On determining the presence of signals in noise

The concept of the existence probability of a signal which may or may not be present in a background of noise is introduced in place of the more usual idea of detection probability. Existence and nonexistence probabilities describe the state of mind of an ideal observer after receiving a waveform, taking into account his prior knowledge. Average existence probabilities are evaluated for a signal whose form is precisely known, and for a signal consisting of a modulated carrier of unknown phase. These calculations give the additional signal energy required to compensate for the observer´s ignorance of the carrier phase, and for his surprise at an unlikely occurrence. They also show that if the observer knows the precise form of the expected signal all types of signal having the same energy are theoretically equally satisfactory, and that when the signal is a modulated carrier, of known modulation but unknown phase, all forms of modulation of given energy also yield the same mean existence probability. By comparing the average existence probabilities at input and output, the efficiency of any system of recognition may be tested, and it is shown that if the signal is a modulated carrier there is no loss of information when, after suitable filtering, a phase-sensitive rectifier and integrator are used if the phase is known, or when an ordinary detector is used if the phase is unknown. The theory is also applied to a radar signal, and it is shown that the filter giving the optimum range information will also yield the greatest existence information.