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that one can obtain by shifting and scaling a mother wavelet,
ψ
(
t
). Here
a
and
b
are the scale and shift parameters (
a
=
o
). From the admissibility condition,
we can say that
ψ
(
ω
) will always have sucient decay. Because the Fourier
transform is zero at the origin and the spectrum decays at high frequencies,
the wavelet has a bandpass behavior. Normalizing the wavelet to unit energy,
we get
2
=
+
∞
2
dt,
ψ
(
t
)
−∞
|
ψ
(
t
)
|
+
∞
−∞
| ψ
(
t
)
(8.22)
1
2
π
2
dω,
=
|
=1
.
L
2
(
IR
) is then defined
The continuous wavelet transform of a function
ft
)
∈
as
T
cw
(
f
(
a, b
)) =
+
∞
−∞
ψ
a,b
f
(
t
)
dt
=
<ψ
a,b
(
t
)
,f
(
t
)
>.
(8.23)
The inverse of
T
cw
(
f
(
a, b
)) can be written as
+
∞
+
∞
1
C
ψ
T
cw
(
f
(
a, b
))
ψ
a,b
(
t
)
da db
a
2
f
(
t
)=
.
(8.24)
−∞
−∞
L
2
(
IR
) can be written as a superposition of shifted and
Thus, any
f
(
t
)
∈
dilated wavelets.
8.3.2 Properties of Continuous Wavelet Transform
•
Linearity: Since the linearity is satisfied by the inner product, we can write
T
cw
(
f
1
(
a, b
)) +
T
cw
((
f
2
(
a, b
)) =
T
cw
((
f
1
(
a, b
)+
f
2
(
a, b
))
.
•
Shift: If
f
(
t
) has a continuous wavelet transform
T
cw
(
f
(
a, b
)), then the
continuous wavelet transform of
f
(
t
−
k
) is given by
T
cw
(
f
(
a, b
−
k
)). Note
that the wavelet transform of
f
(
t
)is
+
∞
ψ
a,b
f
(
t
)
dt
=
T
cw
(
f
(
a, b
))
.
−∞
Therefore, the wavelet transform of
f
(
t
−
k
)is
+
∞
1
ψ
(
t
−
b
ψ
a,b
f
(
t
−
k
)
dt
=
)
f
(
t
−
k
)
dt,
a
|
a
|
−∞
+
∞
ψ
(
T
+
k
−
b
1
=
)
f
(
T
)
dT,
a
|
a
|
−∞
=
T
cw
(
f
(
a, b
−
k
))
.
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