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Course by: Phil Schniter. E-mail the author

# The Scaling Equation

Module by: Phil Schniter. E-mail the author

Summary: This module covers scaling equations and coefficients for the discrete wavelet transform.

Consider the level-1 subspace and its orthonormal basis:

V 1 =span φ 1 , n t nZ V 1 span φ 1 , n t n
(1)
φ 1 , n t=12φ12tn φ 1 , n t 1 2 φ 1 2 t n
(2)
Since V1V0 V1 V0 (i.e., V 0 V 0 is more detailed than V 1 V 1 ) and since φ 1 , 0 t V 0 φ 1 , 0 t V 0 , there must exist coefficients hn nZ h n n such that
φ 1 , 0 t=n=hn φ 0 , n t φ 1 , 0 t n h n φ 0 , n t
(3)
12φ12t=n=hnφtn 1 2 φ 1 2 t n h n φ t n
(4)

## Scaling Equation

φt=2n=hnφ2tn φ t 2 n h n φ 2 t n
(5)
To be a valid scaling function, φt φ t must obey the scaling equation for some coefficient set hn h n .

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