Engineering Analysis/Wavelets

Wavelets are orthogonal basis functions that only exist for certain windows in time. This is in contrast to sinusoidal waves, which exist for all times t. A wavelet, because it is dependant on time, can be used as a basis function. A wavelet basis set gives rise to wavelet decomposition, which is a 2-variable decomposition of a 1-variable function. Wavelet analysis allows us to decompose a function in terms of time and frequency, while fourier decomposition only allows us to decompose a function in terms of frequency.

Mother Wavelet

If we have a basic wavelet function ψ(t), we can write a 2-dimensional function known as the mother wavelet function as such:

${\psi }_{jk}=2^{j/2}\psi (2^{j}t-k)$

Wavelet Series

If we have our mother wavelet function, we can write out a fourier-style series as a double-sum of all the wavelets:

$f(t)=\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}{a}_{jk}{\psi }_{jk}(t)$

Scaling Function

Sometimes, we can add in an additional function, known as a scaling function:

$f(t)=\sum _{i=0}^{\mathrm{\infty }}{c}_i{\varphi }_i+\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}{a}_{jk}{\psi }_{jk}(t)$

The idea is that the scaling function is larger than the wavelet functions, and occupies more time. In this case, the scaling function will show long-term changes in the signal, and the wavelet functions will show short-term changes in the signal.