Values of g[n] and h[n] for the Haar System2.12003/01/172005/06/10 11:00:28 GMT-5PhilSchniterschniter@ee.eng.ohio-state.eduCharletReedstromcharlet@rice.eduPhilSchniterschniter@ee.eng.ohio-state.eduHaarscaling equation
The coefficients
hn were originally introduced at describe
φ1,0t in terms of the basis for
V0:
φ1,0tnhnφ0,nt.
From the previous equation we find that
φ0,mtφ1,0tφ0,mtnhnφ0,ntnhnφ0,mtφ0,nthm
where
δnmφ0,mtφ0,nt, which gives a way to calculate the coefficients
hm when we know
φk,nt.
In the Haar case
hmtφ0,mtφ1,0ttmm1φ1,0t12m010
since
φ1,0t12 in the interval
02 and zero otherwise. Then choosing
P1 in
gn-1nhPn, we find that
gn1201210
for the Haar system. From the wavelet scaling equation
ψt2ngnφ2tnφ2tφ2t1
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
V0 ():
We expect this because
V−1V0W0.