DETECTION OF SONAR SINUSOIDS OF UNKNOWN FREQUENCY AND KNOWN OR UNKNOWN PHASE

摘要

This Memorandum considers the problem of detecting a constant sine wave of unknown frequency and amplitude in gaussian noise. In particular, it is assumed 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 here. The first assumes that although the frequency is not known, the phase of the signal is known, thus allowing co¬herent detection. The second acknowledges that the initial phase indeed could not be known, and an analysis of the incoherent detector is made. It is shown that for large output signal-to-noise ratios, the problem in both cases becomes that of detecting one of m approxi¬mately orthogonal signals in a noise background. The magnitude of the error in the orthogonality approximation is considered. A physical realization of an approximately optimum detector structure is studied in some detail, and the effect of finite observation time is considered.nThe results indicate that the difference between the two cases is small and quite predictable. Thus, in studies of this kind it can be assumed that the initial phase is known, the gaussian character of the quantities that arise may be retained, and finally, the answers can be adjusted to account for the actual lack of knowledge concerning the phase.