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Values of g[n] and h[n] for the Haar System

Module by: Phil Schniter

The coefficients hn h n were originally introduced at describe φ 1 , 0 t φ 1 , 0 t in terms of the basis for V 0 V 0 : φ 1 , 0 t=∑nhn φ 0 , n t . 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,∑nhn φ 0 , n t>=∑nhn< φ 0 , m t, φ 0 , n t>=hm 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 (1)
where δn-m=< φ 0 , m t, φ 0 , n t> δ n m φ 0 , m t φ 0 , n t , which gives a way to calculate the coefficients hm h m when we know φ k , n t φ k , n t .
In the Haar case
hm=∫-∞∞ φ 0 , m t φ 1 , 0 tdt=∫mm+1 φ 1 , 0 tdt=12ifm∈010otherwise 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 (2)
since φ 1 , 0 t=12 φ 1 , 0 t 1 2 in the interval 02 0 2 and zero otherwise. Then choosing P=1 P 1 in gn=-1nhP-n g n -1 n h P n , we find that gn=12if0-12if10otherwise g n 1 2 0 1 2 1 0 for the Haar system. From the wavelet scaling equation ψt=2∑ngnφ2t-n=φ2t-φ2t-1 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 Figure 1:
haar_wavelet.png
Figure 1
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 V − 1 and V 0 V 0 (Figure 2):
haar_shifts.png
Figure 2
We expect this because V − 1 = V 0 ⊕ W 0 V − 1 V 0 W 0 .

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