In this paper, the state estimation problem of linear continuous-time systems is dealt with by a non-asymptotic state observer,which allows the state estimation error to decay within an arbitrarily-small finite time without resorting to high-gaininjection.By processing the measured input and output signals throughmodulation integrals, a number of auxiliary signals not affectedby the initial conditions are obtained, from which the system state can be computed by simple algebra. The problem of internalinstability of modulation integrals is addressed by resorting to a periodic rescaling mechanism that prevents error accumulationand singularities. We show that the combination of modulation integrals with periodic rescaling can be implemented as ajump-linear system. The robustness of the devised method with respect to additive measurement perturbations on the system’sinput/output is characterized by Input-to-State Stability arguments.