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The resolution of two signals of known shapes F1(t) and F2(t) in white Gaussian noise is treated as a problem in statistical decision theory. The observer must decide which of the signals is present with a minimum probability of error. The optimum system for this decision is specified in terms of filters matched to the two signals, the outputs of which are compared. The error probability is exhibited as a function of the cross-correlation of the two signals and of the signal-to-noise ratio. If the phases of the two signals are unknown, as in radar, and if the signals are of equal strength and equal a priori probability, the optimum system consists of filters matched to each of the signals, each followed by a detector. The observer then bases his decision upon which of the detectors has the larger output. The probability of error is computed for this case also.
Carl W. Helstrom (Sat,) studied this question.