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

Module by: Phil Schniter. E-mail the author

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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 δnm= 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={12  if  m010  otherwise   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 0 2 0 2 and zero otherwise. Then choosing P=1 P 1 in gn=-1nhPn g n -1 n h P n , we find that gn={12  if  012  if  10  otherwise   g n 1 2 0 1 2 1 0 for the Haar system. From the wavelet scaling equation ψt=2ngnφ2tn=φ2tφ2t1 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:

Figure 1
Figure 1 (haar_wavelet.png)

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):

Figure 2
Figure 2 (haar_shifts.png)

We expect this because V 1 = V 0 W 0 V 1 V 0 W 0 .

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