D.S. Grigoryan

Possibility of the several sources parameters measurement, making an additive mix of signals, in problems of the digital spectral analysis is closely connected with the superresolution of these sources on the set parameter. Signal supervision in practice is limited on time, frequency or on space and the sources, which parameters are necessary for estimating, remain in limits Rayleghan resolution element which width is defined as size, return duration of sample on the set parameter. Therefore the increasing urgency is got by methods of the digital spectral analysis on short sequence of the data, allowing to reach superresolution. Occurrence of high-speed processing devices of signals has given a new push to development of the spectral estimation theory with the superresolution. Along with it wish to apply methods of digital spectral estimation in modern radio- and hydro applications has forced to search for possibilities of the targets superresolution on frequency, time and space at low a signal/noise ratios, characteristic for radar supervision conditions by means of the real technical devices having the noise and distortions of complex characteristics. In this connection the question on applicability of those or other methods of the digital spectral analysis in concrete conditions of radio signals supervision became more actual. The increasing urgency in the field of spectral estimation is got by influence questions of the gain distortions of reception channels of a digital antenna array on quality of the close located sources spectral estimates, and also a problem of calibration of not identical reception channels. For more strict estimation of revolting factors influence on efficiency of the sources superresolution it is necessary to search for analytical dependences between parameters of revolting factors and distortions of spectral estimations. However, despite an abundance of the works devoted to the digital spectral analysis, while there are no the strict analytical parities, connecting indignations of parameters, used in spectral estimations, with efficiency of estimation on the set method. Frequently the ambiguous treatment of the fact of the superresolution complicates selection a mathematic of the superresolution borders examination for the equations of spectral analysis methods. Many questions of the spectral estimation theory while remain \"veiled\" by the theory of matrixes indignation and the obvious answer to them it is not visible. Besides, many criteria of the superresolution need a strict substantiation. The general between different criteria that the more values, which characterizes indignations, the worse efficiency of achievement of the superresolution established criterion. In the present article it is accepted the superresolution to name the specral lines division fact, i.e. as criterion of the superresolution presence of several maxima of spectral function in limits Rayleghan resolution element here is used. In work the concept of the superresolution area, as which it is necessary to understand all possible values of the parameters, characterizing revolting factors, at which the superresolution takes place is used. Threshold values of these sizes, at which excess the superresolution comes, are named by superresolution border. The article purpose - to show analytical connection of a weight vector indignations with spectral function, on which maxima in a vicinity of measured parameters points (frequencies, coordinates are shown), which number is equal to operating with the set coordinates sources number. Knowing analytical connection of a weight vector indignations with spectral function, having set by the boundary spectral functions corresponding to set criterion of the superresolution, it is possible to calculate boundary values of indignations of a vector of weight factors. On basis of the linear prediction spectral function analysis the equation of a weight vector indignations connection with indignations of ideal error functions are given. The theorem, showing that for any indignations of a weight vector, which are not belonging to space of ideal indignations, minimums of a error square function have offsets on frequencies, concerning real frequencies of sources is formulated. The prove of the theorem and a consequence from it, showing is resulted that for any frequencies not equal to real frequencies of sources for which there is an ideal vector of weight factors, there are its indignations displacing estimations of sources frequencies from real frequencies to the chosen, is formulated. The mathematical apparatus of boundary spectral functions is entered, having set with which it is possible to calculate indignations of a weight vector, leading to occurrence of several spectral function maximums in Rayleghan resolution element limits. The method of weight factors border indignations estimation on a square of boundary error function, defined by the chosen criterion of decision-making on the superresolution on spectral function is offered.