# Machine Learning Applications for Short Reach Optical Communication
## 3 Machine Learning for Short-Reach Optical Communication System Signal Processing
### Machine-Learning-Based Equalizer for Short-Reach Optical Communications
#### 3.1.1 Feed Forward Neural Network Based Equalizer  
FFNN has been widely used as the equalizer in short-reach optical communication systems. The FFNN used for equalization applications has the same neural network architecture as the ANN shown in Figure 4, but with a unique input data arrangement. The input of FFNN comes from a tapped delay line, which generates a delayed input time series data. Hence, we use FFNN to refer to the ANN with delayed time series data input when discussing ML-based signal equalization. Figure 8 shows the structure of an FFNN designed for signal equalization, in which the input of the ANN is fed with a time-delayed series of data received in the optical communication system. The FFNN consists of the input layer, the hidden layer and the output layer. Instead of feeding received data to the input layer directly, in FFNN, the sampled signals are fed into the input layer via the time delay array. Such an array creates different time delays, ensuring that FFNN understands both the current and the past information to conduct effective equalization. After the input layer, the received signal is further processed by the hidden layer with a nonlinear activation function to compensate for the nonlinearity in the optical communication system. Then, the distorted signals are reconstructed, and equalization coefficients are obtained at the output layer. The mathematical expression of the FFNN process is:  
\[y(k)=f\big{(}X_{k}^{T}W_{ik}+B_{h}\big{)}\cdot W_{ho}+B_{o} \tag{14}\]  
where \(X_{k}\) is the input signal sequence, \(W_{ik}\) is the weight matrix between the input layer and the hidden layer, \(W_{ho}\) is the weight matrix between the hidden layer and the output layer, \(B_{h}\) and \(B_{o}\) are the bias metrics of hidden layer and output layer, and \(f\) is the activation nonlinear function of the hidden layer. A nonlinear equalizer based on FFNN has been proposed in an optical communication system with direct detection receivers. A radial basis function (RBF) network, which is based on the FFNN structure, is utilized to perform signal equalization to compensate the dispersion in the system [114]. The RBF method is a nonlinear equalization technology, and it takes the nonlinear transformation function of the photodetector into consideration. Hence, it is well suited for short-reach optical communications, where low cost square-law direct detection is normally used. In this study, the input function of the RBF model is expressed as:  
\[r=[y(i-1)\quad y(i-2)\quad\dots\quad y(i-m)]^{T} \tag{15}\]  
where \(r\) is a real vector space with \(m\) dimensions expressed as \(R^{m}\). Then, the expression of output layer after hidden layer is obtained by a \(R^{m}\) to \(R\) (scalar) mapping:  
Figure 8: Feedforward neural network structure. \[\hat{y}(i)=\omega_{0}+\sum_{j=1}^{K-1}\Phi_{j}(r(i))\cdot\omega_{j} \tag{16}\]  
where \(\omega_{0}\)... \(\omega_{K-1}\) are the weights of the connections between the hidden layer and the output layer, and the nonlinear activation function is implemented by the hidden nodes \(\Phi_{j}(. )\), \(1\leq j\leq K\). The nonlinear function selected by this work is:  
\[\Phi_{j}(r)=\exp(-\hat{r}^{2}/\alpha_{j}^{2}) \tag{17}\]  
where \(\hat{r}=||r(i)-s_{j}||\) is nonlinear function's input parameter, \(s_{j}\) are the RBF centers, and \(\alpha_{j}\) are positive scalars known as width. In the \(R^{m}\) space, the characteristic of nonlinearity \(\Phi_{j}(. )\) has a response with a radially symmetric shape around the center. In this study, the RBF equalizer is implemented offline and experimentally verified in the on-off keying (OOK) system. Results show that RBF equalizer obtained approximately one order of magnitude lower BER than FFE at 20 dB SNR over up to 100 km fiber length. Whilst the study in [114] demonstrates the capability of FFNN, in a high-speed short-reach system, multi-level PAM is preferred compared with OOK due to the higher spectral efficiency.