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Modeling of probability density function of signal mixture exposed to amplitude distortion and additive noise

Keywords:

V.M. Artyushenko – Dr. Sc. (Eng.), Professor, Head of Department of Information Technology and Management Systems, Technological University (Korolyov, Moscow region) E-mail: artuschenko@mail.ru V.I. Volovach – Dr. Sc. (Eng.), Associate Professor, Head of Department of Information and Electronic Service, Volga Region State University of Service (Togliatti) E-mail: volovach.vi@mail.ru


The issues associated with the impact of additive-multiplicative noise on the processed signal are considered. Mathematical expressions, modeling the density of probability distribution (PDF) of instantaneous values of the mixture of the signal and the noise mentioned above, are obtained and analyzed. It is shown that to solve the tasks of synthesis and analysis of information-measuring systems successfully one should acquire and use mathematical models to realistically simulate PDFs of such mixtures. Statistical characteristics of the PDF of instantaneous values of the signal under the influence of multiplicative (modulating) noise are considered. The expression for the joint PDF of instantaneous values of the signal, the envelope and the phase is given. Expressions for the PDF of instantaneous values the signal are obtained and analyzed for a number of practically important cases: when fluctuations of amplitude and phase are independent at the coincident time points and when only phase distortion is present. Expression for the PDF of instantaneous values of the signal is presented for the case when the amplitude and phase of the signal are independent random functions. It is shown that when amplitude and phase distortions are independent, the PDF of the radio signal hardly depends on the PDF of the phase and is mainly determined by the PDF of instantaneous values of the signal. Expressions for the PDF of the signal are presented when the signal is exposed to modulating noise for uniformly distributed on the interval (0, 2π) and deep phase distortion for the most common PDFs of the envelope. The expression for the PDF of instantaneous values of the signal is presented for the case when fluctuations of the amplitude and the phase have some functional relation. It is shown that when such a relation is present the PDF of the signal can be expressed through the statistical characteristics of the envelope. Expressions to determine initial moments of the k-th order of the PDF of the signal for the above case are found. It is shown that the average value of the signal is mainly determined by the nature of amplitude distortion. Statistical characteristics of the PDF of the sum of additive noise and the signal in the presence of modulating (multiplicative) noise are considered. The expressions for the PDF of a mixture of the signal and additive noise are obtained and analyzed for two practically important cases of the signal/noise ratio and the following conditions: when amplitude and phase distortions are independent, when the law of distribution of the phase is arbitrary and amplitude distortion is absent, when there are uniformly distributed on the interval (0, 2π) phase distortions and deep phase distortions, when there is an arbitrary law of distribution of the envelope PDF of the sum signal, when there is arbitrary distribution of amplitude and phase. PDFs of a mixture of the signal and additive noise are obtained for the case when there is a functional relation between amplitude-phase distortions of the signal for two preceding signal/noise ratios. It is shown that when amplitude-phase distortions are functionally related and when the PDF of a mixture of the signal and additive noise is distributed according to the normal law, weight coefficients are determined by the derivative from the characteristic function of the amplitude distortion (envelope fluctuations). It is shown that the received mathematical expressions allow us to model the PDF of instantaneous values of a mixture of the signal affected by multiplicative (modulating) noise (in other words by amplitude distortion) and additive noise, with arbitrary law of distribution of the envelope of the signal.
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