## 2 Principles of the Proposed ANN Equalizer  
Linear optical distortions induced in the short-haul fiber-optic link are converted into nonlinear impairments in the electrical domain. This is because of losing the phase information due to the square-law detection of the direct detector (DD). A nonlinear electrical equalizer is therefore needed at the digital signal processing (DSP) unit to compensate for the nonlinear distortions, recovering the transmitted data. Neural networks have shown promising modeling performance in various biomedical and chemical engineering, biological, and medical fields and applications [31; 32]. Due to the superior linear and nonlinear modeling ability of artificial neural networks (ANNs), they can be utilized, after a proper training scheme, to compensate for these nonlinear system distortions, inverting the response of both the fiber channel and direct photo-detection system. The schematic of the proposed ANN nonlinear equalizer is shown in Figure 1. Since our objective is to implement an effective but computationally inexpensive nonlinear equalizer, we select the basic ANN structure consisting of an input layer, a single hidden layer, and an output layer. The input layer comprises \(N\) nodes, each of which is triggered by a sample of the received (distorted) signal \(y[n]\). In other words, the input vector at each time instant \(i\) represents the current input sample with its \(K/2\) previous and \(K/2\) subsequent samples, where \(N=2K+1\) is the length of the intersymbol interference (ISI) span. Each line connecting two nodes represents a multiplier that multiplies the output of the previous layer by the corresponding weight. The hidden and output layers include \(m\) and 1 nodes, respectively. Each node in the hidden and the output layer is a computing unit that applies a nonlinear activation function to the summation of its weighted inputs. The relationship between the input vector \(\mathbf{Y}=[y_{-K}\,y_{-K+1}\,y_{-K+3}\cdots y_{K}]\) and the ANN output \(\hat{X}\) is thus expressed as follows:  
\[\hat{X}=f_{\mathrm{o}}\Big{(}\sum_{j=1}^{m}w_{j}^{\mathrm{o}}f_{h}\big{(}\sum _{l=1}^{N}w_{jl}^{h}\,y_{l}\big{)}\Big{)}, \tag{1}\]  
where \(w_{jl}^{h}\) is the weight assigned to the connection between the \(l^{th}\) input \(y_{l}\) and the \(j^{th}\) node of the hidden layer. The weight \(w_{j}^{\mathrm{o}}\) is assigned to the connection between the \(j^{th}\) hidden-layer node and the output-layer node. The scalar functions \(f_{h}(\cdot)\) and \(f_{o}(\cdot)\) are the nonlinear activation functions of the hidden and output layers, respectively. It should be clear that the ANN output \(\hat{X}\) represents the equalized sample at time instant \(i\). Using a proper training scheme, the neural network weights can be adjusted such that the output sequence \(\hat{X}[n]\) represents an approximate estimate of the undistorted transmitted sequence \(x[n]\). The training of the ANN equalizer is performed as follows. A stream of known data is first transmitted through the fiber channel to the receiver, and a set of known input-output training data pairs is formulated. The weights of the ANN equalizers are then adjusted by recursively solving an optimization problem that minimizes the mean square error (MSE) between the equalizer outputs (representing the equalized signal samples) and the desired outputs (representing the transmitted signal samples). The training problem is thus given by  
Figure 1: Structure of the proposed artificial neural network (ANN)-based equalizer. \[\underset{\mathbf{W}}{min}\,E(\mathbf{W})=\frac{1}{N_{tr}}\sum_{k=1}^{N_{tr}}\|\hat{X}_{k }(\mathbf{W},\mathbf{Y}_{k})-X_{k}\|_{2}^{2}, \tag{2}\]  
where \(\mathbf{W}=\left[\mathbf{W}_{h}^{T}\,\mathbf{W}_{o}^{T}\right]^{T}\) is a vector that contains the set of weights connecting the input parameters to the hidden-layer nodes \(\mathbf{W}_{h}\) and the set of weights connecting the hidden-layer nodes to the output-layer node \(\mathbf{W}_{o}\).