We study a broad class of linear continuous-time time-varying systems that contain piecewise continuous disturbances and piecewise constant outputs. Under an observability assumption, we construct a new type of observer to estimate the state of the system in a predetermined finite time in the presence of the disturbances. In contrast to the well-established finite time observer design techniques which estimate the system state using a continuous output, our proposed observer only requires a piecewise constant output. Our simulations illustrate the efficacy of our observer.