  Willems' fundamental lemma enables a trajectory-based characterization of
linear systems through data-based Hankel matrices. However, in the presence of
measurement noise, we ask: Is this noisy Hankel-based model expressive enough
to re-identify itself? In other words, we study the output prediction accuracy
from recursively applying the same persistently exciting input sequence to the
model. We find an asymptotic connection to this self-consistency question in
terms of the amount of data. More importantly, we also connect this question to
the depth (number of rows) of the Hankel model, showing the simple act of
reconfiguring a finite dataset significantly improves accuracy. We apply these
insights to find a parsimonious depth for LQR problems over the trajectory
space.
