Shift-invariant representation of two periodic system classes defined over doubly-infinite continuous time

This paper considers an isomorphism introduced by Bamieh et. al. in [1], to construct shift-invariant representations, over the same `time-lifted´ system class, for two classes of linear periodically time-varying (LPTV) systems. Specifically, we consider the class of finite-dimensional linear time-invariant (LTI) systems, characterised by a rational frequency-domain symbol, and a class of LPTV systems with sampled-data structure. Such systems appear together in problems of sampled-data approximation, for example. We study them here in terms of mappings between finite-energy signals defined over the doubly-infinite continuous-time axis (-∞,∞), which arises when working with the ν-gap metric to gauge approximation error for such problems. A complication within this context stems from the absence of an integral-operator input-output representation for every system in the LPTV classes considered. We resolve this issue by working with the graph representation of the systems involved.