The probability distribution for the filtered output of a multiplier whose inputs are correlated, stationary, Gaussian time-series

In this paper the techniques used by Kac and Siegert and by Emerson for evaluating the probability distribution for the filtered output of a square-law device with a stationary, Gaussian input, have been extended to the case of a multiplier whose inputs are a pair of correlated, stationary, Gaussian time-series. It is shown that in this case the probability distribution is determined by the eigenvalues of a pair of simultaneous, linear, homogeneous, integral equations whose kernels involve only the correlation functions of the inputs and the impulse response of the postmultiplier filter. Explicit solutions for the eigenvalues of these integral equations are obtained both for the case of no postmultiplier filtering and for a simple example system using RC filters. Using these solutions the corresponding probability distributions are discussed and in particular, the way in which the probability distribution of the output tends to Gaussian as the postmultiplier filter time constant is increased, is demonstrated.