The probability density function for the output of a correlator with band-pass input waveforms

A general expression is obtained for the characteristic function for the output of a correlator with narrow-band inputs. This general expression is applicable to cases in which the input signals are sinusoidal and have arbitrary amplitudes and phases, and in which the noise inputs are partially correlated. The probability density function for the output is obtained for three special cases, one of which corresponds to a situation that often arises in practice. This practical situation is that in which the signal inputs have the same amplitude and phase, and the noise inputs have the same rms value but are uncorrelated.