## 5 Results and Discussion
### DMD Convergence Criterion
#### 5.2.2 Convergence of Sampling Period
However, not all modes in DMD decomposition exhibit coherence. The nature of modes with higher growth rate magnitudes, whether they are growing (positive) or decaying (negative), and their relationship to either dissipative or productive turbulent structures remains unclear. Furthermore, DMD is likely to generate spurious and non-physical signals, which poses challenges in physically interpreting its outputs. [PERSON] et al. [57] showed that oversampling results in the appearance of spurious modes, while [PERSON] et al. [58] observed that under-truncation in DMD inevitably produces spurious eigenvalues, as an artifact of \"fitting the noise\". In their real-time identification study of electromechanical oscillation systems, [PERSON] et al. [57] found that physical modes typically have higher energy than spurious ones and that the damping factor influences the energy level. According to  
Figure 21: Mode shapes of coherent structures of \(St=0.08\) with respect to the sampling period: pressure signal at the rear fascia, four sampling periods of \(T^{*}=70,140,280,\) and \(560\) sampled with \(f_{s}=1\) kHz and a cut-off frequency of \(100\) Hz. Figure 16.0 Growth rate (**left**) and RMS (**right**) of mode time dynamic amplitude of Modes with \(St=0.08,\) 0.13, 0.17, and 0.23; pressure signal at rear fascia sampled with \(f_{s}=1\) kHz and a cut-off frequency of \(100\) Hz. [PERSON] et al. [58], in cases where no truncation is applied, as in the present study, there is a likelihood that many of the spurious eigenvalues may lie on the unit circle, making it difficult to distinguish relevant modes from irrelevant ones. This may be in contrast with the observations of [PERSON] and [PERSON] [19] who showed that modes with the same frequency but smaller amplitudes are often spurious. Moreover, according to [PERSON] et al. [59], DMD modes with \(|\lambda|\) far from the unit circle are more likely to be spurious, especially in turbulent flows with continuous time and length scales from large to small structures. Interestingly, our initial investigations into applying truncation in the DMD process align with the findings of [PERSON] et al. [59] and [PERSON] and [PERSON] [19]. Consequently, we decided to remove the truncation steps from our DMD algorithm altogether. Based on the analyses presented, we conclude that there is a strong correlation between the growth/decay rate and the nature of the mode, whether it is coherent, dissipative/productive, or spurious. Next, we investigate the impact of the sampling period and frequency resolution on the energy contents of the DMD-resolved modes. As we know, increasing the sampling period (\(T_{w}\)) provides higher resolution in the frequency domain and decreases frequency intervals (\(\Delta f\)). In DMD, the frequency intervals may not be constant over the frequency spectrum. DMD identifies temporal modes based on the underlying dynamics of the flow rather than through predefined bins, as in a discrete Fourier transform (DFT). In this study, we examined how the choice of sampling period affects the frequency spectrum and the resolution between adjacent frequency bins. We used three different sampling periods of 0.5, 1, and 2 s, each sampled at \(f_{s}=1\) kHz. Then we calculated frequency intervals between adjacent frequency bins using the equation \(\Delta f_{n}=f_{i+1}-f_{i}\). Figure 23 illustrates the normal distribution of coagulated frequency intervals for each sampling period. In the case of \(T_{w}=0.5\) s, the mean of frequency intervals is around 2 Hz with a standard deviation \(\sigma=0.47\). Increasing \(T_{w}\) to 1 s shifts the mean value to 1 Hz and almost halves the standard deviation (\(\sigma=0.21\)). The trend remains the same as the mean of intervals decreases to 0.5 Hz with \(\sigma=0.12\) in the case of \(T_{w}=2\) s. This variability can be considered as an advantage of DMD due to its flexibility in capturing physical modes at exact frequencies, although it can also introduce difficulties in achieving convergence in DMD.