## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.2 Convergence of Sampling Period
In one instance, we maintained a constant sampling period (\(T_{w}=2\) s) while increasing \(N\) by raising the sampling frequency. In another instance, we kept the sampling frequency constant and increased the sampling period. As shown in Figure 17, the mean of \(|\lambda|\) begins to converge around \(N=2000\)-\(2500\) and reaches a plateau after 4000. Figure 18 shows  
Figure 16: Zoomed-in version of power spectral density normalized of pressure drag components at the nose, body, and rear fascia by total energy. Figure 15: Convergence of the standard deviation for the magnitude of DMD eigenvalues with respect to sampling period for two conditioned signals: (**left**) \(f_{c}=100\) with \(f_{s}=1\) kHz, (**right**) \(f_{c}=400\) Hz with \(f_{s}=2\) kHz; P and WS stand for pressure and friction drag components, respectively. that the standard deviation of \(|\lambda|\) begins to converge around 2000-2500. However, it does not achieve a plateau comparable to the mean of \(|\lambda|\). In general, the results presented in Figures 17 and 18 indicate that the convergence of eigenvalue decomposition depends on the sampling period or frequency in relation to the total number of snapshots. Next, we studied the impact of the sampling period on flow reconstruction. As shown in Figure 19, increasing the sampling period helps reduce the reconstruction relative error up to a ratio of \(T^{*}=250\) for both cases, \(f_{c}=100\) and 400 Hz. However, beyond this sampling period, in the case of wall shear stress, DMD tends to diverge. The equivalent cycle number is about 20 corresponding to the slowest global modes with \(St=0.08\) (\(f=11.1\) Hz). [PERSON] et al. (2018) found that beyond 20 cycles of the slowest mode DMD starts to divergence. According to their study, four states of mode convergence were identified: initialization, transition, stabilization, and divergence, depending on the sampling range. Between 1 and 8 cycles, DMD experiences an initialization state; from 8 to 15 cycles, it passes through a  
Figure 17: Variation of the mean of the eigenvalue magnitude (\(|\lambda|\)) with respect to the total number of snapshots (\(N\)). \(f_{c1}\) and \(f_{c2}\) represent cut-off frequencies of 100 Hz and 400 Hz, respectively; P and WS stand for pressure and wall shear stress, respectively. Figure 18: Variation of the standard deviation of the eigenvalue (\(|\lambda|\)) with respect to the total number of snapshots (\(N\)). \(f_{c1}\) and \(f_{c2}\) represent cut-off frequencies of 100 Hz and 400 Hz, respectively. P and WS stand for pressure and wall shear stress, respectively. transition phase; after 15 cycles, it stabilizes up to 20 cycles, and then it begins to diverge. Therefore, while increasing the sampling period can aid in achieving the convergence of eigenvalues, it is crucial to maintain a high time resolution (\(\Delta t\)) to avoid oversampling and the generation of spurious modes. DMD's performance is sensitive to spurious modes. [PERSON] and [PERSON] [19] suggested that the double generation of a mode at a specific frequency can be spurious. Higher frequency domain resolution (\(\Delta f\)) increases the risk of this vulnerability. If DMD assigns a large growth rate to a mode due to increased cycles from an extended sampling period, it can result in a significantly large growth or damping ratio, causing a deviation from the true nature of the structure. For further investigation, we selected five different time windows (\(T^{*}\)= 210, 245, 275, 315, 345). For each window, five different initial times (\(t_{0}^{*}=15,25,40,55,70\)) were chosen to calculate the reconstruction relative error. Figure 20a-d show the reconstruction relative error for two pressure and wall shear stress signals at filtering cut-off frequencies of 100 Hz and 400 Hz. The results demonstrate instability in DMD behavior across the cases, with instability typically appearing around \(T^{*}=250\) and worsening beyond \(T^{*}=300\). This instability is random and does not follow a consistent trend in increasing relative error.