## 5 Results and Discussion
### DMD Convergence Criterion  
#### 5.2.1 Convergence of Sampling Frequency  
In this section, we will discuss the dependence of the DMD-based eigendecomposition, and subsequent signal reconstruction on the sampling frequency for two flow parameters: surface pressure at the rear fascia and wall shear stress (WS) on the body (as shown in  
Figure 7: Variation of the drag coefficient mean value and RMS with respect to the sampling period sampled with \(f_{s}=10\) kHz: (**a**) mean pressure drag coefficient for the full body. (**b**) RMS of the full body pressure drag coefficient. (**c**) The mean value of the full body friction drag coefficient. (**d**) RMS of the full body friction drag coefficient. Figure 1). DMD is a high-dimensional linear regression technique that relies on time derivatives (d\(y\)/d\(x\)) to capture oscillatory structures across different time scales. To resolve these structures accurately, the temporal resolution of the snapshots must be fine enough to distinguish each time scale. In analogy to Fourier analysis, where the Nyquist-Shannon theorem dictates that the sampling frequency must be at least twice the maximum frequency to avoid aliasing (for example, a 20 Hz structure requires a 40 Hz sampling frequency), DMD poses additional challenges. Unlike Fourier transforms, DMD modes are not necessarily orthogonal in space or time and approximate fields using decaying or growing oscillatory modes. This sensitivity to mode growth/decay rates, energy, and time derivatives makes DMD more demanding in terms of temporal resolution. It may require higher sampling rates than those prescribed by the Nyquist criterion to accurately resolve local temporal events and variations in energy levels, ensuring proper dynamic capture. To illustrate the complexities of DMD, let us revisit Figure 6, which shows that the rear-fascia pressure signal and body friction reach 90% of their respective total energy at two distinct frequencies: \(St=0.18\) (\(f=26\) Hz) for the pressure signal and \(St=0.60\) (\(f=84\) Hz) for the body friction. Beyond this range, both signals display several high-energy spikes, with a notable spike at \(St=2.51\) (\(f=349\) Hz) 349, contributing \(\approx\)5% of the total energy. We believe this phenomenon is not due to numerical methods or simulation quality; rather, it likely stems from physical characteristics of the flow structures associated with this particular geometry or possibly an artifact of the DMD approach. However, based on the prior discussion, it is more probable that these spikes are inherent to the geometry, and DMD has effectively revealed them. Clearly, improper characterization of this spike, specifically in terms of frequency, amplitude, and growth rate, will lead to significant errors in flow-field reconstruction when using lower-dimensional representations. To better understand how these structure frequencies and energy levels influence DMD convergence, we examined it using two frequency bandwidths: one spanning \(St=0\)-0.72 (\(f=0\)-100 Hz) and another covering \(St=0\)-2.88 (\(f=0\)-400 Hz). The data were sampled at frequencies higher than each bandwidth to allow comparison, assessing the impact of range and signal energy on DMD convergence. Note that the original simulation data were sampled at 10 kHz, making both cases subject to down-sampling. This down-sampling, performed by an integer factor \(k\) (decimation), can introduce aliasing, where high-frequency components are folded into the lower frequency range. To mitigate this, signals were filtered with cutoff frequencies (\(f_{c}\)) of 100 Hz and 400 Hz, reducing the effects of aliasing from down-sampling. For the 0-100 Hz bandwidth, sampling frequencies (\(f_{s}\)) ranged from 100 Hz to 10 kHz, with a constant sampling window of \(T_{w}=2\) s (\(T^{*}=278\)). For the 0-400 Hz bandwidth, sampling frequencies varied from 400 Hz to 10 kHz, maintaining the same \(T_{w}=2\) s. The 2-s window was chosen based on the observed convergence condition (Figure 7).