A. L. Reznik

Computer-analytical calculation of the probability characteristics of readout of random point images

One analytical relation needed for calculating the probability of errorless readout of random discrete images by integrators with two threshold levels is proved. A specific feature of this study is the fact that the relation proved in this paper has been first obtained through analytical calculations on a computer.

Program-combinatorial approach to solving problems of error-free readout of random point images

A novel approach is proposed for solving probability problems arising in studying reliability of readout of random point structures. The essence of this approach is based on combining specialized program-analytical and discrete-combinatorial methods of research seldom used in academic practice. The efficiency of the proposed approach is demonstrated by an example of solving a particular problem that involves two-threshold readout of random point images.

Errorless readout of random discrete-point fields

A closed analytical relation required to describe the reliability of readout of a random discrete-point field by integrators with two threshold levels is rigorously proved. The corresponding probabilistic problem is solved with the use of two nonstandard tricks: first, codes are developed for performing analytical computations on a computer; second, a new notion of “three-dimensional generalized Catalan numbers” is introduced and successfully applied.

Generalized Catalan numbers in problems of processing of random discrete images

The notion of the Catalan numerical sequence is extended. Problems leading to generalized Catalan numbers are formulated and solved.

Linear signal distortion compensation using harmonic decomposition

Compensating for linear distortions of a signal under uniform sampling using Fourier series is considered.

On the reliable readout of random discrete-point structures

An original approach is proposed to solve rather complex probabilistic problems of the readout of random discrete-point fields; these problems lack an exact analytical solution. Schemes are proposed for the direct, iterative, and combinatorially recursive analytical calculation of multidimensional integral expressions that describe the partial solutions of such problems; these solutions are further used to find general closed analytical dependencies. The huge volume of computations forced us to completely formalize the algorithms and to turn over the entire burden of routine analytical calculations to the computer. This helped us to find (and subsequently to prove) a number of new and previously unknown probability formulas that characterize the reliability of the readout of random discrete-point images when the readout is performed by multilevel integrators. Thus, we managed to carry out (which happens rather rarely in scientific practice) the idea expressed by J. von Neumann: when facing a complex problem that defies solution, the researcher resorts to computer calculations that “prompt” the correct answer to him and then strictly prove this answer. Another important point in our investigations is that we introduce the new concept of “three-dimensional generalized Catalan numbers” and find their explicit form, which is advantageously used for solving problems of registering and analyzing random discrete-point images.

Optimal signal interpolation using the values of signal and its derivative

The effect of the bias in a sequence of signal samples and sequence of signal derivative samples on the signal reconstruction accuracy is considered for a bandwidth-limited signal.

Minimization of the error of reconstruction of a continuous periodic signal with uniform sampling

Various approaches to solving the problem of filtration of a continuous periodic signal distorted by stationary additive non-correlated noise are considered. Particular attention is paid to the case where the known signal period is non-divisible by the signal sampling period.

Analytical computer calculations in problems with random division of an interval

The novelty and originality of the presented studies consists of the fact that the rigorous mathematical proof of one of relationships that describe the probability of an error-free reading of random discrete images contained in them was previously found by the authors using specially created programs for computer analytical calculations.

Sampling functions in nonuniform sampling of a periodic signal

Relations for sampling functions in nonuniform sampling of a trigonometric polynomial and its derivatives are obtained.