Issues concerning the detection of surge and rotating stall in a compression system are considered. It is observed that, when surge or rotating stall happens, the plenum pressure rise and mass flow rate of a compression system exhibit abrupt change while those of its linearized model do not. With this observation, it is shown that the surge and rotating stall can be successfully detected by employing a linear-based fault identification filter (FIDF) design technique. This is achieved by treating the difference between the output of the compression system and that of its linearized model at an unstalled operating point as a fault vector and then investigating the effect of the fault on the designed FIDE Simulation results with regard to Moore and Greitzer's compression model (1986) are given to demonstrate the effectiveness of the proposed approach. The theoretical study presented in this note may provide a guideline of detecting the occurrence of unstable phenomena at the onset so that corrective responses can be made in the practical applications.