The paper proposes a system representation formed by a minimal collection of sufficiently long restricted trajectories generated by an observable discrete time LTI system. Conditions are given under which such a collection is a system representation and also an exhaustive parametrization of these representations is provided. These can be also interpreted as a generalized persistency condition which complements the results encountered for the controllable case. In terms of the proposed representation some system properties are investigated and a controllable-autonomous decomposition is given. Finally it is shown how the representation associated to the inverse system, to the parallel and cascade connection, respectively, can be derived.