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function ψ, the mother wavelet. The family of functions {ψ j,k } k∈Z obtained as dilatations
and translations of ψ is an orthonormal basis of W j and is expressed as :
ψ j,k = √ 1 ( ) t
ψ
2 j 2 j − k (3.31)
The wavelet transform of a function f then provides the approximation coefficients denoted
a j [k] and the details coefficients d j [k]. These are obtained by projecting f on the vector
spaces V j and W j :
a j [k] =< f, φ j,k > (3.32)
d j [k] =< f, ψ j,k > (3.33)
where the symbol < ., . > denotes the scalar product in L 2 (R).
3.7.4 Filter banks
The property (3.27) implies that the vector space V j is included in V j−1 . Then, any
function in V j−1 can be written as a linear combination of a function in V j . If we consider
a given sequence h[k], the function √ 1
2
φ ( t
2)
can be expressed with respect to the family
of functions {φ(t − k)} k∈Z as :
(
1 t
∞∑
√ φ = h[k]φ(t − k) (3.34)
2 2)
where :
k=−∞
h[k] =< √ 1 φ( t ),φ(t − k) > (3.35)
2 2
W j being also a sub-space of V j−1 , the function √ 1
2
ψ ( t
2)
can be expressed with respect to
the family of functions {φ(t − k)} k∈Z and a different sequence g[k] :
where :
1
√
2
ψ
( t
=
2)
∞∑
k=−∞
g[k]φ(t − k) (3.36)
g[k] =< √ 1 ψ( t ),φ(t − k) > (3.37)
2 2
Equations (3.34) and (3.36) are known as the two scale equations and are used to expresss
the inter-scale relationship between the scaling function and the mother wavelet
with respect to their translations and the coefficients of filters h[k] and g[k]. Through the
combination of equations (3.32), (3.33), (3.34) and (3.36) it is shown that wavelet decomposition
and reconstruction are computed as a series of discrete convolutions with filters h
and g. In the decomposition stage, approximations and details coefficients are given with :
∞∑
a j+1 [k] = h[n − 2k]a j [n] =a j ⋆ h[2k] (3.38)
n=−∞