This paper deals with state estimation in linear time-invariant systems subject to unknown impulsive input signals. A solution based on a linear impulsive observer and a finite-memory convolution operator is suggested. The problem arises e.g. in the context of systems with intrinsic pulse-modulated feedback that have recently been applied to mathematical modeling of endocrine systems with pulsatile hormone secretion. Simulation results illustrating the performance of the proposed method are provided.