It is known that for a non-linear dynamical system, periodic and
quasi-periodic attractors can be reconstructed in a discrete sense using
time-delay embedding. Following this argument, it has been shown that even
chaotic non-linear systems can be represented as a linear system with
intermittent forcing. Although it is known that linear models such as those
generated by the Hankel Dynamic Mode Decomposition can - in principle -
reconstruct any ergodic dynamical system, quantitative details such as the
required sampling rate and the number of delays remain unknown. This work
addresses fundamental issues related to the structure and conditioning of
linear time delayed models of non-linear dynamics on an attractor. First, we
prove that, for scalar systems, the minimal number of time delays required for
perfect signal recovery is solely determined by the sparsity in the Fourier
spectrum. For the vector case, we devise a rank test and provide a geometric
interpretation of the necessary and sufficient conditions for the existence of
an accurate linear time delayed model. Further, we prove that the output
controllability index of a certain associated linear system serves as a tight
upper bound on the minimal number of time delays required. An explicit
expression for the exact representation of the linear model in the spectral
domain is also provided. From a computational implementation perspective, the
effect of the sampling rate on the numerical conditioning of the time delayed
model is examined. As a natural extension of Baz\'{a}n's work, an upper bound
on the 2-norm condition number is derived, with the implication that
conditioning can be improved with additional time delays and/or decreasing
sampling rates. Finally, it is explicitly shown that the underlying dynamics
can be accurately recovered using only a partial period of trajectory data.