Biomedical Engineering Reference
In-Depth Information
having the form
ψ
mn
(
t
)
=
2
m
/
2
ψ
mn
(2
m
t
−
n
)
,
n
,
m
∈
Z
,
where
ψ
(
t
) is the “mother wavelet.” Usually it is not constructed directly but
rather from another function called the “scaling function”
φ
(
t
)
∈
L
2
(
R
). The
scaling function
φ
is chosen in such a way that
⎧
⎨
(i)
φ
(
t
)
φ
(
t
−
n
)
dt
=
δ
0
,
n
,
n
∈
Z
,
√
2
c
k
φ
(2
t
−
k
)
,
{
c
k
}
k
∈
Z
∈
l
2
φ
(
t
)
=
−∞
(ii)
,
(5.62)
⎩
(iii) for each
f
∈
L
2
(
R
)
, >
0
,
there is a function
f
m
(
t
)
=
n
=−∞
a
mn
φ
(2
m
t
−
n
) such that
f
m
−
f
<.
These conditions lead to a “multiresolution approximation”
{
V
m
}
m
∈
Z
, consist-
ing of closed subspaces of
L
2
(
R
). The space
V
m
is taken to be the closed linear
span of
{
φ
(2
m
t
−
n
)
}
n
∈
Z
. Because of (5.62) (ii), the
V
m
are nested, i.e.
V
m
⊆
V
m
+
1
and because of (5.62) (iii),
∪
m
V
m
is dense in
L
2
(
R
).
There are many different types of wavelet bases created and employed for
different purposes. They can be classified as time-limited wavelets, such as Haar
wavelets and Daubechies wavelets, band-limited wavelets, such as Shannon
and Meyer wavelets. Another standard prototype is the Haar system in which
φ
(
t
)
=
χ
[0
,
1]
(
t
), where
1
,
x
∈
[0
,
1]
,
χ
[0
,
1]
=
0
,
x
∈
[0
,
1]
is the characteristic function of [0
,
1]
.
It is an easy exercise to show that (5.62)
is satisfied. This prototype has poor frequency localization but good time local-
ization. Most of the other examples found, e.g., in [12] and [66], attempt to get
fairly good time and frequency localization simultaneously.
The various scales are related by the dilation equation of the scaling function
√
2
∞
φ
(
t
)
=
c
n
φ
(2
t
−
n
)
,
(5.63)
n
=−∞
√
2
∞
ψ
(
t
)
=
d
n
φ
(2
t
−
n
)
,
n
=−∞
where
d
n
=
c
1
−
n
(
−
1)
n
.
