Continuous Wavelet Transform

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.