Wavelet is a function that is localized in time and frequency, generally with a zero mean.
It is also a tool for decomposing a signal by location and frequency.
Wavelet decomposition is original in two respects relative to Fourier decomposition. On one hand, sine and cosine functions, which are periodic, are replaced by a mother function, which is quite regular.
The mother wavelet is horizontally and vertically deformed through the effect of the signal. The wavelet coefficients, which are equivalent to the Fourier decomposition amplitudes, are thus a measure of the correlation between the signal shape and the contour of the deformed mother wavelet that follows the signal.
the signal − whether it is a chronicle series or an image is decomposed into one general part and some details representing some irregularities. The general part is then itself decomposed into a new general part and more details, and so on. This is an iteration model similar to that of the construction of a fractal. This connection explains the effectiveness of wavelets in the study of fractals.
Dear J. Rafiee
My goal is to obtain wavelet analysis from the solution of the dynamic system, which is a time series, and finally, analyze the fractal dimension of the system according to the wavelets.
Wavelets are a set of mathematical functions used to decompose a signal continuously into its frequency components, the resolution of each component being equal to its scale.
A wavelet transform is the decomposition of a function based on wavelet functions.
Wavelets are transmitted and scaled samples of a function with finite length and highly damping oscillation. compared to the Fourier transform, we can say that the wavelet has a very good localization property.
For example, the Fourier transform of a sharp peak has a large number of coefficients, because the basic functions, the Fourier transform, are sine and cosine functions whose amplitude is constant over the whole interval, while the wavelet functions are functions whose most energy is concentrated in a small interval. Has been and are rapidly damped.
Therefore, with proper selection of mother wavelets, better compression is performed compared to Fourier transform.
By compressing the wavelets, we seek a way to analyze the fractal dimension of the system.
Check https://www.researchgate.net/publication/261648655_Wavelet_analysis_of_chaotic_systems
Also check https://www.hindawi.com/journals/complexity/2021/2373423/
Kindly look at this paper.
Article Enhanced multiresolution wavelet analysis of complex dynamic...
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