A fault estimator for linear systems affected by disturbances is proposed. Faults appearing explicitly in the state equation and in the system output (actuator faults and sensor faults) are considered. With this design neither the estimation of the state vector nor the estimation of the disturbances is required, implying that the structural conditions are less restrictive than the ones required to design an unknown input observer. Furthermore, the number of unknown inputs (faults plus disturbances) may be greater than the number of outputs. The faults are written as an algebraic expression of a high-order derivative of a function depending on the output. Thus, the reconstruction of the fault signals is carried out by means of a sliding mode high-order differentiator, which requires the derivative of the faults to have a bounded norm.