Transformation of any function into its wavelet transform constitutes following simple steps:
1. take a section of the function and compare it to wavelet (mother wavelet)
2. calculate correlation coefficient
3. shift the wavelet to right
* repeat the steps 1 and 2.
4. scale (stretch ) wavelet function
* repeat steps 1 to 3.
5. Repeat the steps 1 to 4 for all scales.
Admissibility of Wavelets
To be classified a function as wavelet function some mathematical criteria must be satisfied and the criteria is admissibility condition. Two major admissibility conditions are:
1. Wavelet function should be of finite energy.
$$ E= \int_{-\infty}^\infty |\psi(t)|^2 dt < \infty $$
2. Wavelet function should be of non zero frequency components that is zero mean function.
Fourier transform of wavelet is given as:
$$ \psi(f) = \int_{-\infty}^\infty \psi(t) e^{-2 \pi f t}dt $$
then admissibility constant Cg is given as:
$$ Cg = \int_0 ^\infty \frac{|\psi(f)|^2}f df < \infty $$
This implies that wavelet has no zero frequency component ie. zero frequency component should not be exist.
1. take a section of the function and compare it to wavelet (mother wavelet)
2. calculate correlation coefficient
3. shift the wavelet to right
* repeat the steps 1 and 2.
4. scale (stretch ) wavelet function
* repeat steps 1 to 3.
5. Repeat the steps 1 to 4 for all scales.
Admissibility of Wavelets
To be classified a function as wavelet function some mathematical criteria must be satisfied and the criteria is admissibility condition. Two major admissibility conditions are:
1. Wavelet function should be of finite energy.
$$ E= \int_{-\infty}^\infty |\psi(t)|^2 dt < \infty $$
2. Wavelet function should be of non zero frequency components that is zero mean function.
Fourier transform of wavelet is given as:
$$ \psi(f) = \int_{-\infty}^\infty \psi(t) e^{-2 \pi f t}dt $$
then admissibility constant Cg is given as:
$$ Cg = \int_0 ^\infty \frac{|\psi(f)|^2}f df < \infty $$
This implies that wavelet has no zero frequency component ie. zero frequency component should not be exist.
