Processing random processes with generalized Dirichlet distribution

The solution of a number of practical problems, such as detection, recognition, etc., involves the use of signal processing procedures invariant with respect to scale parameters. It should be borne in mind that the amplitude of the input signal is either unknown or changes randomly. These procedures use normalization of sample values to estimate a certain energy parameter. Normalization is carried out either by the AGC system or through arithmetic normalization operations. This operation results in that the probability density function (PDF) of the normalized signal is significantly different from the input non-normalized signal, and the changes concern not only the scale. The article proposes a solution based on the fact that Dirichlet probability density (DPD) corresponds to the case when all quantities included in the expression obey the gamma distribution with the same scale parameter values. As a result, the study analyzes mathematical models and PDF properties of the normalized power of the processed signal at an unknown or case-varying signal amplitude using generalized Dirichlet distribution (GDD). It has been shown that when processing non-stationary random processes, it is necessary to consider a set of quantities with different scale factors, leading to the phenomenon of a wide variety of PDF forms, which is called GDD. The latter, having an external similarity to PDF, is not stable even with the simplest linear transformations. Obtained an expression for the marginal distribution of a random variable from a k-dimensional GDD. The linear conversion of the input signal and the joint PDF of partial sums are described, as well as their PDF. For the case of two-dimensional GDD, expressions of marginal PDF random variable and PDF sum of random variables are found. Conditions are defined under which the marginal and sum distribution curves coincide with the GDD curve, i.e. when it becomes possible to extend the DPD properties to the named generalized distribution. Marginal distributions, sum distributions, partial sums, and conditional distributions are shown to differ from GDD by multipliers containing hypergeometric series. The proposed solutions make it possible to determine the possibilities of using GDD to process an information signal at an unknown or randomly changing signal amplitude. It is also shown that the use of DPD allows you to implement the technical implementation of the transition from processing one set of values to another using only parametric settings.