Values of g[n] and h[n] for the Haar System 2.1 2003/01/17 2005/06/10 11:00:28 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Haar scaling equation 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 .