## 7 Flight Example  
Using the yaw angle signal shown in Figure 1a as an example, we determined the number of hovers and S turns. Let the yaw angle deviation threshold \(c=22.5^{\circ}\) and the number of S-turn decomposed layers \(l=2\); the results are shown in Figures 10-13.  
Figure 9: The general flowchart.  
Figure 10 shows the statistical results for the number of hovers and S turns from the original yaw angle signal \(\Psi_{0}\). Using the statistical algorithm for 360\({}^{\circ}\) hovers, the set \(T_{0}\) of the median, the set \(H\) of hover marks, and the set \(S\) of S turn marks near the median of \(\Psi_{0}\) were counted as shown. The hover marks and S turn marks were counted, and it can be inferred that the number of 360\({}^{\circ}\) hovers of the aircraft during this profile was \(N_{H}=10\), and the number of S turns in this layer was \(N_{S}^{0}=6\).  
Figure 11 uses the median-crossing de-redundancy algorithm in Section 3.2 to eliminate the redundant marks with a yaw angle deviation less than the threshold \(c=22.5^{\circ}\) so as to verify the correctness of the algorithm. The signal selected to illustrate the algorithm was the lower part of the original layer \(\Psi_{0}\), that is \(\Psi_{11}\). In the figure, the set \(T_{0}^{11}\) of the median is shown as circle symbols, and the set \(\widetilde{T}_{0}^{11}\) of the redundancy algorithm as diamond symbols; then, the de-redundant set can be worked out using \(\widetilde{T}_{0}^{11}=T_{0}^{11}\setminus\widetilde{T}_{0}^{11}\).  
Figure 12 gives the statistical results of the number of S turns in the two sublayers \(\Psi_{11}\) and \(\Psi_{12}\) of the first layer. Using the de-redundancy algorithms in Sections 3.2 and 5.2, the sets \(S_{11}\) and \(S_{12}\) of S turn marks were obtained as shown. The number of S turn marks obtained in both the upper and lower halves were \(N_{S}^{11}=N_{S}^{12}=4\), resulting in a total of  
Figure 11: Median-crossing de-redundancy for sublayer.  
Figure 10: The number of hovers and S turns from the original signal.  
Figure 13 shows the four sublayers \(\Psi_{21}\)-\(\Psi_{24}\) and the statistical results of the second layer, where Figure 13a,b comes from \(\Psi_{11}\) and the other two from \(\Psi_{12}\). Although there are still many medians in each sublayer signal compared to the previous layer, there were fewer effective marks for S turns. Using the de-redundancy algorithms in Sections 3.2 and 5.2, the number of S turns with \(N_{5}^{21}=2\) and \(N_{5}^{23}=1\) are shown in Figure 13a,c, respectively, and there were no S turning marks in Figure 13b,d, indicating \(N_{5}^{22}=N_{5}^{24}=0\) or there were no S turning actions in these two sublayers. Therefore, the number of S turns obtained from Figure 13 was  
\[N_{S}^{2}=\sum_{j=1}^{4}N_{5}^{2j}=3.\]  
Summing up the number of S turns in all layers, the total of S turns in this flight profile was  
\[N_{S}=\sum_{i=0}^{2}N_{S}^{i}=17.\]  
Figure 13: The second layer of S turns: (**a**) 1 st sublayer; (**b**) 2 nd sublayer; (**c**) 3 rd sublayer; (**d**) 4 th sublayer.  
Upon observing the yaw angle history in each sublayer, the number of hovers and S turns were also worked out artificially. The same results for the number of hovers and S turns by both the PSD and artificial methods verified the effectiveness and programmability of the PSD algorithm presented in this article.