Graphics Reference
In-Depth Information
8.3 Wavelets
A wavelet
ψ
(
t
) is a function in the
L
2
(
IR
) space over the real line
IR
that it
satisfies the following conditions.
•
The admissibility condition
C
ψ
must remain finite, i.e.,
C
ψ
=
+
∞
−∞
| ψ
(
ω
)
2
|
dω <
∞
,
(8.20)
|
ω
|
where
ψ
(
ω
) is the Fourier transform of
ψ
(
t
).
ψ
(
ω
)=
+
∞
−∞
ψ
(
t
)
e
−iωt
dt
.
| ψ
(
ω
)
2
is the total power contained in
ψ
(
t
)and
C
ψ
is, therefore, the
total power per every frequency component present in
ψ
(
t
).
|
•
Its Fourier transform must be zero when the frequency is zero. This means
when
ω
=0,
ψ
(
ω
)=
ψ
(0) = 0
.
As a result, we obtain
+
∞
−∞
ψ
(
t
)
dt
=0.
ψ
(
t
)
dt
is the area under the
curve
ψ
(
t
). Since it is zero,
ψ
(
t
) must change its sign, i.e.,
ψ
(
t
) must be
oscillatory in nature or will have a wavelike behavior.
Since the sum of the power per every frequency component is finite, we
must have
|
ψ
(
ω
)
2
|
1
|
→
0 when
ω
→
0. Now,
ω
→
0 implies
|
→∞
.
|
ω
|
ω
Therefore, to have
| ψ
(
ω
)
2
|
→
0
,
|
ω
|
| ψ
(
ω
)
|
2
→
we must have
0 with a faster rate. Such a basic wavelet is
called a mother wavelet.
The mother wavelet represents a family of functions with two parameters: one
of them is for position and the other one is for frequency. In other words, the
family of functions is
1
|
ψ
(
t
−
b
ψ
a,b
(
t
)=
)
,
a
a
|
where
a
=0and
b
∈
IR
.
8.3.1 Continuous Wavelet Transform
Let us consider the family of functions
1
ψ
(
t
−
b
ψ
a,b
(
t
)=
)
,
(8.21)
a
|
a
|
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