Estimating disturbances/unknown inputs of a given system is an important technique that finds various engineering and industrial applications. Two such examples are the so-called disturbance-observer-based control and fault detection/isolation. The traditional design of a disturbance/unknown input observer (DOB/UIOB) usually involves utilizing the inverse of the model of the open-loop system. Hence, such design can only be applied to systems with stable, or "minimal phase", zeros. In this manuscript, we propose a novel adaptive observer for estimating the unknown and nearly periodic input of a linear-time-invariant (LTI) single-input-single-output (SISO) discrete-time system. The observer assumes the form of an adaptive FIR filter whose coefficients are obtained based on the given system model via the least-squares algorithm with the covariance matrix reset. An important advantage of the proposed approach is that the approach does not involve inverting the open-loop system model; therefore, it can be successfully applied to both minimum-phase and non-minimum phase systems. The effectiveness of the proposed observer design is assessed by a numerical example.