The sequential detection of a sine-wave carrier of arbitrary duty ratio in Gaussian noise

In this paper the Wald theory of sequential analysis is applied to the detection of a sine-wave carrier of arbitrary duty ratio in Gauss]an noise. This is a generalization of a familiar problem. The detector law for the problem is obtained. In particular, it is specialized to the important cases: 1) arbitrary duty ratio and signal-to-noise ratio less than unity and 2) duty ratio much less than unity and peak-signal-to-noise ratio much greater than unity. For the latter case, it is shown that the best detector law goes over into a Bernoulli detector. In the former case it is shown that the only important parameter in detection is the average signal-power to noise-power ratio. For the case of unity duty ratio the detector law goes over into the familiar characteristic. Furthermore, for threshold signals, it is shown that, in general, a first-order approximation to the logarithm of the likelihood ratio (or to the detector law) does not permit the sequential test to converge at or of the threshold parameters. A second-order approximation is always required. Curves of the operating characteristic function and the average sample number function are given for threshold signals.