This paper is devoted to sensor fault detection in linear systems with observer-based approach. It is assumed that the system has linear dynamics with the presence of uncertainties. The uncertainties are modeled as unknown input (disturbance), while it is assumed that the well-known observer matching condition is not necessarily satisfied. To decouple the unknown-input effects, and distinguish their effects from the fault effects, an equivalent dynamic system is proposed which is independent from the unknown input. The equivalent system is constructed by the use of a unique integral filter. The introduced integral-filter, which is called buffer-based integral filter in this paper, has frequency response similar to the low-pass filter. Hence, the capability of noise filtration will also be provided. The construction of the equivalent dynamic system is achieved from the use of multiple successive buffer-based integrators and the number of successive filters is related to relative degree between the unknown input and the sensor output. Then, an unknown input observer is proposed for the equivalent system, and therefore, a disturbance-decoupled and fault-sensitive with exponential-convergent toward-zero residual vector will be generated. Although, the generated residual vector can be used for sensor and actuator fault diagnosis problems; however, the focus of this paper will be on the sensor fault detection. Finally, the applicability of the proposed method will be investigated via simulation of a simple inverted-pendulum on a horizontal-moving cart.