Quickest detection of a tonal burst

We describe and analyze the performance of a technique for the quickest detection of a sinusoid of unknown frequency, amplitude, and phase in additive white noise. The approach is based on the work of Broder and Schwartz (1989) and relies on asymptotic results, that is, the "signal" to be detected as quickly as possible is assumed to be of vanishingly small amplitude, which is the most difficult (and interesting) situation. In the literature, the relationship between the small-signal Page's test and locally optimal fixed-length detection theory is explored in detail for the case of a known contaminant. Here, these results are extended to the case of a stochastic contaminant (i.e., the unknown sinusoid). We derive the version of Page's (1954) test optimized under the assumptions that the amplitude is small, the data arrives in blocks, and the frequency of the sinusoid is uniformly distributed in a given band, and we verify the performance predictions via simulation. To detect a sinusoid of completely unknown frequency, an ensemble of such detectors is required, and this ensemble is very close to an FFT-based scheme. If FFTs are to be used, however, the best performance is obtained when each is augmented by a half-band-shifted version of itself.