## 5 Results and Discussion
### Mean-Subtracted DMD  
One key takeaway from the discussion in Section 5.2 is that low-frequency, slowly evolving modes are a primary contributor to instability in the DMD reconstruction algorithm. In this context, [PERSON] et al. [61] demonstrated that mean subtraction significantly improved the accuracy and stability of DMD-generated Koopman [62] models. By subtracting the mean flow, DMD becomes more effective at detecting and accurately resolving these modes, which are critical to flow instabilities. This process also enhances convergence by reducing bias toward stationary structures. Additionally, mean-subtracted DMD isolates unsteady, dynamic structures in the flow by removing the steady-state component prior to decomposition. This approach allows for a clearer focus on transient and oscillatory phenomena, such as vortex shedding or wake dynamics, which may otherwise be obscured by the mean flow. With this in mind, in what follows next, we explore the impact of the mean subtraction on DMD convergence. The literature suggests that DMD with data centering, such as mean subtraction, can reliably compute the eigenvalue spectra of system dynamics. Without centering, DMD often fails to accurately represent dynamics, particularly in systems with high effective rank or complex structures [63]. However, there is a risk that mean-subtracted DMD outputs may resemble those of the discrete Fourier transform (DFT). [PERSON] et al. [63] found that DMD with data centering generally produces distinct results from DFT, except in cases of regular periodicity, which is unlikely in complex turbulent flows around vehicles at high Reynolds numbers [61]. Several data-centering methods are available in the literature, including subtracting the mean flow field, a reference point, or an equilibrium state. [PERSON] et al. [64] demonstrated that mean subtraction helps DMD avoid converging to the zero-growth rate unit circle and enables the capture of quasi-periodic standing waves in complex systems like rotating detonation combustors (RDCs). However, this process can sometimes mislead the analysis by equating dissipative structures with coherent ones and focusing on slowly evolving components, potentially missing critical dynamic behaviors. In the following sections, we compare mean-subtracted and non-mean-subtracted DMD to observe how relative error trends vary with sampling frequency (\(f_{\mathrm{s}}\)) and time window (\(T_{\mathrm{tr}}\)) when the mean is removed from the original signal. When mean-subtracted data are input into the DMD algorithm, the magnitude of the eigenvalues is constrained to one, and the growth rate becomes zero. As a result, we exclude \(|\lambda|\) from the convergence analysis for mean-subtracted data and focus instead on the effect of the mean subtraction on the convergence of DMD reconstruction relative error. Figure 25 shows the relative error with respect to sampling frequency for two filtered signals at \(f_{c}=100\) Hz and \(f_{c}=400\) Hz. The results indicate that mean-subtracted data converges more rapidly, exhibiting two to four orders of magnitude lower relative error compared to non-centered data. Furthermore, no instability was observed in the mean-subtracted case, whereas non-centered data showed instability. In the case of incremental \(T_{w}\), as shown in Figure 26 for two filtered signals at \(f_{c}=100\) Hz and \(f_{c}=400\) Hz, mean-subtracted DMD not only performed better but also demonstrated greater robustness compared to non-centered data. Increasing the sample number by extending \(T_{w}\) further reduced the relative error. Since the growth rate and \(|\lambda|\) remain close to zero and one, respectively, the risk of divergence or the generation of spurious modes with unrealistic decay/growth ratios is negligible. This makes the mean-subtracted DMD particularly suitable for applications like coherent structure extraction, where quasi-periodic oscillations and energy conservation over longer periods are expected.