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In-Depth Information
t
−
b
−
1/ 2
+∞
(
Wf
) ( ,
a b
)
=
a
f t
( )
ˈ
dt
(3)
ˈ
a
−∞
Stretching the value of the parameter S of wavelet function
ˈ
()
t
transform to get
t
−
b
−
1/ 2
the function
ˈ
,
()
ab
t
=
a
ˈ
. Equation 3 can also be expressed as
a
+∞
(4)
(
Wf
) ( ,
a b
)
=
f t
( )
ˈ
( )
t dt
=
f
,
ˈ
ˈ
ab
,
ab
,
−∞
If the function
()
fx
satisfies the admissibility conditions of equation[1], then there
exists its inverse transformation. And we can restore the original signal
ft
accu-
()
Wf a b
ˈ
(,)
rately based on the wavelet transform
, the inverse transformation formula
is expressed as equation 5.
1
1
+∞ +∞
f t
()
=
Wf a b dadb
( , )
(5)
a
ˈ
2
C
0
−∞
ˈ
and
ˈˉ
()
+∞
(6)
C
=
d
ˉ
< +∞
ˈ
ˉ
0
Daubechies wavelet is proposed by the French scholar Daubechies, , the function
()
ˈ
is called the p level of Daubechies wavelet in the condition of Equation 7 , and it
can also be referred to db wavelets.
t
p
t
ˈ
( )
t t
=
0,
p
=
0, 1, 2, . . . ,
N
(7)
1finite support, which is affected by parameter N, The greater the value of
N ,the length longer.
()
ˈ ˉ
is with N-order zero at ω = 0 in the frequency domain.
2
ˈ
and its displacement orthonormal integer that satisfies the equation
[8]as shown below.
()
t
3
ˈˈ
()(
t
t
−
k t
)
=
ʴ
(8)
k
The nature of Db wavelet makes it has the characteristics like smaller amount of
calculation and flexible selection, so it has a wide range of applications in signal
analysis.
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