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FFT processing systems have become important and extremely useful implementations of narrow-band signal detectors for signals with unknown phase and frequency. This paper examines the Fourier filter and detector and shows it to be an optimum receiver for a very generalized signal model. The general signal model covers a single sinusoidal pulse, pulse trains, and continuous signals. The model provides for the finite bandwidth encountered in real signals and includes the affect of redundant processing which can be employed with high-speed FFT processors. The paper provides a method for evaluating detection sensitivity for this class of signal detectors and gives experimental results from an FFT processor implementation.