Databases Reference
In-Depth Information
Our expansion using coefficients with respect to these functions is obtained from the inner
product of
f
(
t
)
with the wavelet functions:
∞
−∞
ψ
a
,
b
(
w
a
,
b
=
ψ
a
,
b
(
t
),
f
(
t
)
=
t
)
f
(
t
)
dt
(7)
We can recover the function
f
(
t
)
from the
w
a
,
b
by
∞
∞
1
C
ψ
dadb
a
2
(
)
=
w
a
,
b
ψ
a
,
b
(
)
(8)
f
t
t
−∞
−∞
where
∞
2
|
(ω)
|
C
ψ
=
d
ω
(9)
ω
0
For integral (
8
) to exist, we need
C
ψ
to be finite. For
C
ψ
to be finite, we need
(
0
)
=
0.
Otherwise, we have a singularity in the integrand of (
9
). Note that
(
0
)
is the average value of
ψ(
; therefore, the mother wavelet must have zero mean. The condition that
C
ψ
be finite is
often called the
admissibility condition
. We would also like the wavelets to have finite energy;
that is, we want the wavelets to belong to the vector space
L
2
(see Example 12.3.1). Using
Parseval's relationship, we can write this requirement as
∞
t
)
2
d
|
(ω)
|
ω<
∞
−∞
2
has to decay as
|
(ω)
|
ω
For this to happen,
goes to infinity. These requirements mean that
the energy in
is concentrated in a narrow frequency band, which gives the wavelet its
frequency localization capability.
If
a
and
b
are continuous, then
(ω)
w
a
,
b
is called the
continuous wavelet transform
(CWT). Just
as with other transforms, we will be more interested in the discrete version of this transform.
However, we first obtain a series representation where the basis functions are continuous
functions of timewith discrete scaling and translating parameters
a
and
b
. The discrete versions
of the scaling and translating parameters have to be related to each other, because if the scale is
such that the basis functions are narrow, the translation step should be correspondingly small
and vice versa. There are a number of ways we can choose these parameters. The most popular
approach is to select
a
and
b
according to
a
−
m
0
nb
0
a
−
m
a
=
,
b
=
(10)
0
where
m
and
n
are integers,
a
0
is selected to be 2, and
b
0
has a value of 1. This gives us the
wavelet set
a
m
/
2
0
a
0
t
ψ
m
,
n
(
t
)
=
ψ(
−
nb
0
),
m
,
n
∈
Z
(11)
For
a
0
=
2 and
b
0
=
1, we have
2
m
/
2
2
m
t
ψ
m
,
n
(
t
)
=
ψ(
−
n
)
(12)
