An alternative method for analyzing the double threshold detector

An alternative method for analyzing the performance of a double threshold or M-out-of-N detector is discussed. Detection performance for the suggested method is based on the probability that a return crosses the threshold for the Mth time (a detection is declared) on the kth return or look. It is shown that this formulation has many advantages, as compared with the conventional method of analysis which employs the binomial probability distribution, since the upper limit N is not contained in the resulting probability expressions. It is shown that the probability of detection obtained by the alternate method is the same as that obtained if the detection method were analyzed as a Markov chain with M + 1 states. Use of the method results in simple expressions for the mean and variance of the number of looks before detection, provides an alternative way of estimating the probability of a threshold crossing, and leads to computationally simple bounds for the probability of false alarm.