The paper presents a new method for diagnosis of a process fault which takes the form of an abrupt change in some real parameter of a time-continuous linear system. The abrupt fault in the process real parameter is reflected in step changes in many parameters of the input/output model as well as in step changes in canonical state variables of the system. Detection of these state changes will enable localization of the faulty parameter in the system. For detecting state changes, a special type of exact state observer will be used. The canonical state will be represented by the derivatives of the measured output signal. Hence the exact state observer will play the role of virtual sensors for reconstruction of the derivatives of the output signal. For designing the exact state observer, the model parameters before and after the moment of fault occurrence must be known. To this end, a special identification method with modulating functions will be used. A novel concept presented in this paper concerns the structure of the observer. It will take the form of a double moving window observer which consists of two signal processing windows, each of width