  Willems' Fundamental Lemma provides a powerful data-driven parametrization of
all trajectories of a controllable linear time-invariant system based on one
trajectory with persistently exciting (PE) input. In this paper, we present a
novel proof of this result which is inspired by the classical adaptive control
literature and differs from existing proofs in multiple aspects. The proof
involves a quantitative and directional PE notion, allowing to characterize
robust PE properties via singular value bounds, as opposed to binary rank-based
PE conditions. Further, the proof is constructive, i.e., we derive an explicit
PE lower bound for the generated data. As a contribution of independent
interest, we generalize existing PE results from the adaptive control
literature and reveal a crucial role of the system's zeros.
