  In this article, we consider abstract linear conservative systems and their
time-discrete counterparts. Our main result is a representation formula
expressing solutions of the continuous model through the solution of the
corresponding time-discrete one. As an application, we show how observability
properties for the time continuous model yield uniform (with respect to the
time-step) observability results for its time-discrete approximation
counterparts, provided the initial data are suitably filtered. The main output
of this approach is the estimate on the time under which we can guarantee
uniform observability for the time-discrete models. Besides, using a reverse
representation formula, we also prove that this estimate on the time of uniform
observability for the time-discrete models is sharp. We then conclude with some
general comments and open problems.
