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

**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: