## 2 Operating Principle
### Polar Coordinate Transformation and Feature Extraction  
The signals are transformed into the polar coordinate system by a process expressed below:  
\[r=\sqrt{I^{2}+Q^{2}}, \tag{1}\]  
\[\theta=\arctan\frac{Q}{I}, \tag{2}\]  
where \(I\) and \(Q\) represent the real and imaginary parts of the signal, while \(r\) and \(\theta\) indicate the amplitude and phase after in the polar coordinate system. The constellation diagrams in the rectangular coordinate system and the polar coordinate system are shown in Figure 3.  
After normalization, the range of amplitude \(r\) is from 0 to 2.5, while the range of phase \(\theta\) is from \(-3.2\) to 3.2. Then, the amplitude \(r\) is equally divided into 50 intervals, while the phase \(\theta\) is equally divided into 80 intervals. Based on amplitude, phase, and number of symbols in each grid, the three-dimensional polar coordinate graphs of these different modulation formats are generated as illustrated in Figure 4.  
To clearly illustrate the feature extraction procedure, the vertical view of three-dimensional polar coordinate graphs is shown in Figure 5, where the amplitude levels for QPSK, 16 QAM, 32 QAM, 64 QAM, and 128 QAM are 1, 3, 5, 9, and 13, respectively. Meanwhile, the phase distributions of these five modulation formats are also different. In comparison with global feature extraction, extracting effective local features can reduce the computational complexity for some features with high similarity, thus improving the identification performance. In order to extract effective local features, 20 specific zones are selected, as illustrated in Figure 5. The number of symbols within these zones can be expressed as  
\[T_{k}=\sum_{i=r_{start},k}^{r_{end,k}}\sum_{j=c_{start},k}^{c_{end,k}}M_{i,j},k=1,2,3,\ldots 20, \tag{3}\]  
where \(T_{k}\) represents the number of symbols in the \(k\)-th local zone; \(M_{i,j}\) indicates the number of symbols in the grid of the \(i\)-th row and the \(j\)-th column; \(r_{\text{start}}\) and \(r_{\text{end}}\) denote the starting index and ending index of rows in the \(k\)-th local zone, respectively; and \(c_{\text{start}}\) and \(c_{\text{end}}\) are the starting index and ending index of the column in the \(k\)-th local zone, respectively. Figure 3: The constellation diagrams in (**a**) the rectangular coordinate system and (**b**) the polar coordinate system. To reduce the dimension of features and decrease the entire complexity of this scheme, the number of symbols in the 20 specific zones are grouped into six dimensional features, and can be defined as  
\[Z_{1}=T_{1}+T_{2}, \tag{4}\]  
\[Z_{2}=T_{3}, \tag{5}\]  
\[Z_{3}=T_{4}+T_{5}+T_{6}, \tag{6}\]  
\[\theta=\arctan\frac{Q}{I}, \tag{2}\]0  
\[\theta=\arctan\frac{Q}{I}, \tag{2}\]1  
\[\theta=\arctan\frac{Q}{I}, \tag{2}\]2  
where \(Z_{i}\) is the \(i\)-th dimensional feature. As illustrated in Figure 5, \(Z_{1}\) is the sum of the symbol number in the two local zones, \(T_{1}\) (27:36, 46:54) and \(T_{2}\) (27:34, 66:74). The first two numbers in the parentheses indicate the starting and ending index of the rows, and the last two numbers in the parentheses denote the starting and ending index of the columns. It should be noted that both the rows and columns are counted from the top left corner. These two zones correspond to two clusters of the QPSK signal. \(Z_{2}\) denotes the number of symbols in the local zone \(T_{3}\) (31:35, 14:28), which corresponds to two clusters in the second amplitude of the 16 QAM signal. As illustrated in Figure 5, the polar coordinate graph shows symmetry properties. Due to less clusters, for the low order modulation format (16 QAM), the distribution of clusters in the polar coordinate graph is easily recognizable. Therefore, even if the number of symbols is only extracted in the local zone \(T_{3}\) rather  
\[\theta=\arctan\frac{Q}{I}, \tag{2}\]3 The three-dimensional polar coordinate graph of five modulation formats: (**a**) QPSK, (**b**) 16 QAM, (**c**) 32 QAM, (**d**) 64 QAM, and (**e**) 128 QAM.