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The paper considers the theoretical basics and the specific mathematical techniques having been developed for solving the tasks of the stochastic data analysis within the Rice statistical model in which the output signal’s amplitude is composed as a sum of the sought-for initial value and a random Gaussian noise. The Rician signal’s characteristics such as the average value and the noise dispersion have been shown to depend upon the Rice distribution’s parameters nonlinearly what has become a prerequisite for the development of a new approach to the stochastic Rician data analysis implying the joint signal and noise accurate evaluation. The joint computing of the Rice distribution’s parameters allows efficient reconstruction of the signal’s in-formative component against the noise background. A meaningful advantage of the proposed approach consists in the absence of restrictions connected with any a priori suppositions inherent to the traditional techniques. The results of the numerical experiments are provided confirming the efficiency of the elaborated approach to stochastic data analysis within the Rice statistical model.

The techniques of the stochastic data analysis are obviously based upon and determined by the statistical properties of the data to be analyzed. The Rice statistical distribution has recently become a subject of increasing scientific interest because of a wide circle of stochastic data processing tasks which are adequately described just by the Rice statistical model [

The Rice distribution describes an information processing problems in which an initially determined signal is distorted by an inevitable random noise generated by many independent normally distributed summands of zero mean value, i.e. the output signal is composed as a sum of the sought-for initial signal and the Gaussian noise component. The variable to be measured and analyzed is an amplitude, or an envelope of the resulting signal, which is known to obey the Rice statistical distribution [

The paper provides both the theoretical basics of the elaborated approach based upon the Rician signals’ two-parameter analysis and some results of its computer simulation.

The Rician distribution is known to describe an amplitude of the random value, formed by summing an initially determined complex signal and the Gaussian noise distorting this signal. By virtue of the central limit theorem, this situation is rather a typical one at describing various physical processes.

Let A be a determined value that characterizes the physical process to be considered. This value is inevitably distorted by the Gaussian noise created by a great number of independent noise components, while the measured and analyzed value is an amplitude, or the envelope of the resulting signal. The Gaussian noise distorting the initial determined signal is characterized by a zero mean value and a dispersion

The real

In other words, the Rician random variable x represents the amplitude of the signal with the Gaussian real and imaginary parts.

The Rician probability density function is given by the following formula:

where

Here and below we’ll use the following denotations:

The final purpose of the Rician data processing is evidently the evaluation of value A that characterizes the process under the study and coincides with parameter

Let’s denote the various types of the signal’s averaging by the angular brackets:

The Rician value’s mathematical expectation and dispersion are known to be expressed by the following formulas [

where function

From the mathematical peculiarities of the Rician distribution noticed above it follows that the Rician signal’s mean value (2) and its dispersion (3) depend on the both Rician parameters:

The both Rician parameters have a certain physical sense, namely:

The essence of the conception of two-parameter analysis of the Rician distributed stochastic data consists in understanding the necessity of such a joint evaluation of both unknown Rician parameters

The Rice distribution is known to be mutually connected with two other statistical distributions: the Rayleigh distribution and the Gauss distribution. Namely, it is known that the Rice distribution is to be transformed into the Gauss distribution at increasing the value of the signal-to-noise ratio, being determined by the Rician parameters’ ratio:

By virtue of the specifics of the Rice distribution the analysis of the Rician data demands the development of special methods and the corresponding mathematical apparatus.

It is well known that at the Gaussian data analysis a traditional and efficient filtration tool is the data averaging. However, in contrast to the case of the Gaussian distribution the average value of the Rician signal

Therefore if one applies to the Rician data processing the traditional technique of the filtration by means of the data averaging, then within the range of small values of the signal-to-noise ratio the result will be just a leveling of the real signal values.

Let us consider the interconnection between the Rician signal’s dispersion value and the dispersion of the Gaussian noise forming the Rician signal. As mentioned above, the average value of the Rician signal does not coincide with the initial, un-noised signal. Similarly the dispersion of the Rician signal does not coincide with the dispersion of Gaussian noise forming the Rician signal from the initially determined value. These properties are inherent to the Rice distribution and, in contrast to the Gaussian statistical model, do not allow analyzing data just by simple averaging. Instead the indicated peculiarities of the Rician random variable predetermine the necessity of the development of special theoretical approach for the Rician data analysis and processing.

Taking into account the known properties of the Laguerre polynomial [

signal is much less than the Gaussian noise value, we get

Formula (4) coincides with the known formula for the random variable with the Rayleigh distribution, what is not unexpected in virtue of the interconnection between the Rice and the Rayleigh distributions: at

In another limiting case, when the value of the useful signal significantly exceeds the noise level, i.e. at

These theoretical results are illustrated in

The numerical experiment’s results presented in

The above indicated nonlinearities of the Rice distribution are such that the reconstruction of the initial, un-noised signal against the noise background is possible only by means of estimation of the both Rician parameters, or, in other words, by solving the two-parameter task.

The particular theoretical methods having been developed within the two-parameter analysis of the Rician signal in [

Before turning to the detailed consideration of a mathematical technique the present paper deals with let us provide the brief information about the two-parameter techniques of Rician data analysis elaborated earlier in [

The system of equation for method MM12 looks as follows:

Method MM24 is rather an original and simple in its realization with equations’ system for method ММ24 as:

For ML method we have the following system of equations:

The system of equation for method MM13 is as follows:

A mathematical technique to be considered in detail in the present paper method is based upon combining the maximum likelihood technique and the method of moments. Let us designate as MLM. To get the equations for this technique let us consider a sample of n measurements of the Rician value x. Suppose that the i-th measurement in the sample results in

where function

As the first equation for method MLM we’ll use the maximum likelihood equation having been obtained by equalization to zero of the partial derivative of

logarithmic likelihood function (9) by parameter

Having used the known expression

As the second equation for method MLM under consideration we’ll use Formula (2) for the first moment of Rician random value [

In the first equation of system (12) the denotation

type of the first and the zeroth orders. The properties of function

An important theoretical result consists in the fact that for each above mentioned two-parameter methods the corresponding system of two nonlinear equations for two variables

Below as an example we’ll consider in more detail and provide the final formulas and the numerical testing results for one of the above mentioned techniques, namely―for the combined method MLM.

Equations’ system (12) is an essentially nonlinear system for two unknown parameters:

This variable characterizes the value of the signal-to-noise ratio. Then from (12) we’ll get the following equations’ system for a pair variables

In (14)

an average magnitude of Rician value x. From the second equation of (14) we get for variable

Substituting (15) into the first equation of system (14) one can get the following equation for variable r:

Therefore, the task of solving system (14) of two equations for two unknown variables

So, the calculation of the required value of the useful signal

This section presents some results of the computer simulation of solving the task of the Rician signal and noise parameters evolution at data analysis by means of the above mathematical technique MLM based upon combining the maximum likelihood technique and the method of moments.

At conducting the numerical experiments the data obeying the Rice statistical distribution have been generated on two-dimensional grid with the nodes corresponding to various initial values of the signal and the noise parameters. By using the sampled values of Rician signal in each node of the grid the required values of the signal and noise parameters have been calculated by elaborated algorithm, i.e. by means of solving Equations (15)-(16).

from the straight line characterize the accuracy of the calculations. At the simulation the initial values of parameter ^{3} ÷ 10^{4}). In

The presented graphical data illustrate rather a high efficiency of the elaborated technique for the joint signal and noise parameters’ calculation.

The paper considers an approach to solving the task of the stochastic data analysis within the conditions of the Rice statistical distribution and presents the results of the theoretical study of this distribution’s peculiarities. The nonlinear character of the Rician value’s average and dispersion as dependent of the Rician parameters have been shown to cause the necessity of the special mathematical apparatus development for the Rician stochastic data analysis and processing. The so-called two-parameter approach to solving this task, based upon the joint estimation of the both Rician parameters, has been considered and a few specific mathematical techniques having been developed within this approach are presented. The paper provides the theoretical elaboration of the technique based upon combining the maximum likelihood method and the method of moments. The task of solving a system two nonlinear equations for two unknown variables has been mathematically reduced to solving just one equation for one unknown variable, what essentially decreases the necessary calculating resources. The provided results of the numerical experiments confirm the efficiency of the developed technique for solving the problem of the stochastic Rician data analysis ensuring a high precision of the signal and noise parameters estimation.

The work has been supported by RFBR, project N17-07-00064 within the Fundamental Research program.

The author declares no conflicts of interest regarding the publication of this paper.

Yakovleva, T. (2019) Nonlinear Properties of the Rice Statistical Distribution: Theory and Applications in Stochastic Data Analysis. Journal of Applied Mathematics and Physics, 7, 2767-2779. https://doi.org/10.4236/jamp.2019.711190