A residual generation method for fault diagnosis in deterministic and stochastic linear time-varying systems is proposed in this note. Based on appropriate assumptions on the monitored faults and on some persistent excitation condition, it enables complete diagnosis of any number of faults, regardless of the number of output sensors. In order to generate residuals for fault diagnosis, part of the possible faults are estimated with a recently developed technique of adaptive observer. The sensitivity of the residuals to the monitored faults is rigorously analyzed, as well as their insensitivity to the faults to be ignored.