  Scientific research and engineering practice often require the modeling and
decomposition of nonlinear systems. The Dynamic Mode Decomposition (DMD) is a
novel Koopman-based technique that effectively dissects high-dimensional
nonlinear systems into periodically distinct constituents on reduced-order
subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet
an equal degree of unknown. This serial effort investigates the nuances of DMD
sampling with an engineering-oriented emphasis. This Part I aimed at
elucidating how sampling range and resolution affect the convergence of DMD
modes. We employed the most classical nonlinear system in fluid mechanics as
the test subject--the turbulent free-shear flow over a prism--for optimal
pertinency. We numerically simulated the flow by the dynamic-stress
Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity,
high-fidelity data, we parametrized and identified four global convergence
states: Initialization, Transition, Stabilization, and Divergence with
increasing sampling range. Results showed that the Stabilization is the optimal
state for modal convergence, in which DMD output becomes independent of the
sampling range. The Initialization state also yields sufficient accuracy for
most system reconstruction tasks. Moreover, defying popular beliefs,
over-sampling causes algorithmic instability: as the temporal dimension, n,
approaches and transcends the spatial dimension, m (i.e., m < n), the output
diverges and becomes meaningless. Additionally, the convergence of the sampling
resolution depends on the mode-specific dynamics, such that the resolution of
15 frames per cycle for target activities is suggested for most engineering
implementations. Finally, a bi-parametric study revealed that the convergence
of the sampling range and resolution are mutually independent.
