## Operating Principle
There are one, two, three, five and nine amplitude levels for QPSK, 8 QAM, 16 QAM, 32 QAM and 64 QAM, respectively. Since 8 QAM could be distinguished based on the ratio \(H_{1}\), the purpose of the second partition operation was to identify QPSK from other three modulation formats. As shown in Figure 4, the amplitude histograms of the remaining four modulation formats were divided into parts A, B and C. The numbers of the symbols in parts A, B and C were \(N/4\), \(N/2\) and \(N/4\), respectively. Due to the constant amplitude, the distribution of amplitudes for QPSK still concentrated around 1, even in the presence of noise. As a result, compared with the three other modulation formats, the amplitude distribution of the QPSK signals in part A and C was much closer. However, because of the influence of noise, such a distribution feature gradually grew less obvious for low OSNR cases. To address this  
Figure 4: The second partition operation for the amplitude histograms of QPSK, 16 QAM, 32 QAM and 64 QAM. The amplitude histograms of the four modulation formats are divided into part A, B and C. The number of symbols in part A, B and C are \(N/4\), \(N/2\) and \(N/4\), respectively. Figure 5: The third partition for the amplitude histograms of 16 QAM, 32 QAM and 64 QAM. The first \(N/4\) symbols are divided into three parts. The numbers of the symbols in part A, B and C are \(N/16\), \(N/8\) and \(N/16\), respectively. problem, we conducted a square operation of the number of symbols in each individual bin, and the corresponding ratio \(H_{2}\) was defined as  
\[H_{2}=\frac{H_{\text{C2}}}{H_{A2}} \tag{3}\]  
\[H_{A2}=m_{1}^{2}\times c_{1}+\cdots+m_{i}^{2}\times c_{i}+\left(\frac{N}{4}- \sum\limits_{x=1}^{i}{m_{x}}\right)^{2}\times c_{i+1}\]  
where  
\[H_{\text{C2}}=\left(\sum\limits_{x=1}^{j+1}{m_{x}}-\frac{3N}{4}\right)^{2} \times c_{j+1}+m_{j+2}^{2}\times c_{j+2}+\cdots+m_{nbins}^{2}\times c_{nbins}\]  
\(\sum\limits_{x=1}^{i}{m_{x}}\leq\frac{N}{4}\), \(\sum\limits_{x=1}^{i+1}{m_{x}}>\frac{N}{4}\)  
\(\sum\limits_{x=1}^{j}{m_{x}}\leq\frac{3N}{4}\), \(\sum\limits_{x=1}^{j+1}{m_{x}}>\frac{3N}{4}\)  
Due to the closer amplitude distributions of parts A and C, the ratio \(H_{2}\) for QPSK was smaller than those of the three remaining modulation formats. Thus, QPSK could be identified. After that, the partition of the amplitude histogram remained unchanged. The difference was that only part B and C were taken into consideration. The ratio \(H_{3}\) was introduced, calculated as  
\[H_{3}=\frac{H_{B3}}{H_{\text{C3}}} \tag{4}\]  
where \(H_{B3}=\left(\sum\limits_{x=1}^{i+1}{m_{x}}-\frac{N}{4}\right)^{2}\times c_{i +1}+m_{i+2}^{2}\times c_{i+2}+\cdots+\left(\frac{3N}{4}-\sum\limits_{x=1}^{j} {m_{x}}\right)^{2}\times c_{j+1}\), \(\sum\limits_{x=1}^{i}{m_{x}}\leq\frac{N}{4}\), \(\sum\limits_{x=1}^{i+1}{m_{x}}>\frac{N}{4}\), \(\sum\limits_{x=1}^{i}{m_{x}}\leq\frac{3N}{4}\), \(\sum\limits_{x=1}^{j+1}{m_{x}}>\frac{3N}{4}\), \(H_{\text{C3}}=H_{\text{C2}}\). Due to the square operation of the number of symbols in each bin, the distributions in part B and/or part C were more concentrated. Thus, the values of \(H_{B3}\) and/or \(H_{\text{C3}}\) were greater. As shown clearly in Table 1, for 32 QAM, all of the symbols in part C concentrated around the fifth amplitude level, while the symbols in part B spread over three different amplitude levels. Thus, the ratio \(H_{3}\) for 32 QAM was small. Unlike 32 QAM, the symbols for 64 QAM, in part C, spread over three different amplitude levels. \(H_{\text{C3}}\) of 64 QAM was much smaller than that of 32 QAM. For 16 QAM, the symbols in part B and part C were concentrated around the second and third amplitude levels, respectively. \(H_{B3}\) of 16 QAM was much greater than that of 32 QAM. Without considering noise and/or any other transmission impairments, the theoretical values of \(H_{3}\) for 16 QAM, 32 QAM and 64 QAM were 2.98, 1.19 and 2.21, respectively.