The theory of signal detectability

Abstract
The problem of signal detectability treated in this paper is the following: Suppose an observer is given a voltage varying with time during a prescribed observation interval and is asked to decide whether its source is noise or is signal plus noise. What method should the observer use to make this decision, and what receiver is a realization of that method? After giving a discussion of theoretical aspects of this problem, the paper presents specific derivations of the optimum receiver for a number of cases of practical interest. The receiver whose output is the value of the likelihood ratio of the input voltage over the observation interval is the answer to the second question no matter which of the various optimum methods current in the literature is employed including the Neyman - Pearson observer, Siegert's ideal observer, and Woodward and Davies' "observer." An optimum observer required to give a yes or no answer simply chooses an operating level and concludes that the receiver input arose from signal plus noise only when this level is exceeded by the output of his likelihood ratio receiver. Associated with each such operating level are conditional probabilities that the answer is a false alarm and the conditional probability of detection. Graphs of these quantities called receiver operating characteristic, or ROC, curves are convenient for evaluating a receiver. If the detection problem is changed by varying, for example, the signal power, then a family of ROC curves is generated. Such things as betting curves can easily be obtained from such a family. The operating level to be used in a particular situation must be chosen by the observer. His choice will depend on such factors as the permissable false alarm rate, a priori probabilities, and relative importance of errors. With these theoretical aspects serving as an introduction, attention is devoted to the derivation of explicit formulas for likelihood ratio, and for probability of detection and probability of false alarm, for a number of particular cases. Stationary, band-limited, white Gaussian noise is assumed. The seven special cases which are presented were chosen from the simplest problems in signal detection which closely represent practical situations. Two of the cases form a basis for the best available approximation to the important problem of finding probability of detection when the starting time of the signal, signal frequency, or both, are unknown. Furthermore, in these two cases uncertainty in the signal can be varied, and a quantitative relationship between uncertainty and ability to detect signals is presented for these two rather general cases. The variety of examples presented should serve to suggest methods for attacking other simple signal detection problems and to give insight into problems too complicated to allow a direct solution.

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