This note addresses parameterization issues of nonlinear observer-based fault detection (FD) systems which are composed of a residual generator, a residual evaluator and a threshold. Our study consists of two steps. In the first step, with the aid of nonlinear factorization and input-output operator techniques, we prove that any stable residual generator can be parameterized by a cascade connection of the process kernel representation and a post-filter that represents the parameter system. In the second step, based on the state-space representation of the parameterized residual generators, we investigate the so-called L-infinity- and L-2-classes of observer-based FD systems. This leads to the parameterization of the threshold settings for both classes of FD systems and, associated with them, to the characterization of the existence conditions.