   Next: Fast Wavelet Transform (FWT) Up: wavelets Previous: Wavelet Expansion

# Discrete Wavelet Transform

The discrete signal is a set of N samples taken from a continuous signal for some initial time and sampling period . The basis functions and are also vectors containing elements. We let , , and . Correspondingly the wavelet expansion becomes discrete wavelet transform (DWT). The discrete function is represented as a weighted sum in the space spanned by the bases and : This is the inverse wavelet transform where the summation over is for different scale levels and the summation over is for different translations in each scale level, and the coefficients (weights) are projections of the function onto each of the basis functions:  where is called the approximation coefficient and is called the detail coefficient. These are the forward wavelet transform.

An Example:

Assume -point discrete singal and the discrete Haar scaling and wavelet functions are: The coefficient for : The coefficient for : The two coefficients for :  In matrix form, we have Now the function can be expressed as a linear combination of these basis functions: or in matrix form:    Next: Fast Wavelet Transform (FWT) Up: wavelets Previous: Wavelet Expansion
Ruye Wang 2008-12-16