat{Y}_{n}^{k}\)) after linear equalization from the simulation results. In this layer, the system estimates the probability of receiving \(Y_{n}^{k}\) for the given transmitted symbol \(X_{n}^{k}\), i.e., \(P(Y_{n}^{k}\,|X_{n}^{k})\) by approximating it as the probability of \(\hat{Y}_{n}^{k}\) for given \(\hat{X}_{n}^{k}\) (i.e., \(P(\hat{Y}_{n}^{k}\,|\hat{X}_{n}^{k})\)) using the Monte-Carlo method. The transmitted symbols \(X_{n}^{k}\) could be known to the receiver by sending training symbol sequences during the statistical model extraction process. However, this would interrupt the service. Instead, we approximate \(X_{n}^{k}\) by \(\hat{X}_{n}^{k}\) which is the estimate of \(X_{n}^{k}\) after the FEC. We assume that the BER under FEC threshold is10\({}^{-2}\)[35]. The BER after FEC for such a threshold is between 10\({}^{-15}\) and 10\({}^{-12}\)[35]. We need a certain number of data frames (each frame includes 256 subcarriers) for accurate estimation of \(P(\hat{Y}_{n}^{k}\,|\hat{X}_{n}^{k})\). Figure 12 shows the BER as a function of the number of received OFDM frames used in the calculation of \(P(\hat{Y}_{n}^{k}\,|\hat{X}_{n}^{k})\). As shown, the BER converges to \(2.7\times 10^{-3}\) after 640 OFDM frames for the CDS system with PF = 2. Therefore, it can automatically find the sufficient number of OFDM frames. For example, suppose that the number of transmitted symbols, \(X_{n}^{k}=1-3j\) in a subcarrier is 100,000. Due to channel noise and distortion in the fiber optic link, they could be mapped to any point in Figure 10. Next, suppose that the number of received symbols (after normalization and discretization in layers I and II) with \(Y_{n}^{k}=2.2-1.5j\) is 10,000 when the transmitted symbol is \(X_{n}^{k}=1-3j\). Layer III counts these 10,000 received symbols are counted and calculates \(P(\hat{Y}_{n}^{k}=2.2+1.5j|\hat{X}_{n}^{k}=1-3j)=0.1\). In other words, in Layer III, the number of hits in each of the cells of Figure 10 is calculated and the calculated probability \(P(\hat{Y}_{n}^{k}\,|\hat{X}_{n}^{k})\) is stored in a tensor whose dimension is shown in the second column of Table 2. The probability calculation is done for each type of the transmitted symbol. For example, QAM-16 has 16 constellation points or 16 different types of transmitted symbols. The tensor shown in Table 2 has three dimensions.