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Given a time-invariant linear system, the problem of determining those input wave shapes which result in outputs that are zero after a finite time (pulse outputs) is studied. Conditions characterizing these input, pulse-output pairs associated with a linear system are given in terms of their Laplace transforms and the system function. These conditions are then applied in two cases: 1) the general lumped-parameter system, and 2) the general transmission line, to determine classes of suitable input waves for each of these systems. It is shown that the input itself may be taken to be a pulse for all the systems of case 1, and in case 2 for the distortionless infinite line and the general finite line. Furthermore, in case 1 and also for the general RC finite line, the lengths of the associated input and output pulses can be made arbitrarily small.