## 5 Complexity Analysis  
To evaluate the feasibility of the proposed MFI algorithm for use in practical systems, its computational complexity had to be examined. The detailed complexity analysis of the proposed scheme was thus undertaken. The results are presented in Table 3, where the complexities of the five operations involved in the proposed MFI scheme are shown. For the proposed scheme, the required real multipliers, real adders and comparators were \(N+6\times(nbins+1)\), \(N+7\times nbins+6\) and \(2\times N+(2\times N+10)\times nbins+1\), respectively; here, \(N\) is the number of symbols (6000), and \(nbins\) is the number of histogram bins (40). A complexity comparison between the proposed MFI scheme and schemes using DNN, PCASP and M-PSO is shown in Figure 18. Without considering the complexity of power normalization and amplitude histogram generation, in the testing stage of SVM, O(\(N_{s}\cdot d\cdot N\)) operations were required when RBF kernel was applied [38], where \(N_{s}\), \(d\) and \(N\) are the number of support vectors, the dimension of the input data and the number of testing  
Figure 17: Correct identification rates (blue triangles) of the four modulation formats with different launch powers in long-haul transmission experiments. (**a**) PDM-QPSKsignals over a 2000 km link; (**b**) PDM-8 QAM signals over a 2000 km link; (**c**) PDM-16 QAM signals over a 1040 km link; (**d**) PDM-32 QAM signals over a 400 km link. Green dots indicate the error vector magnitudes (EVM). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operation** & **Multipliers** & **Adders** & **Comparators** \\ \hline Power normalization & \(N\) & \(N\)-1 & 0 \\ Generation of amplitude & 0 & \(2\times nbins\) & \(2\times nbins\times N+2\times N-3\) \\ histograms & 0 & \(nbins\)-1 & \(10\times nbins\) \\ Partition operation & 0 & \(nbins\)-1 & \(10\times nbins\) \\ Calculation of ratios & \(6\times(nbins+1)\) & \(4\times nbins+8\) & 0 \\ Comparing with threshold & 0 & 0 & 4 \\ Total & \(N+6\times(nbins+1)\) & \(N+7\times nbins+6\) & \(2\times N+(2\times N+10)\times nbins+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computational complexity of the proposed MFI scheme. samples, respectively. Since the support vector cannot be fully quantified, the complexity of SVM was not considered, as shown in Figure 18. In the numerical simulations, \(N_{s}\) and d were 805 and 40, respectively. As shown in Figure 18, the required multiplications, additions and comparisons for the M-PSO scheme were much higher than those of the proposed scheme. In addition, 16,000 look-up table (LUT) operations were also required for the M-PSO scheme. When the number of symbols \(N\) equaled 2048, the PCASP scheme needed calculations of 131,143 real multiplications, 108,612 real additions, 14 comparisons and 22 LUT operations. For the proposed scheme, the required real multiplications, real additions and comparisons were 6246, 6286 and 492,401, respectively. It should be noted that the complexity of the comparators was much lower than those related to multipliers and adders [29]. The proposed scheme and the scheme using DNN were both based on amplitude histogram. Without considering the complexities of power normalization and amplitude histogram generation, the required real multipliers, real adders and comparators for the identification process in the proposed scheme were 6 \(\times\) (_nbins_ + 1), 5 \(\times\)_nbins_ + 7 and 10 \(\times\)_nbins_ + 4, respectively. The identification process in DNN, however, required \(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{4}\) + C-1 multipliers, (\(N_{1}\)-1)\(N_{2}+(N_{2}\)-1)\(N_{3}+(N_{3}\)-1)\(N_{4}\) + C-1 adders and \(N_{2}+N_{3}+C\)-1 comparators, respectively [6]; here, \(N_{1}\), \(N_{2}\), \(N_{3}\) and \(N_{4}\) are the numbers of neurons in the input, first hidden, second hidden and output layer, respectively, and C is the number of identified modulation formats. In this paper, \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\) and C were taken to be 40, 40, 10, 5 and 5, respectively.