In this paper, it is shown that, under some mild assumptions, it is possible to design observers for linear time-invariant continuous-time and discrete-time systems by feeding classical linear observers (e.g., the Kalman filters and the Luenberger observer) with the successive integrals and the moving average of the measured output, respectively. The main interest in these observers relies on the fact that both the integral and the moving average exhibit low-pass behaviors, thus allowing the design of observers that are less sensitive to high-frequency noise. Examples are reported all throughout this paper to corroborate the theoretical results and to highlight the improved filtering properties of the proposed observers.