title: ÐÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ Ð²ÐµÐ¹Ð²Ð»ÐµÑÐ¾Ð² Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° ÑÐ¿ÐµÐºÑÑÐ°Ð»ÑÐ½ÑÑ ÑÐ²Ð¾Ð¹ÑÑÐ² Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½Ð¾Ð¹ Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°ÑÐµÐ»ÑÐ½Ð¾ÑÑÐ¸ Ð³Ð°ÑÑÑÐ¾Ð²ÑÐºÐ¸Ñ Ð¿Ð¸ÐºÐ¾Ð²
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Using wavelets to analyze the spectral properties of a nonstationary sequence of Gaussian peaks
Creators: name: Ð ÑÐ±Ð¾ÐºÐ¾Ð½Ñ, ÐÐ½Ð³ÐµÐ»Ð¸Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð²ÐµÐ¹Ð²Ð»ÐµÑ Ð°Ð½Ð°Ð»Ð¸Ð·
Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½ÑÐµ ÑÐ¸Ð³Ð½Ð°Ð»Ñ
Ð³Ð°ÑÑÑÐ¾Ð²ÑÐºÐ¸Ð¹ ÑÐ¸Ð³Ð½Ð°Ð»
ÑÐ¿ÐµÐºÑÑÐ°Ð»ÑÐ½ÑÐµ ÑÐ²Ð¾Ð¹ÑÑÐ²Ð°
Ð²ÐµÐ¹Ð²Ð»ÐµÑ ÐÐ¾ÑÐ»Ðµ
ÑÐ¸Ð³Ð½Ð°Ð» Ñ ÑÐ°ÑÑÐ¾ÑÐ½Ð¾Ð¹ Ð¼Ð¾Ð´ÑÐ»ÑÑÐ¸ÐµÐ¹
Ð½ÐµÐ¿ÑÐµÑÑÐ²Ð½Ð¾Ðµ Ð²ÐµÐ¹Ð²Ð»ÐµÑ Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ
wavelet analysis
non-stationary signals
Gauss signal
spectral properties
wavelet Morlet
frequency-modulated signal
continuos wavelet transform

description: Ð¡Ð¸Ð³Ð½Ð°Ð»Ñ, Ð²ÑÑÑÐµÑÐ°ÑÑÐ¸ÐµÑÑ Ð² Ð¿ÑÐ¸ÑÐ¾Ð´Ðµ, ÑÐ²Ð»ÑÑÑÑÑ Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½ÑÐ¼Ð¸. Ð Ð°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð½ÑÐ¼ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½ÑÑ Ð¿ÑÐ¾ÑÐµÑÑÐ¾Ð², ÑÐ¿ÐµÐºÑÑÐ°Ð»ÑÐ½ÑÐµ ÑÐ²Ð¾Ð¹ÑÑÐ²Ð° ÐºÐ¾ÑÐ¾ÑÑÑ Ð¼ÐµÐ½ÑÑÑÑÑ Ð²Ð¾ Ð²ÑÐµÐ¼ÐµÐ½Ð¸, ÑÐ²Ð»ÑÐµÑÑÑ Ð¾ÐºÐ¾Ð½Ð½Ð¾Ðµ Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¤ÑÑÑÐµ. ÐÐ°Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð¸Ð¼ÐµÐµÑ ÑÐ°ÐºÐ¾Ð¹ Ð½ÐµÐ´Ð¾ÑÑÐ°ÑÐ¾Ðº, ÐºÐ°Ðº ÑÐ¸ÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¹ ÑÐ°Ð·Ð¼ÐµÑ Ð¾ÐºÐ½Ð°: Ð¿ÑÐ¸ Ð½ÐµÐ±Ð¾Ð»ÑÑÐ¾Ð¼ ÑÐ°Ð·Ð¼ÐµÑÐµ Ð¾ÐºÐ½Ð° Ð¿Ð¾Ð»ÑÑÐ°ÐµÑÑÑ Ð±Ð¾Ð»ÐµÐµ ÑÐ¾ÑÐ½Ð°Ñ Ð»Ð¾ÐºÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð¿Ð¾ Ð²ÑÐµÐ¼ÐµÐ½Ð¸, Ð½Ð¾ ÑÐµÑÑÐµÑÑÑ Ð¿Ð¾ ÑÐ°ÑÑÐ¾ÑÐµ, Ð¸ Ð½Ð°Ð¾Ð±Ð¾ÑÐ¾Ñ, Ð¿ÑÐ¸ Ð±Ð¾Ð»ÑÑÐ¾Ð¼ ÑÐ°Ð·Ð¼ÐµÑÐµ Ð¾ÐºÐ½Ð° Ð¿Ð¾Ð»ÑÑÐ°ÐµÑÑÑ Ð¼ÐµÐ½ÐµÐµ ÑÐ¾ÑÐ½Ð°Ñ Ð»Ð¾ÐºÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð¿Ð¾ Ð²ÑÐµÐ¼ÐµÐ½Ð¸, Ð½Ð¾ Ð±Ð¾Ð»ÐµÐµ ÑÐ¾ÑÐ½Ð°Ñ Ð¿Ð¾ ÑÐ°ÑÑÐ¾ÑÐµ. ÐÐ¾ÑÑÐ¾Ð¼Ñ Ð²ÑÐµ ÑÐ°ÑÐµ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÑÑÑ Ð²ÐµÐ¹Ð²Ð»ÐµÑ Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ, ÐºÐ¾ÑÐ¾ÑÐ¾Ðµ Ð¸Ñ ÑÐµÑÐ°ÐµÑ. ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¿ÑÐ¸Ð¼ÐµÐ½Ð¸Ð¼Ð¾ÑÑÐ¸ ÑÐµÐ¾ÑÐ¸Ð¸ Ð²ÐµÐ¹Ð²Ð»ÐµÑÐ¾Ð² Ð´Ð»Ñ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²ÐµÐ½Ð½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ñ Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½Ð¾Ð¹ Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°ÑÐµÐ»ÑÐ½Ð¾ÑÑÐ¸ Ð³Ð°ÑÑÑÐ¾Ð²ÑÐºÐ¸Ñ Ð¿Ð¸ÐºÐ¾Ð², ÑÑÐ¸ÑÑÐ²Ð°ÑÑÐ¸Ñ Ð½Ð°ÑÑÑÐµÐ½Ð¸Ðµ Ð¿ÐµÑÐ¸Ð¾Ð´Ð¸ÑÐ½Ð¾ÑÑÐ¸ Ð¼ÐµÐ¶Ð´Ñ Ð¿Ð¸ÐºÐ°Ð¼Ð¸. ÐÐ±Ð¾ÑÐ½Ð¾Ð²Ð°Ð½Ð° Ð¿ÑÐ¸Ð¼ÐµÐ½Ð¸Ð¼Ð¾ÑÑÑ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½Ð¾Ð³Ð¾ ÑÐ¸Ð³Ð½Ð°Ð»Ð° Ð² Ð²Ð¸Ð´Ðµ ÑÑÐ¿ÐµÑÐ¿Ð¾Ð·Ð¸ÑÐ¸Ð¸ Ð³Ð°ÑÑÑÐ¾Ð²ÑÐºÐ¸Ñ Ð¿Ð¸ÐºÐ¾Ð². Ð Ð°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÐµ Ð²Ð¸Ð´Ñ Ð¿Ð¾Ð»Ð¾Ð¶ÐµÐ½Ð¸Ð¹ Ð³Ð°ÑÑÑÐ¾Ð²ÑÐºÐ¸Ñ Ð¿Ð¸ÐºÐ¾Ð² Ð²Ð¾ Ð²ÑÐµÐ¼ÐµÐ½Ð¸. ÐÑÑÐ¸ÑÐ»ÐµÐ½Ð° Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð²ÐµÐ¹Ð²Ð»ÐµÑ-Ð°Ð½Ð°Ð»Ð¸Ð·Ð° ÑÑÐ½ÐºÑÐ¸Ñ Fmax(t), Ð¿Ð¾ÐºÐ°Ð·ÑÐ²Ð°ÑÑÐ°Ñ ÑÐ»Ð¾Ð¶Ð½ÑÑ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÑ ÑÐ°ÑÑÐ¾ÑÐ½Ð¾Ð¹ Ð¼Ð¾Ð´ÑÐ»ÑÑÐ¸Ð¸.
description: Signals encountered in nature are nonstationary. A common method for studying nonstationary processes whose spectral properties change in time is the window Fourier transform. This method has such a disadvantage as a fixed window size: when the window size is small, a more accurate localization in time is obtained, but a loss in frequency emerges, and vice versa, when the window size is big, a less accurate localization in time is obtained, but more accurate in frequency. Therefore, a wavelet transform which solves them is being used more often. This paper is devoted to the applicability of the wavelet theory to find quantitative parameters for the description of a nonstationary sequence of Gaussian peaks, taking into account the violation of periodicity between peaks. The applicability of the model of nonstationary signal as a superposition of Gaussian peaks is justified. Different types of positions of Gaussian peaks in time are considered. The function Fmax(t), showing the complex dynamics of frequency modulation, has been calculated by means of wavelet analysis.