Detection of Sonar Sinusoids of Unknown Frequency and Known or Unknown Phase

A consideration of the problem of detecting a constant sine wave of unknown frequency and amplitude in gaussian noise. The Memorandum assumes that the sinusoid may appear at any one of a finite number of known frequencies, and the probability of its occurrence at each of these frequencies is assumed to be equal. Two cases are treated: the first assumes that, although the frequency is not known, the phase of the signal is known, thus allowing coherent detection; the second acknowledges that the initial phase could not be known, and an analysis of the incoherent detector is made. The results indicate that the difference between the two cases is small and quite predictable. Thus, it can be assumed that, in studies of this kind, the initial phase is known, the gaussian character of the quantities that arise may be retained, and the answers can be adjusted to account for the actual lack of knowledge concerning the phase.