Calculating Detection Probabilities for Adaptive Thresholds

In calculating detection probabiities for radar and sonar systems it is usually assumed that the threshold required to yield a certain probability of false alarm is known. This is often not the case for real systems and therefore the threshold must be estimated using some measure related to the test statistic. This paper presents a calculation technique that handles estimated (adaptive) thresholds in a general framework that can be applied easily to many detection problems. False alarm and detection probabilities are calculated from the characteristic function of the noise or signal plus noise variate and the characteristic function of the threshold estimate. To illustrate the method the detection performance of overlapped discrete Fourier transforms (DFTs) is calculated for a narrowband Gaussian target signal.