The effect of multiplicative noise on probability density function of signal and additive noise

Abstract

The issues are considered that are associated with mathematical modeling of the probability density function (PDF) of instantaneous values of the signal and additive noise in the presence of multiplicative (modulating) noise. We got the expression to determine the (PDF) of instantaneous values of a signal in the presence of multiplicative noise leading to amplitude and phase distortions. It is shown that if there is a functional relationship between fluctuations of amplitude and phase the PDF of a signal can be expressed through statistical characteristics of its envelope. We obtained the expressions for the PDF when fluctuations of amplitude and phase are independent, and also when only phase distortions are present. The expressions are given that describes the PDF of instantaneous values of a signal and its envelope when amplitude-phase distortions are independent. PDF is determined for two practically important cases, namely when signal/noise ratio is big or small, and also for amplitude-phase distortions that are functionally related to each other. It is shown that in the latter case, when we expand the PDF of a mixture of signal and additive noise by the normal law, the weight multipliers are determined from the derivative of the characteristic function of amplitude distortion (envelope fluctuations). It is shown that if the law of phase distribution in the interval (0, 2π) differs from uniform distribution, then in this case, the signal is a non-stationary random function that tends to stationary with an increase of phase distortion.