The observer design problem is investigated in the context of linear left shift invariant discrete behaviors, whose trajectories have supports on Z(+). Necessary and sufficient conditions for the existence of a dead-beat observer of some relevant variables from some measured ones, in the presence of some unmeasured (and irrelevant) variables, are introduced, and a complete parametrization of all dead-beat observers is given. Equivalent conditions for the existence of causal dead-beat observers are then derived. Finally, several classical problems addressed for state-space models, like state estimation, the design of unknown input observers or the design of fault detectors and identifiers (possibly in the presence of disturbances), are cast in this general framework, and the aforementioned equivalent conditions and parametrizations are specialized to these cases.