Determination of the Probability Density Function of the Mixture Signal and Additive Noise under Influence of Multiplicative Noise

Abstract

Abstract Issues related to effect of multiplicative (modulating) (otherwise amplitude distortions) and additive noises on processed signal are considered. Analysis of statistical characteristics of probability density function (PDF) of instantaneous signal values against background of modulating interference is carried out. Expressions of the joint PDF envelope, phase and instantaneous signal values are obtained, for the case of only phase distortions, as well as the case when fluctuations in amplitude and phase are independent of each other. It is noted that in the second case, with the independence of phase and amplitude distortions, the PDF of the signal, first of all, is determined by the PDF of its instantaneous values and practically does not depend on the PDF of the signal phase. Expressions are obtained for the PDF useful signal and the most common PDF envelope when affecting the signal with modulating noise for deep phase distortions and for its uniform distribution over the interval (0, 2π). In addition, for the case of some functional relationship of phase and amplitude fluctuations, the expression for PDF instantaneous signal values is defined. The presence of such a functional connection allows calculating the PDF of the signal through the statistical characteristics of its envelope. It is obtained that PDF of mixture of signal and additive noise contains arbitrary distribution of phase and amplitude, as well as arbitrary law of distribution of envelope of processed signal. It is obtained that weight coefficients are determined by derivative of characteristic function of amplitude distortion in case when PDFs of said mixture are distributed according to normal law and functional connection exists between amplitude and phase distortions.