  This work addresses fundamental issues related to the structure and
conditioning of linear time-delayed models of non-linear dynamics on an
attractor. While this approach has been well-studied in the asymptotic sense
(e.g. for infinite number of delays), the non-asymptotic setting is not
well-understood. First, we show that the minimal time-delays required for
perfect signal recovery are solely determined by the sparsity in the Fourier
spectrum for scalar systems. For the vector case, we provide a rank test and a
geometric interpretation for 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 linear system induced by the Fourier spectrum
serves as a tight upper bound on the minimal number of time delays required. An
explicit expression for the exact linear model in the spectral domain is also
provided. From a numerical perspective, the effect of the sampling rate and the
number of time delays on numerical conditioning is examined. An upper bound on
the condition number is derived, with the implication that conditioning can be
improved with additional time delays and/or decreasing sampling rates.
Moreover, it is explicitly shown that the underlying dynamics can be accurately
recovered using only a partial period of the attractor. Our analysis is first
validated in simple periodic and quasi-periodic systems, and sensitivity to
noise is also investigated. Finally, issues and practical strategies of
choosing time delays in large-scale chaotic systems are discussed and
demonstrated on 3D turbulent Rayleigh-B\'{e}nard convection.
