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This paper studies the problem of detecting and isolating faults (FDI) in linear discrete-time periodic systems. The aim is to design an observer-based residual generator where each residual is sensitive to one fault, whilst remaining insensitive to the other faults that can be affected the system. Making use of the geometric tools, and in particular of the outer observable subspace notion, the FDI problem is formulated and necessary and solvability conditions are given. An algorithmic procedure is described to determine the solution of the FDI problem.