## 3 Results and Discussion
Under the Gaussian noise assumption, the BER is related to the quality factor by [1,2]:  
\[\text{BER}=\frac{1}{2}erfc\bigg{(}\frac{Q}{\sqrt{2}}\bigg{)}, \tag{4}\]  
where \(erfc(\cdot)\) denotes the complementary error function. The quality factor is obtained using the following formula [1,2]:  
\[Q=\frac{I_{1}-I_{0}}{\sigma_{1}+\sigma_{0}}, \tag{5}\]  
where \(I_{r}\) and \(\sigma_{r}\), \(r\in\{0,1\}\), are the mean and the standard deviation of the received signal, respectively, and \(r\) stands for the type of the transmitted symbol (\({}^{\prime}0^{\prime}\) or \({}^{\prime}1^{\prime}\)). It is worth emphasizing that single-polarized signals are typically used in short-reach optical-fiber communication systems where polarization mode dispersion (PMD) effect is negligible. The metric of BER performance is therefore enough to analyze the overall system performance since the optical-fiber channel is deterministic. Note also that the impact of fiber dispersion can be characterized by the dimensionless parameter: \(B^{2}\beta_{2}L\), where \(B\) is the baud rate, \(L\) is the fiber transmission distance, and \(\beta_{2}\) is the fiber dispersion coefficient. For the given baud rate, if the fiber dispersion \(\beta_{2}\) and/or the distance \(L\) increases, the ISI effect increases, which degrades the transmission performance. On the other hand, if the dispersion is fixed, the feasible transmission distance for the given BER scales inversely with \(B^{2}\). Throughout this section, we show the BER performance versus the received optical power. However, the received power and signal-to-noise ratio (SNR) can be related from the following considerations. The receiver adds shot noise and thermal noise whose variances are given by [2]:  
\[\sigma_{shot}^{2}=2 qRP_{r}B_{e}, \tag{6}\]  
\[\sigma_{thermal}^{2}=4K_{B}TB_{e}/R_{L}, \tag{7}\]  
where \(q\) is the electron charge, \(R\) is the responsivity, \(P_{r}\) is the received power, \(B_{e}\) is the 3 dB bandwidth of the receiver electrical LPF, \(K_{B}\) is the Boltzmann constant, \(T\) is absolute temperature in Kelvin, and \(R_{L}\) is the load resistance. The SNR (without including optical noise) is given then by  
\[SNR=\frac{R^{2}P_{r}^{2}}{\sigma_{shot}^{2}+\sigma_{thermal}^{2}}. \tag{8}\]  
Thus, SNR is directly related to the received optical power. Secondly, using the input-output relationship of the proposed ANN equalizer (i.e., Equation (1)), it can be shown that the computational complexity of the ANN equalizer scales as \(\sim\)\(m\times N\) per symbol, where \(m\) and \(N\) are the numbers of hidden layer's nodes and equalizer's taps, respectively. Thirdly, once the ANN equalizer is trained, its memory storage requirements are essentially limited to the number of its neural network weight parameters \(W\). For an ANN equalizer comprising \(N\) input taps and a single hidden layer with \(m\) nodes, the total number of weights is \((mN+m)\), with \((mN)\) parameters corresponding to the weights connecting between the input and hidden layer, and \(m\) weight parameters assigned to the connection between the hidden and output layer, as illustrated in Figure 1. Note that each real number is stored in the memory as a floating-point number [35]. Since the ANN equalizer weights are generally real numbers, the memory storage requirement of the proposed ANN equalizer is \(\sim\)\(m(N+1)\) floating-point numbers. Unless otherwise specified, the proposed ANN equalizer is trained using the advanced training scheme, described in Section 2. The number of noisy dataset groups is \(n=10\). In other words, the total number of training data used is \(N_{tr}=10\times M^{N+2K}\) symbols, where \(M=2\) for the OOK modulation format. A total of 87.5% of the training dataset is used for training, while 12.5% of the data is utilized as the validation set during each epoch. The maximum allowed number of training epochs is 1000, and the batch size is 256. The hidden and output activation functions of the ANN are rectified linear unit (\(ReLU\)) function and unity function, respectively.