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8.1 Orthonormal wavelet basis

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An orthonormal wavelet basis is an orthonormal basis of the form

B 2 j 2 2 j t k j k
The function t is called the wavelet .

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

Haar wavelet

The Haar basis (described by Haar in 1910) is an orthonormalbasis with wavelet t

t 1 0 t 1 2 -1 1 2 t 1 0
For the Haar wavelet, it is easy to verify that the set B is an orthonormal set ( ).

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Notation: j , k t 2 j 2 2 j t k where j is an index of scale and k is an index of location .

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

Wavelet series

x t j ∞ ∞ k ∞ ∞ d j k j , k t
d j k t ∞ ∞ x t j , k t

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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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