Optimum sequential detection of signals in noise

Abstract

A device which performs a sequential test on a mixture of signal and noise is called a Sequential Detector. With such a device, two thresholds are introduced, each of which is associated with a terminal decision. The length of the detection process (integration time) is not fixed in advance of the experiment but is a random variable, depending on the progress of the test. An optimum form of such a test exists and is characterized by the fact that detection is performed on the average faster than with conventional; i.e., fixed sample size (optimum or non-optimum), devices. The sequential analysis developed by A. Wald is fully applied in this paper, but an important new feature is the treatment of correlated samples and its application to continuous sampling processes. In the introduction, the problem is presented within the framework of Wald's Statistical Decision Theory, and the optimum properties of sequential detectors are discussed accordingly. It is pointed out that a sequential detector is defined in terms of conditional probabilities and hence its operation is essentially independent of a priori information, although the average risk or cost of detection necessarily depends on the a priori signal data. The general theory is illustrated with some cases of special interest. The simplest example of detection involves independent, discrete observations; e.g., the case of a pulsed carrier in normal noise. Here the optimum detector still has the well-known <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\log I_0</tex> structure, but it is shown that the square law approximation for weak signals requires a bias correction due to the fourth order term. Coherent sequential detection of causal signals in normal noise provides another illustration of the theory. An interesting result is that the probabilities of error do not depend on the shape of the filter, provided the proper computer is used. The use of RC-filtered noise illustrates the treatment of continuous detection processes. Finally, the reduction in minimum detectable signal level resulting from the use of a sequential detector is computed. A third example is the sequential detection of random signals in normal noise. It is shown that, although the optimum computer involves the knowledge of the inverted correlation matrix, the average length of the test does not. Hence a curious result is obtained that in this instance detection can be performed in an arbitrarily short time. The paper concludes with a discussion of the practical necessity of truncating the detection process and exact expressions for the error probabilities of such truncated tests are derived and compared with Wald's original approximations.