Quasi-optimal filtering of multiplicative noise

Most scientific studies on optimal nonlinear filtering describe the case of input exposure in the form of a mixture of signal and additive noise. At the same time, it is quite obvious that real radio systems are affected by multiplicative noise, which can lead, even without the influence of additive noise, to significant distortion of the desired signal. In the course of the work, quasi-optimal filtering of multiplicative noise in the receiving device was investigated, including an important case of the simultaneous effect of additive and multiplicative noises on the useful signal. Statistical characteristics of the signal mixture envelopes and various noise in the named device were also obtained and analyzed. The effect of multiplicative noise on the operation of the useful signal filtering system in the receiver was investigated under conditions of fast fading of the useful signal, which was also affected by additive white Gaussian noise (WGN). For description, systems of stochastic differential equations were used, on the basis of which a functional scheme of quasi-optimal nonlinear filtering of a speech message was obtained. The purpose of the main circuit blocks is described. The effect of multiplicative noise on the additive mixture of the signal and WGN leads to the non-linear character nature of the filtering system and to the non-stationary mode of the filtering itself. It is shown that in case of simultaneous impact of additive and multiplicative noises on the receiving path (still in the presence of WGN), the structural scheme of the receiving device is further complicated. Further investigation of the issue of quasi-optimal filtering leads to the determination of the statistical characteristics of the PDF envelopes of a mixture of additive noise and signal, which can be used in the analysis of filtering systems. It has been shown that the PDF of the envelope of a mixture of additive noise and a signal subject to multiplicative noise depends on the parameters that determine the statistical properties of the signal components, as well as on the dispersion of additive noise. In another case, it is shown that the PDF of the envelope of a non-Gaussian signal in the presence of a non-Gaussian additive noise is determined by parameters characterizing the depth of fluctuations of the signal and noise, and parameters inverse to the variances of the signal and noise, and also depends on the average value of the envelope of the signal. For both of these cases, equations were also obtained to determine the corresponding PDF envelope and ν-initial moments of the waveform signal. Special cases of determining density of the envelope probability distribution are given: for a Gaussian signal with a Rayleigh PDF envelope and an envelope of non-Gaussian noise described by the Nakagami distribution; for a mixture of noise envelope and signal also described by Nakagami distribution.