A multi-resolution analysis is defined by scaling function and the corresponding wavelet. From the scaling relations

the scaling function and wavelet can be defined from the scaling coefficient

The coefficients are constrained due to the properties of wavelet and scaling function.

These properties lead to

The 3rd and 4th constrains requires the numbers of non-zero element in are even.

One of the solution is setting

so that we don’t need to worry and the 4th constrain becomes the 3rd constrain, and the 5th constrain is always satisfied. Now, only the 1st, 2nd, and 3rd constrains are needed. This is equivalent to equations with number of non-zero elements in is .

Degree of Freedom | ||
---|---|---|

2 | 2 | 0 |

4 | 3 | 1 |

6 | 4 | 2 |

8 | 5 | 3 |

For size of 4, the solution is

In fact, the coefficient for can be grouped as even and odd, so that

and the constrain 3rd can lead to,

,

which is automatically fulfill.