Digital Signal Processing Reference
In-Depth Information
This equation is called the
fundamental wavelet equation
.Thesetofco-
ecients
is called the wavelet function coecients and behaves
as a high-pass filter. This equation expresses the fact that each wavelet
function in a wavelet family can be written as a weighted sum of scaling
functions at the next finer scale.
The following theorem provides an algorithm for constructing a
wavelet orthonormal basis, given a multiscale analysis.
{
h
1
(
n
)
}
Theorem 2.1:
Let
be a multiscale analysis with scaling function
ϕ
(
t
)and
scaling filter
h
0
(
n
).
Define the wavelet filter
h
1
(
n
)by
{
V
m
}
1)
n+1
h
0
(
N
h
1
(
n
)=(
−
−
1
−
n
)
(2.53)
and the wavelet
ψ
(
t
) by equation (2.52).
Then
{
ψ
mn
(
t
)
}
(2.54)
is a wavelet orthonormal basis on
R
.
Alternatively, given any
L
∈
Z
,
}
n∈Z
{
ϕ
Ln
(
t
)
{
ψ
mn
(
t
)
}
m,n∈Z
(2.55)
is an orthonormal basis on
R
.
The proof can be found in [278]. Some very important facts repre-
senting the key statements of multiresolution follow:
(a)
{
ψ
mn
(
t
)
}
is an orthonormal basis for
W
m
.
=
m
,then
W
m
⊥
(b) If
m
W
m
.
(c)
∀
m
∈
Z
,
V
m
⊥
W
m
where
W
m
is the orthogonal complement of
V
m
in
V
m−1
.
(d) In
stands for orthogonal sum. This means
that the two subspaces are orthogonal and that every function in
V
m−1
is a sum of functions in
V
m
∀
m
∈
Z
,
V
m−1
=
V
m
⊕
W
m
,
⊕
and
W
m
. Thus every function
f
(
t
)
∈
V
m−1
is composed of two subfunctions,
f
1
(
t
)
∈
V
m
and
f
2
(
t
)
∈
W
m
,suchthat
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