## Operating Principle  
The digital signal processing (DSP) procedure of the autonomous digital coherent receiver considered herein is depicted in Figure 1, where the proposed MFI scheme has been highlighted. Before conducting the MFI operation, modulation format-independent algorithms (shown in the purple area) were employed to compensate for the CD impairments and timing-jitter and to achieve preliminary polarization demultiplexing. It should be noted that _m_PSK signals can be completely polarization-demultiplexed by CMA. CMA is also capable of compensating residual CD and PMD for different modulation formats [2; 5; 28; 31]. However, for _m_QAM (_m_ > 4) signals, further polarization demultiplexing algorithms may also be required. After having performed the modulation format-independent operations, the proposed MFI scheme could then be applied. Its outputs were provided to the subsequent modulation format-dependent algorithms (shown in the pink box), which consisted of the multi-modulus algorithm (MMA), frequency offset compensation and carrier phase recovery to compensate for various transmission impairments experienced by the signals. As illustrated in Figure 1, the proposed MFI scheme consisted of three steps: first, the CMA equalized signals were power normalized. Then, the amplitude histograms of the signals were obtained, and calculations of the ratios (defined below) were subsequently undertaken. It should be noted that CMA-equalized signals of only one polarization were required to acquire the amplitude histogram. Finally, based on the thresholds of the ratios \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\), five commonly-adopted modulation formats, including PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM, were identified. As shown in Figure 2, there were one, two, three, five, and nine amplitude levels for QPSK, 8 QAM, 16 QAM, 32 QAM, and 64 QAM, respectively. For each level of these five modulation formats, the amplitude value and its associated probability are presented in  
Figure 1: The DSP architecture with the proposed MFI scheme for autonomous digital coherent receivers. Table 1. Without loss of generality, the symbols of each modulation format were assumed to have a unit average power [37]:  
\[\sum_{n=1}^{m}p_{n}\cdot Z_{n}^{2}=1 \tag{1}\]  
where \(Z_{n}\) is the amplitude value, \(p_{n}\) is the associated probability for the \(n\)-\(th\) level, and \(m\) is the number of total levels for the considered modulation format. The amplitude levels of the different modulation formats can be clearly seen in Figure 2. However, in practical transmission systems, the distributions of the amplitude levels may be indistinct, even if CD, timing-jitter, and partial polarization mixing are compensated for using the modulation format independent algorithms. As such, it could be difficult to identify high-order modulation formats, especially in low OSNR cases. To address this challenge, the amplitude histogram of the CMA-equalized signals was employed to identify these five modulation formats over a considerably widened OSNR range. The amplitude histograms of these five modulation formats are shown in Figure 3, where the OSNR values for the PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM signals \begin{table}
\begin{tabular}{c c c c c} \hline
**QPSK** & **8 QAM** & **16 QAM** & **32 QAM** & **64 QAM** \\ \hline \multirow{8}{*}{1/100\%} & & & & 0.218/6.25\% \\  & & & 0.488/12.5\% \\  & & & 0.316/12.5\% & 0.655/6.25\% \\  & 0.577/50\% & 0.447/25\% & 0.707/25\% & 0.787/12.5\% \\  & 0.577/50\% & 1/50\% & 0.949/12.5\% & 0.900/12.5\% \\  & 1.291/50\% & 1.342/25\% & 1.140/25\% & 1.091/18.75\% \\  & & & 1.304/25\% & 1.175/12.5\% \\  & & & 1.327/12.5\% \\  & & & & 1.528/6.25\% \\ \hline \end{tabular}
\end{table} 
Table 1: Amplitude value and associated probability for each level of the five modulation formats. Figure 2: Constellation diagrams of five widely-employed modulation formats: (**a**) QPSK, (**b**) 8 QAM, (**c**) 16 QAM, (**d**) 32 QAM and (**e**) 64 QAM.