The Detection of a Sine Wave in Gaussian Noise

Abstract

This paper deals with the problem of finding the ``optimum'' method of detecting a sine wave of known frequency and amplitude in the presence of noise. The type of noise considered is the so‐called Gaussian process, which is obtained when thermal noise is passed through an arbitrary linear passive device. The analysis takes into account the fact that in practice only a finite sample of observed signal is available. The optimum detection method is defined as that which maximizes the probability of recognizing the presence of a sine wave if one has actually appeared; while the probability of falsely announcing the presence of a sine wave, if none has actually appeared, does not exceed some prechosen value. It is shown that when the noise has a flat spectrum, all the relevant information is contained in the amplitude and phase of the Fourier transform of the received sample at the frequency of the sine wave. Almost the same result holds in the case where the noise has an exponentially decaying autocorrelation, except that in this case the values of the observed sample at the end points of the sample also play a role.