## 2 Principles of the Proposed ANN Equalizer
The operation \(\|\cdot\|_{2}\) denotes Euclidean norm, \(N_{tr}\) is the number of training symbols, and \(\mathbf{Y}_{k}\) is the input equalizer vector corresponding to the \(k^{th}\) training symbol. The parameters \(\hat{X}_{k}\) and \(X_{k}\) are the equalizer output and the corresponding desired output for the \(k^{th}\) training symbol, respectively. To achieve the most efficient ANN equalization performance, converging to the global optimal weights of the equalizer, it is crucial to determine (i) the number of input-layer nodes \(N\), i.e., the number of equalizer taps, (ii) the number of hidden-layer nodes \(m\), and (iii) the number of training data \(N_{tr}\) and the way to select them. The number of equalizer taps \(N\) is chosen according to the length of the ISI of the channel. In other words, depending on modulation format, optical-fiber length \(L\), and dispersion parameter \(\beta_{2}\) of the optical-fiber channel, the number of neighboring symbols \(K\) that may interfere with the current symbol (i.e., ISI span) is estimated. The number of taps required by the equalizer is then determined as \(N=2K+1\). The number of required hidden nodes \(m\) is obtained through a convergence analysis. First, the hidden nodes number \(m\) is set to a certain small number, e.g., 3, and the training problem (2) is solved, and the terminated MSE is calculated. Then, we gradually increase the value of \(m\) and resolve problem (2) until the terminated MSE value saturates up to a certain accuracy. Typically, choosing the training data and their number \(N_{tr}\) is done randomly. However, it has been shown in [21] that the equalization performance of neural networks-based equalizers could be significantly enhanced utilizing an extensive training scheme that considers all possible patterns to be received by the equalizer later on. Instead of transmitting a random dataset during the training phase, the extensive training scheme uses a complete set of training data formulated as in [21]. For an \(M\)-ary modulated data and ISI span length of \(N\) (i.e., \(K\) interfering symbols), there is a total combination of \(M^{N}\) distinct patterns of the training data samples. In other words, we have \(\left(N\times M^{N}\right)\) different input training vectors and \(\left(1\times M^{N}\right)\) corresponding desired outputs. However, while propagating inside the fiber, the outer symbols in each pattern can be affected by the tail of its successive neighboring patterns. Hence, the equalizer may receive patterns that are not included in those basic \(M^{N}\) patterns. To take into account all possible combination patterns that could be received, \(K\) redundant guard symbols are added around the sides of each basic pattern, as shown in Figure 2a. First, \(M^{N+2K}\) different symbols are generated and transmitted through the fiber. At the DSP unit, the samples corresponding to those redundant guard symbols are discarded, as illustrated in Figure 2a, and only samples with regard to original data are used to train the equalizer. The number of training data used by the extensive training scheme is therefore \(N_{tr}=M^{N+2K}\). In other words, a total of \(\left(N\times M^{N+2K}\right)\) input training vectors and \(\left(1\times M^{N+2K}\right)\) corresponding desired outputs are utilized. It should be clear that this extensive training scheme considers all possible received patterns but in a noise-free channel. In other words, it does not account for the noise induced in the system. Therefore, the equalization performance of an NN-based equalizer trained using the extensive training scheme is severely limited by the system noise. The equalization performance though may be potentially enhanced if noisy data are used to train the equalizer's neural network. To enhance the efficiency and robustness of the ANN equalizer, we propose an advanced training scheme that utilizes multiple noisy sets of the complete patterns used in the extensive training scheme.