## 3 ANN and IVSTF Nonlinear Equalizers  
ANN using nonlinear decision boundaries [21] via the multilayer perceptron has been considered as a promising alternative for combating impairments in wireless communications [22; 23]. In ANN, complex mapping between input and output spaces could be achieved, however, at the cost of complexity [14; 24]. The proposed ANN for ASK-based optical Fast-OFDM signals reduces the complexity of the equalizer since it processes only amplitude levels and not phase data nor complex data. A schematic diagram is depicted in Figure 2b, in which \(s(k)\) FOFDM denotes the training transmitted Fast-OFDM sub-carriers, \(k\) (FOFDM refers to Fast-OFDM). The developed NLE using ANN consists of \(k\) neural-sub-networks in which the inherent neurons are interconnected with the Fast-OFDM weights values \(w\)_FOFDM_ (\(k\),\(i\)). The number of neurons in every neural sub-net is proportional to the level of modulation format; for example, it is 16 for 16-ASK. In the output layer, the output signal sums-up all Fast-OFDM neurons \(\hat{s}(k)_{FOFDM}\). To calculate the error, the minimum mean-square error (MMSE) is estimated, after which the neurons/weights are updated similarly to an adaptive digital filter [14]. Such learning process continues until a very low error value is reached (convergence rate). Equation (1) shows how this error is estimated:  
\[e(k)_{FOFDM}=s(k)_{FOFDM}-\hat{s}(k)_{FOFDM}, \tag{1}\]  
In Equation (1), \(\hat{s}(k)_{FOFDM}\) is the signal after the training stand can be calculated by the following mathematical formula:  
\[\hat{s}(k)_{FOFDM}=\sum_{i=1}^{16}w_{FOFDM}(k,i)\varphi_{FOFDM}(k,i)(s(k)_{FOFDM }), \tag{2}\]  
In Equation (2), \(\varphi_{FOFDM}(k,i)\) represents a nonlinear sigmoid \"activation function\" [14]. It should be also noted that the total number of neurons adopted is also proportional to the number of sub-carriers,  
Figure 1: Optical signal-to-noise ratio (OSNR) vs. bit-error-rate (BER) performance of optical Fast-orthogonal frequency division multiplexing (Fast-OFDM) over an additive white Gaussian noise (AWGN) channel.  
as the Fast-OFDM signal consists of (_k_) number of subcarriers; thus, the ANN equalizer consists of (_k_) sub-neural networks, each sub-network is designated for a different subcarrier.  
The designed ANN for NLE operation in optical Fast-OFDM performs Riedmiller-resilient based back-propagation (RP) [24; 25; 26; 27] for the purpose of updating the weights. The steepest descent algorithm is also used to find a global-data minimization [14]. In ANN numerical demonstration, a differentiable hyperbolic tangent function was employed for the hidden layer (which is only one to further reduce complexity) [26; 27] and the linear function \"purelin\" for the output layer. In Figure 2b the MMSE is processed via the RP algorithm in order for ANN to find the weights that minimize the error in terms of the vectors \(S(n)\)_FOFDM_ and \(\hat{S}(n)\)_FOFDM_:  
\[E(n)_{FOFDM}=\parallel S(n)_{FOFDM-}\hat{S}(n)_{FOFDM}\parallel^{2}, \tag{3}\]  
In the developed NLE using ANN the weights are updated according to the five steps presented in [14; 26]. Figure 2a, illustrates the benchmark received IVSTF based NLE with identical procedure to [11], which is of deterministic nature since it can tackle deterministic linear and nonlinear crosstalk effects. IVSTF was used as a comparative NLE and it was implemented in time domain in contrast to ANN-NLE, which is processed in frequency domain after the DFCT block. It should be noted that the Volterra-based equalizer is of reduced complexity compared to the well-known full-step digital back propagation [11], using up to 3 rd order kernels to account for up to 2 nd order chromatic dispersion. The complexity, however, of IVSTF is much higher than ANN since it consists of multiple FFT & IFFT blocks for fiber dispersion and nonlinearity compensation, respectively.