Wavelet analysis of plasma electrical signals.: What is wavelet analysis?

Wavelet analysis is popular in different areas such as signal processing, communications systems, image processing and so on. Wavelet analysis can be represented as a set of basis function as same as Fourier analysis, however, the expression can be obtained in terms of mother wavelet not trigonometric polynomials. Wavelet analysis provides more accurately localized temporal and frequency information so that it is suitable for the non-stationary, time-varying signals.

Mother Wavelet

Consider a complex-valued function ѱ which satisfies the following relationships:

$\int_{-\infty}^{\infty}|\psi(t)|^{2}\text{d}t<\infty$

$C_{\psi}=2\pi\int_{-\infty}^{\infty}\frac{|\Psi(\omega)|^{2}}{|\omega|}\text{d}\omega<\infty$

where Ѱ is the Fourier transform of ѱ. The first equation represents finite energy of the function ѱ and the second equation means that Ѱ(0)=0 if Ѱ(ω) is smooth so it is called admissibility condition. The function ѱ is called the mother wavelet.

Continuous Wavelet Transform

If ѱ satisfies the above condition, the continuous wavelet transform can be defined as:

$S(b,a)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\psi^{'}(\frac{t-b}{a})s(t)\text{d}t$

where ѱ' represents the complex conjugate of ѱ and the parameter "a" means the scale of the analyzing wavelet while parameter "b" is the time shift. Therefore, the function s(t) in time domain can be mapped into the other domain that described by parameter "a" and "b".

In actual, wavelet can be much more accurately localized in temporal and frequency domain because wavelet transform can be regarded as a microscope to visualize the signal s(t). And parameter "a" called scale parameter represents the magnification and "b" chooses the position to be observed.

What is wavelet analysis?

Mother Wavelet

Continuous Wavelet Transform

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