Further Results on the Detectability of Known Signals in Gaussian Noise

The problem under consideration here is the detection of a completely known signal which may or may not be present in a finite sample of gaussian noise. That is, we imagine a situation similar to Fig. 1 in which a stimulus is made up of cither signal plus noise or noise alone and we ask, given T seconds of this stimulus, how accurately can we decide whether or not the signal is present. The noise is thought of as having been produced by a stochastic process and thus the question is really one of statistical hypothesis testing. This particular problem has been treated rather extensively,1"7 and certain questions, even controversies, have arisen. These concern what 423 424 THE BELL SYSTEM TECHNICAL JOURNAL, GAUSSIAN NOISE N(t) MARCH 1961 CLOSED O TO T SIGNAL + NOISE S(t) + N(t) STIMULUS Y(t) Fig. 1 -- Diagram of problem under consideration.