A system of neural networks in a multistage architecture is used to resolve the components of a transient signal. The design is motivated by a desire to use smaller networks that compute a binary decision allowing the use of the Multilayer Perception Design Algorithm for faster and more effective training, and to cascade these simple networks into a pipeline architecture for efficient implementation. Simulation results are compared with the Multistage Subspace technique that utilizes all of the information in the signal model. The networks are trained with examples of one signal component in noise at a specified noise level. The resulting Multistage Neural Network is able to generalize to different noise levels and multiple signals without additional training. The neural network detector and feature extraction system localizes the arrival time and frequency for each sufficiently strong transient signal present.