Receiver operating characteristic of a thresholded sum-squared coherence detector

Given two signals x(t), y(t) observed on the time interval [0, t], a detector is desired which will decide whether y(t) contains only noise uncorrelated with x(t) (hypothesis H1or noise plus signal linearly related to x(t) (hypothesis H0}). Previous results have given the receiver operating characteristic (ROC) of a coherence detector at each point in frequency. A natural extension of this "point" detector is to band detection, whereby all coherence information over a test band is used to make a decision as to whether or not there is a signal present. The sum of the magnitude squared coherence over the band of test provides a statistic for decision. Under the assumption that this statistic is Gaussian, the ROC is found, compared for the special case to the one-point result, and plotted for typical multipoint situations.