nonparametric and robust methods of solving statistical problems
the Fokker–Plank–Kolmogorov equation
optimal and weak robust criterion of nonparametric detection
nonlinear robust biltering
an adaptive robust filter
The theory of nonlinear operators is extremely various in contradistinction to linear images which are structurally homogeneous enough owing to general methods of solving linear differential equations. Poincare’s graphoanalytical method developed at the beginning of the 20th century is widely used for analyzing nonlinear dynamic and static system. But at present no universal methods of solving nonlinear systems are devised in spite of all the considerable achievements in the field of physical and mathematical sciences. More over neither the achievements of discrete and numerical mathematics nor the high level of hardware technical realization of computing aids can provide the proper simulation of rather extensive objects of nonlinear systems with required effectiveness in many cases.
Now the qualitative designing of nonlinear systems both on the analysis and on the synthesis play an important part in the widely spread information systems but there are a lot of problems concerned with the proper designing. In such nonlinear systems the situation is greatly aggravated by powerful interferences when the credibility functions are multimodal for the most typical practical applications of information systems.
The article deals with the problem of non-stationary signals reception in the presence of interferences and with the problem of the synthesis of robust non-linear detectors functioning on the basis of nonlinear differential equations algorithms. Non-stationary signals or limited samples are widespread real conditions of functioning signal detectors in such practical applications as radiolocation, radio navigation, radio communication, seismology which are socially significant and many others.
Therefore the development of corresponding robust methods being the most advanced ways of solving non-parameter problems on the synthesis of the signal detection algorithms against interferences is a very actual task at present.
The article is devoted to the new method of robust nonlinear filtration and provides with the examples of solving well-known problems with new results obtained that is very important for numerous practical applications. The method is based on the conditional parametric approach for nonparametric problems.
The optimum and weak robust criteria of nonparametric detection are obtained by means of the parametric approach and developed robust variant of the Newman-Pearson lemma and coordination criteria as well as by means of Campbell’s influence function, the theory of robust statistics and the theory of Markov’s conditional processes. These robust criteria are applied for solving nonparametric problems of processing non-stationary signals in the presence of the optional interference background.
The synthesis of robust nonlinear detectors are divided into two steps:
1) initial synthesis of the basic linear structure determining the order of nonlinear differential equations;
2) basic and completing synthesis of the structure and nonlinear parametric dependencies of nonlinear equations.
The features of the robust synthesis methodology and the originality of nonlinear robust detector structures are used for solving such problems of detecting information signals on the noise background as:
a) detection of single impulse low frequency signals and their non-coherent sequences;
b) detection of single radio impulse signals;
c) detection of strongly fluctuating non-coherent pack of radio impulses.
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