Construction of goodness-of-fit criteria for the type of impulse response function

Abstract

The article is devoted to the study of the impulse response function, its estimation and properties, square-Gaussian random variables and processes, the rate of convergence of the unknown impulse response function, testing the hypothesis about the type of impulse response function, building a simulation model. The study showed that the pulse response function is the output signal of the system during signal processing, when the input signal is a short pulse. In a more general form, the impulse response function describes the response or output of the system as a function of time. Also, the impulse response function is considered a property of linear displacement systems. During the study of the estimation of the impulse response function on orthonormal and trigonometric bases, two conditions A, B and remarks to them were formed, which are used in the future to find different coefficients. The study of square-Gaussian random variables and processes has shown the benefits of using them in relation to the impulse response function. A theorem was also presented, which estimated the probability of a large deviation of the square-Gaussian process in the norm of a continuous function. To study the rate of convergence of the unknown impulse response function in the space of continuous functions and in the space L2, a lemma was formed, as well as a theorem that directly showed the rate of convergence of the impulse response function in the space of continuous functions. Zero and alternative hypotheses were formed. The null hypothesis claimed that the impulse response function existed, and the alternative hypothesis suggested the opposite. To test the hypothesis about the form of the impulse response function, a theorem was used by which a criterion was formed. Visual Studio Community 2022 integrated development environment (C ++ programming language) and Wolfram Mathematica computer algebra system for analytical transformations and numerical calculations were used to build the simulation model, which allowed to make mathematical calculations quite accurately.