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Orthonormal Wavelet Basis

Module by: Ivan Selesnick

An orthonormal wavelet basis is an orthonormal basis of the form

B={2j2ψ2jt-k|j∈ℤ∧k∈ℤ} B 2 j 2 ψ 2 j t k j k (1)
The function ψt ψ t is called the wavelet.

The problem is how to find a function ψt 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 ψt ψ t

ψt=1if0≤t≤1/2-1if1/2≤t≤10otherwise 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=2j2ψ2jt-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

xt=∑j=-∞∞∑k=-∞∞djk ψ j , k t x t j k d j k ψ j , k t (3)
djk=∫-∞∞xt ψ j , k tdt d j k t x t ψ j , k t

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