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Using the geometric approach, a solution is given to the problem of detecting faults in linear periodic discrete-time systems. The solution consists on developing an unknown input observer. The existing conditions are provided in terms of new geometric notion of outer observable subspace that is the dual notion of inner reachable subspace. Making use of the new notion, a residual generator is developed, such that the residual is made independent of unknown input disturbance, and its output-free response goes asymptotically to zero.