## 6 Computational Complexity Analysis  
When evaluating and comparing equalizers complexity it is essential to take the nature of the equalizer in to considerations, such as the DBP and IVSTF are essentially different from machine learning-based NLEs such as ANN and SVM. That is, as the IVSTF equalization mechanism is based on the idea of reversing of the effect propagation model; consequently, which makes it dependent on the number of fiber spans and the oversampling rate, therefore complexity, does not depend on other signal parameters such as modulation format levels [13]. As the IVSTF equalization is implemented in the hybrid time and frequency domain, it requires several conversions from time to frequency domain and vice versa, which intern makes the complexity depends on the FFT/IFFT pairs that operate on data blocks of size \(Nblock=K*Nsub\), where \(k\) is the oversampling constant and \(Nsub\) is the number of subcarriers. Consequently, the total number of operations for the IVSTF-based NLE is given by Equation (4), [13]:  
\[N_{IVSTF}=\Big{(}N_{Span}+1\Big{)}8N_{Block}Log_{2}N_{Block}+\Big{(}20N_{Span}-6 \Big{)}N_{Block}+16(N_{Span}+1), \tag{4}\]  
On the contrary to IVSTF, ANN based NLEs complexity depends upon some signal parameters, such as the number of modulation forma level and the number of OFDM subcarriers. The idea behind the ANN concept is based on the imitation of the human brain, as it utilizes a large number of low complexity neurons, which makes their implementation require fewer FLOPs when compared to other approaches like IVSTF. Therefore, the total number of operations performed for processing each Fast OFDM symbol can be obtained by Equation (5) [12; 13]. \[FLOPS_{ANN}=Nop_{IL}+Nop_{HL}+Nop_{OL}=\Big{[}N_{SC}^{2}*M\Big{]}+[\left(N_{SC}* M\right)^{2}*\left(N_{HL}-1\right)]+[N_{SC}^{2}*M], \tag{5}\]  
where the \(Nop_{IL}\), \(Nop_{HL}\) and \(Nop_{OL}\) are the Flop count for input-, output-, and hidden-layers which are illustrated below in Equations (5)-(8) respectively. \[Nop_{IL}=\Big{[}N_{SC}^{2}*M\Big{]}, \tag{6}\]  
\[Nop_{HL}=(N_{SC}*M)^{2}*\left(N_{HL}-1\right), \tag{7}\]  
\[Nop_{OL}=\Big{[}N_{SC}^{2}*M\Big{]}, \tag{8}\]  
where \(N_{SC}\) is the number of subcarriers, \(M\) is modulation format level and \(N_{HL}\) is the number of hidden layers. It is worth noting that in the Fast-OFDM case the complexity is reduced by a factor of two when compared to the conventional Optical-OFDM ANN-NLE case as shown in Equation (9), this is due to the fact that in Optical OFDM signals has real and imaginary components that the ANN-NLE has to deal with, where for the Fast-OFDM case the equalizer only deals with the real data [13]. \[FLOPS_{ANN}=Nop_{IL}+Nop_{HL}+Nop_{OL}=2\Big{[}N_{SC}^{2}*M\Big{]}+2[\left(N_{ SC}*M\right)^{2}*\left(N_{HL}-1\right)]+2[N_{SC}^{2}*M], \tag{9}\]  
Below, Figure 8 illustrates a comparison between ANN-NLE and IVSTF-NLE in terms of subcarriers versus Floating points Operations (FLOPs). As the ANN-NLE is a modulation format dependent, therefore, different modulation formats M-ASK (M = 4, 8, and 16) has been used, moreover for the case of number of spans dependent IVSTF-NLE, the modulation formats has been set to 4 ASK, thereafter the number of span has been changed as follow 1 span, 10 spans, and 20 spans. In terms of computational complexity FLOPS, it is shown that for all the number of subcarriers, ANN-NLE outperforms IVSTF-NLE. This is true even, when the best-case scenario of IVSTF-NLE, at 1 span, is compared to the worst-case scenario of ANN-NLE at 16 ASK, while fixing the number of subcarriers. This superiority is due to the utilization of a large number of low complexity neurons,which results in a low computational complexity when compared to the IVSTF-NLE that requires several FFT/IFFT blocks to perform the equalization. It is worth noting that the proposed nonlinear equalizer can be very useful when applied in cost-sensitive metro-regional and short-reach networks.