Digital Signal Processing Reference
In-Depth Information
obtained. We also have to look into the following problems:
1.
Is the set of discrete wavelets complete in
L
2
(
R
)?
2.
If complete, is the set at the same time also redundant?
3.
If complete, then how coarse must the sampling grid be, such that the
set is minimal or nonredundant?
A response to these questions will be given in this section, and we
also will show that the most compact set is the orthonormal wavelet set.
The sampling grid is defined as follows [4]:
a
=
a
m
0
b
=
nb
0
a
m
(2.30)
0
where
ψ
mn
(
t
)=
a
−m/2
ψ
(
a
−m
t
−
nb
0
)
(2.31)
0
Z
. If we consider this set to be complete in
L
2
(
R
)fora
given choice of
ψ
(
t
)
,a,b
,then
with
m, n
∈
L
2
(
R
)
represents a wavelet synthesis. It recombines the components of a signal
to reproduce the original signal
f
(
t
). If we have a wavelet basis, we can
determine a wavelet series expansion. Thus, any square-integrable (finite
energy) function
f
(
t
) can be expanded in wavelets:
{
ψ
mn
}
is an
a
ne wavelet
.
f
(
t
)
∈
f
(
t
)=
m
d
m,n
ψ
mn
(
t
)
(2.32)
n
The wavelet coecient
d
m,n
can be expressed as the inner product
f
(
t
)
ψ
(
a
−m
0
1
a
m/2
0
d
m,n
=
<f
(
t
)
,ψ
mn
(
t
)
>
=
t
−
nb
0
)
dt
(2.33)
These complete sets are called frames. An analysis frame is a set of
vectors
ψ
mn
such that
2
2
2
A
||
f
||
≤
|
<f,ψ
mn
>
|
≤
B
||
f
||
(2.34)
m
n
with
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