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12.7 Values of g[n] and h[n] For the haar system

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The coefficients h n were originally introduced at describe φ 1 , 0 t in terms of the basis for V 0 : φ 1 , 0 t n h n φ 0 , n t . From the previous equation we find that

φ 0 , m t φ 1 , 0 t φ 0 , m t n h n φ 0 , n t n h n φ 0 , m t φ 0 , n t h m
where δ n m φ 0 , m t φ 0 , n t , which gives a way to calculate the coefficients h m when we know φ k , n t .

In the Haar case

h m t ∞ ∞ φ 0 , m t φ 1 , 0 t t m m 1 φ 1 , 0 t 1 2 m 0 1 0
since φ 1 , 0 t 1 2 in the interval 0 2 and zero otherwise. Then choosing P 1 in g n -1 n h P n , we find that g n 1 2 0 1 2 1 0 for the Haar system. From the wavelet scaling equation ψ t 2 n g n φ 2 t n φ 2 t φ 2 t 1 we can see that the Haar mother wavelet and scaling function look like in :

It is now easy to see, in the Haar case, how integer shifts of the mother wavelet describe the differences between signals in V − 1 and V 0 ( ):

We expect this because V − 1 V 0 W 0 .
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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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