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This technical note investigates the development of fault detection and isolation (FDI) filters for linear impulsive systems. The concept of an unobservability subspace is introduced for linear impulsive systems and an algorithm for its construction is described. The necessary and sufficient conditions for solvability of the fundamental problem of residual generation for linear impulsive systems are obtained by utilizing our introduced unobservability subspace. Simulation results demonstrate the effectiveness of our proposed FDI strategy.