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Orthonormal Wavelet Basis

Module by: Ivan Selesnick. E-mail the author

An orthonormal wavelet basis is an orthonormal basis of the form

B= 2j2⁢ψ⁢2j⁢t−k j∈Z and k∈Z B 2 j 2 ψ 2 j t k j k
(1)
The function ψ⁢t ψ t is called the wavelet.

The problem is how to find a function ψ⁢t ψ t so that the set BB is an orthonormal set.

Example 1: Haar Wavelet

The Haar basis (described by Haar in 1910) is an orthonormal basis with wavelet ψ⁢t ψ t

ψ⁢t={1  if  0≤t≤1/2-1  if  1/2≤t≤10  otherwise   ψ t 1 0 t 12 -1 12 t 1 0
(2)
For the Haar wavelet, it is easy to verify that the set BB is an orthonormal set (Figure 1).

Figure 1
Figure 1 (haar.png)

Notation: ψ j , k ⁢t=2j2⁢ψ⁢2j⁢t−k ψ j , k t 2 j 2 ψ 2 j t k where jj is an index of scale and kk is an index of location.

If BB is an orthonormal set then we have the wavelet series.

Wavelet series

x⁢t=∑j=−∞∞∑k=−∞∞d⁢jk⁢ ψ j , k ⁢t x t j k d j k ψ j , k t
(3)
d⁢jk=∫−∞∞x⁢t⁢ ψ j , k ⁢tdt d j k t x t ψ j , k t

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