**2. Wavelet transform**

Wavelets are functions that satisfy certain mathematical requirements. The wavelet name comes from the fact that they must be oscillatory (a wave), and be well placed, therefore exhibiting short time duration. There are several wavelet types, usually grouped into families, from which the Daubechies is one of the best known.

Wavelets are used to represent data or other functions in a similar way as the Fourier analysis uses sines and cosines. The signal analysis by wavelet transform has advantages over traditional methods using Fourier analysis when the signals have time discontinuities or present a non-stationary oscillatory behavior.

The mathematics main branch leading to wavelet analysis began with Joseph Fourier (1807) with his frequency analysis theory, known as Fourier analysis. The first wavelet mention appears in the appendix of A. Haar's thesis (1909). Paul Levy a 1930's physicist, investigating the Brownian motion, found that the Haar basis functions are superior to the Fourier basis functions for studying small and complicated details in the Brownian motion. In 1980, Grossman and Morlet, broadly defined wavelets in the context of quantum physics, providing a way of thinking about wavelets based on physical intuition. In 1985, Stephane Mallat gave wavelets an additional advance. Through his work in digital signal processing, he discovered some relationships among quadrature mirror filters - QMF, pyramidal algorithm, and orthogonal wavelet basis. Based partially on these results, Y. Meyer built the first non-trivial wavelets, which unlike the Haar wavelet, the Meyer wavelets are continuously differentiable, but do not have compact support. Years later, Ingrid Daubechies used Mallat's work to build a set of wavelets with orthogonal basis functions that have become the cornerstone of wavelet applications today.
