Мathematical model for converting continuous signals to digital form
DOI №______
Abstract
The article deals with the method of converting continuous signals to discrete, and subsequently to digital signals. Each of the three stages of transformation is examined in detail and mathematically. The impossibility to put into practice the delta function has been proved. Which is the basis of transformation. Therefore, in practice, it is replaced by a short rectangular, very short pulse. In solving the problem of sampling (impulse conversion) there are three main questions: from what considerations it is necessary to choose the sampling interval; what is the accuracy of replacing a continuous signal by the sequence of its readings; which is the maximum permissible sampling interval at which it is still possible to restore (if necessary) a continuous signal. A valid mathematical answer to these questions is given in this article. Mathematical expressions for standard errors of transformations are determined. It is proved that the discretization of a continuous signal according to the Kotelnikov theorem is related to an error, which consists of two components one is related to a bounded spectrum, the second to the finite number of terms of the series of decomposition. It is estimated that the optimal system of calculus, which requires a minimum number of elements of the pulse recording, is the system with the basis -. It is substantiated that the choice of the basis of the transformation system is made on the basis of technological solutions with the basis close to. Thus, the article provides a comprehensive mathematical study of the transformation of continuous signals into discrete, then digital, for further analysis by existing methods.
Keywords: transformations; signals; discrete; Kotelnikov theorem.
References
1. Ястребов И. П. Дискретизация непрерывных сигналов во времени. Теорема Котельникова: электронное учеб.-метод. пособие. Нижний Новгород, Нижегородский госуниверситет, 2012. 31 с.
2. Рональд Н. Брейсуэлл. Преобразование Фурье Scientific American. 1989. № 8. С. 48–56 [Електронний ресурс]. URL: http://www.ega-math.narod.ru/Nquant/Fourier.htm (04.06.2019).
3. Практическое применение преобразования Фурье для анализа сигналов. Введение для начинающих [Електронний ресурс]. URL: https://habr.com/ru/post/269991/ (05.07.2019).
4. Oppenheim, A. V., Schafer R. W. Discrete-Time Signal Processing, Prentice-Hall, 1989. Р. 447–448. (русский перевод одного из предыдущих изданий - Оппенгейм А. В., Шафер Р. В. Цифровая обработка сигналов: пер. с англ. / под ред. С. Я. Шаца. Москва: Связь, 1979.
5. Анализ сигналов на основе вейвлет-преобразования: матеріал з Нац. бібліотеки ім. Н. Е. Баумана [Електронний ресурс].URL: https://ru.bmstu.wiki/ (11.06.2019).
