## 2 The Analysis of SDP's Characteristics  
SDP is a method that combines time series signal processing with image analysis, describing the amplitude and frequency changes of time series data in the Cartesian coordinate system in a graphical manner within the polar coordinate system. The SDP transformation was initially proposed for the visual representation of speech. Due to its simplicity in computation and the intuitive and vivid images it produces, which effec  
mirrors \(m\) set to 1 and the aperture gain \(\xi\) to \(60^{\circ}\), the changes in the SDP image of this sine signal with varying lag coefficient \(l\) are shown in Figures 2 and 3. From Figure 2, it is evident that when \(l=0\), the scatter points transformed by SDP all fall on an arc, forming two symmetrical spiral arms. As the lag coefficient \(l\) increases, the margins of the two spiral arms gradually expand, and the area of the spiral arms starts to  
Figure 3: The SDP image when the lag coefficient \(l\) changes during the second half of the cycle. Figure 2: The SDP image when the lag coefficient \(l\) changes during the first half of the cycle. increase from 0, with the intersection points of the two spiral arms and the polar coordinate boundary gradually gathering towards the starting angle \(\Theta^{\prime}(\Theta^{\prime}=0^{\circ})\). When \(l=32\), the area of the spiral arms reaches its maximum. As the lag coefficient \(l\) continues to increase, the area of the spiral arms begins to decrease, but the intersection points of the two spiral arms and the polar coordinate boundary continue to gather towards the starting angle \(\Theta^{\prime}\). When \(l=64\), the two spiral arms turn into arcs, with the area of the spiral arms at 0, at which point the intersection points of the two spiral arms and the polar coordinate boundary are at the starting angle \(\Theta^{\prime}\). Hence, as the lag coefficient \(l\) changes in the first half of the cycle, the area of the two symmetrical spiral arms first increases and then decreases, with the intersection points of the spiral arms and the polar coordinate boundary gradually gathering towards the starting angle \(\Theta^{\prime}\) until they reach it. From Figure 3, it can be seen that the SDP image when the lag coefficient \(l\) changes in the second half of the cycle is symmetrical to the first half, with the area of the two symmetrical spiral arms also first increasing and then decreasing, but the intersection points of the spiral arms and the polar coordinate boundary gradually move away from the starting angle \(\Theta^{\prime}\). Therefore, when searching for optimal parameters, the maximum value of the lag coefficient \(l\) should be half of its period, i.e., \(l\in[0,f_{\rm s}/2f]\). 2. The impact of the aperture gain \(\zeta\) on the SDP transformation effect. Continuing with the sine signal \(y=\sin(200\pi t)\) with a sampling frequency of 12,800 Hz as an example, to achieve a better SDP transformation effect, the lag coefficient when the spiral arm area is largest in Figure 2 is chosen, i.e., the lag coefficient \(l\) is 32. With the number of mirrors \(m\) set to 3, and based on \(\zeta\leq 360^{\circ}/m\), the range for the aperture gain \(\zeta\) in the SDP transformation is \([0^{\circ},120^{\circ}]\). The changes in the SDP image of this sine signal with varying aperture gain \(\zeta\) are shown in Figure 4.  
From Figure 4, it is observed that as the aperture gain \(\zeta\) increases, the area of each spiral arm also gradually increases. When \(\zeta\) is too small, the area of each spiral arm is small, which hinders image feature extraction; when \(\zeta\) is too large, overlapping occurs between adjacent spiral arms of different mirrors, masking features. Hence, choosing an appropriate aperture gain \(\zeta\) within the range \((0^{\circ},360^{\circ}/m]\) can ensure the SDP image better represents the signal without overlap between adjacent spiral arms of different mirrors.